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Learning Objectives
1. Learn that the infant industry argument presumes a market imperfection—the presence of a positive production externality.
2. Recognize that a trade policy can be used to correct for an infant industry production externality imperfection.
3. Learn the first-best and second-best policy options to correct for an infant industry production externality imperfection.
4. Learn the practical implementation problems that can arise when governments attempt to apply infant industry protection.
One of the most notable arguments for protection is known as the infant industry argument. The argument claims that protection is warranted for small new firms, especially in less-developed countries. New firms have little chance of competing head-to-head with the established firms located in the developed countries. Developed country firms have been in business longer and over time have been able to improve their efficiency in production. They have better information and knowledge about the production process, about market characteristics, about their own labor market, and so on. As a result, they are able to offer their product at a lower price in international markets and still remain profitable.
A firm producing a similar product in a less-developed country (LDC), on the other hand, would not have the same production technology available to it. Its workers and management would lack the experience and knowledge of its developed country rivals and thus would most likely produce the product less efficiently. If forced to compete directly with the firms in the developed countries, the LDC firms would be unable to produce profitably and thus could not remain in business.
Protection of these LDC firms, perhaps in the form of an import tariff, would raise the domestic price of the product and reduce imports from the rest of the world. If prices are raised sufficiently, the domestic firms would be able to cover their higher production costs and remain in business. Over time, these LDC firms would gain production and management experience that would lower their production costs. Essentially, the firms would follow the same path that the developed country firms had followed to realize their own production efficiency improvements. Protection, then, allows an infant industry time to “grow up.”
Furthermore, since the LDC firms would improve their productive efficiency over time, the protective tariffs could be gradually reduced until eventually, when the tariffs are eliminated, they would compete on an equal footing with the developed country firms.
Many people have argued that this was precisely the industrial development strategy that was pursued by countries like the United States and Germany during their rapid industrial development before the turn of the twentieth century. Both the United States and Germany had high tariffs during their industrial revolution periods. These tariffs helped protect fledgling industries from competition with more-efficient firms in Britain and may have been the necessary requirement to stimulate economic growth.
One counterargument to this theory is that by protecting infant industries, countries are not allocating resources in the short run on the basis of comparative advantage. The Ricardian and Heckscher-Ohlin models of trade show that resources will be allocated most efficiently if countries produce goods whose before-trade prices are lower than those in the rest of the world. This implies that the United States and Germany should have simply imported the cheaper industrial goods from Britain and shifted their own resources to other goods in which they had a comparative advantage if they wished to maximize economic efficiency.
The reason for the discrepancy in policy prescriptions can easily be seen by noting the difference between static comparative advantage and dynamic comparative advantage. The traditional Ricardian theory of comparative advantage identifies the most efficient allocation of resources at one point in time. In this sense, it is a static theory. The policy prescription is based on a snapshot in time.
On the other hand, the infant industry argument is based on a dynamic theory of comparative advantage. In this theory, one asks what is best for a country (i.e., what is most efficient) in the long run. The most efficient long-run strategy may well be different from what is best initially. Here’s why.
The problem faced by many LDCs is that their static comparative advantage goods, in most instances, happen to be agricultural commodities and natural resources. Reliance on production of these two types of goods can be problematic for LDCs. First of all, the prices of agricultural commodities and natural resources have historically been extremely volatile. In some years prices are very high, and in other years the prices are very low. If a country allocates many of its resources to production of goods with volatile prices, then the gross domestic product (GDP) will fluctuate along with the prices. Some years will be very good, and others will be very bad. Although a wealthier country may be able to smooth income by effectively using insurance programs, a poor country might face severe problems, perhaps as severe as famine, in years when the prices of their comparative advantage goods are depressed.
In addition, many people argue that the management and organizational skills necessary to produce agricultural goods and natural resources are not the same as the skills and knowledge needed to build an industrial economy. If true, then concentrating production in one’s static comparative advantage goods would prevent the development of an industrial economy. Thus one of the reasons for protecting an infant industry is to stimulate the learning effects that will improve productive efficiency. Furthermore, these learning effects might spill over into the rest of the economy as managers and workers open new businesses or move to other industries in the economy. To the extent that there are positive spillovers or externalities in production, firms are unlikely to take account of these in their original decisions. Thus, if left alone, firms might produce too little of these types of goods and economic development would proceed less rapidly, if at all.
The solution suggested by the infant industry argument is to protect the domestic industries from foreign competition in order to generate positive learning and spillover effects. Protection would stimulate domestic production and encourage more of these positive effects. As efficiency improves and other industries develop, economic growth is stimulated. Thus by protecting infant industries a government might facilitate more rapid economic growth and a much faster improvement in the country’s standard of living relative to specialization in the country’s static comparative advantage goods.
An Analytical Example
Consider the market for a manufactured good such as textiles in a small, less-developed country.
Suppose that the supply and demand curves in the country are as shown in Figure \(1\). Suppose initially free trade prevails and the world price of the good is \(P_1\). At that price, consumers would demand \(D_1\), but the domestic supply curve is too high to warrant any production. This is the case, then, where domestic producers simply could not produce the product cheaply enough to compete with firms in the rest of the world. Thus the free trade level of imports would be given by the blue line segment, which is equal to domestic demand, \(D_1\).
Suppose that the infant industry argument is used to justify protection for this currently nonexistent domestic industry. Let a specific tariff be implemented that raises the domestic price to \(P_2\). In this case, the tariff would equal the difference between \(P_2\) and \(P_1\)—that is, \(t = P_2 − P_1\). Notice that the increase in domestic price is sufficient to stimulate domestic production of \(S_2\). Demand would fall to \(D_2\) and imports would fall to \(D_2 − S_2\) (the red line segment).
The static (i.e., one-period) welfare effects of the import tariff are shown in Table \(1\).
Table \(1\): Static Welfare Effects of a Tariff
Importing Country
Consumer Surplus − (A + B + C + D)
Producer Surplus + A
Govt. Revenue + C
National Welfare BD
Consumers of textiles are harmed because of the higher domestic price of the good. Producers gain in terms of producer surplus. In addition, employment is created in an industry that did not even exist before the tariff. Finally, the government earns tariff revenue, which benefits some other segment of the population.
The net national welfare effect of the import tariff is negative. Although some segments of the population benefit, two deadweight losses to the economy remain. Area \(B\) represents a production efficiency loss, while area \(D\) represents a consumption efficiency loss.
Dynamic Effects of Infant Industry Protection
Now suppose that the infant industry argument is valid and that by stimulating domestic production with a temporary import tariff, the domestic industry improves its own productive efficiency. We can represent this as a downward shift in the domestic industry supply curve. In actuality, this shift would probably occur gradually over time as the learning effects are incorporated in the production process. For analytical simplicity, we will assume that the effect occurs as follows. First, imagine that the domestic industry enjoys one period of protection in the form of a tariff. In the second period, we will assume that the tariff is removed entirely but that the industry experiences an instantaneous improvement in efficiency such that it can maintain production at its period one level but at the original free trade price. This efficiency improvement is shown as a supply curve shift from \(S\) to \(S′\) in Figure \(2\).
This means that in the second period, free trade again prevails. The domestic price returns to the free trade price of \(P_1\), while domestic demand rises to \(D_1\). Because of the efficiency improvement, domestic supply in free trade is given by \(S_2\) and the level of imports is \(D_1 − S_2\) (the blue segment).
The static (one-period) welfare effects of the tariff removal and efficiency improvement are summarized in Table \(2\). Note that these effects are calculated relative to the original equilibrium before the original tariff was implemented. We do this because we want to identify the welfare effects in each period relative to what would have occurred had the infant industry protection not been provided.
Table \(2\) Static Welfare Effects of Tariff Removal and Efficiency Improvement
Importing Country
Consumer Surplus 0
Producer Surplus + E
Govt. Revenue 0
National Welfare + E
Consumers again face the same free trade price that they would have faced if no protection had been offered. Thus they experience no loss or gain. Producers, however, face a new supply curve that generates a producer surplus of \(+ E\) at the original free trade price. The government tariff is removed, so the government receives no tariff revenue. The net national welfare effect for the second period then is simply the gain in producer surplus.
The overall welfare impact over the two periods relative to no infant industry protection over two periods is simply the sum of each period’s welfare effects. This corresponds to the sum of areas (\(+ E − B − D\)), which could be positive or negative. If the second-period producer surplus gain exceeds the first-period deadweight losses, then the protection has a positive two-period effect on national welfare.
But wait. Presumably the efficiency improvement in the domestic industry would remain, if not improve, in all subsequent periods as well. Thus it is not complete to consider the effects only over two periods. Instead, and for simplicity again, suppose that the new supply curve prevails in all subsequent periods. In this case, the true dynamic national welfare effects would consist of area E multiplied by the number of future periods we wish to consider minus the one-period deadweight losses. Thus even if the costs of the tariff are not made up in the second period, they may well be made up eventually at some point in the future. This would make it even more likely that the temporary protection would be beneficial in the long run.
If, in addition to the direct efficiency effects within the industry, there are spillover efficiency effects on other industries within the domestic economy, then the likelihood that temporary protection is beneficial is enhanced even further. In other words, over time, workers and managers from the protected industries may establish firms or take jobs in other sectors of the economy. Since they will bring their newly learned skills with them, it will cause an improvement in productive efficiency in those sectors as well. In this way, the supply of many manufacturing industries will be increased, allowing these sectors to compete more easily with firms in the rest of the world. Industrialization and GDP growth then is stimulated by the initial protection of domestic industries.
In summary, we have shown the possibility that protection of an infant industry may be beneficial for an economy. At the heart of the argument is the assumption that production experience generates efficiency improvements either directly in the protected industry or indirectly in other industries as a learning spillover ensues. The infant industry argument relies on a dynamic view of the world rather than the static description used in classical trade models. Although protection may be detrimental to national welfare in the short run, it is conceivable that the positive dynamic long-run effects will more than outweigh the short-run (or static) effects.
The Economic Argument against Infant Industry Protection
The main economic argument against infant industry protection is that protection is likely to be a second-best policy choice rather than a first-best policy choice. The key element of the infant industry argument is the presence of a positive dynamic production externality. It is assumed that production experience causes learning, which improves future productive efficiency. Alternatively, it is assumed that these learning effects spill over into other industries and improve those industries’ future productive efficiencies as well.
The theory of the second best states that in the presence of a market distortion, such as a production externality, it is possible to conceive of a trade policy that can improve national welfare. However, in this case, the trade policy—namely, the import tariff—is not the first-best policy because it does not attack the distortion most directly. In this case, the more-efficient policy is a production subsidy targeted at the industries that generate the positive learning effects.
To demonstrate this result, consider the following analytical example. We will use the same supply and demand conditions as depicted in Figure \(2\). The domestic supply and demand curves are given by \(D\) and \(S\), respectively. The initial free trade world price of the good is \(P_1\). At that price, consumers would demand \(D_1\), but the domestic supply curve is too high to warrant any production. Thus the level of imports is given by \(D_1\).
Now suppose that the government implements a specific production subsidy equal to the difference in prices, \(P_2 − P_1\). The subsidy would raise the producer price by the amount of the subsidy to \(P_2\), and hence domestic supply will rise to \(S_2\). The domestic consumer price would remain at \(P_1\), so demand would remain at \(D_1\). Imports would fall to \(D_1 − S_2\).
The static (i.e., one-period) welfare effects of the production subsidy are shown in Table \(3\): Static Welfare Effects of a Production Subsidy.
Table \(3\): Static Welfare Effects of a Production Subsidy
Importing Country
Consumer Surplus 0
Producer Surplus + A
Govt. Revenue − (A + B)
National Welfare B
Consumers of textiles are left unaffected by the subsidy since the domestic price remains the same. Producers gain in terms of producer surplus since the subsidy is sufficient to cause production to begin. In addition, employment is created in an industry. The government, however, must pay the subsidy. Thus someone pays higher taxes to fund the subsidy.
The net national welfare effect of the production subsidy is negative. Although some segments of the population benefit, there remains a production efficiency loss.
Note, however, that relative to an import tariff that generates the same level of domestic production, the subsidy is less costly in the aggregate. The production subsidy causes only a production efficiency loss, while the tariff causes an additional consumption efficiency loss. If the positive dynamic gains in efficiency in subsequent periods are the same, then the production subsidy would generate the same positive stream of benefits but at a lower overall cost to the country. For this reason, the production subsidy is the first-best policy to choose in light of the dynamic production externality. The import tariff remains second best.
For this reason, economists sometimes argue that although an import tariff may indeed be beneficial in the case of infant industries, it does not necessarily mean that protection is appropriate.
Other Arguments against Infant Industry Protection
Political economy problems. Political pressures in democratic economies can make it difficult to implement infant industry protection in its most effective manner. In order for protection to work in the long run, it is important that protection be temporary. There are two main reasons for this. First, it may be that the one-period efficiency improvement is less than the sum of the deadweight costs of protection. Thus if protection is maintained, then the sum of the costs may exceed the efficiency improvements and serve to reduce national welfare in the long run. Second, and more critically, if protection were expected to be long lasting, then the protected domestic firms would have less incentive to improve their productive efficiency. If political pressures are brought to bear whenever the tariffs are scheduled to be reduced or removed, industry representatives might convince legislators that more time is needed to guarantee the intended efficiency improvements. In other words, firms might begin to claim that they need more time to compete against firms in the rest of the world. As long as legislators provide more time to catch up to world efficiency standards, protected firms have little incentive to incur the investment and training costs necessary to compete in a free market. After all, the tariff keeps the price high and allows even relatively inefficient production to produce profits for the domestic firms.
Thus one big problem with applying the infant industry protection is that the protection itself may eliminate the need for the firms to grow up. Without the subsequent efficiency improvements, protection would only generate costs for the economy in the aggregate.
Informational problems. In order for infant industry protection to work, it is important for governments to have reliable information about industries in their economies. They need to know which industries have strong learning effects associated with production and which industries are most likely to generate learning spillover effects to other industries. It would also be useful to know the size of the effects as well as the timing. But governments must decide not only which industries to protect but also how large the protective tariffs should be and over what period of time the tariff should be reduced and eliminated. If the government sets the tariff too low, the protection may be insufficient to generate very much domestic production. If the tariff is set too high, the costs of the tariff might outweigh the long-term efficiency improvements. If the tariff is imposed for too long a period, then firms might not have enough of an incentive to make the changes necessary to improve efficiency. If set for too short a time, then firms may not learn enough to compete with the rest of the world once the tariffs are removed.
Thus in order for infant industry protection to work, it is important to set the tariff for the correct industries, at the correct level, and for the correct period of time. Determining the correct industries, tariff level, and time period is not a simple matter. Indeed, some people argue that it is impossible to answer these questions with a sufficient amount of accuracy to warrant applying these policies.
Failure of import-substitution strategies. One popular development strategy in the 1950s and 1960s was known as import substitution. Essentially, this strategy is just an application of the infant industry argument. However, many of the countries that pursued these kinds of inward-looking strategies, most notably countries in Latin America and Africa, performed considerably less well economically than many countries in Asia. The Asian countries—such as South Korea, Taiwan, Hong Kong, and Japan—pursued what have been labeled export-oriented strategies instead. Since many of these Southeast Asian countries performed so much better economically, it has lent some empirical evidence against the application of infant industry protection.
Key Takeaways
• An import tariff that stimulates infant industry production sufficiently can raise national welfare over time, even for a small importing country.
• An import tariff is a second-best policy to correct for an infant industry production externality imperfection.
• A production subsidy is superior to an import tariff as a policy to correct for an infant industry production externality imperfection.
• In the presence of an infant industry production externality imperfection, a domestic policy is first best, while the best trade policy is second best.
Exercise \(1\)
1. Jeopardy Questions. As in the popular television game show, you are given an answer to a question and you must respond with the question. For example, if the answer is “a tax on imports,” then the correct question is “What is a tariff?”
1. The term used to describe firms in less-developed countries that have a significant cost disadvantage compared with established firms located in the developed countries.
2. The type of comparative advantage that is not present in the short run but that develops in the long run.
3. The first-best policy option for a government that wishes to support an infant industry.
4. A second-best policy option for a government that wishes to support an infant industry.
5. Of increase, decrease, no change, or ambiguous, the effect of infant industry protection on national welfare under standard assumptions in the early periods while protection is in place.
6. Of increase, decrease, no change, or ambiguous, the effect of infant industry protection on national welfare under standard assumptions in the later periods after protection is removed.
7. Of increase, decrease, no change, or ambiguous, the effect of infant industry protection on overall national welfare under standard assumptions over all periods. | textbooks/socialsci/Economics/International_Trade_-_Theory_and_Policy/09%3A_Trade_Policies_with_Market_Imperfections_and_Distortions/9.05%3A_The_Infant_Industry_Argument_and_Dynamic_Comparative_Advantage.txt |
Learning Objectives
1. Learn that a foreign monopoly supplying products to domestic consumers is a type of market imperfection.
2. Recognize that a trade policy can be used to correct for a foreign monopolist imperfection.
3. Learn the first-best and second-best policy options to correct for a foreign monopolist imperfection.
Consider a domestic market supplied by a foreign monopoly firm. The domestic market consists of many consumers who demand the product but has no domestic producers of the product. All supply of the product comes from a single foreign firm.
Although this situation is not very realistic, it is instructive as an application of the theory of the second best. In this case, the market imperfection is that there are not a multitude of firms supplying the market. Rather, we have assumed the extreme opposite case of a monopoly supplier. To make this an international trade story, we simply assume the monopoly happens to be a foreign firm.
Consider the market described in Figure \(1\). Domestic consumer demand is represented by a linear demand curve, \(D\). When demand is linear, it follows that the marginal revenue curve will have twice the slope and will equal demand when the quantity is zero. Let the flat \(MC\) line represent a constant marginal cost in production for the foreign monopolist.
Assuming the monopolist maximizes profit, the profit-maximizing output level is found by setting marginal cost equal to marginal revenue. Why? Profit-maximizing output occurs at the quantity level \(Q_{FT}\). At that quantity, the monopolist would set the price at \(P_{FT}\), the only price that equalizes demand with its supply.
The monopolist’s profit is the difference between total revenue and total cost. Total revenue is given by the product (\(P_{FT}Q_{FT}\)), the yellow area in the graph. Total cost is equal to average cost (\(AC\)) multiplied by output (\(Q_{FT}\)), given by the checkered area. The monopolist’s profit is represented by the uncheckered yellow rectangular area in Figure \(1\).
Strategic Trade Policy
Generally, strategic trade policy refers to cases of advantageous protection when there are imperfectly competitive markets. The case of a foreign monopolist represents one such case.
More specifically, though, the presence of imperfect competition implies that firms can make positive economic profit. Strategic trade policies typically involve the shifting of profits from foreign firms to domestic firms. In this way, national welfare can be improved, although it is often at the expense of foreign countries.
In this example, we shall consider the welfare effects of a specific tariff set equal to \(t\). The tariff will raise the cost of supplying the product to the domestic market by exactly the amount of the tariff. We can represent this in Figure \(2\) by shifting the marginal cost curve upward by the amount of the tariff to \(MC + t\). The monopolist will reduce its profit-maximizing output to \(Q_T\) and raise its price to \(P_T\). Note that the price rises by less than the amount of the tariff.
Table \(1\) provides a summary of the direction and magnitude of the welfare effects to producers, consumers, and the government in the importing country as a result of the import tariff. The aggregate national welfare effects are also shown.
Table \(1\): Welfare Effects of a Tariff
Importing Country
Consumer Surplus − (a + b + c)
Producer Surplus 0
Govt. Revenue + d
National Welfare d − (a + b + c)
Import tariff effects on the importing country’s consumers. Consumers of the product in the importing country suffer a reduction in surplus because of the higher price that prevails. Refer to Table \(1\) and Figure \(2\) to see how the magnitude of the change in producer surplus is represented.
Import tariff effects on the importing country’s producers. It is assumed that there are no domestic producers of the goods; thus there are no producer effects from the tariff.
Import tariff effects on the importing country’s government. The government receives tariff revenue given by the per-unit tax (\(t\)) multiplied by the quantity of imports (\(Q_T\)). Who gains from the tariff revenue depends on how the government spends the money. Presumably these revenues help support the provision of public goods or help sustain transfer payments. In either case, someone in the economy ultimately benefits from the revenue. Refer to Table \(1\) and Figure \(2\) to see how the magnitude of the subsidy payments is represented.
The aggregate welfare effect for the importing country is found by summing the gains and losses to consumers, producers, and the government. The net effect consists of two components: a positive effect on the recipients of the government tariff revenue (\(d\)) and a negative effect on consumers (\(a + b + c\)), who lose welfare due to higher prices.
If demand is linear, it is straightforward to show that the gains to the country will always exceed the losses for some positive nonprohibitive tariff. In other words, there will exist a positive optimal tariff. Thus a tariff can raise national welfare when the market is supplied by a foreign monopolist.
One reason for this positive effect is that the tariff essentially shifts profits away from the foreign monopolist to the domestic government. Note that the original profit level is given by the large blue rectangle shown in Figure \(2\). When the tariff is implemented, the monopolist’s profit falls to a level given by the red rectangle. Thus, in this case, the tariff raises aggregate domestic welfare as it reduces the foreign firm’s profit.
First-Best Policy
Although a tariff can raise national welfare in this case, it is not the first-best policy to correct the market imperfection. A first-best policy must attack the imperfection more directly. In this case, the imperfection is the monopolistic supply of the product to the market. A monopoly maximizes profit by choosing an output level such that marginal revenue is equal to marginal cost. This rule deviates from what a perfectly competitive firm would do—that is, set price equal to marginal cost. When a firm is one among many, it must take the price as given. It cannot influence the price by changing its output level. In this case, the price is its marginal revenue. However, for a monopolist, which can influence the market price, price exceeds marginal revenue. Thus when the monopolist maximizes profit, it sets a price greater than marginal cost. This deviation—that is, \(P > MC\)—is at the core of the market imperfection.
The standard way of correcting this type of imperfection in a domestic context is to regulate the industry. For example, electric utilities are regulated monopolies in the United States. Power can generally be purchased from only one company in any geographical area. To assure that these firms do not set exorbitant prices, the government issues a set of pricing rules that the firms must follow. The purpose is to force the firms to set prices closer, if not equal to, the marginal cost of production.
Now, in the case of utilities, determining the marginal cost of production is a rather difficult exercise, so the pricing rules to optimally regulate the industry are relatively complicated. In the case of a foreign monopolist with a constant marginal cost supplying a domestic market, however, the optimal policy is simple. The domestic government could merely set a price ceiling equal to the firm’s marginal cost in production.
To see why a price ceiling is superior to a tariff, consider Figure \(3\). A second-best policy is the tariff. It would raise national welfare by the area (\(h − a − b − c\)), which as mentioned will be positive for some tariffs and for a linear demand curve. The first-best policy is a price ceiling set equal to the marginal cost at \(P_C\). The price ceiling would force the monopolist to set the price equal to the marginal cost and induce an increase in supply to \(Q_C\). Consumers would experience an increase in consumer surplus, given by the area (\(d + e + f + g + h + i + j + k\)), because of the decline in price. Clearly, in this example, the consumer surplus gain with the price ceiling exceeds the national welfare gain from a tariff.
This shows that although a tariff can improve national welfare, it is not the best policy to correct this market imperfection. Instead, a purely domestic policy—a price ceiling in this case—is superior.
Key Takeaways
• A strategic trade policy attempts to shift foreign profits toward the importing economy.
• An import tariff levied against a foreign monopoly firm supplying domestic demand can raise national welfare.
• An import tariff is a second-best policy to correct for the imperfection of a foreign monopoly firm supplying domestic demand.
• A price ceiling is superior to an import tariff as a policy to correct for the imperfection of a foreign monopoly firm supplying domestic demand.
• In the presence of the imperfection of a foreign monopoly firm supplying domestic demand, a domestic policy is first best, while the best trade policy is second best.
Exercise \(1\)
1. Jeopardy Questions. As in the popular television game show, you are given an answer to a question and you must respond with the question. For example, if the answer is “a tax on imports,” then the correct question is “What is a tariff?”
1. The first-best policy option for a government that faces a foreign monopoly (with constant marginal costs) as the sole firm selling a product in the domestic market.
2. A second-best policy for a government that faces a foreign monopoly (with constant marginal costs) as the sole firm selling a product in the domestic market.
3. The term used to describe a policy that shifts profits from foreign firms toward groups in the domestic economy.
2. Suppose the U.S. market demand for VCRs is given by \(D = 1,000 – 2P\). The U.S. market is supplied by a foreign monopolist with a constant marginal cost of production equal to \$200. The marginal revenue curve faced by the supplier is given by \(MR = 500 – Q\).
1. Calculate the equilibrium price and quantity of imports of VCRs. Depict this equilibrium graphically.
2. Calculate consumer surplus in this market equilibrium.
Suppose the government imposes a specific tariff of \$100.
3. Calculate the new equilibrium price and quantity.
4. Calculate the change in consumer surplus and the tariff revenue.
5. What is the change in national welfare?
6. What is the first-best policy action to raise national welfare in this case? If this policy is applied, what would be the domestic price and quantity imported?
7. Calculate the change in national welfare if the first-best policy is applied rather than the tariff. Compare this with the national welfare effect of the tariff.
8. Briefly explain how to identify first-best policies in general and explain why the policy in this case satisfies the criterion. | textbooks/socialsci/Economics/International_Trade_-_Theory_and_Policy/09%3A_Trade_Policies_with_Market_Imperfections_and_Distortions/9.06%3A_The_Case_of_a_Foreign_Monopoly.txt |
Learning Objectives
1. Learn that monopoly power and monopsony power in trade are types of market imperfections.
2. Recognize that a trade policy can be used to correct for a large-country imperfection.
3. Learn the first-best and second-best policy options to correct for a large-country imperfection.
Perhaps surprisingly, “large” importing countries and “large” exporting countries have a market imperfection present. This imperfection is more easily understood if we use the synonymous terms for “largeness”: monopsony power and monopoly power. Large importing countries are said to have “monopsony power in trade,” while large exporting countries are said to have “monopoly power in trade.” As this terminology suggests, the problem here is that the international market is not perfectly competitive. For complete perfect competition to prevail internationally, we would have to assume that all countries are “small” countries.
Let’s first consider monopoly power. When a large exporting country implements a trade policy, it will affect the world market price for the good. That is the fundamental implication of largeness. For example, if a country imposes an export tax, the world market price will rise because the exporter will supply less. It was shown in Chapter 7: Trade Policy Effects with Perfectly Competitive Markets, Section 7.23: Export Taxes- Large Country Welfare Effects that an export tax set optimally will cause an increase in national welfare due to the presence of a positive terms of trade effect. This effect is analogous to that of a monopolist operating in its own market. A monopolist can raise its profit (i.e., its firm’s welfare) by restricting supply to the market and raising the price it charges its consumers. In much the same way, a large exporting country can restrict its supply to international markets with an export tax, force the international price up, and create benefits for itself with the terms of trade gain. The term monopoly “power” is used because the country is not a pure monopoly in international markets. There may be other countries exporting the product as well. Nonetheless, because its exports are a sufficiently large share of the world market, the country can use its trade policy in a way that mimics the effects caused by a pure monopoly, albeit to a lesser degree. Hence the country is not a monopolist in the world market but has monopoly “power” instead.
Similarly, when a country is a large importer of a good, we say that it has “monopsony power.” A monopsony is a single buyer in a market consisting of many sellers. A monopsony raises its own welfare or utility by restricting its demand for the product and thereby forcing the sellers to lower their price. By buying fewer units at a lower price, the monopsony becomes better off. In much the same way, when a large importing country places a tariff on imports, the country’s demand for that product on world markets falls, which in turn lowers the world market price. It was shown in Chapter 7: Trade Policy Effects with Perfectly Competitive Markets, Section 7.6: The Optimal Tariff that an import tariff, set optimally, will raise national welfare due to the positive terms of trade effect. The effects in these two situations are analogous. We say that the country has monopsony “power” because the country may not be the only importer of the product in international markets, yet because of its large size, it has the “power” of a pure monopsony.
First-Best or Second-Best Trade Policies
It has already been shown that a trade policy can improve a country’s national welfare when that country is either a large importer or a large exporter. The next question to ask is whether the optimal tariff or the optimal export tax, each of which is the very best “trade” policy that can be chosen, will raise national welfare to the greatest extent or whether there is another purely domestic policy that can raise welfare to a larger degree.
Because a formal graphical comparison between the first-best and second-best policies is difficult to construct in this case, we will rely on an intuitive answer based on what has been learned so far. It is argued in Chapter 9: Trade Policies with Market Imperfections and Distortions, Section 9.3: The Theory of the Second Best that the first-best policy will always be that policy that attacks the market imperfection or market distortion most directly. In the case of a large country, it is said that the market imperfection is a country’s monopsony or monopoly power. This power is exercised in “international” markets, however. Since benefits accrue to a country by changing the international terms of trade in a favorable direction, it is through trade that the monopsony or monopoly power can “best” be exercised. This observation clearly indicates that trade policies will be the first-best policy options. When a country is a large importing country, an optimal tariff or import quota will be first best. When a country is a large exporting country, an optimal export tax or voluntary export restraint (VER) will be first best.
Now, of course, this does not mean that a purely domestic policy cannot raise national welfare when a country is “large.” In fact, it was shown in Chapter 8: Domestic Policies and International Trade, Section 8.4: Production Subsidy Effects in a Small Importing Country that an import tariff is equivalent to a domestic production subsidy and a domestic consumption tax set at the same level; thus setting one of these policies at an appropriate level may also be able to raise national welfare. To see that this is true, let’s consider a large importing country initially in free trade. Because it is in free trade, there is a market imperfection present that has not been taken advantage of. Suppose this country’s government implements a production subsidy provided to the domestic import-competing firm. We can work out the effects of this production subsidy in Figure \(1\).
The free trade price is given by \(P_{FT}\). The domestic supply in free trade is \(S_1\), and domestic demand is \(D_1\), which determines imports in free trade as \(D_1 − S_1\) (the red line in Figure \(1\)).
When a specific production subsidy is imposed, the producer’s price rises, at first by the value of the subsidy. The consumer’s price is initially unaffected. This increase in the producer’s price induces the producer to increase its supply to the market. The supply rises along the supply curve and imports begin to fall. However, because the country is a large importer, the decrease in imports represents a decrease in the world demand for the product. As a result, the world price of the good falls, which in turn means that the price paid by consumers in the import market also falls. When a new equilibrium is reached, the producer’s price will have risen (to \(P_P\) in Figure \(1\)), the consumer’s price will have fallen (to \(P_W\)), and the difference between the producer and consumer prices will be equal to the value of the specific subsidy (\(s = P_P − P_W\)). Note that the production subsidy causes an increase in supply from \(S_1\) to \(S_2\) and an increase in demand from \(D_1\) to \(D_2\). Because both supply and demand rise, the effect of the subsidy on imports is, in general, ambiguous.
The welfare effects of the production subsidy are shown in Table \(1\). The letters refer to the area in Figure \(1\).
Table \(1\): Welfare Effects of a Production Subsidy in a Large Country
Consumer Surplus + (e + f + g + h + i + j)
Producer Surplus + a
Govt. Revenue − (a + b + e + f + g)
National Welfare h + i + jb
The first thing to note is that the production subsidy causes welfare improvements for both producers and consumers. All previous policies have these two groups always experiencing opposite effects. It would appear, in this case, we have struck the “mother lode”—finally, a policy that benefits both consumers and producers. Of course, the effects are not all good. To achieve this effect, the government must pay the subsidy to the firms, and that must come from an increase in taxes either now or in the future. So the country must incur a cost in the form of government expenditures. The final effect—that is, the effect on national welfare—is ambiguous. However, it is conceivable that the area given by (\(h + i + j\)) may exceed the area (\(b\)), in which case, national welfare will rise. Of course, if a different subsidy level is set, it is also possible that national welfare will fall. It will depend on the value of the subsidy, and it will vary across every separate market.
In the case that welfare does rise, it will occur because the country is a large importer. The domestic production subsidy allows the country to take advantage of its monopsony power in trade. By stimulating domestic production, the subsidy reduces import demand, which pushes the price of the country’s import good down in the world market. In other words, the country’s terms of trade improves. In this way, a country can take advantage of its monopsony power by implementing a domestic policy, such as a production subsidy to an import-competing industry. Note well, though, that not every subsidy provided will raise national welfare. The subsidy must be set at an appropriate level for the market conditions to assure an increase in national welfare. In general, a relatively small subsidy will achieve this objective. If the subsidy is set too high, the losses from government expenditures will exceed the gains to consumers and producers, and the country will suffer national welfare losses.
Other domestic policies can also be used to raise national welfare in the case of a large importing country. Indeed, any policy that restricts international demand for a product will potentially raise national welfare—only “potentially” because it is necessary to set the policy at the proper level. The other obvious domestic policy that can achieve this result is a domestic consumption tax on the imported product. Recall that a consumption tax is one of the two domestic policies that, when applied together, substitutes for an import tariff. Since the import tariff can raise welfare, so can its constituent parts.
Key Takeaways
• A market imperfection exists whenever a country is “large”: either a large importer, a large exporter, or both.
• In these cases, international perfect competition does not prevail. We say that a large exporting country has monopoly power in trade, while a large importing country has monopsony power in trade.
• Due to the presence of the market imperfection, a trade policy can raise the nation’s welfare above the level possible with free trade.
• Domestic policies, such as production subsidies and consumption taxes, can also raise national welfare when a country is large.
• The first-best policy in the case of a large country is a trade policy.
• A trade policy most directly attacks the market distortion—that is, international imperfect competition.
• If a country is a large importer, the first-best trade policy is the optimal tariff or its equivalent quota.
• If a country is a large exporter, the first-best policy is the optimal export tax or its equivalent VER.
• Domestic policies, used alone, are second-best policy options.
Exercise \(1\)
1. Consider the following imperfect market situation in the table below. From the following list of policy options, identify all types of trade policies and all types of domestic policies that could potentially raise national welfare in the presence of each imperfection. Consider only the partial equilibrium effects of each policy.
Options: An import tariff, an import quota, a voluntary export restraint (VER), an export tax, an export subsidy, a production tax, a production subsidy, a consumption tax, and a consumption subsidy.
Table \(2\): Welfare Improving Policies
Trade Policy Domestic Policy
A large country that imports steel
2. Consider the domestic policy action listed along the top row of the table below. In the empty boxes, use the following notation to indicate the effect of the policy on the variables listed in the first column. Use a partial equilibrium model to determine the answers and assume that the shapes of the supply and demand curves are “normal.” Assume that the policy does not begin with, or result in, prohibitive policies. Use the following notation:
+ the variable increases
the variable decreases
0 the variable does not change
A the variable change is ambiguous (i.e., it may rise, it may fall)
Table \(3\): Effects of a Production Subsidy
Production Subsidy by a Large Importing Country
Domestic Consumer Price
Domestic Producer Price
Domestic Consumer Welfare
Domestic Producer Welfare
Domestic Government Revenue
Domestic National Welfare
Foreign Price
Foreign National Welfare | textbooks/socialsci/Economics/International_Trade_-_Theory_and_Policy/09%3A_Trade_Policies_with_Market_Imperfections_and_Distortions/9.07%3A_Monopoly_and_Monopsony_Power_and_Trade.txt |
Learning Objectives
1. Learn that public goods, which have the features of being nonrival and nonexcludable in consumption, are a type of market imperfection.
2. Recognize that a trade policy can be used to correct for a public good imperfection.
3. Learn the first-best and second-best policy options to correct for a public good imperfection.
One of the oldest and most common arguments supporting protection is the “national security argument,” also called the “national defense argument.” This argument suggests that it is necessary to protect certain industries with a tariff to assure continued domestic production in the event of a war. Many products have been identified as being sufficiently important to warrant protection for this reason. Perhaps the most common industry identified is agriculture. Simply consider the problems that would arise if a nation did not have an adequate food supply when it was at war with the outside world. Low food stocks may induce severe hardships and even famine. A simple solution to avoid this potential problem is to maintain a sufficiently high tariff in order to keep cheap foreign goods out and, in turn, maintain production of the domestic goods.
Similar problems may arise in many other industries. Consider the potential problems for a country’s national security if it could not produce an adequate amount of steel, aluminum, ships, tanks, planes, fuel, and so on in the event of a war. The number of products that could be added to this list is enormous. Indeed, at one time or another in most countries’ histories, it has been argued that almost every product imaginable is important from a national security perspective and thus is deserving of protection. One of the most interesting arguments ever described is that made by the embroidery industry, which once argued for a protective tariff in the United States because embroidered patches on soldiers’ uniforms are essential in maintaining the morale of the troops. Thus it was clear, to them at least, that the embroidery industry needed to be protected for national security reasons.
National Security and Public Goods
We can make better sense of the national security argument if we classify it in the context of the theory of the second best. In this case, we must note that the national security argument is actually incorporating a market imperfection into the story to justify the use of a protective tariff. The market imperfection here is a public good. National security is a public good and public goods are excluded from the standard assumptions of perfect competition. Thus, whenever a product has public good characteristics, we can say that a market imperfection is present. Traditionally, the literature in economics refers to concerns such as national security as noneconomic objectives. The effects that food production may have on the nation’s sense of security, for example, were thought to fall outside the realm of traditional economic markets.
In general, public goods have the following two consumption characteristics: they are nonexcludable and they are nonrival. Nonexcludability means that once the product is produced, it is impossible to prevent people from consuming it. Nonrivalry means that many people can consume the produced product without diminishing its usefulness to others. Here are a few examples to explain the point. First, consider a nonpublic good: soda. A soda is excludable since the producer can put it into a can and require you to pay for it to enjoy its contents. A can of soda is also a rival good. That’s because if you consume the can of soda, there is no way for anyone else to consume the same can. This implies that a can of soda is not a public good. On the other hand, consider oxygen in the atmosphere. (This is an odd example because oxygen in the air is not formally produced, but let’s ignore that for a moment.) Atmospheric oxygen is nonexcludable because once it is there, everyone has free access to its use. It is impossible (or at least very difficult) to prevent some people from enjoying the benefits of the air. Atmospheric oxygen is also nonrival because when one person takes a breath, it does not diminish the usefulness of the atmosphere for others. Thus, if atmospheric oxygen did need to be formally produced, it would be a classic example of a pure public good.
The typical examples of public goods include national security, clean air, lighthouse services, and commercial-free television and radio broadcasts. National security is the public good we are most concerned with in international trade. It is a public good because, once provided, (1) it is difficult to exclude people within the country from the safety and security generated and (2) multiple individuals can enjoy the added safety and security without limiting that received by others.
We know from the theory of the second best that when market imperfections are present, government policies can be used to improve the national welfare. In most cases, trade policies can be used as well. It is well known in economic theory that when a good has public good characteristics, and if private firms are free to supply this good in a free market, then the public good will not be adequately supplied. The main problem occurs because of free ridership. If a person believes that others may pay for a good and if its subsequent provision benefits all people—due to the two public good features—then that person may avoid paying for the good in a private marketplace. If many people don’t pay, then the public good will be insufficiently provided relative to the true demands in the country. It is well known that government intervention can solve this problem. By collecting taxes from the public, and thus forcing everyone to pay some share of the cost, the public good can be provided at an adequate level. Thus national welfare can be increased with government provision of public goods.
A similar logic explains why a trade policy can be used to raise a country’s welfare in the presence of a public good. It is worth pointing out, though, that the goods highlighted above, such as agricultural products and steel production, are not themselves public goods. The public good one wishes to provide in greater abundance is “national security.” And it is through the production of certain types of goods locally that more security can be provided. For example, suppose it is decided that adequate national security is possible only if the nation can provide at least 90 percent of its annual food supplies during wartime. Suppose also that under free trade and laissez-faire domestic policies, the country produces only 50 percent of its annual food supply and imports the remaining 50 percent. Finally, suppose the government believes that it would be very difficult to raise domestic production rapidly in the event that imported products were ever cut off, as might occur during a war. In this case, a government may decide that its imports are too high and thus pose a threat to the country’s national security.
A natural response in this instance is to put high tariffs in place to prevent imports from crowding out domestic production. Surely, a tariff exists that will reduce imports to 10 percent and subsequently cause domestic production to rise to 90 percent. We know from tariff analysis that in the case of a small country, a tariff will cause a net welfare loss for the nation in a perfectly competitive market. These same gains and losses and net welfare effects can be expected to prevail here. However, because of the presence of the public good characteristics of national security, there is more to the story. Although the tariff alone causes a net welfare loss for the economy, the effect is offset with a positive benefit to the nation in the form of greater security. If the added security adds more to national welfare than the economic losses caused by the tariff, then overall national welfare will rise. Thus protectionism can be beneficial for the country.
The national security argument for protection is perfectly valid and sound. It is perfectly logical under these conditions that protectionism can improve the nation’s welfare. However, because of the theory of the second best, many economists remain opposed to the use of protectionism, even in these circumstances. The reason is that protectionism turns out to be a second-best policy option.
Recall that the first-best policy response to a market imperfection is a policy that is targeted as directly as possible at the imperfection itself. Thus, if the imperfection arises because of some production characteristic, a production subsidy or tax should be used. If the problem is in the labor market, a tax or subsidy in that market would be best, and if the market imperfection is associated with international trade, then a trade policy should be used.
In this case, one might argue that the problem is trade related, since one can say that national security is diminished because there are too many imports of, say, agricultural goods. Thus an import tariff should be used. However, this logic is wrong. The actual problem is maintaining an adequate food supply in a time of war. The problem is really a production problem because if imports were to be cut off in an emergency, the level of production would be too low. The most cost-effective way, in this situation, to maintain production at adequate levels will be a production subsidy. The production subsidy will raise domestic production of the good and can be set high enough to assure that an adequate quantity is produced each year. The subsidy will cost the government money and it will generate a net production efficiency loss. Nevertheless, the efficiency loss from a tariff, one that generates the same level of output as a production subsidy, will cause an even greater loss. This is because an import tariff generates both a production efficiency loss and a consumption efficiency loss. Thus, to achieve the same level of production of agricultural goods, a production subsidy will cost less overall than an import tariff. We say, then, that an import tariff is a second-best policy. The first-best policy option is a production subsidy.
Another Case in Which a Trade Policy Is First Best
There is one case in which a trade policy, used to protect or enhance national security, is the first-best policy option. Consider a country that produces goods that could be used by other countries to attack or harm the first country. An example would be nuclear materials. Some countries use nuclear power plants to produce electricity. Some of the products used in this production process, or the knowledge gained by operating a nuclear facility, could be used as an input in the production of more dangerous nuclear weapons. To prevent such materials from reaching countries, especially materials that may potentially threaten a country, export bans are often put into place. The argument to justify an export ban is that preventing certain countries from obtaining materials that may be used for offensive military purposes is necessary to maintain adequate national security.
In the United States, export bans are in place to prevent the proliferation of a variety of products. Many other products require a license from the government to export the product to certain countries. This allows the government to monitor what is being exported to whom and gives them the prerogative to deny a license if it is deemed to be a national security threat. In the United States, licenses are required for goods in short supply domestically; goods related to nuclear proliferation, missile technology, and chemical and biological weapons; and other goods that might affect regional stability, crime, or terrorist activities. In addition, the United States maintains a Special Designated Nationals list, which contains names of organizations to which sales of products are restricted, and a Denied Persons list, which contains names of individuals with whom business is prohibited. In recent years the United States has maintained export bans to several countries, including Cuba, Iran, Syria, and Sudan.
In this case, the export control policy is the first-best policy to enhance national security. This is because the fundamental problem is certain domestic goods getting into the hands of certain foreign nations, groups, or individuals. The problem is a trade problem best corrected with a trade policy. Indeed, there is no effective way to control these sales, and thus to enhance national security, using a purely domestic policy.
Key Takeaways
• The preservation of national security is a common justification for the use of protection.
• The preservation of national security is a type of noneconomic objective.
• Protection can help maintain an adequate domestic supply of materials critical in the event of war, including food, steel, military equipment, and petroleum.
• Export bans can be used to prevent the proliferation of materials that may eventually prove to be threatening to a nation’s security.
• Import tariffs can raise national welfare when increased production of the protected product enhances national security.
• Because national security is a public good and also an imperfection, trade protection can sometimes be beneficial for a country.
• A production subsidy can achieve the same level of production at a lower cost.
• A production subsidy is the first-best policy when increased production of a good enhances national security.
• An import tariff is a second-best policy option.
• An export ban can raise a nation’s welfare when the export of a product reduces national security.
• The export ban, a trade policy, is the first-best policy option when export of a product reduces national security.
Exercise \(1\)
1. Jeopardy Questions. As in the popular television game show, you are given an answer to a question and you must respond with the question. For example, if the answer is “a tax on imports,” then the correct question is “What is a tariff?”
1. The term used to describe policy intentions that are not economic in nature.
2. This is a common justification for import protection of food, steel, shipping, and many other things thought necessary under certain circumstances.
3. This policy is first best if a product in the hands of foreigners could threaten one’s national security.
4. Of a production subsidy or an import tariff, this policy is likely to be first best to protect a nation’s agricultural production.
5. The term describing a “good” like national security that is both nonexcludable and nonrival in consumption. | textbooks/socialsci/Economics/International_Trade_-_Theory_and_Policy/09%3A_Trade_Policies_with_Market_Imperfections_and_Distortions/9.08%3A_Public_Goods_and_National_Security.txt |
Learning Objectives
1. Learn that environmental externalities are a type of market imperfection.
2. Recognize that a trade policy can be used to correct for an environmental imperfection.
3. Learn the first-best and second-best policy options to correct for an environmental imperfection.
One contentious issue in international trade policy discussions concerns the connection between international trade and the environment. Many environmental groups claim that freer trade, as implemented through the World Trade Organization (WTO) agreements or in free trade agreements such as the North American Free Trade Agreement (NAFTA), results in negative environmental outcomes. For example, the Sierra Club argues, “Economic globalization ties the world together as never before. But it also poses serious new threats to our health and the environment. Trade agreements promote international commerce by limiting governments’ ability to act in the public interest. Already food safety, wildlife and pollution control laws have been challenged and weakened under trade rules as illegal ‘barriers to trade.’”Sierra Club, “A Fair Trade Bill of Rights,” Responsible Trade, www.sierraclub.org/trade/ftaa/rights.asp.
In contrast, the WTO, a frequent target for criticism by environmental groups, points to the WTO agreement, which states, “[WTO member] relations in the field of trade and economic endeavor should be conducted with a view to raising standards of living…while allowing for the optimal use of the world’s resources in accordance with the objective of sustainable development, seeking both to protect and preserve the environment and to enhance the means for doing so in a manner consistent with their respective needs and concerns at different levels of economic development.”World Trade Organization, “Environment Issues: Sustainable development,” http://www.wto.org/english/tratop_e/envir_e/sust_dev_e.htm.
Arguably, the stated goals of free trade–oriented groups and environmental groups are very similar, at least as highlighted in the documents produced by both sides. What differ are the methods used to achieve the objectives. For reasons to be elucidated below, the WTO has argued that environmental concerns are not directly within the purview of the WTO agreement, but despite that, environmental policies and international environmental agreements are neither prohibited by nor inconsistent with the WTO accords. In essence, the argument by some has been that the WTO agreement, and free trade agreements more generally, is intended to be about trade and is not intended to solve tangential problems related to the environment. On the other hand, environmental groups have pointed out that sometimes WTO and free trade agreement decisions have a negative effect on environmental outcomes, and thus these agreements should be revised to account for these negative effects.
Below we will consider these issues with respect to one type of environmental concern: pollution caused by consumption of an imported good. Although we will not consider many of the other contested environmental and trade issues, this one example will suffice to establish some important and generalizable conclusions.
Trade Liberalization with Environmental Pollution
Consider a small country importing gasoline with a tariff in place initially such that the domestic tariff-inclusive price is \(P_1\). At this price, domestic supply is \(S_1\), domestic demand (or consumption) is \(D_1\), and the level of imports is (\(D_1 − S_1\)), shown in Figure \(1\).
Suppose that domestic consumption of gasoline causes air pollution. This means that consumption has a negative external effect on all users of air—that is, there is a negative consumption externality.
Let’s assume that the cost to society (in dollar terms) of the air pollution is an increasing function of domestic consumption. In other words, the greater the consumption of gasoline, the greater is the pollution, and the greater is the subsequent harm caused to people in the country. For simplicity, assume the environmental cost, \(EC(D)\), is a linear function of total domestic demand, \(D\). The height of \(EC\) at any level of demand represents the additional dollar cost of an additional gallon of gasoline consumption. This implies the total environmental cost of a consumption level—say, \(D_2\)—is the area under the \(EC\) curve between the origin and \(D_2\).
With the initial tariff in place, domestic demand is \(D_1\), which implies that the total societal cost of pollution is given by the area (\(h + i + j\)). Note that despite the cost of pollution, it does make sense to produce and consume this good if the objective is national welfare. Consumer surplus is given by the area (\(a + b\)) and producer surplus is (\(c + g\)). The sum of these two clearly exceeds the social cost of pollution, (\(h + i + j\)). (Note that these statements are true for Figure \(1\) in particular; they are not true in general. By drawing the \(EC\) curve very steeply, corresponding to a much higher cost of pollution, it might not make sense to produce and consume the good in the market equilibrium.)
Next, suppose that the country agrees to remove the tariff on imported gasoline after signing a trade liberalization agreement. The question we ask is, Can trade liberalization have such a negative effect on the environment that it makes a country worse off? The answer, as we’ll see, is yes.
Suppose the tariff is removed and the price of gasoline falls to \(P_2\). The lower price causes a reduction in production to \(S_2\), an increase in consumption to \(D_2\), and an increase in imports from the blue line segment (\(D_1 − S_1\)) to the red line segment (\(D_2 − S_2\)). Since domestic consumption of gasoline rises, there is also an increase in pollution.
The welfare effects of the tariff elimination are summarized in Table \(1\). The letters refer to the area in Figure \(1\).
Table \(1\): Welfare Effects of a Tariff Elimination with a Negative Environmental Consumption Externality
Importing Country
Consumer Surplus + (c + d + e + f)
Producer Surplus c
Govt. Revenue e
Pollution Effect k
National Welfare (d + f) − k
Consumers of gasoline benefit by the areas (\(c + d + e + f\)) from the lower free trade price. Domestic producers lose (\(c\)) with a reduction in producer surplus. The government also loses tariff revenue (\)e\)). The net total efficiency gains from trade are given by the areas (\(d + f\)). However, the presence of the environmental consumption externality means there is an additional cost (\(k\)) caused by the pollution from higher domestic consumption of gasoline.
The national welfare effect of the tariff elimination is given by (\(d + f − k\)). For a particular level of efficiency gains, the total national effect will depend on the size of the pollution cost. In the graph, the curves are drawn such that area \(k\) is slightly larger than \(d + f\). Thus trade liberalization can cause a reduction in national welfare. The cost of additional pollution may be greater than the efficiency improvements from free trade. However, if the environmental cost of consumption were lower, the \(EC(D)\) line would be flatter and area \(k\) would become smaller. Thus for lower environmental costs, trade liberalization might raise national welfare. The net effect, positive or negative, will depend on the magnitude of the pollution costs relative to the efficiency benefits.
Trade Policy versus Domestic Policy
In general, the theory of the second best suggests that, in the presence of a market imperfection or distortion, a properly chosen trade policy might be found that will raise a small country’s national welfare. However, for most imperfections, a trade policy will be a second-best policy. A better policy, a first-best policy, will always be that policy that attacks the imperfection or distortion most directly. In most instances, the first-best policy will be a domestic policy rather than a trade policy.
In this case, environmental pollution caused by the consumption of gasoline is a market imperfection because gasoline consumption has a negative external effect (via pollution) on others within the society. Economists call this a negative consumption externality. This problem can be corrected with any policy that reduces the negative effect at a cost that is less than the benefit. A tariff on imports is one such policy that could work. However, the most direct policy option, hence the first-best policy choice, is a consumption tax. Below we’ll show the welfare effects of a tariff and a domestic consumption tax and compare the results to demonstrate why a consumption tax is first best while a tariff is second best.
Welfare Effects of a Tariff with Environmental Pollution
First, let’s consider the effects of a tariff when consumption of the import good causes pollution. Consider a small country importing gasoline at the free trade price given by \(P_2\) in Figure \(2\). (Note that this is Figure \(1\) redrawn.) Demand is given by \(D_2\), supply by \(S_2\), and imports are (\(D_2 − S_2\)) (the red line). Suppose that domestic consumption of gasoline causes air pollution. Assume the environmental cost of pollution in dollar terms, \(EC(D)\), is a linear function of total domestic demand, \(D\).
Next, suppose a specific tariff, \(t = P_1 − P_2\), is imposed, thereby raising the domestic price to \(P_1\). Domestic demand for gasoline falls to \(D_1\), supply rises to \(S_1\), and imports fall to (\(D_1 − S_1\)) (the blue line). The welfare effects of the tariff are presented in Table \(2\). The letters refer to the areas in Figure \(2\).
Table \(2\): Welfare Effects of a Tariff with a Negative Environmental Consumption Externality
Importing Country
Consumer Surplus − (c + d + e + f)
Producer Surplus + c
Govt. Revenue + e
Pollution Effect + k
National Welfare k − (d + f)
Consumers of gasoline lose surplus (\(c + d + e + f\)) from the higher domestic price. Domestic producers gain (\(+c\)) with an increase in producer surplus. The government also collects tariff revenue (\(+e\)). The net total efficiency losses from trade are given by the areas (\(d + f\)). However, the presence of the environmental consumption externality means there is an additional benefit caused by the reduced pollution. This benefit is represented by the area \(k\).
The net national welfare effect of the tariff is given by (\(k − d − f\)). Since the curves are drawn such that area \(k\) is slightly larger than \(d + f\), a tariff results in an improvement in national welfare in this example. More generally, we can only say that a tariff may result in an increase in national welfare since it will depend on the shapes of the curves and the size of the tariff.
Welfare Effects of a Consumption Tax with Environmental Pollution
Next, suppose that a consumption tax, \(t = P_1 − P_2\), is imposed instead of a tariff. Refer to Figure \(2\). The tax will raise the consumer’s price to \(P_1\) but will leave the producer’s price at \(P_2\). Domestic producers will not be affected by the consumption tax since continued competition in free trade with firms in the rest of the world will maintain their profit-maximizing price at the world price of \(P_2\). The price changes will cause domestic demand for gasoline to fall to \(D_1\), but supply will remain at \(S_2\). Imports will fall to (\(D_1 − S_2\)) (the yellow line). The welfare effects of the consumption tax are presented in Table \(3\). The letters refer to the area in Figure \(2\).
Table \(1\): Welfare Effects of a Domestic Consumption Tax with a Negative Environmental Consumption Externality
Importing Country
Consumer Surplus − (c + d + e + f)
Producer Surplus 0
Govt. Revenue + c + d + e
Pollution Effect + k
National Welfare kf
Consumers of gasoline lose from the higher price by the area (\(c + d + e + f\)). Domestic producers are unaffected because their price does not change. The government also collects tax revenue, given by (\(c + d + e\)), which is the product of the consumption tax (\(t = P_1 − P_2\)) and the level of consumption (\(D_1\)). The net total efficiency losses from trade are given by the area (\(f\)). However, the presence of the environmental consumption externality means there is an additional benefit caused by the reduced pollution. This benefit is represented by the area \(k\).
The net national welfare effect of the tariff is given by the summation of all effects, (\(k − f\)). Since the curves are drawn such that the area \(k\) is larger than \(f\), a consumption tax results in an improvement in national welfare in this example. More generally, we can only say that a consumption tax may result in an increase in national welfare since it will depend on the shapes of the curves and the size of the tax.
A Comparison: Trade Policy versus Domestic Policy
More interesting is the comparison between the welfare effects of a tariff and those of a consumption tax. Since the two policies are set at identical levels, it is easy to compare the effects. The distributional effects—that is, who wins and who loses—are slightly different in the two cases. First, the effects on consumers are the same since both policies raise the price to the same level. However, domestic producers suffer a loss in producer surplus with a tariff, whereas they are unaffected by the consumption tax. To some, this may look like a bad effect since domestic production of the polluting good is not reduced with the consumption tax. However, it is the net effect that matters. Next, the government collects more revenue with the domestic tax than with the tariff since both taxes are set at the same rate and consumption is greater than imports. Finally, the environmental effect is the same for both since consumption is reduced to the same level.
The net welfare effect of the consumption tax (\(NW^C = k − f\)) clearly must exceed the net welfare effect of a tariff (\(NW^T = k − d − f\))—that is, \(NW^C > NW^T\). The reason is that the tariff incurs two separate costs on society to receive the environmental benefit, whereas the consumption tax incurs only one cost for the same benefit. Specifically, the tariff causes a loss in both consumption and production efficiency (\(d\) and \(f\)), while the consumption tax only causes a consumption efficiency loss (\(f\)). For this reason, we say it is more efficient (i.e., less costly) to use a domestic consumption tax to correct for a negative consumption externality such as pollution than to use a trade policy, even though the trade policy may improve national welfare.
A Source of Controversy
For many environmental advocates, trade liberalization, or globalization more generally, clearly has the potential to cause environmental damage to many ecosystems. Concerns include pollution from industrial production, pollution from consumption, clear-cutting of tropical forests, extinction of plant and animal species, and global warming, among others. Although only one type of environmental problem is addressed above, the principles of the theory of the second best will generally apply to all these concerns.
The analysis above accepts the possibility that consumption causes pollution and that pollution is bad for society. The model shows that under these assumptions, a trade policy can potentially be used to improve environmental outcomes and can even be in society’s overall interest. However, a trade policy is not the most efficient means to achieve the end. Instead, resources will be better allocated if a domestic policy, such as a consumption tax, is used instead. Since the domestic policy attacks the distortion most directly, it minimizes the economic cost. For this reason, a properly chosen consumption tax will always do better than any tariff.
With respect to other types of environmental problems, a similar conclusion can be reached. The best way to correct for most pollution and other environmental problems will be to use a domestic policy intervention such as a production tax, consumption tax, factor-use tax, or another type of domestic regulation. Trade policies, although potentially beneficial, are not the most efficient policy instruments to use.
It is worth emphasizing that the goal of most economic analysis should in many instances be aligned with the goal of environmentalists. It is the extraction and use of natural resources that contributes to environmental damage. At the same time, it is the extraction and use of natural resources that is necessary to produce the goods and services needed to raise human standards of living to acceptable levels. Thus, if we minimize the use of resources to produce a particular level of output, we can achieve both the economist’s goal of maximizing efficiency and the environmentalist’s goal of minimizing damage to the environment.
Understanding the WTO’s Position on Trade and the Environment
In October 1999, the WTO Committee on Trade and Environment, a committee set up during the Uruguay Round to consider the linkages between these two concerns, issued its Trade and Environment report. The report argued that “there is no basis for the sweeping generalizations that are often heard in the public debate, arguing that trade is either good for the environment, or bad for the environment. The real world linkages are a little bit of both.”World Trade Organization, “Trade liberalization reinforces the need for environmental cooperation,” press release, October 8, 1999, http://www.wto.org/english/news_e/pres99_e/pr140_e.htm.
Some of the main findings of the report are listed here with a brief explanation of how these statements relate to the theory of the second best.
Most environmental problems result from polluting production processes, certain kinds of consumption, and the disposal of waste products—trade as such is rarely the root cause of environmental degradation, except for the pollution associated with transportation of goods.World Trade Organization, “Trade liberalization reinforces the need for environmental cooperation,” press release, October 8, 1999, http://www.wto.org/english/news_e/pres99_e/pr140_e.htm.
This statement relates to the theory of the second best by highlighting that the root cause of most environmental problems is the production, consumption, and disposal processes rather than trade. The one exception is pollution caused by ships, trucks, trains, and planes transporting goods across borders, but this is a relatively minor source of global pollution. Recall that first-best solutions are those that attack the root cause of a problem most directly.
Environmental degradation occurs because producers and consumers are not always required to pay for the costs of their actions.World Trade Organization, “Trade liberalization reinforces the need for environmental cooperation,” press release, October 8, 1999, http://www.wto.org/english/news_e/pres99_e/pr140_e.htm.
This statement means that environmental problems are a negative externality in either production or consumption. If producers and consumers had to pay for the environmental effects of their actions, that would mean there is a market for pollution. In a market, the costs and benefits are internalized in the decision-making process. However, in the absence of a market, producer and consumer effects occur “external” to the market, hence the term “externality.”
However, this statement exaggerates one thing if it suggests that environmental degradation would not occur if consumers and producers were required to pay for their actions. In actuality, if a market for pollution existed, producers and consumers would continue to pollute up to the level where the costs of additional pollution exceeded the benefits. This undoubtedly would occur at some positive level of pollution and environmental degradation. As demonstrated in every environmental economics course, the socially optimal level of pollution is not zero.
Environmental degradation is sometimes accentuated by policy failures, including subsidies to polluting and resource-degrading activities—such as subsidies to agriculture, fishing and energy.World Trade Organization, “Trade liberalization reinforces the need for environmental cooperation,” press release, October 8, 1999, http://www.wto.org/english/news_e/pres99_e/pr140_e.htm.
This statement points out that many environmental problems are made worse by government interventions designed to serve some other purpose. For example, subsidies to agricultural production, designed to support the income of farmers, can have the unintended effect of encouraging the greater use of pesticides and fertilizers, thus causing a negative environmental effect. Again, this suggests that the source of environmental problems is typically not international trade.
Trade would unambiguously raise welfare if proper environmental policies were in place.World Trade Organization, “Trade liberalization reinforces the need for environmental cooperation,” press release, October 8, 1999, http://www.wto.org/english/news_e/pres99_e/pr140_e.htm.
Here, “proper environmental policies” means first-best domestic policies targeted at the environmental market failures and the elimination of other domestic policies with the unintended environmental consequences mentioned above. If these domestic policies were in place, then free trade would unambiguously be the first-best trade policy.
Trade barriers generally make for poor environmental policy.World Trade Organization, “Trade liberalization reinforces the need for environmental cooperation,” press release, October 8, 1999, http://www.wto.org/english/news_e/pres99_e/pr140_e.htm.
Why? Because of the theory of the second best. It is generally better to correct environmental externality problems using first-best domestic taxes, subsidies, or regulations than to use second-best trade policies. Thus, although trade policies can have favorable environmental effects, governments can achieve the same results more efficiently—that is, at a lower resource cost—by using domestic policies instead.
This is one of the strongest arguments for excluding an explicit link between environment and trade in the WTO accords and more generally in free trade area agreements. Linking the two together in a trade agreement will surely lead to the avoidance of trade liberalization in some sectors in order to secure a favorable environmental outcome, and this will mean using trade barriers as a tool for environmental policy.
So what can or should be done? First, it is important to recognize that the WTO agreement does not prohibit countries from setting their own environmental standards. What the WTO accord does require is that countries apply most-favored nation (MFN) and national treatment in their application of environmental laws. For example, the WTO agreement does not allow a country to set one environmental standard with respect to goods imported from Argentina and another for goods from Mexico. This would violate MFN. Also, the WTO agreement would not allow a country to treat imported goods differently from goods produced at home. This would violate national treatment.
In fact, most of the WTO dispute settlement rulings (if not all) identified by environmental groups as forcing countries to change (and make more lenient) their environmental laws were not decisions to force a particular environmental standard on countries. Instead, they were decisions to enforce MFN or national treatment. Countries could have complied with any of these rulings by strengthening environmental regulations just as long as they did not discriminate internationally in their application.
Lastly, countries are not prohibited by the WTO agreement from negotiating and implementing international environmental agreements. A prime example is the Kyoto Protocol. This agreement would require signatory countries to reduce their domestic carbon emissions to agreed-on levels within a specified period of time in order to mitigate an important source of global warming. The mechanism used to reduce emissions in this case would be purely domestic policies implemented simultaneously by all signatory countries. As such, this would more likely be a first-best method to correct for global warming and would dominate any type of trade policy to solve the same problem.
One Final Issue: Measurement Problems
In the previous analysis, we assumed the environmental costs of consumption are measurable in dollar terms. However, obtaining these costs is not a simple exercise since there is no market in which pollution is traded. It may be relatively easy to measure the average amount of pollutants (carbon dioxide, sulfur, etc.) caused by each gallon of gasoline consumed, but translating that into a dollar equivalent is not a simple task. Ideally, we would want to know how much people would be willing to pay to prevent the pollution caused by each gallon of consumed gasoline. Environmental economists have tried to measure these types of costs using “contingent valuation” techniques. However, these methods are still in their infancy in terms of providing an accurate and believable measure of environmental cost.
Without good information concerning environmental costs, it becomes almost impossible to set policies appropriately. Although welfare-improving tariffs and domestic policies can raise national welfare, they must be set at correct levels to achieve a welfare-enhancing effect. To obtain the optimal levels requires accurate information about both the economic costs and the benefits of price changes and the environmental effects as well. Without good information, it becomes more likely policies will not achieve the intended effect.
An alternative method to measure costs is for the government to require permits that allow one to pollute. If these permits were tradable, the market price of a permit would provide a reasonable estimate of the pollution cost to society. In essence, this creates a market for pollution. These programs have been applied to control industrial pollutants but have not been used in consumer markets. In addition, to most noneconomists, providing permits that allow pollution seems anathema. However, because these programs attempt to correct for problems related to the measurement of environmental costs, they may be even more efficient even than using domestic taxes.
In the end, we must recognize that our theoretical analysis can only suggest the possibility that trade liberalization will make a country worse off due to increases in pollution. The model shows that this is logically possible. However, the model also shows it is logically possible for trade liberalization to raise national welfare despite increases in pollution. It then becomes an empirical question of what the effect of trade liberalization will be. For this reason, many environmental groups, such as Sierra Club, have proposed that an environmental impact statement (EIS) be prepared for every trade agreement. An EIS would assess the environmental costs of the agreement and thereby make environmental concerns a criterion in the decision process. Presumably, these studies could prevent environmentally unfriendly trade agreements from being ratified.
Many proponents of freer trade have objected to this proposal. Jagdish Bhagwati, for one, in his book In Defense of Globalization, suggests that the ability to measure the environmental costs may be as difficult as, or perhaps more difficult than, measuring the economic effects of a trade agreement.
Key Takeaways
• Environmental problems generally correspond to negative production or consumption externalities. Thus these issues represent market imperfections.
• This section presents a model in which domestic consumption of an import good causes environmental pollution (e.g., gasoline consumption). This is the case of a negative consumption externality.
• The model is used to show that trade liberalization may cause a reduction in national welfare if the additional pollution caused by increased consumption is greater than the efficiency benefits that arise from freer trade. Thus concerns that trade liberalization may cause environmental damage are consistent with economic theory.
• However, the theory of the second best suggests that when market imperfections exist, trade policy corrections may be second-best, not first-best, policy choices.
• Both an import tariff and a domestic consumption tax will reduce domestic consumption of the import good and lead to a reduction in pollution. However, the domestic consumption tax achieves the result at a lower economic cost than the import tariff. Thus we say that the domestic consumption tax is a first-best policy, while the import tariff is a second-best policy.
• The previous result corresponds to the general theory of the second best, which says that the first-best policy will be the policy that targets a distortion or imperfection most directly. In most cases, a domestic policy will be better than a trade policy. In this example, a domestic consumption tax is clearly superior to a trade policy.
Exercise \(1\)
1. Consider a perfectly competitive market for steel in a small exporting country. Suppose that steel production causes local air and water pollution. Assume that the larger is steel output and the higher is the social cost of pollution; thus, steel production creates a negative externality.
1. Explain what type of trade tax or subsidy policy could be used to reduce the negative effects of pollution. Demonstrate the welfare effects using a partial equilibrium diagram. Assume that your policy reduces social costs by R dollars. Under what condition would the policy raise national welfare?
2. Explain what type of purely domestic policy could be used to reduce the pollution. Use a partial equilibrium diagram to demonstrate the welfare effects of this policy. Again assume that your policy reduces social costs by R dollars.
3. Explain why the purely domestic policy may be superior to the trade policy. | textbooks/socialsci/Economics/International_Trade_-_Theory_and_Policy/09%3A_Trade_Policies_with_Market_Imperfections_and_Distortions/9.09%3A_Trade_and_the_Environment.txt |
Learning Objectives
1. Distinguish the different types of economic integration.
2. Learn the effects of trade creation and trade diversion.
3. Understand how free trade area formation can make a country worse off in terms of the theory of the second best.
For a variety of reasons, it often makes sense for nations to coordinate their economic policies. Coordination can generate benefits that are not possible otherwise. A clear example of this is shown in the discussion of trade wars among large countries in Chapter 7: Trade Policy Effects with Perfectly Competitive Markets, Section 7.9: Retaliation and Trade Wars. There it is shown that if countries cooperate and set zero tariffs against each other, then both countries are likely to benefit relative to the case when both countries attempt to secure short-term advantages by setting optimal tariffs. This is just one advantage of cooperation. Benefits may also accrue to countries that liberalize labor and capital movements across borders, that coordinate fiscal policies and resource allocation toward agriculture and other sectors, and that coordinate their monetary policies.
Any type of arrangement in which countries agree to coordinate their trade, fiscal, or monetary policies is referred to as economic integration. There are many different degrees of integration.
Preferential Trade Agreement
A preferential trade agreement (PTA) is perhaps the weakest form of economic integration. In a PTA, countries would offer tariff reductions, though perhaps not eliminations, to a set of partner countries in some product categories. Higher tariffs, perhaps nondiscriminatory tariffs, would remain in all other product categories. This type of trade agreement is not allowed among World Trade Organization (WTO) members, who are obligated to grant most-favored nation (MFN) status to all other WTO members. Under the MFN rule, countries agree not to discriminate against other WTO member countries. Thus, if a country’s low tariff on bicycle imports, for example, is 5 percent, then it must charge 5 percent on imports from all other WTO members. Discrimination or preferential treatment for some countries is not allowed. The country is free to charge a higher tariff on imports from non-WTO members, however. In 1998, the United States proposed legislation to eliminate tariffs on imports from the nations in sub-Saharan Africa. This action represents a unilateral preferential trade agreement since tariffs would be reduced in one direction but not the other. (Note that a PTA is also used more generally to describe all types of economic integration since they all incorporate some degree of “preferred” treatment.)
Free Trade Area
A free trade area (FTA) occurs when a group of countries agrees to eliminate tariffs among themselves but maintain their own external tariff on imports from the rest of the world. The North American Free Trade Agreement (NAFTA) is an example of an FTA. When NAFTA is fully implemented, tariffs of automobile imports between the United States and Mexico will be zero. However, Mexico may continue to set a different tariff than the United States on automobile imports from non-NAFTA countries. Because of the different external tariffs, FTAs generally develop elaborate “rules of origin.” These rules are designed to prevent goods from being imported into the FTA member country with the lowest tariff and then transshipped to the country with higher tariffs. Of the thousands of pages of text that make up NAFTA, most of them describe rules of origin.
Customs Union
A customs union occurs when a group of countries agrees to eliminate tariffs among themselves and set a common external tariff on imports from the rest of the world. The European Union (EU) represents such an arrangement. A customs union avoids the problem of developing complicated rules of origin but introduces the problem of policy coordination. With a customs union, all member countries must be able to agree on tariff rates across many different import industries.
Common Market
A common market establishes free trade in goods and services, sets common external tariffs among members, and also allows for the free mobility of capital and labor across countries. The EU was established as a common market by the Treaty of Rome in 1957, although it took a long time for the transition to take place. Today, EU citizens have a common passport, can work in any EU member country, and can invest throughout the union without restriction.
Economic Union
An economic union typically will maintain free trade in goods and services, set common external tariffs among members, allow the free mobility of capital and labor, and also relegate some fiscal spending responsibilities to a supranational agency. The EU’s Common Agriculture Policy (CAP) is an example of a type of fiscal coordination indicative of an economic union.
Monetary Union
A monetary union establishes a common currency among a group of countries. This involves the formation of a central monetary authority that will determine monetary policy for the entire group. The Maastricht treaty, signed by EU members in 1992, proposed the implementation of a single European currency (the Euro) by 1999.
Perhaps the best example of an economic and monetary union is the United States. Each U.S. state has its own government that sets policies and laws for its own residents. However, each state cedes control, to some extent, over foreign policy, agricultural policy, welfare policy, and monetary policy to the federal government. Goods, services, labor, and capital can all move freely, without restrictions among the U.S. states, and the nation sets a common external trade policy.
Multilateralism versus Regionalism
In the post–World War II period, many nations pursued the objective of trade liberalization. One device used to achieve this was the General Agreement on Tariffs and Trade (GATT) and its successor, the WTO. Although the GATT began with less than 50 member countries, the WTO now claims 153 members as of 2010. Since GATT and WTO agreements commit all member nations to reduce trade barriers simultaneously, the agreements are sometimes referred to as a multilateral approach to trade liberalization.
An alternative method used by many countries to achieve trade liberalization includes the formation of preferential trade arrangements, free trade areas, customs unions, and common markets. Since many of these agreements involve geographically contiguous countries, these methods are sometimes referred to as a regional approach to trade liberalization.
The key question of interest concerning the formation of preferential trade arrangements is whether these arrangements are a good thing. If so, under what conditions? If not, why not?
One reason supporters of free trade may support regional trade arrangements is because they are seen to represent movements toward free trade. Indeed, Section 24 of the original GATT allows signatory countries to form free trade agreements and customs unions despite the fact that preferential agreements violate the principle of nondiscrimination. When a free trade area or customs union is formed between two or more WTO member countries, they agree to lower their tariffs to zero between each other but will maintain their tariffs against other WTO countries. Thus the free trade area is a discriminatory policy. Presumably, the reason these agreements are tolerated within the WTO is because they represent significant commitments to free trade, which is another fundamental goal of the WTO.
However, there is also some concern among economists that regional trade agreements may make it more difficult, rather than easier, to achieve the ultimate objective of global free trade.
The fear is that although regional trade agreements will liberalize trade among their member countries, the arrangements may also increase incentives to raise protectionist trade barriers against countries outside the area. The logic here is that the larger the regional trade area relative to the size of the world market, the larger will be that region’s market power in trade. The more market power, the higher would be the region’s optimal tariffs and export taxes. Thus the regional approach to trade liberalization could lead to the formation of large “trade blocs” that trade freely among members but choke off trade with the rest of the world. For this reason, some economists have argued that the multilateral approach to trade liberalization, represented by the trade liberalization agreements in successive WTO rounds, is more likely to achieve global free trade than the regional or preferential approach.
Much has been written on this subject recently. Here we have merely scratched the surface.
In what follows, we present the economic argument regarding trade diversion and trade creation. These concepts are used to distinguish between the effects of free trade area or customs union formation that may be beneficial and those that are detrimental. As mentioned, preferential trade arrangements are often supported because they represent a movement in the direction of free trade. If free trade is economically the most efficient policy, it would seem to follow that any movement toward free trade should be beneficial in terms of economic efficiency. It turns out that this conclusion is wrong. Even if free trade is most efficient, it is not true that a step in that direction necessarily raises economic efficiency. Whether a preferential trade arrangement raises a country’s welfare and raises economic efficiency depends on the extent to which the arrangement causes trade diversion versus trade creation.
Trade Creation and Trade Diversion
In this section, we present an analysis of trade diversion and trade creation. The analysis uses a partial equilibrium framework, which means that we consider the effects of preferential trade liberalization with respect to a representative industry. Later in the section we consider how the results from the representative industry cases can be extended to consider trade liberalization that covers all trade sectors.
We assume in each case that there are three countries in the world: Countries A, B, and C. Each country has supply and demand for a homogeneous good in the representative industry. Countries A and B will form a free trade area. (Note that trade diversion and creation can occur regardless of whether a preferential trade agreement, a free trade area, or a customs union is formed. For convenience, we’ll refer to the arrangement as a free trade area [FTA].) The attention in this analysis will be on Country A, one of the two FTA members. We’ll assume that Country A is a small country in international markets, which means that it takes international prices as given. Countries B and C are assumed to be large countries (or regions). Thus Country A can export or import as much of a product as desired with Countries B and C at whatever price prevails in those markets.
We assume that if Country A were trading freely with either B or C, it would wish to import the product in question. However, Country A initially is assumed not to be trading freely. Instead, the country will have an MFN-specific tariff (i.e., the same tariff against both countries) applied on imports from both Countries B and C.
In each case below, we will first describe an initial tariff-ridden equilibrium. Then, we will calculate the price and welfare effects that would occur in this market if Countries A and B form an FTA. When the FTA is formed, Country A maintains the same tariff against Country C, the non-FTA country.
Trade Diversion
In general, a trade diversion means that a free trade area diverts trade away from a more-efficient supplier outside the FTA and toward a less-efficient supplier within the FTA. In some cases, trade diversion will reduce a country’s national welfare, but in some cases national welfare could improve despite the trade diversion. We present both cases below.
Figure \(1\) depicts the case in which trade diversion is harmful to a country that joins an FTA. The graph shows the supply and demand curves for Country A. \(P_B\) and \(P_C\) represent the free trade supply prices of the good from Countries B and C, respectively. Note that Country C is assumed to be capable of supplying the product at a lower price than Country B. (Note that in order for this to be possible, Country B must have tariffs or other trade restrictions on imports from Country C, or else all of B’s market would be supplied by C.)
We assume that A has a specific tariff \(t^B = t^C = t^*\) set on imports from both Countries B and C. The tariff raises the domestic supply prices to \(P_T^B\) and \(P_T^C\), respectively. The size of the tariff is denoted by the green dotted lines in Figure \(1\), which show that \(t^* = P_T^B − P^B = P_T^C − P^C\).
Since, with the tariff, the product is cheaper from Country C, Country A will import the product from Country C and will not trade initially with Country B. Imports are given by the red line, or by the distance \(D_1 − S_1\). Initial tariff revenue is given by the area (\(c + e\)), the tariff rate multiplied by the quantity imported.
Next, assume Countries A and B form an FTA and A eliminates the tariff on imports from Country B. Now, \(t^B = 0\), but \(t^C\) remains at \(t^*\). The domestic prices on goods from Countries B and C are now \(P^B\) and \(P_T^C\), respectively. Since \(P^B < P_T^C\), Country A would import all the product from Country B after the FTA and would import nothing from Country C. At the lower domestic price, \(P^B\), imports would rise to \(D_2 − S_2\), denoted by the blue line. Also, since the nondistorted (i.e., free trade) price in Country C is less than the price in Country B, trade is said to be diverted from a more-efficient supplier to a less-efficient supplier.
The welfare effects are summarized in Table \(1\).
Table \(1\): Welfare Effects of Free Trade Area Formation- Trade Diversion Cases
Country A
Consumer Surplus + (a + b + c + d)
Producer Surplus a
Govt. Revenue − (c + e)
National Welfare + (b + d) − e
Free trade area effects on Country A’s consumers. Consumers of the product in the importing country benefit from the free trade area. The reduction in the domestic price of both the imported goods and the domestic substitutes raises consumer surplus in the market. Refer to Table \(1\) and Figure \(1\) to see how the magnitude of the change in consumer surplus is represented.
Free trade area effects on Country A’s producers. Producers in the importing country suffer losses as a result of the free trade area. The decrease in the price of their product on the domestic market reduces producer surplus in the industry. The price decrease also induces a decrease in the output of existing firms (and perhaps some firms will shut down), a decrease in employment, and a decrease in profit, payments, or both to fixed costs. Refer to Table \(1\) and Figure \(1\) to see how the magnitude of the change in producer surplus is represented.
Free trade area effects on Country A’s government. The government loses all the tariff revenue that had been collected on imports of the product. This reduces government revenue, which may in turn reduce government spending or transfers or raise government debt. Who loses depends on how the adjustment is made. Refer to Table \(1\) and Figure \(1\) to see how the magnitude of the tariff revenue is represented.
Free trade area effects on Country A’s national welfare. The aggregate welfare effect for the country is found by summing the gains and losses to consumers, producers, and the government. The net effect consists of three components: a positive production efficiency gain (\(b\)), a positive consumption efficiency gain (\(d\)), and a negative tariff revenue loss (\(e\)). Notice that not all the tariff revenue loss (\(c + e\)) is represented in the loss to the nation. That’s because some of the total losses (area \(c\)) are, in effect, transferred to consumers. Refer to Table \(1\) and Figure \(1\) to see how the magnitude of the change in national welfare is represented.
Because there are both positive and negative elements, the net national welfare effect can be either positive or negative. Figure \(1\) depicts the case in which the FTA causes a reduction in national welfare. Visually, it seems obvious that area \(e\) is larger than the sum of \(a\) and \(b\). Thus, under these conditions, the FTA with trade diversion would cause national welfare to fall.
If conditions were different, however, the national welfare change could be positive. Consider Figure \(2\). This diagram differs from Figure \(1\) only in that the free trade supply price offered by Country B, \(P^B\), is lower and closer to Country C’s free trade supply price, \(P^C\). The description earlier concerning the pre- and post-FTA equilibria remains the same, and trade diversion still occurs. The welfare effects remain the same in direction but differ in magnitude. Notice that the consumer surplus gain is now larger because the drop in the domestic price is larger. Also notice that the net national welfare effect, (\(b + d − e\)), visually appears positive. This shows that in some cases, formation of an FTA that causes a trade diversion may have a positive net national welfare effect. Thus a trade diversion may be, but is not necessarily, welfare reducing.
Generally speaking, the larger the difference between the nondistorted prices in the FTA partner country and in the rest of the world, the more likely it is that trade diversion will reduce national welfare.
Trade Creation
In general, trade creation means that a free trade area creates trade that would not have existed otherwise. As a result, supply occurs from a more-efficient producer of the product. In all cases, trade creation will raise a country’s national welfare.
Figure \(3\) depicts a case of trade creation. The graph shows the supply and demand curves for Country A. \(P^B\) and \(P^C\) represent the free trade supply prices of the good from Countries B and C, respectively. Note that Country C is assumed to be capable of supplying the product at a lower price than Country B. (Note that in order for this to be possible, Country B must have tariffs or other trade restrictions on imports from Country C, or else all of B’s market would be supplied by C.)
We assume that A has a specific tariff, \(t^B = t^C = t^*\), set on imports from both Countries B and C. The tariff raises the domestic supply prices to \(P_T^B\) and \(P_T^C\), respectively. The size of the tariff is denoted by the green dotted lines in Figure \(3\) which show that \(t^* = P_T^B − P^B = P_T^C − P^C\).
Since, with the tariffs, the autarky price in Country A, labeled \(P^A\) in Figure \(3\), is less than the tariff-ridden prices \(P_T^B\) and \(P_T^C\), the product will not be imported. Instead, Country A will supply its own domestic demand at \(S_1 = D_1\). In this case, the original tariffs are prohibitive.
Next, assume Countries A and B form an FTA and A eliminates the tariff on imports from Country B. Now \(t_B = 0\), but \(t^C\) remains at \(t^*\). The domestic prices on goods from Countries B and C are now \(P^B\) and \(P_T^C\), respectively. Since \(P^B < P^A\), Country A would now import the product from Country B after the FTA. At the lower domestic price \(P^B\), imports would rise to the blue line distance, or \(D_2 − S_2\). Since trade now occurs with the FTA and it did not occur before, trade is said to be created.
The welfare effects are summarized in Table \(2\).
Table \(2\): Welfare Effects of Free Trade Area Formation- Trade Diversion Cases
Country A
Consumer Surplus + (a + b + c)
Producer Surplus a
Govt. Revenue 0
National Welfare + (b + c)
Free trade area effects on Country A’s consumers. Consumers of the product in the importing country benefit from the free trade area. The reduction in the domestic price of both imported goods and the domestic substitutes raises consumer surplus in the market. Refer to Table \(2\) and Figure \(3\) to see how the magnitude of the change in consumer surplus is represented.
Free trade area effects on Country A’s producers. Producers in the importing country suffer losses as a result of the free trade area. The decrease in the price of their product in the domestic market reduces producer surplus in the industry. The price decrease also induces a decrease in output of existing firms (and perhaps some firms will shut down), a decrease in employment, and a decrease in profit, payments, or both to fixed costs. Refer to Table \(2\) and Figure \(3\) to see how the magnitude of the change in producer surplus is represented.
Free trade area effects on Country A’s government. Since initial tariffs were prohibitive and the product was not originally imported, there was no initial tariff revenue. Thus the FTA induces no loss of revenue.
Free trade area effects on Country A’s national welfare. The aggregate welfare effect for the country is found by summing the gains and losses to consumers and producers. The net effect consists of two positive components: a positive production efficiency gain (\(b\)) and a positive consumption efficiency gain (\(c\)). This means that if trade creation arises when an FTA is formed, it must result in net national welfare gains. Refer to Table \(2\) and Figure \(3\)to see how the magnitude of the change in national welfare is represented.
Aggregate Welfare Effects of a Free Trade Area
The analysis above considers the welfare effects on participants in one particular market in one country that is entering into a free trade area. However, when a free trade area is formed, presumably many markets and multiple countries are affected, not just one. Thus, to analyze the aggregate effects of an FTA, one would need to sum up the effects across markets and across countries.
The simple way to do that is to imagine that a country entering an FTA may have some import markets in which trade creation would occur and other markets in which trade diversion would occur. The markets with trade creation would definitely generate national welfare gains, while the markets with trade diversion may generate national welfare losses. It is common for economists to make the following statement: “If the positive effects of trade creation are larger than the negative effects of trade diversion, then the FTA will improve national welfare.” A more succinct statement, though also somewhat less accurate, is that “if an FTA causes more trade creation than trade diversion, then the FTA is welfare improving.”
However, the converse statement is also possible—that is, “if an FTA causes more trade diversion than trade creation, then the FTA may be welfare reducing for a country.” This case is actually quite interesting since it suggests that a movement to free trade by a group of countries may actually reduce the national welfare of the countries involved. This means that a movement in the direction of a more-efficient free trade policy may not raise economic efficiency. Although this result may seem counterintuitive, it can easily be reconciled in terms of the theory of the second best.
Free Trade Areas and the Theory of the Second Best
One might ask, if free trade is economically the most efficient policy, how can it be that a movement to free trade by a group of countries can reduce economic efficiency? The answer is quite simple once we put the story of FTA formation into the context of the theory of the second best. Recall that the theory of the second best suggested that when there are distortions or imperfections in a market, then the addition of another distortion (like a trade policy) could actually raise welfare or economic efficiency. In the case of an FTA, the policy change is the removal of trade barriers rather than the addition of a new trade policy. However, the second-best theory works much the same in reverse.
Before a country enters an FTA, it has policy-imposed distortions already in place in the form of tariff barriers applied on imports of goods. This means that the initial equilibrium can be characterized as a second-best equilibrium. When the FTA is formed, some of these distortions are removed—that is, the tariffs applied to one’s FTA partners. However, other distortions remain—that is, tariffs applied against the nonmember countries. If the partial tariff removal substantially raises the negative effects caused by the remaining tariff barriers with the non-FTA countries, then the efficiency improvements caused by free trade within the FTA could be outweighed by the negative welfare effects caused by the remaining barriers outside the FTA, and national welfare could fall.
This is, in essence, what happens in the case of trade diversion. Trade diversion occurs when an FTA shifts imports from a more-efficient supplier to a less-efficient supplier, which by itself causes a reduction in national welfare. Although the economy also benefits through the elimination of the domestic distortions, if these benefits are smaller than the supplier efficiency loss, then national welfare falls. In general, the only way to assure that trade liberalization will lead to efficiency improvements is if a country removes its trade barriers against all countries.
Key Takeaways
• Countries can integrate by reducing barriers to trade under multilateral arrangements like the WTO or by entering into regional arrangements, including preferential trade agreements, free trade agreements, customs unions, common markets, or monetary unions.
• The formation of a free trade area can lead to trade creation or trade diversion.
• Trade creation involves new trade that would not exist without the FTA and is always beneficial for the countries in terms of national welfare.
• Trade diversion involves the shifting of trade away from one country toward one’s free trade partner and is sometimes detrimental to the countries in terms of national welfare.
• Losses caused by trade diversion can be understood in terms of the theory of the second best; because one market distortion remains when another is removed, welfare can fall.
Exercise \(1\)
1. Jeopardy Questions. As in the popular television game show, you are given an answer to a question and you must respond with the question. For example, if the answer is “a tax on imports,” then the correct question is “What is a tariff?”
1. An arrangement in which a group of countries agrees to eliminate tariffs among themselves but maintain their own external tariff on imports from the rest of the world.
2. The term used to describe a change in the pattern of trade in response to trade liberalization in which a country begins to import from a less-efficient supplier.
3. The term used to describe a change in the pattern of trade in response to trade liberalization in which a country begins to import from a more-efficient supplier. | textbooks/socialsci/Economics/International_Trade_-_Theory_and_Policy/09%3A_Trade_Policies_with_Market_Imperfections_and_Distortions/9.10%3A_Economic_Integration-_Free_Trade_Areas%2C_Trade_Creation%2C_and_Trade_Diversion.txt |
Trade policy analysis is often conducted from the implicit vantage point of a benevolent dictator poised to choose the best policies for a country. However, decisions about which policies to apply are rarely made by a sovereign but instead are usually made via a democratic political process. Whenever we consider how the political process affects economic decision making, we call it political economy.
The political economy aspects of trade policymaking are studied briefly in this chapter. Most important is how the concentrations and dispersion of the costs and benefits of trade policies tend to affect the decisions.
10: Political Economy and International Trade
Learning Objectives
1. Understand the motivations of a government in determining the policies that affect international trade.
In most economic models, it is assumed that consumers maximize utility, firms maximize profit, and governments maximize national welfare. Although one can reasonably object to any one of these assumptions, perhaps the one least likely to hold is the assumption about a government’s behavior. Governments are rarely comprised of a solitary decision maker whose primary interest is the maximum well-being of the nation’s constituents. Such a person, if he or she existed, could be labeled a “benevolent dictator.” Although historically some nations have been ruled almost single-handedly by dictators, most dictators could hardly be called benevolent.
The assumption that governments behave as if they had a benevolent dictator may have developed out of the philosophical traditions of utilitarianism. Utilitarianism, whose roots date to writings by Jeremy Bentham in the early 1800s, suggests that the objective of society should be to produce the greatest good for the greatest number. The objective of individuals is to obtain utility (happiness, satisfaction, well-being, etc.). In economic analysis, we presume that individuals obtain all their utility from the consumption of goods and services, and this motivates the behavioral assumption that consumers maximize utility. The assumption that firms maximize profit is based on the same logic. Profit affects the income of firm owners. The greater one’s income, the greater will be one’s consumption possibilities and thus the higher will be one’s utility. Thus profit is merely a means to an end, the end being greater utility. It is not unreasonable, then, that if the objective of individuals and firms is maximum utility, then the objective of a government might be to maximize utility for everyone.
But even if governments do not seek to maximize national welfare, it is still a valid exercise to investigate which policies would lead to maximum utility. Indeed, most of the analysis of trade policies does just this. Policy analysis identifies the differential welfare effects of various policies and points out which of these will lead to the greatest overall utility or welfare.
If one prescribes policies that also maximize national welfare, then one is making the value judgment that maximum national welfare is the appropriate goal for a government. If one presumes that governments do indeed seek to maximize national welfare, then the task is to explain why the choices that governments make are explainable as the outcome of a national welfare maximization exercise. An alternative approach is to consider other reasons for the choices made by governments. This is essentially the task of political economy models.
Political economy is a term that reflects the interaction between the economic system and the political system. Many traditional models of the economy make simplifying assumptions about the behavior of governments. Keeping the model simple is one reason for the assumption of a benevolent dictator. Political economy models attempt to explain more carefully the decision-making process of governments. Today, most governments can be best described as representative democracies. This means that government officials are elected, through some voting procedure, to “represent” the interests of their constituents in making government decisions.
The key issue in political economy and trade models is to explain how political features in democratic economies affect the choice of a trade policy. Among the key questions are the following:
1. Why do countries choose protection so often, especially given that economists have been emphasizing the advantages of free trade for three hundred or more years? In other words, if free trade is as good as economists say, then why do nations choose to protect?
2. In discussions of trade policies, why is so much attention seemingly given to a policy’s effects on businesses or firms and so little attention given to the effects on consumers?
3. Why do political discussions, even today, have a mercantilist spirit, wherein exports are hailed as beneficial, while imports are treated as harmful to the country?
Key Takeaways
• Economic modelers often seem to assume that governments will choose policies to maximize the nation’s welfare.
• Instead, most government policies arise from complex decision making in a representative democracy.
• “Political economy” is a term used to describe the process of government decisions on economic policies.
Exercise \(1\)
1. Jeopardy Questions. As in the popular television game show, you are given an answer to a question and you must respond with the question. For example, if the answer is “a tax on imports,” then the correct question is “What is a tariff?”
1. The name for the philosophical ideas suggesting that the purpose of society is to create the greatest good for the greatest number.
2. The name for a solitary leader of a country whose intention is to maximize the well-being of the nation’s constituents.
3. The term used to describe the interaction of the political system and the economic system.
4. The term used to describe a contention that exports are good and imports are bad for a country. | textbooks/socialsci/Economics/International_Trade_-_Theory_and_Policy/10%3A_Political_Economy_and_International_Trade/10.1%3A_Chapter_Overview.txt |
Learning Objectives
1. Understand how and why lobbying is used to influence the policy decisions of a government.
A government represents the interests of its citizens. As Abraham Lincoln said in the Gettysburg Address, a democratic government is meant to be by the people and for the people. Thus, in a representative democracy, government officials are entrusted to take actions that are in the interests of their constituents. Periodic elections allow citizens to vote for individuals they believe will best fulfill their interests. If elected officials do not fulfill the interests of their constituents, then those constituents eventually have a chance to vote for someone else. Thus, if elected officials are perceived as good representatives of their constituents’ interests, they are likely to be reelected. If they follow their own individual agenda, and if that agenda does not match the general interests of their constituents, then they may lose a subsequent bid for reelection.
Citizens in democratic societies are traditionally granted the right to free speech. It is generally accepted that people should be allowed to voice their opinions about anything in front of others. In particular, people should be free to voice their opinions about government policies and actions without fear of reprisal. Criticisms of, as well as recommendations for, government policy actions must be allowed if a truly representative government is to operate effectively.
The Nature of Lobbying
We can define lobbying as the activity wherein individual citizens voice their opinions to the government officials about government policy actions. It is essentially an information transmission process. By writing letters and speaking with officials, individuals inform the government about their preferences for various policy options under consideration. We can distinguish two types of lobbying: casual lobbying and professional lobbying.
Casual lobbying occurs when a person uses his leisure time to petition or inform government officials of his point of view. Examples of casual lobbying are when people express their opinions at a town meeting or when they write letters to their Congress members. In these cases, there is no opportunity cost for the economy in terms of lost output, although there is a cost to the individual because of the loss of leisure time. Casual lobbying, then, poses few economic costs except to the individual engaging in the activity.
Professional lobbying occurs when an individual or company is hired by someone to advocate a point of view before the government. An example is a law firm hired by the steel industry to help win an antidumping petition. In this case, the law firm will present arguments to government officials to try to affect a policy outcome. The law firm’s fee will come from the extra revenue expected by the steel industry if it wins the petition. Since, in this case, the law firm is paid to provide lobbying services, there is an opportunity cost represented by the output that could have been produced had the lawyers engaged in an alternative productive activity. When lawyers spend time lobbying, they can’t spend time writing software programs, designing buildings, building refrigerators, and so on. (This poses the question, What would lawyers do if they weren’t lawyering?) The lawyers’ actions with this type of lobbying are essentially redistributive in nature, since the lawyers’ incomes will derive from the losses that will accrue to others in the event that the lobbying effort is successful. If the lobbying effort is not successful, the lawyers will still be paid, only this time the losses will accrue to the firm that hired the lawyers. For this reason, lobbying is often called rent seeking because the fees paid to the lobbyists come from a pool of funds (rents) that arise when the lobbying activity is successful. Another name given to professional lobbying in the economics literature is a directly unproductive profit-seeking (DUP) activity .
Lobbying is necessary for the democratic system to work. Somehow information about preferences and desires must be transmitted from citizens to the government officials who make policy decisions. Since everyone is free to petition the government, lobbying is the way in which government officials can learn about the desires of their constituents. Those who care most about an issue will be more likely to voice their opinions. The extent of the lobbying efforts may also inform the government about the intensity of the preferences as well.
Key Takeaways
• In a representative democracy, citizens have the right to both elect their representatives and discuss policy options with their elected representatives.
• Lobbying is the process of providing information to elected officials to influence the policies that are implemented.
• A directly unproductive profit-seeking (DUP) activity is any action that by itself does not directly produce final goods and services consumed by a country’s consumers.
Exercise \(1\)
1. Jeopardy Questions. As in the popular television game show, you are given an answer to a question and you must respond with the question. For example, if the answer is “a tax on imports,” then the correct question is “What is a tariff?”
1. The term for a democratic system in which government agents are entrusted to take actions in the best interests of the voting public.
2. The term used to describe activities to petition the government for particular policies.
3. This type of lobbying does not incur an opportunity cost of forgone production.
4. This type of lobbying does incur an opportunity cost of forgone production.
5. The term used to describe the extra revenues earned because of successful lobbying efforts.
6. The term describing the purposeful effort to direct money away from others and toward oneself.
7. Economic activities defined by the acronym DUP. | textbooks/socialsci/Economics/International_Trade_-_Theory_and_Policy/10%3A_Political_Economy_and_International_Trade/10.2%3A_Some_Features_of_a_Democratic_Society.txt |
Learning Objectives
1. Depict numerical values for the welfare effects of a tariff by a small country.
Consider the market for blue jeans in a small importing country, depicted in Figure $1$. Suppose a sudden increase in the world supply of jeans causes the world market price to fall from $35 to$30. The price decrease causes an increase in domestic demand from nine to ten million pairs of jeans, a decrease in domestic supply from eight to six million pairs, and an increase in imports from one to four million.
Because of these market changes, suppose that the import-competing industry uses its trade union to organize a petition to the government for temporary protection. Let’s imagine that the industry calls for a $5 tariff so as to reverse the effects of the import surge. Note that this type of action is allowable to World Trade Organization (WTO) member countries under the “escape clause” or “safeguards clause.” We can use the measures of producer surplus and consumer surplus to calculate the effects of a$5 tariff. These effects are summarized in Table $1$. The dollar values are calculated from the respective areas on the graph in Figure $1$.
Table $1$: Welfare Effects of an Import Tariff
Area on Graph $Value Consumer Surplus − (a + b + c + d) −$47.5 million
Producer Surplus + a + $35 million Govt. Revenue + c +$5 million
National Welfare − (b + d) − $7.5 million Notice that consumers lose more than the gains that accrue to the domestic producers and the government combined. This is why national welfare is shown to decrease by$7.5 million.
In order to assess the political ramifications of this potential policy, we will make some additional assumptions. In most markets, the number of individuals that makes up the demand side of the market is much larger than the number of firms that makes up the domestic import-competing industry. Suppose, then, that the consumers in this market are made up of millions of individual households, each of which purchases, at most, one pair of jeans. Suppose the domestic blue jeans industry is made up of thirty-five separate firms.
Key Takeaways
• With quantities, prices, and the tariff rate specified, actual values for the changes in consumer and producer surplus and government revenue can be determined.
Exercise $1$
1. Suppose the supply and demand curves for bottles of Coke are given by,
$S = 10P - 7 \nonumber$
$D = 13 – 5P \nonumber$
where $P$ is the price of Coke per bottle, $D$ is the quantity of Coke demand (in millions of bottles), and $S$ is the quantity of Coke supply (in millions of bottles). Suppose the free trade price of Coke is $1.00 and that a tariff of$0.20 is being considered by the government. If the country is a small importer calculate the following:
1. The value of the increase in producer surplus expected due to the tariff.
2. The value of the decrease in consumer surplus expected due to the tariff.
3. The value of the tariff revenue expected due to the tariff.
4. The value of the change in national welfare expected due to the tariff. | textbooks/socialsci/Economics/International_Trade_-_Theory_and_Policy/10%3A_Political_Economy_and_International_Trade/10.3%3A_The_Economic_Effects_of_Protection%3A_An_Example.txt |
Learning Objectives
1. Learn the lobbying implications of the widely dispersed costs of protection to consumers.
If the \$5 tariff is implemented, it will raise the price from \$30 to \$35. Consumption will fall from ten million to nine million pairs of jeans. Because of our simplifying assumption of one household per pair of jeans, one million households will decide not to purchase jeans because of the higher price. They will use the \$35 to buy something else they think is more valuable than jeans. The other nine million households will pay the extra \$5. This means that, at most, a household has to pay an extra \$5 for the same pair of jeans. In terms of consumer surplus loss, nine million consumers lose \$5 each for a total of \$45 million (area \(a + b + c\)), while the remaining one million lose a total of \$2.5 million (area \(d\)).
We can now ask whether a household would be willing to lobby the government to oppose the blue jeans tariff because of the extra cost they would incur. The likely answer is no. For most households, such a small price increase would hardly be noticed. Most consumers do not purchase blue jeans frequently. Also, blue jeans with different styles and brand names typically differ considerably in price. Consumers, who rarely keep track of events affecting particular markets, are unlikely to know that a tariff has even been implemented on the product considered or discussed.
If a person did know of an impending tariff, then presumably \$5 is the maximum a household would be willing to pay toward a lobbying effort, since that is the most one can gain if a tariff is prevented. One might argue that if even a fraction of the \$5 could be collected from some portion of the ten million consumer households, millions of dollars could be raised to contribute to an opposition lobbying effort. However, collecting small contributions from such a large group would be very difficult to do effectively.
Consider the problems one would face in spearheading a consumer lobbying effort to oppose the blue jeans tariff in this example. A seemingly reasonable plan would be to collect a small amount of money from each household hurt by the tariff and use those funds to pay for a professional lobbying campaign directed at the key decision makers. The first problem faced is how to identify which households are likely to be affected by the tariff. Perhaps many of these households purchased blue jeans last year, but many others may be new to the market in the upcoming year. Finding the right people to solicit money from would be a difficult task.
Even if you could identify them, you would have to find a way to persuade them that they ought to contribute. Time spent talking to each household has an opportunity cost to the household member since that person could be doing something else. Suppose that a person values her time at the hourly wage rate that she earns at her job. If she makes \$20 per hour, then you’ll have less than fifteen minutes to convince her to contribute to the lobbying effort since fifteen minutes is worth the \$5 you are trying to save for her. The point here is that even learning about the problem is costly for the household. For small savings, a lobbying group will have to convince its contributors very quickly.
Suppose we knew the names and addresses of the ten million affected households. Perhaps we could send a letter to each of them with a stamped return envelope asking to return it with a \$2 or \$3 contribution to the lobbying effort. With this plan, even purchasing the stamps to mail the envelopes would cost \$3,400,000. One would need to get over half of the households to send in \$3 each just to cover the cost of the mailing. Recipients of the letters will reasonably question the trustworthiness of the solicitation. Will the money really be put to good use? The chances of getting any more than a small return from this kind of solicitation is highly unlikely.
If contributions can be collected, the lobbying group will face another problem that arises with large groups: free ridership. Free riding occurs when someone enjoys the benefits of something without paying for it. The lobbying effort, if successful, will benefit all blue jeans consumers regardless of whether they contributed to the lobbying campaign. In economic terms, we say that the lobbying effort is a public good because individual households cannot be excluded from the benefits of successful lobbying. One of the key problems with public good provision is that individuals may be inclined to free ride—that is, to obtain the benefit without having contributed to its provision. Those who do not contribute also get the added benefit of the full \$5 surplus if the lobbying campaign is successful.
The main point of this discussion, though, is that despite the fact that \$47.5 million dollars will be lost to consumers of blue jeans if the \$5 tariff is implemented, it is very unlikely that this group will be able to form a lobbying campaign to oppose the tariff. Since each household will lose \$5 at most, it is extremely unlikely for any reasonable person to spend sufficient time to mount a successful lobbying campaign. Even if one person or group decided to spearhead the effort and collect contributions from others, the difficulties they would face would likely be insurmountable. In the end, government decision makers would probably hear very little in the way of opposition to a proposed tariff.
Many of the arguments are discussed in detail in Mancur Olson’s well-known book The Logic of Collective Action. One of the book’s key points is that large groups are much less effective than small groups in applying effective lobbying pressure on legislators.
Key Takeaways
• Although the loss of consumer surplus is the largest welfare effect of a tariff, because the number of consumers affected is also very large, the effect on each consumer is relatively small.
• Because a large number of consumers are affected to a small degree, it is difficult to identify precisely which consumers are affected by the tariff.
• Because the per-consumer cost of the tariff is low, most individual lobbying efforts to protest the tariff will cost the individual more than the cost of the tariff.
• Large groups are much less effective than small groups in applying effective lobbying pressure on legislators.
Exercise \(1\)
1. Jeopardy Questions. As in the popular television game show, you are given an answer to a question and you must respond with the question. For example, if the answer is “a tax on imports,” then the correct question is “What is a tariff?”
1. The term used to describe when a person receives a benefit, especially a public good, without contributing a fair share to pay for it.
2. Of concentrated or dispersed, this is how to describe the typical losses that accrue to consumers because of an import tariff.
3. The amount of money lost by each consumer of coffee due to a \$0.25 tariff if \$35 million is lost in consumer surplus in a market of seventy million consumers.
4. Of small or large, this sized group is more likely to form an effective lobby. | textbooks/socialsci/Economics/International_Trade_-_Theory_and_Policy/10%3A_Political_Economy_and_International_Trade/10.4%3A_The_Consumers%27_Lobbying_Decision.txt |
Learning Objectives
1. Learn the lobbying implications of the concentrated benefits of protection to producer interests.
On the producers’ side, let’s assume that there are thirty-five separate, and equally sized, firms. If a \$5 tariff is implemented, producers as a group would gain \$35 million in producer surplus. That means each firm stands to gain \$1 million. Domestic producers would also supply two million additional pairs of jeans, and that would require expansion of the industry labor force. Clearly, the tariff would be beneficial to the firm owners and to industry workers. The potential to expand production, add workers, and increase profits by \$1 million per firm will provide a strong motivation to participate in a lobbying effort. In the case of the firms, however, organization of a lobbying effort will be much easier than the opposing effort by consumers.
First of all, the \$1 million surplus accruing to each firm is pure gravy. Payments to workers and other factors are not a part of the \$1 million additional surplus; thus it is money over and above the marginal costs of additional production. For this reason, profit received in this manner is often referred to as “economic rents.” Since the rents are concentrated in a small number of firms, with \$1 million going to each, each firm will have a strong incentive to participate in a lobbying campaign. But who’s going to spearhead the effort?
Organization of a lobbying campaign will probably be easier for firms than for consumers. First, the industry may have an industry association that maintains continual links with policymakers in state and federal governments. The workers in the industry might also belong to a trade union, which would also have interests in supporting a lobbying effort. Or a few of the industry leaders could take it upon themselves to begin the effort (although that is assumed away in the example). Second, as a smaller group, it is easy to identify the likely beneficiaries from the tariff and to solicit contributions. The lobbying group should easily be able to collect millions of dollars to support an extensive lobbying effort. A mere contribution of \$50,000 per firm would generate \$1.75 million that could be used to hire a professional lobbying team. Even if the chances of a successful outcome are small, it may still be practical for the firms to contribute to a lobbying effort. The return on that \$50,000 “investment” would be \$1 million if successful. That’s a 2,000 percent rate of return—much higher than any brick-and-mortar investment project that might be considered. Free riding would also be less likely to occur since with only thirty-five firms to keep track of, contributors would probably learn who is not participating. Nonparticipation would establish a poor reputation for a firm and could have unpleasant consequences in its future industry association dealings.
With a well-financed lobbying effort, it would not be difficult to make decision makers aware that there is resounding support for the tariff within the industry community. Newspaper and television ads could be purchased to raise public awareness. Interested parties could be flown to the capitol to speak with key decision makers. In this way, the chances of obtaining the tariff may be increased substantially.
The Mancur Olson result applies in reverse to small groups. Small groups are much more effective than large groups in applying effective lobbying pressure on legislators.
Key Takeaways
• Although the increase in producer surplus is a small welfare effect in the example, because the number of producers affected is relatively small, the effect of the tariff on each producer is relatively large.
• Because a small number of producers are affected to a sizeable degree, it is easy to identify who is positively affected by the tariff.
• Because the per-producer benefit of the tariff is high, firm lobbying efforts to promote the tariff will likely be a worthwhile investment.
• Small groups are much more effective than large groups in applying effective lobbying pressure on legislators.
Exercise \(1\)
1. Jeopardy Questions. As in the popular television game show, you are given an answer to a question and you must respond with the question. For example, if the answer is “a tax on imports,” then the correct question is “What is a tariff?”
1. Of concentrated or dispersed, this is how to describe the typical benefits that accrue to producers because of an import tariff.
2. The amount of money gained by each producer of coffee due to a \$0.25 tariff if \$20 million is gained in producer surplus in a market of twenty producing firms.
3. The rate of return on a \$50,000 lobbying expense if that lobbying results in a \$0.30 tariff on coffee and nets the firm an additional \$3,000,000 in profit.
4. Of consumer groups or producer groups, this group is more likely to form an effective lobby. | textbooks/socialsci/Economics/International_Trade_-_Theory_and_Policy/10%3A_Political_Economy_and_International_Trade/10.5%3A_The_Producers%27_Lobbying_Decision.txt |
Learning Objectives
1. Learn how the lobbying and tariff revenue implications of a tariff affect the decision of the government.
How the government decides whether to offer the \$5 tariff, and who decides, will depend on the procedural rules of the democratic country in question. The tariff might be determined as a part of an administered procedure, such as a safeguards action or an antidumping action. Or the tariff may be determined as a part of a bill to be voted on by the legislature and approved, or not, by the executive. Rather than speaking about a particular type of government action, however, we shall consider the motivations of the government more generically.
The first thing the government may notice when being petitioned to consider raising the tariff is that government revenues will rise by \$5 million. Relative to many government budgets, this is a small amount, and so it may have very little influence on policymakers’ decision. However, it will help reduce a budget deficit or add to the monies available for spending on government programs. Thus it could have a small influence.
In a democratic society, governments are called on to take actions that are in the interests of their constituents. If government officials, in this example, merely listen to their constituents, one thing should be obvious. The arguments of the industry seeking protection will surely resonate quite loudly, while the arguments of the consumers who should be opposed to the tariff will hardly even be heard. If a government official bases his or her decision solely on the “loudness” of the constituents’ voices, then clearly he or she would vote for the tariff. This is despite the fact that the overall cost of the tariff to consumers outweighs the benefits to the industry and the government combined.
Notice that the decision to favor the tariff need not be based on anything underhanded or illegal on the part of the industry lobbyists. Bribes need not be given to secure votes. Nor does the industry lobby need to provide false or misleading information. Indeed, the lobby group could provide flawlessly accurate information and still win the support of the officials. Here’s why.
It would be natural for the industry lobby group to emphasize a number of things. First, jobs would be saved (or created) as a result of the tariff. If a number can be attached, it will be. For example, suppose the industry supported 25,000 jobs in the initial equilibrium, when eight million pairs of jeans were produced by the domestic industry. That averages to 320 jeans produced per worker. Thus, when the industry cuts production by two million units, it amounts to 6,250 jobs. The lobby group could then frequently state the “fact” that the tariff will create 6,250 jobs. Second, the lobby would emphasize how the tariff would restore the vitality of the industry. If a surge of imports contributed to the problem, then the lobby would undoubtedly blame foreign firms for taking jobs away from hardworking domestic citizens. Finally, the lobby would emphasize the positive government budget effects as a result of the tariff revenue. All of this information clearly would be quite true.
If the lobby mentioned the higher prices that would result from the tariff, surely it would argue it is a small price to pay to save so many jobs. The lobby might even convince consumers of blue jeans that it is worth paying extra for jeans because it will save domestic jobs. After all, perhaps their own jobs will one day be in jeopardy due to imports. Plus, it is such a small price to pay: at only \$5 extra, no one will even notice!
For a politician facing potential reelection, there is another reason to support the industry over the consumers, even with full information about the effects. Support of the industry will probably generate more future votes. Here’s why.
First, since industry members—management and workers—have a bigger stake in the outcome, they will be more likely to remember the politician’s support (or lack of support) on this issue at election time. Second, the politician can use his support for the industry in his political ads. Consider this political ad if he supports the industry: “I passed legislation that created over six thousand jobs!” Compare it with this truthful ad if he doesn’t support the industry: “By opposing protectionist legislation, I saved you five bucks on blue jeans!” Which one do you think sounds better?
Key Takeaways
• If representatives in a democracy base policy choices on the interests of their constituents and if industry lobbyists are more organized and “vocal” in their demands than consumers, then governments will more likely choose policies like tariffs.
Exercise \(1\)
1. Jeopardy Questions. As in the popular television game show, you are given an answer to a question and you must respond with the question. For example, if the answer is “a tax on imports,” then the correct question is “What is a tariff?”
1. Of consumer voices or producer voices, these are more likely to be heard by government officials in a representative democracy.
2. Of consumer interests or producer interests, governments in representative democracies are more likely to implement policies favoring these.
10.7: The Lobbying Problem in a Democracy
Learning Objectives
1. Recognize some of the problems and pitfalls of policy choice in a representative democracy.
There is a real problem with the lobbying process in democratic societies. Even though lobbying is a legitimate process of information transfer between constituents and government decision makers, it also produces some obvious disparities. Whenever policy actions generate concentrated benefits and dispersed costs, the incentives and abilities to lobby are significantly different across groups. Potential beneficiaries can often use the advantage of small group size and large potential windfalls to wield disproportionate influence on decision makers. Potential losers, whose numbers are large and whose expected costs per person are quite small, have almost no ability to lobby the government effectively. Thus, in a democratic society in which lobbying can influence decisions, decisions are likely to be biased in the favor of those policies that generate concentrated benefits and dispersed losses.
Unfortunately, and perhaps coincidentally, most policy actions taken produce concentrated benefits and dispersed losses. In the case of trade policies, most protectionist actions will cause concentrated benefits to accrue to firms, whereas losses will be dispersed among millions of consumers. This means that protectionist policies are more likely to win political support, especially when lobbying can directly affect legislated actions. Protectionism can easily occur even though the sum total effects of the policy may be negative.
In many countries, a protectionist tendency is reflected in the type of trade policy procedures that are available by law. Escape clause, antisubsidy, and antidumping policies are examples of laws designed to protect firms and industries in particular situations. In evaluating these types of petitions in the United States, there is no requirement that effects on consumers be considered in reaching a decision. Clearly, these laws are designed to protect the concentrated interests of producing firms. It would not be surprising, and indeed it seems likely, that the concentrated interests of businesses affected the ways in which the laws were originally written. The absence of a consumer lobby would also explain why consumer effects are never considered in these actions.
Key Takeaways
• Democratic governments are more likely to choose policies that generate concentrated benefits and dispersed losses, regardless of whether the sum total effects are positive or negative.
Exercise \(1\)
1. Suppose a small country implements a tariff on chicken imports. In the table below indicate whether each group is a winner or loser and whether the effects on that group are concentrated or dispersed.
Table \(1\): Political Economy Effects of a Tariff
Name of Group Winners or Losers Concentrated or Dispersed
Chicken Producers
Chicken Consumers
Taxpayers or Recipients of Government Benefits
2. Suppose a small country implements an export subsidy on soybeans. In the table below indicate whether each group is a winner or loser and whether the effects on that group are concentrated or dispersed.
Table \(2\): Political Economy Effects of an Export Subsidy
Name of Group Winners or Losers Concentrated or Dispersed
Soybean Producers
Soybean Consumers
Taxpayers or Recipients of Government Benefits | textbooks/socialsci/Economics/International_Trade_-_Theory_and_Policy/10%3A_Political_Economy_and_International_Trade/10.6%3A_The_Government%27s_Decision.txt |
Perhaps the most important policy issue of an international trade course is to answer the question “Should a country pursue free trade or some type of selected protection?” Academics, philosophers, policy analysts, and legislators have addressed this question for hundreds of years. And unfortunately, there is still no definitive answer.
The reason is that both free trade and selected protection have both positive and negative aspects. No one policy choice is clearly superior. Nonetheless, economists who have studied trade theory and policy tend to support free trade more so than just about any other contentious economic policy under public consideration. The reasons for this near consensus are complex and poorly understood by the general public. This chapter explains the economic case for free trade through the lens of trade theory and argues that even though free trade may not be “optimal,” it is nonetheless the most pragmatic policy option a country can follow.
• 11.1: Introduction
The original arguments for free trade began to supplant mercantilist views in the early to mid-eighteenth century. Many of these original ideas were based on simple exchange or production models that suggested that free trade would be in everyone’s best interests and surely in the national interest. During the nineteenth and twentieth centuries, however, a series of objections were raised suggesting that free trade was not in everyone’s interest and perhaps was not even in the national interest.
• 11.2: Economic Efficiency Effects of Free Trade
The main source of support for free trade lies in the positive production and consumption efficiency effects. In every model of trade, there is an improvement in aggregate production and consumption efficiency when an economy moves from autarky to free trade. This is equivalent to saying that there is an increase in national welfare.
• 11.3: Free Trade and the Distribution of Income
Although most trade models suggest that aggregate economic efficiency is raised with free trade, these same models do not indicate that every individual in the economy will share in the benefits. Indeed, most trade models demonstrate that movements to free trade will cause a redistribution of income between individuals within the economy. In other words, some individuals will gain from free trade while others will lose.
• 11.4: The Case for Selected Protection
An argument for selected protection arises in the presence of imperfectly competitive markets, market distortions, or both. In these cases, it is often possible to show that an appropriately targeted trade policy (selected protection) can raise aggregate economic efficiency. In other words, free trade need not always be the best policy choice when the objective is to maximize national welfare.
• 11.5: The Economic Case against Selected Protection
The economic case against selected protectionism does not argue that the reasons for protection are conceptually or theoretically invalid. Indeed, there is general acceptance among economists that free trade is probably not the best policy in terms of maximizing economic efficiency in the real world.
• 11.6: Free Trade as the "Pragmatically Optimal" Policy Choice
The economic argument in support of free trade is a sophisticated argument that is based on the interpretation of results from the full collection of trade theories developed over the past two or three centuries. These theories, taken as a group, do not show that free trade is the best policy for every individual in all situations. Instead, the theories show that there are valid arguments supporting both free trade and protectionism.
11: Evaluating the Controversy between Free Trade and Protectionism
Learning Objectives
1. Understand the basis for the modern support for free trade among economists.
For hundreds of years, at least since Adam Smith’s publication of The Wealth of Nations, the majority of economists have been strong supporters of free trade among nations. Paul Krugman once wrote that if there were an economist’s creed, it would surely contain the affirmation, “I advocate free trade.”See Paul Krugman, “Is Free Trade Passe?” Journal of Economic Perspectives 1, no. 2 (1987): 131–44.
The original arguments for free trade began to supplant mercantilist views in the early to mid-eighteenth century. Many of these original ideas were based on simple exchange or production models that suggested that free trade would be in everyone’s best interests and surely in the national interest. During the nineteenth and twentieth centuries, however, a series of objections were raised suggesting that free trade was not in everyone’s interest and perhaps was not even in the national interest. The most prominent of these arguments included the infant industry argument, the terms of trade argument, arguments concerning income redistribution, and more recently, strategic trade policy arguments. Although each of these arguments might be thought of as weakening the case for free trade, instead, each argument brought forth a series of counterarguments that have acted to reassert the position of free trade as a favored policy despite these objections. The most important of these counterarguments include the potential for retaliation, the theory of the second best, the likelihood of incomplete or imperfect information, and the presence of lobbying in a democratic system.
What remains today is a modern, sophisticated argument in support of free trade among nations. It is an argument that recognizes that there are numerous exceptions to the notion that free trade is in everyone’s best interests. The modern case for free trade does not contend, however, that these exceptions are invalid or illogical. Rather, it argues that each exception supporting government intervention in the form of a trade policy brings with it additional implementation problems that are likely to make the policy impractical.
Before presenting the modern argument, however, it is worth deflecting some of the criticisms that are sometimes leveled against the economic theory of free trade. For example, the modern argument for free trade is not based on a simplistic view that everyone benefits from free trade. Indeed, trade theory, and experience in the real world, teaches us that free trade, or trade liberalization, is likely to generate losers as well as winners.
The modern argument for free trade is not based on unrealistic assumptions that lead to unrealistic conclusions. Although it is true that many assumptions contained within any given trade model do not accurately reflect many realistic features of the world, the modern argument for free trade is not based on the results from any one model. Instead, the argument is based on a collection of results from numerous trade models, which are interpreted in reference to realistic situations. If one considers the collection of all trade models jointly, it is much more difficult to contend that they miss realistic features of the world. Trade theory (as a collection of models) does consider imperfectly competitive markets, dynamic effects of trade, externalities in production and consumption, imperfect information, joint production, and many other realistic features. Although many of these features are absent in any one model, they are not absent from the joint collection of models, and it is this “extended model” that establishes the argument for free trade.Ideally, we would create a supermodel of the world economy that simultaneously incorporates all realistic features of the world and avoids what are often called “simplifying assumptions.” Unfortunately, this is not a realistic possibility. As anyone who has studied models of the economy knows, even models that are very simple in structure can be extremely difficult to comprehend, much less solve. As a result, we are forced to “interpret” the results of simple models as we apply them to the complex real world.
Key Takeaways
• The modern support for free trade by most economists is based on a collection of results from a collection of models that incorporate many realistic features of the world into the analysis.
Exercise \(1\)
1. Jeopardy Questions. As in the popular television game show, you are given an answer to a question and you must respond with the question. For example, if the answer is “a tax on imports,” then the correct question is “What is a tariff?”
1. The statement suggested by Paul Krugman as being an element of the economist’s creed—if ever there were such a thing.
2. This is who will benefit from free trade according to a simplistic view held by some free trade advocates.
3. This is what causes unrealistic conclusions in trade theory according to some free trade opponents.
4. The conclusions of one model of international trade or many models of international trade are best used to make trade policy prescriptions. | textbooks/socialsci/Economics/International_Trade_-_Theory_and_Policy/11%3A_Evaluating_the_Controversy_between_Free_Trade_and_Protectionism/11.1%3A_Introduction.txt |
Learning Objectives
1. Learn the major source of support for free trade across a variety of trade models.
The main source of support for free trade lies in the positive production and consumption efficiency effects. In every model of trade, there is an improvement in aggregate production and consumption efficiency when an economy moves from autarky to free trade. This is equivalent to saying that there is an increase in national welfare. This result was demonstrated in the Ricardian model, the immobile factor model, the specific factor model, the Heckscher-Ohlin model, the simple economies-of-scale model, and the monopolistic competition model. The result can also be shown if there are differences in demand between countries. Each of these models shows that a country is likely to have greater national output and superior choices available in consumption as a result of free trade.
Production Efficiency
Improvements in production efficiency mean that countries can produce more goods and services with the same amount of resources. In other words, productivity increases for the given resource endowments available for use in production.
In order to achieve production efficiency improvements, resources must be shifted between industries within the economy. This means that some industries must expand while others contract. Exactly which industries expand and contract will depend on the underlying stimulus or basis for trade. Different trade models emphasize different stimuli for trade. For example, the Ricardian model emphasizes technological differences between countries as the basis for trade, the factor proportions model emphasizes differences in endowments, and so on. In the real world, it is likely that each of these stimuli plays some role in inducing the trade patterns that are observed.
Thus as trade opens, either the country specializes in the products in which it has a comparative technological advantage, or production is shifted to industries that use the country’s relatively abundant factors most intensively, or production is shifted to products in which the country has relatively less demand compared with the rest of the world, or production shifts to products that exhibit economies of scale in production.
If production shifts occur for any of these reasons, or for some combination of these reasons, then trade models suggest that total production would rise. This would be reflected empirically in an increase in the country’s gross domestic product (GDP). This means that free trade would cause an increase in the level of the country’s national output and income.
Consumption Efficiency
Consumption efficiency improvements arise for an individual when changes in the relative prices of goods and services allow the consumer to achieve a higher level of utility. Since the change in prices alters the choices a consumer has, we can say that consumption efficiency improvements imply that more satisfying choices become available. When multiple varieties of goods are available in a product category, as in the monopolistic competition model, then consumption efficiency improvements can mean that the consumer is able to consume greater varieties or is able to purchase a variety that is closer to his ideal.
Although improvements in consumption efficiency are easy to describe for an individual consumer, it is much more difficult to describe consumption efficiency conceptually for the aggregate economy. Nevertheless, when aggregate indifference curves are used to describe the gains from trade, it is possible to portray an aggregate consumption efficiency improvement. One must be careful to interpret this properly, though. The use of an aggregate indifference curve requires the assumptions that (1) all consumers have identical preferences and (2) there is no redistribution of income as a result of the changes in the economy. We have seen, however, that in most trade models income redistribution will occur as an economy moves to free trade, and it may be impossible to redistribute afterward. It is also likely that individuals have different preferences for goods, which also weakens the results using aggregate indifference curves.
Key Takeaways
• The main sources of support for free trade are the positive production and consumption efficiency effects that arise in numerous models when countries trade freely.
• Production efficiency improvements mean that countries produce more goods and services with the same amount of resources.
• Consumption efficiency improvements mean that countries consume a more satisfying mix of goods and services.
Exercise \(1\)
1. Jeopardy Questions. As in the popular television game show, you are given an answer to a question and you must respond with the question. For example, if the answer is “a tax on imports,” then the correct question is “What is a tariff?”
1. The term often used as a synonym for an improvement in economic efficiency.
2. The type of efficiency improvement in which productivity rises for the given resource endowments available for use in production.
3. The type of efficiency improvement relating to consumer choice adjustments in response to a policy change.
4. The enhancement of this is what many economic models show will arise by moving to free trade. | textbooks/socialsci/Economics/International_Trade_-_Theory_and_Policy/11%3A_Evaluating_the_Controversy_between_Free_Trade_and_Protectionism/11.2%3A_Economic_Efficiency_Effects_of_Free_Trade.txt |
Learning Objectives
1. Recognize that a movement to free trade will cause a redistribution of income within the country.
2. Understand how compensation can relieve the problems caused by income redistribution.
A valid criticism of the case for free trade involves the issue of income distribution. Although most trade models suggest that aggregate economic efficiency is raised with free trade, these same models do not indicate that every individual in the economy will share in the benefits. Indeed, most trade models demonstrate that movements to free trade will cause a redistribution of income between individuals within the economy. In other words, some individuals will gain from free trade while others will lose. This was seen in the immobile factor model, the specific factor model, the Heckscher-Ohlin model, and the partial equilibrium analysis of trade liberalization.
There have been two general responses by economists concerning the income distribution issue. Some have argued that the objective of economics is solely to determine the most efficient policy choices. Introductory textbooks often suggest that the objective of the economics discipline is to determine how to allocate scarce resources toward production and consumption. Economists describe an allocation as “optimal” when it achieves the maximum level of aggregate economic efficiency. Put in these terms, economic analysis is “positive” in nature. Positive economics refers to studies that seek to answer questions pertaining to how things work in the economy and the subsequent effects. Positive economic analysis does not intend to explain what “should” be done. Issues pertaining to income distribution are commonly thought of as “normative” in nature, in that the concern is often over what the distribution “should” be. If we apply this reasoning to international trade, then, issues such as the appropriate income distribution are beyond the boundaries of the discipline and should be left to policymakers, government officials, or perhaps philosophers to determine.
Perhaps a more common response by economists concerning the income distribution issue is to invoke the compensation principle. A substantial amount of work by economists has been done to show that because free trade causes an increase in economic efficiency, it is generally possible to redistribute income from the winners to the losers such that, in the end, every individual gains from trade. The basic reason this is possible is that because of the improvement in aggregate efficiency, the sum of the gains to the winners exceeds the sum of the losses to the losers. This implies that it is theoretically possible for the potential winners from free trade to bribe the losers and leave everyone better off as a result of free trade. This allows economists to argue that free trade, coupled with an appropriate compensation package, is preferable to some degree of protectionism.
One major practical problem with compensation, however, is the difficulty of implementing a workable compensation package. In order to achieve complete compensation, one must be able to identify not only who the likely winners and losers will be but also how much they will win and lose and when in time the gains and losses will accrue. Although this is relatively simple to do in the context of a single trade model, such as the Heckscher-Ohlin model, it would be virtually impossible to do in practice given the complexity of the real world. The real world consists of tens of thousands of different industries producing millions of products using thousands of different factors of production. The sources of trade are manifold, including differences in technology, endowments, and demands, as well as the presence of economies of scale. Each source of trade, in turn, stimulates a different pattern of income redistribution when trade liberalization occurs. In addition, the pattern of redistribution over time is likely to be affected by the degree of mobility of factors between industries as the adjustment to free trade occurs. This was seen in the context of simple trade models, from the immobile factor model to the specific factor model to the Heckscher-Ohlin model.
Even in the context of simple trade models, a workable compensation mechanism is difficult to specify. An obvious solution would seem to be for the government to use taxes and subsidies to facilitate compensation. For example, the government could place taxes on those who would gain from free trade (or trade liberalization) and provide subsidies to those who would lose. However, if this were implemented in the context of many trade models, then the taxes and subsidies would change the production and consumption choices made in the economy and would act to reduce or eliminate the efficiency gains from free trade. The government taxes and subsidies, in this case, represent a policy-imposed distortion that, by itself, reduces aggregate economic efficiency. If the compensation package reduces efficiency more than the movement to free trade enhances efficiency, then it is possible for the nation to be worse off in free trade when combined with a tax/subsidy redistribution scheme.Dixit and Norman (1980) showed that under some conditions it is possible to specify a tax and subsidy policy that would guarantee an increase in aggregate economic efficiency with free trade. See A. Dixit and V. Norman, Theory of International Trade: A Dual General Equilibrium Approach (Cambridge: Cambridge University Press, 1980). The simple way to eliminate this problem, conceptually, is to suggest that the redistribution take place as a “lump-sum” redistribution. A lump-sum redistribution is one that takes place after the free trade equilibrium is reached—that is, after all production and consumption decisions are made but before the actual consumption takes place. Then, as if in the middle of the night when all are asleep, goods are taken away from those who have gained from free trade and left at the doors of those who had lost. Lump-sum redistributions are analogous to Robin Hood stealing from the rich and giving to the poor. As long as this redistribution takes place after the consumption choices have been made and without anyone expecting a redistribution to occur, then the aggregate efficiency improvements from free trade are still realized. Of course, although lump-sum redistributions are a clever conceptual or theoretical way to “have your cake and eat it too,” it is not practical or workable in the real world.
This implies that although compensation can solve the problem of income redistribution at the theoretical level, it is unlikely that it will ever solve the problem in the real world. Although some of the major gains and losses from free trade may be identifiable and quantifiable, it is unlikely that analysts would ever be able to identify all who would gain and lose in order to provide compensation and assure that everyone benefits. This means that free trade is extremely likely to cause uncompensated losses to some individuals in the economy. To the extent that these individuals expect these losses and can measure their expected value (accurately or not), then there will also likely be continued resistance to free trade and trade liberalization. This resistance is perfectly valid. After all, trade liberalization involves a government action that will cause injury to some individuals for which they do not expect to be adequately compensated. Furthermore, the economic efficiency argument will not go very far to appease these groups. Would you accept the argument that your expected losses are justifiable because others will gain more than you lose?
One final argument concerning the compensation issue is that compensation to the losers may not even be justifiable. This argument begins by noting that those who would lose from free trade are the same groups who had gained from protectionism. Past protectionist actions represent the implementation of government policies that had generated benefits to certain selected groups in the economy. When trade liberalization occurs, then, rather than suggesting that some individuals lose, perhaps it is more accurate to argue that the special benefits are being eliminated for those groups. On the other hand, those groups that benefit from free trade are the same ones that had suffered losses under the previous regime of protectionism. Thus their gains from trade can be interpreted as the elimination of previous losses. Furthermore, since the previous protectionist actions were likely to have been long lasting, one could even argue that the losers from protection (who would gain from free trade) deserve to be compensated for the sum total of their past losses. This would imply that upon moving to free trade, a redistribution ought to be made not from the winners in trade to the losers but from the losers in trade to the winners. Only in this way could one make up for the transgressions of the past. As before, though, identifying who lost and who gained and by how much would be virtually impossible to achieve, thus making this compensation scheme equally unworkable.
Key Takeaways
• One major problem with movements to free trade is the redistribution of income described in many trade models. This means that although some individuals will benefit from free trade, many others will lose.
• One way to deflect the redistribution concern is to argue that economic analysis provides the positive results of trade policies and is not intended to answer the normative questions of what should be done.
• Another way to deflect the concern about income redistribution is to support compensation from the winners to the losers to assure that all parties benefit from free trade.
• Because compensation requires an enormous amount of information about who wins and loses from trade, how much they win and lose, and when they win and lose, it is impractical to impossible to completely compensate the losers from free trade in a real-world setting.
Exercise \(1\)
1. Jeopardy Questions. As in the popular television game show, you are given an answer to a question and you must respond with the question. For example, if the answer is “a tax on imports,” then the correct question is “What is a tariff?”
1. A principle that, if applied in practice, could eliminate the negative impacts of income redistribution that may arise with free trade.
2. This is what many trade models show will happen to national income because of trade liberalization.
3. This type of compensation can avoid affecting consumption and production decisions.
4. The compensation using these two government policies is likely to affect production and consumption decisions.
5. The name of the mythical character best associated with lump-sum compensation.
6. Of a little or a lot, this is how much information the government needs to make compensation effective. | textbooks/socialsci/Economics/International_Trade_-_Theory_and_Policy/11%3A_Evaluating_the_Controversy_between_Free_Trade_and_Protectionism/11.3%3A_Free_Trade_and_the_Distribution_of_Income.txt |
Learning Objectives
1. Identify the cases in which the implementation of selected protectionism, targeted at particular industries with particular goals in mind, could raise national welfare.
An argument for selected protection arises in the presence of imperfectly competitive markets, market distortions, or both. In these cases, it is often possible to show that an appropriately targeted trade policy (selected protection) can raise aggregate economic efficiency. In other words, free trade need not always be the best policy choice when the objective is to maximize national welfare. Numerous examples found in the trade literature demonstrate that selected protectionism applied under certain circumstances can raise national welfare. These results are in contrast with the standard trade models, which show that free trade is the best policy to maximize economic efficiency. The reason for the conflict is that the standard trade models, in most cases, explicitly assume that markets are perfectly competitive and implicitly assume there are no market distortions.
This general criticism of the standard case for free trade begins by noting that the real world is replete with examples of market imperfections and distortions. These include the presence of externalities both static and dynamic, both positive and negative, and in both production and consumption; markets in which production takes place with monopolistic or oligopolistic firms making positive profits; markets that do not clear, as when unemployment arises; the presence of public goods; the presence of imperfect or asymmetric information; the presence of distorting government policies and regulations; and the presence of national market power in international markets. When these features are included in trade models, it is relatively easy to identify trade policies that can sufficiently correct the market imperfection or distortion so as to raise aggregate efficiency.
For example, an optimal tariff or optimal quota set by a country that is large in an international import market can allow the nation to take advantage of its monopsony power in trade and cause an increase in national welfare. Similarly, an optimal export tax or voluntary export restraint (VER) set by a large country in an international export market will allow it to take advantage of its monopoly power in trade and generate an increase in welfare. This argument for protection is known as the “terms of trade argument.”
A tariff applied to protect an import-competing industry from a surge in foreign imports may reduce or eliminate the impending unemployment in the industry. If the cost of unemployment to the affected workers is greater than the standard net national welfare effect of the tariff, then the tariff may improve national welfare.
A tariff used to restrict imports of goods from more-efficient foreign firms may sufficiently stimulate learning effects within an industry to cause an increase in productivity that, in time, may allow the domestic firms to compete with foreign firms—even without continued protection. These learning effects—in organizational methods, in management techniques, in cost-cutting procedures—might in turn spill over to other sectors in the economy, stimulating efficiency improvements in many other industries. All together, the infant industry protection may cause a substantial increase in the growth of the gross domestic product (GDP) relative to what might have occurred otherwise and thus act to improve national welfare.
A tariff used to stimulate domestic production of a high-technology good might spill over to the research and development division and cause more timely innovations in next-generation products. If these firms turn into industry leaders in these next-generation products, then they will enjoy the near-monopoly profits that accrue to the original innovators. As long as these long-term profits outweigh the short-term costs of protection, national welfare may rise.
An import tariff applied against a foreign monopoly supplying the domestic market can effectively shift profits from the foreign firm to the domestic government. Despite the resulting increase in the domestic price, national welfare may still rise. Also, export subsidies provided to domestic firms that are competing with foreign firms in an oligopoly market may raise domestic firms’ profits by more than the cost of the subsidy, especially if profits can be shifted away from the foreign firms. These two cases are examples of a strategic trade policy.
If pollution, a negative production externality, caused by a domestic import-competing industry is less than the pollution caused by firms in the rest of the world, then a tariff that restricts imports may sufficiently raise production by the domestic firm relative to foreign firms and cause a reduction in world pollution. If the benefits that accrue due to reduced worldwide pollution are greater than the standard cost of protection, then the tariff will raise world welfare.
Alternatively, if pollution is caused by a domestic export industry, then an export tax would reduce domestic production along with the domestic pollution that the production causes. Although the export tax may act to raise production and pollution in the rest of the world, as long as the domestic benefits from the pollution reduction outweigh the costs of the export tax, domestic national welfare may rise.
If certain domestically produced high-technology goods could wind up in the hands of countries that are our potential enemies, and if these goods would allow those countries to use the products in a way that undermines our national security, then the government could be justified to impose an export prohibition on those goods to those countries. In this case, if free trade were allowed in these products, it could reduce the provision of a public good, namely, national security. As long as the improvement in national security outweighs the cost of the export prohibition, national welfare would rise.
These are just some of the examples (many more are conceivable) in which the implementation of selected protectionism, targeted at particular industries with particular goals in mind, could act to raise national welfare, or aggregate economic efficiency. Each of these arguments is perfectly valid conceptually. Each case arises because of an assumption that some type of market imperfection or market distortion is present in the economy. In each case, national welfare is enhanced because the trade policy reduces or eliminates the negative effects caused by the presence of the imperfection or distortion and because the reduction in these effects can outweigh the standard efficiency losses caused by the trade policy.
It would seem from these examples that a compelling case can certainly be made in support of selected protectionism. Indeed, Paul Krugman (1987) wrote that “the case for free trade is currently more in doubt than at any time since the 1817 publication of [David] Ricardo’s Principles of Political Economy.”See Paul Krugman, “Is Free Trade Passe?” Journal of Economic Perspectives 1, no. 2 (1987): 131–44. Many of the arguments showing the potential for welfare-improving trade policies described above have been known for more than a century. The infant industry argument can be traced in the literature as far back as a century before Adam Smith argued against it in The Wealth of Nations (1776). The argument was later supported by writers such as Friedrich List in The National System of Political Economy (1841)See Friedrich List, The National System of Political Economy, McMaster University Archive for the History of Economic Thought, socserv2.socsci.mcmaster.ca:80/~econ/ugcm/3ll3/list/index.html. and John Stuart Mill in his Principles of Political Economy (1848).See John Stuart Mill, Principles of Political Economy, McMaster University Archive for the History of Economic Thought, socserv2.socsci.mcmaster.ca:80/~econ/ugcm/3ll3/mill/index.html. The terms of trade argument was established by Robert Torrens in 1844 in The Budget: On Commercial and Colonial Policy.See Robert Torrens, The Budget: On Commercial and Colonial Policy (London: Smith, Elder, 1844). Frank Graham, in his 1923 article “Some Aspects of Protection Further Considered,” noted the possibility that free trade would reduce welfare if there were variable returns to scale in production.See Frank Graham, “Some Aspects of Protection Further Considered,” The Quarterly Journal of Economics 37, no. 2 (February 1923): 199–227. During the 1950s and 1960s, market distortions such as factor-market imperfections and externality effects were introduced and studied in the context of trade models. The strategic trade policy arguments are some of the more recent formalizations showing how market imperfections can lead to welfare-improving trade policies. Despite this long history, economists have generally continued to believe that free trade is the best policy choice. The main reason for this almost unswerving support for free trade is because as arguments supporting selected protectionism were developed, equally if not more compelling counterarguments were also developed.
Key Takeaways
• In the presence of market imperfections or distortions, selected protection can often raise a country’s national welfare.
• Because real-world markets are replete with market imperfections and distortions, free trade is not the optimal policy to improve national welfare.
Exercise \(1\)
1. Jeopardy Questions. As in the popular television game show, you are given an answer to a question and you must respond with the question. For example, if the answer is “a tax on imports,” then the correct question is “What is a tariff?”
1. The term used to describe market conditions that open up the possibility for welfare-improving trade policies.
2. The term used to describe a market equilibrium in which market imperfections or distortions are present.
3. Of very many or very few, this is the amount of market imperfections likely to be present in modern national economies.
4. Of true or false, a tariff can raise a nation’s welfare when it is a large importing country.
5. Of true or false, a tariff can raise national welfare in the presence of an infant industry.
6. Of true or false, a tariff can raise national welfare if all markets are perfectly competitive and if there are no market imperfections or distortions.
2. Identify a trade policy that can potentially raise national welfare in each of the following situations.
1. When a foreign monopoly supplies the domestic market with no import-competing producers.
2. When a domestic negative production externality is caused by a domestic industry that exports a portion of its production to the rest of the world.
3. When a positive production externality is caused by a domestic industry that competes with imports.
4. When a domestic negative consumption externality is caused by domestic consumers in a market in which the country exports a portion of its production to the rest of the world.
5. When a country is large in an export market. | textbooks/socialsci/Economics/International_Trade_-_Theory_and_Policy/11%3A_Evaluating_the_Controversy_between_Free_Trade_and_Protectionism/11.4%3A_The_Case_for_Selected_Protection.txt |
Learning Objectives
1. Learn the valid counterarguments to the use of selected protection when market imperfections or distortions are present.
The economic case against selected protectionism does not argue that the reasons for protection are conceptually or theoretically invalid. Indeed, there is general acceptance among economists that free trade is probably not the best policy in terms of maximizing economic efficiency in the real world. Instead, the counterarguments to selected protectionism are based on four broad themes:
1. potential reactions by others in response to one country’s protection,
2. the likely presence of superior policies to raise economic efficiency relative to a trade policy,
3. information deficiencies that can inhibit the implementation of appropriate policies, and
4. problems associated with lobbying within democratic political systems. We shall consider each of these issues in turn.
The Potential for Retaliation
One of the problems with using some types of selected protection arises because of the possibility of retaliation by other countries using similar policies. For example, it was shown that whenever a large country in the international market applies a policy that restricts exports or imports (optimally), its national welfare will rise. This is the terms of trade argument supporting protection. However, it was also shown that the use of an optimal trade policy in this context always reduces national welfare for the country’s trade partners. Thus the use of an optimal tariff, export tax, import quota, or voluntary export restraint (VER) is a “beggar-thy-neighbor” policy—one country benefits only by harming others. For this reason, it seems reasonable, if not likely, that the countries negatively affected by the use of such policies, if they are also large in international markets, would retaliate by setting optimal trade policies restricting their exports and imports to the rest of the world. In this way, the retaliating country could generate benefits for itself in some markets to compensate for its losses in others.
However, the final outcome after retaliation occurs is very likely to be a reduction in national welfare for both countries.Harry Johnson (1953) showed the possibility that one country might still improve its national welfare even after a trade war (i.e., optimal protection followed by optimal retaliation); however, this seems an unlikely outcome in real-world cases. Besides, even if one country did gain, it would still do so at the expense of its trade partners, which remains an unsavory result. See Harry G. Johnson, “Optimum Tariffs and Retaliation,” Review of Economic Studies 21, no. 2 (1953): 142–53. This occurs because each trade policy action results in a decline in world economic efficiency. The aggregate losses that accrue to one country as a result of the other’s trade policy will always exceed the benefits that accrue to the policy-setting country. When every large country sets optimal trade policies to improve its terms of trade, the subsequent reduction in world efficiency dominates any benefits that accrue due to its unilateral actions.
What this implies is that although a trade policy can be used to improve a nation’s terms of trade and raise national welfare, it is unlikely to raise welfare if other large countries retaliate and pursue the same policies. Furthermore, retaliation seems a likely response because maintenance of a free trade policy in light of your trade partner’s protection would only result in national aggregate efficiency losses.Indeed, Robert Torrens, the originator of the terms of trade argument, was convinced that a large country should maintain protective barriers to trade when its trade partners maintained similar policies. The case for unilateral free trade even when one’s trade partners use protective tariffs is only valid when a country is small in international markets.
Perhaps the best empirical support for this result is the experience of the world during the Great Depression of the 1930s. After the United States imposed the Smoot-Hawley Tariff Act of 1930, raising its tariffs to an average of 60 percent, approximately sixty countries retaliated with similar increases in their own tariff barriers. As a result, world trade in the 1930s fell to one-quarter of the level attained in the 1920s. Most economists agree that these tariff walls contributed to the length and severity of the economic depression. That experience also stimulated the design of the reciprocal trade liberalization efforts embodied in the General Agreement on Tariffs and Trade (GATT).
The issue of retaliation also arises in the context of strategic trade policies. In these cases, a trade policy can be used to shift profits from foreign firms to the domestic economy and raise domestic national welfare. The policies work in the presence of monopolistic or oligopolistic markets by raising the international market share for one’s own firms. The benefits to the policy-setting country arise only by reducing the profits of foreign firms and subsequently reducing those countries’ national welfare.One exception arises in the model by J. Eaton and G. Grossman, “Optimal Trade and Industrial Policy under Oligopoly,” Quarterly Journal of Economics 101, no. 2 (1986): 383–406. Thus one country’s gains are other country’s losses, and strategic trade policies can rightfully be called beggar-thy-neighbor policies. Since foreign firms would lose from our country’s policies, as before, it is reasonable to expect retaliation by the foreign governments. However, because these policies essentially just reallocate resources among profit-making firms internationally, it is unlikely for a strategic trade policy to cause an improvement in world economic efficiency. This implies that if the foreign country did indeed retaliate, the likely result would be reductions in national welfare for both countries.
Retaliations would only result in losses for both countries when the original trade policy does not raise world economic efficiency. However, some of the justifications for protection that arise in the presence of market imperfections or distortions may actually raise world economic efficiency because the policy acts to eliminate some of the inefficiencies caused by the distortions. In these cases, retaliation would not pose the same problems. There are other problems, though.
The Theory of the Second Best
One of the more compelling counterarguments to potentially welfare-improving trade policies relies on the theory of the second best. This theory shows that when private markets have market imperfections or distortions present, it is possible to add another (carefully designed) distortion, such as a trade policy, and improve economic efficiency both domestically and worldwide. The reason for this outcome is that the second distortion can correct the inefficiencies of the first distortion by more than the inefficiencies caused by the imposed policy. In economist’s jargon, the original distorted economy is at a second-best equilibrium. In this case, the optimal trade policy derived for an undistorted economy (most likely free trade) no longer remains optimal. In other words, policies that would reduce national welfare in the absence of distortions can now improve welfare when there are other distortions present.
This argument, then, begins by accepting that trade policies (protection) can be welfare improving. The problem with using trade policies, however, is that in most instances they are a second-best policy choice. In other words, there will likely be another policy—a domestic policy—that could improve national welfare at a lower cost than any trade policy. The domestic policy that dominates would be called a first-best policy. The general rule used to identify first-best policies is to use that policy that “most directly” attacks the market imperfection or distortion. It turns out that these are generally domestic production, consumption, or factor taxes or subsidies rather than trade policies. The only exceptions occur when a country is large in international markets or when trade goods affect the provision of a public good such as national security.
Thus the counterargument to selected protection based on the theory of the second best is that first-best rather than second-best policies should be chosen to correct market imperfections or distortions.
Since trade policies are generally second best while purely domestic policies are generally first best, governments should not use trade policies to correct market imperfections or distortions. Note that this argument does not contend that distortions or imperfections do not exist, nor does it assume that trade policies could not improve economic efficiency in their presence. Instead, the argument contends that governments should use the most efficient (least costly) method to reduce inefficiencies caused by the distortions or imperfections, and this is unlikely to be a trade policy.
Note that this counterargument to protection is also effective when the issue is income distribution. Recall that one reason countries may use trade policies is to achieve a more satisfying income distribution (or to avoid an unsatisfactory distribution). However, it is unlikely that trade policies would be the most effective method to eliminate the problem of an unsatisfactory income distribution. Instead, there will likely be a purely domestic policy that could improve income distribution more efficiently.
In the cases where a trade policy is first best, as when a country is large in international markets, this argument does not act as a counterargument to protection. However, retaliation remains a valid counterargument in many of these instances.
Information Deficiencies
The next counterargument against selected protectionism concerns the likely informational constraints faced by governments. In order to effectively provide infant industry protection, or to eliminate negative externality effects, stimulate positive externality effects, or shift foreign profits to the domestic economy, the government would need substantial information about the firms in the market, their likely cost structures, supply and demand elasticities indicating the effects on supply and demand as a result of price changes, the likely response by foreign governments, and much more. Bear in mind that although it was shown that selected protection could generate an increase in national welfare, it does not follow that any protection would necessarily improve national welfare. The information requirements arise at each stage of the government’s decision-making process.
First, the government would need to identify which industries possess the appropriate characteristics. For example, in the case of infant industries, the government would need to identify which industries possess the positive learning externalities needed to make the protection work. Presumably, some industries would generate these effects, while others would not. In the case of potential unemployment in a market, the government would need to identify in which industries facing a surge of imports the factor immobility was relatively high. In the case of a strategic trade policy, the government would have to identify which industries are oligopolistic and exhibit the potential to shift foreign profits toward the domestic economy.
Second, the government would need to determine the appropriate trade policy to use in each situation and set the tariff or subsidy at the appropriate level. Although this is fairly straightforward in a simple theoretical model, it may be virtually impossible to do correctly in a real-world situation. Consider the case of an infant industry. If the government identified an industry with dynamic intertemporal learning effects, it would then need to measure how the level of production would influence the size of the learning effects in all periods in the future. It would also need to know how various tariff levels would affect the level of domestic production. To answer this requires information about domestic and foreign supply and demand elasticities. Of course, estimates of past elasticities may not work well, especially if technological advances or preference changes occur in the future. All of this information is needed to determine the appropriate level of protection to grant as well as a timetable for tariff reduction. If the tariff is set too low or for too short a time, the firms might not be sufficiently protected to induce adequate production levels and stimulate the required learning effects. If the tariff is set too high or for too long a period, then the firms might become lazy. Efficiency improvements might not be made and the learning effects might be slow in coming. In this case, the production and consumption efficiency losses from the tariff could outweigh the benefits accruing due to learning.
This same information deficiency problem arises in every example of selected protection. Of course, the government would not need pinpoint accuracy to assure a positive welfare outcome. As demonstrated in the case of optimal tariffs, there would be a range of tariff levels that would raise national welfare above the level attained in free trade. A similar range of welfare-improving protection levels would also hold in all the other cases of selected protection.
However, there is one other informational constraint that is even ignored in most economic analyses of trade policies. This problem arises when there are multiple distortions or imperfections present in the economy simultaneously (exactly what we would expect to see in the real world). Most trade policy analyses incorporate one economic distortion into a model and then analyze what the optimal trade policy would be in that context. Implicitly, this assumes either that there are no other distortions in the economy or that the market in which the trade policy is being considered is too small to have any external effects on other markets. The first assumption is clearly not satisfied in the world, while the second is probably not valid for many large industries.
The following example suggests the nature of the informational problem. Suppose there are two industries that are linked together because their products are substitutable in consumption to some degree. Suppose one of these industries exhibits a positive dynamic learning externality and is having difficulty competing with foreign imports (i.e., it is an infant industry). Assume the other industry heavily pollutes the domestic water and air (i.e., it exhibits a negative production externality). Now suppose the government decides to protect the infant industry with an import tariff. This action would, of course, stimulate domestic production of the good and also stimulate the positive learning effects for the economy. However, the domestic price of this good would rise, reducing domestic consumption. These higher prices would force consumers to substitute other products in consumption. Since the other industry’s products are assumed to be substitutable, demand for that industry’s goods will rise. The increase in demand would stimulate production of that good and, because of its negative externality, cause more pollution to the domestic environment. If the negative effects to the economy from additional pollution are greater than the positive learning effects, then the infant industry protection could reduce rather than improve national welfare.
The point of this example, however, is to demonstrate that in the presence of multiple distortions or imperfections in interconnected markets (i.e., in a general equilibrium model), the determination of optimal policies requires that one consider the intermarket effects. The optimal infant industry tariff must take into account the effects of the tariff on the polluting industry. Similarly, if the government wants to set an optimal environmental policy, it would need to account for the effects of the policy on the industry with the learning externality.
This simple example suggests a much more serious informational problem for the government. If the real economy has numerous market imperfections and distortions spread out among numerous industries that are interconnected through factor or goods market competition, then to determine the true optimal set of policies that would correct or reduce all the imperfections and distortions simultaneously would require the solution to a dynamic general equilibrium model that accurately describes the real economy not only today but also in all periods in the future. This type of model, or its solution, is simply not achievable today with any high degree of accuracy. Given the complexity, it seems unlikely that we would ever be capable of producing such a model.
The implication of this informational problem is that trade policy will always be like a shot in the dark. There is absolutely no way of knowing with a high degree of accuracy whether any policy will improve economic efficiency. This represents a serious blow to the case for government intervention in the form of trade policies. If the intention of government is to set trade policies that will improve economic efficiency, then since it is impossible to know whether any policy would actually achieve that goal, it seems prudent to avoid the use of any such policy. Of course, the goal of government may not be to enhance economic efficiency, and that brings us to the last counterargument against selected protection.
Political Economy Issues: The Problem with Democratic Processes
In democratic societies, government representatives and officials are meant to carry out the wishes of the general public. As a result, decisions by the government are influenced by the people they represent. Indeed, one of the reasons “free speech” is so important in democratic societies is to assure that individuals can make their attitudes toward government policies known without fear of reproach. Individuals must be free to inform the government of which policies they approve and of which they disapprove if the government is truly to be a representative of the people. The process by which individuals inform the government of their preferred policies is generally known as lobbying.
In a sense, one could argue that lobbying can help eliminate some of the informational deficiencies faced by governments. After all, much of the information the government needs to make optimal policies is likely to be better known by its constituent firms and consumers. Lobbying offers a process through which information can be passed from those directly involved in production and consumption activities to the officials who determine policies. However, this process may turn out to be more of a problem than a solution.
One of the results of trade theory is that the implementation of trade policies will likely affect income distribution. In other words, all trade policies will generate income benefits to some groups of individuals and income losses to other groups. Another outcome, though, is that the benefits of protection would likely be concentrated—that is, the benefits would accrue to a relatively small group. The losses from protection, however, would likely be dispersed among a large group of individuals.
This outcome was seen clearly in the partial equilibrium analysis of a tariff. When a tariff is implemented, the beneficiaries would be the import-competing firms, which would face less competition for their product, and the government, which collects tariff revenue. The losses would accrue to the thousands or millions of consumers of the product in the domestic economy.
For example, consider a tariff on textile imports being considered by the government of a small, perfectly competitive economy. Theory shows that the sum of the benefits to the government and the firms will be exceeded by the losses to consumers. In other words, national welfare would fall. Suppose the beneficiaries of protection are one hundred domestic textile firms that would each earn an additional \$1 million in profit as a result of the tariff. Suppose the government would earn \$50 million in additional tariff revenue. Thus the total benefits from the tariff would be \$150 million. Suppose consumers as a group would lose \$200 million, implying a net loss to the economy of \$50 million. However, suppose there are one hundred million consumers of the products. That implies that each individual consumer would lose only \$2.
Now, if the government bases its decision for protection on input from its constituents, then it is very likely that protection will be granted even though it is not in the nation’s best interest. The reason is that textile firms would have an enormous incentive to lobby government officials in support of the policy. If each firm expects an extra \$1 million, it would make sense for the firms to hire a lobbying firm to help make their case before the government. The arguments to be used, of course, are (1) the industry will decline and be forced to lay off workers without protection, thus protection will create jobs; (2) the government will earn additional revenues that can be used for important social programs; and (3) the tax is on foreigners and is unlikely to affect domestic consumers (number 3 isn’t correct, of course, but the argument is often used anyway). Consumers, on the other hand, have very little individual incentive to oppose the tariff. Even writing a letter to your representative is unlikely to be worth the \$2 potential gain. Plus, consumers would probably hear (if they hear anything at all) that the policy will create some jobs and may not affect the domestic price much anyway (after all, the tax is on foreigners).
The implication of this problem is that the lobbying process may not accurately relate to the government the relative costs and benefits that will arise due to the implementation of a trade policy. As a result, the government would likely implement policies that are in the special interests of those groups who stand to accrue the concentrated benefits from protection, even though the policy may generate net losses to the economy as a whole. Thus by maintaining a policy of free trade, an economy could avoid national efficiency losses that could arise with lobbying in a democratic system.
Key Takeaways
• Selected protection may fail to raise national welfare when foreign country retaliations occur. This is a potential problem when many countries are large in international markets.
• Selected protection with a trade policy is typically second best. A purely domestic policy to correct the market imperfection is often the better, or first-best, policy.
• Selected protection requires detailed information in order to set the policy at a level that will assure an improvement in national welfare. Because the necessary information is often lacking, getting selected protection right may be impossible.
• Selected protection can be captured by special interests in the lobbying process in representative democracies, thereby making it less likely that maximum national welfare will be achieved.
Exercise \(1\)
1. Jeopardy Questions. As in the popular television game show, you are given an answer to a question and you must respond with the question. For example, if the answer is “a tax on imports,” then the correct question is “What is a tariff?”
1. The term used to describe a potentially welfare-reducing reaction to beggar-thy-neighbor trade policies.
2. The term used to describe the lowest-cost policy action that corrects for market distortions or imperfections.
3. The often overlooked deficiencies that affect the ability of government to set effective policies.
4. The term used to describe the process by which individuals inform the government of their preferred policies.
5. Economists applying the theory of the second best would argue that free trade is appropriate in spite of market imperfections because these types of policies are usually first best. | textbooks/socialsci/Economics/International_Trade_-_Theory_and_Policy/11%3A_Evaluating_the_Controversy_between_Free_Trade_and_Protectionism/11.5%3A_The_Economic_Case_against_Selected_Protection.txt |
Learning Objectives
1. Understand the modern argument for free trade as a “pragmatically optimal” policy choice.
In summary, the economic argument in support of free trade is a sophisticated argument that is based on the interpretation of results from the full collection of trade theories developed over the past two or three centuries. These theories, taken as a group, do not show that free trade is the best policy for every individual in all situations. Instead, the theories show that there are valid arguments supporting both free trade and protectionism. To choose between the two requires a careful assessment of the pros and cons of each policy regime.
The argument for free trade presented here accepts the notion that free trade may not always be optimal in terms of maximizing economic efficiency. The argument also accepts that free trade may not generate the most preferred distribution of income. In theory, there are numerous cases in which selected protectionism can improve aggregate welfare or could establish a more equal distribution of income. Nevertheless, despite these theoretical possibilities, it remains unclear and perhaps unlikely that selected protectionism could achieve the intended results. First, in many instances, a trade policy is not the best way to achieve the intended improvement in economic efficiency, nor is it likely to be the most efficient way to achieve a more satisfactory distribution of income. Instead, purely domestic tax and subsidy policies dominate. Second, even when a trade policy is the best policy choice, the possibility of retaliations and the likelihood of informational deficiencies or distortions caused by the lobbying process are sufficiently large as to make the intended outcomes unknowable.
In addition, the process of information collection, lobbying, and policy implementation is a costly economic activity. Labor and capital resources are allocated by interest groups attempting to affect policies favorable to them. The government also must expend resources to gather information, to implement and administer policies, and to monitor the effectiveness of these policies. In the United States, the following agencies and groups devote at least some of their time to trade policy implementation: the Office of the United States Trade Representative, the International Trade Commission, the Department of Commerce, the Federal Trade Commission, the Department of Justice, the Congress, and the president, among others. One must wonder whether the cost of this bureaucracy, together with the cost to the private sector to influence the decisions of the government, is worth it, especially when the outcomes are virtually unknowable.
Thus the conclusion reached by many economists is that while free trade may not be “technically optimal,” it remains “pragmatically optimal.” That is, given our informational deficiencies and the other problems inherent in any system of selected protectionism, free trade remains the policy most likely to produce the highest level of economic efficiency attainable.
Key Takeaways
• While free trade may not be “technically optimal,” it remains “pragmatically optimal”—that is, free trade remains the policy most likely to produce the highest level of economic efficiency that is practically attainable.
Exercise \(1\)
1. Jeopardy Question. As in the popular television game show, you are given an answer to a question and you must respond with the question. For example, if the answer is “a tax on imports,” then the correct question is “What is a tariff?”
1. The term used to describe a policy that is relatively easy to implement and has strong positive characteristics but may not be best in all conceivable circumstances. | textbooks/socialsci/Economics/International_Trade_-_Theory_and_Policy/11%3A_Evaluating_the_Controversy_between_Free_Trade_and_Protectionism/11.6%3A_Free_Trade_as_the_Pragmatically_Optimal_Policy_Choice.txt |
Learning ObjectiveS
1. How is economics used?
2. What is an economic theory?
3. What is a market?
Economic analysis serves two main purposes. The first is to understand how goods and services, the scarce resources of the economy, are actually allocated in practice. This is a positive analysis, like the study of electromagnetism or molecular biology; it aims to understand the world without value judgments. The development of this positive theory, however, suggests other uses for economics. Economic analysis can predict how changes in laws, rules, and other government policies will affect people and whether these changes are socially beneficial on balance. Such predictions combine positive analysis—predicting the effects of changes in rules—with studies that make value judgments known as normative analyses. For example, a gasoline tax to build highways harms gasoline buyers (who pay higher prices) but helps drivers (by improving the transportation system). Since drivers and gasoline buyers are typically the same people, a normative analysis suggests that everyone will benefit. Policies that benefit everyone are relatively uncontroversial.
In contrast, cost-benefit analysis weighs the gains and losses to different individuals to determine changes that provide greater benefits than harm. For example, a property tax to build a local park creates a benefit to park users but harms property owners who pay the tax. Not everyone benefits, since some taxpayers don’t use the park. Cost-benefit analysis weighs the costs against the benefits to determine if the policy is beneficial on balance. In the case of the park, the costs are readily measured in monetary terms by the size of the tax. In contrast, the benefits are more difficult to estimate. Conceptually, the benefits are the amount the park users would be willing to pay to use the park. However, if there is no admission charge to the park, one must estimate a willingness-to-pay, the amount a customer is willing and able to pay for a good. In principle, the park provides greater benefits than costs if the benefits to the users exceed the losses to the taxpayers. However, the park also involves transfers from one group to another.
Welfare analysis is another approach to evaluating government intervention into markets. It is a normative analysis that trades off gains and losses to different individuals. Welfare analysis posits social preferences and goals, such as helping the poor. Generally a welfare analysis requires one to perform a cost-benefit analysis, which accounts for the overall gains and losses but also weighs those gains and losses by their effects on other social goals. For example, a property tax to subsidize the opera might provide more value than costs, but the bulk of property taxes are paid by lower- and middle-income people, while the majority of operagoers are wealthy. Thus, the opera subsidy represents a transfer from relatively low-income people to wealthy people, which contradicts societal goals of equalization. In contrast, elimination of sales taxes on basic food items like milk and bread has a greater benefit to the poor, who spend a much larger percentage of their income on food, than do the rich. Thus, such schemes are desirable primarily for their redistribution effects. Economics is helpful for providing methods to determining the overall effects of taxes and programs, as well as the distributive impacts. What economics can’t do, however, is advocate who ought to benefit. That is a matter for society to decide.
KEY TAKEAWAYS
• A positive analysis, analogous to the study of electromagnetism or molecular biology, involves only the attempt to understand the world around us without value judgments.
• Economic analyses employing value judgments are known as normative analyses. When everyone is made better off by a change, recommending that change is relatively uncontroversial.
• A cost-benefit analysis totals the gains and losses to different individuals in dollars and suggests carrying out changes that provide greater benefits than harm. A cost-benefit analysis is a normative analysis.
• Welfare analysis posits social preferences and goals, permitting an optimization approach to social choice. Welfare analysis is normative.
• Economics helps inform society about the consequences of decisions, but the valuation of those decisions is a matter for society to choose. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/01%3A_What_Is_Economics/1.01%3A_Normative_and_Positive_Theories.txt |
Learning ObjectiveS
1. What is opportunity cost?
2. How is it computed?
3. What is its relationship to the usual meaning of cost?
Economists think of cost in a slightly quirky way that makes sense, however, once you think about it for a while. We use the term opportunity cost to remind you occasionally of our idiosyncratic notion of cost. For an economist, the cost of buying or doing something is the value that one forgoes in purchasing the product or undertaking the activity of the thing. For example, the cost of a university education includes the tuition and textbook purchases, as well as the wages that were lost during the time the student was in school. Indeed, the value of the time spent in acquiring the education is a significant cost of acquiring the university degree. However, some “costs” are not opportunity costs. Room and board would not be a cost since one must eat and live whether one is working or at school. Room and board are a cost of an education only insofar as they are expenses that are only incurred in the process of being a student. Similarly, the expenditures on activities that are precluded by being a student—such as hang-gliding lessons, or a trip to Europe—represent savings. However, the value of these activities has been lost while you are busy reading this book.
Opportunity cost is defined by the following:
The opportunity cost is the value of the best forgone alternative.
This definition emphasizes that the cost of an action includes the monetary cost as well as the value forgone by taking the action. The opportunity cost of spending \$19 to download songs from an online music provider is measured by the benefit that you would have received had you used the \$19 instead for another purpose. The opportunity cost of a puppy includes not just the purchase price but the food, veterinary bills, carpet cleaning, and time value of training as well. Owning a puppy is a good illustration of opportunity cost, because the purchase price is typically a negligible portion of the total cost of ownership. Yet people acquire puppies all the time, in spite of their high cost of ownership. Why? The economic view of the world is that people acquire puppies because the value they expect exceeds their opportunity cost. That is, they reveal their preference for owning the puppy, as the benefit they derive must apparently exceed the opportunity cost of acquiring it.
Even though opportunity costs include nonmonetary costs, we will often monetize opportunity costs, by translating these costs into dollar terms for comparison purposes. Monetizing opportunity costs is valuable, because it provides a means of comparison. What is the opportunity cost of 30 days in jail? It used to be that judges occasionally sentenced convicted defendants to “thirty days or thirty dollars,” letting the defendant choose the sentence. Conceptually, we can use the same idea to find out the value of 30 days in jail. Suppose you would pay a fine of \$750 to avoid the 30 days in jail but would serve the time instead to avoid a fine of \$1,000. Then the value of the 30-day sentence is somewhere between \$750 and \$1,000. In principle there exists a critical price at which you’re indifferent to “doing the time” or “paying the fine.” That price is the monetized or dollar cost of the jail sentence.
The same process of selecting between payment and action may be employed to monetize opportunity costs in other contexts. For example, a gamble has a certainty equivalent, which is the amount of money that makes one indifferent to choosing the gamble versus the certain payment. Indeed, companies buy and sell risk, and the field of risk management is devoted to studying the buying or selling of assets and options to reduce overall risk. In the process, risk is valued, and the riskier stocks and assets must sell for a lower price (or, equivalently, earn a higher average return). This differential, known as a risk premium, is the monetization of the risk portion of a gamble.
Buyers shopping for housing are presented with a variety of options, such as one- or two-story homes, brick or wood exteriors, composition or shingle roofing, wood or carpet floors, and many more alternatives. The approach economists adopt for valuing these items is known as hedonic pricing. Under this method, each item is first evaluated separately and then the item values are added together to arrive at a total value for the house. The same approach is used to value used cars, making adjustments to a base value for the presence of options like leather interior, GPS system, iPod dock, and so on. Again, such a valuation approach converts a bundle of disparate attributes into a monetary value.
The conversion of costs into dollars is occasionally controversial, and nowhere is it more so than in valuing human life. How much is your life worth? Can it be converted into dollars? Some insight into this question can be gleaned by thinking about risks. Wearing seatbelts and buying optional safety equipment reduce the risk of death by a small but measurable amount. Suppose a \$400 airbag reduces the overall risk of death by 0.01%. If you are indifferent to buying the airbag, you have implicitly valued the probability of death at \$400 per 0.01%, or \$40,000 per 1%, or around \$4,000,000 per life. Of course, you may feel quite differently about a 0.01% chance of death compared with a risk 10,000 times greater, which would be a certainty. But such an approach provides one means of estimating the value of the risk of death—an examination of what people will, and will not, pay to reduce that risk.
KEY TAKEAWAYS
• The opportunity cost is the value of the best-forgone alternative.
• Opportunity cost of a purchase includes more than the purchase price but all of the costs associated with a choice.
• The conversion of costs into dollar terms, while sometimes controversial, provides a convenient means of comparing costs. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/01%3A_What_Is_Economics/1.02%3A_Opportunity_Cost.txt |
Learning ObjectiveS
1. How do economists reason?
2. What is comparative static?
3. What assumptions are commonly made by economists about human behavior?
4. What do economists mean by marginal?
What this country needs is some one-armed economists.
—Harry S. Truman
Economic reasoning is rather easy to satirize. One might want to know, for instance, what the effect of a policy change—a government program to educate unemployed workers, an increase in military spending, or an enhanced environmental regulation—will be on people and their ability to purchase the goods and services they desire. Unfortunately, a single change may have multiple effects. As an absurd and tortured example, government production of helium for (allegedly) military purposes reduces the cost of children’s birthday balloons, causing substitution away from party hats and hired clowns. The reduction in demand for clowns reduces clowns’ wages and thus reduces the costs of running a circus. This cost reduction increases the number of circuses, thereby forcing zoos to lower admission fees to compete with circuses. Thus, were the government to stop subsidizing the manufacture of helium, the admission fees of zoos would likely rise, even though zoos use no helium. This example is superficially reasonable, although the effects are miniscule.
To make any sense of all the effects of a change in economic conditions, it is helpful to divide up the effects into pieces. Thus, we will often look at the effects of a change in relation to “other things equal,” that is, assuming nothing else has changed. This isolates the effect of the change. In some cases, however, a single change can lead to multiple effects; even so, we will still focus on each effect individually. A gobbledygook way of saying “other things equal” is to use Latin and say “ceteris paribus.” Part of your job as a student is to learn economic jargon, and that is an example. Fortunately, there isn’t too much jargon.
We will make a number of assumptions that you may find implausible. Not all of the assumptions we make are necessary for the analysis, but instead are used to simplify things. Some, however, are necessary and therefore deserve an explanation. There is a frequent assumption in economics that the people we will talk about are exceedingly selfish relative to most people we know. We model the choices that people make, presuming that they select on the basis of their own welfare only. Such people—the people in the models as opposed to real people—are known as “homo economicus.” Real people are indubitably more altruistic than homo economicus, because they couldn’t be less: homo economicus is entirely selfish. (The technical term is self-interested behavior.) That doesn’t necessarily invalidate the conclusions drawn from the theory, however, for at least four reasons:
1. People often make decisions as families or households rather than as individuals, and it may be sensible to consider the household as the “consumer.” Identifying households as fairly selfish is more plausible perhaps than identifying individuals as selfish.
2. Economics is mostly silent on why consumers want things. You may wish to make a lot of money to build a hospital or endow a library, which would be altruistic. Such motives are not inconsistent with self-interested behavior.
3. Corporations are expected to serve their shareholders by maximizing share value, thus inducing self-interested behavior on the part of the corporation. Even if corporations could ignore the interests of their shareholders, capital markets would require them to consider shareholder interests as necessary condition for raising funds to operate and invest. In other words, people choosing investments for high returns will force corporations to seek a high return.
4. There are good, as well as bad, consequences that follow from people acting in their self-interest, and it is important for us to know what they are.
Thus, while the theory of self-interested behavior may not be universally descriptive, it is nonetheless a good starting point for building a framework to study the economics of human behavior.
Self-interested behavior will often be described as “maximizing behavior,” where consumers maximize the value they obtain from their purchases, and firms maximize their profits. One objection to this economic methodology is that people rarely carry out the calculations necessary to literally maximize anything. However, that is not a fatal flaw to the methodology. People don’t consciously do the physics calculations to throw a baseball or thread a needle, yet they somehow accomplish these tasks. Economists often consider that people act “as if” they maximize an objective, even though no explicit calculation is performed. Some corporations in fact use elaborate computer programs to minimize costs or maximize profits, and the field of operations research creates and implements such maximization programs. Thus, while individuals don’t necessarily calculate the consequences of their behavior, some companies do.
A good example of economic reasoning is the sunk cost fallacy. Once one has made a significant nonrecoverable investment, there is a psychological tendency to invest more, even when subsequent investment isn’t warranted. France and Britain continued to invest in the Concorde (a supersonic aircraft no longer in production) long after they realized that the project would generate little return. If you watch a movie to the end, even after you know it stinks, you haven fallen prey to the sunk cost fallacy. The fallacy is attempting to make an investment that has gone bad turn out to be good, even when it probably won’t. The popular phrase associated with the sunk cost fallacy is “throwing good money after bad.” The fallacy of sunk costs arises because of a psychological tendency to make an investment pay off when something happens to render it obsolete. It is a mistake in many circumstances.
Casinos often exploit the fallacy of sunk costs. People who lose money gambling hope to recover their losses by gambling more. The sunk “investment” to win money may cause gamblers to invest even more in order to win back what has already been lost. For most games like craps, blackjack, and one-armed bandits, the house wins on average, so that the average gambler (and even the most skilled slot machine or craps player) loses on average. Thus, for most, trying to win back losses is to lose more on average.
The way economics performs is by a proliferation of mathematical models, and this proliferation is reflected in this book. Economists reason with models. Models help by removing extraneous details from a problem or issue, which allows one more readily to analyze what remains. In some cases the models are relatively simple, like supply and demand. In other cases, the models are more complex. In all cases, the models are constructed to provide the simplest analysis possible that allows us to understand the issue at hand. The purpose of the model is to illuminate connections between ideas. A typical implication of a model is “when A increases, B falls.” This “comparative static” prediction lets us determine how A affects B, at least in the setting described by the model. The real world is typically much more complex than the models we postulate. That doesn’t invalidate the model, but rather by stripping away extraneous details, the model is a lens for focusing our attention on specific aspects of the real world that we wish to understand.
One last introductory warning before we get started. A parody of economists talking is to add the word marginal before every word. Marginal is just economists’ jargon for “the derivative of.” For example, marginal cost is the derivative of cost; marginal value is the derivative of value. Because introductory economics is usually taught to students who have not yet studied calculus (or can’t be trusted to remember it), economists avoid using derivatives and instead refer to the value of the next unit purchased, or the cost of the next unit, in terms of the marginal value or cost. This book uses “marginal” frequently because we wish to introduce the necessary jargon to students who want to read more advanced texts or take more advanced classes in economics. For an economics student not to know the word marginal would be akin to a physics student who does not know the word mass. The book minimizes jargon where possible, but part of the job of a principled student is to learn the jargon, and there is no getting around that.
KEY TAKEAWAYS
• It is often helpful to break economic effects into pieces.
• A common strategy is to examine the effects of a change in relation to “other things equal,” that is, assuming nothing else has changed, which isolates the effect of the change. “Ceteris paribus” means “other things equal.”
• Economics frequently models the choices that people make by assuming that they make the best choice for them. People in a model are known occasionally as “homo economicus.” Homo economicus is entirely selfish. The technical term is acting in one’s self-interest.
• Self-interested behavior is also described as “maximizing behavior,” where consumers maximize the net value they obtain from their purchases, and firms maximize their profits.
• Once one has made a significant nonrecoverable investment, there is a psychological tendency to invest more, even when the return on the subsequent investment isn’t worthwhile, known as the sunk cost fallacy.
• Economists reason with models. By stripping out extraneous details, the model represents a lens to isolate and understand aspects of the real world.
• Marginal is just economists’ jargon for “the derivative of.” For example, marginal cost is the derivative of cost; marginal value is the derivative of value. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/01%3A_What_Is_Economics/1.03%3A_Economic_Reasoning_and_Analysis.txt |
Thumbnail: The price P of a product is determined by a balance between production at each price (supply S) and the desires of those with purchasing power at each price (demand D). The diagram shows a positive shift in demand from D1 to D2, resulting in an increase in price (P) and quantity sold (Q) of the product. (CC BY-SA 3.0; Paweł Zdziarski via Wikipedia)
02: Supply and Demand
Learning Objectives
• What is demand?
• What is the value to buyers of their purchases?
• What assumptions are commonly made about demand?
• What causes demand to rise or fall?
• What is a good you buy only because you are poor?
• What are goods called that are consumed together?
• How does the price of one good influence demand for other goods?
Eating a french fry makes most people a little bit happier, and most people are willing to give up something of value—a small amount of money or a little bit of time—to eat one. The personal value of the french fry is measured by what one is willing to give up to eat it. That value, expressed in dollars, is the willingness to pay for french fries. So, if you are willing to give up 3 cents for a single french fry, your willingness to pay is 3 cents. If you pay a penny for the french fry, you’ve obtained a net of 2 cents in value. Those 2 cents—the difference between your willingness to pay and the amount you pay—is known as consumer surplus. Consumer surplus is the value of consuming a good, minus the price paid.
The value of items—like french fries, eyeglasses, or violins—is not necessarily close to what one must pay for them. For people with bad vision, eyeglasses might be worth $10,000 or more in the sense that people would be willing to pay this amount or more to wear them. Since one doesn’t have to pay nearly this much for eyeglasses means that the consumer surplus derived from eyeglasses is enormous. Similarly, an order of french fries might be worth$3 to a consumer, but since they are available for $1, the consumer obtains a surplus of$2 from purchase.
How much is a second order of french fries worth? For most of us, the first order is worth more than the second one. If a second order is worth $2, we would still gain from buying it. Eating a third order of fries is worth less still, and at some point we’re unable or unwilling to eat any more fries even when they are free, that implies that the value of additional french fries becomes zero eventually. We will measure consumption generally as units per period of time, for example, french fries consumed per month. Many, but not all, goods have this feature of diminishing marginal value—the value of the last unit declines as the number consumed rises. If we consume a quantity q, that implies the marginal value, denoted by v (q), falls as the number of units rise.When marginal value falls, which may occur with beer consumption, constructing demand takes some additional effort, which isn’t a great deal of consequence. Buyers will still choose to buy a quantity where marginal value is decreasing. An example is illustrated in Figure 2.1, where the value is a straight line, declining in the number of units. Figure 2.1 The demand curve Demand needn’t be a straight line, and indeed could be any downward-sloping curve. Contrary to the usual convention, the quantity demanded for any price is represented by the vertical axis whereas the price is plotted along the horizontal. It is often important to distinguish the demand curve—the relationship between price and quantity demanded—from the quantity demanded. Typically, “demand” refers to the curve, while “quantity demanded” is a point on the curve. For a price p, a consumer will buy units q such that v (q) > p since those units are worth more than they cost. Similarly, a consumer would not buy units for which v (q) < p. Thus, the quantity q0 that solves the equation v (q0) = p indicates the quantity the consumer will buy. This value is illustrated in Figure 2.1.We will treat units as continuous, even though they are discrete units. This simplifies the mathematics; with discrete units, the consumer buys those units with value exceeding the price and doesn’t buy those with value less than the price, just as before. However, since the value function isn’t continuous, much less differentiable, it would be an accident for marginal value to equal price. It isn’t particularly difficult to accommodate discrete products, but it doesn’t enhance the model so we opt for the more convenient representation. Another way of expressing this insight is that the marginal value curve is the inverse of the demand function, where the demand function gives the quantity purchased at a given price. Formally, if x (p) is the quantity a consumer buys at price p, then v(x(p))=p. But what is the marginal value curve? Suppose the total value of consumption is u(q). A consumer who pays u(q) for the quantity q is indifferent to receiving nothing and paying nothing. For each quantity, there should exist one and only one price that makes the consumer indifferent between purchasing and receiving nothing. If the consumer is just willing to pay u (q), any additional amount exceeds what the consumer should be willing to pay. The consumer facing price p receives consumer surplus of $$C S=u(q)-p q$$. In order to obtain the maximal benefit, the consumer chooses q to maximize u (q) – pq. When the function CS is maximized, its derivative is zero. This implies that the quantity maximizing the consumer surplus must satisfy $0=d d q(u(q)-p q)=u^{\prime}(q)-p$ Thus, $$v(q)=u^{\prime}(q)$$ ; implying that the marginal value is the derivative of the total value. Consumer surplus is the value of the consumption minus the amount paid, and it represents the net value of the purchase to the consumer. Formally, it is u (q) – pq. A graph of consumer surplus is generated by the following identity: $CS= max q ( u(q)−pq )=u( q 0 )−p q 0 = ∫ 0 q 0 ( u ′ (x)−p )dx = ∫ 0 q 0 ( v(x)−p )dx .$ This expression shows that consumer surplus can be represented as the area below the demand curve and above the price, as illustrated in Figure 2.2. The consumer surplus represents the consumer’s gains from trade, the value of consumption to the consumer net of the price paid. Figure 2.2 Consumer surplus The consumer surplus can also be expressed using the demand curve, by integrating from the price up to where the demand curve intersects with the price axis. In this case, if x(p) is demand, we have CS= ∫ p ∞ x(y) dy . When you buy your first car, you experience an increase in demand for gasoline because gasoline is pretty useful for cars and not so much for other things. An imminent hurricane increases the demand for plywood (to protect windows), batteries, candles, and bottled water. An increase in demand is represented by a movement of the entire curve to the northeast (up and to the right), which represents an increase in the marginal value v (movement up) for any given unit, or an increase in the number of units demanded for any given price (movement to the right). Figure 2.3 illustrates a shift in demand. Similarly, the reverse movement represents a decrease in demand. The beauty of the connection between demand and marginal value is that an increase in demand could, in principle, have meant either more units demanded at a given price or a higher willingness to pay for each unit, but those are in fact the same concept. Both changes create a movement up and to the right. For many goods, an increase in income increases the demand for the good. Porsche automobiles, yachts, and Beverly Hills homes are mostly purchased by people with high incomes. Few billionaires ride the bus. Economists aptly named goods whose demand doesn’t increase with income inferior goods, with the idea that people substitute to better quality, more expensive goods as their incomes rise. When demand for a good increases with income, the good is called a normal good. It would have been better to call such goods superior, but it is too late to change such a widely accepted convention. Figure 2.3 An increase in demand Another factor that influences demand is the price of related goods. The dramatic fall in the price of computers over the past 20 years has significantly increased the demand for printers, monitors, and Internet access. Such goods are examples of complements. Formally, for a given good x, a complement is a good whose consumption increases the value of x. Thus, the use of computers increases the value of peripheral devices like printers and monitors. The consumption of coffee increases the demand for cream for many people. Spaghetti and tomato sauce, national parks and hiking boots, air travel and hotel rooms, tables and chairs, movies and popcorn, bathing suits and sunscreen, candy and dentistry—all are examples of complements for most people. Consumption of one increases the value of the other. The complementary relationship is typically symmetric—if consumption of x increases the value of y, then consumption of y must increase the value of x.The basis for this insight can be seen by denoting the total value in dollars of consuming goods x and y as u(x, y). Then the demand for x is given by the partial ∂u ∂x . The statement that y is a complement means that the demand for x rises as y increases; that is, $$\partial 2 u \partial x \partial y>0$$. But then with a continuous second derivative, $$∂ 2 u ∂y∂x >0$$, which means the demand for y, ∂u ∂y , increases with x. From this we can predict that if the price of good y decreases, then the amount good y, a complementary good to x, will decline. Why, you may ask? The reason is that consumers will purchase more of good x when its price decreases. This will make good y more valuable, and hence consumers will also purchase more of good y as a result. The opposite case of a complement is a substitute. For a given good x, a substitute is a good whose consumption decreases the value of x. Colas and root beer are substitutes, and a fall in the price of root beer (resulting in an increase in the consumption of root beer) will tend to decrease the demand for colas. Pasta and ramen, computers and typewriters, movies (in theaters) and sporting events, restaurants and dining at home, spring break in Florida versus spring break in Mexico, marijuana and beer, economics courses and psychology courses, driving and bicycling—these are all examples of substitutes for most people. An increase in the price of a substitute increases the demand for a good; and, conversely, a decrease in the price of a substitute decreases demand for a good. Thus, increased enforcement of the drug laws, which tends to increase the price of marijuana, leads to an increase in the demand for beer. Much of demand is merely idiosyncratic to the individual—some people like plaids, some like solid colors. People like what they like. People often are influenced by others—tattoos are increasingly common, not because the price has fallen but because of an increased acceptance of body art. Popular clothing styles change, not because of income and prices but for other reasons. While there has been a modest attempt to link clothing style popularity to economic factors,Skirts are allegedly shorter during economic booms and lengthen during recessions. by and large there is no coherent theory determining fads and fashions beyond the observation that change is inevitable. As a result, this course, and economics generally, will accept preferences for what they are without questioning why people like what they like. While it may be interesting to understand the increasing social acceptance of tattoos, it is beyond the scope of this text and indeed beyond most, but not all, economic analyses. We will, however, account for some of the effects of the increasing acceptance of tattoos through changes in the number of parlors offering tattooing, changes in the variety of products offered, and so on. Key Takeaways • Demand is the function that gives the number of units purchased as a function of the price. • The difference between your willingness to pay and the amount you pay is known as consumer surplus. Consumer surplus is the value in dollars of a good minus the price paid. • Many, but not all, goods have the feature of diminishing marginal value—the value of the last unit consumed declines as the number consumed rises. • Demand is usually graphed with price on the vertical axis and quantity on the horizontal axis. • Demand refers to the entire curve, while quantity demanded is a point on the curve. • The marginal value curve is the inverse of demand function. • Consumer surplus is represented in a demand graph by the area between demand and price. • An increase in demand is represented by a movement of the entire curve to the northeast (up and to the right), which represents an increase in the marginal value v (movement up) for any given unit, or an increase in the number of units demanded for any given price (movement to the right). Similarly, the reverse movement represents a decrease in demand. • Goods whose demand doesn’t increase with income are called inferior goods, with the idea that people substitute to better quality, more expensive goods as their incomes rise. When demand for a good increases with income, the good is called normal. • Demand is affected by the price of related goods. • For a given good x, a complement is a good whose consumption increases the value of x. The complementarity relationship is symmetric—if consumption of x increases the value of y, then consumption of y must increase the value of x. • The opposite case of a complement is a substitute. An increase in the consumption of a substitute decreases the value for a good. EXERCISES 1. A reservation price is is a consumer’s maximum willingness to pay for a good that is usually bought one at a time, like cars or computers. Graph the demand curve for a consumer with a reservation price of$30 for a unit of a good.
2. Suppose the demand curve is given by $$x(p)=1-p$$. The consumer’s expenditure is $$p * x(p)=p(1-p)$$. Graph the expenditure. What price maximizes the consumer’s expenditure?
3. For demand $$x(p)=1-p$$, compute the consumer surplus function as a function of p.
4. For demand $$x(p)=p^{-} \varepsilon$$, for ε > 1, find the consumer surplus as a function of p. (Hint: Recall that the consumer surplus can be expressed as $$CS= ∫ p ∞ x(y) dy$$. )
5. Suppose the demand for wheat is given by $$qd = 3 – p$$ and the supply of wheat is given by $$qs = 2p$$, where p is the price.
1. Solve for the equilibrium price and quantity.
2. Graph the supply and demand curves. What are the consumer surplus and producer profits?
3. Now suppose supply shifts to $$qs = 2p + 1$$. What are the new equilibrium price and quantity?
6. How will the following affect the price of a regular cup of coffee, and why?
1. Droughts in Colombia and Costa Rica
2. A shift toward longer work days
3. The price of milk falls
4. A new study that shows many great health benefits of tea
7. A reservation price is a consumer’s maximum willingness to pay for a good that is usually bought one at a time, like cars or computers. Suppose in a market of T-shirts, 10 people have a reservation price of $10 and the 11th person has a reservation price of$5. What does the demand “curve” look like?
8. In Exercise 7, what is the equilibrium price if there were 9 T-shirts available? What if there were 11 T-shirts available? How about 10?
9. A consumer’s value for slices of pizza is given by the following table. Graph this person’s demand for slices of pizza.
Slices of pizza Total value
0 0
1 4
2 7
3 10
4 12
5 11 | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/02%3A_Supply_and_Demand/2.01%3A_Demand_and_Consumer_Surplus.txt |
Learning Objectives
• What is supply?
• What are gains made by sellers called?
• What assumptions are commonly made about supply?
• What causes supply to rise or fall?
• What are goods produced together called?
• How do the prices of one good influence supply for other goods?
The term supply refers to the function that gives the quantity offered for sale as a function of price. The supply curve gives the number of units that will be supplied on the horizontal axis, as a function of the price on the vertical axis; Figure 2.4 illustrates a supply curve. Generally, supply is upward sloping, because if it is a good deal for a supplier to sell 50 units of a product at a price of $10, then it is an even better deal to supply those same 50 at a price of$11. The seller might choose to sell more than 50, but if the first 50 aren’t worth keeping at a price of $10, then it remains true at$11.This is a good point at which to remind the reader that the economists’ familiar assumption of “other things equal” is still in effect. If the increased price is an indication that prices might rise still further, or a consequence of some other change that affects the seller’s value of items, then of course the higher price might not justify sale of the items. We hold other things equal to focus on the effects of price alone, and then will consider other changes separately. The pure effect of an increased price should be to increase the quantity offered, while the effect of increased expectations may be to decrease the quantity offered.
The seller with cost c (q) of selling q units obtains a profit, at price p per unit, of $$p q-c(q)$$. The quantity that maximizes profit for the seller is the quantity q* satisfying $0=d d q $p q-c(q)$=p-c^{\prime}\left(q^{*}\right)$
Thus, “price equals marginal cost” is a characteristic of profit maximization; the supplier sells all the units whose cost is less than price, and doesn’t sell the units whose cost exceeds price. In constructing the demand curve, we saw that it was the inverse of the marginal value. There is an analogous property of supply: The supply curve is the inverse function of marginal cost. Graphed with the quantity supplied on the horizontal axis and price on the vertical axis, the supply curve is the marginal cost curve, with marginal cost on the vertical axis.
Figure 2.4 The supply curve
Analogous to consumer surplus with demand, profit is given by the difference of the price and marginal cost.
$\text { Profit }=\max q p q-c(q)=p q^{*}-c\left(q^{*}\right)=\int 0 q^{*}\left(p-c^{\prime}(x)\right) d x$
This area is shaded in Figure 2.5.
The relationship of demand and marginal value exactly parallels the relationship of supply and marginal cost, for a somewhat hidden reason. Supply is just negative demand; that is, a supplier is just the possessor of a good who doesn’t keep it but instead, offers it to the market for sale. For example, when the price of housing goes up, one of the ways people demand less is by offering to rent a room in their house—that is, by supplying some of their housing to the market. Similarly, the marginal cost of supplying a good already produced is the loss of not having the good—that is, the marginal value of the good. Thus, with exchange, it is possible to provide the theory of supply and demand entirely as a theory of net demand, where sellers are negative demanders. There is some mathematical economy in this approach, and it fits certain circumstances better than separating supply and demand. For example, when the price of electricity rose very high in the western United States in 2003, several aluminum smelters resold electricity that they had purchased in long-term contracts; in other words, demanders became suppliers.
Figure 2.5 Supplier profits
However, the “net demand” approach obscures the likely outcomes in instances where the sellers are mostly distinct from the buyers. Moreover, while there is a theory of complements and substitutes for supply that is exactly parallel to the equivalent theory for demand, the nature of these complements and substitutes tends to be different. For these reasons, and also for the purpose of being consistent with common economic usage, we will distinguish supply and demand.
Figure 2.6 An increase in supply
Anything that increases costs of production will tend to increase marginal cost and thus reduce the supply. For example, as wages rise, the supply of goods and services is reduced because wages are the input price of labor. Labor accounts for about two thirds of all input costs, and thus wage increases create supply reductions (a higher price is necessary to provide the same quantity) for most goods and services. Costs of materials, of course, increase the price of goods using those materials. For example, the most important input into the manufacture of gasoline is crude oil, and an increase of $1 in the price of a 42-gallon barrel of oil increases the price of gasoline about 2 cents—almost one-for-one by volume. Another significant input in many industries is capital and, as we will see, interest is the cost of capital. Thus, increases in interest rates increase the cost of production, and thus tend to decrease the supply of goods. Analogous to complements in demand, a complement in supply to a good x is a good y such that an increase in the production of y increases the supply of x. In demand, a complement in supply is a good whose cost falls as the amount produced of another good rises. Complements in supply are usually goods that are jointly produced. In producing lumber (sawn boards), a large quantity of wood chips and sawdust are also produced as a by-product. These wood chips and sawdust are useful in the manufacture of paper. An increase in the price of lumber tends to increase the quantity of trees sawn into boards, thereby increasing the supply of wood chips. Thus, lumber and wood chips are complements in supply. It turns out that copper and gold are often found in the same kinds of rock—the conditions that give rise to gold compounds also give rise to copper compounds. Thus, an increase in the price of gold tends to increase the number of people prospecting for gold and, in the process, increases not just the quantity of gold supplied to the market but also the quantity of copper. Thus, copper and gold are complements in supply. The classic supply–complement is beef and leather—an increase in the price of beef increases the slaughter of cows, thereby increasing the supply of leather. The opposite of a complement in supply is a substitute in supply. This is a good whose cost rises as the amount produced of another good rises. Military and civilian aircraft are substitutes in supply—an increase in the price of military aircraft will tend to divert resources used in the manufacture of aircraft toward military aircraft and away from civilian aircraft, thus reducing the supply of civilian aircraft. Wheat and corn are also substitutes in supply. An increase in the price of wheat will lead farmers whose land is well suited to producing either wheat or corn to substitute wheat for corn, thus increasing the quantity of wheat and decreasing the quantity of corn. Agricultural goods grown on the same type of land are usually substitutes. Similarly, cars and trucks, tables and desks, sweaters and sweatshirts, horror movies and romantic comedies are all examples of substitutes in supply. Complements and substitutes are important because they are common and have predictable effects on demand and supply. Changes in one market spill over to the other market through the mechanism of complements or substitutes. Key Takeaways • The supply curve gives the number of units as a function of the price that will be supplied for sale to the market. • Price equals marginal cost is an implication of profit maximization; the supplier sells all the units whose cost is less than price and doesn’t sell the units whose cost exceeds price. • The supply curve is the inverse function of marginal cost. Graphed with the quantity supplied on the horizontal axis and price on the vertical axis, the supply curve is the marginal cost curve, with marginal cost on the vertical axis. • Profit is given by the difference of the price and marginal cost. • Supply is negative demand. • An increase in supply refers to either more units available at a given price or a lower price for the supply of the same number of units. Thus, an increase in supply is graphically represented by a curve that is lower or to the right, or both—that is, to the southeast. A decrease in supply is the reverse case, a shift to the northwest. • Anything that increases costs of production will tend to increase marginal cost and thus reduce the supply. • A complement in supply to a good x is a good y such that an increase in the price of y increases the supply of x. • The opposite of a complement in supply is a substitute in supply. EXERCISES 1. A typist charges$30 per hour and types 15 pages per hour. Graph the supply of typed pages.
2. An owner of an oil well has two technologies for extracting oil. With one technology, the oil can be pumped out and transported for $5,000 per day, and 1,000 barrels per day are produced. With the other technology, which involves injecting natural gas into the well, the owner spends$10,000 per day and $5 per barrel produced, but 2,000 barrels per day are produced. What is the supply? Graph it. (Hint: Compute the profits, as a function of the price, for each of the technologies. At what price would the producer switch from one technology to the other? At what price would the producer shut down and spend nothing?) 3. An entrepreneur has a factory that produces Lα widgets, where α < 1, when L hours of labor are used. The cost of labor (wage and benefits) is w per hour. If the entrepreneur maximizes profit, what is the supply curve for widgets? (Hint: The entrepreneur’s profit, as a function of the price, is $$\mathrm{pL}^{\mathrm{a}}-\mathrm{wL}$$. The entrepreneur chooses the amount of labor to maximize profit. Find the amount of labor that maximizes profit, which is a function of p, w, and α. The supply is the amount of output produced, which is Lα.) 4. In the above exercise, suppose now that more than 40 hours entails a higher cost of labor (overtime pay). Let w be$20/hr for under 40 hours, and \$30/hr for each hour over 40 hours, and α = ½. Find the supply curve.
(Hint: Let L(w, p) be the labor demand when the wage is w (no overtime pay) and the price is p. Now show that, if $$$$L(20, p)$$< 40$$, the entrepreneur uses $$$L(20, p)$$$hours. This is shown by verifying that profits are higher at $$L(20, p)$$ than at $$L(30, p)$$. If $$L(30, p)$ > 40$, the entrepreneur uses $$L(30, p)$$hours. Finally, if $$$L(20, p)$$> 40 > $L(30, p)$$, the entrepreneur uses 40 hours. Labor translates into supply via Lα.)
5. In the previous exercise, for what range of prices does employment equal 40 hours? Graph the labor demanded by the entrepreneur.
(Hint: The answer involves 10 . )
6. Suppose marginal cost, as a function of the quantity q produced, is mq. Find the producer’s profit as a function of the price p. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/02%3A_Supply_and_Demand/2.02%3A_Supply_and_Profit.txt |
Learning Objectives
• How are individual demands and supplies aggregated to create a market?
Individuals with their own supply or demand trade in a market, where prices are determined. Markets can be specific or virtual locations—the farmers’ market, the New York Stock Exchange, eBay—or may be an informal or more amorphous market, such as the market for restaurant meals in Billings, Montana, or the market for roof repair in Schenectady, New York.
Figure 2.7 Market demand
Example: If the demand of Buyer 1 is given by $$q = max {0, 10 – p}$$, and the demand of Buyer 2 is given by $$q = max {0, 20 – 4p}$$, what is market demand for the two participants?
Solution: First, note that Buyer 1 buys zero at a price of 10 or higher, while Buyer 2 buys zero at a price of 5 or higher. For a price above 10, market demand is zero. For a price between 5 and 10, market demand is Buyer 1’s demand, or $$10 – p$$. Finally, for a price between zero and 5, the market quantity demanded is $$10 – p$ + 20 – 4p = 30 – 5p$.
Market supply is similarly constructed—the market supply is the horizontal (quantity) sum of all the individual supply curves.
Example: If the supply of Firm 1 is given by $$q = 2p$$, and the supply of Firm 2 is given by $$q = max {0, 5p – 10}$$, what is market supply for the two participants?
Solution: First, note that Firm 1 is in the market at any price, but Firm 2 is in the market only if price exceeds 2. Thus, for a price between zero and 2, market supply is Firm 1’s supply, or 2p. For $$p > 2$$, market supply is $$5p – 10 + 2p = 7p – 10.$$
Key Takeaways
• The market demand gives the quantity purchased by all the market participants—the sum of the individual demands—for each price. This is sometimes called a “horizontal sum” because the summation is over the quantities for each price.
• The market supply is the horizontal (quantity) sum of all the individual supply curves.
EXERCISES
1. Is the consumer surplus for market demand the sum of the consumer surpluses for the individual demands? Why or why not? Illustrate your conclusion with a figure like Figure 2.7.
2. Suppose the supply of firm i is αi p, when the price is p, where i takes on the values $$1, 2, 3, …, n$$. What is the market supply of these n firms?
3. Suppose consumers in a small town choose between two restaurants, A and B. Each consumer has a value vA for A’s meal and a value vB for B’s meal, and each value is a uniform random draw from the [0, 1] interval. Consumers buy whichever product offers the higher consumer surplus. The price of B’s meal is 0.2. In the square associated with the possible value types, identify which consumers buy from A. Find the demand, which is the area of the set of consumers who buy from A in the diagram below. [Hint: Consumers have three choices—buy nothing [value 0], buy from A [value vA – pA], and buy from B [value vB – pB = vB – 0.2).] Draw the lines illustrating which choice has the highest value for the consumer.
Figure 2.8 | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/02%3A_Supply_and_Demand/2.03%3A_Market_Demand_and_Supply.txt |
Learning Objectives
• How are prices determined?
• What happens when price is too low?
• What happens when price is too high?
• When will price remain constant?
Economists use the term equilibrium in the same way that the word is used in physics: to represent a steady state in which opposing forces are balanced so that the current state of the system tends to persist. In the context of supply and demand, equilibrium occurs when the pressure for higher prices is balanced by the pressure for lower prices, and so that rate of exchange between buyers and sellers persists.
When the current price is above the equilibrium price, the quantity supplied exceeds the quantity demanded, and some suppliers are unable to sell their goods because fewer units are purchased than are supplied. This condition, where the quantity supplied exceeds the quantity demanded, is called a surplus. The suppliers failing to sell have an incentive to offer their good at a slightly lower price—a penny less—to make a sale. Consequently, when there is a surplus, suppliers push prices down to increase sales. In the process, the fall in prices reduces the quantity supplied and increases the quantity demanded, thus eventually eliminating the surplus. That is, a surplus encourages price-cutting, which reduces the surplus, a process that ends only when the quantity supplied equals the quantity demanded.
Similarly, when the current price is lower than the equilibrium price, the quantity demanded exceeds the quantity supplied, and a shortage exists. In this case, some buyers fail to purchase, and these buyers have an incentive to offer a slightly higher price to make their desired purchase. Sellers are pleased to receive higher prices, which tends to put upward pressure on the price. The increase in price reduces the quantity demanded and increases the quantity supplied, thereby eliminating the shortage. Again, these adjustments in price persist until the quantity supplied equals the quantity demanded.
Figure 2.9 Equilibration
Similarly, when the price is below p*, the quantity supplied qs is less than the quantity demanded qd. This causes some buyers to fail to find goods, leading to higher asking prices and higher bid prices by buyers. The tendency for the price to rise is illustrated using three arrows pointing up. The only price that doesn’t lead to price changes is p*, the equilibrium price in which the quantity supplied equals the quantity demanded.
The logic of equilibrium in supply and demand is played out daily in markets all over the world—from stock, bond, and commodity markets with traders yelling to buy or sell, to Barcelona fish markets where an auctioneer helps the market find a price, to Istanbul’s gold markets, to Los Angeles’s real estate markets.
The equilibrium of supply and demand balances the quantity demanded and the quantity supplied so that there is no excess of either. Would it be desirable, from a social perspective, to force more trade or to restrain trade below this level?
There are circumstances where the equilibrium level of trade has harmful consequences, and such circumstances are considered in the chapter on externalities. However, provided that the only people affected by a transaction are the buyer and the seller, the equilibrium of supply and demand maximizes the total gains from trade.
This proposition is quite easy to see. To maximize the gains from trade, clearly the highest value buyers must get the goods. Otherwise, if a buyer that values the good less gets it over a buyer who values it more, then gains can arise from them trading. Similarly, the lowest-cost sellers must supply those goods; otherwise we can increase the gains from trade by replacing a higher-cost seller with a lower-cost seller. Thus, the only question is how many goods should be traded to maximize the gains from trade, since it will involve the lowest-cost suppliers selling to the highest-value buyers. Adding a trade increases the total gains from trade when that trade involves a buyer with value higher than the seller’s cost. Thus, the gains from trade are maximized by the set of transactions to the left of the equilibrium, with the high-value buyers buying from the low-cost sellers.
In the economist’s language, the equilibrium is efficient in that it maximizes the gains from trade under the assumption that the only people affected by any given transaction are the buyers and the seller.
Key Takeaways
• The quantity supplied of a good or service exceeding the quantity demanded is called a surplus.
• If the quantity demanded exceeds the quantity supplied, a shortage exists.
• The equilibrium price is the price in which the quantity supplied equals the quantity demanded.
• The equilibrium of supply and demand maximizes the total gains from trade.
EXERCISES
1. If demand is given by $$q^{d} \mathrm{a}(p)=a-b p$$, and supply is given by $$q^{s}(p)=c p$$, solve for the equilibrium price and quantity. Find the consumer surplus and producer profits.
2. If demand is given by $$q^{d} \operatorname{ap} \varepsilon=a p^{-\varepsilon}$$, and supply is given by $$q^{s}(p)=b p^{n}$$, where all parameters are positive numbers, solve for the equilibrium price and quantity. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/02%3A_Supply_and_Demand/2.04%3A_Equilibrium.txt |
Learning Objectives
• What are the effects of changes in demand and supply?
What are the effects of an increase in demand? As the population of California has grown, the demand for housing has risen. This has pushed the price of housing up and also spurred additional development, increasing the quantity of housing supplied as well. We see such a demand increase illustrated in Figure 2.10, which represents an increase in the demand. In this figure, supply and demand have been abbreviated S and D. Demand starts at D1 and is increased to D2. Supply remains the same. The equilibrium price increases from p1* to p2*, and the quantity rises from q1* to q2*.
Figure 2.10 An increase in demand
A decrease in demand—which occurred for typewriters with the advent of computers, or buggy whips as cars replaced horses as the major method of transportation—has the reverse effect of an increase and implies a fall in both the price and the quantity traded. Examples of decreases in demand include products replaced by other products—VHS tapes were replaced by DVDs, vinyl records were replaced by CDs, cassette tapes were replaced by CDs, floppy disks (oddly named because the 1.44 MB “floppy,” a physically hard product, replaced the 720 KB, 5 ¼-inch soft floppy disk) were replaced by CDs and flash memory drives, and so on. Even personal computers experienced a fall in demand as the market was saturated in 2001.
An increase in supply comes about from a fall in the marginal cost: recall that the supply curve is just the marginal cost of production. Consequently, an increased supply is represented by a curve that is lower and to the right on the supply–demand graph, which is an endless source of confusion for many students. The reasoning—lower costs and greater supply are the same thing—is too easily forgotten. The effects of an increase in supply are illustrated in Figure 2.11. The supply curve goes from S1 to S2, which represents a lower marginal cost. In this case, the quantity traded rises from q1* to q2* and price falls from p1* to p2*.
Figure 2.11 An increase in supply
An important source of supply and demand changes can be found in the markets of complements. A decrease in the price of a demand–complement increases the demand for a product; and, similarly, an increase in the price of a demand–substitute increases the demand for a product. This gives two mechanisms to trace through effects from external markets to a particular market via the linkage of demand substitutes or complements. For example, when the price of gasoline falls, the demand for automobiles (a complement) should increase overall. As the price of automobiles rises, the demand for bicycles (a substitute in some circumstances) should rise. When the price of computers falls, the demand for operating systems (a complement) should rise. This gives an operating system seller like Microsoft an incentive to encourage technical progress in the computer market in order to make the operating system more valuable.
Figure 2.12 Price of storage
An increase in the price of a supply–substitute reduces the supply of a product (by making the alternative good more attractive to suppliers); and, similarly, a decrease in the price of a supply-complement reduces the supply of a good. By making the by-product less valuable, the returns to investing in a good are reduced. Thus, an increase in the price of DVD-R disks (used for recording DVDs) discourages investment in the manufacture of CD-R disks, which are a substitute in supply, leading to a decrease in the supply of CD-Rs. This tends to increase the price of CD-Rs, other things equal. Similarly, an increase in the price of oil increases exploration for oil, which increases the supply of natural gas, which is a complement in supply. However, since natural gas is also a demand substitute for oil (both are used for heating homes), an increase in the price of oil also tends to increase the demand for natural gas. Thus, an increase in the price of oil increases both the demand and the supply of natural gas. Both changes increase the quantity traded, but the increase in demand tends to increase the price, while the increase in supply tends to decrease the price. Without knowing more, it is impossible to determine whether the net effect is an increase or decrease in the price.
When the price of gasoline goes up, people curtail their driving to some extent but don’t immediately scrap their SUVs to buy more fuel-efficient automobiles or electric cars. Similarly, when the price of electricity rises, people don’t immediately replace their air conditioners and refrigerators with the most modern, energy-saving ones. There are three significant issues raised by this kind of example. First, such changes may be transitory or permanent, and people react differently to temporary changes than to permanent changes. Second, energy is a modest portion of the cost of owning and operating an automobile or refrigerator, so it doesn’t make sense to scrap a large capital investment over a small permanent increase in cost. Thus, people rationally continue to operate “obsolete” devices until their useful life is over, even when they wouldn’t buy an exact copy of that device. This situation, in which past choices influence current decisions, is called hysteresis. Third, a permanent increase in energy prices leads people to buy more fuel-efficient cars and to replace their old gas-guzzlers more quickly. That is, the effects of a change are larger over a time period long enough that all inputs can be changed (which economists call the long run) than over a shorter time interval where not all inputs can be changed, or the short run.
A striking example of such delays arose when oil quadrupled in price in 1973 and 1974, caused by a reduction in sales by the cartel of oil-producing nations, OPEC, which stands for the Organization of Petroleum Exporting Countries. The increased price of oil (and consequent increase in gasoline prices) caused people to drive less and to lower their thermostats in the winter, thus reducing the quantity of oil demanded. Over time, however, they bought more fuel-efficient cars and insulated their homes more effectively, significantly reducing the quantity demanded still further. At the same time, the increased prices for oil attracted new investments into oil production in Alaska, the North Sea between Britain and Norway, Mexico, and other areas. Both of these effects (long-run substitution away from energy and long-run supply expansion) caused the price to fall over the longer term, undoing the supply reduction created by OPEC. In 1981, OPEC further reduced output, sending prices still higher; but again, additional investments in production, combined with energy-saving investments, reduced prices until they fell back to 1973 levels (adjusted for inflation) in 1986. Prices continued to fall until 1990 (reaching an all-time low level) when Iraq’s invasion of Kuwait and the resulting first Iraqi war sent them higher again.
Short-run and long-run effects represent a theme of economics, with the major conclusion of the theme being that substitution doesn’t occur instantaneously, which leads to predictable patterns of prices and quantities over time.
It turns out that direct estimates of demand and supply are less useful as quantifications than notions of percentage changes, which have the advantage of being unit-free. This observation gives rise to the concept of elasticity, the next topic.
Key Takeaways
• An increase in the demand increases both the price and quantity traded.
• A decrease in demand implies a fall in both the price and the quantity traded.
• An increase in the supply decreases the price and increases the quantity traded.
• A decrease in the supply increases the price and decreases the quantity traded.
• A change in the supply of a good affects its price. This price change will in turn affect the demand for both demand complements and demand subsitutes.
• People react less to temporary changes than to permanent changes. People rationally continue to operate “obsolete” devices until their useful life is over, even when they wouldn’t buy an exact copy of that device, an effect called hysteresis.
• Short-run and long-run effects represent a theme of economics, with the major conclusion that substitution doesn’t occur instantaneously, which leads to predictable patterns of prices and quantities over time.
EXERCISES
1. Video games and music CDs are substitutes in demand. What is the effect of an increase in supply of video games on the price and quantity traded of music CDs? Illustrate your answer with diagrams for both markets.
2. Electricity is a major input into the production of aluminum, and aluminum is a substitute in supply for steel. What is the effect of an increase in price of electricity on the steel market?
3. Concerns about terrorism reduced demand for air travel and induced consumers to travel by car more often. What should happen to the price of Hawaiian hotel rooms? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/02%3A_Supply_and_Demand/2.05%3A_Changes_in_Demand_and_Supply.txt |
Learning Objectives
• What is the best way of measuring the responsiveness of demand?
• What is the best way of measuring the responsiveness of supply?
Let $x (p)$ represent the quantity purchased when the price is p, so that the function x represents demand. How responsive is demand to price changes? One might be tempted to use the derivative, $x′$, to measure the responsiveness of demand, since it measures how much the quantity demanded changes in response to a small change in price. However, this measure has two problems. First, it is sensitive to a change in units. If I measure the quantity of candy in kilograms rather than in pounds, the derivative of demand for candy with respect to price changes even if the demand itself is unchanged. Second, if I change price units, converting from one currency to another, again the derivative of demand will change. So the derivative is unsatisfactory as a measure of responsiveness because it depends on units of measure. A common way of establishing a unit-free measure is to use percentages, and that suggests considering the responsiveness of demand to a small percentage change in price in percentage terms. This is the notion of elasticity of demand. The concept of elasticity was invented by Alfred Marshall (1842–1924) in 1881 while sitting on his roof. The elasticity of demand is the percentage decrease in quantity that results from a small percentage increase in price. Formally, the elasticity of demand, which is generally denoted with the Greek letter epsilon, ε, (chosen mnemonically to indicate elasticity) is
$\varepsilon=-d x x d p p=-p x d x d p=-p x^{\prime}(p) x(p)$
The minus sign is included in the expression to make the elasticity a positive number, since demand is decreasing. First, let’s verify that the elasticity is, in fact, unit free. A change in the measurement of x doesn’t affect elasticity because the proportionality factor appears in both the numerator and denominator. Similarly, a change in the measure of price so that p is replaced by r = ap, does not change the elasticity, since as demonstrated below,
$\varepsilon=-r d d r x(r / a) x(r / a)=-r x^{\prime}(r / a) 1 a x(r / a)=-p x^{\prime}(p) x(p)$
the measure of elasticity is independent of a, and therefore not affected by the change in units.
How does a consumer’s expenditure, also known as (individual) total revenue, react to a change in price? The consumer buys x(p) at a price of p, and thus total expenditure, or total revenue, is TR = px(p). Thus,
$d \text { dp } p x(p)=x(p)+p x^{\prime}(p)=x(p)\left(1+p x^{\prime}(p) x(p)\right)=x(p)(1-\varepsilon)$
Therefore,
$d dp TR 1 p TR =1−ε. \nonumber$
In other words, the percentage change of total revenue resulting from a 1% change in price is one minus the elasticity of demand. Thus, a 1% increase in price will increase total revenue when the elasticity of demand is less than one, which is defined as an inelastic demand. A price increase will decrease total revenue when the elasticity of demand is greater than one, which is defined as an elastic demand. The case of elasticity equal to one is called unitary elasticity, and total revenue is unchanged by a small price change. Moreover, that percentage increase in price will increase revenue by approximately 1 – ε percent. Because it is often possible to estimate the elasticity of demand, the formulae can be readily used in practice. Table 3.1 provides estimates on demand elasticities for a variety of products.
Table 3.1 Various Demand Elasticities
Product Product ε
Salt Movies 0.9
Matches Shellfish, consumed at home 0.9
Toothpicks Tires, short-run 0.9
Airline travel, short-run Oysters, consumed at home 1.1
Residential natural gas, short-run Private education 1.1
Gasoline, short-run Housing, owner occupied, long-run 1.2
Automobiles, long-run Tires, long-run 1.2
Coffee Radio and television receivers 1.2
Legal services, short-run Automobiles, short-run 1.2-1.5
Tobacco products, short-run Restaurant meals 2.3
Residential natural gas, long-run Airline travel, long-run 2.4
Fish (cod) consumed at home Fresh green peas 2.8
Physician services Foreign travel, long-run 4.0
Taxi, short-run Chevrolet automobiles 4.0
Gasoline, long-run Fresh tomatoes 4.6
From http://www.mackinac.org/archives/1997/s1997-04.pdf; cited sources: James D. Gwartney and Richard L. Stroup,, Economics: Private and Public Choice, 7th ed., 1995; 8th ed., 1997; Hendrick S. Houthakker and Lester D. Taylor, Consumer Demand in the United States, 1929–1970 (1966; Cambridge: Harvard University Press, 1970); Douglas R. Bohi, Analyzing Demand Behavior (Baltimore: Johns Hopkins University Press, 1981); Hsaing-tai Cheng and Oral Capps, Jr., "Demand for Fish," American Journal of Agricultural Economics, August 1988; and U.S. Department of Agriculture.
When demand is linear, $$x(p)=a-b p$$, the elasticity of demand has the form
$\varepsilon=\mathrm{bp} \mathrm{a}-\mathrm{bp}=\mathrm{p} \text { a } \mathrm{b}-\mathrm{p}$
This case is illustrated in Figure 3.1.
Figure 3.1 Elasticities for linear demand
If demand takes the form x(p) = a * p−ε, then demand has constant elasticity, and the elasticity is equal to ε. In other words, the elasticity remains at the same level while the underlying variables (such as price and quantity) change.
The elasticity of supply is analogous to the elasticity of demand in that it is a unit-free measure of the responsiveness of supply to a price change, and is defined as the percentage increase in quantity supplied resulting from a small percentage increase in price. Formally, if s(p) gives the quantity supplied for each price p, the elasticity of supply, denoted by η (the Greek letter “eta,” chosen because epsilon was already taken) is
$\eta=\operatorname{ds} \mathrm{s} \mathrm{dp} \mathrm{p}=\mathrm{p} \mathrm{s} \mathrm{ds} \mathrm{dp}=\mathrm{ps}^{\prime}(\mathrm{p}) \mathrm{s}(\mathrm{p})$
Again, similar to demand, if supply takes the form s(p) = a * pη, then supply has constant elasticity, and the elasticity is equal to η. A special case of this form is linear supply, which occurs when the elasticity equals one.
Key Takeaways
• The elasticity of demand is the percentage decrease in quantity that results from a small percentage increase in price, which is generally denoted with the Greek letter epsilon, ε.
• The percentage change of total revenue resulting from a 1% change in price is one minus the elasticity of demand.
• An elasticity of demand that is less than one is defined as an inelastic demand. In this case, increasing price increases total revenue.
• A price increase will decrease total revenue when the elasticity of demand is greater than one, which is defined as an elastic demand.
• The case of elasticity equal to one is called unitary elasticity, and total revenue is unchanged by a small price change.
• If demand takes the form x (p) = a * p−ε , then demand has constant elasticity, and the elasticity is equal to ε.
• The elasticity of supply is defined as the percentage increase in quantity supplied resulting from a small percentage increase in price.
• If supply takes the form s (p) = a * p η, then supply has constant elasticity, and the elasticity is equal to η.
EXERCISES
1. Suppose a consumer has a constant elasticity of demand ε, and demand is elastic $$(ε > 1)$$. Show that expenditure increases as price decreases.
2. Suppose a consumer has a constant elasticity of demand ε, and demand is inelastic $$(ε < 1)$$. What price makes expenditure the greatest?
3. For a consumer with constant elasticity of demand $$(ε > 1)$$, compute the consumer surplus.
4. For a producer with constant elasticity of supply, compute the producer profits. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/03%3A_Quantification/3.01%3A_Elasticity.txt |
Learning Objectives
1. What are the effects of changes in supply and demand on price and quantity?
2. What is a useful approximation of these changes?
When the price of a complement changes—what happens to the equilibrium price and quantity of the good? Such questions are answered by comparative statics, which are the changes in equilibrium variables when other things change. The use of the term “static” suggests that such changes are considered without respect to dynamic adjustment; instead, one just focuses on the changes in the equilibrium level. Elasticities will help us quantify these changes.
How much do the price and quantity traded change in response to a change in demand? We begin by considering the constant elasticity case, which allows us to draw conclusions for small changes for general demand functions. We will denote the demand function by $$\mathrm{q}_{\mathrm{d}}(\mathrm{p})=\mathrm{a} * \mathrm{p}^{-\varepsilon}$$ and supply function by $$\mathrm{q}_{\mathrm{s}}(\mathrm{p})=\mathrm{bp}^{\mathrm{n}}$$. The equilibrium price p* is determined at the point where the quantity supplied equals to the quantity demanded, or by the solution to the following equation:
$q d\left(p^{*}\right)=q s\left(p^{*}\right)$
Substituting the constant elasticity formulae,
$ap * (−ε) = q d (p*)= q s (p*)=bp * (η) .$
Thus,
$a b=p * \varepsilon+\eta$
or
$p^{*}=(a b) 1 \varepsilon+\eta$
The quantity traded, q*, can be obtained from either supply or demand, and the price:
$q^{*}=q s\left(p^{*}\right)=b p^{*} \eta=b(a b) \eta \varepsilon+\eta=a \eta \varepsilon+\eta b \varepsilon \varepsilon+\eta$
There is one sense in which this gives an answer to the question of what happens when demand increases. An increase in demand, holding the elasticity constant, corresponds to an increase in the parameter a. Suppose we increase a by a fixed percentage, replacing a by $$a(1+\Delta)$$. Then price goes up by the multiplicative factor $$(1+\Delta) 1 \varepsilon+\eta$$ and the change in price, as a proportion of the price, is $\Delta p^{*} p^{*}=$(1+\Delta) 1 \varepsilon+\eta$-1$. Similarly, quantity rises by $$\Delta q^{*} q^{*}=(1+\Delta) \eta \varepsilon+\eta-1$$
These formulae are problematic for two reasons. First, they are specific to the case of constant elasticity. Second, they are moderately complicated. Both of these issues can be addressed by considering small changes—that is, a small value of ∆. We make use of a trick to simplify the formula. The trick is that, for small ∆,
$(1+\Delta) \mathbf{r} \approx 1+r \Delta$
The squiggly equals sign ≅ should be read, “approximately equal to.”The more precise meaning of ≅ is that, as ∆ gets small, the size of the error of the formula is small even relative to δ. That is, $$(1+\Delta) r \approx 1+r \Delta$$ means $$(1+\Delta) r-(1+r \Delta) \Delta \rightarrow \Delta \rightarrow 00$$. Applying this insight, we have the following:
For a small percentage increase ∆ in demand, quantity rises by approximately ηΔ ε+η percent and price rises by approximately Δ ε+η percent.
The beauty of this claim is that it holds even when demand and supply do not have constant elasticities because the effect considered is local and, locally, the elasticity is approximately constant if the demand is “smooth.”
Key Takeaways
• For a small percentage increase ∆ in demand, quantity rises by approximately $$\eta $\Delta \varepsilon+\eta$$$ percent and price rises by approximately $$\Delta \varepsilon+\eta$$ percent.
• For a small percentage increase ∆ in supply, quantity rises by approximately $$\varepsilon $\Delta \varepsilon+\eta$$$ percent and price falls by approximately $$\Delta \varepsilon+\eta$$ percent.
EXERCISES
1. Show that, for a small percentage increase ∆ in supply, quantity rises by approximately εΔ ε+η percent and price falls by approximately Δ ε+η percent.
2. If demand is perfectly inelastic (ε = 0), what is the effect of a decrease in supply? Apply the formula and then graph the solution.
3. Suppose demand and supply have constant elasticity equal to 3. What happens to equilibrium price and quantity when the demand increases by 3% and the supply decreases by 3%?
4. Show that elasticity can be expressed as a constant times the change in the log of quantity divided by the change in the log of price (i.e., show $$ε=A dlnx(p) dlnp$ )$. Find the constant A.
5. A car manufacturing company employs 100 workers and has two factories, one that produces sedans and one that makes trucks. With m workers, the sedan factory can make m2 sedans per day. With n workers, the truck factory can make 5n3 trucks per day. Graph the production possibilities frontier.
6. In Exercise 5, assume that sedans sell for $20,000 and trucks sell for$25,000. What assignment of workers maximizes revenue? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/03%3A_Quantification/3.02%3A_Supply_and_Demand_Changes.txt |
An important aspect of economics is economic statistics, and an army of economists collects and analyzes these statistics. This chapter presents an overview of the economic activity of the United States. How much do you need to know about these statistics? It would be ridiculous to memorize them. At the same time, it would be undesirable to be ignorant of how we are changing and how we are not.I apologize to those using the book in foreign countries; this chapter is about the United States not because it is more important but because I know it better. Encourage your professor to write a chapter on your country! All of the statistics in this chapter come from Fedstats, www.fedstats.gov/, from FRED, http://research.stlouisfed.org/fred2/, and from the NBER, http://www.nber.org/.
Learning Objectives
• Who lives in the United States?
There are about 300 million people in the United States, up from 76 million in 1900.
Figure 4.1 U.S. resident population
During the last century, the U.S. population has become primarily an urban population, growing from 40% to 80% urban. The population is primarily white, with 12%–13% African American and 4% classified as other. These proportions are relatively stable over the century, with the white population falling from 89% to 83%. The census is thought to understate minority populations because of greater difficulties in contacting minorities. The census does not attempt to classify people but instead accepts people’s descriptions of their own race.
Figure 4.2 U.S. urban and white population
The U.S. population has been aging significantly, with the proportion of seniors (over 65 years of age) tripling over the past century, and the proportion of young people dropping by over one-third. Indeed, the proportion of children between 0 and 5 years old has dropped from 12.1% of the population to under 7%.
Figure 4.3 Population proportions by age group
Figure 4.4 Proportion of population under age 5
The aging of the American population is a consequence of greater life expectancy. When social security was created in 1935, the average American male lived to be slightly less than 60 years old. The social security benefits, which didn’t start until age 65, thus were not being paid to a substantial portion of the population.
Figure 4.5 U.S. life expectancy at birth
Figure 4.6 U.S. immigrant population (percentages) by continent of origin
It is said that the United States is a country of immigrants, and a large fraction of the population had ancestors who came from elsewhere. Immigration into this United States, however, has been increasing after a long decline, and the fraction of the population that was born in foreign countries is about 11%—one in nine.
Figure 4.7 National origin of immigrants, 1900–2000
One hears a lot about divorce rates in the United States, with statements like “Fifty percent of all marriages end in divorce.” Although it has grown, the divorced population is actually a small fraction of the population of the United States.
Figure 4.8 Male marital status (percentages)
Marriage rates have fallen, but primarily because the “never married” category has grown. Some of the “never married” probably represent unmarried couples, since the proportion of children from unmarried women has risen fairly dramatically. Even so, marriage rates are greater than they were a century ago. However, a century ago there were more unrecorded and common-law marriages than there probably are today.
Figure 4.10 Births to unwed mothers (percentages)
Figure 4.11 Births to women age 19 or less (percentages)
Key Takeaways
• No one in his or her right mind memorizes the takeaways of this chapter; the goal is to have a sense of one’s nation.
• There are about 300 million people in the United States, up from 76 million in 1900.
• The U.S. population has become primarily an urban population, growing from 40% to 80% urban in the past century.
• The population is primarily white, with 12%–13% African American.
• The U.S. population has aged, with the proportion of seniors (over 65 years of age) tripling over the past century, and the proportion of young people dropping by over one-third.
• The baby boom was a dramatic increase in births for the years 1946 to 1964.
• The aging of the American population is a consequence of greater life expectancy.
• About 11% of Americans were born in foreign countries.
• The divorced population is about 10%.
• Marriage rates have fallen, but primarily because the “never married” category has grown.
4.02: Education
Learning Objectives
• Who goes to school and how much?
Why are the Western nations rich and many other nations poor? What creates the wealth of the developed nations? Modern economic analysis attributes much of the growth of the United States and other developed nations to its educated workforce, and not to natural resources. Japan, with a relative scarcity of natural resources but a highly educated workforce, is substantially richer than Brazil, with its abundance of natural resources.
Figure 4.12 Educational attainment in years (percentage of population)
Graduation rates are somewhat below the number of years completed, so that slightly less than three quarters of the U.S. population actually obtain their high school degree. Of those obtaining a high school degree, nearly half obtain a university or college degree.
Figure 4.13 Graduation rates (percentages)
As the number of high school students rose, the portion of high school graduates going to university fell, meaning that a larger segment of the population became high school educated. This increase represents the creation of the U.S. middle class; previously, high school completion and university attendance was in large part a sign of wealth. The creation of a large segment of the population who graduated from high school but didn’t attend university led to a population with substantial skills and abilities but no inherited wealth and they became the middle class.
High school completion has been declining for 30 years. This is surprising given the high rate of financial return to education in the United States. Much of the reduction in completion can be attributed to an increase in General Education Development (GED) certification, which is a program that grants diplomas (often erroneously thought to be a “General Equivalent Degree”) after successfully passing examinations in five subject areas. Unfortunately, those people who obtain GED certification are not as successful as high school graduates, even marginal graduates, and indeed the GED certification does not seem to help students succeed, in comparison with high school graduation.In performing this kind of analysis, economists are very concerned with adjusting for the type of person. Smarter people are more likely to graduate from high school, but one doesn’t automatically become smart by attending high school. Thus, care has been taken to hold constant innate abilities, measured by various measures like IQ scores and performance on tests, so that the comparison is between similar individuals, some of whom persevere to finish school, some of whom don’t. Indeed, some studies use identical twins.
Key Takeaways
• An estimated 85% of the U.S. population completes 12 years of schooling, not counting kindergarten.
• One quarter of the population completes at least 4 years of university.
• High school graduates comprise the bulk of the middle class.
• High school completion has been declining for 30 years. This is surprising given the high rate of financial return to education in the United States. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/04%3A_The_U.S._Economy/4.01%3A_Basic_Demographics.txt |
Learning Objectives
• How much stuff do we have?
There are approximately 100 million households—a group of people sharing living quarters—in the United States. The number of residents per household has consistently shrunk during this century, from over four to under three, as illustrated in Figure 4.14.
Figure 4.14 Household occupancy
Figure 4.15 Proportion of households by type
Figure 4.16 Percentage of incarcerated residents
Ten percent of households do not have an automobile, and 97.6% have a telephone. So-called land line telephones may start to fall as apartment dwellers, especially students, begin to rely exclusively on cell phones. Just under 99% of households have complete plumbing facilities (running water, bath or shower, flush toilet), up from 54.7% in 1940.
How much income do these households make? What is the distribution of income? One way of assessing the distribution is to use quintiles to measure dispersion. A quintile is one fifth, or 20%, of a group. Thus the top income quintile represents the top 20% of income earners, the next represents those ranking 60%–80%, and so on. Figure 4.17 shows the earnings of the top, middle, and bottom quintiles.
Figure 4.17 Income shares for three quintiles
Figure 4.19 Family income, cumulative percentage change
Real income gains in percentage terms have been larger for richer groups, even though the poor have also seen substantially increased incomes.
If the poor have fared less well than the rich in percentage terms, how have African Americans fared? After World War II, African American families earned about 50% of white family income. This ratio has risen gradually, noticeably in the 1960s after the Civil Rights Actlegislation that prohibited segregation based on race in schools, public places, and employment—that is credited with integrating workplaces throughout the southern United States. African American family income lagged white income growth throughout the 1980s but has been rising again, a trend illustrated in Figure 4.20.
Figure 4.20 Black family income as a percentage of white income
There have been three major inflations in the past century. Both World War I and World War II, with a large portion of the goods and services diverted to military use, saw significant inflations. In addition, there was a substantial inflation during the 1970s, after the Vietnam War in the 1960s. The price level fell during the Great Depression, a prolonged and severe economic downturn from 1929 to 1939. Falling price levels create investment problems because inflation-adjusted interest rates, which must adjust for deflation, are forced to be high, since unadjusted interest rates cannot be negative. Changes in the absolute price level are hard to estimate, so the change is separately graphed in Figure 4.22.
Figure 4.21 Consumer price index (1982 = 100)
Moreover, a much greater fraction of expenditures on food are spent away from home, a fraction that has risen from under 15% to 40%.
Figure 4.23 Food expenditure as percentage of income, and proportion spent out
Figure 4.24 After-tax income shares
Key Takeaways
• There are approximately 100 million households in the United States.
• The number of residents per household has shrunk, from over four to under three, over the past 100 years.
• About 60% of households live in single-family detached homes.
• Slightly less than 0.5% of the population is incarcerated in state and federal prisons. This represents a four-fold increase over 1925 to 1975.
• Ten percent of households do not have an automobile, and 97.6% have a telephone.
• Just under 99% of households have complete plumbing facilities (running water, bath or shower, flush toilet), up from 55% in 1940.
• A quintile (or fifth) is a group of size 20%.
• The earnings of the top quintile fell slightly until the late 1960s, when it began to rise. All other quintiles lost income share to the top quintile starting in the mid-1980s. Figures like these suggest that families are getting poorer, except for an elite few. However, families are getting richer, just not as fast as the top quintile.
• Just after World War II, African American families earned about 50% of white family income. This ratio has risen gradually, noticeably in the 1960s after the 1964 Civil Rights Act.
• The consumer price index (CPI), which adjusts for what it costs to buy a “standard” bundle of food, clothing, housing, electricity, and other items, is the most common price index.
• There have been three major inflations in the past century, associated with World War I, World War II, and the 1970s. The price level fell during the Great Depression (1929–1939).
• The cost of food has fallen quite dramatically over the past century.
EXERCISES
1. Have prices actually risen? Economists generally agree that the meaning of “prices have risen” is that you would prefer past prices to current prices. What makes this challenging is that the set of available products change over time. Cars have gone up significantly in price but are also more reliable. Would you be better off with your current income in 1913 than today? You would be very rich with current average income in 1913 but would not have access to modern medicine, television, electronics, refrigeration, highways, and many other technologies. If you made \$40,000 annually in 1913, how would you live and what would you buy? (Do some research.)
2. Compare a \$40,000 income in 1980 to the present. What differences are there in available products? In the quality of products? How rich does \$40,000 make you in each time period? In which period would you choose to live, and why? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/04%3A_The_U.S._Economy/4.03%3A_Households_and_Consumption.txt |
Learning Objectives
• What do we make?
We learned something about where we live and what we buy. Where do we get our income? Primarily, we earn by providing goods and services. Nationally, we produce about $11 trillion worth of goods and services. Broadly speaking, we spend that$11 trillion on personal consumption of goods and services, savings, and government. This, by the way, is often expressed as Y = C + I + G, which states that income (Y) is spent on consumption (C), investment (I, which comes from savings), and government (G). One can consume imports as well, so the short-term constraint looks like Y + M = C + I + G + X, where M is imports and X is exports.
How much does the United States produce? Economists measure output with the gross domestic product (GDP), which is the value of traded goods and services produced within the borders of the United States. GDP thus excludes output of Japanese factories owned by Americans but includes the output of U.S. factories owned by the Japanese.
Importantly, GDP excludes nontraded goods and services. Thus, unpaid housework is not included. If you clean your own home, and your neighbor cleans his or her home, the cleaning does not contribute to GDP. On the other hand, if you and your neighbor pay each other to clean each other’s homes, GDP goes up by the payments, even though the actual production of goods and services remains unchanged. Thus, GDP does not measure our total output as a nation, because it neglects unpaid services. Why does it neglect unpaid services? Primarily because we can’t readily measure them. Data on transactions are generated by tax information and reporting requirements imposed on businesses. For the same reason, GDP neglects illegal activities as well, such as illegal drug sales and pirated music sales. Thus, GDP is not a perfect measure of the production of our society. It is just the best measure we have.
Figure 4.25 shows the growth in GDP and its components of personal consumption, government expenditures, and investment. The figures are expressed in constant 1996 dollars—that is, adjusted for inflation. The figure for government includes the government’s purchases of goods and services—weapons, highways, rockets, pencils—but does not include transfer payments like social security and welfare programs. Transfer payments are excluded from this calculation because actual dollars are spent by the recipient, not by the government. The cost of making the transfer payments (e.g., printing and mailing the checks), however, is included in the cost of government.
Figure 4.25 Output, consumption, investment, and government
$\% \Delta x=x t-x t-1 \times t-1$
Then
$\log (x t)=\log (x t-1)+\log (x t x t-1)=\log (x t-1)+\log (1+\% \Delta x)$
Thus, if the percentage change is constant over time, log(xt) will be a straight line over time. Moreover, for small percentage changes,
$\log (1+\% \Delta x) \approx \% \Delta x$
so that the slope is approximately the growth rate.The meaning of ≈ throughout this book is “to the first order.” Here that means lim %Δx→0 log( 1+%Δx )−%Δx %Δx =0. Moreover, in this case the errors of the approximation are modest up to about 25% changes. Figure 4.26 shows these statistics with a logarithmic scale.
Figure 4.26 Major GDP components in log scale
Immediately noticeable is the approximately constant growth rate from 1950 to the present, because a straight line with a log scale represents a constant growth rate. In addition, government has grown much more slowly (although recall that transfer payments, another aspect of government, aren’t shown). A third feature is the volatility of investment—it shows much greater changes than output and consumption. Indeed, during the Great Depression (1929–1939), income fell somewhat, consumption fell less, government was approximately flat, and investment plunged to 10% of its former level.
Some of the growth in the American economy has arisen because there are more of us. Double the number of people, and consume twice as many goods, and individually we aren’t better off. How much are we producing per capita, and how much are we consuming?
U.S. output of goods, services, and consumption has grown substantially over the past 75 years, a fact illustrated in Figure 4.27. In addition, consumption has been a steady percentage of income. This is more clearly visible when income shares are plotted in Figure 4.28.
Figure 4.27 Per capita income and consumption
Consumption was a very high portion of income during the Great Depression because income itself fell. Little investment took place. The wartime economy of World War II reduced consumption to below 50% of output, with government spending a similar fraction as home consumers. Otherwise, consumption has been a relatively stable 60%–70% of income, rising modestly during the past 20 years, as the share of government shrank and net imports grew. Net imports rose to 4% of GDP in 2001.
The most basic output of our economic system is food, and the U.S. economy does a remarkable job producing food. The United States has about 941 million acres under cultivation to produce food, which represents 41.5% of the surface area of the United States. Land use for agriculture peaked in 1952, at 1,206 million acres, and has been dwindling ever since, especially in the northeast where farms are being returned to forest through disuse. Figure 4.29 shows the output of agricultural products in the United States, adjusted to 1982 prices.
Figure 4.29 U.S. agricultural output, 1982 constant dollars
Figure 4.30 Agricultural output, total and per worker (1982 dollars, log scale)
Figure 4.31 Major nonagricultural sectors of U.S. economy (% GDP)
Mining has diminished as a major factor in the U.S. economy, a consequence of the growth of other sectors and the reduction in the prices for raw materials. Contrary to many popular predictions, the prices of raw materials have fallen even as output and population have grown. We will see later in this book that the fall in prices of raw materials—ostensibly in fixed supply given the limited capacity of the earth—means that people expect a relative future abundance, either because of technological improvements in their use or because of large as yet undiscovered pools of the resources. An example of technological improvements is the substitution of fiber optic cable for copper wires. An enormous amount of copper has been recovered from telephone lines, and we can have more telephone lines and use less copper than was used in the past.
Manufacturing has become less important for several reasons. Many manufactured goods cost less, pulling down the overall value. In addition, we import more manufactured goods than in the past. We produce more services. T&PU stands for transportation and public utilities, and includes electricity and telephone services and transportation including rail and air travel. This sector has shrunk as a portion of the entire economy, although the components have grown in absolute terms. For example, the number of airplane trips has grown dramatically, as illustrated in Figure 4.32.
Figure 4.32 Air travel per capita
Figure 4.33 Electricity production (M kwh)
Figure 4.34 Energy use (quadrillion BTUs)
Figure 4.35 Cars per thousand population and miles driven per capita
The cost of selling goods—wholesale and retail costs—remains relatively stable, as does “FIRE,” which stands for finance, insurance, and real estate costs. Other services, ranging from restaurants to computer tutoring, have grown substantially. This is the so-called service economy that used to be in the news frequently, but is less so these days.
A bit more than 60% of the population works, with the historical percentage graphed in Figure 4.36. The larger numbers in recent years are partially a reflection of the baby boom’s entry into working years, reducing the proportion of elderly and children in American society. However, it is partially a reflection of an increased propensity for households to have two income earners.
Figure 4.36 Percentage of population employed (military and prisoners excluded)
Another sector of the economy that has been of focus in the news is national defense. How much do we spend on the military? In this century, the large expenditure occurred during World War II, when about 50% of GDP was spent by the government and 37% of GDP went to the armed forces. During the Korean War, we spent about 15% of GDP on military goods and less than 10% of GDP during the Vietnam War. The military buildup during Ronald Reagan’s presidency (1980–1988) increased our military expenditures from about 5.5% to 6.5% of GDP—a large percentage change in military expenditures, but a small diversion of GDP. The fall of the Soviet Union led the United States to reduce military expenditures, in what was called the “peace dividend,” but again the effects were modest, as illustrated in Figure 4.38.
Figure 4.38 Defense as a percentage of GDP
Historically, defense represents the largest expenditure by the federal government. However, as we see, defense has become a much smaller part of the economy overall. Still, the federal government plays many other roles in the modern U.S. economy.
Key Takeaways
• Economists measure output with the gross domestic product (GDP), which is the value of traded goods and services produced within the borders of the United States.
• Importantly, GDP excludes nontraded goods and services. Thus, GDP is not a perfect measure of the production of our society. It is just the best measure we have.
• Economists often use a logarithmic scale rather than a dollar scale. On a logarithmic scale, a straight line gives constant percentage growth.
• Economists divide production into goods and services. Goods are historically divided into mining, construction, and manufacturing.
• The prices of raw materials have fallen even as output and population have grown.
• Manufacturing has become less important for several reasons. Many manufactured goods cost less, pulling down the overall value. In addition, we import more manufactured goods than in the past. We produce more services.
• Electricity production has risen dramatically.
• The number of automobiles per capita in the United States peaked in the early 1980s, but we still drive more than ever, suggesting the change is actually an increase in the reliability of automobiles.
• The cost of selling goods—wholesale and retail costs—remains relatively stable, as does “FIRE” (finance, insurance, and real estate costs). Other services have grown substantially.
• A bit more than 60% of the population works.
• Female participation in the labor force has risen quite dramatically in the United States.
• Military expenditures peaked during World War II, when about 50% of GDP was spent by the government, and 37% of GDP went to the armed forces. During the Korean War, we spent about 15% of GDP on the military and less than 10% of GDP during the Vietnam War. The military buildup during Ronald Reagan’s presidency (1980–1988) increased our military expenditures from about 5.5% to 6.5% of GDP—a large percentage change in military expenditures, but a small diversion of GDP. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/04%3A_The_U.S._Economy/4.04%3A_Production.txt |
Learning Objectives
• How big is government and what does the government spend money on?
With a budget over \$2 trillion, the federal government represents just under 20% of the U.S. economy. It is one of the largest organizations in the world; only nations are larger organizations, and only a handful of nations are larger.
The size of the federal government, as a percentage of GDP, is shown in Figure 4.39. Federal expenditures boomed during World War II (1940–1945), but shrank back to nearly prewar levels shortly afterward, with much of the difference attributable to veterans’ benefits and continuing international involvement. Federal expenditures, as a percentage of GDP, continued to grow until Ronald Reagan’s presidency in 1980, when they began to shrink slightly after an initial growth. Figure 4.39 also shows federal revenues, and the deficit—the difference between expenditures and revenues—is apparent, especially for World War II and 1970 to 1998.
Figure 4.39 Federal expenditures and revenues (percent of GDP)
Much has been written about the federal government’s “abdication” of various services, which are pushed onto state and local government. Usually this behavior is attributed to the Reagan presidency (1980–1988). There is some evidence of this behavior in the postwar data, but the effect is very modest and long term. Most of the growth in state and local government occurred between 1947 and 1970, well before the Reagan presidency; state and local government has been stable since then. Moreover, the expenditure of the federal government, which shows ups and downs, has also been fairly stable. In any event, such effects are modest overall.
Figure 4.40 Federal, state, and local level, and total government receipts (% GDP)
Figure 4.41 Federal, regional, and total expenditures (% GDP)
Figure 4.42 Federal expenditures, on and off budget (% GDP)
During the 1980s, the public became aware of off-budget items. Political awareness made off-budget items cease to work as a device for evading balanced-budget requirements, and few new ones were created, although they continue to be debated. Sporadically, there are attempts to push social security off-budget.
Figure 4.43 Federal and regional government employment
Federal employees include two major categories: uniformed military personnel and the executive branch. State and local government is much larger and has tripled in size since 1962, a fact illustrated in Figure 4.43. The biggest growth areas involve public school teachers, police, corrections (prisons), and hospitals. About 850,000 of the federal employees work for the postal service.
Figure 4.44 Major expenditures of the federal government
Figure 4.46 Social security revenue and expenditure (\$ millions)
The social security administration has been ostensibly investing money and has a current value of approximately \$1.5 trillion, which is a bit less than four times the current annual expenditure on social security. Unfortunately, this money is “invested” in the federal government, and thus is an obligation of the federal government, as opposed to an investment in the stock market. Consequently, from the perspective of someone who is hoping to retire in, say, 2050, this investment isn’t much comfort, since the investment won’t make it easier for the federal government to make the social security payments. The good news is that the government can print money. The bad news is that when the government prints a lot of money and the obligations of the social security administration are in the tens of trillions of dollars, it isn’t worth very much.
Figure 4.47 Federal debt, total and percent of GDP
Starting in the late 1970s, the United States began accumulating debt faster than it was growing, and the debt began to rise. That trend wasn’t stabilized until the 1990s, and then only because the economy grew at an extraordinary rate by historical standards. The expenditures following the September 11, 2001, terrorist attacks, combined with a recession in the economy, have sent the debt rising dramatically, wiping out the reduction of the 1990s.
The national debt isn’t out of control, yet. At 4% interest rates on federal borrowing, we spend about 2.5% of GDP on interest servicing the federal debt. The right evaluation of the debt is as a percentage of GDP; viewed as a percentage, the size of the debt is of moderate size—serious but not critical. The serious side of the debt is the coming retirement of the baby boom generation, which is likely to put additional pressure on the government.
An important distinction in many economic activities is one between a stock and a flow. A stock is the current amount of some material; flow represents the rate of change in the amount of some material that exists from one instant to the next. Your bank account represents a stock of money; expenditures and income represent a flow. The national debt is a stock; the deficit is the addition to the debt and is a flow. If you think about a lake with incoming water and evaporation, the amount of water in the lake is the stock of water, while the incoming stream minus evaporation is the flow.
Table 4.1 Expenditures on agencies as percent of non-transfer expenditures
Department or Agency 1977 1990 2002
Legislative 0.4 0.4 0.5
Judiciary 0.2 0.3 0.6
Agriculture 2.1 2.2 2.7
Commerce 3.2 0.7 0.7
Education 3.9 3.8 6.7
Energy 3.1 3.2 2.9
Health 3.7 4.6 8.3
Defense 43.8 59.2 46.9
Homeland Security - - 4.1
Housing & Urban Dev. 13.4 2.9 4.3
Interior 1.6 1.3 1.4
Justice 1.0 1.7 2.7
Labor 6.1 1.7 1.7
State 0.6 0.9 1.3
Transportation 2.2 2.6 2.1
Treasury 1.7 1.6 1.4
Veterans 2.3 2.6 3.3
Corps of Engineers 1.0 0.6 0.6
Environmental P.A. 1.1 1.1 1.1
Fed Emergency M.A. 0.2 0.4 0.0
GSA 0.2 0.5 0.0
Intl Assistance 2.8 2.7 1.9
NASA 1.6 2.5 2.0
NSF 0.3 0.4 0.7
Table 4.1 gives the expenditures on various agencies, as a percentage of the discretionary expenditures, where discretionary is a euphemism for expenditures that aren’t transfers. Transfers, which are also known as entitlements, include social security, Medicare, aid to families with dependent children, unemployment insurance, and veteran’s benefits. Table 4.1 provides the expenditures by what is sometimes known as the “Alphabet Soup” of federal agencies (DOD, DOJ, DOE, FTC, SEC, …).
The National Science Foundation (NSF) provides funding for basic research. The general idea of government-funded research is that it is useful for ideas to be in the public domain and, moreover, that some research isn’t commercially viable but is valuable nevertheless. Studying asteroids and meteors produces little, if any, revenue but could, perhaps, save humanity one day in the event that we needed to deflect a large incoming asteroid. (Many scientists appear pessimistic about actually deflecting an asteroid.) Similarly, research into nuclear weapons might be commercially viable; but, as a society, we don’t want firms selling nuclear weapons to the highest bidder. In addition to the NSF, the National Institutes of Health, also a government agency, funds a great deal of research. How much does the government spend on research and development (R&D)? Figure 4.48 shows the history of R&D expenditures. The 1960s “space race” competition between the United States and the Soviet Union led to the greatest federal expenditure on R&D, and it topped 2% of GDP. There was a modest increase during the Reagan presidency (1980–1988) in defense R&D, which promptly returned to earlier levels.
Figure 4.48 Federal spending on R&D (% GDP)
Figure 4.49 Sources of federal government revenue
An important aspect of tax collection is that income taxes, like the federal income tax as well as social security and Medicare taxes, are very inexpensive to collect relative to sales taxes and excise taxes. Income taxes are straightforward to collect even relative to corporate income taxes. Quite reasonably, corporations can deduct expenses and the costs of doing business and are taxed on their profits, not on revenues. What is an allowable deduction, and what is not, makes corporate profits complicated to administer. Moreover, from an economic perspective, corporate taxes are paid by consumers in the form of higher prices for goods, at least when industries are competitive.
Key Takeaways
• With a budget over \$2 trillion, the federal government represents just under 20% of the U.S. economy.
• Federal employees include military personnel and the executive branch. State and local government employment is much larger than federal employment and has tripled in size since 1962.
• Transfers to individuals represent almost 50% of federal expenditures. Such transfers include social security, Medicare, Medicaid, unemployment insurance, and veteran’s benefits. Transfers are also known as entitlements, and other expenditures are called discretionary spending.
• The social security program has paid out approximately what it took in and is not an investment program.
• The federal government runs deficits, spending more than it earned. Starting in the late 1970s, the United States began accumulating debt faster than it was growing, and the debt began to rise. That trend wasn’t stabilized until the 1990s, and then only because the economy grew at an extraordinary rate by historical standards. The expenditures following the September 11, 2001, terrorist attacks, combined with a recession in the economy, have sent the debt rising dramatically.
• The best way to evaluate the debt is as a percentage of GDP.
• An important distinction in many economic activities is one between a stock and a flow. Your bank account represents a stock of money; expenditures and income represent a flow. The national debt is a stock; the deficit is the addition to the debt and is a flow.
• Government funded research and development represents about 1% of GDP, divided about equally between military and civilian research.
• The federal income tax currently produces just under 50% of federal revenue. Social security and Medicare taxes produce the next largest portion, about one third of revenue. The rest comes from corporate profits’ taxes (about 10%) and excise taxes like those imposed on liquor and cigarettes (under 5%). | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/04%3A_The_U.S._Economy/4.05%3A_Government.txt |
Learning Objectives
• What do we trade with other nations?
The United States is a major trading nation. Figure 4.50 represents total U.S. imports and exports, including foreign investments and earnings (e.g., earnings from U.S.-owned foreign assets). As is clear from this figure, the net trade surplus ended in the 1970s, and the United States now runs substantial trade deficits, around 4% of the GDP. In addition, trade is increasingly important in the economy.
Figure 4.50 Total imports and exports as a proportion of GDP
Figure 4.51 U.S. trade in goods and services
Figure 4.52 Income and payments (% GDP)
Table 4.2 Top U.S. trading partners and trade volumes (\$ billions)
Rank Country Exports Year-to-Date Imports Year-to-Date Total Percent
All Countries 533.6 946.6 1,480.2 100.0%
Top 15 Countries 400.7 715.4 1,116.2 75.4%
1 Canada 123.1 167.8 290.9 19.7%
2 Mexico 71.8 101.3 173.1 11.7%
3 China 22.7 121.5 144.2 9.7%
4 Japan 36.0 85.1 121.0 8.2%
5 Germany 20.4 50.3 70.8 4.8%
6 United Kingdom 23.9 30.3 54.2 3.7%
7 Korea, South 17.5 29.6 47.1 3.2%
8 Taiwan 14.0 22.6 36.5 2.5%
9 France 13.4 20.0 33.4 2.3%
10 Italy 6.9 18.6 25.5 1.7%
11 Malaysia 7.2 18.0 25.2 1.7%
12 Ireland 5.2 19.3 24.5 1.7%
13 Singapore 13.6 10.1 23.7 1.6%
14 Netherlands 15.7 7.9 23.6 1.6%
15 Brazil 9.3 13.2 22.5 1.5%
Who does the United States trade with? Table 4.2 details the top 15 trading partners and the share of trade. The United States and Canada remain the top trading countries of all pairs of countries. Trade with Mexico has grown substantially since the enactment of the 1994 North American Free Trade Act (NAFTA), which extended the earlier U.S.–Canada agreement to include Mexico, and Mexico is the second largest trading partner of the United States. Together, the top 15 account for three quarters of U.S. foreign trade.
Key Takeaways
• The United States is a major trading nation, buying about 16% of GDP and selling about 12%, with a 4% trade deficit. Income from investments abroad is roughly balanced with foreign earnings from U.S. investments; these are known as the capital accounts.
• The United States and Canada remain the top trading countries of all pairs of countries. Mexico is the second largest trading partner of the United States. China and Japan are third and fourth. Together, the top 15 account for three quarters of U.S. international trade.
4.07: Fluctuations
Learning Objectives
• What is a recession?
The U.S. economy has recessions, a term that refers to a period marked by a drop in gross domestic output. Recessions are officially called by the National Bureau of Economic Research, which keeps statistics on the economy and engages in various kinds of economic research. Generally, a recession is called whenever output drops for one-half of a year.
Figure 4.53 Postwar industrial production and recessions
These fluctuations in output are known as the business cycle, which is not an exact periodic cycle but instead a random cycle.
Figure 4.54 Percentage of the population employed
An important aspect of the business cycle is that many economic variables move together, or covary. Some economic variables vary less with the business cycle than others. Investment varies very strongly with the business cycle, while overall employment varies weakly. Interest rates, inflation, stock prices, unemployment, and many other variables also vary systematically over the business cycle. Recessions are clearly visible in the percentage of the population employed, as illustrated in Figure 4.54
Some economic variables are much more variable than others. For example, investment, durable goods purchases, and utilization of production capacity vary more dramatically over the business cycle than consumption and employment. Figure 4.55 shows the percentage of industrial capacity utilized to produce manufactured goods. This series is more volatile than production itself and responds more strongly to economic conditions.
Figure 4.55 Industrial factory capacity utilization
Source: FRED.
Most of the field of macroeconomics is devoted to understanding the determinants of growth and of fluctuations, but further consideration of this important topic is beyond the scope of a microeconomics text.
Key Takeaways
• The U.S. economy has recessions, a term that refers to a drop in gross domestic output. Recessions are officially called by the National Bureau of Economic Research, which keeps statistics on the economy and engages in various kinds of economic research. Generally a recession is called whenever output drops for one half of a year.
• Prior to World War II, a normal boom lasted 2½ years and the longest boom was 4 years, but they have been much longer since 1960. Recessions have historically lasted for 1½ to 2 years, a pattern that continues.
• Fluctuations in output are known as the business cycle, which is not an exact periodic cycle but instead a random cycle.
• An important aspect of the business cycle is that many economic variables move together, or covary. Investment varies very strongly with the business cycle, while overall employment varies weakly. Interest rates, inflation, stock prices, unemployment, and many other variables also vary systematically over the business cycle.
• Some economic variables are much more variable than others. Investment, durable goods purchases, and utilization of production capacity vary more dramatically over the business cycle than consumption and employment. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/04%3A_The_U.S._Economy/4.06%3A_Trade.txt |
We have so far focused on unimpeded markets, and we saw that markets may perform efficiently.The standard term for an unimpeded market is a free market, which is free in the sense of “free of external rules and constraints.” In this terminology, eBay is a free market, even though it charges for the use of the market. In this and subsequent chapters, we examine impediments to the efficiency of markets. Some of these impediments are imposed on otherwise efficiently functioning markets, as occurs with taxes. Others, such as monopoly or pollution, impede efficiency in some circumstances, and government may be used to mitigate the problems that arise.
This chapter analyzes taxes. There are a variety of types of taxes, such as income taxes, property taxes, ad valorem (percentage of value) taxes, and excise taxes (taxes on a specific good like cigarettes or gasoline). Here, we are primarily concerned with sales taxes, which are taxes on goods and services sold at retail. Our insights into sales taxes translate naturally into some other taxes.
Learning Objectives
• How do taxes affect equilibrium prices and the gains from trade?
Consider first a fixed, per-unit tax such as a 20-cent tax on gasoline. The tax could either be imposed on the buyer or the supplier. It is imposed on the buyer if the buyer pays a price for the good and then also pays the tax on top of that. Similarly, if the tax is imposed on the seller, the price charged to the buyer includes the tax. In the United States, sales taxes are generally imposed on the buyer—the stated price does not include the tax—while in Canada, the sales tax is generally imposed on the seller.
An important insight of supply and demand theory is that it doesn’t matter—to anyone—whether the tax is imposed on the supplier or the buyer. The reason is that ultimately the buyer cares only about the total price paid, which is the amount the supplier gets plus the tax; and the supplier cares only about the net to the supplier, which is the total amount the buyer pays minus the tax. Thus, with a a 20-cent tax, a price of \$2.00 to the buyer is a price of \$1.80 to the seller. Whether the buyer pays \$1.80 to the seller and an additional 20 cents in tax, or pays \$2.00, produces the same outcome to both the buyer and the seller. Similarly, from the seller’s perspective, whether the seller charges \$2.00 and then pays 20 cents to the government, or charges \$1.80 and pays no tax, leads to the same profit.There are two minor issues here that won’t be considered further. First, the party who collects the tax has a legal responsibility, and it could be that businesses have an easier time complying with taxes than individual consumers. The transaction costs associated with collecting taxes could create a difference arising from who pays the tax. Such differences will be ignored in this book. Second, if the tax is percentage tax, it won’t matter to the outcome; but the calculations are more complicated because a 10% tax on the seller at a seller’s price of \$1.80 is different from a 10% tax on a buyer’s price of \$2.00. Then the equivalence between taxes imposed on the seller and taxes imposed on the buyer requires different percentages that produce the same effective tax level. In addition, there is a political issue: Imposing the tax on buyers makes the presence and size of taxes more transparent to voters.
First, consider a tax imposed on the seller. At a given price p, and tax t, each seller obtains pt, and thus supplies the amount associated with this net price. Taking the before-tax supply to be SBefore, the after-tax supply is shifted up by the amount of the tax. This is the amount that covers the marginal value of the last unit, plus providing for the tax. Another way of saying this is that, at any lower price, the sellers would reduce the number of units offered. The change in supply is illustrated in Figure 5.1.
Figure 5.1 Effect of a tax on supply
Figure 5.2 Effect of a tax on demand
Figure 5.3 Effect of a tax on equilibrium
Also noteworthy in this figure is that the price the buyer pays rises, but generally by less than the tax. Similarly, the price that the seller obtains falls, but by less than the tax. These changes are known as the incidence of the tax—a tax mostly borne by buyers, in the form of higher prices, or by sellers, in the form of lower prices net of taxation.
There are two main effects of a tax: a fall in the quantity traded and a diversion of revenue to the government. These are illustrated in Figure 5.4. First, the revenue is just the amount of the tax times the quantity traded, which is the area of the shaded rectangle. The tax raised, of course, uses the after-tax quantity qA* because this is the quantity traded once the tax is imposed.
Figure 5.4 Revenue and deadweight loss
In addition, a tax reduces the quantity traded, thereby reducing some of the gains from trade. Consumer surplus falls because the price to the buyer rises, and producer surplus (profit) falls because the price to the seller falls. Some of those losses are captured in the form of the tax, but there is a loss captured by no party—the value of the units that would have been exchanged were there no tax. The value of those units is given by the demand, and the marginal cost of the units is given by the supply. The difference, shaded in black in the figure, is the lost gains from trade of units that aren’t traded because of the tax. These lost gains from trade are known as a deadweight loss. That is, the deadweight loss is the buyer’s values minus the seller’s costs of units that are not economic to trade only because of a tax or other interference in the market. The net lost gains from trade (measured in dollars) of these lost units are illustrated by the black triangular region in the figure.
The deadweight loss is important because it represents a loss to society much the same as if resources were simply thrown away or lost. The deadweight loss is value that people don’t enjoy, and in this sense can be viewed as an opportunity cost of taxation; that is, to collect taxes, we have to take money away from people, but obtaining a dollar in tax revenue actually costs society more than a dollar. The costs of raising tax revenues include the money raised (which the taxpayers lose), the direct costs of collection, like tax collectors and government agencies to administer tax collection, and the deadweight loss—the lost value created by the incentive effects of taxes, which reduce the gains for trade. The deadweight loss is part of the overhead of collecting taxes. An interesting issue, to be considered in the subsequent section, is the selection of activities and goods to tax in order to minimize the deadweight loss of taxation.
Without more quantification, only a little more can be said about the effect of taxation. First, a small tax raises revenue approximately equal to the tax level times the quantity, or tq. Second, the drop in quantity is also approximately proportional to the size of the tax. Third, this means the size of the deadweight loss is approximately proportional to the tax squared. Thus, small taxes have an almost zero deadweight loss per dollar of revenue raised, and the overhead of taxation, as a percentage of the taxes raised, grows when the tax level is increased. Consequently, the cost of taxation tends to rise in the tax level.
Key Takeaways
• Imposing a tax on the supplier or the buyer has the same effect on prices and quantity.
• The effect of the tax on the supply-demand equilibrium is to shift the quantity toward a point where the before-tax demand minus the before-tax supply is the amount of the tax.
• A tax increases the price a buyer pays by less than the tax. Similarly, the price the seller obtains falls, but by less than the tax. The relative effect on buyers and sellers is known as the incidence of the tax.
• There are two main economic effects of a tax: a fall in the quantity traded and a diversion of revenue to the government.
• A tax causes consumer surplus and producer surplus (profit) to fall.. Some of those losses are captured in the tax, but there is a loss captured by no party—the value of the units that would have been exchanged were there no tax. These lost gains from trade are known as a deadweight loss.
• The deadweight loss is the buyer’s values minus the seller’s costs of units that are not economic to trade only because of a tax (or other interference in the market efficiency).
• The deadweight loss is important because it represents a loss to society much the same as if resources were simply thrown away or lost.
• Small taxes have an almost zero deadweight loss per dollar of revenue raised, and the overhead of taxation, as a percentage of the taxes raised, grows when the tax level is increased.
EXERCISES
1. Suppose demand is given by qd (p) = 1 – p and supply qs (p) = p, with prices in dollars. If sellers pay a 10-cent tax, what is the after-tax supply? Compute the before-tax equilibrium price and quantity, the after-tax equilibrium quantity, and buyer’s price and seller’s price.
2. Suppose demand is given by qd (p) = 1 – p and supply qs (p) = p, with prices in dollars. If buyers pay a 10-cent tax, what is the after-tax demand? Do the same computations as the previous exercise, and show that the outcomes are the same.
3. Suppose demand is given by qd (p) = 1 – p and supply qs (p) = p, with prices in dollars. Suppose a tax of t cents is imposed, t ≤1. What is the equilibrium quantity traded as a function of t? What is the revenue raised by the government, and for what level of taxation is it highest? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/05%3A_Government_Interventions/5.01%3A_Effects_of_Taxes.txt |
Learning Objectives
• Who bears the largest burden of a tax-buyers or sellers?
How much does the quantity fall when a tax is imposed? How much does the buyer’s price rise and the price to the seller fall? The elasticities of supply and demand can be used to answer this question. To do so, we consider a percentage tax t and employ the methodology introduced in Chapter 2, assuming constant elasticity of both demand and supply. Let the equilibrium price to the seller be ps and the equilibrium price to the buyer be pb. As before, we will denote the demand function by qd(p) = ap and supply function by qs(p) = bpη. These prices are distinct because of the tax, and the tax determines the difference:
$p_{b}=(1+t) p_{s}$
Equilibrium requires
$\text { a } p d-\varepsilon=q d(p b)=q s(p s)=b p s \eta$
Thus,
$a ( (1+t) p s ) −ε =a p d −ε = q d ( p b )= q s ( p s )=b p s η .$
This solves for
$p d=(1+t) p s=(a b) 1 n+\varepsilon(1+t) \eta \eta+\varepsilon$
Recall the approximation $$(1+t) r \approx 1+r t$$.
Thus, a small proportional tax increases the price to the buyer by approximately η t ε+η and decreases the price to the seller by ε t ε+η . The quantity falls by approximately η ε t ε+η . Thus, the price effect is mostly on the “relatively inelastic party.” If demand is inelastic, ε is small; then the price decrease to the seller will be small and the price increase to the buyer will be close to the entire tax. Similarly, if demand is very elastic, ε is very large, and the price increase to the buyer will be small and the price decrease to the seller will be close to the entire tax.
We can rewrite the quantity change as $$\eta \varepsilon t \varepsilon+\eta=t 1 \varepsilon+1 \eta$$. Thus, the effect of a tax on quantity is small if either the demand or the supply is inelastic. To minimize the distortion in quantity, it is useful to impose taxes on goods that either have inelastic demand or inelastic supply.
For example, cigarettes are a product with very inelastic demand and moderately elastic supply. Thus, a tax increase will generally increase the price by almost the entire amount of the tax. In contrast, travel tends to have relatively elastic demand, so taxes on travel—airport, hotel, and rental car taxes—tend not to increase the final prices so much but have large quantity distortions.
Key Takeaways
• A small proportional tax t increases the price to the buyer by approximately $$\eta t \varepsilon+\eta$$ and decreases the price to the seller by $$\varepsilon t \varepsilon+\eta$$. The quantity falls by approximately $\eta $\varepsilon t \varepsilon+\eta$$.
• The price effect is mostly on the “relatively inelastic party.”
• The effect of a tax on quantity is small if either the demand or the supply is inelastic. To minimize the distortion in quantity, it is useful to impose taxes on goods that either have inelastic demand or inelastic supply.
EXERCISE
1. For the case of constant elasticity (of both supply and demand), what tax rate maximizes the government’s revenue? How does the revenue-maximizing tax rate change when demand becomes more inelastic? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/05%3A_Government_Interventions/5.02%3A_Incidence_of_Taxes.txt |
Learning Objectives
• How does a tax affect the gains from trade?
The presence of the deadweight loss implies that raising $1 in taxes costs society more than$1. But how much more? This idea—that the cost of taxation exceeds the taxes raised—is known as the excess burden of taxation, or just the excess burden. We can quantify the excess burden with a remarkably sharp formula.
To start, we will denote the marginal cost of the quantity q by c(q) and the marginal value by v(q). The elasticities of demand and supply are given by the standard formulae
$\varepsilon=-\text { dq } q \text { dv } v=-v(q) q v^{\prime}(q)$
and
$\eta=\operatorname{dq} q \operatorname{dc} c=c(q) q c^{\prime}(q)$
Consider an ad valorem (at value) tax that will be denoted by t, meaning a tax on the value, as opposed to a tax on the quantity. If sellers are charging c(q), the ad valorem tax is tc(q), and the quantity q* will satisfy $$v\left(q^{*}\right)=(1+t) c\left(q^{*}\right)$$.
From this equation, we immediately deduce
$d q^{*} d t=c\left(q^{*}\right) v^{\prime}\left(q^{*}\right)-(1+t) c^{\prime}\left(q^{*}\right)=c\left(q^{*}\right)-v\left(q^{*}\right) \varepsilon q^{*}-(1+t)\left(q^{*}\right) \eta q^{*}=-q^{*}(1+t)(1 \varepsilon+1 \eta)=-q^{*} \varepsilon_{\mathrm{n}}(1+t)(\varepsilon+\eta)$
Tax revenue is given by Tax = tc(q*)q*.
The effect on taxes collected, Tax, of an increase in the tax rate t is
$dTax dt =c(q*)q*+t( c(q*)+q* c ′ (q*) ) dq* dt =c(q*)( q*−t( 1+ 1 η ) q*εη (1+t)( ε+η ) )= c(q*)q* (1+t)( ε+η ) ( (1+t)( ε+η )−t( 1+η )ε )= c(q*)q* (1+t)( ε+η ) ( ε+η−tη(ε−1) ).$
Thus, tax revenue is maximized when the tax rate is tmax, given by
$t_{\max }=\varepsilon+\eta \eta(\varepsilon-1)=\varepsilon \varepsilon-1(1 \eta+1 \varepsilon)$
The value ε ε−1 is the monopoly markup rate, which we will meet when we discuss monopoly. Here it is applied to the sum of the inverse elasticities.
The gains from trade (including the tax) is the difference between value and cost for the traded units, and thus is
$G F T=\int 0 g^{*} v(q)-c(q) d q$
Thus, the change in the gains from trade as taxes increase is given by
$dGFT dTax = ∂GFT ∂t ∂Tax ∂t = ( v(q*)−c(q*) ) dq* dt c(q*)q* (1+t)( ε+η ) ( ε+η−tη(ε−1) ) =− ( v(q*)−c(q*) ) q*εη (1+t)( ε+η ) c(q*)q* (1+t)( ε+η ) ( ε+η−tη(ε−1) )=− ( tc(q*) )εη c(q*)( ε+η−tη(ε−1) ) =− εηt ε+η−tη(ε−1) =− ε ε−1 t t max −t .$
The value tmax is the value of the tax rate t that maximizes the total tax taken. This remarkable formula permits the quantification of the cost of taxation. The minus sign indicates that it is a loss—the deadweight loss of monopoly, as taxes are raised, and it is composed of two components. First, there is the term ε ε−1 , which arises from the change in revenue as quantity is changed, thus measuring the responsiveness of revenue to a quantity change. The second term provides for the change in the size of the welfare loss triangle. The formula can readily be applied in practice to assess the social cost of taxation, knowing only the tax rate and the elasticities of supply and demand.
The formula for the excess burden is a local formula—it calculates the increase in the deadweight loss associated with raising an extra dollar of tax revenue. All elasticities, including those in tmax, are evaluated locally around the quantity associated with the current level of taxation. The calculated value of tmax is value given the local elasticities; if elasticities are not constant, this value will not necessarily be the actual value that maximizes the tax revenue. One can think of tmax as the projected value. It is sometimes more useful to express the formula directly in terms of elasticities rather than in terms of the projected value of tmax, in order to avoid the potential confusion between the projected (at current elasticities) and actual (at the elasticities relevant to tmax) value of tmax. This level can be read directly from the derivation shown below:
$dGFT dTax =− εηt ε+η−η(ε−1)t$
Key Takeaways
• The cost of taxation that exceeds the taxes raised is known as the excess burden of taxation, or just the excess burden.
• Tax revenue is maximized when the tax rate is $t max = ε ε−1 ( 1 η + 1 ε ).$
• The change in the gains from trade as taxes increase is given by $\mathrm{dGFT} \text { dTax }=-\varepsilon \varepsilon-1 \text { t } \mathrm{t} \text { max }-\mathrm{t}$ .
EXERCISES
1. Suppose both demand and supply are linear, qD= (a – b) p and qS = (c + d) p. A quantity tax is a tax that has a constant value for every unit bought or sold. Determine the new equilibrium supply price pS and demand price pD when a quantity tax of amount t is applied.
2. An ad valorem tax is a proportional tax on value, like a sales tax. Repeat the previous exercise for an ad valorem tax t.
3. Let supply be given by p = q and demand by p = 1 – q. Suppose that a per-unit tax of 0.10 is applied.
1. What is the change in quantity traded?
2. Compute the tax revenue and deadweight loss. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/05%3A_Government_Interventions/5.03%3A_Excess_Burden_of_Taxation.txt |
Learning Objectives
• What happens when the government, not a market, sets the price?
A price floor is a minimum price at which a product or service is permitted to sell. Many agricultural goods have price floors imposed by the government. For example, tobacco sold in the United States has historically been subject to a quota and a price floor set by the Secretary of Agriculture. Unions may impose price floors as well. For example, the Screen Actors Guild (SAG) imposes minimum rates for guild members, generally pushing up the price paid for actors above what would prevail in an unconstrained market. (The wages of big-name stars aren’t generally affected by SAG because these are individually negotiated.) The most important example of a price floor is the minimum wage, which imposes a minimum amount that a worker can be paid per hour.
A price ceiling is a maximum price that can be charged for a product or service. Rent control imposes a maximum price on apartments (usually set at the historical price plus an adjustment for inflation) in many U.S. cities. Taxi fares in New York, Washington, DC, and other cities are subject to maximum legal fares. During World War II, and again in the 1970s, the United States imposed price controls to limit inflation, imposing a maximum price for the legal sale of many goods and services. For a long time, most U.S. states limited the legal interest rate that could be charged (these are called usury laws), and this is the reason why so many credit card companies are located in South Dakota. South Dakota was the first state to eliminate such laws. In addition, ticket prices for concerts and sporting events are often set below the equilibrium price. Laws prohibiting scalping then impose a price ceiling. Laws preventing scalping are usually remarkably ineffective in practice, of course.
The theory of price floors and ceilings is readily articulated with simple supply and demand analysis. Consider a price floor—a minimum legal price. If the price floor is low enough—below the equilibrium price—there are no effects because the same forces that tend to induce a price equal to the equilibrium price continue to operate. If the price floor is higher than the equilibrium price, there will be a surplus because, at the price floor, more units are supplied than are demanded. This surplus is illustrated in Figure 5.5.
In Figure 5.5, the price floor is illustrated with a horizontal line and is above the equilibrium price. Consequently, at the price floor, a larger quantity is supplied than is demanded, leading to a surplus. There are units that are socially efficient to trade but aren’t traded—because their value is less than the price floor. The gains from trade associated with these units, which is lost due to the price floor, represent the deadweight loss.
The price increase created by a price floor will increase the total amount paid by buyers when the demand is inelastic, and otherwise will reduce the amount paid. Thus, if the price floor is imposed in order to be of benefit to sellers, we would not expect to see the price increased to the point where demand becomes elastic, for otherwise the sellers receive less revenue. Thus, for example, if the minimum wage is imposed in order to increase the average wages to low-skilled workers, then we would expect to see the total income of low-skilled workers rise. If, on the other hand, the motivation for the minimum wage is primarily to make low-skilled workers a less effective substitute for union workers, and hence allow union workers to increase their wage demands, then we might observe a minimum wage that is in some sense “too high” to be of benefit to low-skilled workers.
Figure 5.5 A price floor
However, this is the minimum loss to society associated with a price floor. Generally there will be other losses. In particular, the loss given above assumes that suppliers who don’t sell, don’t produce. As a practical matter, some suppliers who won’t sell in the end may still produce because they hope to sell. In this case, additional costs are incurred and the deadweight loss will be larger to reflect these costs.
Example: Suppose both supply and demand are linear, with the quantity supplied equal to the price and the quantity demanded equal to one minus the price. In this case, the equilibrium price and the equilibrium quantity are both ½. A price floor of p > ½ induces a quantity demanded of 1 – p. How many units will suppliers offer, if a supplier’s chance of trading is random? Suppose $$q \geq 1-p$$ units are offered. A supplier’s chance of selling is 1−p q . Thus, the marginal supplier (who has a marginal cost of q by assumption) has a probability 1−p q of earning p, and a certainty of paying q. Exactly q units will be supplied when this is a break-even proposition for the marginal supplier—that is, $$1-p q p-q=0, \text { or } q=p(1-p)$$.
The deadweight loss then includes not just the triangle illustrated in the previous figure, but also the cost of the $$p(1-p)-(1-p)$$unsold units.
The SAG, a union of actors, has some ability to impose minimum prices (a price floor) for work on regular Hollywood movies. If the SAG would like to maximize the total earnings of actors, what price should they set in the linear demand and supply example?
The effects of a price floor include lost gains from trade because too few units are traded (inefficient exchange), units produced that are never consumed (wasted production), and more costly units produced than necessary (inefficient production).
A price ceiling is a maximum price. Analogous to a low price floor, a price ceiling that is larger than the equilibrium price has no effect. Tell me that I can’t charge more than a billion dollars for this book (which is being given away for free), and it won’t affect the price charged or the quantity traded. Thus, the important case of a price ceiling is one that is less than the equilibrium price.
In this case, which should now look familiar, the price is forced below the equilibrium price and too few units are supplied, while a larger number are demanded, leading to a shortage. The deadweight loss is illustrated in Figure 5.7, and again represents the loss associated with units that are valued at more than they cost but aren’t produced.
Figure 5.7 A price ceiling
Analogous to the case of a price floor, there can be additional losses associated with a price ceiling. In particular, some lower-value buyers may succeed in purchasing, denying the higher-value buyers the ability to purchase. This effect results in buyers with high values failing to consume, and hence their value is lost.
In addition to the misallocation of resources (too few units and units not allocated to those who value them the most), price ceilings tend to encourage illegal trade as people attempt to exploit the prohibited gains from trade. For example, it became common practice in New York to attempt to bribe landlords to offer rent-controlled apartments, and such bribes could exceed \$50,000. In addition, potential tenants expended a great deal of time searching for apartments, and a common strategy was to read the obituaries late at night when the New York Times had just come out, hoping to find an apartment that would be vacant and available for rent.
An important and undesirable by-product of price ceilings is discrimination. In a free or unconstrained market, discrimination against a particular group, based on race, religion, or other factors, requires transacting not based on price but on another factor. Thus, in a free market, discrimination is costly—discrimination entails, for instance, not renting an apartment to the highest bidder but to the highest bidder of the favored group. In contrast, with a price ceiling, there is a shortage; and sellers can discriminate at lower cost, or even at no cost. That is, if there are twice as many people seeking apartments as there are apartments available at the price ceiling, landlords can “pick and choose” among prospective tenants and still get the maximum legal rent. Thus, a price ceiling has the undesirable by-product of reducing the cost of discrimination.
Key Takeaways
• A price floor is a minimum price at which a product or service is permitted to sell. Many agricultural goods have price floors imposed by the government. The most important example of a price floor is the minimum wage.
• A price ceiling is a maximum price that can be charged for a product or service. Rent control imposes a maximum price on apartments in many U.S. cities. A price ceiling that is larger than the equilibrium price has no effect.
• If a price floor is low enough—below the equilibrium price—there are no effects. If the price floor is higher than the equilibrium price, there will be a surplus.
• The deadweight loss of a price floor is the difference between the value of the units not traded—and value is given by the demand curve—and the cost of producing these units. This is the minimum loss to society associated with a price floor.
• The effects of a price floor include lost gains from trade because too few units are traded (inefficient exchange), units produced that are never consumed (wasted production), and more costly units produced than necessary (inefficient production).
• When a price ceiling is below the equilibrium price, the price is forced below the equilibrium price and a shortage results.
• In addition to underproduction, a price ceiling may also lead to inefficient allocation. Price ceilings tend to encourage illegal trade and discrimination.
EXERCISES
1. In Example, show that the quantity produced is less than the equilibrium quantity, which is ½. Compute the gains from trade, given the overproduction of suppliers. What is the deadweight loss of the price floor?
2. Suppose that units aren’t produced until after a buyer has agreed to purchase, as typically occurs with services. What is the deadweight loss in this case? (Hint: What potential sellers will offer their services? What is the average cost of supply of this set of potential sellers?)
3. Adapt the price floor example above to the case of a price ceiling, with p < ½, and compute the lost gains from trade if buyers willing to purchase are all able to purchase with probability qS/qD. (Hint: Compute the value of qD units; the value realized by buyers collectively will be that amount times the probability of trade.) | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/05%3A_Government_Interventions/5.04%3A_Price_Floors_and_Ceilings.txt |
Learning Objectives
• Why does the government meddle in markets?
Both demand and supply tend to be more elastic in the long run. This means that the quantity effects of price floors and ceilings tend to be larger over time. An extreme example of this is rent control, a maximum price imposed on apartments.
Rent control is usually imposed in the following way: As a prohibition or limitation on price increases. For example, New York City’s rent control, imposed during World War II, prevented landlords from increasing rent, even when their own costs increased, such as when property taxes increased. This law was softened in 1969 to be gradually replaced by a rent-stabilization law that permitted modest rent increases for existing tenants.
Figure 5.8 Rent control, initial effect
We start with a rent-control law that has little or no immediate effect because it is set at current rents. Thus, in the near term, tenants’ fears of price increases are eased and there is little change in the apartment rental market. This is not to say that there is zero effect—some companies considering construction of an apartment building on the basis of an expectation of higher future rents may be deterred, and a few marginal apartments may be converted to other uses because the upside potential for the owner has been removed, but such effects are modest at best.
Figure 5.9 Rent control, long-run effect
The shortage is created by two separate factors—demand is increasing as incomes and population rise, and supply is decreasing as costs rise. This reduces the quantity of available housing units supplied and increases the demand for those units.
How serious is the threat that units will be withdrawn from the market? In New York City, over 200,000 apartment units were abandoned by their owners, usually because the legal rent didn’t cover the property taxes and legally mandated maintenance. In some cases, tenants continued to inhabit the buildings even after the electricity and water were shut off. It is fair to say that rent control devastated large areas of New York City, such as the Bronx. So why would New York City, and so many other communities, impose rent control on itself?
The politics of rent control are straightforward. First, rent control involves a money transfer from landlords to tenants, because tenants pay less than they would absent the law, and landlords obtain less revenue. In the short run, due to the inelastic short-run supply, the effect on the quantity of apartments is small, so rent control is primarily just a transfer from landlords to tenants.
In a city like New York, the majority of people rent. A tiny fraction of New Yorkers are landlords. Thus, it is easy to attract voters to support candidates who favor rent control—most renters will benefit, while landlords don’t. The numbers, of course, don’t tell the whole story because, while landlords are small in number, they are wealthier on average, and thus likely have political influence beyond the number of votes they cast. However, even with their larger economic influence, the political balance favors renters. In the 100ab zip codes of Manhattan (the first three digits are 100), 80% of families were renters in the year 2000. Thus, a candidate who runs on a rent-control platform appeals to a large portion of the voters.
Part of the attraction of rent control is that there is little economic harm in the short run, and most of that harm falls on new residents of New York City. As new residents generally haven’t yet voted in New York, potential harm to them has only a small effect on most existing New Yorkers, and thus isn’t a major impediment to getting voter support for rent control. The slow rate of harm to the city is important politically because the election cycle encourages a short time horizon—if successful at lower office, a politician hopes to move on to higher office and is unlikely to be blamed for the long-run damage to New York City by rent control.
Rent control is an example of a political situation sometimes called the tyranny of the majority, where a majority of the people have an incentive to confiscate the wealth of a minority. But there is another kind of political situation that is in some sense the reverse, where a small number of people care a great deal about something, and the majority are only slightly harmed on an individual basis. No political situation appears more extreme in this regard than that of refined sugar. There are few U.S. cane sugar producers (nine in 1997), yet the U.S. imposes quotas that raise domestic prices much higher than world prices, in some years tripling the price that Americans pay for refined sugar. The domestic sugar producers benefit, while consumers are harmed. But consumers are harmed by only a small amount each—perhaps 12 to 15 cents per pound—which is not enough to build a consensus to defeat politicians who accept donations from sugar producers. This is a case where concentrated benefits and diffused costs determine the political outcome. A small number of people with strong incentives are able to expropriate a small amount per person from a large number of people. Because there aren’t many sugar producers, it is straightforward for them to act as a single force. In contrast, it is pretty hard for consumers to become passionate about 12 cents per pound increase in the domestic sugar price when they consume about 60 pounds per year of sugar.
Key Takeaways
• Both demand and supply tend to be more elastic in the long run.
• Rent control is usually imposed as a prohibition or limitation on price increases. The nature of rent control is that it begins with, at most, minor effects because it doesn’t bind until the equilibrium rent increases. Thus, the cost of rent control tends to be in the future, and ill effects worsen over time.
• A candidate who runs on a rent-control platform appeals to a large portion of the voters as there are more renters than landlords.
• Rent control is an example of a political situation sometimes called the tyranny of the majority, where a majority of people have an incentive to confiscate the wealth of a minority.
• Concentrated benefits and diffused costs are the opposite of tyranny of the majority. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/05%3A_Government_Interventions/5.05%3A_The_Politics_of_Price_Controls.txt |
Learning Objectives
• How is a price support different from a price floor?
A price support is a combination of two programs—a minimum price, or price floor, and government purchase of any surplus. Thus, a price support is different from a price floor because, with a price floor, any excess production by sellers is a burden on the sellers. In contrast, with a price support, any excess production is a burden on the government.
The U.S. Department of Agriculture operates a price support for cheese and has possessed warehouses full of cheese in the past. There are also price supports for milk and other agricultural products.
Figure 5.10 Price supports
There are two kinds of deadweight loss in a price-support program. First, consumers who would like to buy at the equilibrium price are deterred by the higher prices, resulting in the usual deadweight loss, illustrated by the lighter shading. In addition, however, there are goods produced that are then either destroyed or put in warehouses and not consumed, which means the costs of production of those goods is also lost, resulting in a second deadweight loss. That loss is the cost of production, which is given by the supply curve, and is the area under the supply curve for the government purchases. It is shaded in a horizontal fashion. The total deadweight loss of the price support is the sum of these two individual losses. Unlike the case of a price floor or ceiling, a price support creates no ambiguity about what units are produced, or which consumers are willing and able to buy. Thus, the rationing aspect of a price floor or ceiling is not present for a price support, nor is the incentive to create a black market other than one created by selling the warehouse full of product.
Key Takeaways
• A price support is a combination of two programs: a price floor and government purchase of surplus. Excess production is a burden on the government.
• A price support above the equilibrium price leads to a surplus.
• The deadweight loss of price supports involves the usual deadweight loss plus the entire cost of unconsumed goods.
5.07: Quantity Restrictions and Quotas
Learning Objectives
• What is a quota?
The final common way that governments intervene in market transactions is to impose a quota. A quota is a maximal production quantity, usually set based on historical production. In tobacco, peanuts, hops, California oranges, and other products, producers have production quotas based on their historical production. Tobacco quotas were established in the 1930s, and today a tobacco farmer’s quota is a percentage of the 1930s level of production. The Secretary of Agriculture sets the percentage annually. Agricultural products are not the only products with quotas. The right to drive a taxi in New York requires a medallion issued by the city, and there are a limited number of medallions. This is a quota. Is it a restrictive quota? The current price of a New York taxi medallion—the right to drive a taxi legally in New York City—is \$413,000 (as of 2008). This adds approximately \$30,000 to \$40,000 annually to the cost of operating a taxi in New York, using a risk-adjusted interest rate.
What are the effects of a quota? A quota restricts the quantity below what would otherwise prevail, forcing the price up, which is illustrated in Figure 5.11. It works like a combination of a price floor and a prohibition on entry.
Generally, the immediate effects of a quota involve a transfer of money from buyers to sellers. The inefficient production and surplus of the price floor are avoided because a production limitation created the price increase. This transfer has an undesirable and somewhat insidious attribute. Because the right to produce is a capital good, it maintains a value, which must be captured by the producer. For example, an individual who buys a taxi medallion today, and pays \$400,000, makes no economic profits—he captures the forgone interest on the medallion through higher prices but no more than that. The individuals who receive the windfall gain are those who were driving taxis and were grandfathered in to the system and issued free medallions. Those people who were driving taxis 70 years ago—and are mostly dead at this point—received a windfall gain from the establishment of the system. Future generations pay for the program, which provides no net benefits to the current generation. All the benefits were captured by people long since retired.
Figure 5.11 A quota
Does this mean that it is harmless to eliminate the medallion requirement? Unfortunately, not. The current medallion owners who, if they bought recently, paid a lot of money for their medallions would see the value of these investments destroyed. Thus, elimination of the program would harm current medallion owners.
If the right to produce is freely tradable, the producers will remain the efficient producers, and the taxi medallions are an example of this. Taxi medallions can be bought and sold. Moreover, a medallion confers the right to operate a taxi, but doesn’t require that the owner of the medallion actually drive the taxi. Thus, a “medallion owning company” can lease the right to drive a taxi to an efficient driver, thereby eliminating any inefficiency associated with the person who drives the taxi.
In contrast, because tobacco-farming rights aren’t legally tradable across county lines, tobacco is very inefficiently grown. The average size of a burley tobacco farm is less than 5 acres, so some are much smaller. There are tobacco farms in Florida and Missouri, which only exist because of the value of the quota—if they could trade their quota to a farm in North Carolina or Kentucky, which are much better suited to producing cigarette tobacco, it would pay to do so. In this case, the quota, which locked in production rights, also locked in production that gets progressively more inefficient as the years pass.
Quotas based on historical production have the problem that they don’t evolve in ways that production methods and technology do, thus tending to become progressively more inefficient. Tradable quotas eliminate this particular problem but continue to have the problem that future generations are harmed with no benefits.
Key Takeaways
• A quota is a maximum production quantity, usually set based on historical production.
• A quota restricts the quantity below what would otherwise prevail, forcing the price up.
• A quota transfers wealth from buyers to sellers. No surplus arises because of the production limitation. Future generations pay for the program, which provides future sellers no benefits.
• Quotas based on historical production have the problem that they don’t evolve in ways that production methods and technology do, thus tending to become progressively more inefficient. Tradable quotas eliminate this particular problem, but continue to have the problem that future generations are harmed with no benefits.
EXERCISE
1. Suppose demand for a product is qd = 1 – p, and the marginal cost of production is c. A quota at level Q ≤ 1 – c is imposed. What is the value of the quota, per unit of production? Use this to derive the demand for the quota as a function of the level of quota released to the market. If the government wishes to sell the quota, how much should it sell to maximize the revenue on the product? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/05%3A_Government_Interventions/5.06%3A_Price_Supports.txt |
Supply and demand offers one approach to understanding trade, and it represents the most important and powerful concept in the toolbox of economists. However, for some issues, especially those of international trade, another related tool is very useful: the production possibilities frontier. Analysis using the production possibilities frontier was made famous by the “guns and butter” discussions of World War II. From an economic perspective, there is a trade-off between guns and butter—if a society wants more guns, it must give up something, and one thing to give up is butter. While the notion of getting more guns might lead to less butter often seems mysterious, butter is, after all, made with cows, and indirectly with land and hay. But the manufacture of butter also involves steel containers, tractors to turn the soil, transportation equipment, and labor, all of which either can be directly used (steel, labor) or require inputs that could be used (tractors, transportation) to manufacture guns. From a production standpoint, more guns entail less butter (or other things).
Learning Objectives
• What can we produce, and how does that relate to cost?
Formally, the set of production possibilities is the collection of “feasible outputs” of an individual, group or society, or country. You could spend your time cleaning your apartment, or you could study. The more time you devote to studying, the higher your grades will be, but the dirtier your apartment will be. This is illustrated, for a hypothetical student, in Figure 6.1.
The production possibilities set embodies the feasible alternatives. If you spend all your time studying, you could obtain a 4.0 (perfect) grade point average (GPA). Spending an hour cleaning reduces the GPA, but not by much; the second hour reduces it by a bit more, and so on.
The boundary of the production possibilities set is known as the production possibilities frontier. This is the most important part of the production possibilities set because, at any point strictly inside the production possibilities set, it is possible to have more of everything, and usually we would choose to have more.To be clear, we are considering an example with two goods: cleanliness and GPA. Generally there are lots of activities, like sleeping, eating, teeth brushing, and so on; the production possibilities frontier encompasses all of these goods. Spending all your time sleeping, studying, and cleaning would still represent a point on a three-dimensional frontier. The slope of the production possibilities frontier reflects opportunity cost because it describes what must be given up in order to acquire more of a good. Thus, to get a cleaner apartment, more time or capital, or both, must be spent on cleaning, which reduces the amount of other goods and services that can be had. For the two-good case in Figure 6.1, diverting time to cleaning reduces studying, which lowers the GPA. The slope dictates how much lost GPA there is for each unit of cleaning.
Figure 6.1 The production possibilities frontier
Consider two people, Ann and Bob, getting ready for a party. One is cutting up vegetables; the other is making hors d’oeuvres. Ann can cut up 2 ounces of vegetables per minute, or make one hors d’oeuvre in a minute. Bob, somewhat inept with a knife, can cut up 1 ounce of vegetables per minute, or make 2 hors d’oeuvres per minute. Ann and Bob’s production possibilities frontiers are illustrated in Figure 6.2, given that they have an hour to work.
Since Ann can produce 2 ounces of chopped vegetables in a minute, if she spends her entire hour on vegetables, she can produce 120 ounces. Similarly, if she devotes all her time to hors d’oeuvres, she produces 60 of them. The constant translation between the two means that her production possibilities frontier is a straight line, which is illustrated on the left side of Figure 6.2. Bob’s is the reverse—he produces 60 ounces of vegetables or 120 hors d’oeuvres, or something on the line in between.
Figure 6.2 Two production possibilities frontiers
For Ann, the opportunity cost of an ounce of vegetables is half of one hors d’oeuvre—to get one extra ounce of vegetables, she must spend 30 extra seconds on vegetables. Similarly, the cost of one hors d’oeuvre for Ann is 2 ounces of vegetables. Bob’s costs are the inverse of Ann’s—an ounce of vegetables costs him two hors d’oeuvres.
Figure 6.3 Joint PPF
Now change the hypothetical slightly. Suppose that Bob and Ann are putting on separate dinner parties, each of which will feature chopped vegetables and hors d’oeuvres in equal portions. By herself, Ann can only produce 40 ounces of vegetables and 40 hors d’oeuvres if she must produce equal portions. She accomplishes this by spending 20 minutes on vegetables and 40 minutes on hors d’oeuvres. Similarly, Bob can produce 40 of each, but by using the reverse allocation of time.
By working together, they can collectively have more of both goods. Ann specializes in producing vegetables, and Bob specializes in producing hors d’oeuvres. This yields 120 units of each, which they can split equally to have 60 of each. By specializing in the activity in which they have lower cost, Bob and Ann can jointly produce more of each good.
Moreover, Bob and Ann can accomplish this by trading. At a “one for one” price, Bob can produce 120 hors d’oeuvres, and trade 60 of them for 60 ounces of vegetables. This is better than producing the vegetables himself, which netted him only 40 of each. Similarly, Ann produces 120 ounces of vegetables and trades 60 of them for 60 hors d’oeuvres. This trading makes them both better off.
The gains from specialization are potentially enormous. The grandfather of economics, Adam Smith, wrote about specialization in the manufacture of pins:
One man draws out the wire; another straights it; a third cuts it; a fourth points it; a fifth grinds it at the top for receiving the head; to make the head requires two or three distinct operations; to put it on is a peculiar business; to whiten the pins is another; it is even a trade by itself to put them into the paper; and the important business of making a pin is, in this manner, divided into about eighteen distinct operations, which, in some manufactories, are all performed by distinct hands, though in others the same man will sometimes perform two or three of them.Adam Smith, An Inquiry into the Nature and Causes of the Wealth of Nations, originally published in 1776, released by the Gutenberg project, 2002.
Smith goes on to say that skilled individuals could produce at most 20 pins per day acting alone; but that, with specialization, 10 people could produce 48,000 pins per day, 240 times as many pins per capita.
Key Takeaways
• The production possibilities set is the collection of “feasible outputs” of an individual or group.
• The boundary of the production possibilities set is known as the production possibilities frontier.
• The principle of diminishing marginal returns implies that the production possibilities frontier is concave toward the origin, which is equivalent to increasing opportunity cost.
• Efficiencies created by specialization create the potential for gains from trade.
EXERCISES
1. The Manning Company has two factories, one that makes roof trusses and one that makes cabinets. With m workers, the roof factory produces m trusses per day. With n workers, the cabinet plant produces 5 n . The Manning Company has 400 workers to use in the two factories. Graph the production possibilities frontier. (Hint: Let T be the number of trusses produced. How many workers are used to make trusses?)
2. Alarm & Tint, Inc., has 10 workers working a total of 400 hours per week. Tinting takes 2 hours per car. Alarm installation is complicated, however, and performing A alarm installations requires A2 hours of labor. Graph Alarm & Tint’s production possibilities frontier for a week.
3. Consider two consumers and two goods, x and y. Consumer 1 has utility $$u_{1}\left(x_{1}, y_{1}\right)=x_{1}+y_{1}$$ and consumer 2 has utility $$\mathrm{u}_{2}\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right)=\min \left\{\mathrm{x}_{2}, \mathrm{y}_{2}\right\}$$. Consumer 1 has an endowment of (1, 1/2) and consumer 2’s endowment is (0, 1/2).
1. Draw the Edgeworth box for this economy.
2. Find the contract curve and the individually rational part of it. (You should describe these in writing and highlight them in the Edgeworth box.)
3. Find the prices that support an equilibrium of the system and the final allocation of goods under those prices.
For Questions 4 to 7, consider an orange juice factory that uses, as inputs, oranges and workers. If the factory uses x pounds of oranges and y workers per hour, it produces gallons of orange juice; $$T=20 x^{0.25} y^{0.5}$$.
4. Suppose oranges cost $1 and workers cost$10. What relative proportion of oranges and workers should the factory use?
5. Suppose a gallon of orange juice sells for \$1. How many units should be sold, and what is the input mix to be used? What is the profit?
6. Generalize the previous exercise for a price of p dollars per gallon of orange juice.
7. What is the supply elasticity? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/06%3A_Trade/6.01%3A_Production_Possibilities_Frontier.txt |
Learning Objectives
• Who can produce more?
• How does that relate to cost?
• Can a nation be cheaper on all things?
Ann produces chopped vegetables because her opportunity cost of producing vegetables, at half of one hors d’oeuvre, is lower than Bob’s. When one good has a lower opportunity cost over another, it is said to have a comparative advantage. That is, Ann gives up less to produce chopped vegetables than Bob, so in comparison to hors d’oeuvres, she has an advantage in the production of vegetables. Since the cost of one good is the amount of another good forgone, a comparative advantage in one good implies a comparative disadvantage—a higher opportunity cost—in another. If you are better at producing butter, you are necessarily worse at something else—and, in particular, the thing you give up less of to get more butter.
To illustrate this point, let’s consider another party planner. Charlie can produce one hors d’oeuvre or 1 ounce of chopped vegetables per minute. His production is strictly less than Ann’s; that is, his production possibilities frontier lies inside of Ann’s. However, he has a comparative advantage over Ann in the production of hors d’oeuvres because he gives up only 1 ounce of vegetables to produce an hors d’oeuvre, while Ann must give up 2 ounces of vegetables. Thus, Ann and Charlie can still benefit from trade if Bob isn’t around.
When one production possibilities frontier lies outside another, the larger is said to have an absolute advantage—it can produce more of all goods than the smaller. In this case, Ann has an absolute advantage over Charlie—she can, by herself, have more—but not over Bob. Bob has an absolute advantage over Charlie, too; but again, not over Ann.
Diminishing marginal returns implies that the more of a good that a person produces, the higher the cost is (in terms of the good given up). That is to say, diminishing marginal returns means that supply curves slope upward; the marginal cost of producing more is increasing in the amount produced.
Trade permits specialization in activities in which one has a comparative advantage. Moreover, whenever opportunity costs differ, potential gains from trade exist. If Person 1 has an opportunity cost of c1 of producing good x (in terms of y, that is, for each unit of x that Person 1 produces, Person 1 gives up c1 units of y), and Person 2 has an opportunity cost of c2, then there are gains from trade whenever c1 is not equal to c2 and neither party has specialized.If a party specialized in one product, it is a useful convention to say that the marginal cost of that product is now infinite, since no more can be produced. Suppose c1 < c2. Then by having Person 1 increase the production of x by Δ, c1Δ less of the good y is produced. Let Person 2 reduce the production of x by Δ so that the production of x is the same. Then there is c2Δ units of y made available, for a net increase of (c2c1)Δ. The net changes are summarized in Table 6.1.
Table 6.1 Construction of the gains from trade
1 2 Net Change
Change in x –Δ 0
Change in y c1Δ c2Δ (c2c1
Whenever opportunity costs differ, there are gains from reallocating production from one producer to another, gains which are created by having the low-cost producers produce more, in exchange for greater production of the other good by the other producer, who is the low-cost producer of this other good. An important aspect of this reallocation is that it permits production of more of all goods. This means that there is little ambiguity about whether it is a good thing to reallocate production—it just means that we have more of everything we want.If you are worried that more production means more pollution or other bad things, rest assured. Pollution is bad, so we enter the negative of pollution (or environmental cleanliness) as one of the goods we would like to have on hand. The reallocation dictated by differences in marginal costs produces more of all goods. Now with this said, we have no reason to believe that the reallocation will benefit everyone—there may be winners and losers.
How can we guide the reallocation of production to produce more goods and services? It turns out that, under some circumstances, the price system does a superb job of creating efficient production. The price system posits a price for each good or service, and anyone can sell at the common price. The insight is that such a price induces efficient production. To see this, suppose we have a price p, which is the number of units of y that one has to give to get a unit of x. (Usually prices are in currency, but we can think of them as denominated in goods, too.) If I have a cost c of producing x, which is the number of units of y that I lose to obtain a unit of x, I will find it worthwhile to sell x if p > c, because the sale of a unit of x nets me pc units of y, which I can either consume or resell for something else I want. Similarly, if c > p, I would rather buy x (producing y to pay for it). Either way, only producers with costs less than p will produce x, and those with costs greater than p will purchase x, paying for it with y, which they can produce more cheaply than its price. (The price of y is 1/p—that is, the amount of x one must give to get a unit of y.)
Thus, a price system, with appropriate prices, will guide the allocation of production to ensure the low-cost producers are the ones who produce, in the sense that there is no way of reallocating production to obtain more goods and services.
Key Takeaways
• A lower opportunity cost creates a comparative advantage in production.
• A comparative advantage in one good implies a comparative disadvantage in another.
• It is not possible to have a comparative disadvantage in all goods.
• An absolute advantage means the ability to produce more of all goods.
• Diminishing marginal returns implies that the more of a good that a person produces, the higher is the cost (in terms of the good given up). That is to say, diminishing marginal returns means that supply curves slope upward; the marginal cost of producing more is increasing in the amount produced.
• Trade permits specialization in activities in which one has a comparative advantage.
• Whenever opportunity costs differ, potential gains from trade exist.
• Trade permits production of more of all goods.
• A price system, with appropriate prices, will guide the allocation of production to ensure the low-cost producers are the ones who produce, in the sense that there is no way of reallocating production to obtain more goods and services.
EXERCISES
1. Graph the joint production possibilities frontier for Ann and Charlie, and show that collectively they can produce 80 of each if they need the same number of each product. (Hint: First show that Ann will produce some of both goods by showing that, if Ann specializes, there are too many ounces of vegetables. Then show that, if Ann devotes x minutes to hors d’oeuvres, $$60+x=2(60-x)$$.)
2. Using Manning’s production possibilities frontier in Exercise 1 in Section 6.1, compute the marginal cost of trusses in terms of cabinets.
3. Using Alarm & Tint’s production possibilities frontier in Exercise 2 in Section 6.1, compute the marginal cost of alarms in terms of window tints. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/06%3A_Trade/6.02%3A_Comparative_and_Absolute_Advantage.txt |
Learning Objectives
• How does the abundance or rarity of inputs to production affect the advantage of nations?
Production possibilities frontiers provide the basis for a rudimentary theory of international trade. To understand the theory, it is first necessary to consider that there are fixed and mobile factors. Factors of production is jargon for inputs to the production process. Labor is generally considered a fixed factor because most countries don’t have borders that are wide open to immigration, although of course some labor moves across international borders. Temperature, weather, and land are also fixed—Canada is a high-cost citrus grower because of its weather. There are other endowments that could be exported but are expensive to export because of transportation costs, including water and coal. Hydropower—electricity generated from the movement of water—is cheap and abundant in the Pacific Northwest; and, as a result, a lot of aluminum is smelted there because aluminum smelting requires lots of electricity. Electricity can be transported, but only with losses (higher costs), which gives other regions a disadvantage in the smelting of aluminum. Capital is generally considered a mobile factor because plants can be built anywhere, although investment is easier in some environments than in others. For example, reliable electricity and other inputs are necessary for most factories. Moreover, comparative advantage may arise from the presence of a functioning legal system, the enforcement of contracts, and the absence of bribery. This is because enforcement of contracts increase the return on investment by increasing the probability that the economic return to investment isn’t taken by others.
Fixed factors of production are factors that are not readily moved and thus give particular regions a comparative advantage in the production of some kinds of goods and not in others. Europe, the United States, and Japan have a relative abundance of highly skilled labor and have a comparative advantage in goods requiring high skills like computers, automobiles, and electronics. Taiwan, South Korea, Singapore, and Hong Kong have increased the available labor skills and now manufacture more complicated goods like DVDs, computer parts, and the like. Mexico has a relative abundance of middle-level skills, and a large number of assembly plants operate there, as well as clothing and shoe manufacturers. Lower-skilled Chinese workers manufacture the majority of the world’s toys. The skill levels of China are rising rapidly.
The basic model of international trade, called Ricardian theory, was first described by David Ricardo (1772–1823). It suggests that nations, responding to price incentives, will specialize in the production of goods in which they have a comparative advantage, and purchase the goods in which they have a comparative disadvantage. In Ricardo’s description, England has a comparative advantage of manufacturing cloth and Portugal similarly in producing wine, leading to gains from trade from specialization.
The Ricardian theory suggests that the United States, Canada, Australia, and Argentina should export agricultural goods, especially grains that require a large land area for the value generated (they do). It suggests that complex technical goods should be produced in developed nations (they are) and that simpler products and natural resources should be exported by the lesser-developed nations (they are). It also suggests that there should be more trade between developed and underdeveloped nations than between developed and other developed nations. The theory falters on this prediction—the vast majority of trade is between developed nations. There is no consensus for the reasons for this, and politics plays a role—the North American Free Trade Act (NAFTA) vastly increased the volume of trade between the United States and Mexico, for example, suggesting that trade barriers may account for some of the lack of trade between the developed and the underdeveloped world. Trade barriers don’t account for the volume of trade between similar nations, which the theory suggests should be unnecessary. Developed nations sell each other such products as mustard, tires, and cell phones, exchanging distinct varieties of goods they all produce.
Key Takeaways
• The term “factors of production” is jargon for inputs to the production process.
• Labor is generally considered a fixed or immobile factor because most countries don’t have borders that are wide open to immigration. Temperature, weather, and land are also fixed factors.
• Fixed factors of production give particular regions a comparative advantage in the production of some kinds of goods, and not in others.
• The basic model of international trade, known as the Ricardian theory, suggests that nations, responding to price incentives, will specialize in the production of goods in which they have a comparative advantage, and purchase the goods in which they have a comparative disadvantage. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/06%3A_Trade/6.03%3A_Factors_of_Production.txt |
Learning Objectives
• How does trade affect domestic prices for inputs and goods and services?
The Ricardian theory emphasizes that the relative abundance of particular factors of production determines comparative advantage in output, but there is more to the theory. When the United States exports a computer to Mexico, American labor, in the form of a physical product, has been sold abroad. When the United States exports soybeans to Japan, American land (or at least the use of American land for a time) has been exported to Japan. Similarly, when the United States buys car parts from Mexico, Mexican labor has been sold to the United States; and when Americans buy Japanese televisions, Japanese labor has been purchased. The goods that are traded internationally embody the factors of production of the producing nations, and it is useful to think of international trade as directly trading the inputs through the incorporation of inputs into products.
If the set of traded goods is broad enough, factor price equalization predicts that the value of factors of production should be equalized through trade. The United States has a lot of land, relative to Japan; but by selling agricultural goods to Japan, it is as if Japan has more land, by way of access to U.S. land. Similarly, by buying automobiles from Japan, it is as if a portion of the Japanese factories were present in the United States. With inexpensive transportation, the trade equalizes the values of factories in the United States and Japan, and also equalizes the value of agricultural land. One can reasonably think that soybeans are soybeans, wherever they are produced, and that trade in soybeans at a common price forces the costs of the factors involved in producing soybeans to be equalized across the producing nations. The purchase of soybeans by the Japanese drives up the value of American land and drives down the value of Japanese land by giving an alternative to its output, leading toward equalization of the value of the land across the nations.
Factor price equalization was first developed by Paul Samuelson (1915–) and generalized by Eli Heckscher (1879–1952) and Bertil Ohlin (1899–1979). It has powerful predictions, including the equalization of wages of equally skilled people after free trade between the United States and Mexico. Thus, free trade in physical goods should equalize the price of such items as haircuts, land, and economic consulting in Mexico City and New York City. Equalization of wages is a direct consequence of factor price equalization because labor is a factor of production. If economic consulting is cheap in Mexico, trade in goods embodying economic consulting—boring reports, perhaps—will bid up the wages in the low-wage area and reduce the quantity in the high-wage area.
An even stronger prediction of the theory is that the price of water in New Mexico should be the same as in Minnesota. If water is cheaper in Minnesota, trade in goods that heavily use water—for example, paper—will tend to bid up the value of Minnesota water while reducing the premium on scarce New Mexico water.
It is fair to say that if factor price equalization works fully in practice, it works very, very slowly. Differences in taxes, tariffs, and other distortions make it a challenge to test the theory across nations. On the other hand, within the United States, where we have full factor mobility and product mobility, we still have different factor prices—electricity is cheaper in the Pacific Northwest. Nevertheless, nations with a relative abundance of capital and skilled labor export goods that use these intensively; nations with a relative abundance of land export land-intensive goods like food; nations with a relative abundance of natural resources export these resources; and nations with an abundance of low-skilled labor export goods that make intensive use of this labor. The reduction of trade barriers between such nations works like Ann and Bob’s joint production of party platters. By specializing in the goods in which they have a comparative advantage, there is more for all.
Key Takeaways
• Goods that are traded internationally embody the factors of production of the producing nations.
• It is useful to think of international trade as directly trading the inputs through the incorporation of inputs into products.
• If the set of traded goods is broad enough, the value of factors of production should be equalized through trade. This prediction is known as factor price equalization. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/06%3A_Trade/6.04%3A_International_Trade.txt |
When the person sitting next to you lights up a cigarette, he gets nicotine and the cigarette company gets some of his money. You just suffer, with no compensation. If your neighbor’s house catches fire because he fell asleep with that cigarette burning in his hand, your house may burn to the ground. The neighbor on the other side who plays very loud music late into the night—before your big economics test—enjoys the music, and the music distributors gets his money. You flunk out of college and wind up borrowing \$300,000 to buy a taxi medallion. Drunk drivers, cell phones ringing in movie theaters, loud automobiles, polluted air, and rivers polluted to the point that they catch fire, like Cleveland’s Cuyahoga did, are all examples where a transaction between two parties harmed other people. These are “external effects.”
But external effects are not necessarily negative. The neighbor who plants beautiful flowers in her yard brightens your day. Another person’s purchase of an electric car reduces the smog you breathe. Your neighbor’s investment in making his home safe from fire conveys a safety advantage to you. Indeed, even your neighbor’s investment in her own education may provide an advantage to you—you may learn useful things from your neighbor. Inventions and creations, whether products or poetry, produce value for others. The creator of a poem or a mathematical theorem provides a benefit to others.
Learning Objectives
• How can society stop people from doing annoying things and encourage them to do pleasing things?
These effects are called external effects, or externalities. An externality is any effect on people not involved in a particular transaction. Pollution is the classic example. When another person buys and smokes cigarettes, there is a transaction between the cigarette company and the smoker. But if you are sitting near the smoker, you are an affected party who is not directly compensated from the transaction, at least before taxes were imposed on cigarettes. Similarly, you pay nothing for the benefits you get from viewing your neighbor’s flowers, nor is there a direct mechanism to reward your neighbor for her efforts.
Externalities will generally cause competitive markets to behave inefficiently from a social perspective, absent a mechanism to involve all the affected parties. Without such a mechanism, the flower planter will plant too few beautiful flowers, for she has no reason to take account of your preferences in her choices. The odious smoker will smoke too much and too close to others, and the loud neighbor will play music much too late into the night. Externalities create a market failure—that is, a situation where a competitive market does not yield the socially efficient outcome.
Education is viewed as creating an important positive externality. Education generates many externalities, including more—and better—employment, less crime, and fewer negative externalities of other kinds. It is widely believed that educated voters elect better politicians.This is a logical proposition, but there is scant evidence in favor of it. There is evidence that educated voters are more likely to vote, but little evidence that they will vote for better candidates. Educated individuals tend to make a society wealthy, an advantage to all of society’s members. As a consequence, most societies subsidize education in order to promote it.
A major source of externalities arises in communicable diseases. Your vaccination not only reduces the likelihood that you will contract a disease but also makes it less likely that you will infect others with the disease.
Let’s consider pollution as a typical example. A paper mill produces paper, and a bad smell is an unfortunate by-product of the process. Each ton of paper produced increases the amount of bad smells produced. The paper mill incurs a marginal cost, associated with inputs like wood and chemicals and water. For the purposes of studying externalities, we will refer to the paper mill’s costs as a private cost, the cost borne by the supplier (in this case, the paper mill itself). In addition, there are external costs, which are the costs borne by third parties, that arise in this case from the smell. Adding the private costs and the external costs yield the total costs for all parties, or the social costs. These costs, in their marginal form, are illustrated in Figure 7.1.
Figure 7.1 A negative externality
Left to its own devices, the paper market would equate the marginal private cost and the marginal benefit to produce the competitive quantity sold at the competitive price. Some of these units—all of those beyond the quantity labeled “Socially Efficient Quantity”—are bad from a social perspective: They cost more to society than they provide in benefits. This is because the social cost of these units includes pollution, but paper buyers have no reason to worry about pollution or even to know that it is being created in the process of manufacturing paper.
The deadweight loss of these units is shown as a shaded triangle in the figure. The loss arises because the marginal social cost of the units exceeds the benefit, and the difference between the social cost and the benefits yields the loss to society. This is a case where too much is produced because the market has no reason to account for all the costs; some of the costs are borne by others.
Figure 7.2 External costs and benefits
The marginal private benefit—the benefit obtained by the buyer—and the marginal private cost give the demand and supply of a competitive market, and hence the competitive quantity results from the intersection of these two. The marginal social benefit and the marginal social cost give the value and cost from a social perspective; equating these two generates the socially efficient outcome. This can be either greater or less than the competitive outcome depending on which externality is larger.
Consider a town on a scenic bay that is filled with lobsters. The town members collect and eat the lobsters, and over time the size of the lobsters collected falls, until they are hardly worth searching for. This situation persists indefinitely. Few large lobsters are caught, and it is barely worth one’s time attempting to catch them. This sort of overuse of a resource due to lack of ownership is known as the tragedy of the commons.
The tragedy of the commons is a problem with a common resource shared by many people—in this case, the lobster bay. Catching lobsters creates an externality by lowering the productivity of other lobster catchers. The externality leads to overfishing, since individuals don’t take into account the negative effect they have on each other, ultimately leading to a nearly useless resource and potentially driving the lobsters to extinction. As a consequence, the lobster catching is usually regulated.
Key Takeaways
• An externality is any effect on people not involved in a particular transaction.
• Pollution is the classic negative externality.
• Externalities will generally cause competitive markets to behave inefficiently from a social perspective. Externalities create a market failure—that is, a competitive market does not yield the socially efficient outcome.
• Education is viewed as creating an important positive externality.
• A major source of externalities arises in communicable diseases. Your vaccination not only reduces the likelihood that you will contract a disease but also makes it less likely that you will infect others with the disease.
• Private costs are those borne by the parties to a transaction; external costs are costs borne by others not party to the transaction; and social costs represent the sum of private and external costs.
• Private benefits are those enjoyed by the parties to a transaction; external benefits are those enjoyed by others not party to the transaction; and social benefits represent the sum of private and external benefits.
• Demand is marginal private benefit; supply is marginal private cost.
• The social optimum arises at the quantity where marginal social benefit equals marginal social cost. A different quantity than the social optimum creates a deadweight loss equal to the difference of marginal social benefit and cost.
• The tragedy of the commons is the overuse of a resource arising because users ignore the harmful effects of their use on other users.
EXERCISES
1. A child who is vaccinated against polio is more likely to contract polio (from the vaccine) than an unvaccinated child. Does this fact imply that programs forcing vaccination of schoolchildren are ill-advised? Include with your answer a diagram illustrating the negative marginal benefit of vaccination, and use the horizontal axis to represent the proportion of the population vaccinated.
2. The total production from an oil field generally depends on the rate at which the oil is pumped, with faster rates leading to lower total production but earlier production. Suppose two different producers can pump from the field. Illustrate—using an externality diagram where the horizontal axis is the rate of production for one of the producers—the difference between the socially efficient outcome and the equilibrium outcome. Like many other states, Texas’s law requires that, when multiple people own land over a single oil field, the output is shared among the owners, with each owner obtaining a share equal to a proportion of the field under their land. This process is called unitization. Does it solve the problem of externalities in pumping and yield an efficient outcome? Why or why not?
3. Imagine that many students are bothered by loud music playing at 7 a.m. near their dorm. Using economic analysis, how would you improve the situation?
4. A local community uses revenue from its property taxes to build an expressway. Explain how this could give rise to free riders and how to solve the free-rider problem. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/07%3A_Externalities/7.01%3A_External_Effects.txt |
Learning Objectives
• Can society regulate annoying behavior with taxes?
Arthur Cecil Pigou (1877–1959) proposed a solution to the problem of externalities that has become a standard approach. This simple idea is to impose a per-unit tax on a good, thereby generating negative externalities equal to the marginal externality at the socially efficient quantity. This is known as a Pigouvian tax. Thus, if at the socially efficient quantity, the marginal external cost is \$1, then a \$1 per-unit tax would lead to the right outcome. This is illustrated in Figure 7.3.
The tax that is added is the difference, at the socially efficient quantity, between the marginal social cost and the marginal private cost, which equals the marginal external cost. The tax level need not equal the marginal external cost at other quantities, and the figure reflects a marginal external cost that is growing as the quantity grows. Nevertheless, the new supply curve created by the addition of the tax intersects demand (the marginal benefit) at the socially efficient quantity. As a result, the new competitive equilibrium, taking account of the tax, is efficient.
The case of a positive externality is similar. Here, a subsidy is needed to induce the efficient quantity. It is left as an exercise.
Figure 7.3 The Pigouvian tax
Taxes and subsidies are fairly common instruments to control externalities. We subsidize higher education with state universities, and the federal government provides funds for research and limited funds for the arts. Taxes on cigarettes and alcoholic beverages are used to discourage these activities, perhaps because smoking and drinking alcoholic beverages create negative externalities. (Cigarettes and alcohol also have inelastic demands, which make them good candidates for taxation since there is only a small distortion of the quantity.) However, while important in some arenas, taxes and subsidies are not the most common approach to regulation of externalities.
Key Takeaways
• A Pigouvian tax is a per-unit tax on a good, thereby generating negative externalities equal to the marginal externality at the socially efficient quantity.
• Imposition of a Pigouvian tax leads to a competitive equilibrium, taking account of the tax, which is efficient.
• In the case of a positive externality, a subsidy can be used to obtain efficiency.
• Taxes and subsidies are fairly common instruments to control externalities.
EXERCISES
1. Identify the tax revenue produced by a Pigouvian tax in Figure 7.3. What is the relationship between the tax revenue and the damage produced by the negative externality? Is the tax revenue sufficient to pay those damaged by the external effect an amount equal to their damage? (Hint: Is the marginal external effect increasing or decreasing?)
2. Identify, by using a diagram, the Pigouvian subsidy needed to induce the efficient quantity in the case of a positive externality. When is the subsidy expended smaller than the total external benefit?
3. Use the formulae for estimating the effect of a tax on quantity to deduce the size of the tax needed to adjust for an externality when the marginal social cost is twice the marginal private cost.
7.03: Quotas
Learning Objectives
• Can society regulate annoying behavior by just telling people what to do?
The Pigouvian tax and subsidy approach to dealing with externalities has several problems. First, it requires knowing the marginal value or cost of the external effect, and this may be a challenge to estimate. Second, it requires the imposition of taxes and permits the payment of subsidies, which encourages what might be politely termed as “misappropriation of funds.” That is, once a government agency is permitted to tax some activities and subsidize others, there will be a tendency to tax things people in the agency don’t like and subsidize “pet” projects, using the potential for externalities as an excuse rather than a real reason. U.S. politicians have been especially quick to see positive externalities in oil, cattle, and the family farm—externalities that haven’t been successfully articulated. (The Canadian government, in contrast, sees externalities in filmmaking and railroads.)
An alternative to the Pigouvian tax or subsidy solution is to set a quota, which is a limit on the activity. Quotas can be maxima or minima, depending on whether the activity generates negative or positive externalities. We set maximum levels on many pollutants rather than tax them, and ban some activities, like lead in gasoline or paint, or chlorofluorocarbons outright (a quota equal to zero). We set maximum amounts on impurities, like rat feces, in foodstuffs. We impose minimum educational attainment (eighth grade or age 16, whichever comes first), minimum age to drive, and minimum amount of rest time for truck drivers and airline pilots. A large set of regulations governs electricity and plumbing, designed to promote safety, and these tend to be “minimum standards.” Quotas are a much more common regulatory strategy for dealing with externalities than taxes and subsidies.
The idea behind a quota is to limit the quantity to the efficient level. If a negative externality in pollution means our society pollutes too much, then impose a limit or quantity restriction on pollution. If the positive externality of education means individuals in our society receive too little education from the social perspective, force them to go to school.
As noted, quotas have the advantage of addressing the problem without letting the government spend more money, limiting the government’s ability to misuse funds. On the other hand, quotas have the problem of identifying who should get the quota; quotas will often misallocate the resource. Indeed, a small number of power plants account for almost half of the man-made sulfur dioxide (SO2) pollution emitted into the atmosphere, primarily because these plants historically emitted a lot of pollution and their pollution level was set by their historical levels. Quotas tend to harm new entrants compared to existing firms and discourage the adoption of new technology. Indeed, the biggest polluters must stay with old technology in order to maintain their right to pollute.
Key Takeaways
• An alternative to the Pigouvian tax or subsidy solution is to set a quota. Quotas can be maxima or minima, depending on whether the activity generates negative or positive externalities.
• Quotas are a much more common regulatory strategy for dealing with externalities than taxes and subsidies.
• The goal of a quota is to limit the quantity to the efficient level.
• Quotas tend to harm new entrants compared to existing firms and discourage the adoption of new technology.
EXERCISES
1. If a quota is set to the socially efficient level, how does the value of a quota right compare to the Pigouvian tax?
2. Speeding (driving fast) creates externalities by increasing the likelihood and severity of automobile accidents, and most countries put a limit on speed; but one could instead require fast drivers to buy a permit to speed. Discuss the advantages and disadvantages of “speeding permits.” | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/07%3A_Externalities/7.02%3A_Pigouvian_Taxes.txt |
Learning Objectives
• Is there a better way to regulate annoying behavior than either taxes or quotas?
A solution to inefficiencies in the allocation of quota rights is to permit trading them. Tradable permits are quotas for pollution that can be exchanged to create a market in the right to pollute, and thereby create a tax on polluting. The emission of pollution requires the purchase of permits to pollute, and the price of these permits represents a tax on pollution. Thus, tradable permits represent a hybrid of a quota system and a Pigouvian taxation system—a quota determines the overall quantity of pollution as in a quota system, determining the supply of pollution rights, but the purchase of pollution rights acts like a tax on pollution, a tax whose level is determined by the quota supply and demand.
Figure 7.4 SO2 permit prices
The major advantage of a tradable permits system is that it creates the opportunity for efficient exchange—one potential polluter can buy permits from another, leaving the total amount of pollution constant. Such exchange is efficient because it uses the pollution in a manner creating the highest value, eliminating a bias toward “old” sources. Indeed, a low-value polluter might sell its permits and just shut down if the price of pollution was high enough.
A somewhat unexpected advantage of tradable permits has been the purchase of permits by environmental groups like the Sierra Club. Environmental groups can buy permits and then not exercise them, as a way of cleaning the air. In this case, the purchase of the permits creates a major positive externality on the rest of society, since the environmental group expends its own resources to reduce pollution of others.
Tradable permits offer the advantages of a taxation scheme—efficient use of pollution—without needing to estimate the social cost of pollution directly. This is especially valuable when the strategy is to set a quantity equal to the current quantity, and then gradually reduce the quantity in order to reduce the effects of the pollution. The price of permits can be a very useful instrument in assessing the appropriate time to reduce the quantity, since high permit prices, relative to likely marginal external costs, suggest that the quantity of the quota is too low, while low prices suggest that the quantity is too large and should be reduced.
Key Takeaways
• A solution to inefficiencies in the allocation of quota rights is to permit trading them.
• Tradable permits represent a hybrid of a quota system and a Pigouvian taxation system. The quota determines the overall quantity of pollution, while the purchase of pollution rights acts like a tax on pollution.
• The United States has permitted the trading of permits for some pollutants, like sulfur dioxide.
• The major advantage of a tradable permits system is that it creates the opportunity for efficient exchange.
• A somewhat unexpected advantage of tradable permits has been the purchase of permits by environmental groups, as a way of buying cleaner air.
• Tradable permits offer the advantages of a taxation scheme—efficient use of pollution—without needing to estimate the social cost of pollution directly.
• The price of permits can be a very useful instrument in assessing the appropriate time to reduce the quantity, since high permit prices, relative to likely marginal external costs, suggest that the quantity of the quota is too low, while low prices suggest that the quantity is too large and should be reduced. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/07%3A_Externalities/7.04%3A_Tradable_Permits_and_Auctions.txt |
Learning Objectives
• Can I just bribe my neighbor to stop being annoying?
The negative externality of a neighbor playing loud music late at night is not ordinarily solved with a tax or with a quota but instead through an agreement. When there aren’t many individuals involved, the individuals may be able to solve the problem of externalities without involving a government but through negotiation. This insight was developed by Nobel laureate Ronald Coase (1910– ), and is sometimes known as Coasian bargaining.
Coase offered the example of a cattle ranch next to a farm. There is a negative externality in that the cattle tend to wander over to the farm and eat the crops, rather than staying on the ranch. What happens next depends on property rights, which are the rights that come with ownership.
One of three things might be efficient from a social perspective. It might be efficient to erect a fence to keep the cows away from the crops. It might be efficient to close down the farm. Finally, it might be efficient to close down the ranch, if the farm is valuable enough and if the fence costs more than the value of the ranch.
If the farmer has a right not to have his crops eaten and can confiscate the cows if they wander onto the farm, then the rancher will have an incentive to erect a fence to keep the cows away, if that is the efficient solution. If the efficient solution is to close down the ranch, then the rancher will do that, since the farmer can confiscate the cows if they go over to the farm and it isn’t worth building the fence by hypothesis. Finally, if the efficient solution to the externality is to close down the farm, then the rancher will have an incentive to buy the farm in order to purchase the farm’s rights so that he can keep the ranch in operation. Since it is efficient to close down the farm only if the farm is worth less than the ranch, there is enough value in operating the ranch to purchase the farm at its value and still have money left over; that is, there are gains from trade from selling the farm to the rancher. In all three cases, if the farmer has the property rights, then the efficient outcome is reached.
Now suppose instead that the rancher has the rights and that the farmer has no recourse if the cows eat his crops. If shutting down the farm is efficient, then the farmer has no recourse but to shut it down. Similarly, if building the fence is efficient, then the farmer will build the fence to protect his crops. Finally, if shutting down the ranch is efficient, the farmer will buy the ranch from the rancher in order to be able to continue to operate the more valuable farm. In all cases, the efficient solution is reached through negotiation.
Coase argued that bargaining can generally solve problems of externalities and that the real problem is ill-defined property rights. If the rancher and the farmer can’t transfer their property rights, then the efficient outcome may not arise. In the Coasian view of externalities, if an individual owned the air, air pollution would not be a problem because the owner would charge for the use and wouldn’t permit an inefficient level of pollution. The case of air pollution demonstrates some of the limitations of the Coasian approach because ownership of the air, or even the more limited right to pollute into the air, would create an additional set of problems; a case where the cure is likely to be worse than the disease.
Bargaining to solve the problem of externalities is often feasible when a small number of people are involved. When a large number of people are potentially involved, as with air pollution, bargaining is unlikely to be successful in addressing the problem of externalities, and a different approach is required.
Key Takeaways
• When there aren’t many individuals involved, the individuals may be able to solve the problem of externalities without involving a government, but through negotiation.
• Nobel laureate Ronald Coase argued that bargaining can generally solve problems of externalities and that the real problem is ill-defined property rights.
• Bargaining to solve the problem of externalities is often feasible when a small number of people are involved. When a large number of people are potentially involved, as with air pollution, bargaining is unlikely to be successful in addressing the problem of externalities. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/07%3A_Externalities/7.05%3A_Coasian_Bargaining.txt |
Learning Objectives
• Is extinction really an economic phenomenon?
• Why do we overfish?
Consider an unregulated fishing market like the lobster market considered previously, and let S be the stock of fish. The purpose of this example is illustrative of the logic, rather than an exact accounting of the biology of fish populations, but is not unreasonable. Let S be the stock of a particular species of fish. Our starting point is an environment without fishing: How does the fish population change over time? Denote the change over time in the fish population by $$S^{\cdot}\left(S^{\cdot}\right.$$ is notation for the derivative with respect to time, notation that dates back to Sir Isaac Newton). We assume that population growth follows the logistic equation $$S^{\cdot}=r S(1-S)$$. This equation reflects two underlying assumptions. First, mating and reproduction are proportional to the stock of fish S. Second, survival is proportional to the amount of available resources 1 – S, where 1 is set to be the maximum sustainable population. (Set the units of the number of fish so that 1 is the full population.)
The dynamics of the number of fish are illustrated in Figure 7.5. On the horizontal axis is the number of fish, and on the vertical axis is the change in S. When $$S^{\cdot}>0$$, S is increasing over time, and the arrows on the horizontal axis reflect this. Similarly, if $$S^{\cdot}<0$$, S is decreasing.
Absent fishing, the value 1 is a stable steady state of the fish population, in which the variables stay constant and forces are balanced. It is a steady state because, if $$S = 1, S ˙ = 0$$ ; that is, there is no change in the fish population. It is stable because the effect of a small perturbation—S near but not exactly equal to 1—is to return to 1. (In fact, the fish population is very nearly globally stable. Start with any population other than zero and the population returns to 1.)It turns out that there is a closed form solution for the fish population: $$S(t)=S(0) S(0)+(1-S(0)) e-r t$$.
Figure 7.5 Fish population dynamics
Now we introduce a human population and turn to the economics of fishing. Suppose that a boat costs b to launch and operate and that it captures a fixed fraction a of the total stock of fish S; that is, each boat catches aS. Fish sell for a price $$p=Q-1 \varepsilon$$, where the price arises from the demand curve, which in this case has constant elasticity ε, and Q is the quantity of fish offered for sale. Suppose there are n boats launched; then the quantity of fish caught is Q = naS. Fishers enter the market as long as profits are positive, which leads to zero profits for fishers; that is, $$b=(Q n) p(Q)$$. This equation makes a company just indifferent to launching an additional boat because the costs and revenues are balanced. These two equations yield two equations in the two unknowns n and $$Q: n=Q p(Q) b=1 \text { b } Q \varepsilon-1 \varepsilon$$, and $$Q=n a S$$. These two equations solve for the number of fish caught, , $$Q=(\text { as b }) \varepsilon$$ and the number of boats, $$n=a \varepsilon-1 \text { b } \varepsilon \operatorname{S~} \varepsilon-1$$.
Subtracting the capture by humans from the growth in the fish population yields
$S^{\cdot}=r S(1-S)-(a S b) \varepsilon$
Thus, a steady state satisfies $$0=S^{\cdot}=r S(1-S)-(a S b) \varepsilon$$.
Figure 7.6 Fish population dynamics with fishing
The dark curve represents $$$S^{\cdot}$$$, and thus for S between 0 and the point labeled $$$$S^{*}$$$$, $$$S^{\cdot}$$$ is positive and so S is increasing over time. Similarly, to the right of $$$$S^{*}$$$$, S is decreasing. Thus, $$$$S^{*}$$$$ is stable under small perturbations in the stock of fish and is an equilibrium.
We see that if demand for fish is elastic, fishing will not drive the fish to extinction. Even so, fishing will reduce the stock of fish below the efficient level because individual fishers don’t take account of the externality they impose—their fishing reduces the stock for future generations. The level of fish in the sea converges to S* satisfying
$0=r S^{*}\left(1-S^{*}\right)-\left(a S^{*} b\right) \varepsilon$
In contrast, if demand is inelastic, fishing may drive the fish to extinction. For example, if r = 2 and a = b = 1, and ε = 0.7, extinction is necessary, as is illustrated in Figure 7.7.
Figure 7.7 Fish population dynamics: extinction
It is possible, even with inelastic demand, for there to be a stable fish population. Not all parameter values lead to extinction. Using the same parameters as before, but with ε = 0.9, we obtain a stable outcome as illustrated in Figure 7.8.
Figure 7.8 Possibility of multiple equilibria
In addition to the stable equilibrium outcome, there is an unstable steady state, which may converge either upward or downward. A feature of fishing with inelastic demand is that there exists a region where extinction is inevitable because, when the stock is near zero, the high demand price induced by inelasticity forces sufficient fishing to ensure extinction.
As a consequence of the fishing externality, nations attempt to regulate fishing, both by extending their own reach 200 miles into the sea and by treaties limiting fishing in the open sea. These regulatory attempts have met with only modest success at preventing overfishing.
What is the efficient stock of fish? This is a challenging mathematical problem, but some insight can be gleaned via a steady-state analysis. A steady state arises when $$S^{\cdot}=0$$. If a constant amount Q is removed, a steady state in the stock must occur at $$0=S^{\cdot}=r S(1-S)-Q$$. This maximum catch then occurs at S = ½ and Q = ¼ r. This is not the efficient level, for it neglects the cost of boats, and the efficient stock will actually be larger. More generally, it is never efficient to send the population below the maximum point on the survival curve plotted in Figure 7.5.
Conceptually, fishing is an example of the tragedy of the commons externality already discussed. However, the threat of a permanent extinction and alluring possibility of solving dynamic models make it a particularly dramatic example.
Key Takeaways
• Extinction arises from the interaction of two systems: one biological and one economic.
• When demand is elastic, extinction should not arise.
• When demand is inelastic, population decreases reduce the quantity, which increase total revenue, which leads to more investment in fishing. When demand is sufficiently inelastic, the heightened investment leads to proportionally more fish caught and the fish go extinct.
• Fishing is an example of the tragedy of the commons externality.
EXERCISE
1. Suppose ε = 1. For what parameter values are fish necessarily driven to extinction? Can you interpret this condition to say that the demand for caught fish exceeds the production via reproduction? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/07%3A_Externalities/7.06%3A_Fishing_and_Extinction.txt |
Consider a company offering a fireworks display. Pretty much anyone nearby can watch the fireworks, and people with houses in the right place have a great view of them. The company that creates the fireworks can’t compel those with nearby homes to pay for the fireworks, and so a lot of people get to watch them without paying. This will make it difficult or impossible for the fireworks company to make a profit.
Learning Objectives
• What are people who just use public goods without paying called, and what is their effect on economic performance?
A public good is a good that has two attributes: Nonexcludability, which means the producer can’t prevent the use of the good by others, and nonrivalry, which means that many people can use the good simultaneously. The classic example of a public good is national defense. National defense is clearly nonexcludable because, if we spend the resources necessary to defend our national borders, it isn’t going to be possible to defend everything except one apartment on the second floor of a three-story apartment building on East Maple Street. Once we have kept our enemies out of our borders, we’ve protected everyone within the borders. Similarly, the defense of the national borders exhibits a fair degree of nonrivalry, especially insofar as the strategy of defense is to deter an attack in the first place. That is, the same expenditure of resources protects all. It is theoretically possible to exclude some people from the use of a poem or a mathematical theorem, but exclusion is generally quite difficult. Both poems and theorems are nonrivalrous. Similarly, technological and software inventions are nonrivalrous, even though a patent grants the right to exclude the use by others. Another good that permits exclusion at a cost is a highway. A toll highway shows that exclusion is possible on the highways. Exclusion is quite expensive, partly because the tollbooths require staffing, but mainly because of the delays imposed on drivers associated with paying the tolls—the time costs of toll roads are high. Highways are an intermediate case where exclusion is possible only at a significant cost, and thus should be avoided if possible. Highways are also rivalrous at high-congestion levels, but nonrivalrous at low-congestion levels. That is, the marginal cost of an additional user is essentially zero for a sizeable number of users, but then marginal cost grows rapidly in the number of users. With fewer than 700 cars per lane per hour on a four-lane highway, generally the flow of traffic is unimpeded.The effect of doubling the number of lanes from two to four is dramatic. A two-lane highway generally flows at 60 mph or more provided there are fewer than 200 cars per lane per hour, while a four-lane highway can accommodate 700 cars per lane per hour at the same speed. As congestion grows beyond this level, traffic slows down and congestion sets in. Thus, west Texas interstate highways are usually nonrivalrous, while Los Angeles’s freeways are usually very rivalrous.
Like highways, recreational parks are nonrivalrous at low-use levels, becoming rivalrous as they become sufficiently crowded. Also like highways, it is possible, but expensive, to exclude potential users, since exclusion requires fences and a means for admitting some but not others. (Some exclusive parks provide keys to legitimate users, while others use gatekeepers to charge admission.)
Take the example of a neighborhood association that is considering buying land and building a park in the neighborhood. The value of the park is going to depend on the size of the park, and we suppose for simplicity that the value in dollars of the park to each household in the neighborhood is S b n −a , where n is the number of park users, S is the size of the park, and a and b are parameters satisfying $$0<a \leq b<1$$. This functional form builds in the property that larger parks provide more value at a diminishing rate, but there is an effect from congestion. The functional form gives a reason for parks to be public—it is more efficient for a group of people to share a large park than for each individual to possess a small park, at least if $$b>a$$, because the gains from a large park exceed the congestion effects. That is, there is a scale advantage—doubling the number of people and the size of the park increases each individual’s enjoyment.
How much will selfish individuals voluntarily contribute to the building of the park? That of course depends on what they think others will contribute. Consider a single household, and suppose that each household, i, thinks the others will contribute S-1 to the building of the park. Given this expectation, how much should each household, i, contribute? If the household contributes s, the park will have size $$S=S_{-1}+s$$, which the household values at $$(S-1+s) b n-a$$. Thus, the net gain to a household that contributes s when the others contribute $$S_{-1} \text {is }(S-1+s) \text { b } n-a-s$$.
Exercise 1 shows that individual residents gain from their marginal contribution if and only if the park is smaller than $S 0=(b n-a) 11-b$ . Consequently, under voluntary contributions, the only equilibrium park size is S0. That is, for any park size smaller than S0, citizens will voluntarily contribute to make the park larger. For any larger size, no one is willing to contribute.
Under voluntary contributions, as the neighborhood grows in number, the size of the park shrinks. This makes sense—the benefits of individual contributions to the park mostly accrue to others, which reduces the payoff to any one contributor.
How large should the park be? The total value of the park of size S to the residents together is n times the individual value, which gives a collective value of S b n 1−a ; and the park costs S, so from a social perspective the park should be sized to maximize S b n 1−a −S, which yields an optimal park of size $$S*= (b n 1−a ) 1 1−b$$. Thus, as the neighborhood grows, the park should grow, but as we saw, the park would shrink if the neighborhood has to rely on voluntary contributions. This is because people contribute individually, as if they were building the park for themselves, and don’t account for the value they provide to their neighbors when they contribute. Under individual contributions, the hope that others contribute leads individuals not to contribute. Moreover, use of the park by others reduces the value of the park to each individual, so that the size of the park shrinks as the population grows under individual contributions. In contrast, the park ought to grow faster than the number of residents grows, as the per capita park size is $$S* n = b 1 1−b n b−a 1−b$$, which is an increasing function of n.Reminder: In making statements like should and ought, there is no conflict in this model because every household agrees about the optimal size of the park, so that a change to a park size of S*, paid with equal contributions, maximizes every household’s utility.
The lack of incentive for individuals to contribute to a social good is known as a free-rider problem. The term refers to the individuals who don’t contribute to the provision of a public good, who are said to be free riders, that is, they ride freely on the contributions of others. There are two aspects of the free-rider problem apparent in this simple mathematical model. First, the individual incentive to contribute to a public good is reduced by the contributions of others, and thus individual contributions tend to be smaller when the group is larger. Put another way, the size of the free-rider problem grows as the community grows larger. Second, as the community grows larger, the optimal size of the public good grows. The market failure under voluntary contributions is greater as the community is larger. In the theory presented, the optimal size of the public good is $$$S*= (b n 1−a ) 1 1−b$$$, and the actual size under voluntary contributions is $$$S*= (b n 1−a ) 1 1−b$$$, a gap that gets very large as the number of people grows.
The upshot is that people will voluntarily contribute too little from a social perspective, by free riding on the contributions of others. A good example of the provision of public goods is a coauthored term paper. This is a public good because the grade given to the paper is the same for each author, and the quality of the paper depends on the sum of the efforts of the individual authors. Generally, with two authors, both work pretty hard on the manuscript in order to get a good grade. Add a third author, and it is a virtual certainty that two of the authors will think the third didn’t work as hard and is a free rider on the project.
The term paper example also points to the limitations of the theory. Many people are not as selfish as the theory assumes and will contribute more than would be privately optimal. Moreover, with small numbers, bargaining between the contributors and the division of labor (each works on a section) may help to reduce the free-rider problem. Nevertheless, even with these limitations, the free-rider problem is very real; and it gets worse the more people are involved. The theory shows that if some individuals contribute more than their share in an altruistic way, the more selfish individuals contribute even less, undoing some of the good done by the altruists.
Key Takeaways
• A public good has two attributes: nonexcludability, which means the producer can’t prevent the use of the good by others; and nonrivalry, which means that many people can use the good simultaneously.
• Examples of public goods include national defense, fireworks displays, and mathematical theorems.
• Nonexcludability implies that people don’t have to pay for the good; nonrivalry means that the efficient price is zero.
• A free rider is someone who doesn’t pay for a public good.
• Generally voluntary contributions lead to too little provision of public goods.
• In spite of some altruism, the free-rider problem is very real, and it gets worse the more people are involved.
EXERCISES
1. Verify that individual residents gain from contributing to the park if $$S< (b n −a ) 1 1−b$$ and gain from reducing their contributions if $$S> (b n −a ) 1 1−b .$$.
2. For the model presented in this section, compute the elasticity of the optimal park size with respect to the number of residents—that is, the percentage change in S* for a small percentage change in n. [Hint: Use the linear approximation trick $$(1+Δ) r ≈rΔ$$ for Δ near zero.]
3. For the model presented in this section, show that an individual’s utility when the park is optimally sized and the expenses are shared equally among the n individuals is $u=( b b 1−b − b 1 1−b ) n b−a 1−b .$
Does this model predict an increase in utility from larger communities?
4. Suppose two people, Person 1 and Person 2, want to produce a playground to share between them. The value of the playground of size S to each person is S , where S is the number of dollars spent to build it. Show that, under voluntary contributions, the size of the playground is ¼ and that the efficient size is 1.
5. For the previous exercise, now suppose Person 1 offers “matching funds”—that is, offers to contribute an equal amount to the contributions of Person 2. How large a playground will Person 2 choose? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/08%3A_Public_Goods/8.01%3A_Free_Riders.txt |
Learning Objectives
• If people won’t pay for public goods, can society tax them instead?
Faced with the fact that voluntary contributions produce an inadequate park, the neighborhood turns to taxes. Many neighborhood associations or condominium associations have taxing authority and can compel individuals to contribute. One solution is to require each resident to contribute the amount 1, resulting in a park that is optimally sized at n, as clearly shown in the example from the previous section. Generally it is possible to provide the correct size of the public good using taxes to fund it. However, this is challenging in practice, as we illustrate in this slight modification of the previous example.
Let individuals have different strengths of preferences, so that individual i values the public good of size S at an amount v i S b n −a that is expressed in dollars. (It is useful to assume that all people have different v values to simplify arguments.) The optimal size of the park for the neighborhood is $$n −a 1−b ( b ∑ i=1 n v i ) 1 1−b = (b v ¯ ) 1 1−b n 1−a 1−b$$, where $$v ¯ = 1 n ∑ i=1 n v$$ i is the average value. Again, taxes can be assessed to pay for an optimally sized park, but some people (those with small v values) will view that as a bad deal, while others (with large v) will view it as a good deal. What will the neighborhood choose to do?
If there are an odd number of voters in the neighborhood, we predict that the park size will appeal most to the median voter.The voting model employed here is that there is a status quo, which is a planned size of S. Anyone can propose to change the size of S, and the neighborhood then votes yes or no. If an S exists such that no replacement gets a majority vote, that S is an equilibrium under majority voting. This is the voter whose preferences fall in the middle of the range. With equal taxes, an individual obtains v i S b n −a − S n . If there are an odd number of people, n can be written as 2k + 1. The median voter is the person for whom there are k values vi larger than hers and k values smaller than hers. Consider increasing S. If the median voter likes it, then so do all the people with higher v’s, and the proposition to increase S passes. Similarly, a proposal to decrease S will get a majority if the median voter likes it. If the median voter likes reducing S, all the individuals with smaller vi will vote for it as well. Thus, we can see the preferences of the median voter are maximized by the vote, and simple calculus shows that this entails $$S= (b v k ) 1 1−b n 1−a 1−b$$.
Unfortunately, voting does not result in an efficient outcome generally and only does so when the average value equals the median value. On the other hand, voting generally performs much better than voluntary contributions. The park size can either be larger or smaller under median voting than is efficient.The general principle here is that the median voting will do better when the distribution of values is such that the average of n values exceeds the median, which in turn exceeds the maximum divided by n. This is true for most empirically relevant distributions.
Key Takeaways
• Taxation—forced contribution—is a solution to the free-rider problem.
• An optimal tax rate is the average marginal value of the public good.
• Voting leads to a tax rate equal to the median marginal value, and hence does not generally lead to efficiency, although it outperforms voluntary contributions.
EXERCISES
1. Show for the model of this section that, under voluntary contributions, only one person contributes, and that person is the person with the largest vi. How much do they contribute? [Hint: Which individual i is willing to contribute at the largest park size? Given the park that this individual desires, can anyone else benefit from contributing at all?]
2. Show that, if all individuals value the public good equally, voting on the size of the good results in the efficient provision of the public good. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/08%3A_Public_Goods/8.02%3A_Provision_With_Taxation.txt |
Learning Objectives
• What can we do if we disagree about the optimal level of public goods?
The example in the previous section showed the challenges to a neighborhood’s provision of public goods created by differences in the preferences. Voting does not generally lead to the efficient provision of the public good and does so only rarely when all individuals have the same preferences.
A different solution was proposed by TieboutCharles Tiebout, 1919–1962. His surname is pronounced “tee-boo.” in 1956, which works only when the public goods are local. People living nearby may or may not be excludable, but people living farther away can be excluded. Such goods that are produced and consumed in a limited geographical area are local public goods. Schools are local—more distant people can readily be excluded. With parks it is more difficult to exclude people from using the good; nonetheless, they are still local public goods because few people will drive 30 miles to use a park.
Suppose that there are a variety of neighborhoods, some with high taxes, better schools, big parks, beautifully maintained trees on the streets, frequent garbage pickup, a first-rate fire department, extensive police protection, and spectacular fireworks displays, and others with lower taxes and more modest provision of public goods. People will move to the neighborhood that fits their preferences. As a result, neighborhoods will evolve with inhabitants that have similar preferences for public goods. Similarity among neighbors makes voting more efficient, in turn. Consequently, the ability of people to choose their neighborhoods to suit their preferences over taxes and public goods will make the neighborhood provision of public goods more efficient. The “Tiebout theory” shows that local public goods tend to be efficiently provided. In addition, even private goods such as garbage collection and schools can be efficiently publicly provided when they are local goods, and there are enough distinct localities to offer a broad range of services.
Key Takeaways
• When public goods are local—people living nearby may or may not be excludable, whereas people living farther away may be excluded—the goods are “local public goods.”
• Specialization of neighborhoods providing in distinct levels of public goods, when combined with households selecting their preferred neighborhood, can lead to efficient provision of public goods.
EXERCISES
1. Consider a babysitting cooperative, where parents rotate supervision of the children of several families. Suppose that, if the sitting service is available with frequency Y, a person’s i value is $$viY$$ and the costs of contribution $$y=(1 / 2) \mathrm{ny}^{2}$$, where y is the sum of the individual contributions and n is the number of families. Rank $$v_{1} \geq v_{2} \geq \ldots \geq v_{n}$$.
2. What is the size of the service under voluntary contributions?
(Hint: Let $$yi$$ be the contribution of family i. Identify the payoff of family j as $$v j ( y j + ∑ i≠j y i )−½n ( y j ) 2$$.
What value of $$yj$$maximizes this expression?)
3. What contributions maximize the total social value
4. $$( ∑ j=1 n v j )( ∑ j=1 n y j )−½n ∑ i=1 n ( y j ) 2$$ ?
5. $$yi i$$
6. Let $$μ= 1 n ∑ j=1 n v j$$ and $$σ 2 = 1 n ∑ j=1 n ( v j −μ ) 2$$. Conclude that, under voluntary contributions, the total value generated by the cooperative is $$n 2 ( μ 2 − σ 2 )$$ .
(Hint: It helps to know that $$σ 2 = 1 n ∑ j=1 n ( v j −μ ) 2 = 1 n ∑ j=1 n v j 2 − 2 n ∑ j=1 n μ v j + 1 n ∑ j=1 n μ 2 = 1 n ∑ j=1 n v j 2 − μ 2$$. ) | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/08%3A_Public_Goods/8.03%3A_Local_Public_Goods.txt |
The basic theory of the firm regards the firm as a mechanism for transforming productive inputs into final and intermediate goods and services. This process is known as production. For instance, the smelting of copper or gold removes impurities and makes the resulting product into a more valuable substance. Silicon Valley transforms silicon, along with a thousand other chemicals and metals, into computer chips used in everything from computers to toasters. Cooking transforms raw ingredients into food, by adding flavor and killing bacteria. Moving materials to locations where they have higher value is a form of production. Moving stone to a building site to construct an exterior wall, bringing the King Tut museum exhibit to Chicago, or qualifying a basketball team for the league playoffs are all examples of different types of production. According to this simple view, a firm is comprised of a technology or set of technologies for transforming materials into valuable goods and to maximize profits. This “production function” view of the firm is appropriate for some environments, when products and services are standardized and technologies are widely available. However, in other settings, this way of thinking about the firm is misleading, especially when the internal organization of the firm is important. Nevertheless, the “production function” model of the firm is a natural starting point to begin our investigation of competition.
Learning Objectives
• What types of companies exist?
There are four varieties of firms created in law, although these types have several subtypes. At one end is the proprietorship, which is a firm owned by a single individual (the proprietor) or perhaps by a family. The family farm and many “mom-and-pop” restaurants and convenience stores are operated proprietorships. Debts accrued by the proprietorship are the personal responsibility of the proprietor. Legal and accounting firms are often organized as partnerships. Partnerships are firms owned by several individuals who share profits as well as liabilities of the firm according to a specified formula that varies by the relative contribution and potential cost of each partner in the firm. Thus, if a partner in a law firm steals a client’s money and disappears, the other partners may be responsible for absorbing some portion of the loss. In contrast, a corporation is treated legally as a single entity owned by shareholders. Like a person, a corporation can incur debt and is therefore responsible for repayment. This stands in contrast to a partnership where particular individuals may be liable for debts incurred. Hence, when the energy trader company Enron collapsed, the Enron shareholders lost the value of their stock; however, they were not responsible for repaying the debt that the corporation had incurred. Moreover, company executives are also not financially responsible for debts of the corporation, provided they act prudentially. If a meteor strikes a manufacturer and destroys the corporation, the executives are not responsible for the damage or for the loans that may not be repaid as a result. On the other hand, executives are not permitted to break the law, as was the case with corporate officers at Archer Daniels Midland, the large agricultural firm who colluded to fix the price of lysine and were subsequently fined and jailed for their misdeeds, as was the corporation.
Corporations who shield company executives and shareholders from fines and punishments are said to offer “limited liability.” So why would anyone in his or her right mind organize a firm as a proprietorship or a partnership? The explanation is that it is costly to incorporate businesses—about \$1,000 per year at the time of this writing—and corporations are taxed, so that many small businesses find it less costly to be organized as proprietorships. Moreover, it may not be possible for a family-owned corporation to borrow money to open a restaurant; for example, potential lenders may fear not being repaid in the event of bankruptcy, so they insist that owners accept some personal liability. So why are professional groups organized as partnerships and not as corporations? The short answer is that a large variety of hybrid, organizational forms exist instead. The distinctions are blurred, and organizations like “Chapter S Corporations” and “Limited Liability Partnerships” offer the advantages of partnerships (including avoidance of taxation) and corporations. The disadvantages of these hybrids are the larger legal fees and greater restrictions on ownership and freedom to operate that exist in certain states and regions.
Usually proprietorships are smaller than partnerships, and partnerships are smaller than corporations, though there are some very large partnerships (e.g., the Big Four accounting firms) as well as some tiny corporations. The fourth kind of firm may be of any size. It is distinguished primarily by its source of revenue and not by how it is internally organized. The nonprofit firm is prohibited from distributing a profit to its owners. Religious organizations, academic associations, environmental groups, most zoos, industry associations, lobbying groups, many hospitals, credit unions (a type of bank), labor unions, private universities, and charities are all organized as nonprofit corporations. The major advantage of nonprofit firms is their tax-free status. In exchange for avoiding taxes, nonprofits must be engaged in government-approved activities, suggesting that nonprofits operate for the social benefit of some segment of society. So why can’t you establish your own nonprofit that operates for your personal benefit in order to avoid taxes? Generally, you alone isn’t enough of a socially worthy purpose to meet the requirements to form a nonprofit.Certainly some of the nonprofit religious organizations created by televangelists suggest that the nonprofit established for the benefit of a single individual isn’t too far-fetched. Moreover, you can’t establish a nonprofit for a worthy goal and not serve that goal but just pay yourself all the money the corporation raises, because nonprofits are prohibited from overpaying their managers. Overpaying the manager means not serving the worthy corporate goal as well as possible. Finally, commercial activities of nonprofits are taxable. Thus, when the nonprofit zoo sells stuffed animals in the gift shop, generally the zoo collects sales tax and is potentially subject to corporate taxes.
The modern corporation is a surprisingly recent invention. Prior to World War I, companies were typically organized as a pyramid with a president at the top and vice presidents who reported to him at the next lower level, and so on. In a pyramid, there is a well-defined chain of command, and no one is ever below two distinct managers of the same level. The problem with a pyramid is that two retail stores that want to coordinate have to contact their managers, and possibly their manager’s managers, and so on up the pyramid until a common manager is reached. There are circumstances where such rigid decision making is unwieldy, and the larger the operation of a corporation, the more unwieldy it gets.
Four companies—Sears, DuPont, General Motors, and Standard Oil of New Jersey (Exxon)—found that the pyramid didn’t work well for them. Sears found that its separate businesses of retail stores and mail order required a mix of shared inputs (purchased goods) but distinct marketing and warehousing of these goods. Consequently, retail stores and mail order needed to be separate business units, but the purchasing unit has to service both of them. Similarly, DuPont’s military business (e.g., explosives) and consumer chemicals were very different operations serving different customers yet often selling the same products so that again the inputs needed to be centrally produced and to coordinate with two separate corporate divisions. General Motors’s many car divisions are “friendly rivals,” in which technology and parts are shared across the divisions, but the divisions compete in marketing their cars to consumers. Again, technology can’t be housed under just one division, but instead is common to all. Finally, Standard Oil of New Jersey was attempting to create a company that managed oil products from oil exploration all the way through to pumping gasoline into automobile gas tanks. With such varied operations all over the globe, Standard Oil of New Jersey required extensive coordination and found that the old business model needed to be replaced. These four companies independently invented the modern corporation, which is organized into separate business units. These business units run as semiautonomous companies themselves, with one business unit purchasing inputs at a negotiated price from another unit and selling outputs to a third unit. The study of the internal organization of firms and its ramifications for competitiveness is fascinating but beyond the scope of this book.If you want to know more about organization theory, I happily recommend Competitive Solutions: The Strategist’s Toolkit, by R. Preston McAfee, Princeton: Princeton University Press, 2002.
Key Takeaways
• The most basic theory of the firm regards the firm as a means of transforming materials into other, more intermediate and final goods. This is known as production.
• A proprietorship is a firm owned by an individual (the proprietor) or perhaps a family and operated by a small number of people. The family farm and many restaurants and convenience stores are operated in this way. Debts accrued by the proprietorship are the personal responsibility of the proprietor.
• Attorneys and accountants are often organized as partnerships. Partnerships share profits according to a formula and are usually liable for losses incurred by the partnership.
• A corporation is a legal entity that may incur debt, and has the responsibility for repayment of that debt. The owners or officers of the corporation are not liable for the repayment.
• A large variety of hybrid, organizational forms exist.
• The nonprofit firm is prohibited from distributing a profit to its owners. In exchange for avoiding taxes, nonprofits must be engaged in government-approved activities that operate for the benefit of some segment of society. Commercial activities of nonprofits are taxable.
• Large corporations are often organized into business units that are semiautonomous affiliated companies. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/09%3A_Producer_Theory-_Costs/9.01%3A_Types_of_Firms.txt |
Learning Objectives
• How is the output of companies modeled?
• Are there any convenient functional forms?
The firm transforms inputs into outputs. For example, a bakery takes inputs like flour, water, yeast, labor, and heat and makes loaves of bread. An earth moving company combines capital equipment, ranging from shovels to bulldozers with labor in order to digs holes. A computer manufacturer buys parts “off-the-shelf” like disk drives and memory, with cases and keyboards, and combines them with labor to produce computers. Starbucks takes coffee beans, water, some capital equipment, and labor to brew coffee.
Many firms produce several outputs. However, we can view a firm that is producing multiple outputs as employing distinct production processes. Hence, it is useful to begin by considering a firm that produces only one output. We can describe this firm as buying an amount x1 of the first input, x2 of the second input, and so on (we’ll use xn to denote the last input), and producing a quantity of the output. The production function that describes this process is given by $$y=f\left(x_{1}, x_{2}, \ldots, x_{n}\right)$$.
The production function is the mapping from inputs to an output or outputs.
For the most part we will focus on two inputs in this section, although the analyses with more than inputs is “straightforward.”
Example: The Cobb-Douglas production function is the product of each input, x, raised to a given power. It takes the form $$f\left(x_{1}, x_{2}, \ldots, x_{n}\right)$$= a 0 x 1 a 1 x 2 a 2 … x n a n .
The constants a1 through an are typically positive numbers less than one. For example, with two goods, capital K and labor L, the Cobb-Douglas function becomes a0KaLb. We will use this example frequently. It is illustrated, for $$a_{0}=1, a=1 / 3, \text { and } b=2 / 3$$, in Figure 9.1.
Figure 9.1 Cobb-Douglas isoquants
Figure 9.2 The production function
The fixed-proportions production function comes in the form $$f\left(x_{1}, x_{2}, \ldots, x_{n}\right)$$ = Min { a 1 x 1 , a 2 x 2 , …, a n x n }.
The fixed-proportions production function is a production function that requires inputs be used in fixed proportions to produce output. It has the property that adding more units of one input in isolation does not necessarily increase the quantity produced. For example, the productive value of having more than one shovel per worker is pretty low, so that shovels and diggers are reasonably modeled as producing holes using a fixed-proportions production function. Moreover, without a shovel or other digging implement like a backhoe, a barehanded worker is able to dig so little that he is virtually useless. Ultimately, the size of the holes is determined by min {number of shovels, number of diggers}. Figure 9.3 illustrates the isoquants for fixed proportions. As we will see, fixed proportions make the inputs “perfect complements.”
Figure 9.3 Fixed-proportions and perfect substitutes
The marginal product of an input is just the derivative of the production function with respect to that input.This is a partial derivative, since it holds the other inputs fixed. Partial derivatives are denoted with the symbol δ. An important property of marginal product is that it may be affected by the level of other inputs employed. For example, in the Cobb-Douglas case with two inputsThe symbol α is the Greek letter “alpha.” The symbol β is the Greek letter “beta.” These are the first two letters of the Greek alphabet, and the word alphabet itself originates from these two letters. and for constant A,
$f(K, L)=A K a L \beta$
the marginal product of capital is
$∂f ∂K (K,L)=αA K α−1 L β .$
If α and β are between zero and one (the usual case), then the marginal product of capital is increasing in the amount of labor, and it is decreasing in the amount of capital employed. For example, an extra computer is very productive when there are many workers and a few computers, but it is not so productive where there are many computers and a few people to operate them.
The value of the marginal product of an input is the marginal product times the price of the output. If the value of the marginal product of an input exceeds the cost of that input, it is profitable to use more of the input.
Some inputs are easier to change than others. It can take 5 years or more to obtain new passenger aircraft, and 4 years to build an electricity generation facility or a pulp and paper mill. Very skilled labor such as experienced engineers, animators, and patent attorneys are often hard to find and challenging to hire. It usually requires one to spend 3 to 5 years to hire even a small number of academic economists. On the other hand, it is possible to buy shovels, telephones, and computers or to hire a variety of temporary workers rapidly, in a day or two. Moreover, additional hours of work can be obtained from an existing labor force simply by enlisting them to work “overtime,” at least on a temporary basis. The amount of water or electricity that a production facility uses can be varied each second. A dishwasher at a restaurant may easily use extra water one evening to wash dishes if required. An employer who starts the morning with a few workers can obtain additional labor for the evening by paying existing workers overtime for their hours of work. It will likely take a few days or more to hire additional waiters and waitresses, and perhaps several days to hire a skilled chef. You can typically buy more ingredients, plates, and silverware in one day, whereas arranging for a larger space may take a month or longer.
The fact that some inputs can be varied more rapidly than others leads to the notions of the long run and the short run. In the short run, only some inputs can be adjusted, while in the long run all inputs can be adjusted. Traditionally, economists viewed labor as quickly adjustable and capital equipment as more difficult to adjust. That is certainly right for airlines—obtaining new aircraft is a very slow process—for large complex factories, and for relatively low-skilled, and hence substitutable, labor. On the other hand, obtaining workers with unusual skills is a slower process than obtaining warehouse or office space. Generally speaking, the long-run inputs are those that are expensive to adjust quickly, while the short-run factors can be adjusted in a relatively short time frame. What factors belong in which category is dependent on the context or application under consideration.
Key Takeaways
• Firms transform inputs into outputs.
• The functional relationship between inputs and outputs is the production function.
• The Cobb-Douglas production function is the product of the inputs raised to powers and comes in the form $$f( x 1 , x 2 ,…, x n )= a 0 x 1 a 1 x 2 a 2 … x n a n$$ for positive constants $$a_{1}, \ldots, \text { a_{n}. }$$.
• An isoquant is a curve or surface that traces out the inputs leaving the output constant.
• The fixed-proportions production function comes in the form $$f( x 1 , x 2 ,…, x n )=min { a 1 x 1 , a 2 x 2 , …, a n x n }$$.
• Fixed proportions make the inputs “perfect complements.”
• Two inputs K and L are perfect substitutes in a production function f if they enter as a sum; that is, $$f\left(K, L, x_{3}, \ldots, x_{n}\right)$$ = $$g\left(K + cL, x_{3}, \ldots, x_{n}\right)$$, for a constant c.
• The marginal product of an input is just the derivative of the production function with respect to that input. An important aspect of marginal products is that they are affected by the level of other inputs.
• The value of the marginal product of an input is just the marginal product times the price of the output. If the value of the marginal product of an input exceeds the cost of that input, it is profitable to use more of the input.
• Some inputs are more readily changed than others.
• In the short run, only some inputs can be adjusted, while in the long run all inputs can be adjusted.
• Traditionally, economists viewed labor as quickly adjustable and capital equipment as more difficult to adjust.
• Generally speaking, the long-run inputs are those that are expensive to adjust quickly, while the short-run factors can be adjusted in a relatively short time frame.
EXERCISE
1. For the Cobb-Douglas production function, suppose there are two inputs K and L, and the sum of the exponents is one. Show that, if each input is paid the value of the marginal product per unit of the input, the entire output is just exhausted. That is, for this production function, show $$K ∂f ∂K +L ∂f ∂L =f(K,L)$$. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/09%3A_Producer_Theory-_Costs/9.02%3A_Production_Functions.txt |
Learning Objectives
• If firms maximize profits, how will they behave?
Consider an entrepreneur who would like to maximize profit, perhaps by running a delivery service. The entrepreneur uses two inputs, capital K (e.g., trucks) and labor L (e.g., drivers), and rents the capital at cost r per dollar of capital. The wage rate for drivers is w. The production function is F (K, L)—that is, given inputs K and L, the output is F (K, L). Suppose p is the price of the output. This gives a profit of Economists often use the symbol $$π$$, the Greek letter “pi,” to stand for profit. There is little risk of confusion because economics doesn’t use the ratio of the circumference to the diameter of a circle very often. On the other hand, the other two named constants, Euler’s e and i, the square root of -1, appear fairly frequently in economic analysis.
$π=pF(K,L)−rK−wL.$
First, consider the case of a fixed level of K. The entrepreneur chooses L to maximize profit. The value L* of L that maximizes the function π must satisfy
$0= ∂π ∂L =p ∂F ∂L (K,L*)−w$.
This expression is known as a first-order condition, a mathematical condition for optimization stating that the first derivative is zero.It is possible that L = 0 is the best that an entrepreneur can do. In this case, the derivative of profit with respect to L is not necessarily zero. The first-order condition instead would be either $$0= ∂π ∂L$$, or L = 0, and $$0≥ ∂π ∂L$$. The latter pair of conditions reflects the logic that either the derivative is zero and we are at a maximum, or L = 0, in which case a small increase in L must not cause π to increase. The first-order condition recommends that we add workers to the production process up to the point where the last worker’s marginal product is equal to his wage (or cost).
Figure 9.4 Profit-maximizing labor input
The second property is known as the second-order condition, a mathematical condition for maximization stating that the second derivative is nonpositive.The orders refer to considering small, but positive, terms Δ, which are sent to zero to reach derivatives. The value $$\Delta^{2}$$, the second-order term, goes to zero faster than Δ, the first-order term. It is expressed as
$0≥ ∂ 2 π (∂L) 2 =p ∂ 2 F (∂L) 2 (K,L*).$
This is enough of a mathematical treatment to establish comparative statics on the demand for labor. Here we treat the choice L* as a function of another parameter—the price p, the wage w, or the level of capital K. For example, to find the effect of the wage on the labor demanded by the entrepreneur, we can write
$0=p ∂F ∂L (K,L*(w))−w.$
This expression recognizes that the choice L* that the entrepreneur makes satisfies the first-order condition and results in a value that depends on w. But how does it depend on w? We can differentiate this expression to obtain
$0=p ∂ 2 F (∂L) 2 (K,L*(w))L * ′ (w)−1,$
or
$L * ′ (w)= 1 p ∂ 2 F (∂L) 2 (K,L*(w)) ≤0.$
The second-order condition enables one to sign the derivative. This form of argument assumes that the choice L* is differentiable, which is not necessarily true.
Digression: In fact, there is a form of argument that makes the point without calculus and makes it substantially more general. Suppose w1 < w2 are two wage levels and that the entrepreneur chooses L1 when the wage is w1 and L2 when the wage is w2. Then profit maximization requires that these choices are optimal. In particular, when the wage is w1, the entrepreneur earns higher profit with L1 than with L2:
$\mathrm{pf}(\mathrm{K}, \mathrm{L} 1)-\mathrm{rK}-\mathrm{w} 1 \mathrm{L} 1 \geq \mathrm{pf}(\mathrm{K}, \mathrm{L} 2)-\mathrm{rK}-\mathrm{w} 1 \mathrm{L} 2$
When the wage is w2, the entrepreneur earns higher profit with L2 than with L1:
$\mathrm{pf}(\mathrm{K}, \mathrm{L} 2)-\mathrm{rK}-\mathrm{w} 2 \mathrm{L} 2 \geq \mathrm{pf}(\mathrm{K}, \mathrm{L} 1)-\mathrm{rK}-\mathrm{w} 2 \mathrm{L} 1$.
The sum of the left-hand sides of these two expressions is at least as large as the sum of the right-hand side of the two expressions:
$\mathrm{pf}(\mathrm{K}, \mathrm{L} 1)-\mathrm{rK}-\mathrm{w} 1 \mathrm{L} 1+\mathrm{pf}(\mathrm{K}, \mathrm{L} 2)-\mathrm{rK}-\mathrm{w} 2 \mathrm{L} 2 \geq \mathrm{pf}(\mathrm{K}, \mathrm{L} 1)-\mathrm{rK}-\mathrm{w} 2 \mathrm{L} 1+\mathrm{pf}(\mathrm{K}, \mathrm{L} 2)-\mathrm{rK}-\mathrm{w} 1 \mathrm{L} 2$
A large number of terms cancel to yield the following:
$-w 1 L 1-w 2 L 2 \geq-w 2 L 1-w 1 L 2$
This expression can be rearranged to yield the following:
$(w 1-w 2)(L 2-L 1) \geq 0$
This shows that the higher labor choice must be associated with the lower wage. This kind of argument, sometimes known as a revealed preference kind of argument, states that choice implies preference. It is called “revealed preference” because choices by consumers were the first place the type of argument was applied. It can be very powerful and general, because issues of differentiability are avoided. However, we will use the more standard differentiability-type argument, because such arguments are usually more readily constructed.
The effect of an increase in the capital level K on the choice by the entrepreneur can be calculated by considering L* as a function of the capital level K:
$0=p \partial F \partial L\left(K, L^{*}(K)\right)-w$
Differentiating this expression with respect to K, we obtain
$0=p \partial 2 F \partial K \partial L\left(K, L^{*}(K)\right)+p \partial 2 F(\partial L) 2\left(K, L^{*}(K)\right) L^{*}(K)$
or
$L^{* \prime}(K)=-\partial 2 F \partial K \partial L\left(K, L^{*}(K)\right) \partial 2 F(\partial L) 2\left(K, L^{*}(K)\right)$
We know the denominator of this expression is not positive, thanks to the second-order condition, so the unknown part is the numerator. We then obtain the conclusion that
an increase in capital increases the labor demanded by the entrepreneur if $$∂ 2 F ∂K∂L (K,L*(K))>0$$, and decreases the labor demanded if $$\partial 2 \mathrm{F} \partial \mathrm{K} \partial \mathrm{L}\left(\mathrm{K}, \mathrm{L}^{*}(\mathrm{K})\right)<0$$.
This conclusion looks like gobbledygook but is actually quite intuitive. Note that $$$\partial 2 \mathrm{F} \partial \mathrm{K} \partial \mathrm{L}\left(\mathrm{K}, \mathrm{L}^{*}(\mathrm{K})\right)>0$$$ means that an increase in capital increases the derivative of output with respect to labor; that is, an increase in capital increases the marginal product of labor. But this is, in fact, the definition of a complement! That is, $$$\partial 2 \mathrm{F} \partial \mathrm{K} \partial \mathrm{L}\left(\mathrm{K}, \mathrm{L}^{*}(\mathrm{K})\right)>0$$$ means that labor and capital are complements in production—an increase in capital increases the marginal productivity of labor. Thus, an increase in capital will increase the demand for labor when labor and capital are complements, and it will decrease the demand for labor when labor and capital are substitutes.
This is an important conclusion because different kinds of capital may be complements or substitutes for labor. Are computers complements or substitutes for labor? Some economists consider that computers are complements to highly skilled workers, increasing the marginal value of the most skilled, but substitutes for lower-skilled workers. In academia, the ratio of secretaries to professors has fallen dramatically since the 1970s as more and more professors are using machines to perform secretarial functions. Computers have increased the marginal product of professors and reduced the marginal product of secretaries, so the number of professors rose and the number of secretaries fell.
The revealed preference version of the effect of an increase in capital is to posit two capital levels, K1 and K2, with associated profit-maximizing choices L1 and L2. The choices require, for profit maximization, that
$\mathrm{pF}(\mathrm{K} 1, \mathrm{L} 1)-\mathrm{r} \mathrm{K} 1-\mathrm{w} \mathrm{L} 1 \geq \mathrm{pF}(\mathrm{K} 1, \mathrm{L} 2)-\mathrm{r} \mathrm{K} 1-\mathrm{w} \mathrm{L} 2$
and
$\mathrm{pF}(\mathrm{K} 2, \mathrm{L} 2)-\mathrm{r} \mathrm{K} 2-\mathrm{w} \mathrm{L} 2 \geq \mathrm{pF}(\mathrm{K} 2, \mathrm{L} 1)-\mathrm{r} \mathrm{K} 2-\mathrm{w} \mathrm{L} 1$
Again, adding the left-hand sides together produces a result at least as large as the sum of the right-hand sides:
$\operatorname{pf}(K 1, L 1)-K K-w L 1+p F(K 2, L 2)-r K 2-w L 2 \geq p F(K 2, L 1)-r K 2-w L 1+p F(K 1, L 2)-r K 1-w L 2$
Eliminating redundant terms yields
$\mathrm{pF}(\mathrm{K} 1, \mathrm{L} 1)+\mathrm{pF}(\mathrm{K} 2, \mathrm{L} 2) \geq \mathrm{pF}(\mathrm{K} 2, \mathrm{L} 1)+\mathrm{pF}(\mathrm{K} 1, \mathrm{L} 2)$
or
$\mathrm{F}(\mathrm{K} 2, \mathrm{L} 2)-\mathrm{F}(\mathrm{K} 1, \mathrm{L} 2) \geq \mathrm{F}(\mathrm{K} 2, \mathrm{L} 1)-\mathrm{F}(\mathrm{K} 1, \mathrm{L} 1)$
or
$$∫ K 1 K 2 ∂F ∂K (x, L 2 )dx ≥ ∫ K 1 K 2 ∂F ∂K (x, L 1 )dx$$,Here we use the standard convention that $$\int a b \ldots d x=-\int b a \ldots d x$$.
or
$∫ K 1 K 2 ∂F ∂K (x, L 2 )− ∂F ∂K (x, L 1 )dx≥0 ,$
and finally
$∫ K 1 K 2 ∫ L 1 L 2 ∂ 2 F ∂K∂L (x,y) dy dx≥0 .$
Thus, if $$\mathrm{K}_{2}>\mathrm{K}_{1}$$ and $$∂ 2 F ∂K∂L (K,L)>0$$ for all K and L, then $$\mathrm{L}_{2} \geq \mathrm{L}_{1}$$ ; that is, with complementary inputs, an increase in one input increases the optimal choice of the second input. In contrast, with substitutes, an increase in one input decreases the other input. While we still used differentiability of the production function to carry out the revealed preference argument, we did not need to establish that the choice L* was differentiable to perform the analysis.
Example (Labor demand with the Cobb-Douglas production function): The Cobb-Douglas production function has the form $$F(K,L)=A K α L β$$, for constants A, α, and β, all positive. It is necessary for β < 1 for the solution to be finite and well defined. The demand for labor satisfies
$0=p ∂F ∂L (K,L*(K))−w=p β A K α L * β−1 −w,$
or
$L^{*}=(p \beta A K a w) 11-\beta$
When $$α + β = 1$$, L is linear in capital. Cobb-Douglas production is necessarily complementary; that is, an increase in capital increases labor demanded by the entrepreneur.
Key Takeaways
• Profit maximization arises when the derivative of the profit function with respect to an input is zero. This property is known as a first-order condition.
• Profit maximization arises with regards to an input when the value of the marginal product is equal to the input cost.
• A second characteristic of a maximum is that the second derivative is negative (or nonpositive). This property is known as the second-order condition.
• Differentiating the first-order condition permits one to calculate the effect of a change in the wage on the amount of labor hired.
• Revealed preference arguments permit one to calculate comparative statics without using calculus, under more general assumptions.
• An increase in capital will increase the demand for labor when labor and capital are complements, and it will decrease the demand for labor when labor and capital are substitutes.
• Cobb-Douglas production functions are necessarily complements; hence, any one input increases as the other inputs increase.
EXERCISES
1. For the fixed-proportions production function min {K, L}, find labor demand when capital is fixed at K.
2. The demand for hamburgers has a constant elasticity of 1 of the form $$x(p)=8,000 p-1$$. Each entrant in this competitive industry has a fixed cost of $2,000 and produces x hamburgers per year, where x is the amount of meat in pounds. 1. If the price of meat is$2 per pound, what is the long-run supply of hamburgers?
2. Compute the equilibrium number of firms, the quantity supplied by each firm, and the market price of hamburgers.
3. Find the short-run industry supply. Does it have constant elasticity?
3. A company that produces software needs two inputs, programmers (x) at a price of p and computers (y) at a price of r. The output is given by $$T=4 x^{1 / 3} y^{1 / 3}$$, measured in pages of code.
1. What is the marginal cost?
2. Now suppose each programmer needs two computers to do his job. What ratio of p and r would make this input mix optimal?
4. A toy factory costs \$2 million to construct, and the marginal cost of the qth toy is $$\max \left[10, q^{2} / 1,000\right]$$.
1. What is average total cost?
2. What is short-run supply?
3. What is the long-run competitive supply of toys? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/09%3A_Producer_Theory-_Costs/9.03%3A_Profit_Maximization.txt |
Learning Objectives
• If a firm faces constraints on its behavior, how can we measure the costs of these constraints?
When capital K can’t be adjusted in the short run, it creates a constraint, on the profit available, to the entrepreneur—the desire to change K reduces the profit available to the entrepreneur. There is no direct value of capital because capital is fixed. However, that doesn’t mean we can’t examine its value. The value of capital is called a shadow value, which refers to the value associated with a constraint. Shadow value is well-established jargon.
What is the shadow value of capital? Let’s return to the constrained, short-run optimization problem. The profit of the entrepreneur is
$π=pF(K,L)−rK−wL.$
The entrepreneur chooses the value L* to maximize profit; however, he is constrained in the short run with the level of capital inherited from a past decision. The shadow value of capital is the value of capital to profit, given the optimal decision L*. Because $$0= ∂π ∂L =p ∂F ∂L (K,L*)−w$$, the shadow value of capital is $$dπ(K,L*) dK = ∂π(K,L*) ∂K =p ∂F ∂K (K,L*)−r.$$
Note that this value could be negative if the entrepreneur might like to sell some capital but can’t, perhaps because it is installed in the factory.
Every constraint has a shadow value. The term refers to the value of relaxing the constraint. The shadow value is zero when the constraint doesn’t bind; for example, the shadow value of capital is zero when it is set at the profit-maximizing level. Technology binds the firm; the shadow value of a superior technology is the increase in profit associated with it. For example, parameterize the production technology by a parameter a, so that aF(K, L) is produced. The shadow value of a given level of a is, in the short run,
$dπ(K,L*) da = ∂π(K,L*) ∂a =pF(K,L*).$
A term is vanishing in the process of establishing the shadow value. The desired value L* varies with the other parameters like K and a, but the effect of these parameters on L* doesn’t appear in the expression for the shadow value of the parameter because $$0= ∂π ∂L$$ at L*.
Key Takeaways
• When an input is fixed, its marginal value is called a shadow value.
• A shadow value can be negative when an input is fixed at too high a level.
• Every constraint has a shadow value. The term refers to the value of relaxing a constraint. The shadow value is zero when the constraint doesn’t bind.
• The effect of a constraint on terms that are optimized may be safely ignored in calculating the shadow value.
9.05: Input Demand
Learning Objectives
• How much will firms buy?
• How do they respond to input price changes?
Over a long period of time, an entrepreneur can adjust both the capital and the labor used at the plant. This lets the entrepreneur maximize profit with respect to both variables K and L. We’ll use a double star, **, to denote variables in their long-run solution. The approach to maximizing profit over two c separately with respect to each variable, thereby obtaining the conditions
$0=p \partial F \partial L\left(K^{* *}, L^{* *}\right)-w$
and
$0=\mathrm{p} \partial \mathrm{F} \partial \mathrm{K}\left(\mathrm{K}^{* *}, \mathrm{L}^{* *}\right)-\mathrm{r}$
We see that, for both capital and labor, the value of the marginal product is equal to the purchase price of the input.
It is more challenging to carry out comparative statics exercises with two variables, and the general method won’t be developed here.If you want to know more, the approach is to arrange the two equations as a vector with x = (K, L), z = (r/p, w/p), so that 0= F ′ (x**)−z, and then differentiate to obtain $$d x^{* *}=\left(F^{\prime \prime}\left(x^{* *}\right)\right)-1 d z$$, which can then be solved for each comparative static. However, we can illustrate one example as follows.
Example: The Cobb-Douglas production function implies choices of capital and labor satisfying the following two first-order conditions:It is necessary for $$a+\beta<1$$ for the solution to be finite and well defined.
0=\mathrm{p} \partial \mathrm{F} \partial \mathrm{L}\left(\mathrm{K}^{* *}, \mathrm{L}^{* *}\right)-\mathrm{w}=\mathrm{p} \beta \mathrm{AK}^{* *} \mathrm{a} \mathrm{L}^{* *} \beta-1-\mathrm{w},
0=\mathrm{p} \partial \mathrm{F} \partial \mathrm{K}\left(\mathrm{K}^{* *}, \mathrm{L}^{* *}\right)-\mathrm{r}=\mathrm{pa} A \mathrm{K}^{* *} \mathrm{a}-1 \mathrm{L}^{* *} \beta-\mathrm{r}
To solve these expressions, first rewrite them to obtain
$\mathrm{w}=\mathrm{p} \beta \mathrm{AK}^{* *} \mathrm{a} \mathrm{L}^{* *} \beta-1$
and
$r=p q A K^{* *} a-1 L^{* *} \beta$
Then divide the first expression by the second expression to yield
$\mathrm{w} \mathrm{r}=\beta \mathrm{K}^{* *} \mathrm{al}^{* *}$
or
$\mathrm{K}^{* *}=\mathrm{a} \mathrm{w} \mathrm{Br} \mathrm{L}^{* *}$
This can be substituted into either equation to obtain
$L^{* *}=(\text { Ap a a } \beta 1-a \text { r a } w 1-a \text { ) } 11-a-\beta$
and
$\mathrm{K}^{* *}=(\mathrm{Ap} \mathrm{a} 1-\beta \beta \beta \mathrm{r} 1-\beta \mathrm{w} \beta) 11-\mathrm{a}-\beta$ .
While these expressions appear complicated, the dependence on the output price p, and the input prices r and w, is quite straightforward.
How do equilibrium values of capital and labor respond to a change in input prices or output price for the Cobb-Douglas production function? It is useful to cast these changes in percentage terms. It is straightforward to demonstrate that both capital and labor respond to a small percentage change in any of these variables with a constant percentage change.
An important insight of profit maximization is that it implies minimization of costs of yielding the chosen output; that is, profit maximization entails efficient production.
The logic is straightforward. The profit of an entrepreneur is revenue minus costs, and the revenue is price times output. For the chosen output, then, the entrepreneur earns the revenue associated with the output, which is fixed since we are considering only the chosen output, minus the costs of producing that output. Thus, for the given output, maximizing profits is equivalent to maximizing a constant (revenue) minus costs. Since maximizing –C is equivalent to minimizing C, the profit-maximizing entrepreneur minimizes costs. This is important because profit-maximization implies not being wasteful in this regard: A profit-maximizing entrepreneur produces at least cost.
Figure 9.5 Tangency and Isoquants
What point on an isoquant minimizes total cost? The answer is the point associated with the lowest (most southwest) isocost that intersects the isoquant. This point will be tangent to the isoquant and is denoted by a star. At any lower cost, it isn’t possible to produce the desired quantity. At any higher cost, it is possible to lower cost and still produce the quantity.
The fact that cost minimization requires a tangency between the isoquant and the isocost has a useful interpretation. The slope of the isocost is minus the ratio of input prices. The slope of the isoquant measures the substitutability of the inputs in producing the output. Economists call this slope the marginal rate of technical substitution, which is the amount of one input needed to make up for a decrease in another input while holding output constant. Thus, one feature of cost minimization is that the input price ratio equals the marginal rate of technical substitution.
Key Takeaways
• In the long run, all inputs can be optimized, which leads to multiple first-order conditions.
• The solution can be illustrated graphically and computed explicitly for Cobb-Douglas production functions.
• An important implication of profit maximization is cost minimization—output is produced by the most efficient means possible.
• Cost minimization occurs where the ratio of the input prices equals the slope of the isocost curve, known as the marginal rate of technical substitution, which is the amount of one input needed to make up for a decrease in another input and hold output constant.
EXERCISE
For the Cobb-Douglas production function $$F(K, L)=A K \text { a } L \beta$$, show that $$r L^{* *} \partial L^{* *} \partial r=-\alpha 1-\alpha-\beta$$, $$w L^{* *} \partial L^{* *} \partial w=-1-\alpha 1-\alpha-\beta$$, $$p L^{* *} \partial L^{* *} \partial p=11-\alpha-\beta, r K^{* *} \partial K^{* *} \partial r=-1-\beta 1-\alpha-\beta$$, $$w {K}^* * \partial {K }^* * \partial w=-\beta 1-a-\beta$$ and $$p K^{* *} \partial K^{* *} \partial p=11-\alpha-\beta$$. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/09%3A_Producer_Theory-_Costs/9.04%3A_The_Shadow_Value.txt |
Learning Objectives
• What are the different ways of measuring costs, and how are they related to the amount of time the firm has to change its behavior?
How much does it cost to produce a given quantity q? We already have a detailed answer to this question, but now we need to focus less on the details and more on the “big picture.” First, let’s focus on the short run and suppose L is adjustable in the short run but K is not. Then the short-run total cost of producing q,—that is, the total cost of output with only short-term factors varying—given the capital level, is
$\operatorname{SRTC}(\mathrm{q} | \mathrm{K})=\min \mathrm{L} \text { rK+wL, over all L satisfying } \mathrm{F}(\mathrm{K}, \mathrm{L}) \geq \mathrm{q}$
In words, this equation says that the short-run total cost of the quantity q, given the existing level K, is the minimum cost, where L gets to vary (which is denoted by “min over L”) and where the L considered is large enough to produce q. The vertical line | is used to indicate a condition or conditional requirement; here |K indicates that K is fixed. The minimum lets L vary but not K. Finally, there is a constraint $$F(K, L) \geq q$$, which indicates that one must be able to produce q with the mix of inputs because we are considering the short-run cost of q.
The short-run total cost of q, given K, has a simple form. First, since we are minimizing cost, the constraint F(K, L) ≥ q will be satisfied with equality, F(K, L) = q. This equation determines L, since K is fixed; that is, $$$\mathrm{F}(\mathrm{K}, \mathrm{L} \mathrm{S}(\mathrm{q}, \mathrm{K}))=\mathrm{q}$$$ gives the short-run value of L, LS(q, K). Finally, the cost is then $$$\mathrm{F}(\mathrm{K}, \mathrm{L} \mathrm{S}(\mathrm{q}, \mathrm{K}))=\mathrm{q}$$$.
The short-run marginal cost, given K, is just the derivative of short-run total cost with respect to output, q. To establish the short-run marginal cost, note that the equation $$F(K, L)=q$$ implies
$$\operatorname{aF} \partial \mathrm{L}(\mathrm{K}, \mathrm{L} \mathrm{S}(\mathrm{q}, \mathrm{K})) \mathrm{d} \mathrm{L}=\mathrm{d} \mathrm{q}$$
or
$\text { dL dq } | \mathrm{F}=\mathrm{q}=1 \text { dF } \partial \mathrm{L}(\mathrm{K}, \mathrm{L} \mathrm{S}(\mathrm{q}, \mathrm{K}))$
The tall vertical line, subscripted with F = q, is used to denote the constraint F(K, L) = q that is being differentiated. Thus, the short-run marginal cost is
$\operatorname{SRMC}(\mathrm{q} | \mathrm{K})=\operatorname{SRT} \mathrm{C}^{\prime}(\mathrm{q})=\mathrm{d} \mathrm{dq}(\mathrm{rK}+\mathrm{wL})=\mathrm{w} \text { dL } \mathrm{dq} | \mathrm{F}=\mathrm{q}=\mathrm{w} \partial \mathrm{F} \partial \mathrm{L}(\mathrm{K}, \mathrm{L} \mathrm{S}(\mathrm{q}, \mathrm{K}))$
There are three other short-run costs we require to complete the analysis. First, there is the short-run average cost of production that we obtain by dividing the total cost by the quantity produced:
$\operatorname{SRAC}(q | \mathrm{K})=\operatorname{SRTC}(\mathrm{q} | \mathrm{K}) \mathrm{q}$
Second, there is the short-run variable cost that is the total cost minus the cost of producing zero units—that is, minus the fixed cost—which in this case is rK. Finally, we need one more short-run cost: the short-run average variable cost. The short-run average variable cost is the short-run variable cost divided by quantity, which is given,
$\operatorname{SRAVC}(\mathrm{q} | \mathrm{K})=\operatorname{SRT}(\mathrm{q} | \mathrm{K})-\operatorname{SRTC}(0 | \mathrm{K}) \mathrm{q}=\mathrm{w} \mathrm{L} \mathrm{S}(\mathrm{q} | \mathrm{K}) \mathrm{q}$
The short-run average variable cost is the average cost ignoring the investment in capital equipment.
The short-run average cost could also be called the short-run average total cost, since it is the average of the total cost per unit of output, but “average total” is a bit of an oxymoron.An oxymoron is a word or phrase that is self-contradictory, like “jumbo shrimp,” “stationary orbit,” “virtual reality,” “modern tradition,” or “pretty ugly.” Oxymoron comes from the Greek “oxy,” meaning sharp, and “moros,” meaning dull. Thus, oxymoron is itself an oxymoron, so an oxymoron is self-descriptive. Another word that is self-descriptive is “pentasyllabic.” Consequently, when total, fixed, or variable is not specified, the convention is to mean total. Note that the marginal variable cost is the same as the marginal total costs, because the difference between variable cost and total cost is a constant—the cost of zero production, also known as the short-run fixed cost of production.
At this point, we have identified four distinct costs, all of which are relevant to the short run. These are the total cost, the marginal cost, the average cost, and the average variable cost. In addition, all of these costs may be considered in the long run as well. There are three differences in the long run. First, the long run lets all inputs vary, so the long-run total cost is the total cost of output with all factors varying. In this case, $$\text { LRTC(q) }=\min \mathrm{L}, \mathrm{K} \mathrm{rK}+\mathrm{wL}$$ over all L and K combinations satisfying $$F(K, L) \geq q$$.
Second, since all inputs can vary, the long-run cost isn’t conditioned on K. Finally, the long-run average variable cost is the long-run total cost divided by output; it is also known as the long-run average total cost. Since a firm could use no inputs in the long run and thus incur no costs, the cost of producing zero is zero. Therefore, in the long run, all costs are variable, and the long-run average variable cost is the long-run average total cost divided by quantity.
Note that the easiest way to find the long-run total cost is to minimize the short-run total cost over K. Since this is a function of one variable, it is straightforward to identify the K that minimizes cost, and then plug that K into the expression for total cost.
One might want to distinguish the very short run from the short run, the medium run, the long run, and the very long run.. But a better approach is to view adjustment as a continuous process, with a gradual easing of the constraints. Faster adjustment costs more. Continuous adjustment is a more advanced topic, requiring an Euler equation approach.
Key Takeaways
• The short-run total cost is the minimum cost of producing a given quantity minimized over the inputs variable in the short run. Sometimes the word total is omitted.
• The short-run fixed cost is the short-run total cost at a zero quantity.
• The short-run marginal cost, given K, is just the derivative of the short-run total cost with respect to quantity.
• The short-run average cost is the short-run total cost divided by quantity.
• The short-run average variable cost is the short-run total cost minus the short-run fixed cost, all divided by quantity.
• Marginal variable cost is the same as the marginal total costs, because the difference between total cost and variable cost is the fixed cost of production, a constant.
• The long-run total cost is the minimum cost of producing a given quantity minimized over all inputs. Sometimes the word total is omitted.
• The long-run fixed cost is zero.
• The long-run marginal cost is the derivative of the long-run total cost with respect to quantity.
• The long-run average cost is the long-run total cost divided by quantity.
• The long-run average variable cost equals the long-run average cost.
EXERCISES
1. For the Cobb-Douglas production function $$\mathrm{F}(\mathrm{K}, \mathrm{L})=\mathrm{A} \mathrm{K} \mathrm{a} \mathrm{L} \beta$$, with $$a+\beta<1$$, K fixed in the short run but not in the long run, and cost r of capital and w for labor, show
$\operatorname{SRTC}(\mathrm{q} | \mathrm{K})=\mathrm{rK}+\mathrm{w}(\mathrm{q} \mathrm{A} \mathrm{K} \mathrm{a}) 1 \beta$ ,
$\operatorname{SRAVC}(\mathrm{q} | \mathrm{K})=\mathrm{w} \mathrm{q} 1-\beta \beta(\mathrm{A} \mathrm{K} \mathrm{a}) 1 \beta$ ,
$\operatorname{SRMC}(\mathrm{q} | \mathrm{K})=\mathrm{w} \quad \mathrm{q} 1-\beta \beta \beta(\mathrm{AK} \mathrm{a}) 1 \beta$,
$\angle R T C(q | K)=((a \beta) \beta a+\beta+(\beta a) a a+\beta) w \beta a+\beta r a a+\beta(q A) 1 a+\beta$ .
2. Consider a cost function of producing an output q of the form $$c(a)=q^{2}+2 q+16$$. Determine the following:
1. Marginal cost
2. Average cost
3. Average variable cost
Graph the long-run supply curve, assuming the cost function is for a single plant and can be replicated without change. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/09%3A_Producer_Theory-_Costs/9.06%3A_Myriad_Costs.txt |
Learning Objectives
• How does a competitive firm respond to price changes?
In this section, we consider a competitive firm (or entrepreneur) that can’t affect the price of output or the prices of inputs. How does such a competitive firm respond to price changes? When the price of the output is p, the firm earns profits $$π=pq−c(q|K)$$, where $$c (q|K )$$ is the total cost of producing, given that the firm currently has capital K. Assuming that the firm produces at all, it maximizes profits by choosing the quantity qs satisfying $$0=p-c^{\prime}(q s | K)$$; which is the quantity where price equals marginal cost. However, this is a good strategy only if producing a positive quantity is desirable, so that $$\text { p } q s-c(q s | K) \geq-c(0, K)$$, which maybe rewritten as $$p \geq c(q s | K)-c(0, K) q s$$. The right-hand side of this inequality is the average variable cost of production, and thus the inequality implies that a firm will produce, provided price exceeds the average variable cost. Thus, the profit-maximizing firm produces the quantity qs, where price equals marginal cost, provided price is as large as minimum average variable cost. If price falls below minimum average variable cost, the firm shuts down.
The behavior of the competitive firm is illustrated in Figure 10.1. The thick line represents the choice of the firm as a function of the price, which is on the vertical axis. Thus, if the price is below the minimum average variable cost (AVC), the firm shuts down. When price is above the minimum average variable cost, the marginal cost gives the quantity supplied by the firm. Thus, the choice of the firm is composed of two distinct segments: the marginal cost, where the firm produces the output such that price equals marginal cost; and shutdown, where the firm makes a higher profit, or loses less money, by producing zero.
Figure 10.1 also illustrates the average total cost, which doesn’t affect the short-term behavior of the firm but does affect the long-term behavior because, when price is below average total cost, the firm is not making a profit. Instead, it would prefer to exit over the long term. That is, when the price is between the minimum average variable cost and the minimum average total cost, it is better to produce than to shut down; but the return on capital was below the cost of capital. With a price in this intermediate area, a firm would produce but would not replace the capital, and thus would shut down in the long term if the price were expected to persist. As a consequence, minimum average total cost is the long-run “shutdown” point for the competitive firm. (Shutdown may refer to reducing capital rather than literally setting capital to zero.) Similarly, in the long term, the firm produces the quantity where the price equals the long-run marginal cost.
Figure 10.1 Short-run supply
$0=\mathrm{d} \mathrm{dq} \quad \mathrm{C}(\mathrm{q}) \mathrm{q}=\mathrm{C}^{\prime}(\mathrm{q}) \mathrm{q}-\mathrm{C}(\mathrm{q}) \mathrm{q} 2$
But this can be rearranged to imply $$C^{\prime}(q)=C(q) q$$, where marginal cost equals average cost at the minimum of average cost.
The long-run marginal cost has a complicated relationship to short-run marginal cost. The problem in characterizing the relationship between long-run and short-run marginal costs is that some costs are marginal in the long run that are fixed in the short run, tending to make long-run marginal costs larger than short-run marginal costs. However, in the long run, the assets can be configured optimally, While some assets are fixed in the short run, and this optimal configuration tends to make long-run costs lower.
Instead, it is more useful to compare the long-run average total costs and short-run average total costs. The advantage is that capital costs are included in short-run average total costs. The result is a picture like Figure 10.2.
Figure 10.2 Average and marginal costs
In Figure 10.3, the quantity produced is larger than the quantity that minimizes long-run average total cost. Consequently, as is visible in the figure, the quantity where short-run average cost equals long-run average cost does not minimize short-run average cost. What this means is that a factory designed to minimize the cost of producing a particular quantity won’t necessarily minimize short-run average cost. Essentially, because the long-run average total cost is increasing, larger plant sizes are getting increasingly more expensive, and it is cheaper to use a somewhat “too small” plant and more labor than the plant size with the minimum short-run average total cost. However, this situation wouldn’t likely persist indefinitely because, as we shall see, competition tends to force price to the minimum long-run average total cost. At this point, then, we have the three-way equality between long-run average total cost, short-run average total cost, and short-run marginal cost.
Figure 10.3 Increased plant size
Key Takeaways
• The profit-maximizing firm produces the quantity where price equals marginal cost, provided price is as large as minimum average variable cost. If price falls below minimum average variable cost, the firm shuts down.
• When price falls below short-run average cost, the firm loses money. If price is above average variable cost, the firm loses less money than it would by shutting down; once price falls below short-run average variable cost, shutting down entails smaller losses than continuing to operate.
• The minimum of average cost occurs at the point where marginal cost equals average cost.
• If price is below long-run average cost, the firm exits in the long run.
• Every point on long-run average total cost must be equal to a point on some short-run average total cost.
• The quantity where short-run average cost equals long-run average cost need not minimize short-run average cost if long-run average cost isn’t constant.
EXERCISE
1. Suppose a company has total cost given by $$r K+q 22 K$$, where capital K is fixed in the short run. What is short-run average total cost and what is marginal cost? Plot these curves. For a given quantity q0, what level of capital minimizes total cost? What is the minimum average total cost of q0? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/10%3A_Producer_Theory-_Dynamics/10.01%3A_Reactions_of_Competitive_Firms.txt |
Learning Objectives
• When firms get bigger, when do average costs rise or fall?
• How does size relate to profit?
An economy of scale—that larger scale lowers cost—arises when an increase in output reduces average costs. We met economies of scale and its opposite, diseconomies of scale, in the previous section, with an example where long-run average total cost initially fell and then rose, as quantity was increased.
What makes for an economy of scale? Larger volumes of productions permit the manufacture of more specialized equipment. If I am producing a million identical automotive taillights, I can spend \$50,000 on an automated plastic stamping machine and only affect my costs by 5 cents each. In contrast, if I am producing 50,000 units, the stamping machine increases my costs by a dollar each and is much less economical.
Indeed, it is somewhat more of a puzzle to determine what produces a diseconomy of scale. An important source of diseconomies is managerial in nature—organizing a large, complex enterprise is a challenge, and larger organizations tend to devote a larger percentage of their revenues to management of the operation. A bookstore can be run by a couple of individuals who rarely, if ever, engage in management activities, where a giant chain of bookstores needs finance, human resource, risk management, and other “overhead” type expenses just in order to function. Informal operation of small enterprises is replaced by formal procedural rules in large organizations. This idea of managerial diseconomies of scale is reflected in the aphorism “A platypus is a duck designed by a committee.”
In his influential 1975 book The Mythical Man-Month, IBM software manager Fred Books describes a particularly severe diseconomy of scale. Adding software engineers to a project increases the number of conversations necessary between pairs of individuals. If there are n engineers, there are $$½n (n – 1)$$ pairs, so that communication costs rise at the square of the project size. This is pithily summarized in Brooks’ Law: “Adding manpower to a late software project makes it later.”
Another related source of diseconomies of scale involves system slack. In essence, it is easier to hide incompetence and laziness in a large organization than in a small one. There are a lot of familiar examples of this insight, starting with the Peter Principle, which states that people rise in organizations to the point of their own incompetence, meaning that eventually people cease to do the jobs that they do well.Laurence Johnston Peter (1919–1990). The notion that slack grows as an organization grows implies a diseconomy of scale.
Generally, for many types of products, economies of scale from production technology tend to reduce average cost, up to a point where the operation becomes difficult to manage. Here the diseconomies tend to prevent the firm from economically getting larger. Under this view, improvements in information technologies over the past 20 years have permitted firms to get larger and larger. While this seems logical, in fact firms aren’t getting that much larger than they used to be; and the share of output produced by the top 1,000 firms has been relatively steady; that is, the growth in the largest firms just mirrors world output growth.
Related to an economy of scale is an economy of scope. An economy of scope is a reduction in cost associated with producing several distinct goods. For example, Boeing, which produces both commercial and military jets, can amortize some of its research and development (R&D) costs over both types of aircraft, thereby reducing the average costs of each. Scope economies work like scale economies, except that they account for advantages of producing multiple products, where scale economies involve an advantage of multiple units of the same product.
Economies of scale can operate at the level of the individual firm but can also operate at an industry level. Suppose there is an economy of scale in the production of an input. For example, there is an economy of scale in the production of disk drives for personal computers. This means that an increase in the production of PCs will tend to lower the price of disk drives, reducing the cost of PCs, which is a scale economy. In this case, it doesn’t matter to the scale economy whether one firm or many firms are responsible for the increased production. This is known as an external economy of scale, or an industry economy of scale, because the scale economy operates at the level of the industry rather than in the individual firm. Thus, the long-run average cost of individual firms may be flat, while the long-run average cost of the industry slopes downward.
Even in the presence of an external economy of scale, there may be diseconomies of scale at the level of the firm. In such a situation, the size of any individual firm is limited by the diseconomy of scale, but nonetheless the average cost of production is decreasing in the total output of the industry, through the entry of additional firms. Generally there is an external diseconomy of scale if a larger industry drives up input prices; for example, increasing land costs. Increasing the production of soybeans significantly requires using land that isn’t so well suited for them, tending to increase the average cost of production. Such a diseconomy is an external diseconomy rather than operating at the individual farmer level. Second, there is an external economy if an increase in output permits the creation of more specialized techniques and a greater effort in R&D is made to lower costs. Thus, if an increase in output increases the development of specialized machine tools and other production inputs, an external economy will be present.
An economy of scale arises when total average cost falls as the number of units produced rises. How does this relate to production functions? We let $$y = f (x1, x2, … , xn)$$ be the output when the n inputs $$x_{1}, x_{2}, \ldots, x n$$ are used. A rescaling of the inputs involves increasing the inputs by a fixed percentage; e.g., multiplying all of them by the constant λ (the Greek letter “lambda”), where λ > 1. What does this do to output? If output goes up by more than λ, we have an economy of scale (also known as increasing returns to scale): Scaling up production increases output proportionately more. If output goes up by less than λ, we have a diseconomy of scale, or decreasing returns to scale. And finally, if output rises by exactly λ, we have constant returns to scale. How does this relate to average cost? Formally, we have an economy of scale if $$f(λ x 1 ,λ x 2 ,…,λ x n )>λf( x 1 , x 2 ,…, x n ) if λ > 1.$$
This corresponds to decreasing average cost. Let w1 be the price of input one, w2 the price of input two, and so on. Then the average cost of producing $$y = f(x1, x2, … , xn)$$ is $$A V C=w 1 \times 1+w 2 \times 2+\ldots+w n \times n f(x 1, x 2, \dots, x n)$$
What happens to average cost as we scale up production by λ > 1? Call this AVC(λ).
$AVC(λ) = w 1 λ x 1 + w 2 λ x 2 +…+ w n λ x n f(λ x 1 ,λ x 2 ,…,λ x n ) =λ w 1 x 1 + w 2 x 2 +…+ w n x n f(λ x 1 ,λ x 2 ,…,λ x n )= λf( x 1 , x 2 ,…, x n ) f(λ x 1 ,λ x 2 ,…,λ x n ) AVC(1)$
Thus, average cost falls if there is an economy of scale and rises if there is a diseconomy of scale.
Another insight about the returns to scale concerns the value of the marginal product of inputs. Note that if there are constant returns to scale, then
$x 1 ∂f ∂ x 1 + x 2 ∂f ∂ x 2 +…+ x n ∂f ∂ x n = d dλ f(λ x 1 ,λ x 2 ,…,λ x n ) | λ→1= lim λ→1 f(λ x 1 ,λ x 2 ,…,λ x n )−f( x 1 , x 2 ,…, x n ) λ−1 =f( x 1 , x 2 ,…, x n ) .$
The value ∂f ∂ x 1 is the marginal product of input x1, and similarly ∂f ∂ x 2 is the marginal product of the second input, and so on. Consequently, if the production function exhibits constant returns to scale, it is possible to divide up output in such a way that each input receives the value of the marginal product. That is, we can give x 1 ∂f ∂ x 1 to the suppliers of input one, x 2 ∂f ∂ x 2 to the suppliers of input two, and so on; and this exactly uses up all of the output. This is known as “paying the marginal product,” because each supplier is paid the marginal product associated with the input.
If there is a diseconomy of scale, then paying the marginal product is feasible; but there is generally something left over, too. If there are increasing returns to scale (an economy of scale), then it is not possible to pay all the inputs their marginal product; that is, $$x 1 ∂f ∂ x 1 + x 2 ∂f ∂ x 2 +…+ x n ∂f ∂ x n >f( x 1 , x 2 ,…, x n ).$$
Key Takeaways
• An economy of scale arises when an increase in output reduces average costs.
• Specialization may produce economies of scale.
• An important source of diseconomies is managerial in nature—organizing a large, complex enterprise is a challenge, and larger organizations tend to devote a larger percentage of their revenues to management of the operation.
• An economy of scope is a reduction in cost associated with producing several related goods.
• Economies of scale can operate at the level of the individual firm but can also operate at an industry level. At the industry level, scale economies are known as an external economies of scale or an industry economies of scale.
• The long-run average cost of individual firms may be flat, while the long-run average cost of the industry slopes downward.
• Generally there is an external diseconomy of scale if a larger industry drives up input prices. There is an external economy if an increase in output permits the creation of more specialized techniques and a greater effort in R&D is made to lower costs.
• A production function has increasing returns to scale if an increase in all inputs by a constant factor λ increases output by more than λ.
• A production function has decreasing returns to scale if an increase in all inputs by a constant factor λ increases output by less than λ.
• The production function exhibits increasing returns to scale if and only if the cost function has an economy of scale.
• When there is an economy of scale, the sum of the values of the marginal product exceeds the total output. Consequently, it is not possible to pay all inputs their marginal product.
• When there is a diseconomy of scale, the sum of the values of the marginal product is less than the total output. Consequently, it is possible to pay all inputs their marginal product and have something left over for the entrepreneur.
EXERCISES
1. Given the Cobb-Douglas production function $$f(x 1, x 2, \ldots, x n)=x 1 \text { a } 1 \times 2 \text { a } 2 \ldots x \text { n an }$$, show that there is constant returns to scale if $$a 1+a 2+\ldots+a n=1$$, increasing returns to scale if $$a 1 + a 2 +…+ a n >1$$, and decreasing returns to scale if $$a 1 + a 2 +…+ a n <1.$$
2. Suppose a company has total cost given by $$[K+g 22 K]$$, where capital K can be adjusted in the long run. Does this company have an economy of scale, diseconomy of scale, or constant returns to scale in the long run?
3. A production function f is homogeneous of degree r if $$f(λ x 1 ,λ x 2 ,…,λ x n )= λ r f( x 1 , x 2 ,…, x n )$$. Consider a firm with a production function that is homogeneous of degree r. Suppose further that the firm pays the value of marginal product for all of its inputs. Show that the portion of revenue left over is $$1 – r$$. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/10%3A_Producer_Theory-_Dynamics/10.02%3A_Economies_of_Scale_and_Scope.txt |
Learning Objectives
• How do changes in demand or cost affect the short- and long-run prices and quantities traded?
Having understood how a competitive firm responds to price and input cost changes, we consider how a competitive market responds to demand or cost changes.
Figure 10.4 Long-run equilibrium
As drawn, the industry is in equilibrium, with price equal to P0, which is the long-run average total cost, and also equates short-run supply and demand. That is, at the price of P0, and industry output of Q0, no firm wishes to shut down, no firm can make positive profits from entering, there is no excess output, and no consumer is rationed. Thus, no market participant has an incentive to change his or her behavior, so the market is in both long-run and short-run equilibrium. In long-run equilibrium, long-run demand equals long-run supply, and short-run demand equals short-run supply, so the market is also in short-run equilibrium, where short-run demand equals short-run supply.
Now consider an increase in demand. Demand might increase because of population growth, or because a new use for an existing product is developed, or because of income growth, or because the product becomes more useful. For example, the widespread adoption of the Atkins diet increased demand for high-protein products like beef jerky and eggs. Suppose that the change is expected to be permanent. This is important because the decision of a firm to enter is based more on expectations of future demand than on present demand.
Figure 10.5 reproduces the equilibrium figure, but with the curves “grayed out” to indicate a starting position and a darker, new demand curve, labeled D1.
Figure 10.5 A shift in demand
The initial effect of the increased demand is that the price is bid up, because there is excess demand at the old price, P0. This is reflected by a change in both price and quantity to P1 and Q1, to the intersection of the short-run supply (SRS) and the new demand curve. This is a short-run equilibrium, and persists temporarily because, in the short run, the cost of additional supply is higher.
At the new, short-run equilibrium, price exceeds the long-run supply (LRS) cost. This higher price attracts new investment in the industry. It takes some time for this new investment to increase the quantity supplied, but over time the new investment leads to increased output, and a fall in the price, as illustrated in Figure 10.6.
As new investment is attracted into the industry, the short-run supply shifts to the right because, with the new investment, more is produced at any given price level. This is illustrated with the darker short-run supply, SRS2. The increase in price causes the price to fall back to its initial level and the quantity to increase still further to Q2.
Figure 10.6 Return to long-run equilibrium
In Figure 10.7, we start at the long-run equilibrium where LRS and D0 and SRS0 all intersect. If demand falls to D1, the price falls to the intersection of the new demand and the old short-run supply, along SRS0. At that point, exit of firms reduces the short-run supply and the price rises, following along the new demand D1.
Figure 10.7 A decrease in demand
Figure 10.8 A big decrease in demand
Figure 10.9 A decrease in supply
The case of a change in supply is more challenging because both the long-run supply and the short-run supply are shifted. But the logic—start at a long-run equilibrium, then look for the intersection of current demand and short-run supply, then look for the intersection of current demand and long-run supply—is the same whether demand or supply have shifted.
Key Takeaways
• A long-run equilibrium occurs at a price and quantity when the demand equals the long-run supply, and the number of firms is such that the short-run supply equals the demand.
• At long-run equilibrium prices, no firm wishes to shut down, no firm can make positive profits from entering, there is no excess output, and no consumer is rationed.
• An increase in demand to a system in long-run equilibrium first causes a short-run increase in output and a price increase. Then, because entry is profitable, firms enter. Entry shifts out short-run supply until the system achieves long-run equilibrium, decreasing prices back to their original level and increasing output.
• A decrease in demand creates a short-run equilibrium where existing short-run supply equals demand, with a fall in price and output. If the price fall is large enough (to average variable cost), some firms shut down. Then as firms exit, supply contracts, prices rise, and quantity contracts further.
• The case of a change in supply is more challenging because both the long-run supply and the short-run supply are shifted. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/10%3A_Producer_Theory-_Dynamics/10.03%3A_Dynamics_With_Constant_Costs.txt |
Learning Objectives
• If long-run costs aren’t constant, how do changes in demand or costs affect short- and long-run prices and quantities traded?
The previous section made two simplifying assumptions that won’t hold in all applications of the theory. First, it assumed constant returns to scale, so that long-run supply is horizontal. A perfectly elastic long-run supply means that price always eventually returns to the same point. Second, the theory didn’t distinguish long-run from short-run demand. But with many products, consumers will adjust more over the long-term than immediately. As energy prices rise, consumers buy more energy-efficient cars and appliances, reducing demand. But this effect takes time to be seen, as we don’t immediately scrap our cars in response to a change in the price of gasoline. The short-run effect is to drive less in response to an increase in the price, while the long-run effect is to choose the appropriate car for the price of gasoline.
To illustrate the general analysis, we start with a long-run equilibrium. Figure 10.10 reflects a long-run economy of scale, because the long-run supply slopes downward, so that larger volumes imply lower cost. The system is in long-run equilibrium because the short-run supply and demand intersection occurs at the same price and quantity as the long-run supply and demand intersection. Both short-run supply and short-run demand are less elastic than their long-run counterparts, reflecting greater substitution possibilities in the long run.
Figure 10.10 Equilibrium with external scale economy
Figure 10.12 Long-run after a decrease in demand
There are four basic permutations of the dynamic analysis—demand increase or decrease and a supply increase or decrease. Generally, it is possible for long-run supply to slope down—this is the case of an economy of scale—and for long-run demand to slope up.The demand situation analogous to an economy of scale in supply is a network externality, in which the addition of more users of a product increases the value of the product. Telephones are a clear example—suppose you were the only person with a phone—but other products like computer operating systems and almost anything involving adoption of a standard represent examples of network externalities. When the slope of long-run demand is greater than the slope of long-run supply, the system will tend to be inefficient, because an increase in production produces higher average value and lower average cost. This usually means that there is another equilibrium at a greater level of production. This gives 16 variations of the basic analysis. In all 16 cases, the procedure is the same. Start with a long-run equilibrium and shift both the short-run and long-run levels of either demand or supply. The first stage is the intersection of the short-run curves. The system will then go to the intersection of the long-run curves.
An interesting example of competitive dynamics’ concepts is the computer memory market, which was discussed previously. Most of the costs of manufacturing computer memory are fixed costs. The modern DRAM plant costs several billion dollars; the cost of other inputs—chemicals, energy, labor, silicon wafers—are modest in comparison. Consequently, the short-run supply is vertical until prices are very, very low; at any realistic price, it is optimal to run these plants 100% of the time.The plants are expensive, in part, because they are so clean—a single speck of dust falling on a chip ruins the chip. The Infineon DRAM plant in Virginia stopped operations only when a snowstorm prevented workers and materials from reaching the plant. The nature of the technology has allowed manufacturers to cut the costs of memory by about 30% per year over the past 40 years, demonstrating that there is a strong economy of scale in production. These two features—vertical short-run supply and strong economies of scale—are illustrated in Figure 10.13. The system is started at the point labeled with the number 0, with a relatively high price, and technology that has made costs lower than this price. Responding to the profitability of DRAM, short-run supply shifts out (new plants are built and die-shrinks permit increasing output from existing plants). The increased output causes prices to fall relatively dramatically because short-run demand is inelastic, and the system moves to the point labeled 1. The fall in profitability causes DRAM investment to slow, which allows demand to catch up, boosting prices to the point labeled 2. (One should probably think of Figure 10.13 as being in a logarithmic scale.)
Figure 10.13 DRAM market
Figure 10.14 DRAM revenue cycle
KEY TAKEAWAY
• In general, both demand and supply may have long-run and short-run curves. In this case, when something changes, initially the system moves to the intersections of the current short-run supply and demand for a short-run equilibrium, then to the intersection of the long-run supply and demand. The second change involves shifting short-run supply and demand curves.
EXERCISES
1. Land close to the center of a city is in fixed supply, but it can be used more intensively by using taller buildings. When the population of a city increases, illustrate the long- and short-run effects on the housing markets using a graph.
2. Emus can be raised on a wide variety of ranch land, so that there are constant returns to scale in the production of emus in the long run. In the short run, however, the population of emus is limited by the number of breeding pairs of emus, and the supply is essentially vertical. Illustrate the long- and short-run effects of an increase in demand for emus. (In the late 1980s, there was a speculative bubble in emus, with prices reaching \$80,000 per breeding pair, in contrast to \$2,000 or so today.)
3. There are long-run economies of scale in the manufacture of computers and their components. There was a shift in demand away from desktop computers and toward notebook computers around the year 2001. What are the short- and long-run effects? Illustrate your answer with two diagrams, one for the notebook market and one for the desktop market. Account for the fact that the two products are substitutes, so that if the price of notebook computers rises, some consumers shift to desktops. (To answer this question, start with a time 0 and a market in long-run equilibrium. Shift demand for notebooks out and demand for desktops in.) What happens in the short run? What happens in the long run to the prices of each? What does that price effect do to demand? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/10%3A_Producer_Theory-_Dynamics/10.04%3A_General_Long-run_Dynamics.txt |
Learning Objectives
• How is a stream of payments or liabilities evaluated?
The promise of $1 in the future is not worth$1 today. There are a variety of reasons why a promise of future payments is not worth the face value today, some of which involve risk that the money may not be paid. Let’s set aside such risk for the moment. Even when the future payment is perceived to occur with negligible risk, most people prefer $1 today to$1 payable a year hence. One way to express this is by the present value: The value today of a future payment of a dollar is less than a dollar. From a present value perspective, future payments are discounted.
From an individual perspective, one reason that one should value a future payment less than a current payment is due to arbitrage.Arbitrage is the process of buying and selling in such a way as to make a profit. For example, if wheat is selling for $3 per bushel in New York but$2.50 per bushel in Chicago, one can buy in Chicago and sell in New York, profiting by $0.50 per bushel minus any transaction and transportation costs. Such arbitrage tends to force prices to differ by no more than transaction costs. When these transaction costs are small, as with gold, prices will be about the same worldwide. Suppose you are going to need$10,000 one year from now to put a down payment on a house. One way of producing $10,000 is to buy a government bond that pays$10,000 a year from now. What will that bond cost you? At current interest rates, a secure bondEconomists tend to consider U.S. federal government securities secure, because the probability of such a default is very, very low. will cost around $9,700. This means that no one should be willing to pay$10,000 for a future payment of $10,000 because, instead, one can have the future$10,000 by buying the bond, and will have $300 left over to spend on cappuccinos or economics textbooks. In other words, if you will pay$10,000 for a secure promise to repay the $10,000 a year hence, then I can make a successful business by selling you the secure promise for$10,000 and pocketing $300. This arbitrage consideration also suggests how to value future payments: discount them by the relevant interest rate. Example (Auto loan): You are buying a$20,000 car, and you are offered the choice to pay it all today in cash, or to pay $21,000 in one year. Should you pay cash (assuming you have that much in cash) or take the loan? The loan is at a 5% annual interest rate because the repayment is 5% higher than the loan amount. This is a good deal for you if your alternative is to borrow money at a higher interest rate; for example, on (most) credit cards. It is also a good deal if you have savings that pay more than 5%—if buying the car with cash entails cashing in a certificate of deposit that pays more than 5%, then you would be losing the difference. If, on the other hand, you are currently saving money that pays less than 5% interest, paying off the car is a better deal. The formula for present value is to discount by the amount of interest. Let’s denote the interest rate for the next year as r1, the second year’s rate as r2, and so on. In this notation,$1 invested would pay $$\left(1+r_{1}\right) \text { next year, or } \left(1+r_{1}\right) \times\left(1+r_{2}\right) \text { after } 2 \text { years, or } \left(1+r_{1}\right) \times\left(1+r_{2}\right) \times\left(1+r_{3}\right)$$ after 3 years. That is, ri is the interest rate that determines the value, at the end of year i, of $1 invested at the start of year i. Then if we obtain a stream of payments A0 immediately, A1 at the end of year one, A2 at the end of year two, and so on, the present value of that stream is $\mathrm{PV}=\mathrm{A} 0+\mathrm{A} 11+\mathrm{r} 1+\mathrm{A} 2(1+\mathrm{r} 1)(1+\mathrm{r} 2)+\mathrm{A} 2(1+\mathrm{r} 1)(1+\mathrm{r} 2)(1+\mathrm{r} 3)+\ldots$ Example (Consolidated annuities or consols): What is the value of$1 paid at the end of each year forever, with a fixed interest rate r? Suppose the value is v. ThenThis development uses the formula that, for $$-1<a<1,11-a=1+a+a 2+\ldots$$, which is readily verified. Note that this formula involves an infinite series.
$v= 1 1+r + 1 (1+r) 2 + 1 (1+r) 3 +…= 1 1− 1 1+r −1= 1 r .$
At a 5% interest rate, $1 million per year paid forever is worth$20 million today. Bonds that pay a fixed amount every year forever are known as consols; no current government issues consols.
Example (Mortgages): Again, fix an interest rate r, but this time let r be the monthly interest rate. A mortgage implies a fixed payment per month for a large number of months (e.g., 360 for a 30-year mortgage). What is the present value of these payments over n months? A simple way to compute this is to use the consol value, because
$M= 1 1+r + 1 (1+r) 2 + 1 (1+r) 3 +…+ 1 (1+r) n = 1 r − 1 (1+r) n+1 − 1 (1+r) n+2 − 1 (1+r) n+3 −…= 1 r − 1 (1+r) n ( 1 (1+r) + 1 (1+r) 2 + 1 (1+r) 3 +… )= 1 r − 1 (1+r) n 1 r = 1 r ( 1− 1 (1+r) n ).$
Thus, at a monthly interest rate of 0.5%, paying $1 per month for 360 months produces a present value M of 1 0.005 ( 1− 1 (1.005) 360 )=$166.79 . Therefore, to borrow $100,000, one would have to pay$100,000 $166.79 =$599.55 per month. It is important to remember that a different loan amount just changes the scale: Borrowing $150,000 requires a payment of$150,000 $166.79 =$899.33 per month, because $1 per month generates$166.79 in present value.
Example (Simple and compound interest): In the days before calculators, it was a challenge to actually solve interest-rate formulas, so certain simplifications were made. One of these was simple interest, which meant that daily or monthly rates were translated into annual rates by incorrect formulas. For example, with an annual rate of 5%, the simple interest daily rate is 5% 365 =0.07692%. The fact that this is incorrect can be seen from the calculation ( 1+ .05 365 ) 365 =1.051267%, which is the compound interest calculation. Simple interest increases the annual rate, so it benefits lenders and harms borrowers. (Consequently, banks advertise the accurate annual rate on savings accounts—when consumers like the number to be larger—but not on mortgages, although banks are required by law to disclose, but not to advertise widely, actual annual interest rates on mortgages.)
Example (Obligatory lottery): You win the lottery, and the paper reports that you’ve won $20 million. You’re going to be paid$20 million, but is it worth $20 million? In fact, you get$1 million per year for 20 years. However, in contrast to our formula, you get the first million right off the bat, so the value is
$\mathrm{PV}=1+11+\mathrm{r}+1(1+\mathrm{r}) 2+1(1+\mathrm{r}) 3+\ldots+1(1+\mathrm{r}) 19=1+1 \mathrm{r}(1-1(1+\mathrm{r}) 19)$
Table 11.1 computes the present value of our $20 million dollar lottery, listing the results in thousands of dollars, at various interest rates. At 10% interest, the value of the lottery is less than half the “number of dollars” paid; and even at 5%, the value of the stream of payments is 65% of the face value. Table 11.1 Present value of$20 million
r 3% 4% 5% 6% 7% 10%
PV (000s) $15,324$14,134 $13,085$12,158 $11,336$9,365
The lottery example shows that interest rates have a dramatic impact on the value of payments made in the distant future. Present value analysis is the number one tool used in MBA programs, where it is known as net present value, or NPV, analysis. It is accurate to say that the majority of corporate investment decisions are guided by an NPV analysis.
Example (Bond prices): A standard Treasury bill has a fixed future value. For example, it may pay $10,000 in one year. It is sold at a discount off the face value, so that a one-year,$10,000 bond might sell for $9,615.39, producing a 4% interest rate. To compute the effective interest rate r, the formula relating the future value FV, the number of years n, and the price is $( 1+r ) n = FV Price$ or $r= ( FV Price ) 1 n −1.$ We can see from either formula that Treasury bill prices move inversely to interest rates—an increase in interest rates reduces Treasury prices. Bonds are a bit more complicated. Bonds pay a fixed interest rate set at the time of issue during the life of the bond, generally collected semiannually, and the face value is paid at the end of the term. These bonds were often sold on long terms, as much as 30 years. Thus, a three-year,$10,000 bond at 5% with semiannual payments would pay $250 at the end of each half year for 3 years, and pay$10,000 at the end of the 3 years. The net present value, with an annual interest rate r, is
$\text { NPV }=\ 250(1+r) 12+\ 250(1+r) 1+\ 250(1+r) 32+\ 250(1+r) 2+\ 250(1+r) 52+\ 250(1+r) 3+\ 10000(1+r) 3$
The net present value will be the price of the bond. Initially, the price of the bond should be the face value, since the interest rate is set as a market rate. The U.S. Treasury quit issuing such bonds in 2001, replacing them with bonds in which the face value is paid and then interest is paid semiannually.
Key Takeaways
• Capital goods change slowly, in part because they are durable.
• The acquisition of goods that will be used over time, whether they be factories, homes, or televisions, is known as investment.
• The promise of $1 in the future is not worth$1 today. The difference is a discount on future payments.
• Arbitrage involves buying and selling such that a positive surplus is earned.
• Arbitrage is possible unless future payments are discounted by the appropriate interest rate.
• “Simple” interest means that daily, monthly, or annual rates are translated into daily, monthly, or annual rates by incorrect formulas. Accurate calculations are known as compound interest.
• A standard Treasury bill has a fixed future value. Treasury bill prices move inversely to interest rates—an increase in interest rates reduces Treasury prices.
• Bonds pay a fixed interest rate set at the time of issue during the life of the bond, generally collected semiannually, and the face value is paid at the end of the term.
EXERCISES
1. At a 7% annual interest rate, what is the present value of $100 paid at the end of one year, and$200 paid at the end of the second year?
2. Compute the NPV of the 3-year, $10,000 bond with$250 semiannual payments semiannually, at an interest rate of 4%.
3. You can finance your $20,000 car with a straight 5% loan paid monthly over 5 years, or a loan with one year interest-free followed by 4 years of 7% interest. (Hint: In both cases, figure out the fixed monthly payments that produce an NPV equal to$20,000.)
4. You win the lottery. At what interest rate should you accept $7 million today over 20 annual payments of$500,000?
5. An investor discounts future profits at 5% per year. Suppose a stock pays $1 in dividends after one year, growing 1% each year thereafter. How much should the stock be worth today? 6. You are buying a$20,000 car. If you make monthly payments of \$1,000, how long will it take you to pay off the debt if the interest rate is 1% per month? How does this change when the interest rate drops to 0.5%? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/11%3A_Investment/11.01%3A_Present_Value.txt |
Learning Objectives
• How do I evaluate an investment opportunity?
A simple investment project requires an investment, I, followed by a return over time. If you dig a mine, drill an oil well, build an apartment building or a factory, or buy a share of stock, you spend money now, hoping to earn a return in the future. We will set aside the very important issue of risk until the next subsection, and ask how one makes the decision to invest.
The NPV approach involves assigning a rate of return r that is reasonable for the specific project and then computing the corresponding present value of the expected stream of payments. Since the investment is initially expended, it is counted as negative revenue. This yields an expression that looks like
$\mathrm{NPV}=-\mathrm{I}+\mathrm{R} 11+\mathrm{r}+\mathrm{R} 2(1+\mathrm{r}) 2+\mathrm{R} 3(1+\mathrm{r}) 3+\ldots$
where R1 represents first-year revenues, R2 represents second-year revenues, and so on.The most common approach treats revenues within a year as if they are received at the midpoint, and then discounts appropriately for that mid-year point. The present discussion abstracts from this practice. The investment is then made when NPV is positive—because this would add to the net value of the firm.
Carrying out an NPV analysis requires two things. First, investment and revenues must be estimated. This is challenging, especially for new products where there is no direct way of estimating demand, or with uncertain outcomes like oil wells or technological research.The building of the famed Sydney Opera House, which looks like billowing sails over Sydney Harbor in Australia, was estimated to cost $7 million and actually cost$105 million. A portion of the cost overrun was due to the fact that the original design neglected to install air conditioning. When this oversight was discovered, it was too late to install a standard unit, which would interfere with the excellent acoustics, so instead an ice hockey floor was installed as a means of cooling the building. Second, an appropriate rate of return must be identified. The rate of return is difficult to estimate, mostly because of the risk associated with the investment payoffs. Another difficulty is recognizing that project managers have an incentive to inflate the payoffs and minimize the costs to make the project appear more attractive to upper management. In addition, most corporate investment is financed through retained earnings, so that a company that undertakes one investment is unable to make other investments, so the interest rate used to evaluate the investment should account for opportunity cost of corporate funds. As a result of these factors, interest rates of 15%–20% are common for evaluating the NPV of projects of major corporations.
Example (Silver mine): A company is considering developing a silver mine in Mexico. The company estimates that developing the mine requires building roads and opening a large hole in the ground, which would cost $4 million per year for 4 years, during which time the mines generates zero revenue. Starting in year 5, the expenses would fall to$2 million per year, and $6 million in net revenue would accrue from the silver that is mined for the next 40 years. If the company cost of funds is 18%, should it develop the mine? The earnings from the mine are calculated in the table below. First, the NPV of the investment phase during years 0, 1, 2, and 3 is $\mathrm{NPV}=-4+-41.18+-4(1.18) 2+-4(1.18) 3=-12.697$ A dollar earned in each of years 4 through 43 has a present value of $1(1+r) 4+1(1+r) 5+1(1+r) 6+\ldots+1(1+r) 43=1(1+r) 3 \times 1 r(1-1(1+r) 40)=13.377$ The mine is just profitable at 18%, in spite of the fact that its$4 million payments are made in 4 years, after which point $4 million in revenue is earned for 40 years. The problem in the economics of mining is that 18% makes the future revenue have quite modest present values. Year Earnings ($M)/yr PV ($M) 0–3 –4 –12.697 4–43 4 13.377 Net 0.810 There are other approaches for deciding to take an investment. In particular, the internal rate of return (IRR) approach solves the equation NPV = 0 for the interest rate. Then the project is undertaken if the rate of return is sufficiently high. This approach is flawed because the equation may have more than one solution—or no solutions—and the right thing to do in these events is not transparent. Indeed, the IRR approach gets the profit-maximizing answer only if it agrees with NPV. A second approach is the payback period, which asks calculates the number of years a project must be run before profitability is reached. The problem with the payback period is deciding between projects—if I can only choose one of two projects, the one with the higher NPV makes the most money for the company. The one with the faster payback may make a quite small amount of money very quickly, but it isn’t apparent that this is a good choice. When a company is in risk of bankruptcy, a short payback period might be valuable, although this would ordinarily be handled by employing a higher interest rate in an NPV analysis. NPV does a good job when the question is whether or not to undertake a project, and it does better than other approaches to investment decisions. For this reason, NPV has become the most common approach to investment decisions. Indeed, NPV analysis is more common than all other approaches combined. NPV does a poor job, however, when the question is whether to undertake a project or to delay the project. That is, NPV answers “yes or no” to investment, but when the choice is “yes or wait,” NPV requires an amendment. Key Takeaways • The NPV approach involves assigning a rate of return r that is reasonable for, and specific to, the project and then computing the present value of the expected stream of payments. The investment is then made when NPV is positive—since this would add to the net value of the firm. • Carrying out an NPV analysis requires estimating investment and revenues and identifying an appropriate rate of return. • Interest rates of 15%–20% are common for evaluating the NPV of projects of major corporations. EXERCISES 1. Suppose that, without a university education, you’ll earn$25,000 per year. A university education costs $20,000 per year, and you forgo the$25,000/year that you would have earned for 4 years. However, you earn $50,000 per year for the following 40 years. At 7%, what is the NPV of the university education? 2. Now that you’ve decided to go to the university based on the previous answer, suppose that you can attend East State U, paying$3,000 per year for 4 years and earning $40,000 per year when you graduate, or you can attend North Private U, paying$22,000 per year for the 4 years and earning $50,000 per year when you graduate. Which is the better deal at 7%? 3. A bond is a financial instrument that pays a fixed amount, called the face value, at a maturity date. Bonds can also pay out fixed payments, called coupons, in regular intervals up until the maturity date. Suppose a bond with face value$1,000 sells for $900 on the market and has annual coupon payments starting a year from today up until its maturity date 10 years from now. What is the coupon rate? Assume r = 10%. 4. The real return on stocks averages about 4% annually. Over 40 years, how much will$1,000 invested today grow?
5. You have made an invention. You can sell the invention now for $1 million and work at something else, producing$75,000 per year for 10 years. (Treat this income as received at the start of the year.) Alternatively, you can develop your invention, which requires working for 10 years, and it will net you $5 million in 10 years hence. For what interest rates are you better off selling now? (Please approximate the solution.) 6. A company is evaluating a project with a start-up fee of$50,000 but pays \$2,000 every second year thereafter, starting 2 years from now. Suppose that the company is indifferent about taking on the project—or not. What discount rate is the company using? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/11%3A_Investment/11.02%3A_Investment.txt |
Learning Objectives
• What is the effect of risk on investment?
• What is an option?
Risk has a cost, and people and corporations buy insurance against financial risk.For example, NBC spent $6 million to buy an insurance policy against U.S. nonparticipation in the 1980 Moscow Summer Olympic Games—and the United States didn’t participate (because of the Soviet invasion of Afghanistan)—and NBC was paid$94 million from the policy. The standard approach to investment under uncertainty is to compute an NPV, using a “risk-adjusted” interest rate to discount the expected values received over time. The interest rate is increased or lowered depending on how risky the project is.
For example, consider a project like oil exploration. The risks are enormous. Half of all underwater tracts in the Gulf Coast near Louisiana and Texas that are leased are never drilled, because they turn out to be a bad bet. Half of all the tracts that are drilled are dry. Hence, three quarters of the tracts that are sold produce zero or negative revenue. To see how the economics of such a risky investment might be developed, suppose that the relevant rate of return for such investments is 18%. Suppose further that the tract can be leased for $500,000 and the initial exploration costs$1 million. If the tract has oil (with a 25% probability), it produces $1 million per year for 20 years and then runs dry. This gives an expected revenue of$250,000 per year. To compute the expected net present value, we first compute the returns:
Table 11.2 Oil tract return
Expected revenue EPV
0 –$1.5M –$1.5M
1–20 $0.25M$1.338M
Net –$0.162 At 18%, the investment is a loss—the risk is too great given the average returns. A very important consideration for investment under uncertainty is the choice of interest rate. It is crucial to understand that the interest rate is specific to the project and not to the investor. This is perhaps the most important insight of corporate finance generally: The interest rate should adjust for the risk associated with the project and not for the investor. For example, suppose hamburger retailer McDonald’s is considering investing in a cattle ranch in Peru. McDonald’s is overall a very low-risk firm, but this particular project is quite risky because of local conditions. McDonald’s still needs to adjust for the market value of the risk it is undertaking, and that value is a function of the project risk, not the risk of McDonald’s other investments. This basic insight of corporate finance (the study of funding of operations of companies)—the appropriate interest rate is determined by the project, not by the investor—is counterintuitive to most of us because it doesn’t apply to our personal circumstances. For individuals, the cost of borrowing money is mostly a function of our own personal circumstances, and thus the decision of whether to pay cash for a car or borrow the money is not so much a function of the car that is being purchased but of the wealth of the borrower. Even so, personal investors borrow money at distinct interest rates. Mortgage rates on houses are lower than interest rates on automobiles, and interest rates on automobiles are lower than on credit cards. This is because the “project” of buying a house has less risk associated with it: The percentage loss to the lender in the event of borrower default is lower on a house than on a car. Credit cards carry the highest interest rates because they are unsecured by any asset. One way of understanding why the interest rate is project-specific, but not investor-specific, is to think about undertaking the project by creating a separate firm to make the investment. The creation of subsidiary units is a common strategy, in fact. This subsidiary firm created to operate a project has a value equal to the NPV of the project using the interest rate specific to the subsidiary, which is the interest rate for the project, independent of the parent. For the parent company, owning such a firm is a good thing if the firm has positive value, but not otherwise.It may seem that synergies between parent and subsidiary are being neglected here, but synergies should be accounted for at the time they produce value—that is, as part of the stream of revenues of the subsidiary. Investments in oil are subject to another kind of uncertainty: price risk. The price of oil fluctuates. Moreover, oil pumped and sold today is not available for the future. Should you develop and pump the oil you have today, or should you hold out and sell in the future? This question, known as the option value of investment, is generally somewhat challenging and arcane, but a simple example provides a useful insight. An option is the right to buy or sell at a price determined in advance. To develop this example, let’s set aside some extraneous issues first. Consider a very simple investment, in which either C is invested or not.This theory is developed in striking generality by Avinash Dixit and Robert Pindyck, Investment Under Uncertainty, Princeton University Press, 1994. If C is invested, a value V is generated. The cost C is a constant; it could correspond to drilling or exploration costs or, in the case of a stock option, the strike price of the option, which is the amount one pays to obtain the share of stock. The value V, in contrast, varies from time to time in a random fashion. To simplify the analysis, we assume that V is uniformly distributed on the interval [0, 1], so that the probability of V falling in an interval $[a, b] is (b – a) \text{ if } 0 ≤a ≤ b ≤ 1$. The option only has value if C < 1, which we assume for the rest of this section. The first thing to note is that the optimal rule to make the investment is a cutoff value—that is, to set a level V0 and exercise the option if, and only if, VV0. This is because—if you are willing to exercise the option and generate value V—you should be willing to exercise the option and obtain even more value. The NPV rule simply says V0 = C; that is, invest whenever it is profitable. The purpose of the example developed below is to provide some insight into how far wrong the NPV rule will be when option values are potentially significant. Now consider the value of option to invest, given that the investment rule VV0 is followed. Call this option value J(V0). If the realized value V exceeds V0, one obtains VC. Otherwise, one delays the investment, producing a discounted level of the same value. This logic says, $J(\vee 0)=(1-\vee 0)(1+\vee 02-C)+V 0(11+r](\vee 0))$ This expression for J (V0) is explained as follows. First, the hypothesized distribution of V is uniform on [0, 1]. Consequently, the value of V will exceed V0 with probability 1 – V0. In this event, the expected value of V is the midpoint of the interval [V0, 1], which is ½(V0 + 1). The value ½(V0 + 1) – C is the average payoff from the strategy of investing whenever VV0, which is obtained with probability 1 – V0. Second, with probability V0, the value falls below the cutoff level V0. In this case, no investment is made and, instead, we wait until the next period. The expected profits of the next period are J (V0), and these profits are discounted in the standard way. The expression for J is straightforward to solve: $\mathrm{J}(\mathrm{V} 0)=(1-\mathrm{V} 0)(1+\mathrm{V} 02-\mathrm{C}) 1-\mathrm{V} 01+\mathrm{r}$ Rudimentary calculus shows $\mathrm{J}^{\prime}(\mathrm{V} 0)=1+2 \mathrm{rC}+\mathrm{V} 02-2(1+\mathrm{r}) \vee 02(1+\mathrm{r})(1-\mathrm{V} 01+\mathrm{r}) 2$ First, note that $$\mathrm{J}^{\prime}(\mathrm{C})>0 \text { and } \mathrm{J}^{\prime}(1)<0$$, which together imply the existence of a maximum at a value V0 between C and 1, satisfying J ′ ( V 0 )=0. Second, the solution occurs at $\mathrm{V} 0=(1+r)-(1+r) 2-(1+2 r C)=(1+r)-r 2+2 r(1-C)$ The positive root of the quadratic has V0 > 1, which entails never investing, and hence is not a maximum. The profit-maximizing investment strategy is to invest whenever the value exceeds V0 given by the negative root in the formula. There are a couple of notable features about this solution. First, at r = 0, V0 = 1. This is because r = 0 corresponds to no discounting, so there is no loss in holding out for the highest possible value. Second, as r → ∞, V0C. As r → ∞, the future is valueless, so it is worth investing if the return is anything over costs. These are not surprising findings, but rather quite the opposite: They should hold in any reasonable formulation of such an investment strategy. Moreover, they show that the NPV rule, which requires V0 = C, is correct only if the future is valueless. How does this solution behave? The solution is plotted as a function of r, for C = 0, 0.25, and 0.5, in Figure 11.1. The horizontal axis represents interest rates, so this figure shows very high interest rates by current standards, up to 200%. Even so, V0 remains substantially above C. That is, even when the future has very little value, because two-thirds of the value is destroyed by discounting each period, the optimal strategy deviates significantly from the NPV strategy. Figure 11.2 shows a close-up of this graph for a more reasonable range of interest rates, for interest rates of 0%–10%. Figure 11.1 Investment strike price given interest rate r in percent For C = 0.25, at 10% the cutoff value for taking an investment is 0.7, nearly three times the actual cost of the investment. Indeed, the cutoff value incorporates two separate costs: the actual expenditure on the investment C and the lost opportunity to invest in the future. The latter cost is much larger than the expenditure on the investment in many circumstances and, in this example, can be quantitatively much larger than the actual expenditure on the investment. Some investments can be replicated. There are over 13,000 McDonald’s restaurants in the United States, and building another one doesn’t foreclose building even more. For such investments, NPV analysis gets the right answer, provided that appropriate interest rates and expectations are used. Other investments are difficult to replicate or logically impossible to replicate—having pumped and sold the oil from a tract, that tract is now dry. For such investments, NPV is consistently wrong because it neglects the value of the option to delay the investment. A correct analysis adds a lost value for the option to delay the cost of the investment—a value that can be quantitatively large—as we have seen. Figure 11.2 Investment strike price given interest rate r in percent Example: When should you refinance a mortgage? Suppose you are paying 10% interest on a$100,000 mortgage, and it costs $5,000 to refinance; but refinancing permits you to lock in a lower interest rate, and hence pay less. When is it a good idea? To answer this question, we assume that the$5,000 cost of refinancing is built into the loan so that, in essence, you borrow $105,000 at a lower interest rate when you refinance. This is actually the most common method of refinancing a mortgage. To simplify the calculations, we will consider a mortgage that is never paid off; that is, one pays the same amount per year forever. If the mortgage isn’t refinanced, one pays 10% of the$100,000 face value of the mortgage each year, or $10,000 per year. If one refinances at interest rate r, one pays r ×$105,000 per year, so the NPV of refinancing is
$\mathrm{NPY}=\ 10,000-r \times \ 105,000$
Thus, NPV is positive whenever $$r<10105=9.52 \%$$
Should you refinance when the interest rate drops to this level? No. At this level, you would exactly break even, but you would also be carrying a $105,000 mortgage rather than a$100,000 mortgage, making it harder to benefit from any further interest-rate decreases. The only circumstance in which refinancing at 9.52% is sensible is if interest rates can’t possibly fall further.
When should you refinance? That depends on the nature and magnitude of the randomness governing interest rates, preferences over money today versus money in the future, and attitudes to risk. The model developed in this section is not a good guide to answering this question, primarily because the interest rates are strongly correlated over time. However, an approximate guide to implementing the option theory of investment is to seek an NPV of twice the investment, which would translate into a refinance point of around 8.5%.
Key Takeaways
• The standard approach to investment under uncertainty is to compute an NPV, with the interest rate used adjusted to compensate for the risk.
• The most important thing to understand is that the interest rate is specific to the project and not to the investor.
• The option value of investment includes the value of decisions that have not yet been made. Building a factory today forecloses the opportunity of building the factory a year later, when better information concerning market conditions has been realized, but also creates the option of selling over the next year.
• NPV gets the right answer when investments can be replicated.
• An important example of the option value is refinancing a home.
EXERCISE
1. You are searching for a job. The net value of jobs that arise is uniformly distributed on the interval [0, 1]. When you accept a job, you must stop looking at subsequent jobs. If you can interview with one employer per week, what jobs should you accept? Use a 7% annual interest rate.
(Hint: Relate the job-search problem to the investment problem, where accepting a job is equivalent to making the investment. What is c in the job-search problem? What is the appropriate interest rate?) | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/11%3A_Investment/11.03%3A_Investment_Under_Uncertainty.txt |
Learning Objectives
• How much of a limited resource should be consumed today, and how much should be saved for future consumption?
For the past 60 years, the world has been “running out of oil.” There are news stories about the end of the reserves being only 10, 15, or 20 years away. The tone of these stories is that, at this time, we will run out of oil completely and prices will be extraordinarily high. Industry studies counter that more oil continues to be found and that the world is in no danger of running out of oil.
If you believe that the world will run out of oil, what should you do? You should buy and hold. That is, if the price of oil in 20 years is going to be $1,000 per barrel, then you can buy oil at$40 per barrel, hold it for 20 years, and sell it at \$1,000 per barrel. The rate of return from this behavior is the solution to (1+r) 20 = 1000 40 .
This equation solves for r = 17.46%, which represents a healthy rate of return on investment. This substitution is part of a general conclusion known as the Ramsey rule:The solution to this problem is known as Ramsey pricing, after the discoverer Frank Ramsey (1903–1930). For resources in fixed supply, prices rise at the interest rate. With a resource in fixed supply, owners of the resource will sell at the point maximizing the present value of the resource. Even if they do not, others can buy the resource at the low present value of price point, resell at the high present value, and make money.
The Ramsey rule implies that prices of resources in fixed supply rise at the interest rate. An example of the Ramsey rule in action concerns commodities that are temporarily fixed in supply, such as grains after the harvest. During the period between harvests, these products rise in price on average at the interest rate, where the interest rate includes storage and insurance costs, as well as the cost of funds.
Example: Let time be $$t=0,1, \ldots$$, and suppose the demand for a resource in fixed supply has constant elasticity: $$\mathrm{p}(\mathrm{Q})=\mathrm{a} \mathrm{Q}-1 \varepsilon$$. Suppose that there is a total stock R of the resource, and the interest rate is fixed at r. What is the price and consumption of the resource at each time?
Solution: Let Qt represent the quantity consumed at time t. Then the arbitrage condition requires
$\text { a } Q 0-1 \varepsilon(1+r) t=p(Q 0)(1+r) t=p(Q t)=a Q t-1 \varepsilon$
Thus, $$Q t=Q 0(1+r)-t \varepsilon$$. Finally, the resource constraint implies
$R=(Q 0+Q 1+Q 2+\ldots)=Q 0(1+(1+r)-\varepsilon+(1+r)-2 \varepsilon+\ldots)=Q 01-(1+r)-\varepsilon$
This solves for the initial consumption Q0. Consumption in future periods declines geometrically, thanks to the constant elasticity assumption.
Market arbitrage ensures the availability of the resource in the future and drives up the price to ration the good. The world runs out slowly, and the price of a resource in fixed supply rises on average at the interest rate.
Resources like oil and minerals are ostensibly in fixed supply—there is only so much oil, gold, bauxite, or palladium in the earth. Markets, however, behave as if there is an unlimited supply, and with good reason. People are inventive and find substitutes. England’s wood shortage of 1651 didn’t result in England being cold permanently, nor was England limited to the wood it could grow as a source of heat. Instead, coal was discovered. The shortage of whale oil in the mid-19th century led to the development of oil resources as a replacement. If markets expect that price increases will lead to substitutes, then we rationally should use more today, trusting that technological developments will provide substitutes.Unlike oil and trees, whales were overfished and there was no mechanism for arbitraging them into the future—that is, no mechanism for capturing and saving whales for later use. This problem, known as the tragedy of the commons, results in too much use (Garett Hardin, Science, 1968, Tragedy of the Commons). Trees have also been overcut, most notably on Easter Island. Thus, while some believe that we are running out of oil, most investors are betting that we are not, and that energy will not be very expensive in the future—either because of continued discovery of oil or because of the creation of alternative energy sources. If you disagree, why not invest and take the bet? If you bet on future price increases, that will tend to increase the price today, encouraging conservation today, and increase the supply in the future.
Key Takeaways
• The Ramsey rule holds that, for resources in fixed supply, prices rise at the interest rate.
• With constant elasticity, consumption of a resource in fixed supply declines geometrically.
• Market arbitrage ensures the availability of the resource in the future and drives up the price to ration the good. The world runs out slowly, and the price of a resource in fixed supply rises on average at the interest rate.
• Substitutes mitigate the fixed supply aspect of natural resources; for example, fiber optic cable substitutes for copper.
EXERCISE
1. With an elasticity demand of two, compute the percentage of the resource that is used each year if the interest rate is 10%. If the interest rate falls, what happens to the proportion quantity used? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/11%3A_Investment/11.04%3A_Resource_Extraction.txt |
Learning Objectives
• How are the prices of renewable resources determined?
• When should trees be harvested?
A tree grows slowly but is renewable, so the analysis of Section 11.4 doesn’t help us to understand when it is most profitable to cut down the tree. Consider harvesting for pulp and paper use. In this case, the amount of wood chips is what matters to the profitability of cutting down the tree, and the biomass of the tree provides a direct indication of this. Suppose the biomass sells for a net price p, which has the costs of harvesting and replanting deducted from it, and the biomass of the tree is b(t) when the tree is t years old. It simplifies the analysis slightly to use continuous time discounting $$\mathrm{e}-\rho \mathrm{t}=(11+r) \mathrm{t}, \text { where } \rho=\log (1+r)$$
Consider the policy of cutting down trees when they are T years old. This induces a cutting cycle of length T. A brand new tree will produce a present value of profits of
$e −ρT pb(T)+ e −2ρT pb(T)+ e −3ρT pb(T)+…= e −ρT pb(T) 1− e −ρT = pb(T) e ρT −1 .$
This profit arises because the first cut occurs at time T, with discounting e-ρT, and produces a net gain of pb(T). The process then starts over, with a second tree cut down at time 2T, and so on.
Profit maximization gives a first-order condition on the optimal cycle length T of
$0=\mathrm{d} \mathrm{dT} \mathrm{pb}(\mathrm{T}) \mathrm{e} \rho \mathrm{T}-1=\mathrm{p} \mathrm{b}^{\prime}(\mathrm{T}) \mathrm{e} \rho \mathrm{T}-1-\mathrm{pb}(\mathrm{T}) \rho \text { e } \rho \mathrm{T}(\mathrm{e} \rho \mathrm{T}-1) 2$
This can be rearranged to yield
$\mathrm{b}^{\prime}(\mathrm{T}) \mathrm{b}(\mathrm{T})=\rho 1-\mathrm{e}-\rho \mathrm{T}$
The left-hand side of this equation is the growth rate of the tree. The right-hand side is approximately the continuous-time discount factor, at least when T is large, as it tends to be for trees, which are usually on a 20- to 80-year cycle, depending on the species. This is the basis for a conclusion: Cut down the tree slightly before it is growing at the interest rate. The higher that interest rates are, the shorter the cycle for which the trees should be cut down.
The pulp and paper use of trees is special, because the tree is going to be ground up into wood chips. What happens when the object is to get boards from the tree, and larger boards sell for more? In particular, it is more profitable to get a 4 × 4 than two 2 × 4s. Doubling the diameter of the tree, which approximately raises the biomass by a factor of six to eight, more than increases the value of the timber by the increase in the biomass.
It turns out that our theory is already capable of handling this case. The only adaptation is a change in the interpretation of the function b. Now, rather than representing the biomass, b(t) must represent the value in boards of a tree that is t years old. (The parameter p may be set to one.) The only amendment to the rule for cutting down trees is as follows: The most profitable point in time to cut down the tree occurs slightly before the time when the value (in boards) of the tree is growing at the interest rate.
For example, lobsters become more valuable as they grow. The profit-maximizing time to harvest lobsters is governed by the same equation, where b (T) is the value of a lobster at age T. Prohibiting the harvest of lobsters under age T is a means of insuring the profit-maximizing capture of lobsters and preventing overfishing.
The implementation of the formula is illustrated in Figure 11.3. The dashed line represents the growth rate b ′ (T) b(T) , while the solid line represents the discount rate, which was set at 5%. Note that the best time to cut down the trees is when they are approximately 28.7 years old and, at that time, they are growing at 6.5%. Figure 11.3 also illustrates another feature of the optimization—there may be multiple solutions to the optimization problem, and the profit-maximizing solution involves $$b^{\prime}(T) b(T)$$ cutting $$\rho 1-\mathrm{e}-\rho \mathrm{T}$$ from above.
Figure 11.3 Optimal solution for T
The U.S. Department of the Interior is in charge of selling timber rights on federal lands. The department uses the policy of maximum sustainable yield to determine the specific time that the tree is cut down. Maximum sustainable yield maximizes the long-run average value of the trees cut down; that is, it maximizes b(T) T .
Maximum sustainable yield is actually a special case of the policies considered here, and arises for a discount factor of 0. It turns out (thanks to a formula known variously as l’Hôpital’s or l’Hospital’s Rule) that the $$\lim \rho \rightarrow 0 \rho 1-\mathrm{e}-\rho \mathrm{T}=1 \mathrm{T}$$.
Thus, the rule $$\mathrm{b}^{\prime}(T) \mathrm{b}(T)=\rho 1-\mathrm{e}-\rho \mathrm{T} \rightarrow 1 \mathrm{T} \text { as } \rho \rightarrow 0$$, and this is precisely the same rule that arises under maximum sustainable yield.
Thus, the Department of the Interior acts as if the interest rate is zero when it is not. The justification given is that the department is valuing future generations at the same level as current generations—that is, increasing the supply for future generations while slightly harming the current generation of buyers. The major consequence of the department’s policy of maximum sustainable yield is to force cutting of timber even when prices are low during recessions.
Key Takeaways
• Renewable resources are harvested slightly earlier than the point at which they grow at the interest rate, because earlier planting of the next generation has value.
• Maximum sustainable yield maximizes the long-run average value of the trees cut down, which is the optimal policy only when the interest rate is zero.
EXERCISES
1. Show that maximum sustainable yield results in cutting down a tree when it is T years old, where T satisfies $$b^{\prime}(T) b(T)=1 T$$.
2. Suppose that the growth rate of trees satisfies $$b^{\prime}(T) b(T)=t e-t$$. Numerically approximate the efficient time to cut down a tree if ρ = 0.1. How does this compare to the solution of maximum sustainable yield? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/11%3A_Investment/11.05%3A_A_Time_to_Harvest.txt |
Learning Objectives
• How are the prices of collectibles determined?
• What is the investment value of collectibles?
• How fast do the prices rise?
Many people purchase durable goods as investments, including Porsche Speedsters (see Figure 11.4), Tiffany lamps, antique telephones, postage stamps and coins, baseball cards, original Barbie dolls, antique credenzas, autographs, original rayon Hawaiian shirts, old postcards, political campaign buttons, old clocks, and even Pez dispensers. How is the value of, say, a 1961 Porsche Speedster or a $500 bill from the Confederacy, which currently sells for over$500, determined?
The theory of resource prices can be adapted to cover these items, which are in fixed supply. There are four major differences that are relevant. First, using the item doesn’t consume it: The goods are durable. I can own an “I Like Ike” campaign button for years, and then sell the same button. Second, these items may depreciate. Cars wear out even when they aren’t driven, and the brilliant color of Pez dispensers fades. Every time that a standard 27.5-pound gold bar, like the kind in the Fort Knox depository, is moved, approximately $5 in gold wears off the bar. Third, the goods may cost something to store. Fourth, the population grows, and some of the potential buyers are not yet born. Figure 11.4 The Porsche Speedster To understand the determinants of the prices of collectibles, it is necessary to create a major simplification to perform the analysis in continuous time. Let t, ranging from zero to infinity, be the continuous-time variable. If the good depreciates at rate δ, and q0 is the amount available at time 0, the quantity available at time t is q(t)= q 0 e −δ t . For simplicity, assume that there is constant elasticity of demand ε. If g is the population growth rate, the quantity demanded, for any price p, is given by x d (p,t)=a e gt p −ε , for a constant a that represents the demand at time 0. This represents demand for the good for direct use, but neglects the investment value of the good—the fact that the good can be resold for a higher price later. In other words, xd captures the demand for looking at Pez dispensers or driving Porsche Speedsters, but does not incorporate the value of being able to resell these items. The demand equation can be used to generate the lowest use value to a person owning the good at time t. That marginal use value v arises from the equality of supply and demand: $\text { q } 0 \text { e }-\delta t=q(t)=x d(v, t)=a \text { e } g t v-\varepsilon$ or $v \varepsilon=a q 0 e(\delta+g) t$ Thus, the use value to the marginal owner of the good at time t satisfies $$\mathrm{v}=(\text { a } \mathrm{q} 0) 1 \varepsilon \text { e } \delta+\mathrm{g} \varepsilon \mathrm{t}$$ An important aspect of this development is that the value to the owner is found without reference to the price of the good. The reason this calculation is possible is that the individuals with high values will own the good, and the number of goods and the values of people are assumptions of the theory. Essentially, we already know that the price will ration the good to the individuals with high values, so computing the lowest value individual who holds a good at time t is a straightforward “supply equals demand” calculation. Two factors increase the marginal value to the owner—there are fewer units available because of depreciation, and there are more high-value people demanding them because of population growth. Together, these factors make the marginal use value grow at the rate $$\delta+g \varepsilon$ . Assume that s is the cost of storage per unit of time and per unit of the good, so that storing x units for a period of length Δ costs sxΔ. This is about the simplest possible storage cost technology. The final assumption we make is that all potential buyers use a common discount rate r, so that the discount of money or value received Δ units of time in the future is e-rΔ. It is worth a brief digression to explain why it is sensible to assume a common discount rate, when it is evident that many people have different discount rates. Different discount rates induce gains from trade in borrowing and lending, and create an incentive to have banks. While banking is an interesting topic to study, this section is concerned with collectibles, not banks. If we have different discount factors, then we must also introduce banks, which would complicate the model substantially. Otherwise, we would intermingle the theory of banking and the theory of collectibles. It is probably a good idea to develop a joint theory of banking and collectibles, given the investment potential of collectibles, but it is better to start with the pure theory of either one before developing the joint theory. Consider a person who values the collectible at v. Is it a good thing for this person to own a unit of the good at time t ? Let p be the function that gives the price across time, so that p(t) is the price at time t. Buying the good at time t and then selling what remains (recall that the good depreciates at rate δ) at time t + Δ gives a net value of \($∫ 0 Δ e −ru (v−s )du−p(t)+ e −rΔ e −δΔ p(t+Δ).$$ For the marginal person—that is, the person who is just indifferent to buying or not buying at time t—this must be zero at every moment in time, for Δ = 0. If v represents the value to a marginal buyer (indifferent to holding or selling) holding the good at time t, then this expression should come out to be zero. Thus, dividing by Δ, $$0= lim Δ→0 1 Δ ∫ 0 Δ e −ru (v−s )du− p(t) Δ + e −(r+δ)Δ p(t+Δ) Δ= lim Δ→0 v−s+ p(t+Δ)−p(t) Δ − 1− e −(r+δ)Δ Δ p(t+Δ)=v−s+ p ′ (t)−(r+δ)p(t).$$ Recall that the marginal value is $$v= ( a q 0 ) 1 ε e δ+g ε t$$, which gives $$p ′ (t)=(r+δ)p(t)+s−v=(r+δ)p(t)+s− ( a q 0 ) 1 ε e δ+g ε t$$. The general solution to this differential equation is $p(t)= e (r+δ)t ( p(0)+ 1− e −(r+δ)t (r+δ) s− ( a q 0 ) 1 ε 1− e −( r+δ− δ+g ε )t r+δ− δ+g ε ) .$ It turns out that this equation only makes sense if $$r+δ− δ+g ε >0$$, for otherwise the present value of the marginal value goes to infinity, so there is no possible finite initial price. Provided that demand is elastic and discounting is larger than growth rates (which is an implication of equilibrium in the credit market), this condition will be met. What is the initial price? It must be the case that the present value of the price is finite, for otherwise the good would always be a good investment for everyone at time 0, using the “buy and hold for resale” strategy. That is, $lim t→∞ e −rt p(t)<∞.$ This condition implies that $$\lim t \rightarrow \infty \text { e } \delta t(p(0)+1-e-(r+\delta) t(r+\delta) s-(a q 0) 1 \varepsilon 1-e-(r+\delta-\delta+g \varepsilon) t r+\delta-\delta+g \varepsilon)<\infty$$, and thus $$p(0)+ 1 (r+δ) s− ( a q 0 ) 1 ε 1 r+δ− δ+g ε =0.$$ This equation may take on two different forms. First, it may be solvable for a nonnegative price, which happens if $p(0)= ( a q 0 ) 1 ε 1 r+δ− δ+g ε − 1 (r+δ) s≥0$ Second, it may require destruction of some of the endowment of the good. Destruction must happen if the quantity of the good q0 at time 0 satisfies $( a q 0 ) 1 ε 1 r+δ− δ+g ε − 1 (r+δ) s<0 .$ In this case, there is too much of the good, and an amount must be destroyed to make the initial price zero. Since the initial price is zero, the good is valueless at time zero, and destruction of the good makes sense—at the current quantity, the good is too costly to store for future profits. Enough is destroyed to ensure indifference between holding the good as a collectible and destroying it. Consider, for example, the$500 Confederate bill shown in Figure 11.5. Many of these bills were destroyed at the end of the U.S. Civil War, when the currency became valueless and was burned as a source of heat. Now, an uncirculated version retails for $900. Figure 11.5$500 Confederate States bill
The amount of the good that must be destroyed is such that the initial price is zero. As q0 is the initial (predestruction) quantity, the amount at time zero after the destruction is the quantity q(0) satisfying
$0=p(0)= ( a q(0) ) 1 ε 1 r+δ− δ+g ε − 1 (r+δ) s .$
Given this construction, we have that
$p(0)+ 1 (r+δ) s− ( a q(0) ) 1 ε 1 r+δ− δ+g ε =0$
where either $$q(0) = q0 and p(0) ≥ 0, or q(0) < q0 and p(0) = 0.$$
Destruction of a portion of the stock of a collectible, followed by price increases, is actually a quite common phenomenon. In particular, consider the “Model 500” telephone by Western Electric illustrated in Figure 11.6. This ubiquitous classic phone was retired as the United States switched to tone dialing and push-button phones in the 1970s, and millions of phones—perhaps over 100 million—wound up in landfills. Now the phone is a collectible, and rotary phone enthusiasts work to keep them operational.
Figure 11.6 Western Electric Model 500 telephone
The solution for p(0) dramatically simplifies the expression for p(t):
$p(t)= e (r+δ)t ( p(0)+ 1− e −(r+δ)t (r+δ) s− ( a q(0) ) 1 ε 1− e −( r+δ− δ+g ε )t r+δ− δ+g ε )= e (r+δ)t ( − e −(r+δ)t (r+δ) s+ ( a q(0) ) 1 ε e −( r+δ− δ+g ε )t r+δ− δ+g ε )= ( a q(0) ) 1 ε e δ+g ε t r+δ− δ+g ε − s r+δ$
This formula enables one to compare different collectibles. The first insight is that storage costs enter linearly into prices, so that growth rates are approximately unaffected by storage costs. The fact that gold is easy to store—while stamps and art require control of humidity and temperature in order to preserve value, and are hence more expensive to store—affects the level of prices but not the growth rate. However, depreciation and the growth of population affect the growth rate, and they do so in combination with the demand elasticity. With more elastic demand, prices grow more slowly and start at a lower level.
Key Takeaways
• The price of collectibles includes two distinct sources of value—use value and investment value. The relevant use value is that of the marginal user, a value that rises as the quantity falls or the population grows.
• The use value to the marginal owner is found without reference to the price of the good. It grows at the destruction rate plus the population growth rate, all divided by the demand elasticity.
• The investment value net of storage must equal the interest rate.
• If storage costs are high, equilibrium pricing may first involve destruction of some quantity of units.
• Storage costs enter linearly into prices, so that growth rates are approximately unaffected by storage costs. The fact that gold is easy to store—while stamps and art require control of humidity and temperature in order to preserve value, and are hence more expensive to store—affects the level of prices but not the growth rate. However, depreciation and the growth of population affect the growth rate, and they do so in combination with the demand elasticity. With more elastic demand, prices grow more slowly and start at a lower level. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/11%3A_Investment/11.06%3A_Collectibles.txt |
Learning Objectives
• How are the prices of storable goods—apples, potatoes, or wheat—determined?
Typically, wheat harvested in the fall has to last until the following harvest. How should prices evolve over the season? If I know that I need wheat in January, should I buy it at harvest time and store it myself, or wait and buy it in January? We can use a theory analogous to the theory of collectibles developed in Section 11.6 to determine the evolution of prices for commodities like wheat, corn, orange juice, and canola oil.
Unlike collectibles, buyers need not hold commodities for their personal use, since there is no value in admiring the wheat in your home. Let p(t) be the price at time t, and suppose that the year has length T. Generally there is a substantial amount of uncertainty regarding the size of wheat harvests, and most countries maintain an excess inventory as a precaution. However, if the harvest were not uncertain, there would be no need for a precautionary holding. Instead, we would consume the entire harvest over the course of a year, at which point the new harvest would come in. It is this such model that is investigated in this section.
Let δ represent the depreciation rate (which, for wheat, includes the quantity eaten by rodents), and let s be the storage cost. Buying at time t and reselling at t + Δ should be a break-even proposition. If one purchases at time t, it costs p (t) to buy the good. Reselling at t + Δ, the storage cost is about sΔ. (This is not the precisely relevant cost; but rather it is the present value of the storage cost, and hence the restriction to small values of Δ.) The good depreciates to only have e −δΔ left to sell, and discounting reduces the value of that amount by the factor e −rΔ . For this to be a break-even proposition, for small Δ,
$0=e-r \Delta e-\delta \Delta p(t+\Delta)-s \Delta-p(t)$
or
$p(t+\Delta)-p(t) \Delta=1-e-(r+\delta) \Delta \Delta p(t+\Delta)+s$
taking the limit as $$\Delta \rightarrow 0, \mathrm{p}^{\prime}(\mathrm{t})=(\mathrm{r}+\delta) \mathrm{p}(\mathrm{t})+\mathrm{s}$$
This arbitrage condition ensures that it is a break-even proposition to invest in the good; the profits from the price appreciation are exactly balanced by depreciation, interest, and storage costs. We can solve the differential equation to obtain
$p(t)=e(r+\delta) t(p(0)+1-e-(r+\delta) t r+\delta s)=e(r+\delta) t p(0)+e(r+\delta) t-1 r+\delta s$
The unknown is p (0). The constraint on p (0), however, is like the resource extraction problem—p (0) is determined by the need to use up the harvest over the course of the year.
Suppose demand has constant elasticity ε. Then the quantity used comes in the form $$x(t)=\operatorname{ap}(t)-\varepsilon$$. Let z(t) represent the stock at time t. Then the equation for the evolution of the stock is $$z^{\prime}(t)=-x(t)-\delta z(t)$$. This equation is obtained by noting that the flow out of stock is composed of two elements: depreciation, δz, and consumption, x. The stock evolution equation solves for
$z(t)= e −δt ( q(0)− ∫ 0 t e δu x(u)du ) .$
Thus, the quantity of wheat is consumed exactly if
$\int 0 \text { T e } \delta u x(u) d u=q(0)$
But this equation determines the initial price through
$q(0)= ∫ 0 T e δu x(u)du = ∫ 0 T e δu ap (u) −ε du = ∫ 0 T e δu a ( e (r+δ)u p(0)+ e (r+δ)u −1 r+δ s ) −ε du .$
This equation doesn’t lead to a closed form for p(0) but is readily estimated, which provides a practical means of computing expected prices for commodities in temporarily fixed supply.
Figure 11.7 Prices over a cycle for seasonal commodities
Figure 11.8 Log of price of gold over time
Key Takeaways
• There is a seasonal pattern to goods that are produced periodically. The price equation produces a “sawtooth” pattern. The increasing portion is an exponential.
• Gold prices show evidence of exponential growth predicted by the theory.
EXERCISE
1. Consider a market for a commodity that can be stored with zero cost from winter to summer, but cannot be stored from summer to winter. The winter demand and supply are Qwd = 50 – 2Pw and Qws = 3Pw, and the summer demand and supply are Qsd = 100 – 3Ps and Qss = Ps. Compute Pw, Ps, Qw, and Qs, and the amount of hoarding from winter to summer. (Set discounting to zero.) | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/11%3A_Investment/11.07%3A_Summer_Wheat.txt |
Learning Objectives
• How do economists model consumer choice?
Economists use the term utility in a peculiar and idiosyncratic way. Utility refers not to usefulness but to the flow of pleasure or happiness that a person enjoys—some measure of the satisfaction a person experiences. Usefulness might contribute to utility, but so does style, fashion, or even whimsy.
The term utility is unfortunate, not just because it suggests usefulness but because it makes the economic approach to behavior appear more limited than it actually is. We will make very few assumptions about the form of utility that a consumer might have. That is, we will attempt to avoid making value judgments about the preferences a consumer holds—whether he or she likes to smoke cigarettes or eat only carrots, watch Arnold Schwarzenegger movies, or spend time with a hula-hoop. Consumers like whatever it is that they like; the economic assumption is that they attempt to obtain the goods that they enjoy. It is the consequences of the pursuit of happiness that comprise the core of consumer theory.
In this chapter, we will focus on two goods. In many cases, the generalization to an arbitrary number of goods is straightforward. Moreover, in most applications it won’t matter because we can view one of the goods as a “composite good,” reflecting consumption of a bunch of other goods.Thus, for example, savings for future consumption, providing for descendants, or giving to your alma mater are all examples of consumption. Our consumer will, in the end, always spend all of his or her income, although this happens because we adopt a very broad notion of spending. In particular, savings are “future spending.”
As a starting point, suppose there are two goods, X and Y. To distinguish the quantity of the good from the good itself, we’ll use capital letters to indicate the good, and lowercase letters to indicate the quantity of that good that is consumed. If X is rutabagas, a consumer who ate three of them would have x = 3. How can we represent preferences for this consumer? To fix ideas, suppose the consumer is both hungry and thirsty, and the goods are pizza and beer. The consumer would like more of both, reflected in greater pleasure for greater consumption. Items that one might consume are generally known as “bundles,” as in bundles of goods and services, and less frequently as “tuples,” a short form for the “n-tuple,” meaning a list of n quantities. Since we will focus on two goods, both of these terms are strained in the application—a bundle because a bundle of two things isn’t much of a bundle, and a tuple because what we have here is a “two-tuple,” also known as a pair. But part of the job of studying economics is to learn the language of economics, so bundles it is.
One might naturally consider measuring utility on some kind of physical basis (production of dopamine in the brain, for example) but it turns out that the actual quantities of utility don’t matter for the theory we develop. What matters is whether a bundle produces more than another, less, or the same. Let u(x, y) represent the utility that a consumer gets from consuming x units of beer and y units of pizza. The function u guides the consumer’s choice in the sense that, if the consumer can choose either $$\left(x_{1}, y_{1}\right) \text { or }\left(x_{2}, y_{2}\right)$$, we expect him to choose $$\left(x_{1}, y_{1}\right) \text { if } u\left(x_{1}, y_{1}\right)>u\left(x_{2}, y_{2}\right)$$ But notice that a doubling of u would lead to the same choices because
$u\left(x_{1}, y_{1}\right)>u\left(x_{2}, y_{2}\right) \text { if and only if } 2 u\left(x_{1}, y_{1}\right)>2 u\left(x_{2}, y_{2}\right)$
Thus, doubling the utility doesn’t change the preferences of the consumer. But the situation is more extreme than this. Even exponentiating the utility doesn’t change the consumer’s preferences because $$u\left(x_{1}, y_{1}\right)>u\left(x_{2}, y_{2}\right)$$ if and only if e $$u\left(x_{1}, y_{1}\right)>e u\left(x_{2}, y_{2}\right)$$
In other words, there are no natural units for utility, at least until such time as we are able to measure pleasure in the brain.
It is possible to develop the theory of consumer choice without supposing that a utility function exists at all. However, it is expedient to begin with utility in order to simplify the analysis for introductory purposes.
Key Takeaways
• Consumer theory is to demand as producer theory is to supply.
• Consumer theory is based on the premise that we can infer what people like from the choices they make.
• Utility refers not to usefulness but to the flow of pleasure or happiness that a person enjoys—some measure of the satisfaction a person experiences.
• There are no natural units for utility; any increasing transformation is acceptable. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/12%3A_Consumer_Theory/12.01%3A_Utility_Maximization.txt |
Learning Objectives
• How does income limit choice?
Suppose that a consumer has a fixed amount of money to spend, M. There are two goods X and Y, with associated prices pX and pY. The feasible choices that the consumer can make satisfy p X x+ p Y y≤M. In addition, we will focus on consumption and rule out negative consumption, so x ≥ 0 and y ≥ 0. This gives a budget set or feasible set, as illustrated in Figure 12.1. The budget set is the set of goods a consumer can afford to purchase.
The budget line is the boundary of the budget set, and it consists of the goods that just exhaust the consumer’s budget
Figure 12.1 Budget set
An increase in the price of one good pivots or rotates the budget line. Thus, if the price of X increases, the endpoint M p Y remains the same, but M p X falls. This is illustrated in Figure 12.2.
Figure 12.2 Effect of an increase in price on the budget
Figure 12.3 An increase in income
An increase in both prices by the same proportional factor has an effect identical to a decrease in income. Thus, one of the three financial values—the two prices and income—is redundant. That is, we can trace out all of the possible budget lines with any two of the three parameters. This can prove useful. We can arbitrarily set pX to be the number one without affecting the generality of the analysis. When setting a price to one, that related good is called the numeraire, and essentially all prices are denominated with respect to that one good.
A real-world example of a numeraire occurred when the currency used was based on gold, so that the prices of other goods were denominated in terms of the value of gold.
Money is not necessarily the only constraint on the consumption of goods that a consumer faces. Time can be equally important. One can own all of the compact disks in the world, but they are useless if one doesn’t actually have time to listen to them. Indeed, when we consider the supply of labor, time will be a major issue—supplying labor (working) uses up time that could be used to consume goods. In this case, there will be two kinds of budget constraints—a financial one and a temporal one. At a fixed wage, time and money translate directly into one another, and the existence of the time constraint won’t present significant challenges to the theory. The conventional way to handle the time constraint is to use, as a baseline, working “full out,” and then to view leisure as a good that is purchased at a price equal to the wage. Thus, if you earn \$20 an hour, we would set your budget at \$480 a day, reflecting 24 hours of work; but we would then permit you to buy leisure time, during which eating, sleeping, brushing your teeth, and every other nonwork activity could be accomplished at a price equal to \$20 per hour.
Key Takeaways
• The budget set or feasible set is the set of goods that the consumer can afford to purchase.
• The budget line is the pair of goods that exactly spend the budget. The budget line shifts out when income rises and pivots when the price of one good changes.
• Increasing prices and income by the same multiplicative factor leaves the feasible set unchanged.
EXERCISES
1. Graph the budget line for apples and oranges with prices of \$2 and \$3, respectively, and \$60 to spend. Now increase the price of apples from \$2 to \$4, and draw the budget line.
2. Suppose that apples cost \$1 each. Water can be purchased for 50 cents per gallon up to 20,000 gallons, and 10 cents per gallon for each gallon beyond 20,000 gallons. Draw the budget constraint for a consumer who spends \$200 on apples and water.
3. Graph the budget line for apples and oranges with prices of \$2 and \$3, respectively, and \$60 to spend. Now increase the expenditure to \$90, and draw the budget line. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/12%3A_Consumer_Theory/12.02%3A_Budget_or_Feasible_Set.txt |
Learning Objectives
• What is an isoquant?
• Why does it help to analyze consumer choice?
With two goods, we can graphically represent utility by considering the contour map of utility. Utility contours are known as isoquants, meaning “equal quantity,” and are also known as indifference curves, since the consumer is indifferent between points on the line. In other words an indifference curve, also known as an iso-utility curve, is the set of goods that produce equal utility.
We have encountered this idea already in the description of production functions, where the curves represented input mixes that produced a given output. The only difference here is that the output being produced is consumer “utility” instead of a single good or service.
Figure 12.4 Utility isoquants
${\partial u}{\partial x}{dx} + {\partial u}{\partial y}{dy} = 0$
$\mathrm{dy} \mathrm{dx} | \mathrm{u}=\mathrm{u} 0=-\partial \mathrm{u} \partial \mathrm{x} \partial \mathrm{u} \partial \mathrm{y}$
This slope is known as the marginal rate of substitution and reflects the trade-off, from the consumer’s perspective, between the goods. That is to say, the marginal rate of substitution (of Y for X) is the amount of Y that the consumer is willing to lose in order to obtain an extra unit of X.
An important assumption concerning isoquants is reflected in the figure: “Midpoints are preferred to extreme points.” Suppose that the consumer is indifferent between $$\left(x_{1}, y_{1}\right) \text { and }\left(x_{2}, y_{2}\right)$$ ; that is, u (x1, y1) = u (x2, y2). Then we can say that preferences are convex if any point on the line segment connecting (x1, y1) and (x2, y2) is at least as good as the extremes. Formally, a point on the line segment connecting (x1, y1) and (x2, y2) comes in the form $$\left(\alpha x_{1}+(1-\alpha) x_{2}, \alpha y_{1}+(1-\alpha) y_{2}\right)$$ for α between zero and one. This is also known as a “convex combination” between the two points. When α is zero, the segment starts at (x2, y2) and proceeds in a linear fashion to (x1, y1) at α = 1. Preferences are convex if, for any α between 0 and 1, $$u\left(x_{1}, y_{1}\right)=u\left(x_{2}, y_{2}\right)$$ implies $$u\left(\alpha x_{1}+(1-\alpha) x_{2},\left(y_{1}+(1-\alpha)\right.\right.$$$$\left.y_{2}\right) \geq u\left(x_{1}, y_{1}\right)$$
This property is illustrated in Figure 12.5. The line segment that connects two points on the indifference curve lies to the northeast of the indifference curve, which means that the line segment involves strictly more consumption of both goods than some points on the indifference curve. In other words, it is preferred to the indifference curve. Convex preferences mean that a consumer prefers a mix to any two equally valuable extremes. Thus, if the consumer likes black coffee and also likes drinking milk, then the consumer prefers some of each—not necessarily mixed—to only drinking coffee or only drinking milk. This sounds more reasonable if you think of the consumer’s choices on a monthly basis. If you like drinking 60 cups of coffee and no milk per month as much as you like drinking 30 glasses of milk and no coffee, convex preferences entail preferring 30 cups of coffee and 15 glasses of milk to either extreme.
How does a consumer choose which bundle to select? The consumer is faced with the problem of maximizing u (x, y) subject to p X x+ p Y y≤M.We can derive the solution to the consumer’s problem as follows. First, “solve” the budget constraint p X x+ p Y y≤M for y, to obtain y≤ M− p X x p Y . If Y is a good, this constraint will be satisfied with equality, and all of the money will be spent. Thus, we can write the consumer’s utility as
$u(x, M-p X \times p Y)$
The first-order condition for this problem, maximizing it over x, has
$0=\mathrm{d} \mathrm{d} \mathrm{x} \mathrm{u}(\mathrm{x}, \mathrm{M}-\mathrm{p} \mathrm{X} \times \mathrm{p} \mathrm{Y})=\partial \mathrm{u} \partial \mathrm{x}-\mathrm{p} \mathrm{X} \mathrm{p} \mathrm{Y} \partial \mathrm{u} \partial \mathrm{y}$ .
This can be rearranged to obtain the marginal rate of substitution (MRS):
$\mathrm{p} \mathrm{X} \mathrm{p} \mathrm{Y}=\text { du } \partial \mathrm{x} \text { \partial u } \partial \mathrm{y}=-\mathrm{dy} \mathrm{dx} | \mathrm{u}=\mathrm{u} 0=\mathrm{MRS}$
The marginal rate of substitution (MRS) is the extra amount of one good needed to make up for a decrease in another good, staying on an indifference curve
The first-order condition requires that the slope of the indifference curve equals the slope of the budget line; that is, there is a tangency between the indifference curve and the budget line. This is illustrated in Figure 12.6. Three indifference curves are drawn, two of which intersect the budget line but are not tangent. At these intersections, it is possible to increase utility by moving “toward the center,” until the highest of the three indifference curves is reached. At this point, further increases in utility are not feasible, because there is no intersection between the set of bundles that produce a strictly higher utility and the budget set. Thus, the large black dot is the bundle that produces the highest utility for the consumer.
It will later prove useful to also state the second-order condition, although we won’t use this condition now:
$0 \geq \mathrm{d} 2(\mathrm{dx}) 2 \mathrm{u}(\mathrm{x}, \mathrm{M}-\mathrm{p} \mathrm{X} \times \mathrm{p} \mathrm{Y})=\partial 2 \mathrm{u}(\partial \mathrm{x}) 2-\mathrm{p} \mathrm{X} \mathrm{p} \mathrm{Y} \partial 2 \mathrm{u} \partial \mathrm{x} \partial \mathrm{y}+(\mathrm{p} \mathrm{X} \mathrm{p} \mathrm{Y}) 2 \partial 2 \mathrm{u}(\partial \mathrm{y}) 2$
Note that the vector $$(u 1, u 2)=(\partial u \partial x, \partial u \partial y)$$ is the gradient of u, and the gradient points in the direction of steepest ascent of the function u. Second, the equation that characterizes the optimum,
$0=\mathrm{p} \times \partial \mathrm{u} \partial \mathrm{y}-\mathrm{p} \mathrm{Y} \partial \mathrm{u} \partial \mathrm{x}=(\partial \mathrm{u} \partial \mathrm{x}, \partial \mathrm{u} \partial \mathrm{y}) \cdot(-\mathrm{p} \mathrm{Y}, \mathrm{p} \mathrm{X})$,
Figure 12.6 Graphical utility maximization
where • is the “dot product” that multiplies the components of vectors and then adds them, says that the vectors (u1, u2) and (–pY, pX) are perpendicular and, hence, that the rate of steepest ascent of the utility function is perpendicular to the budget line.
When does this tangency approach fail to solve the consumer’s problem? There are three ways that it can fail. First, the utility might not be differentiable. We will set aside this kind of failure with the remark that fixing points of nondifferentiability is mathematically challenging but doesn’t lead to significant alterations in the theory. The second failure is that a tangency doesn’t maximize utility. Figure 12.7 illustrates this case. Here there is a tangency, but it doesn’t maximize utility. In Figure 12.7 , the dotted indifference curve maximizes utility given the budget constraint (straight line). This is exactly the kind of failure that is ruled out by convex preferences. In Figure 12.7, preferences are not convex because, if we connect two points on the indifference curves and look at a convex combination, we get something less preferred, with lower utility—not more preferred as convex preferences would require.
Figure 12.7 “Concave” preferences: Prefer boundaries
The third failure is more fundamental: The derivative might fail to be zero because we’ve hit the boundary of x = 0 or y = 0. This is a fundamental problem because, in fact, there are many goods that we do buy zero of, so zeros for some goods are not uncommon solutions to the problem of maximizing utility. We will take this problem up in a separate section, but we already have a major tool to deal with it: convex preferences. As we shall see, convex preferences ensure that the consumer’s maximization problem is “well behaved.”
Key Takeaways
• Isoquants, meaning “equal quantity,” are also known as indifference curves and represent sets of points holding utility constant. They are analogous to production isoquants.
• Preferences are said to be convex if any point on the line segment connecting a pair of points with equal utility is preferred to the endpoints. This means that whenever the consumer is indifferent between two points, he or she prefers a mix of the two.
• The first-order conditions for the maximizing utility involve equating the marginal rate of substitution and the price ratio.
• At the maximum, the rate of steepest ascent of the utility function is perpendicular to the budget line.
• There are two main ways that the first-order conditions fail to characterize the optimum: the consumer doesn’t have convex preferences, or the optimum involves a zero consumption of one or more goods. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/12%3A_Consumer_Theory/12.03%3A_Isoquants.txt |
Learning Objectives
• Are there any convenient functional forms for analyzing consumer choice?
The Cobb-Douglas utility function comes in the form u( x,y )= x α y 1−α . Since utility is zero if either of the goods is zero, we see that a consumer with Cobb-Douglas preferences will always buy some of each good. The marginal rate of substitution for Cobb-Douglas utility is
$-d y d x | u=u 0=\partial u \partial x \partial u \partial y=a y(1-a) x$
Thus, the consumer’s utility maximization problem yields
$p X p Y=-d y d x | u=u 0=\partial u \partial x \partial u \partial y=a y(1-a) x$
Thus, using the budget constraint, $$(1-a) \times p X=a y p Y=a(M-x p X)$$ This yields
$\mathrm{x}=\mathrm{aMp} \mathrm{X}, \quad \mathrm{y}=(1-\mathrm{a}) \mathrm{M} \mathrm{p} \mathrm{Y}$
The Cobb-Douglas utility results in constant expenditure shares. No matter what the price of X or Y, the expenditure xpX on X is αM. Similarly, the expenditure on Y is (1 – α)M. This makes the Cobb-Douglas utility very useful for computing examples and homework exercises.
When two goods are perfect complements, they are consumed proportionately. The utility that gives rise to perfect complements is in the form $$u(x, y)=\min \{x, \beta y\}$$ for some constant β (the Greek letter “beta”). First observe that, with perfect complements, consumers will buy in such a way that x = βy. The reason is that, if x > βy, some expenditure on x is a waste since it brings in no additional utility; and the consumer gets higher utility by decreasing x and increasing y. This lets us define a “composite good” that involves buying some amount y of Y and also buying βy of X. The price of this composite commodity is $$\beta \mathrm{p} \mathrm{X}+\mathrm{pY}$$, and it produces utility $$u=M \beta p X+p Y$$. In this way, perfect complements boil down to a single good problem.
If the only two goods available in the world were pizza and beer, it is likely that satiation—the point at which increased consumption does not increase utility—would set in at some point. How many pizzas can you eat per month? How much beer can you drink? (Don’t answer that.)
Figure 12.8 Isoquants for a bliss point
Key Takeaways
• The Cobb-Douglas utility results in constant expenditure shares.
• When two goods are perfect complements, they are consumed proportionately. Perfect complements boil down to a single good problem.
• A bliss point, or satiation, is a point at which further increases in consumption reduce utility.
EXERCISES
1. Consider a consumer with utility $$u(x, y)=x y$$. If the consumer has $100 to spend, and the price of X is$5 and the price of Y is \$2, graph the budget line; and then find the point that maximizes the consumer’s utility given the budget. Draw the utility isoquant through this point. What are the expenditure shares?
2. Consider a consumer with utility $$u(x, y)=x y$$. Calculate the slope of the isoquant directly by solving u(x,y)= u 0 for y as a function of x and the utility level u0. What is the slope $$-d y d x | u=u 0$$ ? Verify that it satisfies the formula given above.
3. Consider a consumer with utility $$u(x, y)=(x y) 2$$. Calculate the slope of the isoquant directly by solving u(x,y)= u 0 for y as a function of x and the utility level u0. What is the slope − dy dx | u= u 0 ? Verify that the result is the same as in the previous exercise. Why is it the same?
4. The case of perfect substitutes arises when all that matters to the consumer is the sum of the products—for example, red shirts and green shirts for a colorblind consumer. In this case, $$u(x, y)=x+y$$. Graph the isoquants for perfect substitutes. Show that the consumer maximizes utility by spending his or her entire income on whichever product is cheaper.
5. Suppose u( x,y )= x α + y α for α < 1. Show that $$x=M p X(1+(p Y p X) a)$$ and $$y=M p Y(1+(p X p Y) a)$$.
6. Suppose that one consumer has the utility function u (which is always a positive number), and a second consumer has utility w. Suppose, in addition, that for any x, y, $$w(x, y)=(u(x, y))^{2}$$; that is, the second person’s utility is the square of the first person’s. Show that these consumers make the same choices—that is, $$\left.u \geq x_{a}, y_{a}\right) \geq u\left(x_{b}, y_{b}\right)$$ if and only if $$w\left(x_{a}, y_{a}\right) \geq w\left(x_{b}, y_{b}\right)$$ | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/12%3A_Consumer_Theory/12.04%3A_Examples.txt |
Learning Objectives
• When prices change, how do consumers change their behavior?
It would be a simpler world if an increase in the price of a good always entailed buying less of it. Alas, it isn’t so, as Figure 12.9 illustrates. In this figure, an increase in the price of Y causes the budget line to pivot around the intersection on the x-axis, since the amount of X that can be purchased hasn’t changed. In this case, the quantity y of Y demanded rises.
Figure 12.9 Substitution with an increase in price
At first glance, this increase in the consumption of a good in response to a price increase sounds implausible, but there are examples where it makes sense. The primary example is leisure. As wages rise, the cost of leisure (forgone wages) rises. But as people feel wealthier, they choose to work fewer hours. The other examples given, which are hotly debated in the “tempest in a teapot” kind of way, involve people subsisting on a good like potatoes but occasionally buying meat. When the price of potatoes rises, they can no longer afford meat and buy even more potatoes than before.
Thus, the logical starting point on substitution—what happens to the demand for a good when the price of that good increases—does not lead to a useful theory. As a result, economists have devised an alternative approach based on the following logic. An increase in the price of a good is really a composition of two effects: an increase in the relative price of the good and a decrease in the purchasing power of money. As a result, it is useful to examine these two effects separately. The substitution effect considers the change in the relative price, with a sufficient change in income to keep the consumer on the same utility isoquant.Some authors instead change the income enough to make the old bundle affordable. This approach has the virtue of being readily computed, but the disadvantage is that the substitution effect winds up increasing the utility of the consumer. Overall the present approach is more economical for most purposes. The income effect changes only income.
Figure 12.10 Substitution effect
We can readily see that the substitution effect of a price increase in Y is to decrease the consumption of Y and increase the consumption of X.To construct a formal proof, first show that if pY rises and y rises, holding utility constant, the initial choice prior to the price increase is feasible after the price increase. Use this to conclude that, after the price increase, it is possible to have strictly more of both goods, contradicting the hypothesis that utility was held constant. The income effect is the change in consumption resulting from the change in income. The effect of any change in price can be decomposed into the substitution effect, which holds utility constant while changing and the income effect, which adjusts for the loss of purchasing power arising from the price increase.
Example (Cobb-Douglas): Recall that the Cobb-Douglas utility comes in the form u( x,y )= x α y 1−α . Solving for x, y we obtain
$$x=a M p X$$, $$y=(1-a) M p Y$$
and
$u(x, y)=a a(1-a) 1-a M p X a p Y 1-a$
Thus, consider a multiplicative increase Δ in $$p_{Y}$$ that is, multiplying pY by Δ > 1. In order to leave the utility constant, M must rise by Δ1α. Thus, x rises by the factor Δ1α and y falls by the factor Δα < 1. This is the substitution effect.
What is the substitution effect of a small change in the price pY for any given utility function, not necessarily Cobb-Douglas? To address this question, it is helpful to introduce some notation. We will subscript the utility to indicate partial derivative; that is,
$$u 1=\partial u \partial x$$, $$\mathrm{u} 2=\partial \mathrm{u} \partial \mathrm{y}$$.
Note that, by the definition of the substitution effect, we are holding the utility constant, so u(x, y) is being held constant. This means, locally, that 0=du= u 1 dx+ u 2 dy. Writing dx for an unknown infinitesimal change in x can be put on a formal basis. The easiest way to do so is to think of dx as representing the derivative of x with respect to a parameter, which will be pY.
In addition, we have $$\mathrm{M}=\mathrm{p} \mathrm{X} \mathrm{x}+\mathrm{p} \mathrm{Y} \mathrm{y}, \text { so } \mathrm{dM}=\mathrm{p} \mathrm{X} \mathrm{d} \mathrm{x}+\mathrm{p} \mathrm{Y} \mathrm{dy}+\mathrm{yd} \mathrm{p} \mathrm{Y}$$.
Finally, we have the optimality condition p X p Y = ∂u ∂x ∂u ∂y , which is convenient to write as p X u 2 = p Y u 1 . Differentiating this equation, and lettingu 11 = ∂ 2 u (∂x) 2 , u 12 = ∂ 2 u ∂x∂y , and u 22 = ∂ 2 u (∂y) 2 ,we havep X ( u 12 dx+ u 22 dy)= u 1 d p Y + p Y ( u 11 dx+ u 12 dy).
For a given dpY, we now have three equations in three unknowns: dx, dy, and dM. However, dM only appears in one of the three. Thus, the effect of a price change on x and y can be solved by solving two equations: 0= u 1 dx+ u 2 dy and p X ( u 12 dx+ u 22 dy)= u 1 d p Y + p Y ( u 11 dx+ u 12 dy) for the two unknowns, dx and dy. This is straightforward and yields
$\text { dxd } p Y=-p Y u 1 p \times 2 u 11+2 p X p Y u 12+p Y 2 u 22 d y d p Y=p Y u 2 p \times 2 u 11+2 p X p Y u 12+p Y 2 u 22$
These equations imply that x rises and y falls.This is a consequence of the fact that p X 2 u 11 +2 p X p Y u 12 + p Y 2 u 22 <0 , which follows from the already stated second-order condition for a maximum of utility. We immediately see that
$\text { dy } \mathrm{d} \mathrm{p} \mathrm{Y} \mathrm{d} \mathrm{x} \mathrm{d} \mathrm{p} \mathrm{Y}=-\mathrm{u} 1 \mathrm{u} 2=-\mathrm{p} \mathrm{X} \mathrm{p} \mathrm{Y}$
Thus, the change in (x, y ) follows the budget line locally. (This is purely a consequence of holding utility constant.)
To complete the thought while we are embroiled in these derivatives, note that $$\mathrm{p} \times \mathrm{u} 2=\mathrm{p} \mathrm{Y} \text { u } 1$$ implies that $$p X d x+p Y d y=0$$
Thus, the amount of money necessary to compensate the consumer for the price increase, keeping utility constant, can be calculated from our third equation:
$\mathrm{dM}=\mathrm{p} \mathrm{X} \mathrm{d} \mathrm{x}+\mathrm{p} \mathrm{Y} \text { dy }+\mathrm{yd} \mathrm{p} \mathrm{Y}=\mathrm{yd} \mathrm{p} \mathrm{Y}$
The amount of income necessary to ensure that the consumer makes no losses from a price increase in Y is the amount that lets him or her buy the bundle that he or she originally purchased; that is, the increase in the amount of money is precisely the amount needed to cover the increased price of y. This shows that locally there is no difference between a substitution effect that keeps utility constant (which is what we explored) and one that provides sufficient income to permit purchasing the previously purchased consumption bundle, at least when small changes in prices are contemplated.
Key Takeaways
• An increase in the price of a good is really a composition of two effects: an increase in the relative price of the good and a decrease in the purchasing power of money. It is useful to examine these two effects separately. The substitution effect considers the change in the relative price, with a sufficient change in income to keep the consumer on the same utility isoquant. The income effect changes only income.
• The substitution effect is the change in consumption resulting from a price change keeping utility constant. The substitution effect always involves a reduction in the good whose price increased.
• The amount of money required to keep the consumer’s utility constant from an infinitesimal price increase is precisely the amount required to let him or her buy his or her old bundle at the new prices. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/12%3A_Consumer_Theory/12.05%3A_Substitution_Effects.txt |
Learning Objectives
• How do consumers change their purchases when their income rises or falls?
Wealthy people buy more caviar than poor people. Wealthier people buy more land, medical services, cars, telephones, and computers than poorer people because they have more money to spend on goods and services and, overall, buy more of them. But wealthier people also buy fewer of some goods, too. Rich people buy fewer cigarettes and processed cheese foods. You don’t see billionaires waiting in line at McDonald’s, and that probably isn’t because they have an assistant to wait in line for them. For most goods, at a sufficiently high income, the purchase tends to trail off as income rises.
When an increase in income causes a consumer to buy more of a good, that good is called a normal good for that consumer. When the consumer buys less, the good is called an inferior good, which is an example of sensible jargon that is rare in any discipline. That is, an inferior good is any good whose quantity demanded falls as income rises. At a sufficiently low income, almost all goods are normal goods, while at a sufficiently high income, most goods become inferior. Even a Ferrari is an inferior good against some alternatives, such as Lear jets.
The curve that shows the path of consumption as income changes, holding prices constant, is known as an Engel curve.The Engel curve is named for Ernst Engel (1821–1896), a statistician—not for Friedrich Engels, who wrote with Karl Marx. An Engel curve graphs (x(M), y(M)) as M varies, where x(M) is the amount of X chosen with income M, and similarly y(M) is the amount of Y. An example of an Engel curve is illustrated in Figure 12.11.
Figure 12.11 Engel curve
Example (Cobb-Douglas): Since the equations $$x=a M p X$$, $$y=(1-a) M p Y$$ define the optimal consumption, the Engel curve is a straight line through the origin with slope $$(1-a) p X a p Y$$.
An inferior good will see the quantity fall as income rises. Note that, with two goods, at least one is a normal good—they can’t both be inferior goods because otherwise, when income rises, less of both would be purchased. An example of an inferior good is illustrated in Figure 12.12. Here, as income rises, the consumption of x rises, reaches a maximum, and then begins to decline. In the declining portion, X is an inferior good.
The definition of the substitution effect now permits us to decompose the effect of a price change into a substitution effect and an income effect. This is illustrated in Figure 12.13.
What is the mathematical form of the income effect? This is actually more straightforward to compute than the substitution effect computed above. As with the substitution effect, we differentiate the conditions $$M=p X x+p Y y$$ and $$\mathrm{p} \mathrm{Xu} 2=\mathrm{p} \mathrm{Y} \text { u } 1$$, holding pX and pY constant, to obtain $$\mathrm{dM}=\mathrm{p} \mathrm{X} \mathrm{d} \mathrm{x}+\mathrm{p} \mathrm{Y} \mathrm{d} \mathrm{y}$$ and $$p X(u 12 d x+u 22 d y)=p Y(u 11 d x+u 12 d y)$$.
Figure 12.12 Backward bending—inferior good
The second condition can also be written as $$\text { dy } \mathrm{d} \mathrm{x}=\mathrm{p} \mathrm{Y} \text { u } 11-\mathrm{p} \mathrm{X} \text { u } 12 \mathrm{p} \mathrm{X} \text { u } 22-\mathrm{p} \mathrm{Y} \text { u } 12$$
This equation alone defines the slope of the Engel curve without determining how large a change arises from a given change in M. The two conditions together can be solved for the effects of M on X and Y. The Engel curve is given by
$\mathrm{dx} \mathrm{dM}=\mathrm{p} \times 2 \mathrm{u} 11-2 \mathrm{p} \mathrm{X} \mathrm{u} 12+\mathrm{p} \times 2 \mathrm{u} 22 \mathrm{p} \mathrm{X} \mathrm{u} 22-\mathrm{p} \mathrm{Y} \text { u } 12$
and
$\text { dy } \mathrm{dM}=\mathrm{p} \text { Y } 2 \text { u } 11-2 \mathrm{p} \mathrm{X} \text { u } 12+\mathrm{p} \times 2 \text { u } 22 \text { p } \mathrm{Y} \text { u } 11-\mathrm{p} \mathrm{X} \text { u } 12$
Note (from the second-order condition) that good Y is inferior if $$p Y u 11-p X u 12>0$$, or if u 11 u 1 − u 12 u 2 >0, or u 1 u 2 is increasing in x. Since u 1 u 2 is locally constant when M increases, equaling the price ratio, and an increase in y increases u 1 u 2 (thanks to the second-order condition), the only way to keep u 1 u 2 equal to the price ratio is for x to fall. This property characterizes an inferior good—an increase in the quantity of the good increases the marginal rate of substitution of that good for another good.
Key Takeaways
• When an increase in income causes a consumer to buy more of a good, that good is called a normal good for that consumer. When the consumer buys less, the good is called an inferior good. At a sufficiently high income, most goods become inferior.
• The curve that shows the path of consumption as income rises is known as an Engel curve.
• For the Cobb-Douglas utility, Engel curves are straight lines through the origin.
• Not all goods can be inferior.
• The effect of a price increase decomposes into two effects: a decrease in real income and a substitution effect from the change in the price ratio. For normal goods, a price increase decreases quantity. For inferior goods, a price increase decreases quantity only if the substitution effect is larger than the income effect.
EXERCISES
1. Show that, in the case of perfect complements, the Engel curve does not depend on prices.
2. Compute the substitution effect and income effect associated with a multiplicative price increase Δ in $$p_{Y}$$ that is, multiplying pY by Δ > 1 for the case of the Cobb-Douglas utility $$u(x, y)=x \text { a } y 1-a$$ | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/12%3A_Consumer_Theory/12.06%3A_Income_Effects.txt |
Learning Objectives
• Are there important details that haven’t been addressed in the presentation of utility maximization?
• What happens when consumers buy none of a good?
Let us revisit the maximization problem considered in this chapter to provide conditions under which local maximization is global. The consumer can spend M on either or both of two goods. This yields a payoff of $$h(x)=u(x, M-p X x p Y)$$. When is this problem well behaved? First, if h is a concave function of x, which implies h ″ (x)≤0, The definition of concavity is such that h is concave if 0 < a < 1 and for all x, y, $$h(a x+(1-a) y) \geq a h(x)+(1-a) h(y)$$. It is reasonably straightforward to show that this implies the second derivative of h is negative; and if h is twice differentiable, the converse is true as well. then any solution to the first-order condition is, in fact, a maximum. To see this, note that h ″ (x)≤0 entails h ′ (x) is decreasing. Moreover, if the point x* satisfies h ′ (x*)=0, then for $$x \leq x^{*}, h^{\prime}(x) \geq 0 ; \text { and for } x \geq x^{*}, h^{\prime}(x) \leq 0$$, because h ′ (x) gets smaller as x gets larger, and h ′ (x*)=0. Now consider x ≤ x*. Since h ′ (x)≥0, h is increasing as x gets larger. Similarly, for x ≥ x*, h ′ (x)≤0, which means that h gets smaller as x gets larger. Thus, h is concave and h ′ (x*)=0 means that h is maximized at x*.
Thus, a sufficient condition for the first-order condition to characterize the maximum of utility is that h ″ (x)≤0 for all x, pX, pY, and M. Letting z= p X p Y , this is equivalent to u 11 −2z u 12 + z 2 u 22 ≤0 for all z > 0.
In turn, we can see that this requires (i) u11 ≤ 0 (z = 0), (ii) u22 ≤ 0 (z→∞), and (iii) u 11 u 22 − u 12 ≥0 ( z= u 11 u 22 ). In addition, since
$-(u 11+2 z u 12+z 2 u 22)=(-u 11-z-u 22) 2+2 z(u 11 u 22-u 12)$
(i), (ii), and (iii) are sufficient for u11 +2zu 12 + z2u 22 ≤0.
Therefore, if (i) u11 ≤ 0, (ii) u22 ≤ 0, and (iii) u 11 u 22 − u 12 ≥0, a solution to the first-order conditions characterizes utility maximization for the consumer.
When will a consumer specialize and consume zero of a good? A necessary condition for the choice of x to be zero is that the consumer doesn’t benefit from consuming a very small x ; that is, h ′ (0)≤0. This means that
$h^{\prime}(0)=u 1(0, M p Y)-u 2(0, M p Y) p X p Y \leq 0$
or
$\text { u } 1(0, \mathrm{MpY}) \text { u } 2(0, \mathrm{MpY}) \leq \mathrm{pX} \mathrm{pY}$
Moreover, if the concavity of h is met, as assumed above, then this condition is sufficient to guarantee that the solution is zero. To see this, note that concavity of h implies h ′ is decreasing. Combined with h ′ (0)≤0, this entails that h is maximized at 0. An important class of examples of this behavior is quasilinear utility. Quasilinear utility comes in the form u(x, y) = y + v(x), where v is a concave function ( v ″ (x)≤0 for all x). That is, quasilinear utility is utility that is additively separable.
Figure 12.14 Quasilinear isoquants
The procedure for dealing with corners is generally this. First, check concavity of the h function. If h is concave, we have a procedure to solve the problem; when h is not concave, an alternative strategy must be devised. There are known strategies for some cases that are beyond the scope of this text. Given h concave, the next step is to check the endpoints and verify that h ′ (0)>0 (for otherwise x = 0 maximizes the consumer’s utility) and h ′ ( M p X )<0 (for otherwise y = 0 maximizes the consumer’s utility). Finally, at this point we seek the interior solution h ′ (x)=0. With this procedure, we can ensure that we find the actual maximum for the consumer rather than a solution to the first-order conditions that don’t maximize the consumer’s utility.
Key Takeaways
• Conditions are available that ensure that the first-order conditions produce a utility maximum.
• With convex preferences, zero consumption of one good arises when utility is decreasing in the consumption of one good, spending the rest of income on the other good.
EXERCISE
1. Demonstrate that the quasilinear consumer will consume zero X if and only if $$v^{\prime}(0) \leq p x p y$$, and that the consumer instead consumes zero Y if $$v^{\prime}(M p X) \geq p x p y$$. The quasilinear utility isoquants, for $$v(x)=(x+0.03) 0.3$$, are illustrated in Figure 12.14. Note that, even though the isoquants curve, they are nonetheless parallel to each other. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/12%3A_Consumer_Theory/12.07%3A_Mathematical_Cleanup.txt |
Learning Objectives
• If we want people to work more, should we pay them more or will that cause them to work less?
Consider a taxi driver who owns a car, a convenience store owner, or anyone else who can set his or her own hours. Working has two effects on this consumer: more goods consumption but less leisure consumption. To model this, we let x be the goods consumption, L the amount of nonwork time or leisure, and working time T – L, where T is the amount of time available for activities of all kinds. The variable L includes a lot of activities that aren’t necessarily fun—like trips to the dentist, haircuts, and sleeping—but for which the consumer isn’t paid and which represent choices. One could argue that sleeping isn’t really a choice, in the sense that one can’t choose zero sleep; but this can be handled by adjusting T to represent “time available for chosen behavior” so that T – L is work time and L is chosen nonwork activities. We set L to be leisure rather than labor supply because it is leisure that is the good thing, whereas most of us view working as something that we are willing to do provided we’re paid for it.
Labor supply is different from other consumption because the wage enters the budget constraint twice: first as the price of leisure, and second as income from working. One way of expressing this is to write the consumer’s budget constraint as $$p x+w L=M+w T$$.
Here, M represents nonwork income, such as gifts, government transfers, and interest income. We drop the subscript on the price of X and use w as the wage. Finally, we use a capital L for leisure because a lowercase L looks like the number one. The somewhat Dickensian idea is that the consumer’s maximal budget entails working the total available hours T, and any nonworked hours are purchased at the wage rate w. Alternatively, one could express the budget constraint so as to reflect that expenditures on goods px equal the total money, which is the sum of nonwork income M and work income $$w(T-L), \text { or } p x=M+w(T-L)$$.
These two formulations of the budget constraint are mathematically equivalent.
The strategy for solving the problem is also equivalent to the standard formulation, although there is some expositional clarity used by employing the budget constraint to eliminate x. That is, we write the utility $$u(x, L) \text { as } h(L)=u(M+w(T-L) p L)$$.
As before, we obtain the first-order condition $$0=h^{\prime}\left(L^{*}\right)=-u 1(w p)+u 2$$, where the partial derivatives u1 and u2 are evaluated at $$\left(M+w\left(T-L^{*}\right) p, L^{*}\right)$$. Note that the first-order condition is the same as the standard two-good theory developed already. This is because the effect, so far, is merely to require two components to income: M and wT, both of which are constant. It is only when we evaluate the effect of a wage increase that we see a difference.
To evaluate the effect of a wage increase, differentiate the first-order condition to obtain
$0=(u 11(w p) 2-2 u 12(w p)+u 22) d L d w-u 1 p-(w p) u 11 T-L p+u 12 T-L p$
Since $$\text { u } 11(\text { w } p) 2-2 \text { u } 12(w p)+u 22<0$$
by the standard second-order condition,
dL dw >0 if and only if u 1 p +( w p ) u 11 T−L p − u 12 T−L p <0;
that is, these expressions are equivalent to one another. Simplifying the latter, we obtain −( w p ) u 11 T−L p + u 12 T−L p u 1 p >1, or (T−L) −( w p ) u 11 + u 12 u 1 >1, or ∂ ∂L Log( u 1 )> 1 T−L =− ∂ ∂L Log(T−L), or ∂ ∂L Log( u 1 )+ ∂ ∂L Log(T−L)>0, or ∂ ∂L Log( u 1 (T−L) )>0.
Since the logarithm is increasing, this is equivalent to u 1 (T−L) being an increasing function of L. That is, L rises with an increase in wages and a decrease in hours worked if the marginal utility of goods times the hours worked is an increasing function of L, holding constant everything else, but evaluated at the optimal values. The value u1 is the marginal value of an additional good, while the value T – L represents the hours worked. Thus, in particular, if goods and leisure are substitutes, so that an increase in L decreases the marginal value of goods, then an increase in wages must decrease leisure, and labor supply increases in wages. The case where the goods are complements holds a hope for a decreasing labor supply, so we consider first the extreme case of complements.
Example (Perfect complements): $$u(x, L)=\min \{x, L\}$$
In this case, the consumer will make consumption and leisure equal to maximize the utility, so $$\mathrm{M}+\mathrm{w}\left(\mathrm{T}-\mathrm{L}^{*}\right) \mathrm{p}=\mathrm{L}^{*} \text { or } \mathrm{L}^{*}=\mathrm{M}+\mathrm{wT} \mathrm{p} 1+\mathrm{w} \mathrm{p}=\mathrm{M}+\mathrm{wT} \mathrm{p}+\mathrm{w}$$.
Thus, L is increasing in the wages if pT > M; that is, if M is sufficiently small so that one can’t buy all of one’s needs and not work at all. (This is the only reasonable case for this utility function.) With strong complements between goods and leisure, an increase in wages induces fewer hours worked.
Example (Cobb-Douglas): $$h(L)=(M+w(T-L) p) a L 1-a$$.
The first-order condition gives $$0=h^{\prime}(L)=-a(M+w(T-L) p) a-1\lfloor 1-a w p+(1-a)(M+w(T-L) p) a L-a$$
or $$\text { al } w p=(1-a) M+w(T-L) p$$,
$$\mathrm{w} \mathrm{p} \mathrm{L}=(1-\mathrm{a}) \mathrm{M}+\mathrm{wT} \mathrm{p}$$
or $$\mathrm{L}=(1-\mathrm{a})(\mathrm{M} \mathrm{w}+\mathrm{T})$$
If M is high enough, the consumer doesn’t work but takes L = T; otherwise, the equation gives the leisure, and labor supply is given by T−L=Max{ 0,αT−(1−α)( M w ) }.
Labor supply increases with the wage, no matter how high the wage goes.
The wage affects not just the price of leisure but also the income level. This makes it possible for the income effect of a wage increase to dominate the substitution effect. Moreover, we saw that this is more likely when the consumption of goods takes time; that is, the goods and leisure are complements.
Figure 13.1 Hours per week
A number of physicists have changed careers to become researchers in finance or financial economics. Research in finance pays substantially better than research in physics, and yet requires many of the same mathematical skills like stochastic calculus. Physicists who see their former colleagues driving Porsches and buying summerhouses are understandably annoyed that research in finance—which is intellectually no more difficult or challenging than physics—pays so much better. Indeed, some physicists are saying that other fields—such as finance, economics, and law—“shouldn’t” pay more than physics.
The difference in income between physics’ researchers and finance researchers is an example of a compensating differential. A compensating differential is income or costs that equalize different choices. There are individuals who could become either physicists or finance researchers. At equal income, too many choose physics and too few choose finance, in the sense that there is a surplus of physicists and a shortage of finance researchers. Finance salaries must exceed physics’ salaries in order to induce some of the researchers who are capable of doing either one to switch to finance, which compensates those individuals for doing the less desirable task.
Jobs that are dangerous or unpleasant must pay more than jobs requiring similar skills but without the bad attributes. Thus, oil-field workers in Alaska’s North Slope, well above the Arctic Circle, earn a premium over workers in similar jobs in Houston, Texas. The premium—or differential pay—must be such that the marginal worker is indifferent between the two choices: The extra pay compensates the worker for the adverse working conditions. This is why it is known in economics’ jargon by the phrase of a compensating differential.
The high salaries earned by professional basketball players are not compensating differentials. These salaries are not created because of a need to induce tall people to choose basketball over alternative jobs like painting ceilings, but instead are payments that reflect the rarity of the skills and abilities involved. Compensating differentials are determined by alternatives, not by direct scarcity. Professional basketball players are well paid for the same reason that Picasso’s paintings are expensive: There aren’t very many of them relative to demand.
A compensating differential is a feature of other choices as well as career choices. For example, many people would like to live in California for its weather and scenic beauty. Given the desirability of California over, for example, Lincoln, Nebraska, or Rochester, New York, there must be a compensating differential for living in Rochester; and two significant ones are air quality and housing prices. Air quality worsens as populations rise, thus tending to create a compensating differential. In addition, the increase in housing prices also tends to compensate—housing is inexpensive in Rochester, at least compared with California.There are other compensations, besides housing, for living in Rochester—cross-country skiing and proximity to mountains and lakes, for example. Generally, employment is only a temporary factor that might compensate, because employment tends to be mobile, too, and move to the location that the workers prefer, when possible. It is not possible on Alaska’s North Slope.
Housing prices also compensate for location within a city. For most people, it is more convenient—both in commuting time and for services—to be located near the central business district than in the outlying suburbs. The main compensating differentials are school quality, crime rates, and housing prices. We illustrate the ideas with a simple model of a city in the next section.
Key Takeaways
• Leisure—time spent not working—is a good like other goods, and the utility cost of working is less leisure.
• Labor supply is different from other goods because the wage enters the budget constraint twice—first as the price of leisure, and second as income from working.
• If goods and leisure are substitutes, so that an increase in L decreases the marginal value of goods, then an increase in wages must decrease leisure, and labor supply increases in wages.
• With strong complements between goods and leisure, an increase in wages induces fewer hours worked.
• Complementarity between goods and leisure is reasonable because it takes time to consume goods.
• For most developed nations, increases in wages are associated with fewer hours worked.
• A compensating differential is income or costs that equalize different choices.
• Jobs that are dangerous or unpleasant must pay more than jobs requiring similar skills but without the bad attributes.
• The premium—or differential pay—must be such that the marginal worker is indifferent between the two choices: The extra pay compensates the worker for the adverse working conditions.
• City choice is also subject to compensating differentials, and significant differentials include air quality, crime rates, tax rates, and housing prices.
EXERCISES
1. A thought question: Does a bequest motive—the desire to give money to others—change the likelihood that goods and leisure are complements?
2. Show that an increase in the wage increases the consumption of goods; that is, x increases when the wage increases. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/13%3A_Applied_Consumer_Theory/13.01%3A_Labor_Supply.txt |
Learning Objectives
• How are the prices of suburban ranch houses, downtown apartments, and rural ranches determined?
An important point to understand is that the good, in limited supply in cities, is not a physical structure like a house, but the land on which the house sits. The cost of building a house in Los Angeles is quite similar to the cost of building a house in Rochester, New York. The big difference is the price of land. A $1 million house in Los Angeles might be a$400,000 house sitting on a $600,000 parcel of land. The same house in Rochester might be$500,000—a $400,000 house on a$100,000 parcel of land.
Usually, land is what fluctuates in value rather than the price of the house that sits on the land. When a newspaper reports that house prices rose, in fact what rose were land prices, for the price of housing has changed only at a slow pace, reflecting increased wages of house builders and changes in the price of lumber and other inputs. These do change, but historically the changes have been small compared to the price of land.
We can construct a simple model of a city to illustrate the determination of land prices. Suppose the city is constructed on a flat plane. People work at the origin (0, 0). This simplifying assumption is intended to capture the fact that a relatively small, central portion of most cities involves business, with a large area given over to housing. The assumption is extreme, but not unreasonable as a description of some cities.
Suppose commuting times are proportional to distance from the origin. Let c(t) be the cost to the person of a commute of time t, and let the time taken be t = λr, where r is the distance. The function c should reflect both the transportation costs and the value of time lost. The parameter λ accounts for the inverse of the speed in commuting, with a higher λ indicating slower commuting. In addition, we assume that people occupy a constant amount of land. This assumption is clearly wrong empirically, and we will consider making house size a choice variable later.
A person choosing a house priced at p (r), at distance r, thus pays $$$c(\lambda r)+p(r)$$$ for the combination of housing and transportation. People will choose the lowest cost alternative. If people have identical preferences about housing and commuting, then house prices p will depend on distance and will be determined by $$$c(\lambda r)+p(r)$$$ equal to a constant, so that people are indifferent to the distance from the city’s center—decreased commute time is exactly compensated by increased house prices.
The remaining piece of the model is to figure out the constant. To do this, we need to figure out the area of the city. If the total population is N, and people occupy an area of one per person, then the city size rmax satisfies $$\mathrm{N}=\Pi \mathrm{r} \max 2$$, and thus $$r \max =N \Pi$$.
At the edge of the city, the value of land is given by some other use, like agriculture. From the perspective of the determinant of the city’s prices, this value is approximately constant. As the city takes more land, the change in agricultural land is a very small portion of the total land used for agriculture. Let the value of agricultural land be v per housing unit size. Then the price of housing is $$p\left(r_{\max }\right)=v$$, because this is the value of land at the edge of the city. This lets us compute the price of all housing in the city:
$$c(\lambda r)+p(r)=c(\lambda r \max )+p(r \max )=c(\lambda r \max )+v=c(\lambda N n)+v$$
or
$$p(r)=c(\lambda N n)+v-c(\lambda r)$$
This equation produces housing prices like those illustrated in Figure 13.2, where the peak is the city’s center. The height of the figure indicates the price of housing.
Figure 13.2 House price gradient
It is straightforward to verify that house prices increase in the population N and the commuting time parameter λ, as one would expect. To quantify the predictions, we consider a city with a population of 1,000,000; a population density of 10,000 per square mile; and an agricultural use value of $6 million per square mile. To translate these assumptions into the model’s structure, first note that a population density of 10,000 per square mile creates a fictitious “unit of measure” of about 52.8 feet, which we’ll call a purlong, so that there is one person per square purlong (2,788 square feet). Then the agricultural value of a property is v =$600 per square purlong. Note that this density requires a city of radius rmax equal to 564 purlongs, which is 5.64 miles.
The only remaining structure left to identify in the model is the commuting cost c. To simplify the calculations, let c be linear. Suppose that the daily cost of commuting is $2 per mile (roundtrip), so that the present value of daily commuting costs in perpetuity is about$10,000 per mile.This amount is based upon 250 working days per year, for an annual cost of about $500 per mile, yielding a present value at 5% interest of$10,000. See Section 11.1. With a time value of $25 per hour and an average speed of 40 mph (1.5 minutes per mile), the time cost is 62.5 cents per minute. Automobile costs (such as gasoline, car depreciation, and insurance) are about 35–40 cents per mile. Thus the total is around$1 per mile, which doubles with roundtrips. This translates into a cost of commuting of $100 per purlong. Thus, we obtain $$\mathrm{p}(\mathrm{r})=\mathrm{c}(\lambda \mathrm{N} \mathrm{n})+\mathrm{v}-\mathrm{c}(\lambda \mathrm{r})=\ 100(\mathrm{N} \mathrm{n}-\mathrm{r})+\ 600=\ 57,000-\ 100 \mathrm{r}$$ Thus, the same 2,788-square-foot property at the city’s edge sells for$600 versus $57,000, less than six miles away at the city’s center. With reasonable parameters, this model readily creates dramatic differences in land prices, based purely on commuting time. As constructed, a quadrupling of population approximately doubles the price of land in the central city. This probably understates the change, since a doubling of the population would likely increase road congestion, increasing λ and further increasing the price of central city real estate. As presented, the model contains three major unrealistic assumptions. First, everyone lives on an identically sized piece of land. In fact, however, the amount of land used tends to fall as prices rise. At$53 per square foot, most of us buy a lot less land than at 20 cents per square foot. As a practical matter, the reduction of land per capita is accomplished both through smaller housing units and through taller buildings, which produce more housing floor space per acre of land. Second, people have distinct preferences and the disutility of commuting, as well as the value of increased space, varies with the individual. Third, congestion levels are generally endogenous—the more people who live between two points, the greater the traffic density and, consequently, the higher the level of λ. Problems arise with the first two assumptions because of the simplistic nature of consumer preferences embedded in the model, while the third assumption presents an equilibrium issue requiring consideration of transportation choices.
This model can readily be extended to incorporate different types of people, different housing sizes, and endogenous congestion. To illustrate such generalizations, consider making the housing size endogenous. Suppose preferences are represented by the utility function $$\mathrm{u}=\mathrm{H} \mathrm{a}-\lambda \mathrm{r}-\mathrm{p}(\mathrm{r}) \mathrm{H}$$, where H is the house size that the person chooses, and r is the distance that he or she chooses. This adaptation of the model reflects two issues. First, the transport cost has been set to be linear in distance for simplicity. Second, the marginal value of housing decreases in the house size, but the value of housing doesn’t depend on distance from the center. For these preferences to make sense, $$a<1$$ (otherwise either zero or an infinite house size emerges). A person with these preferences would optimally choose a house size of $$\mathrm{H}=(\mathrm{ap}(\mathrm{r})) 11-\mathrm{a}$$, resulting in utility
$$u^{*}=(a \text { a } 1-a-a 11-a) p(r)-a 1-a-\lambda r$$. Utility at every location is constant, so $$\left(a a 1-a-a 11-a u^{*}+\lambda r\right) 1-a a=p(r)$$.
A valuable attribute of the form of the equation for p is that the general form depends on the equilibrium values only through the single number u*. This functional form produces the same qualitative shapes as shown in Figure 13.2. Using the form, we can solve for the housing size H:
H(r)= ( α p(r) ) 1 1−α = α 1 1−α ( u*+λr α α 1−α − α 1 1−α ) 1 α = ( u*+λr α −1 −1 ) 1 α = ( α 1−α (u*+λr) ) 1 α .
The space in the interval $$[r, r+\Delta] \text { is } n\left(2 r \Delta+\Delta^{2}\right)$$. In this interval, there are approximately $$n(2 r \Delta+\Delta 2) H(r)=n(2 r \Delta+\Delta 2)\left(1-a a\left(u^{*}+\lambda r\right)\right) 1$$ α people. Thus, the number of people within rmax of the city’s center is $$\int 0 \text { r } \max 2 \pi r\left(1-a a\left(u^{*}+\lambda r\right)\right) 1 \text { a dr }=N$$
This equation, when combined with the value of land on the periphery $$\mathrm{v}=\mathrm{p}(\mathrm{r} \max )=\left(\mathrm{a} \mathrm{a} 1-\mathrm{a}-\mathrm{a} 11-\mathrm{a} \mathrm{u}^{*}+\lambda \mathrm{r} \max \right) 1-\mathrm{a} \mathrm{a}$$, jointly determines rmax and u*.
When different people have different preferences, the people with the highest disutility of commuting will tend to live closer to the city’s center. These tend to be people with the highest wages, since one of the costs of commuting is time that could have been spent working.
Key Takeaways
• An important point to understand is that, in cities, houses are not in limited supply; but it is the land on which the houses sit that is.
• The circular city model involves people who work at a single point but live dispersed around that point. It is both the size of the city and the housing prices that are determined by consumers who are indifferent to commuting costs—lower housing prices at a greater distance balance the increased commuting costs.
• Substituting plausible parameters into the circular city model produces dramatic house price differentials, explaining much of the price differences within cities.
• A quadrupling of population approximately doubles the price of land in the central city. This likely understates the actual change since an increase in population slows traffic.
EXERCISE
1. For the case of $$a=1 / 2$$, solve for the equilibrium values of u* and rmax. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/13%3A_Applied_Consumer_Theory/13.02%3A_Urban_Real_Estate_Prices.txt |
Learning Objectives
• How much should you save, and how do interest rate changes affect spending?
The consumption of goods doesn’t take place in a single instance, but over time. How does time enter into choice? We’re going to simplify the problem a bit and focus only on consumption, setting aside working for the time being. Let x1 be consumption in the first period and x2 in the second period. Suppose the value of consumption is the same in each period, so that $$u\left(x_{1}, x_{2}\right)=v\left(x_{1}\right)+\delta v\left(x_{2}\right)$$, where δ is called the rate of “pure” time preference. The consumer is expected to have income M1 in the first period and M2 in the second. There is a market for lending and borrowing, which we assume has a common interest rate r.
The consumer’s budget constraint, then, can be written $$(1+r)\left(M_{1}-x_{1}\right)=x_{2}-M_{2}$$
This equation says that the net savings in Period 1, plus the interest on the net savings in Period 1, equals the net expenditures in Period 2. This is because whatever is saved in Period 1 earns interest and can then be spent in Period 2; alternatively, whatever is borrowed in Period 1 must be paid back with interest in Period 2. Rewriting the constraint: $$(1+r) x_{1}+x_{2}=(1+r) M_{1}+M_{2}$$
This equation is known as the intertemporal budget constraint—that is, the budget constraint that allows for borrowing or lending. It has two immediate consequences. First, 1 + r is the price of Period 2 consumption in terms of Period 1 consumption. Thus, the interest rate gives the relative prices. Second, the relevant income is the permanent income, which is the present value of the income stream. Clearly a change in income that leaves the present value of income the same should have no effect on the choice of consumption.
Once again, as with the labor supply, a change in the interest rate affects not just the price of consumption but also the budget for consumption. In other words, an increase in the interest rate represents an increase in budget for net savers but a decrease in budget for net borrowers.
As always, we rewrite the optimization problem to eliminate one of the variables, to obtain $$u=v(x 1)+\delta v((1+r)(M 1-x 1)+M 2)$$
Thus, the first-order conditions yield $$0=v^{\prime}(x 1)-(1+r) \delta v^{\prime}(x 2)$$
This condition says that the marginal value of consumption in Period 1, v ′ ( x 1 ), equals the marginal value of consumption in Period 2, δ v ′ ( x 2 ), times the interest factor. That is, the marginal present values are equated. Note that the consumer’s private time preference, δ, need not be related to the interest rate. If the consumer values Period 1 consumption more than does the market, so that $$\delta(1+r)<1$$, then $$v^{\prime}(x 1)<v^{\prime}(x 2)$$; that is, the consumer consumes more in Period 1 than in Period 2.As usual, we are assuming that utility is concave, which in this instance means that the second derivative of v is negative, and thus the derivative of v is decreasing. In addition, to ensure an interior solution, it is useful to require what are called the Inada conditions: $$v^{\prime}(0)=\infty \text { and } v^{\prime}(\infty)=0$$ Similarly, if the consumer’s discount of future consumption is exactly equal to the market discount, $$\delta(1+r)=1$$, the consumer will consume the same amount in both periods. Finally, if the consumer values Period 1 consumption less than the market, $$\delta(1+r)>1$$, the consumer will consume more in Period 2. In this case, the consumer is more patient than the market.
Figure 13.3 Borrowing and lending
The effect of an interest rate increase is to pivot the budget constraint around the point (M1, M2). Note that this point is always feasible—that is, it is feasible to consume one’s own endowment. The effect of an increase in the interest rate is going to depend on whether the consumer is a borrower or a lender. As Figure 13.4 illustrates, the net borrower borrows less in the first period—the price of first-period consumption has risen and the borrower’s wealth has fallen. It is not clear whether the borrower consumes less in the second period because the price of second-period consumption has fallen, even though wealth has fallen, too—two conflicting effects.
An increase in interest rates is a benefit to a net lender. The lender has more income, and the price of Period 2 consumption has fallen. Thus the lender must consume more in the second period, but only consumes more in the first period (lends less) if the income effect outweighs the substitution effect. This is illustrated in Figure 13.5.
Figure 13.4 Interest rate change
The government, from time to time, will rebate a portion of taxes to “stimulate” the economy. An important aspect of the outcome of such a tax rebate is the effect to which consumers will spend the rebate, versus save the rebate, because the stimulative effects of spending are thought to be larger than the stimulative effects of saving.This belief shouldn’t be accepted as necessarily true; it was based on a model that has since been widely rejected by the majority of economists. The general idea is that spending creates demand for goods, thus encouraging business investment in production. However, saving encourages investment by producing spendable funds, so it isn’t at all obvious whether spending or saving has a larger effect. The theory suggests how people will react to a “one-time” or transitory tax rebate, compared to a permanent lowering of taxes. In particular, the budget constraint for the consumer spreads lifetime income over the lifetime. Thus, for an average consumer who might spend a present value of $750,000 over a lifetime, a$1,000 rebate is small potatoes. On the other hand, a $1,000 per year reduction is worth$20,000 or so over the lifetime, which should have 20 times the effect of the transitory change on the current expenditure.
Tax rebates are not the only way that we receive one-time payments. Money can be found, or lost, and we can have unexpected costs or windfall gifts. From an intertemporal budget constraint perspective, these transitory effects have little significance; and thus the theory suggests that people shouldn’t spend much of a windfall gain in the current year, or cut back significantly when they have a moderately sized, unexpected cost.
As a practical matter, most individuals can’t borrow at the same rate at which they lend. Many students borrow on credit cards at very high interest rates and obtain a fraction of that in interest on savings. That is to say, borrowers and lenders face different interest rates. This situation is readily identified in Figure 13.6. The cost of a first-period loan is a relatively high loss of x2, and similarly the value of first-period savings is a much more modest increase in second-period consumption. Such effects tend to favor “neither a borrower nor a lender be,” as Shakespeare recommends, although it is still possible for the consumer to optimally borrow in the first period (e.g., if M1 = 0) or in the second period (if M2 is small relative to M1).
Figure 13.6 Different rates for borrowing and lending
Figure 13.7 The effect of a transitory income increase
Key Takeaways
• The intertemporal budget constraint takes into account the fact that savings obtain interest, producing additional money. The price of early consumption is one plus the interest rate.
• The relevant income is “permanent income” rather than “current income.” A change in income that leaves the present value of income the same should have no effect on the choice of consumption.
• A change in the interest rate affects not just the price of consumption but also the budget for consumption. An increase in the interest rate represents an increase in budget for savers but a decrease in budget for borrowers.
• If the consumer values early consumption more than the market, the consumer consumes more early rather than later, and conversely.
• Whether the consumer is a lender or borrower depends not just on the preference for earlier versus later consumption but also on the timing of their incomes.
• The effect of an interest rate increase is to pivot the budget constraint around the income point. The effect of an increase in the interest rate on consumption is going to depend on whether the consumer is a borrower or a lender.
• An increase in interest rates is a benefit to a net lender. The lender must continue to lend in the present and will consume more in the future.
• People should react less to a “one-time” or transitory tax rebate than to a permanent lowering of taxes.
• Most individuals can’t borrow at the same rate at which they lend. Interest rate differentials favor spending all of one’s income.
• Differences in borrowing and lending interest rates cause transitory changes in income to have larger effects than the intertemporal budget constraint would suggest. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/13%3A_Applied_Consumer_Theory/13.03%3A_Dynamic_Choice.txt |
Learning Objectives
• How should you evaluate gambles?
• How is risk priced?
There are many risks in life, even if one doesn’t add to these risks by intentionally buying lottery tickets. Gasoline prices go up and down, the demand for people trained in your major fluctuates, and house prices change. How do people value gambles? The starting point for the investigation is the von Neumann-MorgensternJohn von Neumann (1903–1957) and Oskar Morgenstern (1902–1977) are the authors of Theory of Games and Economic Behavior (Princeton, NJ: Princeton University Press, 1944). utility function. The idea of a von Neumann-Morgenstern utility function for a given person is that, for each possible outcome x, there is a value v(x) assigned by the person, and the average value of v is the value the person assigns to the risky outcome. In other words, the von Neumann-Morgenstern utility function is constructed in such a way that a consumer values gambles as if they were the expected utility
This is a “state-of-the-world” approach, in the sense that each of the outcomes is associated with a state of the world, and the person maximizes the expected value of the various possible states of the world. Value here doesn’t mean a money value, but a psychic value or utility.
To illustrate the assumption, consider equal probabilities of winning $100 and winning$200. The expected outcome of this gamble is $150—the average of$100 and $200. However, the actual value of the outcome could be anything between the value of$100 and the value of $200. The von Neumann-Morgenstern utility is $$1 / 2 v(\ 100)+1 / 2 v(\ 200)$$ The von Neumann-Morgenstern formulation has certain advantages, including the logic that what matters is the average value of the outcome. On the other hand, in many tests, people behave in ways not consistent with the theory.For example, people tend to react more strongly to very unlikely events than is consistent with the theory. Nevertheless, the von Neumann approach is the prevailing model of behavior under risk. To introduce the theory, we will consider only money outcomes, and mostly the case of two money outcomes. The person has a von Neumann-Morgenstern utility function v of these outcomes. If the possible outcomes are x1, x2, … , xn and these occur with probability π1, π2, … , πn respectively, the consumer’s utility is $u= π 1 v( x 1 )+ π 2 v( x 2 )+…+ π n v( x n )= ∑ i=1 n π i v( x i ) .$ This is the meaning of “having a von Neumann-Morgenstern utility function”—that utility can be written in this weighted sum form. The first insight that flows from this definition is that an individual dislikes risk if v is concave. To see this, note that the definition of concavity posits that v is concave if, for all π in [0, 1] and all values x1 and x2, $$v(n \times 1+(1-n) \times 2) \geq \pi v(\times 1)+(1-n) v(\times 2)$$ For smoothly differentiable functions, concavity is equivalent to a second derivative that is not positive. Using induction, the definition of concavity can be generalized to show $$v( π 1 x 1 + π 2 x 2 +…+ π n x n )≥ π 1 v( x 1 )+ π 2 v( x 2 )+…+ π n v( x n ).$$. Figure 13.8 Expected utility and certainty equivalents A useful concept is the certainty equivalent of a gamble. The certainty equivalent is an amount of money that provides equal utility to the random payoff of the gamble. The certainty equivalent is labeled CE in the figure. Note that CE is less than the expected outcome, if the person is risk averse. This is because risk-averse individuals prefer the expected outcome to the risky outcome. The risk premium is defined to be the difference between the expected payoff (this is expressed as πx1 + (1 – π)x2 in the figure) and the certainty equivalent. This is the cost of risk—it is the amount of money an individual would be willing to pay to avoid risk. This means as well that the risk premium is the value of insurance. How does the risk premium of a given gamble change when the base wealth is increased? It can be shown that the risk premium falls as wealth increases for any gamble, if and only if $$-v^{n}(x) v^{\prime}(x)$$ is decreasing. The measure $$\rho(x)=-v^{\prime \prime}(x) v^{\prime}(x)$ (x)$ is known as the Arrow-Pratt measure of risk aversionThe measure was named after its discoverers, Nobel laureate Kenneth Arrow and John Pratt., and also as the measure of absolute risk aversion. It is a measure of risk aversion computed as the negative of the ratio of the second derivative of utility divided by the first derivative of utility. To get an idea about why this measure matters, consider a quadratic approximation to v. Let μ be the expected value, and let δ2 be the expected value of $$(x-\mu)^{2}$ ). Then we can approximate v(CE) two different ways. $\mathrm{v}(\mu)+\mathrm{v}^{\prime}(\mu)(\mathrm{CE}-\mu) \approx \mathrm{v}(\mathrm{CE})=\mathrm{E}\{\mathrm{v}(\mathrm{x})\} \approx \mathrm{E}\left\{\mathrm{v}(\mu)+\mathrm{v}^{\prime}(\mu)(\mathrm{x}-\mu)+1 / 2 \mathrm{v}^{\prime \prime}(\mu)(\mathrm{x}-\mu) 2\right\}$ Thus, $\mathrm{v}(\mu)+\mathrm{v}^{\prime}(\mu)(\mathrm{CE}-\mu) \approx \mathrm{E}\left\{\mathrm{v}(\mu)+\mathrm{v}^{\prime}(\mu)(\mathrm{x}-\mu)+1 / 2 \mathrm{v}^{\prime \prime}(\mu)(\mathrm{x}-\mu) 2\right\}$ Canceling v(μ) from both sides and noting that the average value of x is μ, so E(x – μ) = 0, we have \($\mathbf{v}^{\prime}(\mu)(\mathrm{CE}-\mu) \approx 1 / 2 \mathrm{v}^{\prime \prime}(\mu) \sigma 2$$ Then, dividing by $$\mathrm{v}^{\prime}(\mathrm{x}), \mu-\mathrm{CE} \approx 12 \mathrm{v}(\mu) \mathrm{v}^{\prime}(\mu) \sigma 2=12 \rho(\mu) \sigma 2$$ That is, the risk premium—the difference between the average outcome and the certainty equivalent—is approximately equal to the Arrow-Pratt measure times half the variance, at least when the variance is small. The translation of risk into dollars, by way of a risk premium, can be assessed even for large gambles if we are willing to make some technical assumptions. If a utility has constant absolute risk aversion (CARA), the measure of risk aversion doesn’t change with wealth; that is ρ=− v ″ (x) v ′ (x) is a constant. This turns out to imply, after setting the utility of zero to zero, that $$v(x)=1 \rho(1-e-\rho x)$$. (This formulation is derived by setting v(0) = 0, handling the case of ρ = 0 with appropriate limits.) Now also assume that the gamble x is normally distributed with mean μ and variance δ2. Then the expected value of v(x) is $$\operatorname{Ev}(\mathrm{x})=1 \rho(1-\mathrm{e}-\rho(\mu-\rho 2 \sigma 2))$$ It is an immediate result from this formula that the certainty equivalent, with CARA preferences and normal risks, is μ− ρ 2 σ 2 . Hence, the risk premium of a normal distribution for a CARA individual is ρ 2 σ 2 . This formulation will appear when we consider agency theory and the challenges of motivating a risk averse employee when outcomes have a substantial random component. An important aspect of CARA with normally distributed risks is that the preferences of the consumer are linear in the mean of the gamble and the variance. In fact, given a choice of gambles, the consumer selects the one with the highest value of μ− ρ 2 σ 2 . Such preferences are often called mean variance preferences, and they describe people who value risk linearly with the expected return. Such preferences comprise the foundation of modern finance theory. Key Takeaways • The von Neumann-Morgenstern utility function for a given person is a value v(x) for each possible outcome x, and the average value of v is the value the person assigns to the risky outcome. Under this theory, people value risk at the expected utility of the risk. • The von Neumann approach is the prevailing model of behavior under risk, although there are numerous experiment-based criticisms of the theory. • An individual dislikes risk if v is concave. • For smoothly differentiable functions, concavity is equivalent to a second derivative that is not positive. • People with concave von Neumann-Morgenstern utility functions are known as risk-averse people. • The certainty equivalent of a gamble is an amount of money that provides equal utility to the random payoff of the gamble. The certainty equivalent is less than the expected outcome if the person is risk averse. • The risk premium is defined to be the difference between the expected payoff and the certainty equivalent. • The risk premium falls as wealth increases for any gamble, if and only if − v ″ (x) v ′ (x) is decreasing. • The measure ρ(x)=− v ″ (x) v ′ (x) is known as the Arrow-Pratt measure of risk aversion, and also as the measure of absolute risk aversion. • The risk premium is approximately equal to the Arrow-Pratt measure times half the variance when the variance is small. • Constant absolute risk aversion provides a basis for “mean variance preferences,” the foundation of modern finance theory. EXERCISES 1. Use a quadratic approximation on both sides of the equation to sharpen the estimate of the risk premium. First, note that $$v(μ)+ v ′ (μ)(CE−μ)+½ v ″ (μ) (CE−μ) 2 ≈v(CE) =E{v(x)}≈E{v(μ)+ v ′ (μ)(x−μ)+½ v ″ (μ) (x−μ) 2 }.$$ 2. Conclude that $$\mu-\mathrm{CE} \approx 1 \rho(1+\rho 2 \sigma 2-1)$$. This approximation is exact to the second order. 3. Suppose that $$u(x)=x^{0.95}$$ for a consumer with a wealth level of$50,000. Consider a gamble, with equal probability of winning $100 and losing$100, and compute the risk premium associated with the gamble.
4. Suppose that $$u(x)=x^{0.99}$$ for a consumer with a wealth level of $100,000. A lottery ticket costs$1 and pays $5,000,000 with the probability 1 10,000,000 . Compute the certainty equivalent of the lottery ticket. 5. The return on U.S. government treasury investments is approximately 3%. Thus, a$1 investment returns $1.03 after one year. Treat this return as risk-free. The stock market (S&P 500) returns 7% on average and has a variance that is around 16% (the variance of return on a$1 investment is $0.16). Compute the value of ρ for a CARA individual. What is the risk premium associated with equal probabilities of a$100 gain or loss given the value of ρ?
6. A consumer has utility $$u(x)=x^{7 / 8}$$ and a base wealth of $100,000. She is about to take part in a gamble that will give her$10,000 (bringing her to $110,000) if a fair die rolls less than 3 (probability 1/3), but will cost her$5,000 (leaving her with \$95,000) otherwise.
1. What is the certainty equivalent of participating in this gamble?
2. How much would she be willing to pay in order to avoid this gamble? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/13%3A_Applied_Consumer_Theory/13.04%3A_Risk_Aversion.txt |
Learning Objectives
• How should a consumer go about finding the lowest price when available prices are random?
In most communities, grocery stores advertise sale prices every Wednesday in a newspaper insert, and these prices can vary from week to week and from store to store. The price of gasoline can vary as much as 15 cents per gallon in a one-mile radius. Should you decide that you want to buy a specific Sony television, you may see distinct prices at Best Buy and other electronics retailers. For many goods and services, there is substantial variation in prices, which implies that there are gains for buyers to search for the best price.
The theory of consumer search behavior is just a little bit arcane, but the basic insight will be intuitive enough. The general idea is that, from the perspective of a buyer, the price that is offered is random and has a probability density function f (p). If a consumer faces a cost of search (e.g., if you have to visit a store—in person, telephonically, or virtually—the cost includes your time and any other costs necessary to obtain a price quote), the consumer will set a reservation price, which is a maximum price that he or she will pay without visiting another store. That is, if a store offers a price below p*, the consumer will buy; otherwise, he or she will visit another store, hoping for a better price.
Call the reservation price p*, and suppose that the cost of search is c. Let J(p*) represent the expected total cost of purchase (including search costs). Then J must equal $$J(p*)=c+ ∫ 0 p* pf(p)dp + ∫ p* ∞ J(p*)f(p)dp$$.
This equation arises because the current draw (which costs c) could either result in a price less than p*, in which case observed price, with density f, will determine the price paid p; or the price will be too high, in which case the consumer is going to take another draw, at cost c, and on average get the average price J(p*). It is useful to introduce the cumulative distribution function F, with $$F(x)=\int 0 \times f(p) d p$$. Note that something has to happen, so F(∞) = 1.
We can solve the equality for $$J(p*), J(p*)= ∫ 0 p* pf(p)dp +c F(p*)$$.
This expression has a simple interpretation. The expected price J(p*) is composed of two terms. The first is the expected price, which is $$\int 0 \mathrm{p}^{*} \mathrm{p} \mathrm{f}(\mathrm{p}) \mathrm{F}\left(\mathrm{p}^{*}\right) \mathrm{dp}$$. This has the interpretation of the average price conditional on that price being less than p*. This is because f(p) F(p*) is, in fact, the density of the random variable which is the price given that the price is less than p*. The second term is c F(p*) . This is the expected search cost, and it arises because 1 F(p*) is the expected number of searches. This arises because the odds of getting a price low enough to be acceptable is F(p*). There is a general statistical property underlying the number of searches. Consider a basketball player who successfully shoots a free throw with probability y. How many balls, on average, must he throw to sink one basket? The answer is 1/y. To see this, note that the probability that exactly n throws are required is $$(1-y)^{n-1}$$ y. This is because n are required means that n – 1 must fail (probability (1 – y)n–1) and then the remaining one goes in, with probability y. Thus, the expected number of throws is
$y + 2(1-y)y + 3 (1-y) 2 y + 4 (1-y) 3 y+…=y(1 + 2(1-y) + 3(1-y) 2 + 4 (1-y) 3 +…)=y( (1 + (1-y) + (1-y) 2 + (1-y) 3 +…)+(1−y)(1 + (1-y) + (1-y) 2 + (1-y) 3 +…)= (1−y) 2 (1 + (1-y) + (1-y) 2 + (1-y) 3 +…)+ (1−y) 3 (1 + (1-y) + (1-y) 2 +…)+…=y( 1 y +(1−y) 1 y + (1−y) 2 1 y + (1−y) 3 1 y +… )= 1 y .$
Our problem has the same logic—where a successful basketball throw corresponds to finding a price less than p*.
The expected total cost of purchase, given a reservation price p*, is given by $$J(p*)= ∫ 0 p* pf(p)dp +c F(p*) .$$
But what value of p* minimizes cost? Let’s start by differentiating
$J ′ (p*)=p* f(p*) F(p*) − f(p*) ∫ 0 p* pf(p)dp +c F (p*) 2= f(p*) F(p*) ( p*− ∫ 0 p* pf(p)dp +c F(p*) )= f(p*) F(p*) ( p*−J(p*) ).$
Thus, if p* < J(p*), J is decreasing, and it lowers cost to increase p*. Similarly, if p* > J(p*), J is increasing in p*, and it reduces cost to decrease p*. Thus, minimization occurs at a point where p* = J(p*).
Moreover, there is only one such solution to the equation p* = J(p*) in the range where f is positive. To see this, note that at any solution to the equation p* = J(p*), J ′ (p*)=0 and
$J ″ (p*)= d dp* ( f(p*) F(p*) ( p*−J(p*) ) )=( d dp* f(p*) F(p*) )(p*−J(p*))+ f(p*) F(p*) (1− J ′ (p*))= f(p*) F(p*) > 0$
This means that J takes a minimum at this value, since its first derivative is zero and its second derivative is positive, and that is true about any solution to p* = J(p*). Were there to be two such solutions, J ′ would have to be both positive and negative on the interval between them, since J is increasing to the right of the first (lower) one, and decreasing to the left of the second (higher) one. Consequently, the equation p* = J(p*) has a unique solution that minimizes the cost of purchase.
Consumer search to minimize cost dictates setting a reservation price equal to the expected total cost of purchasing the good, and purchasing whenever the price offered is lower than that level. That is, it is not sensible to “hold out” for a price lower than what you expect to pay on average, although this might be well useful in a bargaining context rather than in a store searching context.
Example (Uniform): Suppose that prices are uniformly distributed on the interval [a, b]. For p* in this interval,
$J(p*)= ∫ 0 p* pf(p)dp +c F(p*) = ∫ a p* p dp b−a +c p*−a b−a= ½(p * 2 − a 2 )+c(b−a) p*−a =½(p*+a)+ c(b−a) p*−a .$
Thus, the first-order condition for minimizing cost is $$0=J^{\prime}\left(p^{*}\right)=1 / 2-c(b-a)\left(p^{*}-a\right) 2$$, implying $$p^{*}=a+2 c(b-a)$$.
There are a couple of interesting observations about this solution. First, not surprisingly, as c → 0, p* → α; that is, as the search costs go to zero, one holds out for the lowest possible price. This is sensible in the context of the model, but in real search situations delay may also have a cost that isn’t modeled here. Second, p* < b, the maximum price, if 2c < (b – a). In other words, if the most you can save by a search is twice the search cost, then don’t search, because the expected gains from the search will be half the maximum gains (thanks to the uniform distribution) and the search will be unprofitable.
The third observation, which is much more general than the specific uniform example, is that the expected price is a concave function of the cost of search (second derivative negative). That is, in fact, true for any distribution. To see this, define a function $$H(c)=\min p^{*} J\left(p^{*}\right)=\min p^{*} \int 0 p^{*} p f(p) d p+c F\left(p^{*}\right)$$. Since $$\mathrm{J}^{\prime}\left(\mathrm{p}^{*}\right)=0, \mathrm{H}^{\prime}(\mathrm{c})=\partial \partial \mathrm{c} \mathrm{J}\left(\mathrm{p}^{*}\right)=1 \mathrm{F}\left(\mathrm{p}^{*}\right)$$
From here it requires only a modest effort to show that p* is increasing in c, from which it follows that H is concave. This means that there are increasing returns to decreasing search costs in that the expected total price of search is decreasing at an increasing rate as the cost of search decreases.
Key Takeaways
• For many goods, prices vary across location and across time. In response to price variation, consumers will often search for low prices.
• In many circumstances the best strategy is a reservation price strategy, where the consumer buys whenever offered a price below the reservation price.
• Consumer search to minimize cost dictates setting a reservation price equal to the expected total cost of purchasing the good, and purchasing whenever the price offered is lower than that level.
EXERCISE
1. Suppose that there are two possible prices, one and two, and that the probability of the lower price one is x. Compute the consumer’s reservation price, which is the expected cost of searching, as a function of x and the cost of search c. For what values of x and c should the consumer accept two on the first search or continue searching until the lower price one is found? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/13%3A_Applied_Consumer_Theory/13.05%3A_Search.txt |
Learning Objectives
• How are several prices simultaneously determined?
• What are the efficient allocations?
• Does a price system equilibrium yield efficient prices?
The EdgeworthFrancis Edgeworth (1845–1926) introduced a variety of mathematical tools, including calculus, for considering economics and political issues, and was certainly among the first to use advanced mathematics for studying ethical problems. box considers a two-person, two-good “exchange economy.” That is, two people have utility functions of two goods and endowments (initial allocations) of the two goods. The Edgeworth box is a graphical representation of the exchange problem facing these people and also permits a straightforward solution to their exchange problem.
Figure 14.1 The Edgeworth box
What points are efficient? The economic notion of efficiency is that an allocation is efficient if it is impossible to make one person better off without harming the other person; that is, the only way to improve 1’s utility is to harm 2, and vice versa. Otherwise, if the consumption is inefficient, there is a rearrangement that makes both parties better off, and the parties should prefer such a point. Now, there is no sense of fairness embedded in the notion, and there is an efficient point in which one person gets everything and the other gets nothing. That might be very unfair, but it could still be the case that improving 2 must necessarily harm 1. The allocation is efficient if there is no waste or slack in the system, even if it is wildly unfair. To distinguish this economic notion, it is sometimes called Pareto efficiency.Vilfredo Pareto (1848–1923) was a pioneer in replacing concepts of utility with abstract preferences. His work was later adopted by the economics profession and remains the modern approach.
We can find the Pareto-efficient points by fixing Person 1’s utility and then asking what point, on the indifference isoquant of Person 1, maximizes Person 2’s utility. At that point, any increase in Person 2’s utility must come at the expense of Person 1, and vice versa; that is, the point is Pareto efficient. An example is illustrated in Figure 14.2.
Figure 14.2 An efficient point
This process of identifying the points that are Pareto efficient can be carried out for every possible utility level for Person 1. What results is the set of Pareto-efficient points, and this set is also known as the contract curve. This is illustrated with the thick line in Figure 14.3. Every point on this curve maximizes one person’s utility given the other’s utility, and they are characterized by the tangencies in the isoquants.
The contract curve need not have a simple shape, as Figure 14.3 illustrates. The main properties are that it is increasing and ranges from Person 1 consuming zero of both goods to Person 2 consuming zero of both goods.
Figure 14.3 The contract curve
Example: Suppose that both people have Cobb-Douglas utility. Let the total endowment of each good be one, so that x2 = 1 – x1. Then Person 1’s utility can be written as
$$u_{1}=x^{a} y^{1-a}$$, and 2’s utility is $$u_{2}=(1-x)^{\beta}(1-y)^{1-\beta}$$ Then a point is Pareto efficient if
$αy (1−α)x = ∂ u 1 ∂x ∂ u 1 ∂y = ∂ u 2 ∂x ∂ u 2 ∂y = β(1−y) (1−β)(1−x) .$
Thus, solving for y, a point is on the contract curve if
$y= (1−α)βx (1−β)α+(β−α)x = x (1−β)α (1−α)β + β−α (1−α)β x = x x+( (1−β)α (1−α)β )(1−x) .$
Thus, the contract curve for the Cobb-Douglas case depends on a single parameter (1−β)α (1−α)β . It is graphed for a variety of examples (α and β) in Figure 14.4.
Figure 14.4 Contract curves with Cobb-Douglas utility
Key Takeaways
• The Edgeworth box considers a two-person, two-good “exchange economy.” The Edgeworth box is a graphical representation of the exchange problem facing these people and also permits a straightforward solution to their exchange problem. A point in the Edgeworth box is the consumption of one individual, with the balance of the endowment going to the other.
• Pareto efficiency is an allocation in which making one person better off requires making someone else worse off—there are no gains from trade or reallocation.
• In the Edgeworth box, the Pareto-efficient points arise as tangents between isoquants of the individuals. The set of such points is called the contract curve. The contract curve is always increasing.
EXERCISES
1. If two individuals have the same utility function concerning goods, is the contract curve the diagonal line? Why or why not?
2. For two individuals with Cobb-Douglas preferences, when is the contract curve the diagonal line? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/14%3A_General_Equilibrium/14.01%3A_Edgeworth_Box.txt |
Learning Objectives
• How are prices in the two-person economy determined?
• Are these prices efficient?
The contract curve provides the set of efficient points. What point will actually be chosen? Let’s start with an endowment of the goods. An endowment is just a point in the Edgeworth box that represents the initial ownership of both goods for both people. The endowment is marked with a triangle in Figure 14.5. Note this point indicates the endowment of both Person 1 and Person 2 because it shows the shares of each.
Figure 14.5 Individually rational efficient points
In Figure 14.5, starting at the endowment, the utility of both players is increased by moving in a southeast direction—that is, down and to the right—until the contract curve is reached. This involves Person 1 getting more X (movement to the right) in exchange for giving up some Y (movement down). Thus, we can view the increase in utility as a trade—Person 1 trades some of his Y for some of Person 2’s X.
In principle, any of the darker points on the contract curve, which give both people at least as much as they achieve under autarky, might result from trade. The two people get together and agree on exchange that puts them at any point along this segment of the curve, depending upon the bargaining skills of the players. But there is a particular point, or possibly a set of points, that results from exchange using a price system. A price system involves a specific price for trading Y for X, and vice versa, that is available to both parties. In this figure, prices define a straight line whose slope is the negative of the Y for X price (the X for Y price is the reciprocal).
Figure 14.6 Equilibrium with a price system
The fact that a price line exists, that (i) goes through the endowment and (ii) goes through the contract curve at a point tangent to both people’s utility, is relatively easy to show. Consider lines that satisfy property (ii), and let’s see if we can find one that goes through the endowment. Start on the contract curve at the point that maximizes 1’s utility given 2’s reservation utility, and you can easily see that the price line through that point passes above and to the right of the endowment. The similar price line maximizing 2’s utility given 1’s reservation utility passes below and to the left of the endowment. These price lines are illustrated with dotted lines. Thus, by continuity, somewhere in between is a price line that passes through the endowment.
The point labeled O represents an equilibrium of the price system, in so far as supply and demand are equated for both goods. Note, given the endowment and the price through the endowment, both parties maximize utility by going to the O. To see this, it may help to consider a version of the figure that only shows Person 2’s isoquants and the price line.
Figure 14.7 Illustration of price system equilibrium
In the Edgeworth box, we see that, given an endowment, it is possible to reach some Pareto-efficient point using a price system. Moreover, any point on the contract curve arises as an equilibrium of the price system for some endowment. The proof of this proposition is startlingly easy. To show that a particular point on the contract curve is an equilibrium for some endowment, just start with an endowment equal to the point on the contract curve. No trade can occur because the starting point is Pareto efficient—any gain by one party entails a loss by the other.
Furthermore, if a point in the Edgeworth box represents an equilibrium using a price system (that is, if the quantity supplied equals the quantity demanded for both goods), it must be Pareto efficient. At an equilibrium to the price system, each player’s isoquant is tangent to the price line and, hence, tangent to each other. This implies that the equilibrium is Pareto efficient.
Two of the three propositions are known as the first and second welfare theorems of general equilibrium. The first welfare theorem of general equilibrium states that any equilibrium of the price system is Pareto efficient. The second welfare theorem of general equilibrium states that any Pareto-efficient point is an equilibrium of the price system for some endowment. They have been demonstrated by Nobel laureates Kenneth Arrow and Gerard Debreu, for an arbitrary number of people and goods. They also demonstrated the third proposition—that, for any endowment, there exists an equilibrium of the price system with the same high level of generality.
Key Takeaways
• Autarky means consuming one’s endowment without trade.
• If the endowment is not on the contract curve, there are points on the contract curve that make both people better off.
• A price system involves a specific price for trading Y for X, and vice versa, that is available to both parties. Prices define a straight line whose slope is the negative of the Y for X price (the X for Y price is the reciprocal).
• There is a price that (i) goes through the endowment and (ii) goes through the contract curve at a point tangent to both people’s utility. Such a price represents a supply and demand equilibrium: Given the price, both parties would trade to the same point on the contract curve.
• In the Edgeworth box, it is possible to reach some Pareto-efficient point using a price system. Moreover, any point on the contract curve arises as an equilibrium of the price system for some endowment.
• If a point in the Edgeworth box represents an equilibrium using a price system, it must be Pareto efficient.
• The first and second welfare theorems of general equilibrium are that any equilibrium of the price system is Pareto efficient and any Pareto-efficient point is an equilibrium of the price system for some endowment. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/14%3A_General_Equilibrium/14.02%3A_Equilibrium_With_Price_System.txt |
Learning Objectives
• What happens in a general equilibrium when there are more than two people buying more than two goods?
• Does the Cobb-Douglas case provide insight?
We will illustrate general equilibrium for the case when all consumers have Cobb-Douglas utility in an exchange economy. An exchange economy is an economy where the supply of each good is just the total endowment of that good, and there is no production. Suppose that there are N people, indexed by $$n = 1, 2, … , N$$ There are G goods, indexed by $$g=1,2, \ldots, G$$. Person n has Cobb-Douglas utility, which we can represent using exponents α(n, g), so that the utility of person n can be represented as $$\Pi g=1 G \times(n, g) a(n, g)$$, where x(n, g) is person n’s consumption of good g. Assume that $$a(n, g) \geq 0$$ for all n and g, which amounts to assuming that the products are, in fact, goods. Without any loss of generality, we can require $$\sum g=1 G a(n, g)=1$$ for each n. (To see this, note that maximizing the function U is equivalent to maximizing the function Uβ for any positive β.)
Let y(n, g) be person n’s endowment of good g. The goal of general equilibrium is to find prices p1, p2, … , pG for the goods in such a way that demand for each good exactly equals supply of the good. The supply of good g is just the sum of the endowments of that good. The prices yield a wealth for person n equal to W n = ∑ g=1 G p g y(n,g) .
We will assume that $$\Sigma \mathrm{n}=1 \mathrm{N} \mathrm{a}(\mathrm{n}, \mathrm{g}) \mathrm{y}(\mathrm{n}, \mathrm{i})>0$$ for every pair of goods g and i. This assumption states that for any pair of goods, there is at least one agent that values good g and has an endowment of good i. The assumption ensures that there is always someone who is willing and able to trade if the price is sufficiently attractive. The assumption is much stronger than necessary but useful for exposition. The assumption also ensures that the endowment of each good is positive.
The Cobb-Douglas utility simplifies the analysis because of a feature that we already encountered in the case of two goods, which holds, in general, that the share of wealth for a consumer n on good g equals the exponent α(n, g). Thus, the total demand for good g is $$\mathrm{X} \mathrm{g}=\Sigma \mathrm{n}=1 \mathrm{N} \mathrm{a}(\mathrm{n}, \mathrm{g}) \mathrm{W} \mathrm{n} \mathrm{p} \mathrm{g}$$.
The equilibrium conditions, then, can be expressed by saying that supply (sum of the endowments) equals demand; or, for each good g, $$\mathrm{X} \mathrm{g}=\Sigma \mathrm{n}=1 \mathrm{N} \mathrm{a}(\mathrm{n}, \mathrm{g}) \mathrm{W} \mathrm{n} \mathrm{p} \mathrm{g}$$
We can rewrite this expression, provided that pg > 0 (and it must be, for otherwise demand is infinite), to be
$p g-\Sigma i=1 G p i \Sigma n=1 N y(n, i) a(n, g) \Sigma n=1 N y(n, g)=0$
Let B be the G × G matrix whose (g, i) term is b $$g i=\sum n=1 N y(n, i) a(n, g) \sum n=1 N y(n, g)$$
Let p be the vector of prices. Then we can write the equilibrium conditions as (IB) p = 0, where 0 is the zero vector. Thus, for an equilibrium (other than p = 0) to exist, B must have an eigenvalue equal to 1 and a corresponding eigenvector p that is positive in each component. Moreover, if such an eigenvector–eigenvalue pair exists, it is an equilibrium, because demand is equal to supply for each good.
The actual price vector is not completely identified because if p is an equilibrium price vector, then so is any positive scalar times p. Scaling prices doesn’t change the equilibrium because both prices and wealth (which is based on endowments) rise by the scalar factor. Usually economists assign one good to be a numeraire, which means that all other goods are indexed in terms of that good; and the numeraire’s price is artificially set to be 1. We will treat any scaling of a price vector as the same vector.
The relevant theorem is the Perron-Frobenius theorem.Oskar Perron (1880–1975) and Georg Frobenius (1849–1917). It states that if B is a positive matrix (each component positive), then there is an eigenvalue λ > 0 and an associated positive eigenvector p; and, moreover, λ is the largest (in absolute value) eigenvector of B.The Perron-Frobenius theorem, as usually stated, only assumes that B is nonnegative and that B is irreducible. It turns out that a strictly positive matrix is irreducible, so this condition is sufficient to invoke the theorem. In addition, we can still apply the theorem even when B has some zeros in it, provided that it is irreducible. Irreducibility means that the economy can’t be divided into two economies, where the people in one economy can’t buy from the people in the second economy because they aren’t endowed with anything that the people in the first economy value. If B is not irreducible, then some people may wind up consuming zero of things they value. This conclusion does most of the work of demonstrating the existence of an equilibrium. The only remaining condition to check is that the eigenvalue is in fact 1, so that (IB) p = 0.
Suppose that the eigenvalue is λ. Then λp = Bp. Thus for each g,
$λ p g = ∑ i=1 G ∑ n=1 N α(n,g)y(n,i) ∑ m=1 N y(m,g) p i$
or
$λ p g ∑ m=1 N y(m,g) = ∑ i=1 G ∑ n=1 N α(n,g)y(n,i) p i .$
Summing both sides over g,
$λ ∑ g=1 G p g ∑ m=1 N y(m,g) = ∑ g=1 G ∑ i=1 G ∑ n=1 N α(n,g)y(n,i) p i= ∑ i=1 G ∑ n=1 N ∑ g=1 G α(n,g) y(n,i) p i = ∑ i=1 G ∑ n=1 N y(n,i) p i .$
Thus, λ = 1 as desired.
The Perron-Frobenius theorem actually provides two more useful conclusions. First, the equilibrium is unique. This is a feature of the Cobb-Douglas utility and does not necessarily occur for other utility functions. Moreover, the equilibrium is readily approximated. Denote by Bt the product of B with itself t times. Then for any positive vector v, $$\lim t \rightarrow \infty \text { B } t v=p$$. While approximations are very useful for large systems (large numbers of goods), the system can readily be computed exactly with small numbers of goods, even with a large number of individuals. Moreover, the approximation can be interpreted in a potentially useful manner. Let v be a candidate for an equilibrium price vector. Use v to permit people to calculate their wealth, which for person n is $$\mathrm{W} \mathrm{n}=\Sigma \mathrm{i}=1 \mathrm{G} \mathrm{v} \text { i } \mathrm{y}(\mathrm{n}, \mathrm{i})$$. Given the wealth levels, what prices clear the market? Demand for good g is $$x g (v)= ∑ n=1 N α(n,g) W n = ∑ i=1 G v i ∑ n=1 N α(n,g)y(n,i)$$, and the market clears, given the wealth levels, if $$p g = ∑ i=1 G v i ∑ n=1 N α(n,g)y(n,i) ∑ n=1 N y(n,g)$$, which is equivalent to p = Bv. This defines an iterative process. Start with an arbitrary price vector, compute wealth levels, and then compute the price vector that clears the market for the given wealth levels. Use this price to recalculate the wealth levels, and then compute a new market-clearing price vector for the new wealth levels. This process can be iterated and, in fact, converges to the equilibrium price vector from any starting point.
We finish this section by considering three special cases. If there are two goods, we can let αn = αα(n, 1), and then conclude that α(n, 2) = 1 – an. Then let Y g = ∑ n=1 N y(n,g) be the endowment of good g. Then the matrix B is
$B=( 1 Y 1 ∑ n=1 N y(n,1) a n 1 Y 1 ∑ n=1 N y(n,2) a n 1 Y 2 ∑ n=1 N y(n,1)(1− a n ) 1 Y 2 ∑ n=1 N y(n,2)(1− a n ) )=( 1 Y 1 ∑ n=1 N y(n,1) a n 1 Y 1 ∑ n=1 N y(n,2) a n 1 Y 2 ( Y 1 − ∑ n=1 N y(n,1) a n ) 1− 1 Y 2 ∑ n=1 N y(n,2) a n ).$
The relevant eigenvector of B is $$p=( ∑ n=1 N y(n,2) a n ∑ n=1 N y(n,1)(1− a n ) ) .$$
The overall level of prices is not pinned down—any scalar multiple of p is also an equilibrium price—so the relevant term is the price ratio, which is the price of Good 1 in terms of Good 2, or
$p 1 p 2 = ∑ n=1 N y(n,2) a n ∑ n=1 N y(n,1)(1− a n ) .$
We can readily see that an increase in the supply of Good 1, or a decrease in the supply of Good 2, decreases the price ratio. An increase in the preference for Good 1 increases the price of Good 1. When people who value Good 1 relatively highly are endowed with a lot of Good 2, the correlation between preference for Good 1, an, and endowment of Good 2 is higher. The higher the correlation, the higher is the price ratio. Intuitively, if the people who have a lot of Good 2 want a lot of Good 1, the price of Good 1 is going to be higher. Similarly, if the people who have a lot of Good 1 want a lot of Good 2, the price of Good 1 is going to be lower. Thus, the correlation between endowments and preferences also matters to the price ratio.
In our second special case, we consider people with the same preferences but who start with different endowments. Hypothesizing identical preferences sets aside the correlation between endowments and preferences found in the two-good case. Since people are the same, α(n, g) = Ag for all n. In this case, $$b gi = ∑ n=1 N y(n,i)α(n,g) ∑ n=1 N y(n,g) = A g Y i Y g$$, whereas before $$Y g = ∑ n=1 N y(n,g)$$ is the total endowment of good g. The matrix B has a special structure, and in this case, $$p g=A g Y g$$ is the equilibrium price vector. Prices are proportional to the preference for the good divided by the total endowment for that good.
Now consider a third special case, where no common structure is imposed on preferences, but endowments are proportional to each other; that is, the endowment of person n is a fraction wn of the total endowment. This implies that we can write $$\mathrm{y}(\mathrm{n}, \mathrm{g})=\mathrm{w}_{\mathrm{n}} \mathrm{Y}_{\mathrm{g}}$$, an equation assumed to hold for all people n and goods g. Note that by construction, $$∑ n=1 N w n =1$$, since the value wn represents n’s share of the total endowment. In this case, we have
$b gi = ∑ n=1 N y(n,i)α(n,g) ∑ n=1 N y(n,g) = Y i Y g ∑ n=1 N w n α(n,g)$
These matrices also have a special structure, and it is readily verified that the equilibrium price vector satisfies $$p g = 1 Y g ∑ n=1 N w n α(n,g) .$$
This formula receives a similar interpretation—the price of good g is the strength of preference for good g, where strength of preference is a wealth-weighted average of the individual preference, divided by the endowment of the good. Such an interpretation is guaranteed by the assumption of Cobb-Douglas preferences, since these imply that individuals spend a constant proportion of their wealth on each good. It also generalizes the conclusion found in the two-good case to more goods, but with the restriction that the correlation is now between wealth and preferences. The special case has the virtue that individual wealth, which is endogenous because it depends on prices, can be readily determined.
Key Takeaways
• General equilibrium puts together consumer choice and producer theory to find sets of prices that clear many markets.
• For the case of an arbitrary number of goods and an arbitrary number of consumers—each with Cobb-Douglas utility—there is a closed form for the demand curves, and the price vector can be found by locating an eigenvector of a particular matrix. The equilibrium is unique (true for Cobb-Douglas but not true more generally).
• The actual price vector is not completely identified because if p is an equilibrium price vector, then so is any positive scalar times p. Scaling prices doesn’t change the equilibrium because both prices and wealth (which is based on endowments) rise by the scalar factor.
• The intuition arising from one-good models may fail because of interactions with other markets—increasing preferences for a good (shifting out demand) changes the values of endowments in ways that then reverberate through the system.
EXERCISE
1. Consider a consumer with Cobb-Douglas utility $$U= ∏ i=1 G x i a i$$,where $$\Sigma i=1 \mathrm{G} \mathrm{a} \mathrm{i}=1$$ and facing the budget constraint $$W= ∑ i=1 G p i x i$$.Show that the consumer maximizes utility by choosing $$x i = a i W p i$$ for each good i. (Hint: Express the budget constraint as $$x G = 1 p G ( W− ∑ i=1 G−1 p i x i )$$,and thus utility as $$U=( ∏ i=1 G−1 x i a i ) ( 1 p G ( W− ∑ i=1 G−1 p i x i ) ) a G.)$$ This function can now be maximized in an unconstrained fashion. Verify that the result of the maximization can be expressed as $$p i x i = a i a G p G x G$$, and thus $$W= ∑ i=1 G p i x i$ = ∑ i=1 G a i a G p G x G = p G x G a G$,which yields $$p i x i = a i W$$ | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/14%3A_General_Equilibrium/14.03%3A_General_Equilibrium.txt |
Learning Objectives
• How do monopolies come about?
A monopoly is a firm that faces a downward sloping demand and has a choice about what price to charge—without fearing of chasing all of its customers away to rivals.
There are very few pure monopolies. The U.S. post office has a monopoly in first-class mail but faces competition from FedEx and other express-mail companies, as well as from fax and e-mail providers. Microsoft has a great deal of market power, but a small percentage of personal computer users choose Apple or Linux operating systems. In contrast, there is only one U.S. manufacturer of aircraft carriers.
However, many firms have market power or monopoly power, which means that they can increase their price above marginal cost and sustain sales for a long period of time.These terms are used somewhat differently among authors. Both terms require downward sloping demand and usually some notion of sustainability of sales. Some distinguish the terms by whether they are “large” or not; others by how long the price increase can be sustained. We won’t need such distinctions here. The theory of monopoly is applicable to such firms, although they may face an additional and important constraint: A price increase may affect the behavior of rivals. This behavior of rivals is the subject of the next chapter.
A large market share is not proof of a monopoly, nor is a small market share proof that a firm lacks monopoly power. For example, U.S. Air dominated air traffic to Philadelphia and Pittsburgh but still lost money. Porsche has a small share of the automobile market—or even the high-end automobile market—but still has monopoly power in that market.
There are three basic sources of monopoly. The most common source is to be granted a monopoly by the government, either through patents—in which case the monopoly is temporary—or through a government franchise. Intelsat was a government franchise that was granted a monopoly on satellite communications, a monopoly that ultimately proved quite lucrative. Many cities and towns license a single cable TV company or taxi company, although usually basic rates and fares are set by the terms of the license agreement. New drugs are granted patents that provide the firms monopoly power for a period of time. (Patents generally last 20 years, but pharmaceutical drugs have their own patent laws.) Copyright also confers a limited monopoly for a limited period of time. Thus, the Disney Corporation owns copyrights on Mickey Mouse—copyrights that, by law, should have expired but were granted an extension by Congress each time they were due to expire. Copyrights create monopoly power over music as well as cartoon characters. Time Warner owns the rights to the song “Happy Birthday to You” and receives royalties every time that it is played on the radio or other commercial venue.Fair-use provisions protect individuals with noncommercial uses of copyrighted materials. Many of the Beatles’ songs that Paul McCartney coauthored were purchased by Michael Jackson.
A second source of monopoly is a large economy of scale. The scale economy needs to be large relative to the size of demand. A monopoly can result when the average cost of a single firm serving the entire market is lower than that of two or more firms serving the market. For example, long-distance telephone lines were expensive to install, and the first company to do so, AT&T, wound up being the only provider of long-distance service in the United States. Similarly, scale economies in electricity generation meant that most communities had a single electricity provider prior to the 1980s, when new technology made relatively smaller scale generation more efficient.
The demand-side equivalent of an economy of scale is a network externality. A network externality arises when others’ use of a product makes it more valuable to each consumer. Standards are a common source of network externality. Since AA batteries are standardized, it makes them more readily accessible, helps drive down their price through competition and economies of scale, and thus makes the AA battery more valuable. They are available everywhere, unlike proprietary batteries. Fax machines are valuable only if others have similar machines. In addition to standards, a source of network externality is third-party products. Choosing Microsoft Windows as a computer operating system means that there is more software available than for Macintosh or Linux, as the widespread adoption of Windows has led to the writing of a large variety of software for it. The JVC Video Home System of VCRs came to dominate the Sony Beta system, primarily because there were more movies to rent in the VHS format than in the Beta format at the video rental store. In contrast, recordable DVDs have been hobbled by incompatible standards of DVD+R and DVD-R, a conflict not resolved even as the next generation—50 GB discs such as Sony’s Blu-ray—starts to reach the market. DVDs themselves were slow to be adopted by consumers because few discs were available for rent at video rental stores, which is a consequence of few adoptions of DVD players. As DVD players became more prevalent and the number of discs for rent increased, the market tipped and DVDs came to dominate VHS.
The third source of monopoly is control of an essential, or a sufficiently valuable, input to the production process. Such an input could be technology that confers a cost advantage. For example, software is run by a computer operating system and needs to be designed to work well with the operating system. There have been a series of allegations that Microsoft kept secret some of the “application program interfaces” used by Word as a means of hobbling rivals. If so, access to the design of the operating system itself is an important input.
Key Takeaways
• A monopoly is a firm that faces a downward sloping demand and has a choice about what price to charge—an increase in price doesn’t send most, or all, of the customers away to rivals.
• There are very few pure monopolies. There are many firms that have market power or monopoly power, which means that they can increase their price above marginal cost and sustain sales for a long period of time.
• A large market share is not proof of a monopoly, nor is a small market share proof that a firm lacks monopoly power. There are three basic sources of monopoly: one created by government, like patents; a large economy of scale or a network externality; and control of an essential, or a sufficiently valuable, input to the production process. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/15%3A_Monopoly/15.01%3A_Sources_of_Monopoly.txt |
Learning Objectives
• What are the basic effects of monopoly, compared to a competitive industry?
Even a monopoly is constrained by demand. A monopoly would like to sell lots of units at a very high price, but a higher price necessarily leads to a loss in sales. So how does a monopoly choose its price and quantity?
A monopoly can choose price, or a monopoly can choose quantity and let the demand dictate the price. It is slightly more convenient to formulate the theory in terms of quantity rather than price, because costs are a function of quantity. Thus, we let p(q) be the demand price associated with quantity q, and c(q) be the cost of producing q. The monopoly’s profits are $$π=p(q)q−c(q)$$.
The monopoly earns the revenue pq and pays the cost c. This leads to the first-order condition for the profit-maximizing quantity
$qm: 0= ∂π ∂q =p( q m )+ q m p ′ ( q m )− c ′ ( q m ).$
The term p(q)+q p ′ (q) is known as marginal revenue. It is the derivative of revenue pq with respect to quantity. Thus, a monopoly chooses a quantity qm where marginal revenue equals marginal cost, and charges the maximum price p(qm) that the market will bear at that quantity. Marginal revenue is below demand p(q) because demand is downward sloping. That is, $$\mathrm{p}(\mathrm{q})+\mathrm{q} \mathrm{p}^{\prime}(\mathrm{q})<\mathrm{p}(\mathrm{q})$$
Figure 15.1 Basic monopoly diagram
We can rearrange the monopoly pricing formula to produce an additional insight:
$p(q m)-c^{\prime}(q m)=-q m p^{\prime}(q m) \text { or } p(q m)-c^{\prime}(q m) p(q m)=-q m p^{\prime}(q m) p(q m)=1 \varepsilon$
The left-hand side of this equation (price minus marginal cost divided by price) is known as the price-cost margin or Lerner Index.Abba Lerner (1903–1982). Note that $$1-q m p^{\prime}(q m) p(q m)=-1 q m p^{\prime}(q m) p(q m)=-d q q d p p=\varepsilon$$m, which is used in the derivation. The right-hand side is one divided by the elasticity of demand. This formula relates the markup over marginal cost to the elasticity of demand. It is important because perfect competition forces price to equal marginal cost, so this formula provides a measure of the deviation from competition and, in particular, says that the deviation from competition is small when the elasticity of demand is large, and vice versa.
Marginal cost will always be greater than or equal to zero. If marginal cost is less than zero, the least expensive way to produce a given quantity is to produce more and throw some away. Thus, the price-cost margin is no greater than one; and, as a result, a monopolist produces in the elastic portion of demand. One implication of this observation is that if demand is everywhere inelastic (e.g., $$p(q)=q-a \text { for } a>1$$), the optimal monopoly quantity is essentially zero, and in any event would be no more than one molecule of the product.
In addition, the effects of monopoly are related to the elasticity of demand. If demand is very elastic, the effect of monopoly on prices is quite limited. In contrast, if the demand is relatively inelastic, monopolies will increase prices by a large margin.
We can rewrite the formula to obtain $$\mathrm{p}(\mathrm{q} \mathrm{m})=\varepsilon \varepsilon-1 \mathrm{c}^{\prime}(\mathrm{q} \mathrm{m})$$.
Thus, a monopolist marks up marginal cost by the factor ε ε−1 , at least when ε > 1. This formula is sometimes used to justify a “fixed markup policy,” which means that a company adds a constant percentage markup to its products. This is an ill-advised policy, not justified by the formula, because the formula suggests a markup that depends upon the demand for the product in question, and thus not a fixed markup for all products that a company produces.
Key Takeaways
• Even a monopoly is constrained by demand.
• A monopoly can either choose price, or choose quantity and let the demand dictate the price.
• A monopoly chooses a quantity qm where marginal revenue equals marginal cost, and charges the maximum price p(qm) that the market will bear at that quantity.
• Marginal revenue is below demand p(q) because demand is downward sloping.
• The monopoly price is higher than the marginal cost.
• There is a deadweight loss of monopoly for the same reason that taxes create a deadweight loss: The higher price of the monopoly prevents some units from being traded that are valued more highly than they cost.
• A monopoly restricts output and charges a higher price than would prevail under competition.
• The price-cost margin is the ratio of price minus marginal cost over price and measures the deviation from marginal cost pricing.
• A monopoly chooses a price or quantity that equates the price-cost margin to the inverse of the demand elasticity.
• A monopolist produces in the elastic portion of demand.
• A monopolist marks up marginal cost by the factor ε ε−1 , when the elasticity of demand ε exceeds one.
EXERCISES
1. If demand is linear, $$p(q)=a-b q$$, what is marginal revenue? Plot demand and marginal revenue, and total revenue qp(q) as a function of q.
2. For the case of constant elasticity of demand, what is marginal revenue?
3. If both demand and supply have constant elasticity, compute the monopoly quantity and price.
4. Consider a monopolist with cost c = 3q.
1. If demand is given by $$q=50-2 p$$, what is the monopoly price and quantity? What are the profits?
2. Repeat part (a) for demand given by $$q=10 / p$$
5. The government wishes to impose a tax, of fraction t, on the profits of a monopolist. How does this affect the monopolist’s optimal output quantity?
6. If demand has constant elasticity, what is the marginal revenue of the monopolist?
15.03: Effect of Taxes
Learning Objectives
• How does a monopoly respond to taxes?
A tax imposed on a seller with monopoly power performs differently than a tax imposed on a competitive industry. Ultimately, a perfectly competitive industry must pass on all of a tax to consumers because, in the long run, the competitive industry earns zero profits. In contrast, a monopolist might absorb some portion of a tax even in the long run.
To model the effect of taxes on a monopoly, consider a monopolist who faces a tax rate t per unit of sales. This monopolist earns $$π=p(q)q−c(q)−tq.$$\)
The first-order condition for profit maximization yields $$0= ∂π ∂q =p( q m )+ q m p ′ ( q m )− c ′ ( q m )−t.$$
Viewing the monopoly quantity as a function of t, we obtain $$d q m dt = 1 2 p ′ ( q m )+ q m p ″ ( q m )− c ″ ( q m ) <0$$ with the sign following from the second-order condition for profit maximization. In addition, the change in price satisfies $$p^{\prime}(q m) d q m d t=p^{\prime}(q m) 2 p^{\prime}(q m)+q m p^{\prime \prime}(q m)-c^{\prime \prime}(q m)>0$$
Thus, a tax causes a monopoly to increase its price. In addition, the monopoly price rises by less than the tax if $$\mathrm{p}^{\prime}(\mathrm{q} \mathrm{m}) \mathrm{d} \mathrm{q} \mathrm{m} \mathrm{dt}<1, \text { or } \mathrm{p}^{\prime}(\mathrm{q} \mathrm{m})+\mathrm{q} \mathrm{m} \mathrm{p}^{\prime \prime}(\mathrm{q} \mathrm{m})-\mathrm{c}^{\prime \prime}(\mathrm{q} \mathrm{m})<0$$
This condition need not be true but is a standard regularity condition imposed by assumption. It is true for linear demand and increasing marginal cost. It is false for constant elasticity of demand, ε > 1 (which is the relevant case, for otherwise the second-order conditions fail), and constant marginal cost. In the latter case (constant elasticity and marginal cost), a tax on a monopoly increases price by more than the amount of the tax.
Key Takeaways
• A perfectly competitive industry must pass on all of a tax to consumers because, in the long run, the competitive industry earns zero profits. A monopolist might absorb some portion of a tax even in the long run.
• A tax causes a monopoly to increase its price and reduce its quantity.
• A tax may or may not increase the monopoly markup.
EXERCISES
1. Use a revealed preference argument to show that a per-unit tax imposed on a monopoly causes the quantity to fall. That is, hypothesize quantities qb before the tax and qa after the tax, and show that two facts—the before-tax monopoly preferred qb to qa, and the taxed monopoly made higher profits from qb—together imply that qbqa.
2. When both demand and supply have constant elasticity, use the results of 0 to compute the effect of a proportional tax (i.e., a portion of the price paid to the government). | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/15%3A_Monopoly/15.02%3A_Basic_Analysis.txt |
Learning Objectives
• Do monopolies charge different consumers different prices?
• Why and how much?
Pharmaceutical drugs for sale in Mexico are generally priced substantially below their U.S. counterparts. Pharmaceutical drugs in Europe are also cheaper than in the United States, although not as inexpensive as in Mexico, with Canadian prices usually falling between the U.S. and European prices. (The comparison is between identical drugs produced by the same manufacturer.)
Pharmaceutical drugs differ in price from country to country primarily because demand conditions vary. The formula p( q m )= ε ε−1 c ′ ( q m ) shows that a monopoly seller would like to charge a higher markup over marginal cost to customers with less elastic demand than to customers with more elastic demand because ε ε−1 is a decreasing function of ε, for ε > 1. Charging different prices for the same product to different customers is known as price discrimination. In business settings, it is sometimes known as value-based pricing, which is a more palatable term to relay to customers.
Computer software vendors often sell a “student” version of their software, usually at substantially reduced prices, but require proof of student status to qualify for the lower price. Such student discounts are examples of price discrimination, and students have more elastic demand than business users. Similarly, the student and senior citizen discounts at movies and other venues sell the same thing—a ticket to the show—for different prices, and thus qualify as price discrimination.
In order for a seller to price discriminate, the seller must be able to
• identify (approximately) the demand of groups of customers, and
• prevent arbitrage.
Arbitrage is also known as “buying low and selling high,” and represents the act of being an intermediary. Since price discrimination requires charging one group a higher price than another, there is potentially an opportunity for arbitrage, arising from members of the low-price group buying at the low price and selling at the high price. If the seller can’t prevent arbitrage, arbitrage essentially converts a two-price system to sales at the low price.
Why offer student discounts at the movies? You already know the answer to this: Students have lower incomes on average than others, and lower incomes translate into a lower willingness to pay for normal goods. Consequently, a discount to a student makes sense from a demand perspective. It is relatively simple to prevent arbitrage by requiring that a student identification card be presented. Senior citizen discounts are a bit subtler. Generally, senior citizens aren’t poorer than other groups of customers (in the United States, at least). However, seniors have more free time and therefore are able to substitute to matinee showingsMatinee showings are those early in the day, which are usually discounted. These discounts are not price discrimination because a show at noon isn’t the same product as a show in the evening. or to drive to more distant locations should those offer discounts. Thus, seniors have relatively elastic demand, more because of their ability to substitute than because of their income.
Airlines commonly price discriminate, using “Saturday night stay-overs” and other devices. To see that such charges represent price discrimination, consider a passenger who lives in Dallas but needs to spend Monday through Thursday in Los Angeles for 2 weeks in a row. This passenger could buy two roundtrip tickets:
Trip 1:
First Monday: Dallas → Los Angeles
First Friday: Los Angeles → Dallas
Trip 2:
Second Monday: Dallas → Los Angeles
Second Friday: Los Angeles → Dallas
At the time of this writing, the approximate combined cost of these two flights was \$2,000. In contrast, another way of arranging exactly the same travel is to have two roundtrips, one of which originates in Dallas, while the other originates in Los Angeles:
Trip 1:
First Monday: Dallas → Los Angeles
Second Friday: Los Angeles → Dallas
Trip 2:
First Friday: Los Angeles → Dallas
Second Monday: Dallas → Los Angeles
This pair of roundtrips involves exactly the same travel as the first pair, but costs less than \$500 for both (at the time of this writing). The difference is that the second pair involves staying over Saturday night for both legs, and that leads to a major discount for most U.S. airlines. (American Airlines quoted the fares.)
How can airlines price discriminate? There are two major groups of customers: business travelers and leisure travelers. Business travelers have the higher willingness to pay overall, and the nature of their trips tends to be that they come home for the weekend. In contrast, leisure travelers usually want to be away for a weekend, so a weekend stay over is an indicator of a leisure traveler. It doesn’t work perfectly as an indicator—some business travelers must be away for the weekend—but it is sufficiently correlated with leisure travel that it is profitable for the airlines to price discriminate.
These examples illustrate an important distinction. Senior citizen and student discounts are based on the identity of the buyer, and qualifying for the discount requires that one show an identity card. In contrast, airline price discrimination is not based on the identity of the buyer but rather on the choices made by the buyer. Charging customers based on identity is known as direct price discrimination, while offering a menu or set of prices and permitting customers to choose distinct prices is known as indirect price discrimination.The older and incoherent language for these concepts identified direct price discrimination as “third-degree price discrimination,” while indirect price discrimination was called second-degree price discrimination. In the older language, first-degree price discrimination meant perfect third-degree price discrimination.
Two common examples of indirect price discrimination are coupons and quantity discounts. Coupons offer discounts for products and are especially common in grocery stores, where they are usually provided in a free newspaper section at the front of the store. Coupons discriminate on the basis of the cost of time. It takes time to find the coupons for the products that one is interested in buying. Thus, those with a high value of time won’t find it worth their while to spend 20 minutes to save \$5 (effectively a \$15 per hour return), while those with a low value of time will find that return worthwhile. Since those with a low value of time tend to be more price-sensitive (more elastic demand), coupons offer a discount that is available to all but used primarily by customers with a more elastic demand, and thus increase the profits of the seller.
Quantity discounts are discounts for buying more. Thus, the large size of milk, laundry detergent, and other items often cost less per unit than smaller sizes, and the difference is greater than the savings on packaging costs. In some cases, the larger sizes entail greater packaging costs; some manufacturers “band together” individual units, incurring additional costs to create a larger size that is then discounted. Thus, the “24-pack” of paper towels sells for less per roll than the individual rolls; such large volumes appeal primarily to large families, who are more price-sensitive on average.
Key Takeaways
• A monopoly seller would like to charge a higher markup over marginal cost to customers with less elastic demand than to customers with more elastic demand.
• In order for a seller to price discriminate, the seller must be able to
• identify (approximately) the demand of groups of customers, and
• prevent arbitrage.
• Since price discrimination requires charging one group a higher price than another, there is potentially an opportunity for arbitrage.
• Airlines commonly price discriminate, using “Saturday night stay-overs” and other devices.
• Direct price discrimination is based upon the identity of the buyer, while indirect price discrimination involves several offers and achieves price discrimination through customer choices.
• Two common examples of indirect price discrimination are coupons and quantity discounts.
EXERCISE
1. Determine whether the following items are direct price discrimination, indirect price discrimination, or not price discrimination—and why.
1. Student discounts at local restaurants
2. Financial aid at colleges
3. Matinee discounts at the movies
4. Home and professional versions of Microsoft’s operating system
5. Lower airline fares for weekend flights
6. Buy-one-get-one-free specials | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/15%3A_Monopoly/15.04%3A_Price_Discrimination.txt |
Learning Objectives
• Is price discrimination good or bad for society as a whole?
Is price discrimination a good thing or a bad thing? It turns out that there is no definitive answer to this question. Instead, it depends on circumstances. We illustrate this conclusion with a pair of exercises.
This exercise illustrates a much more general proposition: If a price discriminating monopolist produces less than a nondiscriminating monopolist, then price discrimination reduces welfare. This proposition has an elementary proof. Consider the price discriminating monopolist’s sales, and then allow arbitrage. The arbitrage increases the gains from trade, since every transaction has gains from trade. Arbitrage, however, leads to a common price like that charged by a nondiscriminating monopolist. Thus, the only way that price discrimination can increase welfare is if it leads a seller to sell more output than he or she would otherwise. This is possible, as the next exercise shows.
In Exercise 2, we see that price discrimination that brings in a new group of customers may increase the gains from trade. Indeed, this example involves a Pareto improvement: The seller and Group 2 are better off, and Group 1 is no worse off, than without price discrimination. (A Pareto improvement requires that no one is worse off and at least one person is better off.)
Whether price discrimination increases the gains from trade overall depends on circumstances. However, it is worth remembering that people with lower incomes tend to have more elastic demand, and thus get lower prices under price discrimination than absent price discrimination. Consequently, a ban on price discrimination tends to hurt the poor and benefit the rich, no matter what the overall effect.
A common form of price discrimination is known as two-part pricing. Two-part pricing usually involves a fixed charge and a marginal charge, and thus offers the ability for a seller to capture a portion of the consumer surplus. For example, electricity often comes with a fixed price per month and then a price per kilowatt-hour, which is two-part pricing. Similarly, long distance and cellular telephone companies charge a fixed fee per month, with a fixed number of “included” minutes, and a price per minute for additional minutes. Such contracts really involve three parts rather than two parts, but are similar in spirit.
Figure 15.2 Two-part pricing
Key Takeaways
• If a price discriminating monopolist produces less than a nondiscriminating monopolist, then price discrimination reduces welfare.
• Price discrimination that opens a new, previously unserved market increases welfare.
• A ban on price discrimination tends to hurt the poor and benefit the rich, no matter what the overall effect.
• Two-part pricing involves a fixed charge and a marginal charge.
• The ideal two-part price is to charge marginal cost plus a fixed charge equal to the customer’s consumer surplus, in which case the seller captures the entire gains from trade.
EXERCISES
1. Let marginal cost be zero for all quantities. Suppose that there are two equal-sized groups of customers, Group 1 with demand $$q(p)=12-p$$, and Group 2 with demand $$q(p) = 8 – p$$. Show that a nondiscriminating monopolist charges a price of 5, and the discriminating monopolist charges Group 1 the price of 6 and Group 2 the price of 4. Then calculate the gains from trade, with discrimination and without, and show that price discrimination reduces the gains from trade.
2. Let marginal cost be zero for all quantities. Suppose that there are two equal-sized groups of customers, Group 1 with demand $$q (p) = 12 – p$$, and Group 2 with demand $$q(p) = 4 – p$$. Show that a nondiscriminating monopolist charges a price of 6, and the discriminating monopolist charges Group 1 the price of 6 and Group 2 the price of 2. Then calculate the gains from trade, with discrimination and without, and show that price discrimination increases the gains from trade. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/15%3A_Monopoly/15.05%3A_Welfare_Effects.txt |
Learning Objectives
• When there is a scale economy, what market prices will arise?
• How is the monopoly price constrained by the threat of entry?
A natural monopoly arises when a single firm can efficiently serve the entire market because average costs are lower with one firm than with two firms. An example is illustrated in Figure 15.3. In this case, the average total cost of a single firm is lower than if two firms were to split the output between them. The monopolist would like to price at pm, which maximizes profits.The monopoly price may or may not be sustainable. A monopoly price is not sustainable if it were to lead to entry, thereby undercutting the monopoly. The feasibility of entry, in turn, depends on whether the costs of entering are not recoverable (“sunk”) and how rapidly entry can occur. If the monopoly price is not sustainable, the monopoly may engage in limit pricing, which is jargon for pricing to deter (limit) entry.
Historically, the United States and other nations have regulated natural monopoly products and supplies such as electricity, telephony, and water service. An immediate problem with regulation is that the efficient price—that is, the price that maximizes the gains from trade—requires a subsidy from outside the industry. We see the need for a subsidy in Figure 15.3 because the price that maximizes the gains from trade is p1, which sets the demand (marginal value) equal to the marginal cost. At this price, however, the average total cost exceeds the price, so that a firm with such a regulated price would lose money. There are two alternatives. The product could be subsidized: Subsidies are used with postal and passenger rail services in the United States and historically for many more products in Canada and Europe, including airlines and airplane manufacture. Alternatively, regulation could be imposed to limit the price to p2, the lowest break-even price. This is the more common strategy used in the United States.
Figure 15.3 Natural monopoly
There are two strategies for limiting the price: price-cap regulation, which directly imposes a maximum price, and rate of return regulation, which limits the profitability of firms. Both of these approaches induce some inefficiency of production. In both cases, an increase in average cost may translate into additional profits for the firm, causing regulated firms to engage in unnecessary activities.
Key Takeaways
• A natural monopoly arises when a single firm can efficiently serve the entire market.
• Historically, the United States and other nations have regulated natural monopolies including electricity, telephony, and water service.
• The efficient price is typically unsustainable because of decreasing average cost.
• Efficient prices can be achieved with subsidies that have been used, for example, in postal and passenger rail services in the United States and historically for several products in Canada and Europe, including airlines and airplane manufacture. Alternatively, regulation could be imposed to limit the price to average cost, the lowest break-even price. This is the more common strategy in the United States.
• Two common strategies for limiting the price are price-cap regulation, which directly imposes a maximum price, and rate of return regulation, which limits the profitability of firms. Both of these approaches induce some inefficiency of production. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/15%3A_Monopoly/15.06%3A_Natural_Monopoly.txt |
Learning Objectives
• How do monopolies respond to predictable cost fluctuation as it arises in electricity and hotel markets?
Fluctuations in demand often require holding capacity, which is used only a fraction of the time. Hotels have off-seasons when most rooms are empty. Electric power plants are designed to handle peak demand, usually on hot summer days, with some of the capacity standing idle on other days. Demand for transatlantic airline flights is much higher in the summer than during the rest of the year. All of these examples have the similarity that an amount of capacity—hotel space, airplane seats, electricity generation—will be used over and over, which means that it is used in both high demand and low demand states. How should prices be set when demand fluctuates? This question can be reformulated as to how to allocate the cost of capacity across several time periods when demand systematically fluctuates.
Consider a firm that experiences two costs: a capacity cost and a marginal cost. How should capacity be priced? This issue applies to a wide variety of industries, including pipelines, airlines, telephone networks, construction, electricity, highways, and the Internet.
The basic peak-load pricing problem, pioneered by Marcel Boiteux (1922– ), considers two periods. The firm’s profits are given by
$π= p 1 q 1 + p 2 q 2 −β max { q 1 , q 2 }−mc( q 1 + q 2 ).$
Setting price equal to marginal cost is not sustainable because a firm selling with price equal to marginal cost would not earn a return on the capacity, and thus would lose money and go out of business. Consequently, a capacity charge is necessary. The question of peak-load pricing is how the capacity charge should be allocated. This question is not trivial because some of the capacity is used in both periods.
For the sake of simplicity, we will assume that demands are independent; that is, q1 is independent of p2, and vice versa. This assumption is often unrealistic, and generalizing it actually doesn’t complicate the problem too much. The primary complication is in computing the social welfare when demands are functions of two prices. Independence is a convenient starting point.
Social welfare is $$W= ∫ 0 q 1 p 1 (x)dx + ∫ 0 q 2 p 2 (x)dx −β max { q 1 , q 2 }−mc( q 1 + q 2 ).$$
The Ramsey problem is to maximize W subject to a minimum profit condition. A technique for accomplishing this maximization is to instead maximize L = W + λπ.
By varying λ, we vary the importance of profits to the maximization problem, which will increase the profit level in the solution as λ increases. Thus, the correct solution to the constrained maximization problem is the outcome of the maximization of L, for some value of λ.
A useful notation is 1A, which is known as the indicator function of the set A. This is a function that is 1 when A is true, and zero otherwise. Using this notation, the first-order condition for the maximization of L is
$0= ∂L ∂ q 1 = p 1 ( q q )−β 1 q 1 ≥ q 2 −mc+λ( p 1 ( q q )+ q 1 p 1 ′ ( q 1 )−β 1 q 1 ≥ q 2 −mc ) or p 1 ( q 1 )−β 1 q 1 ≥ q 2 −mc p 1 = λ λ+1 1 ε 1$,
where 1 q 1 ≥ q 2 is the characteristic function of the event q1 ≥ q2. Similarly, $$p 2 ( q 2 )−β 1 q 1 ≤ q 2 −mc p 2 = λ λ+1 1 ε 2$$. Note as before that λ → ∞ yields the monopoly solution.
There are two potential types of solutions. Let the demand for Good 1 exceed the demand for Good 2. Either q1 > q2, or the two are equal.
Case 1 (q1 > q2):
$\text { p } 1(q 1)-\beta-m c p 1=\lambda \lambda+11 \varepsilon 1 \text { and } p 2(q 2)-m c p 2=\lambda \lambda+11 \varepsilon 2$
In Case 1, with all of the capacity charge allocated to Good 1, quantity for Good 1 still exceeds quantity for Good 2. Thus, the peak period for Good 1 is an extreme peak. In contrast, Case 2 arises when assigning the capacity charge to Good 1 would reverse the peak—assigning all of the capacity charge to Good 1 would make Period 2 the peak.
Case 2 (q1 = q2):
The profit equation can be written $$p_{1}(q)-m c+p_{2}(q)-m c=\beta$$. This equation determines q, and prices are determined from demand.
The major conclusion from peak-load pricing is that either the entire cost of capacity is allocated to the peak period or there is no peak period, in the sense that the two periods have the same quantity demanded given the prices. That is, either the prices equalize the quantity demanded or the prices impose the entire cost of capacity only on one peak period.
Moreover, the price (or, more properly, the markup over marginal cost) is proportional to the inverse of the elasticity, which is known as Ramsey pricing.
Key Takeaways
• Fluctuations in demand often require holding capacity, which is used only a fraction of the time. Peak-load pricing allocates the cost of capacity across several time periods when demand systematically fluctuates.
• Important industries with peak-load problems include pipelines, airlines, telephone networks, construction, electricity, highways, and the Internet.
• Under efficient peak-load pricing, either the prices equalize the quantity demanded, or the prices impose the entire cost of capacity only on one peak period. Moreover, the markup over marginal cost is proportional to the inverse of the elasticity.
EXERCISE
1. For each of the following items, state whether you would expect peak-load pricing to equalize the quantity demanded across periods or impose the entire cost of capacity on the peak period. Explain why.
1. Hotels in Miami
2. Electricity | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/15%3A_Monopoly/15.07%3A_Peak-load_Pricing.txt |
Introduction
Competitive theory studies price-taking consumers and firms—that is, people who can’t individually affect the transaction prices. The assumption that market participants take prices as given is justified only when there are many competing participants. We have also examined monopoly, precisely because a monopoly, by definition, doesn’t have to worry about competitors. Strategic behavior involves the examination of the intermediate case, where there are few enough participants that they take each other into account—and their actions individually matter—so that the behavior of any one participant influences choices of the other participants. That is, participants are strategic in their choices of action, recognizing that their choices will affect choices made by others.
The right tool for the job of examining strategic behavior in economic circumstances is game theory, the study of how people play games. Game theory was pioneered by the mathematical genius John von Neumann (1903–1957). Game theory has also been very influential in the study of military strategy; and, indeed, the strategy of the cold war between the United States and the Soviet Union was guided by game-theoretic analyses.An important reference for game theory is John von Neumann (1903–1957) and Oskar Morgenstern (1902–1977), Theory of Games and Economic Behavior (Princeton, NJ: Princeton University Press, 1944). Important extensions were introduced by John Nash (1928–), the mathematician made famous by Sylvia Nasar’s delightful book, A Beautiful Mind (Simon & Schuster, 1998). Finally, applications in the military arena were pioneered by Nobel laureate Thomas Schelling (1921–), The Strategy of Conflict (Cambridge: Cambridge University Press, 1960).
The theory provides a description that fits common games like poker or the board game Monopoly, but will cover many other situations as well. In any game, there is a list of players. Games generally unfold over time; at each moment in time, players have information—possibly incomplete—about the current state of play and a set of actions they can take. Both information and actions may depend on the history of the game prior to that moment. Finally, players have payoffs and are assumed to play in such a way as to maximize their anticipated payoff, taking into account their expectations for the play of others. When the players, their information and available actions, and payoffs have been specified, we have a game.
Learning Objectives
• How are games modeled?
• What is optimal play?
The simplest game is called a matrix payoff game with two players. In a matrix payoff game, all actions are chosen simultaneously. It is conventional to describe a matrix payoff game as played by a row player and a column player. The row player chooses a row in a matrix; the column player simultaneously chooses a column. The outcome of the game is a pair of payoffs where the first entry is the payoff of the row player, and the second is the payoff of the column player. Figure 16.1 provides an example of a “2 × 2” matrix payoff game—the most famous game of all—which is known as the prisoner’s dilemma. In the game, the strategies are to confess or not to confess.
Figure 16.1 The prisoner’s dilemma
In the prisoner’s dilemma, two criminals named Row and Column have been apprehended by the police and are being questioned separately. They are jointly guilty of the crime. Each player can choose either to confess or not. If Row confesses, we are in the top row of the matrix (corresponding to the row labeled “Confess”). Similarly, if Column confesses, the payoff will be in the relevant column. In this case, if only one player confesses, that player goes free and the other serves 20 years in jail. (The entries correspond to the number of years lost to prison. The first entry is always Row’s payoff; the second entry is Column’s payoff.) Thus, for example, if Column confesses and Row does not, the relevant payoff is the first column and the second row.
Figure 16.2 Solving the prisoner’s dilemma
If Column confesses and Row does not, Row loses 20 years, and Column loses no years; that is, it goes free. This is the payoff (–20, 0) in reverse color in Figure 16.2. If both confess, they are both convicted and neither goes free, but they only serve 10 years each. Finally, if neither confesses, there is a 10% chance that they are convicted anyway (using evidence other than the confession), in which case they each average a year lost.
The prisoner’s dilemma is famous partly because it is readily solvable. First, Row has a strict advantage to confessing, no matter what Column is going to do. If Column confesses, Row gets –10 for confessing, –20 for not confessing, and thus is better off confessing. Similarly, if Column doesn’t confess, Row gets 0 for confessing (namely, goes free), –1 for not confessing, and is better off confessing. Either way, no matter what Column does, Row should choose to confess.If Row and Column are friends and care about each other, that should be included as part of the payoffs. Here, there is no honor or friendship among thieves, and Row and Column only care about what they themselves will get. This is called a dominant strategy, a strategy that is optimal no matter what the other players do.
The logic is exactly similar for Column: No matter what Row does, Column should choose to confess. That is, Column also has a dominant strategy to confess. To establish this, first consider what Column’s best action is, when Column thinks Row will confess. Then consider Column’s best action when Column thinks Row won’t confess. Either way, Column gets a higher payoff (lower number of years lost to prison) by confessing.
The presence of a dominant strategy makes the prisoner’s dilemma particularly easy to solve. Both players should confess. Note that this gets them 10 years each in prison, and thus isn’t a very good outcome from their perspective; but there is nothing they can do about it in the context of the game, because for each the alternative to serving 10 years is to serve 20 years. This outcome is referred to as (Confess, Confess), where the first entry is the row player’s choice, and the second entry is the column player’s choice.
Figure 16.3 An entry game
In this case, if both companies enter, Microsoft ultimately wins the market, earning 2 and Piuny loses 2. If either firm has the market to itself, it gets 5 and the other firm gets zero. If neither enters, they both get zero. Microsoft has a dominant strategy to enter: It gets 2 when Piuny enters, 5 when Piuny doesn’t, and in both cases it does better than when it doesn’t enter. In contrast, Piuny does not have a dominant strategy: Piuny wants to enter when Microsoft doesn’t, and vice versa. That is, Piuny’s optimal strategy depends upon Microsoft’s action; or, more accurately, Piuny’s optimal strategy depends upon what Piuny believes Microsoft will do.
Piuny can understand Microsoft’s dominant strategy if it knows the payoffs of Microsoft.It isn’t so obvious that one player will know the payoffs of another player, which often causes players to try to signal that they are going to play a certain way—that is, to demonstrate commitment to a particular advantageous strategy. Such topics are taken up in business strategy and managerial economics. Thus, Piuny can conclude that Microsoft is going to enter, and this means that Piuny should not enter. Thus, the equilibrium of the game is for Microsoft to enter and Piuny not to enter. This equilibrium is arrived at by the iterated elimination of dominated strategies, eliminating strategies by sequentially removing strategies that are dominated for a player. First, we eliminated Microsoft’s dominated strategy in favor of its dominant strategy. Microsoft had a dominant strategy to enter, which means that the strategy of not entering was dominated by the strategy of entering, so we eliminated the dominated strategy. That leaves a simplified game in which Microsoft enters, as shown in Figure 16.4.
Figure 16.4 Eliminating a dominated strategy
In this simplified game, after the elimination of Microsoft’s dominated strategy, Piuny also has a dominant strategy: not to enter. Thus, we iterate and eliminate dominated strategies again—this time eliminating Piuny’s dominated strategies—and wind up with a single outcome: Microsoft enters, and Piuny doesn’t. The iterated elimination of dominated strategies solves the game.A strategy may be dominated not by any particular alternate strategy but by a randomization over other strategies, which is an advanced topic not considered here.
Figure 16.5 shows another game, with three strategies for each player.
Figure 16.5 A 3 x 3 game
Middle dominates Bottom for Row, yielding:
Figure 16.6 Eliminating a dominated strategy
Figure 16.7 Eliminating another dominated strategy
Figure 16.8 Eliminating a third dominated strategy
Figure 16.9 Game solved
The iterated elimination of dominated strategies is a useful concept, and when it applies, the predicted outcome is usually quite reasonable. Certainly it has the property that no player has an incentive to change his or her behavior given the behavior of others. However, there are games where it doesn’t apply, and these games require the machinery of a Nash equilibrium, named for Nobel laureate John Nash (1928–).
Key Takeaways
• Strategic behavior arises where there are few enough market participants that their actions individually matter, and where the behavior of any one participant influences choices of the other participants.
• Game theory is the study of how people play games. A game consists of the players, their information and available actions, and payoffs.
• In a matrix payoff game, all actions are chosen simultaneously. The row player chooses a row in a matrix; the column player simultaneously chooses a column. The outcome of the game is a pair of payoffs where the first entry is the payoff of the row player, and the second is the payoff of the column player.
• In the prisoner’s dilemma, two criminals named Row and Column have been apprehended by the police and are being questioned separately. They are jointly guilty of the crime. Each player can choose either to confess or not. Each player individually benefits from confessing, but together they are harmed.
• A dominant strategy is a strategy that is best for a player no matter what others choose.
• Iterated elimination of dominated strategies first removes strategies dominated by others, then checks if any new strategies are dominated and removes them, and so on. In many cases, iterated elimination of dominated strategies solves a game. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/16%3A_Games_and_Strategic_Behavior/16.01%3A_Matrix_Games.txt |
Learning Objectives
• What is an equilibrium to a game?
In a Nash equilibrium, each player chooses the strategy that maximizes his or her expected payoff, given the strategies employed by others. For matrix payoff games with two players, a Nash equilibrium requires that the row chosen maximize the row player’s payoff (given the column chosen by the column player) and the column, in turn, maximize the column player’s payoff (given the row selected by the row player). Let us consider first the prisoner’s dilemma, which we have already seen. Here it is illustrated once again in Figure 16.10.
Figure 16.10 Prisoner's dilemma again
Given that the row player has chosen to confess, the column player also chooses to confess because –10 is better than –20. Similarly, given that the column player chooses confession, the row player chooses confession because –10 is better than –20. Thus, for both players to confess is a Nash equilibrium. Now let us consider whether any other outcome is a Nash equilibrium. In any other outcome, at least one player is not confessing. But that player could get a higher payoff by confessing, so no other outcome could be a Nash equilibrium.
The logic of dominated strategies extends to Nash equilibrium, except possibly for ties. That is, if a strategy is strictly dominated, it can’t be part of a Nash equilibrium. On the other hand, if it involves a tied value, a strategy may be dominated but still be part of a Nash equilibrium.
The Nash equilibrium is justified as a solution concept for games as follows. First, if the players are playing a Nash equilibrium, no one has an incentive to change his or her play or to rethink his or her strategy. Thus, the Nash equilibrium has a “steady state” in that no one wants to change his or her own strategy given the play of others. Second, other potential outcomes don’t have that property: If an outcome is not a Nash equilibrium, then at least one player has an incentive to change what he or she is doing. Outcomes that aren’t Nash equilibria involve mistakes for at least one player. Thus, sophisticated, intelligent players may be able to deduce each other’s play, and play a Nash equilibrium.
Do people actually play Nash equilibria? This is a controversial topic and mostly beyond the scope of this book, but we’ll consider two well-known games: tic-tac-toe (see, for example, http://www.mcafee.cc/Bin/tictactoe/index.html) and chess. Tic-tac-toe is a relatively simple game, and the equilibrium is a tie. This equilibrium arises because each player has a strategy that prevents the other player from winning, so the outcome is a tie. Young children play tic-tac-toe and eventually learn how to play equilibrium strategies, at which point the game ceases to be very interesting since it just repeats the same outcome. In contrast, it is known that chess has an equilibrium, but no one knows what it is. Thus, at this point, we don’t know if the first mover (white) always wins, or if the second mover (black) always wins, or if the outcome is a draw (neither is able to win). Chess is complicated because a strategy must specify what actions to take, given the history of actions, and there are a very large number of potential histories of the game 30 or 40 moves after the start. So we can be quite confident that people are not (yet) playing Nash equilibria to the game of chess.
The second most famous game in game theory is a coordination game called the battle of the sexes. The battle of the sexes involves a married couple who are going to meet each other after work but haven’t decided where they are meeting. Their options are a Baseball game or the Ballet. Both prefer to be with each other, but the Man prefers the Baseball game and the Woman prefers the Ballet. This gives payoffs as shown in Figure 16.11.
Figure 16.11 The battle of the sexes
The Man would prefer that they both go to the Baseball game, and the Woman prefers that both go to the Ballet. They each get 2 payoff points for being with each other, and an additional point for being at their preferred entertainment. In this game, iterated elimination of dominated strategies eliminates nothing. One can readily verify that there are two Nash equilibria: one in which they both go to the Baseball game and one in which they both go to the Ballet. The logic is this: If the Man is going to the Baseball game, the Woman prefers the 2 points she gets at the Baseball game to the single point she would get at the Ballet. Similarly, if the Woman is going to the Baseball game, the Man gets three points going there versus zero at the Ballet. Hence, going to the Baseball game is one Nash equilibrium. It is straightforward to show that for both to go to the Ballet is also a Nash equilibrium and, finally, that neither of the other two possibilities in which they go to separate places is an equilibrium.
Now consider the game of matching pennies, a child’s game in which the sum of the payoffs is zero. In this game, both the row player and the column player choose heads or tails, and if they match, the row player gets the coins, while if they don’t match, the column player gets the coins. The payoffs are provided in Figure 16.12.
Figure 16.12 Matching pennies
You can readily verify that none of the four possibilities represents a Nash equilibrium. Any of the four involves one player getting –1; that player can convert –1 to 1 by changing his or her strategy. Thus, whatever the hypothesized equilibrium, one player can do strictly better, contradicting the hypothesis of a Nash equilibrium. In this game, as every child who plays it knows, it pays to be unpredictable, and consequently players need to randomize. Random strategies are known as mixed strategies because the players mix across various actions.
Key Takeaways
• In a Nash equilibrium, each player chooses the strategy that maximizes his or her expected payoff, given the strategies employed by others. Outcomes that aren’t Nash equilibria involve mistakes for at least one player.
• The game called “the battle of the sexes” has two Nash equilibria.
• In the game of matching pennies, none of the four possibilities represents a Nash equilibrium. Consequently, players need to randomize. Random strategies are known as mixed strategies because the players mix across various actions. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/16%3A_Games_and_Strategic_Behavior/16.02%3A_Nash_Equilibrium.txt |
Learning Objectives
• What games require or admit randomization as part of their solution?
Let us consider the matching pennies game again, as illustrated in Figure 16.13.
Figure 16.13 Matching pennies again
If $$2p – 1 > 1 – 2p$$, then Row is better off, on average, playing Heads than Tails. Similarly, if $$2p – 1 < 1 – 2p$$, then Row is better off playing Tails than Heads. If, on the other hand, $$2p – 1 = 1 – 2p$$, then Row gets the same payoff no matter what Row does. In this case, Row could play Heads, could play Tails, or could flip a coin and randomize Row’s play.
A mixed strategy Nash equilibrium involves at least one player playing a randomized strategy and no player being able to increase his or her expected payoff by playing an alternate strategy. A Nash equilibrium in which no player randomizes is called a pure strategy Nash equilibrium.
Figure 16.14 Mixed strategy in matching pennies
Note that randomization requires equality of expected payoffs. If a player is supposed to randomize over strategy A or strategy B, then both of these strategies must produce the same expected payoff. Otherwise, the player would prefer one of them and wouldn’t play the other.
Computing a mixed strategy has one element that often appears confusing. Suppose that Row is going to randomize. Then Row’s payoffs must be equal for all strategies that Row plays with positive probability. But that equality in Row’s payoffs doesn’t determine the probabilities with which Row plays the various rows. Instead, that equality in Row’s payoffs will determine the probabilities with which Column plays the various columns. The reason is that it is Column’s probabilities that determine the expected payoffs for Row; if Row is going to randomize, then Column’s probabilities must be such that Row is willing to randomize.
Thus, for example, we computed the payoff to Row of playing Heads, which was 2p – 1, where p was the probability that Column played Heads. Similarly, the payoff to Row of playing Tails was 1 – 2p. Row is willing to randomize if these are equal, which solves for p = ½.
Now let’s try a somewhat more challenging example and revisit the battle of the sexes. Figure 16.15 illustrates the payoffs once again.
Figure 16.15 Mixed strategy in battle of the sexes
Figure 16.16 Full computation of the mixed strategy
For example, if the Man (row player) goes to the Baseball game, he gets 3 when the Woman goes to the Baseball game (probability p), and otherwise gets 1, for an expected payoff of $$3p + 1(1 – p) = 1 + 2p$$. The other calculations are similar, but you should definitely run through the logic and verify each calculation.
A mixed strategy in the battle of the sexes game requires both parties to randomize (since a pure strategy by either party prevents randomization by the other). The Man’s indifference between going to the Baseball game and to the Ballet requires $$1 + 2p = 2 – 2p$$, which yields p = ¼. That is, the Man will be willing to randomize which event he attends if the Woman is going to the Ballet ¾ of the time, and otherwise to the Baseball game. This makes the Man indifferent between the two events because he prefers to be with the Woman, but he also likes to be at the Baseball game. To make up for the advantage that the game holds for him, the Woman has to be at the Ballet more often.
Similarly, in order for the Woman to randomize, the Woman must get equal payoffs from going to the Baseball game and going to the Ballet, which requires $$2q = 3 – 2q$$, or q = ¾. Thus, the probability that the Man goes to the Baseball game is ¾, and he goes to the Ballet ¼ of the time. These are independent probabilities, so to get the probability that both go to the Baseball game, we multiply the probabilities, which yields 3/16. Figure 16.17 fills in the probabilities for all four possible outcomes.
Figure 16.17 Mixed strategy probabilities
Note that more than half of the time (Baseball, Ballet) is the outcome of the mixed strategy and the two people are not together. This lack of coordination is generally a feature of mixed strategy equilibria. The expected payoffs for both players are readily computed as well. The Man’s payoff is $$1+2 p=2-2 p$$, and since p = ¼, the Man obtains 1½. A similar calculation shows that the Woman’s payoff is the same. Thus, both do worse than coordinating on their less preferred outcome. But this mixed strategy Nash equilibrium, undesirable as it may seem, is a Nash equilibrium in the sense that neither party can improve his or her own payoff, given the behavior of the other party.
In the battle of the sexes, the mixed strategy Nash equilibrium may seem unlikely; and we might expect the couple to coordinate more effectively. Indeed, a simple call on the telephone should rule out the mixed strategy. So let’s consider another game related to the battle of the sexes, where a failure of coordination makes more sense. This is the game of “chicken.” In this game, two players drive toward one another, trying to convince the other to yield and ultimately swerve into a ditch. If both swerve into the ditch, we’ll call the outcome a draw and both get zero. If one swerves and the other doesn’t, the driver who swerves loses and the other driver wins, and we’ll give the winner one point.Note that adding a constant to a player’s payoffs, or multiplying that player’s payoffs by a positive constant, doesn’t affect the Nash equilibria—pure or mixed. Therefore, we can always let one outcome for each player be zero, and another outcome be one. The only remaining question is what happens when neither yield, in which case a crash results. In this version, the payoff has been set at four times the loss of swerving, as shown in Figure 16.18, but you can change the game and see what happens.
Figure 16.18 Chicken
This game has two pure strategy equilibria: (Swerve, Don’t) and (Don’t, Swerve). In addition, it has a mixed strategy. Suppose that Column swerves with probability p. Then Row gets $$0p + –1(1 – p)$$ from swerving, $1p + –4(1 – p)$ from not swerving, and Row will randomize if these are equal, which requires $$p = ¾$$. That is, the probability that Column swerves in a mixed strategy equilibrium is ¾. You can verify that the row player has the same probability by setting the probability that Row swerves equal to q and computing Column’s expected payoffs. Thus, the probability of a collision is 1/16 in the mixed strategy equilibrium.
The mixed strategy equilibrium is more likely, in some sense, in this game: If the players already knew who was going to yield, they wouldn’t actually need to play the game. The whole point of the game is to find out who will yield, which means that it isn’t known in advance. This means that the mixed strategy equilibrium is, in some sense, the more reasonable equilibrium.
Figure 16.19 Rock, paper, scissors
Key Takeaways
• A mixed strategy Nash equilibrium involves at least one player playing a randomized strategy and no player being able to increase his or her expected payoff by playing an alternate strategy.
• A Nash equilibrium without randomization is called a pure strategy Nash equilibrium.
• If a player is supposed to randomize over two strategies, then both must produce the same expected payoff.
• The matching pennies game has a mixed strategy and no pure strategy.
• The battle of the sexes game has a mixed strategy and two pure strategies.
• The game of chicken is similar to the battle of the sexes and, like it, has two pure strategies and one mixed strategy.
EXERCISES
1. Let q be the probability that Row plays Heads. Show that Column is willing to randomize, if and only if q = ½. (Hint: First compute Column’s expected payoff when Column plays Heads, and then compute Column’s expected payoff when Column plays Tails. These must be equal for Column to randomize.)
2. Show that in the rock, paper, scissors game there are no pure strategy equilibria. Show that playing all three actions with equal likelihood is a mixed strategy equilibrium.
3. Find all equilibria of the following games:
Figure 16.20
4. If you multiply a player’s payoff by a positive constant, the equilibria of the game do not change. Is this true or false, and why? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/16%3A_Games_and_Strategic_Behavior/16.03%3A_Mixed_Strategies.txt |
Learning Objectives
• How can game theory be applied to the economic settings?
Our first example concerns public goods. In this game, each player can either contribute or not. For example, two roommates can either clean their apartment or not. If they both clean, the apartment is nice. If one cleans, then that roommate does all of the work and the other gets half of the benefits. Finally, if neither cleans, neither is very happy. This suggests the following payoffs as shown in Figure 16.21.
Figure 16.21 Cleaning the apartment
You can verify that this game is similar to the prisoner’s dilemma in that the only Nash equilibrium is the pure strategy in which neither player cleans. This is a game-theoretic version of the tragedy of the commons—even though both roommates would be better off if both cleaned, neither do. As a practical matter, roommates do solve this problem, using strategies that we will investigate when we consider dynamic games.
Figure 16.22 Driving on the right
Figure 16.23 Bank location game
Figure 16.24 Political mudslinging
You have probably had the experience of trying to avoid encountering someone, whom we will call Rocky. In this instance, Rocky is actually trying to find you. Here it is Saturday night and you are choosing which party, of two possible parties, to attend. You like Party 1 better and, if Rocky goes to the other party, you get 20. If Rocky attends Party 1, you are going to be uncomfortable and get 5. Similarly, Party 2 is worth 15, unless Rocky attends, in which case it is worth 0. Rocky likes Party 2 better (these different preferences may be part of the reason why you are avoiding him), but he is trying to see you. So he values Party 2 at 10, Party 1 at 5, and your presence at the party he attends is worth 10. These values are reflected in Figure 16.25.
Figure 16.25 Avoiding Rocky
Figure 16.26 Price cutting game
Key Takeaways
• The free-rider problem of public goods with two players can be formulated as a game.
• Whether to drive on the right or the left is a game similar to battle of the sexes.
• Many everyday situations are reasonably formulated as games.
EXERCISES
1. Verify that the bank location game has no pure strategy equilibria and that there is a mixed strategy equilibrium where each city offers a rebate with probability ½.
2. Show that the only Nash equilibrium of the political mudslinging game is a mixed strategy with equal probabilities of throwing mud and not throwing mud.
3. Suppose that voters partially forgive a candidate for throwing mud in the political mudslinging game when the rival throws mud, so that the (Mud, Mud) outcome has payoff (2.5, 0.5). How does the equilibrium change?
1. Show that there are no pure strategy Nash equilibria in the avoiding Rocky game.
2. Find the mixed strategy Nash equilibria.
3. Show that the probability that you encounter Rocky is 7 12 .
4. Show that the firms in the price-cutting game have a dominant strategy to price low, so that the only Nash equilibrium is (Low, Low). | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/16%3A_Games_and_Strategic_Behavior/16.04%3A_Examples.txt |
Learning Objectives
• How do dynamic games play out?
So far, we have considered only games that are played simultaneously. Several of these games—notably the price cutting and apartment cleaning games—are actually played over and over again. Other games, like the bank location game, may only be played once, but nevertheless are played over time. Recall the bank location game, as illustrated once again in Figure 16.27.
Figure 16.27 Bank location game revisited
In this game, NYC makes the first move and chooses Rebate (to the left) or No Rebate (to the right). If NYC chooses Rebate, LA can then choose Rebate or None. Similarly, if NYC chooses No Rebate, LA can choose Rebate or None. The payoffs [using the standard of (LA, NYC) ordering] are written below the choices.
Figure 16.28 Sequential bank location (NYC payoff listed first)
What NYC would like to do depends upon what NYC believes LA will do. What should NYC believe about LA? (Boy, does that rhetorical question suggest a lot of facetious answers.) The natural belief is that LA will do what is in LA’s best interest. This idea—that each stage of a dynamic game is played in an optimal way—is called subgame perfection.
Subgame perfection requires each player to act in its own best interest, independent of the history of the game.Subgame perfection was introduced by Nobel laureate Reinhard Selten (1930–). This seems very sensible and, in most contexts, it is sensible. In some settings, it may be implausible. Even if I see a player make a particular mistake three times in a row, subgame perfection requires that I must continue to believe that that player will not make the mistake again. Subgame perfection may be implausible in some circumstances, especially when it pays to be considered somewhat crazy.
In the example, subgame perfection requires LA to offer a Rebate when NYC does (since LA gets 20 by rebating vs. 10), and to not offer a Rebate when NYC doesn’t. This is illustrated in the game, as shown in Figure 16.29, using arrows to indicate LA’s choices. In addition, the actions that LA won’t choose have been recolored in a light gray.
Once LA’s subgame perfection choices are taken into account, NYC is presented with the choice of offering a Rebate, in which case it gets 0, or not offering a Rebate, in which case it gets 10. Clearly the optimal choice for NYC is to offer No Rebate, in which case LA doesn’t either; and the result is 30 for LA, and 10 for NYC.
Dynamic games are generally “solved backward” in this way. That is, first establish what the last player does, then figure out—based upon the last player’s expected behavior—what the penultimate player does, and so on.
Figure 16.29 Subgame perfection
Figure 16.30 Can’t avoid Rocky
Since Rocky’s optimal choice eliminates your best outcomes, you make the best of a bad situation by choosing Party 1. Here, Rocky has a second mover advantage: Rocky’s ability to condition on your choice means that by choosing second he does better than he would do in a simultaneous game. In contrast, a first mover advantage is a situation where choosing first is better than choosing simultaneously. First mover advantages arise when going first influences the second mover advantageously.
Key Takeaways
• To decide what one should do in a sequential game, one figures out what will happen in the future, and then works backward to decide what to do in the present.
• Subgame perfection requires each player to act in his or her own best interest, independent of the history of the game.
• A first mover advantage is a situation where choosing first is better than choosing simultaneously. First mover advantages arise when going first influences the second mover advantageously.
• A second mover advantage is a situation where choosing second is better than choosing simultaneously. Second mover advantages arise when going second permits exploiting choices made by others.
EXERCISES
1. Formulate the battle of the sexes as a sequential game, letting the woman choose first. (This situation could arise if the woman were able to leave a message for the man about where she has gone.) Show that there is only one subgame perfect equilibrium, that the woman enjoys a first mover advantage over the man, and that she gets her most preferred outcome.
2. What payoffs would players receive if they played this two-player sequential game below? Payoffs are listed in parentheses, with Player 1’s payoffs always listed first. (Note that choosing “in” allows the other player to make a decision, while choosing “out” ends the game.)
Figure 16.31
3. Consider the following game:
Figure 16.32
1. Find all equilibria of the above game.
2. What is the subgame perfect equilibrium if you turn this into a sequential game, with Column going first? With Row going first?
3. In which game does Column get the highest payoff—the simultaneous game, the sequential game when Column goes first, or the sequential game when Column goes second? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/16%3A_Games_and_Strategic_Behavior/16.05%3A_Subgame_Perfection.txt |
Learning Objectives
• What can happen in games that are repeated over and over?
• What role does the threat of retaliation play?
Some situations, like the price-cutting game or the apartment cleaning game, are played over and over. Such situations are best modeled as a supergame.The supergame was invented by Robert Aumann (1930–) in 1959. A supergame is a game that is played an infinite number of times, where the players discount the future. The game played each time is known as a stage game. Generally supergames are played in times 1, 2, 3, ….
Cooperation may be possible in supergames, if the future is important enough. Consider the price-cutting game introduced previously and illustrated again in Figure 16.33.
Figure 16.33 Price cutting game revisited
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The dominant strategy equilibrium to this game is (Low, Low). It is clearly a subgame perfect equilibrium for the players to just play (Low, Low) over and over again because, if that is what Firm 1 thinks that Firm 2 is doing, Firm 1 does best by pricing Low, and vice versa. But that is not the only equilibrium to the supergame.
Consider the following strategy, called a grim trigger strategy, which involves being nice initially but not nice forever when someone else isn’t cooperative. Price High, until you see your rival price Low. After your rival has priced Low, price Low forever. This is called a trigger strategy because an action of the other player (pricing Low) triggers a change in behavior. It is a grim strategy because it punishes forever.
If your rival uses a grim trigger strategy, what should you do? Basically, your only choice is when to price Low because, once you price Low, your rival will price Low, and then your best choice is also to price Low from then on. Thus, your strategy is to price High up until some point t – 1, and then price Low from time t on. Your rival will price High through t, and price Low from t + 1 on. This gives a payoff to you of 15 from period 1 through t – 1, 25 in period t, and then 5 in period t + 1 on. We can compute the payoff for a discount factor δ:
$V t =15(1+δ+ δ 2 +…+ δ t−1 )+25 δ t +5( δ t+1 + δ t+2 +…)=15 1− δ t 1−δ +25 δ t +5 δ t 1−δ = 15 1−δ − δ t 1−δ (15−25(1−δ)−5δ)= 15 1−δ − δ t 1−δ (−10+20δ).$
If $$–10 + 20δ < 0$$, it pays to price Low immediately, at t = 0, because it pays to price Low; and the earlier that one prices Low, the higher the present value. If $$–10 + 20δ > 0$$, it pays to wait forever to price Low; that is, t = ∞. Thus, in particular, the grim trigger strategy is an optimal strategy for a player when the rival is playing the grim trigger strategy if δ ≥ ½. In other words, cooperation in pricing is a subgame perfect equilibrium if the future is important enough; that is, the discount factor δ is high enough.
The logic of this example is that the promise of future cooperation is valuable when the future itself is valuable, and that promise of future cooperation can be used to induce cooperation today. Thus, Firm 1 doesn’t want to cut price today because that would lead Firm 2 to cut price for the indefinite future. The grim trigger strategy punishes price cutting today with future Low profits.
Supergames offer more scope for cooperation than is illustrated in the price-cutting game. First, more complex behavior is possible. For example, consider the game shown in Figure 16.34:
Figure 16.34 A variation of the price-cutting game
Here, again, the unique equilibrium in the stage game is (Low, Low). But the difference between this game and the previous game is that the total profits of Firms 1 and 2 are higher in either (High, Low) or (Low, High) than in (High, High). One solution is to alternate between (High, Low) and (Low, High). Such alternation can also be supported as an equilibrium, using the grim trigger strategy—that is, if a firm does anything other than what it is supposed to do in the alternating solution, the firms instead play (Low, Low) forever.
The folk theorem says that if the value of the future is high enough, any outcome that is individually rational can be supported as an equilibrium to the supergame. Individual rationality for a player in this context means that the outcome offers a present value of profits at least as high as that offered in the worst equilibrium in the stage game from that player’s perspective. Thus, in the price-cutting game, the worst equilibrium of the stage game offered each player 5, so an outcome can be supported if it offers each player at least a running average of 5.
The simple logic of the folk theorem is this. First, any infinite repetition of an equilibrium of the stage game is itself a subgame perfect equilibrium. If everyone expects this repetition of the stage game equilibrium, no one can do better than to play his or her role in the stage game equilibrium every period. Second, any other plan of action can be turned into a subgame perfect equilibrium merely by threatening any agent who deviates from that plan with an infinite repetition of the worst stage game equilibrium from that agent’s perspective. That threat is credible because the repetition of the stage game equilibrium is itself a subgame perfect equilibrium. Given such a grim trigger–type threat, no one wants to deviate from the intended plan.
The folk theorem is a powerful result and shows that there are equilibria to supergames that achieve very good outcomes. The kinds of coordination failures that we saw in the battle of the sexes, and the failure to cooperate in the prisoner’s dilemma, need not arise; and cooperative solutions are possible if the future is sufficiently valuable.
However, it is worth noting some assumptions that have been made in our descriptions of these games—assumptions that matter but are unlikely to be true in practice. First, the players know their own payoffs. Second, they know their rival’s payoffs. They possess a complete description of the available strategies and can calculate the consequences of these strategies—not just for themselves but also for their rivals. Third, each player maximizes his or her expected payoff; they know that their rivals do the same; they know that their rivals know that everyone maximizes; and so on. The economic language for this is the structure of the game, and the players’ preferences are common knowledge. Few real-world games will satisfy these assumptions exactly. Since the success of the grim trigger strategy (and other strategies we haven’t discussed) generally depends upon such knowledge, informational considerations may cause cooperation to break down. Finally, the folk theorem shows us that there are lots of equilibria to supergames but provides no guidance on which ones will be played. These assumptions can be relaxed, although they may lead to wars on the equilibrium path “by accident”—and a need to recover from such wars—so that the grim trigger strategy becomes suboptimal.
Key Takeaways
• A supergame is a game that is played over and over again without end, where the players discount the future. The game played each time is known as a stage game.
• Playing a “one-shot” Nash equilibrium to the stage game forever is a subgame perfect equilibrium to the supergame.
• A grim trigger strategy involves starting play by using one behavior and, if another player ever does something else, switching to one-shot Nash behavior forever.
• The folk theorem says that if the value of the future is high enough, any outcome that is individually rational can be supported as an equilibrium to the supergame. Individual rationality for a player means that the outcome offers a present value of profits at least as high as that offered in the worst equilibrium in the stage game from that player’s perspective.
• If players are patient, full cooperation is obtainable as one of many subgame perfect equilibria to supergames.
EXERCISE
1. Consider the game in Figure 16.34, and consider a strategy in which Firm 1 prices High in odd-numbered periods and Low in even-numbered periods, while Firm 2 prices High in even-numbered periods and Low in odd-numbered periods. If either deviates from these strategies, both firms price Low from then on. Let δ be the discount factor. Show that these firms have a payoff of $$251-\delta 2 \text { or } 25 \delta 1-\delta 2$$, depending upon which period it is. Then show that the alternating strategy is sustainable if $$10+ 5δ 1−δ ≤ 25δ 1− δ 2$$. This, in turn, is equivalent to $$δ≥ 6 −2$$. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/16%3A_Games_and_Strategic_Behavior/16.06%3A_Supergames.txt |
Learning Objectives
• How do industries with only a few firms behave?
• How is their performance measured?
The CournotAugustus Cournot (1801–1877). oligopoly model is the most popular model of imperfect competition. It is a model in which the number of firms matters, and it represents one way of thinking about what happens when the world is neither perfectly competitive nor a monopoly.
In the Cournot model, there are n firms, who simultaneously set quantities. We denote a typical firm as firm i and number the firms from i = 1 to i = n. Firm i chooses a quantity qi ≥ 0 to sell, and this quantity costs ci (qi). The sum of the quantities produced is denoted by Q. The price that emerges from the competition among the firms is p(Q), and this is the same price for each firm. It is probably best to think of the quantity as really representing a capacity, and competition in prices by the firms determining a market price given the market capacity.
The profit that a firm i obtains is $$π i =p(Q) q i − c i ( q i ).$$
Each firm chooses qi to maximize profit. The first-order conditionsBear in mind that Q is the sum of the firms’ quantities, so that when firm i increases its output slightly, Q goes up by the same amount. give
$0= ∂ π i ∂ q i =p(Q)+ p ′ (Q) q i − c ′ i ( q i ) .$
This equation holds with equality provided qi > 0. A simple thing that can be done with the first-order conditions is to rewrite them to obtain the average value of the price-cost margin:
$p(Q)− c ′ i ( q i ) p(Q) =− p ′ (Q) q i p(Q) =− Q p ′ (Q) p(Q) q i Q = s i ε .$
Here s i = q i Q is firm i’s market share. Multiplying this equation by the market share and summing over all firms i = 1, …, n yields
$$∑ i=1 n p(Q)− c ′ i ( q i ) p(Q) s i = 1 ε ∑ i=1 n s i 2 = HHI ε$$ where $$HHI= ∑ i=1 n s i 2$$ is the Hirschman-Herfindahl Index (HHI).The HHI is named for Albert Hirschman (1915– ), who invented it in 1945, and Orris Herfindahl (1918–1972), who invented it independently in 1950. The HHI has the property that if the firms are identical, so that $$s_{i}=1 / n$$ for all i, then the HHI is also 1/n. For this reason, antitrust economists will sometimes use 1/HHI as a proxy for the number of firms, and describe an industry with “2 ½ firms,” meaning an HHI of 0.4.To make matters more confusing, antitrust economists tend to state the HHI using shares in percent, so that the HHI is on a 0 to 10,000 scale.
We can draw several inferences from these equations. First, larger firms, those with larger market shares, have a larger deviation from competitive behavior (price equal to marginal cost). Small firms are approximately competitive (price nearly equals marginal cost), while large firms reduce output to keep the price higher, and the amount of the reduction, in price-cost terms, is proportional to market share. Second, the HHI reflects the deviation from perfect competition on average; that is, it gives the average proportion by which price equal to marginal cost is violated. Third, the equation generalizes the “inverse elasticity result” proved for monopoly, which showed that the price-cost margin was the inverse of the elasticity of demand. The generalization states that the weighted average of the price-cost margins is the HHI over the elasticity of demand.
Because the price-cost margin reflects the deviation from competition, the HHI provides a measure of how large a deviation from competition is present in an industry. A large HHI means the industry “looks like monopoly.” In contrast, a small HHI looks like perfect competition, holding constant the elasticity of demand.
The case of a symmetric (identical cost functions) industry is especially enlightening. In this case, the equation for the first-order condition can be rewritten as $$0=p(Q)+ p ′ (Q) Q n − c ′ ( Q n ) or p(Q)= εn εn−1 c ′ ( Q n ).$$
Thus, in the symmetric model, competition leads to pricing as if demand was more elastic, and indeed is a substitute for elasticity as a determinant of price.
Key Takeaways
• Imperfect competition refers to the case of firms that individually have some price-setting ability or “market power” but are constrained by rivals.
• The Cournot oligopoly model is the most popular model of imperfect competition.
• In the Cournot model, firms choose quantities simultaneously and independently, and industry output determines price through demand. A Cournot equilibrium is a Nash equilibrium to the Cournot model.
• In a Cournot equilibrium, the price-cost margin of each firm is that firm’s market share divided by the elasticity of demand. Hence the share-weighted average price-cost margin is the sum of market squared market shares divided by the elasticity of demand.
• The Hirschman-Herfindahl Index (HHI) is the weighted average of the price-cost margins.
• In the Cournot model, larger firms deviate more from competitive behavior than do small firms.
• The HHI measures the industry deviation from perfect competition.
• The Cournot model generalizes the “inverse elasticity result” proved for monopoly. The HHI is one with monopoly.
• A large value for HHI means the industry “looks like monopoly.” In contrast, a small HHI looks like perfect competition, holding constant the elasticity of demand.
• With n identical firms, a Cournot industry behaves like a monopoly facing a demand that is n times more elastic. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/17%3A_Imperfect_Competition/17.01%3A_Cournot_Oligopoly.txt |
Learning Objectives
• What happens to quantity-setting firms when there are fixed costs of entry?
How does the Cournot industry perform? Let us return to the more general model, which doesn’t require identical cost functions. We already have one answer to this question: the average price-cost margin is the HHI divided by the elasticity of demand. Thus, if we have an estimate of the demand elasticity, we know how much the price deviates from the perfect competition benchmark.
The general Cournot industry actually has two sources of inefficiency. First, price is above marginal cost, so there is the deadweight loss associated with unexploited gains from trade. Second, there is the inefficiency associated with different marginal costs. This is inefficient because a rearrangement of production, keeping total output the same, from the firm with high marginal cost to the firm with low marginal cost, would reduce the cost of production. That is, not only is too little output produced, but what output is produced is inefficiently produced, unless the firms are identical.
To assess the productive inefficiency, we let c ′ 1 be the lowest marginal cost. The average deviation from the lowest marginal cost, then, is
$$χ= ∑ i=1 n s i ( c ′ i − c ′ 1 ) = ∑ i=1 n s i (p− c ′ 1 −(p− c ′ i )) =p− c ′ 1 − ∑ i=1 n s i (p− c ′ i )=p− c ′ 1 −p ∑ i=1 n s i (p− c ′ i ) p =p− c ′ 1 − p ε ∑ i=1 n s i 2 =p− c ′ 1 − p ε HHI .$$
Thus, while a large HHI means a large deviation from price equal to marginal cost and hence a large level of monopoly power (holding constant the elasticity of demand), a large HHI also tends to indicate greater productive efficiency—that is, less output produced by high-cost producers. Intuitively, a monopoly produces efficiently, even if it has a greater reduction in total output than other industry structures.
There are a number of caveats worth mentioning in the assessment of industry performance. First, the analysis has held constant the elasticity of demand, which could easily fail to be correct in an application. Second, fixed costs have not been considered. An industry with large economies of scale, relative to demand, must have very few firms to perform efficiently, and small numbers should not necessarily indicate the market performs poorly even if price-cost margins are high. Third, it could be that entry determines the number of firms and that the firms have no long-run market power, just short-run market power. Thus, entry and fixed costs could lead the firms to have approximately zero profits, in spite of price above marginal cost.
Using Exercise 1, suppose there is a fixed cost F that must be paid before a firm can enter a market. The number of firms n should be such that firms are able to cover their fixed costs, but add one more cost and they can’t. This gives us a condition determining the number of firms n:
$$(1-c n+1) 2 \geq F \geq(1-c n+2) 2$$
Thus, each firm’s net profits are $$( 1−c n+1 ) 2 −F≤ ( 1−c n+1 ) 2 − ( 1−c n+2 ) 2 = (2n+3) (1−c) 2 (n+1) 2 (n+2) 2 .$$
Note that the monopoly profits $$πm$$ are $$1 / 4(1-c)^{2}$$. Thus, with free entry, net profits are less than $$(2n+3)4 (n+1) 2 (n+2) 2 π m$$, and industry net profits are less than $n$(2n+3)4 (n+1) 2 (n+2) 2 π m$$.
Table 17.1 shows the performance of the constant-cost, linear-demand Cournot industry when fixed costs are taken into account and when they aren’t. With two firms, gross industry profits are 8/9 of the monopoly profits, not substantially different from monopoly. But when fixed costs sufficient to ensure that only two firms enter are considered, the industry profits are at most 39% of the monopoly profits. This percentage—39%—is large because fixed costs could be “relatively” low, so that the third firm is just deterred from entering. That still leaves the two firms with significant profits, even though the third firm can’t profitably enter. As the number of firms increases, gross industry profits fall slowly toward zero. The net industry profits, on the other hand, fall dramatically and rapidly to zero. With 10 firms, the gross profits are still about a third of the monopoly level, but the net profits are only at most 5% of the monopoly level.
Table 17.1 Industry Profits as a Fraction of Monopoly Profits
Number of Firms Gross Industry Profits (%) Net Industry Profits (%)
2 88.9 39.0
3 75.0 27.0
4 64.0 19.6
5 55.6 14.7
10 33.1 5.3
15 23.4 2.7
20 18.1 1.6
The Cournot model gives a useful model of imperfect competition, a model that readily permits assessing the deviation from perfect competition. The Cournot model embodies two kinds of inefficiency: (a) the exercise of monopoly power and (b) technical inefficiency in production. In settings involving entry and fixed costs, care must be taken in applying the Cournot model.
Key Takeaways
• The Cournot industry has two sources of inefficiency: too little output is produced, and what output is produced is inefficiently produced (unless the firms are identical).
• The HHI analysis has held constant the elasticity of demand, which could easily fail to be correct, and fixed costs have not been considered.
• Consideration of fixed costs reduces the apparent inefficiency of Cournot industry.
EXERCISES
1. Suppose the inverse demand curve is $$p(Q)=1-Q$$, and that there are n Cournot firms, each with constant marginal cost c, selling in the market.
1. Show that the Cournot equilibrium quantity and price are $$Q=n(1-c) n+1 \text { and } p(Q)=1+n c n+1$$
2. Show that each firm’s gross profits are $$(1-c n+1) 2$$
2. Suppose the inverse demand curve is $$p(Q)=1-Q$$, and that there are n Cournot firms, each with marginal cost c selling in the market.
1. Find the Cournot equilibrium price and quantity.
2. Determine the gross profits for each firm.
3. What formula from the Cournot model is used in antitrust analysis? How is it used?
4. Consider n identical Cournot firms in equilibrium.
1. Show that the elasticity of market demand satisfies ε>1/n.
2. Is this consistent in the case when n = 1 (monopoly)?
5. The market for Satellite Radio consists of only two firms. Suppose the market demand is given by $$P=250-Q$$, where P is the price and Q is the total quantity, so Q = Q1 + Q2. Each firm has total costs given by $$C\left(Q_{i}\right)=Q_{i}^{2}+5 Q i+200$$
1. What is the market price predicted by the Cournot duopoly model?
2. If the industry produces a total quantity X, what allocation of quantity (with X = Q1 + Q2) between the two companies minimizes total cost? (Your answer should express total cost as a function of X.)
3. If the firms merge with the cost found in b, what is the market price? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/17%3A_Imperfect_Competition/17.02%3A_Cournot_Industry_Performance.txt |
Learning Objectives
• What are the types of differentiated products and how do firms selling differentiated products behave?
Breakfast cereals range from indigestible, unprocessed whole grains to boxes that are filled almost entirely with sugar, with only the odd molecule or two of grain thrown in. Such cereals are hardly good substitutes for each other. Yet similar cereals are viewed by consumers as good substitutes, and the standard model of this kind of situation is the Hotelling model.Hotelling theory is named for Harold Hotelling (1895–1973). Hotelling was the first to use a line segment to represent both the product that is sold and the preferences of the consumers who are buying the products. In the Hotelling model, customers' preferences are located by points on the same line segment. The same line is used to represent products. For example, movie customers are differentiated by age, and we can represent moviegoers by their ages. Movies, too, are designed to be enjoyed by particular ages. Thus, a preteen movie is unlikely to appeal very much to a 6-year-old or to a 19-year-old, while a Disney movie appeals to a 6-year-old, but less to a 15-year-old. That is, movies have a target age, and customers have ages, and these are graphed on the same line.
Figure 17.1 Hotelling Model for Breakfast Cereals
There are two main types of differentiation, each of which can be modeled using the Hotelling line. These types are quality and variety. Quality refers to a situation where consumers agree on which product is better; the disagreement among consumers concerns whether higher quality is worth the cost. In automobiles, faster acceleration, better braking, higher gas mileage, more cargo space, more legroom, and greater durability are all good things. In computers, faster processing, brighter screens, higher resolution screens, lower heat, greater durability, more megabytes of RAM, and more gigabytes of hard drive space are all good things. In contrast, varieties are the elements about which there is not widespread agreement. Colors and shapes are usually varietal rather than quality differentiators. Some people like almond-colored appliances, others choose white, with blue a distant third. Food flavors are varieties, and while the quality of ingredients is a quality differentiator, the type of food is usually a varietal differentiator. Differences in music would primarily be varietal.
Quality is often called vertical differentiation, while variety is horizontal differentiation.
The standard Hotelling model fits two ice cream vendors on a beach. The vendors sell an identical product, and they can choose to locate wherever they wish. For the time being, suppose the price they charge for ice cream is fixed at \$1. Potential customers are also spread randomly along the beach.
We let the beach span an interval from 0 to 1. People desiring ice cream will walk to the closest vendor because the price is the same. Thus, if one vendor locates at x and the other at y, and x < y, those located between 0 and ½ (x + y) go to the left vendor, while the rest go to the right vendor. This is illustrated in Figure 17.2.
Figure 17.2 Sharing the Hotelling Market
Note that the vendor at x sells more by moving toward y, and vice versa. Such logic forces profit-maximizing vendors to both locate in the middle. The one on the left sells to everyone left of ½, while the one on the right sells to the rest. Neither can capture more of the market, so equilibrium locations have been found. (To complete the description of an equilibrium, we need to let the two “share” a point and still have one on the right side and one on the left side of that point.)
This solution is commonly used as an explanation of why U.S. political parties often seem very similar to each other—they have met in the middle in the process of chasing the most voters. Political parties can’t directly buy votes, so the “price” is fixed; the only thing parties can do is locate their platform close to voters’ preferred platform, on a scale of “left” to “right.” But the same logic that a party can grab the middle, without losing the ends, by moving closer to the other party will tend to force the parties to share the same middle-of-the-road platform.
The model with constant prices is unrealistic for the study of the behavior of firms. Moreover, the two-firm model on the beach is complicated to solve and has the undesirable property that it matters significantly whether the number of firms is odd or even. As a result, we will consider a Hotelling model on a circle and let the firms choose their prices.
Key Takeaways
• In the Hotelling model, there is a line, and the preferences of each consumer are represented by a point on this line. The same line is used to represent products.
• There are two main types of differentiation: quality and variety. Quality refers to a situation where consumers agree on which product is better. Varieties are the differentiators about which there is not widespread agreement.
• Quality is often called vertical differentiation, while variety is horizontal differentiation.
• The standard Hotelling model involves two vendors selling an identical product and choosing to locate on a line. Both charge the same price. People along the line buy from the closest vendor.
• The Nash equilibrium for the standard model involves both sellers locating in the middle. This is inefficient because it doesn’t minimize transport costs.
• The standard model is commonly used as a model to represent the positions of political candidates.
EXERCISE
1. Suppose there are four ice cream vendors on the beach, and customers are distributed uniformly. Show that it is a Nash equilibrium for two to locate at ¼ and two to locate at ¾. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/17%3A_Imperfect_Competition/17.03%3A_Hotelling_Differentiation.txt |
Learning Objectives
• Is there a simple, convenient model of differentiated product competition, and how does it perform?
In the circle model, a Hotelling model is set on a circle. There are n firms evenly spaced around the circle whose circumference is 1. Thus, the distance between any firm and each of its closest neighbors is 1/n. Consumers care about two things: how distant the firm they buy from is and how much they pay for the good. Consumers minimize the sum of the price paid and t times the distance between the consumer’s location (also on the circle) and the firm. Each consumer’s preference is uniformly distributed around the circle. The locations of firms are illustrated in Figure 17.3.
Figure 17.3 A Segment of the Circle Model
$$x^{*}=p+t n-r 2 t=12 n+p-r 2 t$$
Thus, consumers who are closer than x* to the firm charging r buy from that firm, and consumers who are further away than x* buy from the alternative firm. Demand for the firm charging r is twice x* (because the firm sells to both sides), so profits are price minus marginal cost times two $$x^{*} ; \text { that } i s,(r-c) 2 x^{*}=(r-c)(1 n+p-r t)$$
The first-order conditionBecause profit is quadratic in r, we will find a global maximum. for profit maximization is $$0=\partial \partial r(r-c)(1 n+p-r t)=(1 n+p-r t)-r-c t$$
We could solve the first-order condition for r. But remember that the question is, when does p represent a Nash equilibrium price? The price p is an equilibrium price if the firm wants to choose r = p. Thus, we can conclude that p is a Nash equilibrium price when $$p=c+t n$$
This value of p ensures that a firm facing rivals who charge p also chooses to charge p. Thus, in the Hotelling model, price exceeds marginal cost by an amount equal to the value of the average distance between the firms because the average distance is 1/n and the value to a consumer for traveling that distance is t. The profit level of each firm is t n 2 , so industry profits are t n .
How many firms will enter the market? Suppose the fixed cost is F. We are going to take a slightly unusual approach and assume that the number of firms can adjust in a continuous fashion, in which case the number of firms is determined by the zero profit condition $$F=t n 2, \text { or } n=t F$$
What is the socially efficient number of firms? The socially efficient number of firms minimizes the total costs, which are the sum of the transportation costs and the fixed costs. With n firms, the average distance a consumer travels is n $$\int-12 n 12 n|x| d x=2 n \int 012 n x \, d x=n(12 n) 2=14 n$$
Thus, the socially efficient number of firms minimizes the transport costs plus the entry costs t 4n +nF. This occurs at n= 1 2 t F . The socially efficient number of firms is half the number of firms that enter with free entry.
Too many firms enter in the Hotelling circle model. This extra entry arises because efficient entry is determined by the cost of entry and the average distance of consumers, while prices are determined by the marginal distance of consumers, or the distance of the marginal consumer. That is, competing firms’ prices are determined by the most distant customer, and that leads to prices that are too high relative to the efficient level; free entry then drives net profits to zero only when it is excess entry.
The Hotelling model is sometimes used to justify an assertion that firms will advertise too much, or engage in too much research and development (R&D), as a means of differentiating themselves and creating profits.
Key Takeaways
• A symmetric Nash equilibrium to the circle model involves a price that is marginal cost plus the transport cost t divided by the number of firms n. The profit level of each firm is t n 2 , so industry profits are t n .
• The socially efficient number of firms is half the number that would enter with free entry.
• The circle model is sometimes used to justify an assertion that firms will advertise too much, or engage in too much R&D, relative to the socially efficient amount. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/17%3A_Imperfect_Competition/17.04%3A_The_Circle_Model.txt |
Learning Objectives
• Can information held by sellers but relevant to buyers be an impediment to trade?
Nobel laureate George Akerlof (1940– ) examined the market for used cars and considered a situation known as the market for lemons, where the sellers are better informed than the buyers. This is quite reasonable because sellers have owned the car for a while and are likely to know its quirks and potential problems. Akerlof showed that this differential information may cause the used car market to collapse; that is, the information possessed by sellers of used cars destroys the market and the opportunities for profitable exchange.
To understand Akerlof’s insight, suppose that the quality of used cars lies on a 0 to 1 scale and that the population of used cars is uniformly distributed on the interval from 0 to 1. In addition, let that quality represent the value a seller places on the car, and suppose buyers put a value that is 50% higher than the seller. Finally, the seller knows the actual quality, while the buyer does not.
Can a buyer and seller trade in such a situation? First, note that trade is a good thing because the buyer values the car more than the seller. That is, both the buyer and seller know that they should trade. But can they agree on a price? Consider a price p. At this price, any seller who values the car less than p will be willing to trade. But because of our uniform distribution assumption, this means the distribution of qualities of cars offered for trade at price p will be uniform on the interval 0 to p. Consequently, the average quality of these cars will be ½ p, and the buyer values these cars 50% more, which yields ¾ p. Thus, the buyer is not willing to pay the price p for the average car offered at price p.
The effect of the informed seller and uninformed buyer produces a “lemons” problem. At any given price, all the lemons and only a few of the good cars are offered, and the buyer—not knowing the quality of the car—isn’t willing to pay as much as the actual value of a high-value car offered for sale. This causes the market to collapse; and only the worthless cars trade at a price around zero. Economists call this situation, where some parties have information that others do not, an informational asymmetry.
In the real world, of course, the market has found partial or imperfect solutions to the lemons problem identified by Akerlof. First, buyers can become informed and regularly hire their own mechanic to inspect a car they are considering. Inspections reduce the informational asymmetry but are costly in their own right. Second, intermediaries offer warranties and certification to mitigate the lemons problem. The existence of both of these solutions, which involve costs in their own right, is itself evidence that the lemons problem is a real and significant problem, even though competitive markets find ways to ameliorate the problems.
An important example of the lemons problem is the inventor who creates an idea that is difficult or impossible to patent. Consider an innovation that would reduce the cost of manufacturing computers. The inventor would like to sell it to a computer company, but she or he can’t tell the computer company what the innovation entails prior to price negotiations because then the computer company could just copy the innovation. Similarly, the computer company can’t possibly offer a price for the innovation in advance of knowing what the innovation is. As a result, such innovations usually require the inventor to enter the computer manufacturing business, rather than selling to an existing manufacturer, entailing many otherwise unnecessary costs.
Key Takeaways
• Information itself can lead to market failures.
• The market for lemons refers to a situation where sellers are better informed than buyers about the quality of the good for sale, like used cars.
• The informational asymmetry—sellers know more than buyers—causes the market to collapse.
• Inspections, warranties, and certification mitigate the lemons problem. The existence of these costly solutions is itself evidence that the lemons problem (informational asymmetry is an impediment to trade) is a real and significant problem.
• An example of the lemons problem is the inventor who creates an idea that is difficult or impossible to patent and cannot be verified without being revealed.
EXERCISE
1. In Akerlof’s market for lemons model, suppose it is possible to certify cars, verifying that they are better than a particular quality q. Thus, a market for cars “at least as good as q” is possible. What price or prices are possible in this market? (Hint: sellers offer cars only if q ≤ quality ≤ p.) What quality maximizes the expected gains from trade? | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/18%3A_Information/18.01%3A_Market_for_Lemons.txt |
Learning Objectives
• Can information about values and costs that is not relevant to the other party be an impediment to trade?
The lemons problem is a situation where the buyers are relatively uninformed and care about the information held by sellers. Lemons problems are limited to situations where the buyer isn’t well-informed, and these problems can be mitigated by making information public. In many transactions, the buyer knows the quality of the product, so lemons concerns aren’t a significant issue. There can still be a market failure, however, if there are a limited number of buyers and sellers.
Consider the case of one buyer and one seller bargaining over the sale of a good. The buyer knows his own value v for the good, but not the seller’s cost. The seller knows her own cost c for the good, but not the buyer’s value. The buyer views the seller’s cost as uniformly distributed on the interval [0,1], and, similarly, the seller views the buyer’s value as uniformly distributed on [0,1].The remarkable fact proved by Roger Myerson and Mark Satterthwaite (“Efficient Mechanisms for Bilateral Trade,” Journal of Economic Theory 28 [1983]: 265–281) is that the distributions don’t matter; the failure of efficient trade is a fully general property. Philip Reny and Preston McAfee (“Correlated Information and Mechanism Design,” Econometrica 60, no. 2 [March 1992]: 395–421) show the nature of the distribution of information matters, and Preston McAfee (“Efficient Allocation with Continuous Quantities,” Journal of Economic Theory 53, no. 1 [February 1991]: 51–74.) showed that continuous quantities can overturn the Myerson-Satterthwaite theorem. Can efficient trade take place? Efficient trade requires that trade occurs whenever v > c, and the remarkable answer is that it is impossible to arrange efficient trade if the buyer and seller are to trade voluntarily. This is true even if a third party is used to help arrange trade, provided the third party isn’t able to subsidize the transaction.
The total gains from trade under efficiency are $$∫ 0 1 ∫ 0 v v−c dc dv= ∫ 0 1 v 2 2 dv= 1 6 .$$
A means of arranging trade, known as a mechanism,A mechanism is a game for achieving an objective, in this case to arrange trades. asks the buyer and seller for their value and cost, respectively, and then orders trade if the value exceeds the cost and dictates a payment p by the buyer to the seller. Buyers need not make honest reports to the mechanism, however, and the mechanisms must be designed to induce the buyer and seller to report honestly to the mechanism so that efficient trades can be arranged.Inducing honesty is without loss of generality. Suppose that the buyer of type v reported the type z(v). Then we can add a stage to the mechanism in which the buyer reports a type, which is converted via the function z to a report, and then that report is given to the original mechanism. In the new mechanism, reporting v is tantamount to reporting z(v) to the original mechanism.
Consider a buyer who actually has value v but reports a value r. The buyer trades with the seller if the seller has a cost less than r, which occurs with probability r.
$u(r, v)=v r-E c p(r, c)$
The buyer gets the actual value v with probability r, and makes a payment that depends on the buyer’s report and the seller’s report. But we can take expectations over the seller’s report to eliminate it (from the buyer’s perspective), and this is denoted Ec p (r, c), which is just the expected payment given the report r. For the buyer to choose to be honest, u must be maximized at r = v for every v; otherwise, some buyers would lie and some trades would not be efficiently arranged. Thus, we can concludeWe maintain an earlier notation that the subscript refers to a partial derivative, so that if we have a function f, f1 is the partial derivative of f with respect to the first argument of $$f. d dv u(v,v)= u 1 (v,v)+ u 2 (v,v)= u 2 (v,v)=r | r=v =v.$$
The value u(v,v) is the gain accruing to a buyer with value v who reports having value v. Because the buyer with value 0 gets zero, the total gain accruing to the average buyer can be computed by integrating by parts $$∫ 0 1 u(v,v) dv=−(1−v)u(v,v) | v=0 1 + ∫ 0 1 (1−v)( du dv ) dv= ∫ 0 1 (1−v)vdv = 1 6 .$$
In the integration by parts, $$dv = d – (1 – v)$$ is used. The remarkable conclusion is that if the buyer is induced to truthfully reveal the buyer’s value, the buyer must obtain the entire gains from trade. This is actually a quite general proposition. If you offer the entire gains from trade to a party, that party is induced to maximize the gains from trade. Otherwise, he or she will want to distort away from maximizing the entire gains from trade, which will result in a failure of efficiency.
The logic with respect to the seller is analogous: the only way to get the seller to report her cost honestly is to offer her the entire gains from trade.
The Myerson-Satterthwaite theorem shows that private information about value may prevent efficient trade. Thus, the gains from trade are insufficient to induce honesty by both parties. (Indeed, they are half the necessary amount.) Thus, any mechanism for arranging trades between the buyer and the seller must suffer some inefficiency. Generally this occurs because buyers act like they value the good less than they do, and sellers act like their costs are higher than they truly are.
It turns out that the worst-case scenario is a single buyer and a single seller. As markets get “thick,” the per capita losses converge to zero, and markets become efficient. Thus, informational problems of this kind are a small-numbers issue. However, many markets do in fact have small numbers of buyers or sellers. In such markets, it seems likely that informational problems will be an impediment to efficient trade.
Key Takeaways
• The Myerson-Satterthwaite theorem shows that the gains from trade are insufficient to induce honesty about values and costs by a buyer and seller. Any mechanism for arranging trades between the buyer and the seller must suffer some inefficiency.
• Generally this inefficiency occurs because buyers act like they value the good less than they do, and sellers act like their costs are higher than they truly are, resulting in an inefficiently low level of trade.
• As markets get “thick,” the per capita losses converge to zero, and markets become efficient. Informational problems of this kind are a small-numbers issue.
EXERCISE
1. Let h(r, c) be the gains of a seller who has cost c and reports r, $$h(r, c) = p(v, r) – (1 – r)c.$$
Noting that the highest cost seller (c = 1) never sells and thus obtains zero profits, show that honesty by the seller implies the expected value of h is 1/16. | textbooks/socialsci/Economics/Introduction_to_Economic_Analysis/18%3A_Information/18.02%3A_Myerson-Satterthwaite_Theorem.txt |
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