Abstract: Thanks to Tim Blomfield and Nicholas Emery for contributions to the literature summaries and organization.

Economic growth under transformative AI

Introduction At least since Herbert Simon's 1960 prediction that artificial intelligence would soon replace all human labor, many economists have understood that there is a possibility that sooner or later artificial intelligence (AI) will dramatically transform the global economy. AI could have a transformative impact on a wide variety of domains; indeed, it could transform market structure, the value of education, the geopolitical balance of power, and practically anything else. We will focus on three of the clearest and best-studied classes of potential transformations in economics: the potential impacts on output growth, on wage growth, and on the labor share, i.e. the share of output paid as wages. On all counts we will focus on long-run impacts rather than transition dynamics. Instead of attempting to predict the future, our focus will be on surveying the vast range of possibilities identified in the economics literature. The potential impact of AI on the growth rate of output could take the form of • a decrease to the growth rate, even perhaps rendering it negative; • a permanent increase to the growth rate, as the Industrial Revolution increased the global growth rate from near zero to something over two percent per year; • a continuous acceleration in growth, with the growth rate growing unboundedly as time tends to infinity; or even • an acceleration in the growth rate rapid enough to produce infinite output in finite time. Basic physics suggests that the last of these scenarios is impossible, of course-as is eternal exponential or super-exponential growth. The relevant possibility is that AI induces a growth path that resembles these benchmarks for some time, until production confronts limiting factors, such as land or energy, that have never or have not recently been important constraints on growth. We will thus also consider how increases in growth may eventually be choked off by such limiting factors. The potential impact of AI on the labor market includes predictions that • wages fall; • wages rise, but less quickly than output, producing a declining labor share; • wages rise in line with output, producing a constant labor share; or even • wages rise more quickly than output, producing a rising labor share. In the respective limiting cases, AI could result in a future in which wages are (literally or asymptotically) zero or near zero; very high in absolute terms but approximately zero percent of total output; or high both in absolute and in relative terms. As this discussion illustrates, the space of possibilities is vast. At the same time, as Simon's failed prediction testifies, transformational impacts from AI are by no means certain to transpire on any particular time horizon. Most studies to date of the economics of AI, therefore, have focused on the most immediate, moderate, and likely impacts of AI. 1 These include marginal shifts in output and factor shares; impacts on marketing and statistical discrimination; impacts on regional inequality; and industry-specific forecasts of AI-induced growth and labor displacement over the next few decades. Empirical estimates of the future economic importance of AI have drawn inferences from foreseeable industry-specific applications of AI, from totals spent on AI R&D, and from comparisons between AI and past technological developments in computing, internet connectivity, or related fields (see e.g.  Chen et al. (2016) ). Industrial and R&D-based inferences may however severely underestimate the field's transformativeness if technological development has substantial external effects, as is often assumed; the importance of the atomic bomb is not well approximated by the cost of the Manhattan project. Even in the absence of externalities, furthermore, inferences from R&D expenditures implicitly discount future output according to the time preference of the research funders, who may assign negligible value to the impact of their technologies on the distant future. And common referenceclass-based projections preclude the possibility that AI ultimately proves to be truly transformational, less like the economic impact of broadband than 1 Summaries of the economic implications of AI have been put out by most major consulting firms and by several governments and academic institutes. A proper review of these reviews would require a document in its own right, but for comparison, the review that appears to go furthest in discussing AI's transformative possibilities is that from Accenture  (Purdy and Daugherty, 2016) . The most radical scenario the authors consider is one in which AI comes to serve as a "new factor of production" complementing both labor and capital. They forecast that, in this scenario, the result will be what they call "a transformative effect on growth", by which they mean a doubling of growth rates in developed countries up to 2035. like that of the Industrial Revolution or the evolution of the human species itself. In recent years, economists have begun to engage earnestly in formal theoretical explorations of a wide array of the transformative possibilities of AI, including those outlined above. We aim to summarize the findings of these explorations.  2  In the process, we hope not only to state the conclusions of various models, but also to give the reader some mathematical intuition for the most important mechanisms at play. (Indeed we have altered some of the models slightly, to clarify their implications regarding transformativeness and their relationships to the other models discussed here.) This document is intended for anyone comfortable with a moderate amount of mathematical notation and interested in understanding the channels through which AI could have a transformative impact on wages and growth. Readers with backgrounds in economics will hopefully come to better understand the possibilities which concern singularitarians, and readers with singularitarian backgrounds will hopefully come to better understand the relevant tools and insights of economics. The rest of this document proceeds as follows. §2 consists of an overview of the economics that will be relevant for understanding the subsequent sections. §3 discusses models in which AI is added to a standard production function. §4 discusses models in which AI is added to a "task-based" production function. §3 and §4 both implicitly take place in a setting of exogenous productivity growth; §5 discusses models in which productivity growth is endogenous and AI can feature in its production. §6 compares the results found in §3-5. Finally, §7 concludes.
 Economics background Literature on the economics of AI extends prior literature on the economics of production and growth. Those without backgrounds in economics may therefore find it helpful to review the latter briefly. Those with backgrounds in economics may want to skim the section to familiarize themselves with the notation and terminology we introduce.
 Production and factor shares Throughout this review, we say that output at a time t is determined by a production function F and the list of input (or "factor") quantities available at t. If for simplicity we categorize all production factors as either labor L or capital K, and think of all output as a single good Y , we can write Y t = F (K t , L t ). We will temporarily abandon the time subscripts and explore production in a static setting; they will return when we explore growth. The production function F is assumed to be continuously differentiable, increasing, and concave in each argument. It is also assumed to exhibit constant returns to scale (CRS); doubling the earth, with all its labor and all its capital, would presumably double output. Finally, its inputs are assumed to be complements: the marginal productivity of each is increasing in the supply of the other.  3  There can be "capital-augmenting" technology, denoted A, and "laboraugmenting" technology, denoted B. Increases in some factor-augmenting technology make the use of the given factor more efficient, so that we can proceed as if we had more of it. With technology, that is, our production function takes the form Y = F (AK, BL). Note that, by CRS, a technological advance that multiplies both A and B by a given factor will multiply output by this factor as well. The "marginal product" of a factor is the derivative of output with respect to that factor. Both inputs are assumed to be paid their marginal products. That is, the wage rate is F L (AK, BL), and the capital rental rate is F K (AK, BL). This makes sense if we imagine that production is taking place in a competitive market, with lots of identical firms facing this production function. If a factor employed at one firm is not being paid its marginal product, another firm will offer to pay more for it. These factor payments will equal total output-i.e. KF K (AK, BL) + LF L (AL, BL) = F (AK, BL) (1) -whenever F is CRS, by Euler's Homogeneous Function Theorem. If output exceeded the sum of factor payments, we would have to explain what was done with the output not paid as wages or rents; and if factor payments exceeded output, we would have to explain how the deficit was filled. The fraction of output paid out as wages-i.e. LF L (AK, BL)/F (AK, BL) -is termed the "labor share". Likewise, the "capital share" is the fraction of output that accrues to the owners of capital. We will assume through most this document that everyone is employed. In reality, of course, many cannot work, or will not work if the wage rate is sufficiently low. Model scenarios in which the wage rate falls dramatically may therefore be more accurately interpreted as scenarios in which unemployment is widespread. For our purposes this will not be an important distinction.
 Substitution Given a production (or utility) function and a list of factor (or consumption good) prices, suppose a purchaser spends a fixed budget so as to maximize output (or utility). The elasticity of substitution between two factors X 1 and X 2 is, intuitively, the value such that, if the relative price of X 1 falls by a small proportion (say, 1%), the relative quantity of X 1 purchased will rise by an -times larger proportion ( %). Conversely, then, given a list of factor quantities, the elasticity of substitution is the value such that, in order for the relative quantity of X 1 sold to increase by a small proportion (say 1%), the relative price of X 1 must fall by a (1/ )-times larger proportion ((1/ )%). Suppose that goods X 1 and X 2 are divided among many owners, and suppose there are many interested purchasers. (The sets of owners and potential purchasers need not be disjoint, but it may be easier to imagine that they are.) The price of X 1 should then equal the market-clearing price: the price such that, given the price of X 2 , precisely the entire supply of X 1 is purchased. If the price of X 1 were higher than this, anyone who owns some of it would do well to sell her unsold goods at just less than the market price. If it were lower, i.e. if purchasers would have been willing to buy more than the entire supply of X 1 at the given price, then anyone who owns some X 1 could charge more than the given price and still sell all she has. Now consider the consequences of exogenously increasing the relative quantity of X 1 by 1%, say by increasing its absolute quantity by 1% across each of its owners and leaving the quantity of X 2 unchanged. The marketclearing relative price of X 1 will then fall by (1/ )%. As we can see, marginally increasing the relative abundance of a good results in smaller relative expenditure on that good-i.e. its owners receive a smaller share of total income-precisely when the elasticity of substitution between it and other goods, on the current margin, is less than 1. For illustration: food and other goods are not very substitutable. When food was much scarcer, its owners were able to command such higher prices for it that people spent larger shares of their incomes on it. On the other hand, industrially produced goods and handmade goods are very substitutable. As the former grew more plentiful, following the Industrial Revolution's explosion in manufacturing, people spent larger shares of their incomes on them. Goods are perfect complements if = 0. In this case, output (or utility) is some constant times the minimum of the goods' quantities, in some ratio. (Consider left shoes and right shoes, in the 1:1 ratio, or bicycle frames and wheels, in the 1:2 ratio.) However the goods' relative prices change, the relative quantities purchased will stay fixed at the given ratio. The case of perfect substitutability is approached in the limit as → ∞. In this case a positive quantity of each good is only purchased if their prices are equal; if their prices differ, only the cheaper is purchased. For ease of notation, let us define the "substitution parameter" ρ ( − 1)/ . Note that the cases < 1, = 1, and > 1 correspond to ρ < 0, ρ = 0, and ρ > 0 respectively. A production function exhibits constant elasticity of substitution (CES) if its elasticity of substitution does not depend on the factor prices and quantities. A two-factor CES production function that is also CRS, with factoraugmenting technology, must take the form Y = [(AK) ρ + (BL) ρ ] 1/ρ (3) if ρ = 0 and Y = (AK) a (BL) 1−a , (4) for some a ∈ (0, 1), if ρ = 0. In the second case, the function is called "Cobb-Douglas". When ρ ≤ 0, output requires strictly positive quantities of both factors. If we have more than two factors i, natural extensions of the above functions look similar, with the Cobb-Douglas exponents a i still summing to 1. When ρ = 0, the share of output paid to factor X, with factor-augmenting technology C, equals (CX/Y ) ρ . When ρ = 0, a factor's share equals the exponent on that factor. In general, the share of factor X is decreasing in CX/Y when ρ < 0, independent of CX/Y when ρ = 0, and increasing in CX/Y when ρ > 0. Naturally a CES and CRS utility function, defined over consumption goods, has the same properties. The notion that a utility function could be CRS may sound absurd, since there is a natural sense in which marginal utility decreases in consumption. But if we are only interested in modeling how an agent with a CES utility function will apportion her spending at a given time, and if we are considering scenarios in which she has no uncertainty, we can without loss of generality invoke a utility function that is CRS as well. This is because, for any utility function in which returns to scale are not constant, there is a monotonic transformation of it-a new utility function ranking all "consumption bundles" (lists of good quantities) in the same order-which is CRS. Observe that there is a deep similarity between cases (a) in which there is a single consumption good, but multiple factors to its production, and (b) in which consumer utility is defined over multiple consumption goods, each of which employs a single production factor. Consumers in the latter cases function like factories in the former cases, as if consumer goods were inputs to the production of utility.
 Exogenous growth In practice, of course, the substitution parameter between labor and capital may not be constant. It may be high when labor is more abundant than capital and low otherwise, or vice-versa, for example. It may also change over time, for reasons independent of factor quantities. That said, it has been estimated in a variety of contexts and times to be substantially negative (see e.g.  Oberfield and Raval (2014)  or  Chirinko and Mallick (2017) ). When exploring preliminary hypotheses about growth, factor shares, and other macroeconomic variables, therefore, it can be helpful to start with a model in which production is CES (and CRS) with ρ < 0. Output per person in the developed world has grown substantially over the past few centuries, following a roughly exponential trajectory. In a two-factor production function without technology growth, the only possible explanation for this would be the capital stock growing more quickly than the population. Capital accumulation cannot be the primary force driving longrun growth, however, for at least two reasons. First: if ρ = 0, capital accumulation can produce unbounded output, albeit at a growth rate that slows to zero; and if ρ > 0 (or if production is not CES but the substitution parameter is permanently bounded above zero), capital accumulation can in principle sustain a positive growth rate. 4 But if ρ < 0 (constant or bounded below), as the stock of capital per unit of labor grows, capital's marginal productivity falls to the point that output growth slows to a halt. That is, a lack of labor per unit of capital definitively constrains the growth of output per person. As we can see from the first equation above, when ρ < 0, the capital term tends to zero as the quantity of capital increases. In the limit, then, if capital is far more plentiful than labor, output tends to BL, and output per person tends to B. Second, historically, the capital share throughout the developed world has been roughly constant (at about 1/3). As noted in §2.2, however, if ρ < 0, an unboundedly increasing stock of capital per unit of labor should decrease the capital share to zero. Capital-augmenting technology growth just increases the effective capital stock, so if ρ < 0 it cannot produce long-run output growth either, for the same reasons. 5 The way to get long-run per-capita output growth in this framework, as shown by  Uzawa (1961) , is to introduce labor-augmenting technology growth. To illustrate this, suppose for simplicity that A is fixed, that B grows at some constant exponential rate g B , and that a constant proportion s of output is saved as capital each period. 6 If s is high enough that capital accumulation can keep up with the growing effective labor force, and if the labor supply is also constant at L, the result is a growth path in which output Y t , capital K t , and "effective labor" B t L all grow at rate g B . (Recall that, by CRS, equal proportional increases to K t and B t L will produce an equal proportional increase to Y .) If s is too small, output will be constrained by capital accumulation. (This is clearest when s = 0.) In this case, Y t eventually approximately equals AK t . Then K t+1 = K t + sY t ≈ K t + sAK t (5) (using discrete-time notation for clarity), so capital and output both grow at asymptotic rate sA. 7 The requirement that capital accumulation keep up with the growing effective labor force is thus the requirement that sA ≥ g B . We will call this condition "sufficient saving". Note that given any fixed g A > 0 and g B , the sufficient saving condition will eventually be met. Letting the labor force grow at some positive rate g L makes no interesting difference. In this case, so long as sA ≥ g B + g L , B t L t grows at rate g B + g L , and Y t and K t do likewise. Regardless of population growth, factor shares are constant over time, 8 so wages and capital rents per person grow at rate g B . The empirical causes of technology growth remain highly uncertain. An exogenous growth model is one that does not attempt to model these causes, but simply takes constant exponential growth in B for granted.
 Endogenous growth On an endogenous growth account, on the other hand, growth in B is modeled as the output of some deliberate effort, such as technological research. That is, technology, like final output, is generated from inputs such as labor, capital, and the stock of existing technology. In the most commonly  Kaldor (1957)  and many since. There are relatively plausible ways to microfound this phenomenon, if we wish to develop our model in more detail, and we will touch on some of these in §3.5. For most of this document, however, we will simply take a constant saving rate for granted. 7 Models without labor, in which Y t = AK t , are termed "AK models". An AK economy is of course only ever constrained by capital accumulation and exhibits growth at rate sA. 8 It follows from the identity sY t = K t+1 − K t that sY t /K t = g K,t . Since in the long run a constant saving rate maintains g Y = g K , the long-run capital share (AK/Y ) ρ equals (sA/g Y ) ρ . If there is insufficient saving (so that g Y = sA < g B ), or in the edge case of sA = g B = g Y , the capital share tends to 1. used research-based growth model, that of  Jones (1995) , the growth of B in absolute terms is given by Ḃt = B φ t (S t L t ) λ (6) for λ > 0 and unrestricted values of φ, where S t denotes the fraction of the labor force working as researchers (or "scientists") at t. Intuitively, λ > 1 corresponds to cases in which researchers complement each other, and λ < 1 corresponds to cases in which some sort of "duplicated work" or "stepping on toes" effect predominates. Output is given by Y t = F (K t , B t (1 − S t )L t ), as before. In this setting, though output is CRS with respect to capital and effective labor at any given time, it exhibits increasing returns to scale in population across time. We therefore cannot continue to assume that all inputs to production are paid their marginal products. In particular, we cannot assume that technological innovators are compensated for all the additional future output that their research produces on the margin; this sum of marginal products would exceed total output! (Indeed,  Nordhaus (2004)  estimates that innovative firms accrue on average only about 2% of the value they produce.) It would be beyond the scope of this section to summarize theories regarding the empirical or optimal number of researchers, or the empirical or optimal level of worker pay. For now, to introduce endogenous technological development without having to consider its interactions with the rest of the framework, we might simply assume that a government sets S t and pays S t L t workers to do research. Their wages are equal to non-research workers' wages, in this stylization, and they are financed by lump-sum taxes levied equally across the population. To maintain a constant rate of output growth, we must maintain a constant rate of labor productivity growth, by the reasoning laid out in §2.3. However, the growth rate of B at t, denoted g B,t , is by definition equal to Ḃt /B t . Holding L and S fixed, therefore, we now have g B,t = B φ−1 t (BL) λ . ( 7 ) If φ < 1, as we can see, g B,t falls to 0 as B t grows. If φ > 1, g B,t rises to infinity. Only in the knife-edge case of φ = 1 do we get exponential growth with a constant number of researchers. 9 As a matter of fact, the number of researchers has grown dramatically over the past few centuries. Both the population and the fraction of the population working in research have grown. For simplicity, and because the number of researchers cannot grow indefinitely in a fixed population, let us here ignore the second trend and suppose that S is fixed, with L t growing at a constant rate g L . In this case labor productivity growth g B is constant over time iff B φ−1 t (SL t ) λ = B φ−1 0 exp(g B (φ − 1)t)S λ L λ 0 exp(g L λt) (8) is constant over time; that is, iff the change in the number of researchers just offsets the change in the difficulty of producing proportional productivity increases. This in turn will obtain iff (φ − 1)g B + λg L = 0, or 10 g B = λg L 1 − φ . (9) Because we are holding 1 − S t fixed, the number of non-research workers will grow at rate g L . So the number of effective non-research workers will grow at rate g B + g L . As we have seen, under a constant saving rate, capital and output will grow at this rate too, and output per person will grow at rate g B . Observe that the steady-state rate of labor productivity growth is here undefined when φ = 1. The calculation also breaks down when φ > 1, absurdly predicting a negative rate. This is because the value assumed to exist in the derivation, namely a steady-state productivity growth rate g B under a growing research workforce, does not exist when φ ≥ 1. When φ = 1, it is straightforward to see that a positive growth rate in the number of researchers produces an increasing rate of labor productivity growth. When φ > 1, even a constant number of researchers produces ever-increasing labor productivity growth as well. The ever-increasing labor productivity growth rate g B,t that follows when φ = 1 and g L > 0 translates into an increasing output growth rate up to the an unexplained process of exponential population growth is needed when φ < 1. Models of this form with φ < 1 are then termed "semi-endogenous". 10 It follows from g B,t = B φ−1 t (SL t ) λ that the corresponding growth path is stable. If B t is "too high", growth subsequently slows, since φ − 1 < 0. Likewise, if B t is "too low", growth accelerates. point that g B,t +g L = sA. At that point capital accumulation cannot keep up with the growth of the effective labor force, and production is constrained by capital. If capital-augmenting technology can be developed in parallel with labor-augmenting technology, however, the two factors can both grow at an increasing rate, and output therefore can as well. That is, we have a Type I growth singularity. When φ > 1, moreover, even a constant number of researchers is enough to produce "infinite output in finite time", i.e. a Type II growth singularity. The intuition for this is easiest to grasp when φ = 2 and (SL) λ = 1, so that we have g B,t = B t . Suppose g B,0 is such that technology doubles every time period. Thus B 1 = 2B 0 , so g B,1 = 2g B,0 . At this doubled growth rate, technology doubles every half-period; B 1.5 = 2B 1 . By repeated applications of the same reasoning, the technology level approaches a vertical asymptote at t = 2. If capital-augmenting technology follows a similar process, output approaches a vertical asymptote at t = 2 as well. The potential for endogenous growth processes to produce explosive growth is striking. However, since the researcher population growth rate has long been positive and the productivity growth rate has long been roughly constant, and in fact declining over recent decades  (Gordon, 2016) , we can infer that at least historically φ < 1. Indeed, the most extensive study of the topic to date-that done by  Bloom et al. (2020) -estimates φ = −2.1. An estimate of φ ∈ (0, 1) would indicate that, when we have access to a large stock of existing technologies, these aid in the development of new technologies, but offer diminishing marginal aid. An estimate of φ < 0 implies that when there is a large stock of existing technologies it is harder to develop new technologies-perhaps because so much of the low-hanging technological fruit has already been developed, with this "fishing out" effect outweighing the effect of technological assistance in technological development.
 AI in basic models of good production
 Capital productivity in isolation At face value, AI promises to make capital more productive. This would most naturally be modeled in the standard framework as an increase to A, which would amount to effective capital accumulation. As  Acemoglu and Restrepo (2018a)  point out, and as we have seen, this on its own would not be predicted to have very transformative economic effects. It would increase output and wages somewhat. But given ρ < 0 and a fixed or only slowgrowing labor supply, labor is the primary bottleneck to output, and any increases to wages would come ever more from an increase in the labor share rather than an increase in output. Indeed, the only "transformative" effect of capital productivity is that, as A → ∞, all else equal, the labor share should rise to 1. This is of course the opposite of the intuitive trend, which is also the observed trend in the labor share in recent decades, especially in the industries that have undergone most automation. The models below, therefore, are all designed to shed light on the consequences of increasing the productivity of capital in combination with various structural changes to the production function that AI might also precipitate.
 Imperfect substitution Nordhaus (2015) explores the transformative possibility of AI in the standard model of good production without adding anything explicit about AI. Instead, he posits that AI changes some of the model's parameters "behind the scenes". This process has two steps. First, suppose that AI raises the substitution parameter between labor and capital (or certain kinds of capital, such as computers) so that it is permanently bounded above 0. In this case, capital accumulation is sufficient for exponential output growth, even without population growth or technological development of any kind. For illustration, consider our CES production function with ρ > 0 and technology represented but held fixed, and supposing the saving rate s is constant. If the capital supply grows more quickly than than the labor supply, Y t will eventually approximately equal AK t , and capital and output will accumulate exponentially at rate sA. More generally, if labor-augmenting technology grows exogenously at some rate g B ≥ 0, the output growth rate following the substitutability change shifts from min(sA, g B ) to max(sA, g B ). The substitutability change thus increases the growth rate as long as sA > g B . Second, suppose that A t grows exogenously at rate g A > 0. It does not matter whether this technology growth is due to AI or to forces that predated (but were less relevant before) the substitutability change. The growth rate of output will then tend to sA t , which, with A t , will itself be growing exponentially. We will thus have a Type I growth singularity. Under both transformative scenarios-the one-time growth rate increase that can occur without capital-augmenting technological development and the growth explosion that occurs with it-capital per worker will grow to infinity. Since ρ > 0, the capital share will now tend to 1 rather than 0. For any fixed value ρ < 1 however, capital and labor are still complements; we still have F LK > 0. Absolute wages will therefore grow rapidly as the effective capital stock grows, as long as ρ is bounded below 1. In fact, with g A > 0, wages will grow superexponentially, though less quickly than output or effective capital. Absolute wages will stagnate or fall only if ρ rapidly rises to 1 (i.e. if rapidly grows to infinity)-or if capital and labor become perfect substitutes, in which case ρ = 1 (i.e. is infinite). The latter case is explored further in §3.3. Nordhaus also discusses an analogous possibility: that AI will transform consumption growth via the "demand side" of the economy, rather than the "supply side". To explore this scenario, instead of dividing the space of goods into two production inputs and one output, let us divide it into one input ("capital" K) and two outputs (which might be called "standard consumption" Y and "computer-produced consumption" Z). Capital grows exogenously at rate g K . Given capital stock K t , the production of the two consumption goods must satisfy Y t + Z t /D t = K t . (10) That is, each unit of capital can produce either 1 unit of standard consumption or D t units of computer-produced consumption per unit time (without being used up). 1/D t is the relative price of Z at t: it is the number of units of Y that must be given up at t per unit of Z. Consumers' utility function U is defined over Y and Z. U has the same features the production function was assumed to have: it is differentiable, increasing, and concave in each argument; it is CRS (recalling the caveat in §2.2); and its inputs are complements. In response to consumer demand, production is allocated between Y and Z to maximize utility. Suppose D t grows exponentially at rate g D . (This might be thought of as Moore's Law: famously, the number of computations that can be purchased with a given amount of capital seems to double approximately every eighteen months.) The relative price of Z then falls exponentially at rate g D . With each proportional fall in this relative price, the relative quantity of Z pro-duced will rise by σg D , where σ denotes the elasticity of substitution between the goods in the consumer utility function on the given margin. As long as σ < 1, therefore, the proportion S of capital allocated to computer-produced consumption falls to 0 over time. As long as σ > 1, S rises to 1.  11  Let C t Y t +Z t denote total consumption. If σ is bounded below 1, then, as t → ∞, approximately all capital is ultimately allocated to producing standard consumption, so g C = g K . If σ is bounded above 1, then, as t → ∞, approximately all of capital is ultimately allocated to producing computer-produced consumption, so g C = g K + g D . The AI-relevant implications are straightforward. If computer-produced consumption is not currently very substitutable for other consumption (σ bounded below 1), but developments in AI render it more substitutable (such that σ is then bounded above 1), then the consumption growth rate could rise from around g K to around g K + g D . This would not be a growth singularity, as we are using the term. But given the speed of Moore's Law, it would be a dramatic shift. The wage and labor share are not defined in this model, since capital is the only factor of production. As should be clear, however, an analogous model with labor would behave similarly, as long as instead of simply positing growth in capital g K , we also posit equal growth g B = g K in labor-augmenting technology. Then the labor share will be constant (by CRS), and consumption-denominated wages will grow at the consumption growth rate. Finally, Nordhaus constructs various tests of the hypothesis that we are headed for a growth increase via the channels discussed above. If we are in fact headed for a supply-driven growth increase, for example, we should expect to find a rising growth rate and a rising capital share. If we are headed for a demand-driven growth increase, we should expect to find a rising share of global income spent on computer-produced goods. A 11 This is not quite correct. From the definition of substitution elasticity, a fall in the relative price of Z by some small amount (say, g D ) will increase demand for Z by a proportional amount (say, σg D ) holding the budget fixed. Here, however, as long as g K > 0, the budget is also increasing. We must therefore impose some further restriction on the utility function if we are to draw any conclusions about what happens to the budget shares over time. For simplicity, we will assume that U is "homothetic", meaning that the budget shares optimally allocated to each good depend only on the goods' relative prices and not on the size of the budget. thorough discussion of his empirical conclusions would be beyond the scope of this survey, but he concludes that on balance the evidence disconfirms these hypotheses. Models in which labor and capital must be combined in more complex ways tend to produce the same broad conclusion. If labor and capital are sufficiently substitutable, then increasing capital productivity can increase the capital share, but it will still increase the absolute wage rate.  Berg et al. (2018)  detail a variety of such models. We will not work through them here.
 Perfect substitution We have seen that the wage-depressing effect of increasing capital productivity is most extreme when capital and labor are highly substitutable, but that for finite elasticities of substitution, absolute wages still tend to grow. Prospects for wages look worse in cases of perfect substitutability. As noted in §2.2, when goods are perfect substitutes toward some end, they are only ever both purchased in positive quantities when their prices are the same. Even if robots were already fully substitutable for human labor, therefore, we would only observe their transformative effects once their rental rate had fallen below what would otherwise have been the wage rate. In other words, perhaps the substitutability does not need to rise; perhaps it is perfect, and all that needs to change is a relative price. To illustrate this dynamic, consider the following simple model, inspired by  Hanson (2001) . Equipment Q, labor L, and land W are employed in a Cobb-Douglas production function, Y = F (Q, L, W ) = Q a L b W 1−a−b . ( 11 ) The output good can be consumed or invested as capital K. Capital can serve either as equipment or as robotics, which functions as labor, whereas the human workforce H is fixed and can only serve as labor. The productivity of capital-the number of units of effective capital generated by one unit of converted output-is denoted A. That is, if p denotes the fraction of capital employed as equipment, output is Y = (pAK) a (H + (1 − p)AK) b W 1−a−b . ( 12 ) A t rises exogenously without bound. For simplicity we will assume that a constant and sufficient fraction s of output is saved as capital. Because the substitution parameter between equipment and labor is not less than (in fact is equal to) 0, the accumulation of effective equipment is enough to sustain output growth. Early in time, when effective capital is scarce, all capital is used as equipment; p = 1. Indeed, at the rate at which capital can be converted from equipment to robotics, it would be valuable instead to use some human labor as equipment, if that were possible. Capital then grows (using discrete-time notation for clarity) such that K t+1 = K t + s(A t K t ) a H b W 1−a−b (13) ⇒ g K,t = (K t+1 − K t )/K t = sA a t K a−1 t H b W 1−a−b . As we can see from the right hand side, capital growth will approach a steady state such that ag A + (a − 1)g K = 0 ⇒ g K = a 1 − a g A . (14) We will thus have output growth of g Y = a(g A + g K ) = g A a/(1 − a). As the equipment stock grows, wages rise. As the productivity of capital rises and effective equipment grows more abundant, however, there comes a time past which it is optimal to split further capital between equipment and robotics. The labor growth rate then jumps to the rate that keeps its marginal productivity equal to that of equipment, and the output growth rate jumps accordingly. 12 In particular, with capital now filling the roles of both equipment and labor, we now have g Y = g K = g A (a + b)/(1 − a − b) , by the same calculation as above.  13  Hanson estimates the growth implications of crossing the robotics cost threshold using a slightly more realistic model with roughly realistic estimates of the parameters involved. The productivity of capital is assumed to double (i.e. the cost of effective capital is assumed to halve) every two years, in a conservative approximation to Moore's Law. Before capital begins to be used as robotics, output in the model grows at a relatively familiar rate of 4.3% per year. After, the growth rate is 45%. In the model above, because the production function is Cobb-Douglas, the labor share-the share of output paid in compensation for human and/or robotic labor-is independent of the factor quantities. As human labor constitutes an ever smaller share of total labor, however, the human labor share falls to zero. Furthermore, even the absolute wage F L falls to zero. To see this, note that in a CRS production function, the marginal productivities of equipment and labor are kept equal (F Q,t = F L,t ) when the quantities of the two factors grow at the same rate. We can thus rearrange our formula regarding competitive CRS factor payments: F L,t Q t + F L,t L t + F W,t W = Y t ⇒ F L,t = Y t − F W,t W Q t + L t . (15) With a constant share of output accruing to land as well, but the quantity of land fixed, land rent per unit of land-i.e. the land rental rate F W -must grow at the same rate as output. Y and F W will thus both grow at g Y , and Q and L will both grow at rate g A + g K = g A + g Y > g Y . The right-hand ratio will then fall to zero. In any CRS production function without labor-augmenting technology, what happens to the marginal productivity of labor, and thus wages, depends on the quantity of effective labor relative to that of the other effective factors of production. This relative quantity need not rise; it could fall, if labor's complements grow productive and plentiful more quickly than its substitutes, or stay fixed if they grow at the same rate. Consider the following model, very similar to the above, but in which technology augments only equipment, not capital used as robotics: Y = F (Q, L, W ) = Q a L b W 1−a−b = (pAK) a (H + (1 − p)K) b W 1−a−b . (16) The growing stock of equipment implies that, as above, wages rise before the substitutability cost threshold is crossed. Furthermore, we will still have g Y = g K , and thus g Y = a(g A + g K ) ⇒ g Y = a 1 − a g A (17) before the threshold is crossed. Finally, the threshold will still eventually be crossed: if all capital were used as equipment indefinitely, the marginal productivity of capital used as equipment AF Q = aY /K would fall below that of labor F L = bY /H. After the threshold is crossed, we will have g Y = a(g A + g K ) + bg K ⇒ g Y = a 1 − a − b g A . ( 18 ) Note that this is still a growth rate increase, though not as large as that in the first model above. To see what happens to wages, however, observe that when p is chosen so that invested output is split optimally between equipment and robotics, it will satisfy AF Q = F L , or aY pK = bY H + (1 − p)K ⇒ p = H + K K a a + b . (19) As K → ∞, we have p → a/(a + b) < 1, and therefore g L = g K = g Y . As above, because the production function is Cobb-Douglas, the labor share is constant. Now, however, the quantity of effective labor grows no more quickly than output, so labor payments per labor quantity-i.e. wages per human worker-merely stagnate.  Korinek and Stiglitz (2019)  offer another illustration of this phenomenon, in the context of a somewhat similar model. As usual we will simplify here to highlight the intuition. Suppose that Y is produced as in the second model of this section (i.e. the model just above), except that the substitution parameter between land and the other two factors is bounded below 0. Though land is in fixed supply, it is at first plentiful enough that its factor share is low. The saving rate is fixed. At first, as capital accumulates, it is split between use as robotic labor and use as equipment, so that the relative quantities of labor and equipment are unchanged. The capital and labor shares are roughly constant, but the absolute wage stagnates, as we have seen. In time, however, land becomes a binding constraint. The share of output received as land rents approaches 1, and the absolute wage falls to 0. As should be clear, the same logic could apply to many more complex models. Embed any production function in a "surrounding" production function with a fixed-supply and low-substitutability resource such as land, and in the long run, even if all of the original production function's resources grow abundant, the resource in fixed supply constrains growth and its owners receive approximately all output.
 Substitability in robotics production Like  Korinek and Stiglitz, Mookherjee and Ray (2017)  develop a model in which capital can replace human labor without technological progress. Unlike Korinek and Stiglitz, they do not simply assume that capital can be used as robotics, but make the robot production function explicit and identify a condition under which human labor replacement can occur. A simplification of their model is as follows. The final good Y is produced using capital K and labor L in a typical two-factor production function F , with a substitution parameter bounded below 0. Labor is supplied by human work H and robotics R, which are perfect substitutes. Robotics is better thought of as the provision of robot services than as robots, because it must be used as it is produced; it cannot be accumulated. If we would like to think of it as a kind of physical capital, we would say that it exhibits full depreciation. Robotics is also produced using capital and labor, using a standard but perhaps different production function f , also with a substitution parameter bounded below 0. Whereas one unit of robotics is defined as that which replaces 1 human worker in the output production function, however, one unit of robotics replaces some D ∈ (0, 1) human workers in the robotics production function. For each input X ∈ {K, H, R}, S X is defined (assuming X > 0) to be the fraction of X that is allocated to the production of robotics rather than the final good. For simplicity, the population of human workers is fixed and there is no technological progress. More formally, output and robotics at t are Y t = F ((1 − S K,t )K t , (1 − S H,t )H + (1 − S R,t )R t ); (20) R t = f (S K,t K t , S H,t H + DS R,t R t ). As usual, a constant fraction of output is saved as capital. Early on, when capital is scarce and human labor relatively plentiful, there may be no reason to produce robotics at all. As capital accumulates and output begins to be constrained by human labor, however, the marginal output productivity of capital falls to zero. It may therefore at some point be worthwhile to allocate some positive fractions S K and S H of available capital and human labor to robotics production. To be precise, it will necessarily start being worthwhile iff f L (k, 0) > 1 given k > 0, i.e. if, given some capital (which is eventually near-worthless in final good production), marginal contributions of labor can create robotics at a ratio of more than 1:1. Let us call this the "robotization condition". Note that it is a relatively weak condition; f L (k, 0) = ∞ given k > 0 if f is CES, for example. If robotics production relies on capital and human labor-i.e. if we set S R = 0-it too will ultimately be constrained by lack of labor: as S K,t K t → ∞, R t → S H,t R, where R lim x→∞ f (x, H). (21) Output in turn is constrained by total labor, despite the possibility of robotics: as (1 − S K,t )K t → ∞, Y t → Ȳ (1 − S H,t )H + R t H + R , (22) where Ȳ lim x→∞ F (x, H + R). Long-run output will be maximized by setting S H = S * H , the value that maximizes (1 − S * H )H + S * H R-i.e. S * H = 1 if the robotization condition is met, because in this case R > H, and S * H = 0 if not. Long-run output will then approach an upper bound of Ȳ ((1−S * H )H +S * H R)/(H + R). The human labor share will approach 1, either because it is the scarce input to output directly or because robotics is the scarce input to output and human labor is the scarce input to robotics. The absolute wage will of course stagnate. In short, robotization can raise the output ceiling, but it cannot on its own produce a sustainably positive growth rate. One might expect that, if we do not fix S R = 0, it will eventually be optimal to use robotics in the production of robotics. In fact, this will only happen if D is large enough that, as the quantity of capital allocated to robotics production grows large, one unit of robotics can produce more than one unit of robotics: that is, if lim k→∞ f L (k, H) > 1/D; or equivalently, because f is CRS, if f L (k, 0) > 1/D for k > 0. The identification of such a condition in their more general setting is Mookherjee and Ray's key insight, and they call the condition the "von Neumann singularity condition", after the work by  Burks and Von Neumann (1966)  on self-replicating automata. It is of course very closely analogous to the robotization condition above, but stronger since we are imposing D < 1. We might take this to be the natural case; robotics production is presumably harder to automate than most other tasks are.  14  Suppose that this condition is met, and that S K K is large enough that f L (S K K, H) > 1/D. Then there is an optimal quantity of robotics to allocate to robotics production, so as to maximize net robotics production. This is the quantity such that use of a marginal unit of robotics on robotics production increases robotics output by exactly one unit. That is, it is optimal to set S R > 0 such that S R satisfies f L (S K K, H + DS * R R) = 1/D. The value of R depends on S R , since more inputs to robotics production will correspond to higher robotic output. Nevertheless we know that a unique S R ∈ (0, 1) satisfying the above equality exists, for a given S K K. To see this, recall that f L (S K K, H + DS R R) > 1/D at S R = 0, by supposition. And we must have lim S R →1 f L (S K K, H + DS R R) < 1/D, or else the quantity of robotics output R and thus also S R R would grow without bound, fixing S K K, as S R → 1; but in this case f L → 0, by the assumption that the substitution parameter in the robotics production function is bounded below 0. By the concavity and continuous differentiability of f , therefore, there is a unique S * R : f L (S K K, H + DS * R R) = 1/D. Under the singularity condition, S K and S R approach constants S * K and S * R , strictly between 0 and 1, as the capital stock grows.  15  Growth proceeds 14 Presumably at least some other tasks are more difficult to automate, however. As Mookherjee and Ray present in the original paper, their central result does not depend on robotics production being more difficult to automate than all other tasks, just on robotics production being sufficiently difficult to automate. 15 Consider a time t at which S R,t > 0, and let m > 1 denote K t /K t for some t > t. From t to t , the capital input to robotics production is multiplied by mS K,t /S K,t . Because f is CRS, to maintain the condition that f L,t = f L,t = 1/D the labor input to robotics production must also be multiplied by mS K,t /S K,t , and robotics production will then also be multiplied by this factor. We thus have (a) H + DS R,t R t = (H + DS R,t R t )mS K,t /S K,t and (b) R t = R t mS K,t /S K,t . Because both inputs to robotics production are multiplied by a common quantity, f K is ECONOMIC GROWTH UNDER TRANSFORMATIVE AI as in an AK model, with the rate of capital accumulation, final good output growth, and robotics output growth all asymptotically constant and proportional to the saving rate.  16  The wage level is constant and lower than it is in the absence of the singularity condition, since the ratio of capital to labor in robotics production is still asymptotically constant but now positive rather than zero. The share of income accruing to human labor falls to zero. As with  Hanson (2001) , moderate tweaks to this model could result in absolute human wages rising or falling, rather than merely stagnating. Also, in the presence of population growth or labor-augmenting technology growth, introducing automation can increase the growth rate of final good output (or final good output per capita) from the rate of effective labor (or labor-augmenting technology) growth to something much higher, given sufficient saving. As we saw in §3.3, when human work must compete with robotics for which it is perfectly substitutable, the standard result is that the human labor share falls to 0 and the wage stagnates or changes (likely falls) as well. Above, however, we saw that modeling robotics production explicitly, rather than stipulating a frictionless conversion of the final output good into robotics, allows for a channel through which human work can remain necessary. A positive human labor share can be maintained, even when robotics can fully constant across periods. It follows that if the capital input to final good production grows proportionally more (less) than the labor input, the marginal productivity of capital in final good production falls (rises), and the marginal contribution of capital to final good production via robot production rises (falls). Thus, to maintain the condition that capital is allocated efficiently, the capital and labor inputs to final good production must be multiplied by a common quantity across periods: (c) m(1 − S K,t )/(1 − S K,t ) = R t (1 − S R,t )/(R t (1 − S R,t )). Substituting (b) into (a) and (c) and solving for S K,t and S R,t , we find that, as m → ∞, •  S K,t → S * K S K,t DR t (1 − S R,t )/(DR t (1 − S R,t ) − (1 − S K,t )H) and • S R,t → S * R S R = Y /K = F ((1-S * K ), (1-S * R )R/K), where R/K likewise satisfies R/K = f (S * K , S * R R/K). Technological progress that allows capital to produce more robotics increases long-run R/K and functionally "increases A", though it will never exceed its upper bound of F (1, ∞). This will be finite, by the assumption that F's substitution parameter is bounded below 0. substitute for human work in the final good production function, when human work cannot be fully substituted for in robotics production.  Korinek (2018)  presents another model in which humans and robots must in some sense compete and in which robots are not simply "capital that can function as labor" but items that must be produced and sustained. He too finds that the human labor share, and in his case also the wage rate, can fall to 0 unless human labor remains necessary for the maintenance of the robot population. But we will not explore his model further here.
 Growth impacts via impacts on saving In some of the models we have considered, saving has been an important determinant of growth. The saving rate, however, has been assumed to be exogenous. This leaves open another channel through which developments in AI could impact growth: by changing the rate of return to saving, more advanced AI could change the rate of saving and thus the growth rate. This scenario can be illustrated most simply using a model from  Korinek and Stiglitz (2019) . Suppose labor and capital are perfectly substitutable. 17 Labor can only be supplied by humans. Activity unfolds in discrete time, and capital depreciates fully every period; it cannot accumulate. (Capital depreciation simplifies the exposition but is not necessary for the central result.) Given saving rate s t , that is, output and capital growth are then Y t = AK t + BL t , K t+1 = s t Y t . ( 23 ) We begin with K = 0. Output per capita is thus B and, without saving, does not grow. If A < 1, there is no incentive to save, and doing so cannot generate growth; foregoing each unit of consumption at t would offer someone only A < 1 additional units of consumption at t + 1, starting from the same baseline of B. For any A > 1, on the other hand, individuals with no or sufficiently low pure time preference will want to save some fraction s t > 0 of their incomes; not to do so would be to miss the opportunity to give up marginal consumption at baseline B in exchange for a larger quantity of marginal consumption also at baseline B. More precisely, it should be clear that positive saving will be optimal, given any pure time discount factor β < 1, as long as A > 1/β. Furthermore, under certain assumptions about the shape of individuals' utility functions, the induced saving rate will be some constant s > 1/A, independent (at least roughly) of the absolute output level. In the long run, as the relative contribution of effective human labor BL t grows negligible, we will have Y t ≈ AK t . And since K t+1 = sY t ≈ sAK t (24) ⇒ (K t+1 − K t )/K t ≈ sA − 1, capital and therefore output will grow at asymptotic rate sA − 1 > 0. In short, an increase in A-induced, perhaps, by AI developments which render robots cost-effective replacements for human labor-can trigger saving and can thus increase the growth rate of output and output per capita. Here, an A-increase raises per capita output growth from zero to a positive number, leaves the wage rate constant at B, and pushes the human labor share to zero; but other impacts on wages, the labor share, and growth are possible. The point is just that, in addition to the ways in which increases in A can sometimes directly impact the growth rate, they can sometimes do so indirectly by impacting s. There is another mechanism through which developments in AI could impact the saving rate. If the saving rate is heterogeneous across the population, then growth will depend on how income is distributed between high-and lowsavers. Developments in AI could thus affect the growth rate by affecting the income distribution. In principle, this effect could have implications for growth in either direction. Here, we will focus on the especially interesting and counterintuitive possibility that AI slows and even reverses growth by transferring wealth from those with low to those with high propensity to consume. This scenario is illustrated most simply by  Sachs and Kotlikoff (2012) , though the same mechanism is explored in more detail by  Sachs et al. (2015) . If some investment goods are sufficiently substitutable for labor, automation raises capital rents but lowers wages. If saving for the future comes disproportionately out of wage income, for whatever reason, then this wage-lowering can cause future output to fall. Consider an overlapping generations (OLG) economy with constant population size. Each person lives for two periods. The young work, investing some of their income; the old live off their investments. More precisely, output is a symmetric Cobb-Douglas function of labor and capital. The output good can be consumed or invested as capital K. Capital can be used either as equipment Q or as robotics, which serves as labor, and it is split between these uses until their marginal products are equal. The human workforce H is fixed and can only serve as labor. Unlike in the Hanson model, however, the productivity of equipment is fixed. A denotes the productivity of robotics only. Formally, if p is the share of capital used as equipment, Y t = F (Q t , L t ) = (p t K t ) 1/2 (H + (1 − p t )A t K t ) 1/2 . ( 25 ) Capital at t is financed by those who were young at t − 1, who put aside half their wage incomes as investment.  18  The old at t consume all their wealth: not only their investment income, F Q,t p t K t + F L,t (1 − p t )A t K t , but even the capital stock K t , which is liquidated after use in production. The economy is in a zero-growth steady state when investment is just replenished each period: that is, when F L,t /2 = K t . Now suppose robotics grows slightly more productive, so that A t+1 > A t . Let G A 1 + g A denote A t+1 /A t . For a single period, total output and the incomes of the old grow. The young see a fall in wages, however, and investment therefore falls as well. This fall in investment outweighs the fact that some of the investment, namely that in robotics, is now more productive. Output therefore falls. The wage rate falls too, due both to the abundance of robotics and to the lack of investment in equipment. More formally: in the new equilibrium, the marginal product of robotics is again equal to that of equipment. Because the "relative cost" of robotics has now been multiplied by 1/G A and because here ρ = 0, the relative quantity of labor must be G A times higher. Letting asterisks denote the new equilibrium outcomes, L * /Q * = G A L t /Q t . So F * L = 1 2 L * Q * −1/2 , F L,t = 1 2 L t Q t −1/2 (26) ⇒ F * L = F L,t G −1/2 A < F L,t . The new rate of return on investment is r * = A t+1 F * L = G A A t F * L , the new wage is F * L , and the saving rate remains 1/2. The consumption of the old 18 Let r t denote the interest rate at t: that is, here, F Q,t+1 , or equally A t+1 F L,t+1 . Suppose that period utility is logarithmic in consumption and that the young choose the saving rate s t to maximize lifetime utility ln((1 − s t )F L,t ) + ln(s t F L,t (1 + r t )). Then the chosen saving rate will always equal 1/2. thus equals half their income while young, times 1 + r * : (1 + G A A t F * L ) F * L 2 = (1 + A t G 1/2 A F L,t ) F L,t 2G 1/2 A (27) = (G −1/2 A + A t F L,t ) F L,t 2 < (1 + A t F L,t ) F L,t 2 . The new equilibrium therefore features lower output in all subsequent periods, and lower consumption for both young and old. If robotics productivity continues to grow at rate g A , the wage rate, and thus the consumption of the young, will continue to fall to 0 at rate G 1/2 A − 1. The consumption of the old (and thus also output) will fall at a falling rate, and will eventually stabilize above 0, as the increasing productivity of invested equipment ever more closely compensates for the falling absolute amount invested.  19  The human labor share thus falls to 0. Again, the direction of these impacts is sensitive to whether the "winners" from advances in AI save more or less than the "losers". As  Berg et al. (2018)  point out, for instance, those who make most of their incomes from wages currently empirically exhibit lower saving rates than those who make most of their incomes from capital rents, so the mechanism identified by Sachs and Kotlikoff should if anything increase output growth. In any event, the key point is just a reiteration of the well-known fact that, in a neoclassical growth model with finitely lived agents and no (or imperfect) intergenerational altruism, the rate of saving is not necessarily optimal. Accordingly, policymakers must always consider not only the impact of a policy or technological development on short-term output, but also its impact on the saving rate. When a given development produces a suboptimal saving rate, it should be counterbalanced by investment subsidies or by transfers from those with high to those with low propensity to consume: in this model, from the old to the young.
 AI in task-based models of good production 4.1 Introducing the task-based framework In §3, we imagined that capital and labor were each employed in a single sector. In the Cobb-Douglas case, we held the exponent a on capital fixed. We then explored the implications of changing ρ, the substitutability of capital and other durable investments for human labor, and of independently changing the growth rates of factor-augmenting technology. In reality, however, capital and labor are of course employed heterogeneously, and this heterogeneity seems likely to shape the economic impacts of developments in AI. Indeed, sectors with high rates of automation have historically experienced stagnating or declining wages  (Acemoglu and Autor, 2012; Acemoglu and Restrepo, 2020) , even as wages on average have increased. Here, therefore, we will explore a model of CES automation from Zeira (1998), which makes room for this sort of heterogeneity. (We will follow the exposition and extension of Zeira's model given by  Aghion et al. (2019) .) As we will see, this model amounts roughly to assuming a fixed substitution parameter ρ and either a changing capital exponent a, in the Cobb-Douglas case, or impacts on factor-augmenting technology which are sensitive to ρ when ρ = 0. Let us begin with the ρ = 0 case. Suppose output is given by a Cobb-Douglas combination of a large number n of factors X i , for i = 1, . . . , n: Y = X a 1 1 • X a 2 2 • • • • • X 1−a 1 −•••−a n−1 n . (28) At such a fine-grained level, these "factors" might better be thought of as intermediate production goods (Zeira, 1998), or even as individual tasks (Acemoglu and Autor, 2011). We will refer to them as tasks. Fraction a of the tasks are automatable, in that they can be performed by capital or labor, and fraction 1−a are not, in that they can only be performed by labor. Given capital and labor stocks K and L, if all automatable tasks are indeed automated (performed exclusively by capital), K/(na) units of capital will be spent on each automatable task and L/(n(1 − a)) units of labor on each non-automated task. With just a little algebra, we have Y = AK a L 1−a , (29) a two-factor Cobb-Douglas production function with an unimportant coefficient A. 20 Now consider a general CES production function with a continuum of production factors Y i from i = 0 to 1, instead of just two: Y = 1 0 Y ρ i di 1/ρ (30) Tasks i ≤ β ∈ (0, 1) are automatable. Let K and L denote the total supplies of capital and labor, and K i and L i the densities of capital and labor allocated to performing some task i (so Y i = K i + L i ). Suppose again that all automated tasks are indeed performed exclusively by capital. Since the tasks are symmetric and the marginal product of each task is diminishing (∂ 2 Y /∂X 2 i < 0), the density of capital applied to each task i ≤ β will be equal (assuming that production proceeds efficiently), as will the density of labor applied to each i > β. And since we must have β 0 K i di = K and 1 β L i di = L, (31) we know that K i = K/β ∀i ≤ β and L i = L/(1 − β) ∀i > β. We can thus write our production function as Y = [β(K/β) ρ + (1 − β)(L/(1 − β)) ρ ] 1/ρ (32) = [β 1−ρ K ρ + (1 − β) 1−ρ L ρ ] 1/ρ = F (AK, BL) = [(AK) ρ + (BL) ρ ] 1/ρ where A = β (1−ρ)/ρ and B = (1 − β) (1−ρ)/ρ . This is simply a two-factor CES production function. We have assumed that all automatable tasks are indeed performed exclusively by capital. This will obtain so long as there is more capital per automatable task than labor per non-automatable task, i.e. as long as K β > L 1 − β . ( 33 ) In this case, even when capital is spread across all automatable tasks, we have F K < F L , so there is no incentive to use labor on a task that capital can perform. Let us call this condition the "automation condition". For any fixed β, if capital accumulates indefinitely and the labor supply stays fixed or grows more slowly, the condition will eventually hold.
 Task automation Let us now explore the implications of task automation in more detail, across the regimes of CES production with ρ = 0 and < 0. The case of ρ > 0 will be covered in §4.3. As we have seen, under Cobb-Douglas production, task automation raises a along the range from 0 to 1. Since a constant saving rate imposes g Y = g K , the growth rate in this case satisfies g Y = a(g A + g K ) + (1 − a)(g B + g L ) ⇒ g Y = a 1 − a g A + g B + g L . (34) The impact of a one-time increase to a, or of increases only up to some bound strictly below 1, is therefore straightforward. The capital share rises with a. If g A > 0, the growth rate increases, ultimately raising the wage rate; otherwise the growth rate is unchanged, and the impact on the wage rate is ambiguous. Given asymptotic complete automation, the model approximates an AK model. If g A = 0, the growth rate rises to s. If g A > 0, the growth rate rises without bound. If 1 − a falls to 0 at a constant exponential rate, wages too rise superexponentially. Automation, as we have defined it, allows capital to perform more tasks. One might therefore imagine that it is equivalent to the development of some sort of capital-augmenting technology.  Aghion et al. (2019)  observe, however, that automation in the above model is actually equivalent to the development of labor -augmenting-and capital-depleting!-technology, as long as ρ < 0 and the automation condition holds. To see this, recall our production function: Y = [(AK) ρ + (BL) ρ ] 1/ρ , (35) where A = β (1−ρ)/ρ and B = (1 − β) (1−ρ)/ρ . As β rises from 0 to 1, therefore, A falls from unboundedly large values to 1, and B in turn rises from 1 without bound. The reason for this result is that, as β rises, capital is spread more thinly across the widened range of automatable tasks, and labor is concentrated more heavily in the narrowed range of non-automatable tasks.  21  Automation therefore allows capital to serve as a better complement to labor. A marginal unit of labor is spread across fewer non-automatable tasks, producing a larger increase to the supply of each; given the abundance of capital, this then produces a larger increase to output. Conversely, under this allocation, labor serves as a worse complement to capital, requiring capital to spread itself over more tasks (and only partially compensating for this effect by supplying the remaining tasks more extensively). As explained in §2.3, when ρ < 0, labor-augmenting technology is the key to sustained output growth. Let us spell that out in this context. Suppose that, by some exogenous process, a constant fraction of the remaining nonautomatable tasks are made automatable each period, so that (1−β t ) → 0 at a constant rate g 1−β < 0. Then B will grow at rate g B = g 1−β (1 − ρ)/ρ > 0. A is asymptotically constant at 1, so g A ≈ 0. If the saving rate s is constant and high enough to maintain the automation condition, we get our familiar "balanced growth path". The capital stock, effective labor supply, and output all grow at asymptotic rate g Y = g B + g L , output per capita grows at rate g B , and the labor share is asymptotically constant and positive.  22  Automation can thus increase the growth rate of output per capita, and have other transformative consequences.  23  In the model above, because the 21 Here we are only considering increases in β up to the point that capital per automatable task no longer exceeds labor per non-automatable task, so that the automation condition is satisfied. 22 The capital share here equals β 1−ρ t (K t /Y t ) ρ . As β → 1, the capital share rises to an upper bound of (K/Y ) ρ , where K/Y is the long-run capital-to-labor ratio, as long as this exists and is finite. It follows from sY t = K t+1 − K t that sY t /K t = g K,t ; and since g K = g Y = g B + g L , we have K/Y = s/(g B + g L ). The labor share will thus fall to 1 − (s/(g B + g L )) ρ . This is nonnegative because sA ≥ g B + g L , by sufficient saving, and A is asymptotically 1; and it is strictly positive as long as we are not in the knife-edge case of sA = g B + g L . 23 One might however take the position that what we have here been calling automation is not a new force on the horizon, promising to augment pre-existing drivers of growth, but a microfoundation for the process of labor-augmenting technological development we have observed for centuries. On this view, advances in AI will continue to push β ever automation rate −g 1−β is the only driver of growth, introducing it increases the growth rate from 0 to g 1−β (1 − ρ)/ρ. In the presence of growth from other sources, automation can increase the growth rate further. Consider for instance what follows if we have B t = D t (1 − β) (1−ρ)/ρ , with β constant but D t growing exogenously at rate g D . Given saving sufficient to maintain the automation condition, output per capita grows at rate g D . The implications of introducing automation at rate −g 1−β then depend on whether saving is still sufficient to maintain the automation condition. If it is, the per-capita growth rate increases to g D + g 1−β (1 − ρ)/ρ, and the labor share falls to an asymptotic positive value, as observed above. For more on a model of task automation with (something close to) direct labor-augmenting technology growth g D > 0, see §4.4. Now suppose again that g D = 0 , but now suppose that saving is not sufficient to maintain the automation condition-as it cannot be if, for instance, all tasks become automatable. In this case some automatable tasks will not be automated. Here, things proceed roughly as in a model of full substitutability. The growth rate is capped at sA, the wage rate equals the capital rental rate and stagnates, and the labor share equals L/(L+K). Assuming sA > g L , the labor share falls at rate g L −g K = g L −sA. Finally, consider the implications of exogenous growth in capitalaugmenting technology A. When not all tasks are automated, increases to A only increase the effective capital stock. If g A > 0 and s > 0, even if the automation condition is not yet met, the growth rate sA will grow until it is met. The capital stock will then grow at the output growth rate, the effective capital stock will grow faster by g A , and the capital share will fall to 0, roughly as explained in §3.1. When all tasks are automated, on the other hand, output Y t asymptotically equals A t K t , and sustained exponential growth in A produces a Type I growth singularity.
 Task creation Let us begin with the  Aghion et al. (2019)  model of automation and introduce a process of task creation. The resulting model will be somewhat akin to that developed by  Hémous and Olsen (2014) . closer to 1, but this process will simply continue the existing trend. As before, output is a CES production of a range of tasks. Each task is performed by labor and/or capital, with tasks above an automation threshold β requiring labor. Now, however, new and initially non-automated tasks can be created. The range of tasks thus runs from i = 0 to N , with tasks i ≤ β automatable, and not only β but also N can be increased. By the same reasoning as in §4.1, if there is enough saving that the automation condition is met, output is Y = [(AK) ρ + (BL) ρ ] 1/ρ ( 36 ) where A = β (1−ρ)/ρ and B = (N − β) (1−ρ)/ρ . If ρ < 0, then increases to β holding N fixed act like labor-augmenting technology, as in  Aghion et al. (2019) . By the same token, however, increases to N holding β fixed act like labor-depleting technology; they require labor to "spread itself too thinly". It will never be productive to create new tasks, and automation and growth will simply proceed as in §4.2. If ρ > 0, on the other hand, it is increases to N , holding β fixed, that function as labor-augmenting technology. In particular, they asymptotically produce g B = g N (1 − ρ)/ρ. As explained in §3.2, growth then proceeds at a rate of max(sA, g B ), where s denotes the saving rate. Increasing the rate of task creation can thus increase the growth rate. More importantly, increases to β, regardless of N , function as advances in capital-augmenting technology. Recall that effective capital accumulation is enough for growth when ρ > 0. By raising the "ceiling" N and allowing for future automation to raise β, task creation can thus have radical effects. To see this, first suppose that N increases exogenously at a constant proportional rate g N , and that N − β is constant. This is essentially the case explored by  Nordhaus (2015)  and summarized in §3.2: given ρ > 0, capital accumulation and capital-augmenting technology growth combine to produce a Type I growth singularity. The labor share will fall to 0, even while wages, like output (though more slowly than output), grow superexponentially. If g A = g B , i.e. if g β = g N so that a constant fraction of tasks is always automated, the outcome is similar; indeed, conditions are even more favorable to labor. Upon endogenizing the task automation and creation processes, the g β = g N condition turns out to hold under relatively natural-seeming circumstances. For a few more words on this, see the end of the following section. Finally, as discussed at the end of §3.3, suppose we embed production function (36) in a "surrounding" production function. That is, suppose that the good denoted Y , which we had been referring to as the final output good, must instead be combined with a fixed-supply resource (such as land W ) in order to produce the final output Z. What follows depends centrally, as we have seen, on the substitution parameter ρ between Y and W . If ρ > 0, essentially nothing changes; the land share is asymptotically 0, growth in Z approximately equals growth in Y , and so forth. If ρ < 0, on the other hand, output approaches an upper bound. Then the relative quantity of Y rises without bound, by capital accumulation or asymptotic task automation; the land share rises to 1; and the wage rate falls to 0. This is more similar to the case explored by Hémous and Olsen, though they take the fixed-supply resource to be skilled labor, whereas we are ignoring skill differences throughout this review. In any event, we will not explore this further here.  Acemoglu and Restrepo (2018b, 2019a ) develop a similar model of task creation, but combine it with a process of task replacement. Here is a simplification.
 Task replacement Instead of ranging from 0 to N , task indices i now range from N − 1 to N . Capital is equally productive at all tasks it can perform, but labor productivity at task i is B i = D i (1 − β) (1−ρ)/ρ , where D i = exp(g D i), for some g D > 0. In an exogenous growth setting, both β ∈ (N − 1, N ) and N grow over time at a constant exogenous absolute rate-let us say, without loss of generality, at one unit per unit time. The fraction of tasks not automatable is thus constant at N − β, but the productivity of human labor at the non-automatable tasks grows at exponential rate g B = g D . With enough saving, all automatable tasks are automated. Output and capital grow at rate g D + g L , in line with the effective labor supply, and wages grow at rate g D . The labor share is constant, as in any CES model with labor-augmenting technology growth and sufficient saving. Moving from asymptotic automation in the original setting of §4.2 to this model of task automation, creation, and replacement thus increases the output growth rate iff g D > g 1−β (1 − ρ)/ρ. As usual, if we imagine starting from a world without task creation and replacement, introducing this pro-cess raises the growth rate from 0 to a positive number; and if we imagine starting from a world with some other source of exogenous growth in labor productivity, introducing this process raises the growth rate, given enough saving to maintain the automation condition. This model is nearly equivalent to a task-based model in which the task-range is fixed at the unit interval, β is fixed, and B grows exogenously at rate g D . Its framing is motivated by the empirical observation that we have long seen the automation of existing tasks go hand-in-hand with the creation of new, high-productivity tasks that, at least temporarily, only humans can perform  (Goldin and Katz, 2009; Acemoglu and Autor, 2012) , and continue to see this pattern in the present  (Autor, 2015; Acemoglu and Restrepo, 2019b) . The result is a near-complete turnover of job types over time, rather than a mere encroachment of automation onto human territory. In this sense, this promises to be a more realistic model of automation than one without task replacement. As we have just seen, more realistic models of this type are also compatible with balanced growth. This balanced growth result is, however, sensitive to the assumption that advances in automation technology (increases in β) and task creation (increases in N ) proceed at the same rate. If task creation oustrips automatability, the labor share rises, and as β − N + 1 → 0 asymptotically, the labor share rises to 1. In this case, we approach a state in which labor performs all tasks. Capital is relegated to an ever-shrinking band of the lowest-labor-productivity tasks. Since output and, given a constant saving rate, capital grow at the same rate as effective labor, while capital is used ever less efficiently, capital rents fall and the capital share falls to 0. In equilibrium, output grows at rate g D + g L , and wages grow at the labor productivity growth rate g D , as before. Now suppose that automatability outstrips task creation, and in particular that it does so at a constant rate g N −β < 0. What follows depends on the extent to which capital accumulation keeps up with this process. If s ≤ g D + g L , the automation condition will not be met in the long run, and capital and effective labor will be perfect substitutes on the margin. Output will thus equal K t + D t L t , and the stock of capital grows at the same rate as that of effective labor when s (K t + D t L t )/K t = g D + g L , i.e. when K t /(K t + D t L t ) = s/(g D + g L ). In the long run we thus have g K = g Y = g D + g L , the labor share and the fraction of tasks not automated approach 1 − s/(g D + g L ) (ranging from 1 at s = 0 to 0 at s = g D + g L ), and wages grow at rate g D . Now suppose that s ∈ (g D + g L , g D + g N −β (1 − ρ)/ρ + g L ], so that capital accumulation outpaces labor and labor productivity growth in isolation but not in combination with automation. Now the fraction of tasks automated grows over time: if it stayed constant, capital per automated task would ultimately exceed effective labor per non-automated task, since s > g D + g L , and it would be profitable to reallocate some capital to automatable but non-automated tasks. But the fraction of tasks automated does not catch up with the automatability frontier: if all automatable tasks were automated, effective labor per non-automated task would ultimately exceed capital per automated task, since s < g D +g N −β (1−ρ)/ρ+g L , and it would be profitable to reallocate some labor to currently automated tasks. Thus the automation condition is not met in this scenario either, and capital and effective labor are still perfect substitutes on the margin. Since the stock of capital grows more quickly than that of effective labor, in the long run output grows at rate s, wages grow at rate g D , and the labor share again falls to 0. Finally, if s > g D + g N −β (1 − ρ)/ρ + g L , the automation condition is met. Growth proceeds at g D + g N −β (1 − ρ)/ρ + g L and the labor share approaches a positive constant, as in the model of §4.2 with direct labor-augmenting technology growth g D > 0. Empirically the saving rate is currently far higher than the growth rate of effective labor, so unless automation accelerates dramatically, this is the most relevant case for consideration. Now let us briefly and informally consider the implications of endogenizing the automation technology and task creation processes. Suppose that, in addition to the labor force, there is a pool of researchers who allocate their efforts between increasing β and increasing N . Upon doing either, they earn a patent right to some of the gains that result. This scenario, as detailed by  Acemoglu and Restrepo (2018b) , produces intuitive equilibrating pressures, suggesting that we might expect to observe automation technology and task creation proceeding at the same rate, without having to assume this ad hoc. Excessive development of automation technology results in tasks that are automatable but not automated, because of an insufficient ratio of capital to effective labor. This eliminates the immediate value of further automation technology. Excessive task creation, on the other hand, increases the value of automation technology, by inefficiently relegating capital to a narrower range of tasks. The full range of possibilities here, however, is essentially the same as in the exogenous growth case. The proportion of tasks automated can fall to 0, if the saving rate is sufficiently low; there can be asymptotically complete automation if the saving rate is sufficiently high; and there is partial automation in intermediate cases. The primary novelty of the endogenous research case is that here, which case obtains can depend on the researchers' productivity at developing automation technology relative to their productivity at task creation. In particular, increases in productivity at developing automation technology, relative to productivity at task creation, can increase the equilibrium automation rate and decrease the equilibrium labor share. Also, the growth rate here is not exogenous but depends on the level of productivity at both researcher tasks, as well as on the size of the researcher population.
 AI in technology production Throughout the discussion so far (except for the brief note at the end of the previous section), technological development has been exogenous, when it has appeared at all. Even in this circumstance, developments in AI have proved capable of delivering transformative economic consequences. When technological development is endogenous, and in particular when more advanced AI can allow it to proceed more quickly, the resulting process of "recursive self-improvement" can generate even more transformative consequences.
 Learning by doing This recursive effect can be seen most simply in a model with Cobb-Douglas production. Let us interpret the production as task-based, with fraction a of tasks automated, as in §4.1. The model below is inspired by the exploration of learning by doing in  Hanson (2001) . The labor supply grows at exogenous rate g L > 0, but capital productivity growth proceeds endogenously. We will use a modification of the endogenous growth model presented in §2.4. In that model, technology growth is a function of existing technology and "researcher effort". Here, instead, we will not introduce research and will say simply that capital productivity grows as a function of the existing technology and output. That is, Y t = (A t K t ) a L 1−a t , (37) where Ȧt = A φ t Y λ t ⇒ g A,t = A φ−1 t Y λ t (38) for some φ < 1 and λ > 0. One might interpret this as a model in which the production process itself contributes to the generation of productivityincreasing ideas. Given a constant saving rate, g K = g Y . So, from our production function, g Y = a(g A + g Y ) + (1 − a)g L g Y = a 1 − a g A + g L . (39) From our formula for g A,t , the steady state of g A (if it exists) will be that which satisfies (φ − 1)g A + λg Y = 0. Substituting for g Y in this expression and solving for g A , we have g A = λ(1 − a) (1 − a)(1 − φ) − λa g L . (40) This exponential growth path will exist as long as the denominator is positive: that is, as long as a < 1 − φ 1 − φ + λ . (41) In this case, output growth will be given by 24 g Y = (1 − a)(1 − φ) (1 − a)(1 − φ) − λa g L . (42) Otherwise, the recursive process by which proportional increases to A t generate proportional increases to Y t , which in turn generate proportional increases to A t+1 (using discrete-time notation for clarity), results in the proportional increases at t + 1 being larger than those at t. The growth rates of A and Y thus increase without bound. The transformative potential of automation is now straightforward. Increases in a increase the long-run growth rate without bound, as a approaches (1 − φ)/(1 − φ + λ). Past this threshold, increases in a trigger a growth singularity. The singularity type can be determined by substituting (37) into expression (38) for g A,t : g A,t = A φ−1 t Y λ t = A φ−1 t A aλ t K aλ t L (1−a)λ t . (43) Since 25 K t ∝ Y t = (A t K t ) a L 1−a t , rearranging gives us K t ∝ A a/(1−a) t L t . Sub- stituting for K t into (43), we have g A,t ∝ A φ−1+λa/(1−a) t L λ t . (44) When a > (1 − φ)/(1 − φ + λ), the exponent on A t is positive, producing a Type II singularity. When a = (1 − φ)/(1 − φ + λ), we have g A,t ∝ L λ t ; the technology growth rate itself grows asymptotically at rate λg L , producing a Type I singularity.
 Automated research Cockburn et al. (  2019 ) taxonomize AI systems as belonging to three broad categories: symbolic reasoning, robotics, and deep learning. Symbolic reasoning systems, they argue, have proven to have few applications. Robotics-by which they broadly mean capital that can substitute for human labor in various ways, instead of complementing it-has of course had many applications. It has also been the subject of a substantial majority of the theoretical literature on the economics of AI, including all that discussed in this survey so far. They propose however that the most transformative possibilities come from deep learning systems, by which they mean systems that can learn as human researchers can: artificial systems that can participate directly in the process of technological development. Citing  Griliches's (1957)  discussion of the implications of "inventing a method of invention", they argue that deep learning systems (in their use of the term) will have qualitatively more radical consequences than mere robotics. As we will see, this appears to be correct. Following  Aghion et al. (2019) , let us focus directly on the implications of technology production by using an even simpler production function than usual: Y t = A t (1 − S)L t . ( 45 ) 25 The "∝" symbol means "is asymptotically proportional to". Labor L is the only factor of production. S is the constant proportion of people who work in research as opposed to final good production. Output technology A, however, is developed using a CES function of both labor and capital K. Building on the standard technology production function from §2.4 (where Ȧt = A φ t (S t L t ) λ , but here assuming in effect that λ = 1), we have Ȧt = A φ t [(C t K t ) ρ + (D t SL t ) ρ ] 1/ρ , ρ = 0; (46) Ȧt = A φ t (C t K t ) a (D t SL t ) 1−a , ρ = 0, for some substitution parameter ρ and, in the Cobb-Douglas case, capital exponent a. C t and D t denote capital-and labor-augmenting technology levels respectively. As usual, we will assume a constant saving rate s. Suppose ρ < 0. Recall that in this case sustained growth in effective capital and in effective research labor are both necessary to sustain growth in output technology, and growth will be driven by whichever factor grows more slowly. Observe that when growth is constrained by effective capital accumulation, we have g A ∝ A φ−1 t C t K t , and that when it is constrained by effective research labor growth, we have g A ∝ A φ−1 t D t L t . Regarding φ: • Recall from the reasoning of §2.4 that, if φ < 1, we have g A = (g C + g K )/(1 − φ) on the capital-constrained path and g A = (g D + g L )/(1 − φ) on the labor-constrained path. In the former case, since g K = g Y = g A + g L , we can substitute for g K and rearrange to get g A = −(g C + g L )/φ. A capital-constrained path with this growth rate exists when φ < 0. Thus, if φ < 0, g A = min(−(g C + g L )/φ, (g D + g L )/(1 − φ)). A onetime increase to C t , D t , L t , or S, as long as S remains below 1, does not affect the output technology growth rate. A permanent increase to g C , g D , or g L , on the other hand, does increase the growth rate of output technology and thereby output per capita. • If φ = 0, the growth path constrained by effective labor growth remains defined as in the case of φ < 0. The above expression for the growth path constrained by effective capital accumulation, however, is now undefined. To find the capital-driven growth path with C and L fixed, observe that we here have g K = g A and sA t (1 − S)L = K t+1 − K t ⇒ g K,t = sA t (1 − S)L/K t , (47) g A,t ≈ CK t /A t = C(K t−1 /A t + s(1 − S)LA t−1 /A t ). ( 48 ) In steady state, K t−1 = K t /(1 + g A ), so K t−1 /A t = (K t /A t )/(1 + g A ). Also, g A = g K = s(1 − S)LA t /K t , so K t /A t = s(1 − S)L/g A ; so K t−1 /A t = s(1 − S)L/(g A (1 + g A )). Finally, A t−1 /A t = 1/(1 + g A ). Substituting these terms for K t−1 /A t and A t−1 /A t into (48), we find g A = sC(1 − S)L. ( 49 ) Thus if g C = g L = 0, technology grows at the minimum of this rate and the growth rate to which it is driven by labor-augmenting technology growth, which given φ = 0 is simply g D . Formally, g A = min( sC(1 − S)L), g D ). If g C > 0 or g L > 0, output technology growth cannot be constrained by capital accumulation, and g A = g D + g L . • If φ ∈ (0, 1), output technology growth cannot be constrained by capital accumulation; such a scenario would imply g A ∝ A φ−1 t C t K t ∝ A φ t C t L t , contradictorily producing superexponential growth in output technology, output, and capital. We have g A = (g D + g L )/(1 − φ). • If φ = 1, suppose labor remains fixed at L and labor-augmenting technology remains fixed at D, while effective capital accumulates. In the long run we then have g A = DSL. A one-time increase to D, S, or L increases the growth rate of output technology and thereby output (again, as long as S remains below 1). If we begin from a state in which g D = g L = 0 and introduce positive labor or labor-augmenting technology growth, the result is a Type I growth singularity. • If φ > 1, we have a Type II growth singularity regardless of the other parameters, as explained in §2.4. Recall from §4.2 that, given ρ < 0, increases to D can be interpreted as increases to the fraction of research tasks that have been automated. Suppose ρ = 0. Technology growth is then g A,t = A φ−1 t (C t K t ) a (D t SL t ) 1−a (50) = A φ−1 0 (C 0 K 0 ) a (D 0 SL 0 ) 1−a e mt , where m (g A (φ − 1) + a(g C + g K ) + (1 − a)(g D + g L )). ( 51 ) From our assumption of a constant saving rate, g K = g Y = g A + g L . So m = g A (φ − 1 + a) + ag C + (1 − a)g D + g L . (52) Regarding φ: • If φ < 1 − a, we will have, in equilibrium, the constant output technology growth rate that sets m = 0. This is g A = (ag C + (1 − a)g D + g L )/(1 − φ − a) . One-time increases to C, D, S, or L do not change the growth rate, but increases to g C , g D , or g L do. • If φ = 1 − a, we have steady growth only if g C = g D = g L = 0, since we are assuming that these growth terms are all nonnegative. Fixing A 0 = K 0 = 1, the output technology growth rate is C a 0 (D 0 SL 0 ) 1−a . A one-time increase to C, D, S, or L increases the growth rate. If g C > 0, g D > 0, or g L > 0, we have a Type I growth singularity. • If φ > 1 − a, we have a Type II growth singularity regardless of the other parameters. Recall from §4.1 that increases to a can be interpreted as increases to the fraction of research tasks that have been automated. They can thus induce Type I and Type II growth singularities, if φ ∈ (0, 1). Suppose ρ > 0. Recall that in this case sustained growth in effective capital or in effective research labor suffice to sustain growth in output technology, and growth will be driven by whichever factor grows more quickly. Now, when growth is driven by effective capital accumulation, we have g A ∝ A φ−1 t C t K t , and when it is driven by effective research labor growth, we have g A ∝ A φ−1 t D t L t . Regarding φ: • If φ < 0, the capital-and labor-driven technology growth rates equal (g C +g K )/(1−φ) and (g D +g L )/(1−φ), respectively. In the former case, since g K = g Y = g A + g L , we can substitute for g K and rearrange to get g A = (g C + g L )/φ. Thus g A = max((gC + gL)/φ, (gD + gL)/(1 − φ)). Growth rate increases require increases to g C , g D , or g L . • If φ = 0, the growth path driven by effective labor growth remains defined as in the case of φ < 0. As found in the discussion of the ρ < 0 regime above, the growth rate on the growth path driven by effective capital growth when φ = 0 exhibits growth rate g A = sC(1 − S)L. Thus if g C = g L = 0, technology grows at the maximum of this rate and the growth rate to which it is driven by labor-augmenting technology growth, which given φ = 0 is simply g D . Formally, g A = max( sC(1 − S)L, g D ). • If φ > 0, we have a Type II growth singularity regardless of the other parameters. An intuition for these results is as follows. As explained briefly in §2.4, the growth of some variable X exhibits a Type II singularity if its growth rate takes the form g X ∝ X ψ for some ψ > 0. When ρ < 0, capital accumulation cannot accelerate technological development, which is bottlenecked by its slower-growing factor, namely effective labor. Output technology growth is then g A ∝ A φ−1 , so the Type II singularity requires φ > 1. When ρ > 0, on the other hand, capital accumulation at rate g Y = g A effectively multiplies g A by a factor of A, so g A ∝ A φ . The Type II singularity therefore requires only φ > 0. Note that our analysis of the ρ > 0 case also covers the case in which the development of output technology is fully automated. Simply use ρ = 1. It also covers a common interpretation of the possibility of "recursive self-improvement". If A represents cognitive ability, and enhanced intelligence (human or artificial) speeds the rate at which intelligence can be improved such that g A,t = A φ−1 t K t , then explosive growth obtains iff φ > 0 (since, again, g K = g A ). If we remove capital accumulation entirely and say that the development of intelligence depends only on the intelligence level, such that g A,t = A φ−1 t , then explosive growth obtains iff φ > 1. Now let us briefly examine the implications of automating good production in combination with automating research. Suppose we replace the laboronly good production function with a CES production function in capital and labor, with substitution parameter ρ Y . Fraction S L of labor and S K of capital is used in technology production. We will ignore factor-augmenting technology in the good production function; output technology A can be thought of as augmenting both, by CRS. Finally, we will assume sufficient saving. If ρ Y < 0, little changes. Output is still bottlenecked by the scarce factor, namely labor (assuming that we do not irrationally have S K,t → 1, in which case output is only the more constrained). Output therefore asymptotically resembles A t (1 − S L )L t , as before. If ρ Y > 0, output asymptotically resembles A t (1−S K )K t . The result is at least a Type I singularity, as long as g D + g L > 0, regardless of the values of ρ A and φ in output technology development. This is because output growth is here given by g Y,t = sA t (where s denotes the saving rate), and A t grows at least exponentially, driven by growth in effective researcher time. If ρ Y = 0, so that output is Cobb-Douglas in capital and labor, the occurrence of a growth singularity is sensitive to the capital exponent a Y . If we interpret increases in a Y as the automation of tasks in output production, as in §4.1, then automation can trigger a singularity in this way. We will not explore this further here.
 AI assistance in research In §4, we discussed several papers which use a microfoundation of the output production function as a basis for exploring the implications of a certain kind of automation. Somewhat analogously, Agrawal et al. (  2019 ) use  Weitzman's (1998)  microfoundation of the process of technological development as a basis for exploring the implications of a certain way in which advances in AI might assist in technological development. Let Y = A(1−S)L, as before, and hold S fixed but posit labor growth g L > 0. Given A existing "technological ideas", a researcher has access to only A φ , for some φ ∈ (0, 1), perhaps due to some sort of cognitive limitation. 26 New ideas are made from combinations of existing ideas. Given access to A φ ideas, a researcher therefore faces 2 A φ idea-combinations. Of these, not all can generate new technological ideas, perhaps due to some other sort of cognitive limitation. Instead, each researcher's idea-generation function is "isoelastic" in ideas available: Ȧ = α (2 A φ ) θ − 1 θ , θ > 0; Ȧ = α ln(2 A φ ) = α ln(2)A φ , θ = 0, (53) for some α > 0 and some θ ∈ [0, 1]. 27 Most transformatively, if AI tools help researchers search through the ever-growing "haystacks" of possible idea-combinations for valuable "needles", they could permanently increase θ.  Agrawal et al. (2019)  argue that this is precisely the sort of activity to which AI systems are best suited: they are already being profitably used to identify promising combinations of chemicals in pharmaceutical development, for example. (See  Agrawal et al. (2018)  for a more thorough defense of this argument.) If this turns out to hold across the board, the result is stark: as shown above, a permanent increase to θ produces a Type II growth singularity.  Agrawal et al. (2019)  also explore the potential impacts of AI assistance in research teams, rather than in assisting individual researchers.  Seeber et al. (2020)  do the same, in the context of a much more applied and less formal inquiry. Neither analysis appears to reveal channels for transformative growth effects substantively different from those presented above.
 Growth impacts via impacts on technology investment Throughout §5, we have taken technology production to be endogenous, in the sense that it has required explicit inputs of capital and labor. Nevertheless we have taken the level of investment in technological developmentthe fraction S of labor, and (in §5.2) the amount of capital, allocated to research-to be exogenous. A final way in which AI could have a transformative impact, therefore, is by changing the levels of investment in, and effort allocated to, technological development. As we have seen, at least in some circumstances, these changes can change the growth rate, or can determine the existence or type of a growth singularity. This pathway to transformative impact is somewhat analogous to the possibility, explored in §3.4, that developments in AI could affect the growth rate by affecting the saving rate, even in an economy without endogenous technological development. As in that case, this change could in principle be positive or negative. (Indeed, one way AI could impact the extent to which resources are devoted to technological development is by affecting the saving rate, as long as capital is modeled as an input to technology production.) Also as in that case, to the extent that the literature has explored this pathway to transformative impact, it has focused on the counterintuitive possibility that AI slows (though not, here, reverses) growth. This could take place by accelerating the "Schumpeterian" process of "creative destruction". On this analysis, the incentive to innovate comes from a temporary monopoly that the innovators enjoy, either by patents or by trade secrets, during which they can extract rents from those who would benefit by using the new technology in production. AI, however, could make it easier for competitors to copy innovations. Relatedly, AI could also ease the rapid development of technologies only negligibly more productivityenhancing than those they replace. Because these technologies would entirely eliminate the markets for those they replace, their rapid development would curtail the incentive for innovation. In the absence of this incentive, technology growth can slow to a halt. This cannot cause output per capita to fall, at least in most models, but it can cause output per capita to stagnate. This dynamic is explored more formally by  Aghion et al. (2019)  in the context of the model of automated research laid out in §5.2, and by  Acemoglu and Restrepo (2018b)  in the context of the model of automation and task replacement laid out in §4.4. We will not work through it here. As with the Sachs and Kotlikoff (2012) observation that AI can do damage by lowering the saving rate, the insight here is not primarily an insight about artificial intelligence. It is primarily a special case of the well-known fact, mentioned briefly in §2.4, that though free and competitive markets can generally be expected to appropriately compensate production factors for a final good in a static setting, the same cannot be said about the inputs to technological development. Policymakers interested in growth must always consider the impact of structural economic changes on the incentives for technological innovation, therefore, and must adjust their funding or subsidization of basic research in light of such changes as they unfold.
 Overview of the possibilities The table below summarizes the transformative scenarios we have considered. They have been rearranged slightly for clarity, and some near-redundant possibilities have been removed, but they primarily follow the order in which they are presented in §3-5. Relevant literature is cited below each scenario. Note that, in keeping with the presentation so far, the cited literature introduces the models that allow for the scenarios in question, but does not always discuss the transformative scenarios on which we have focused. We have not considered all possible AI scenarios, as this table makes clear. Nevertheless we have hopefully sampled the possibilities thoroughly enough that the reader is now comfortable filling some of the gaps.
 Scenario 28 Growth PS in production & capital-augmenting tech growth I → 0 L 28 "PS", "HS", "MS", and "LS" stand for perfect, high, moderate, and low substitutability, and refer to substitution parameters ρ = 1, > 0, = 0, and < 0 respectively. Unless otherwise noted, the "HS" case allows for perfect substitutability. In the scenarios with endogenous growth, "negative", "low", "intermediate", and "high [research] feedback" refer to research/learning feedback exponents φ < 0, < 1, = 1, and > 1. 29 + andrefer to cases in which AI shifts the output path up or down without changing the growth rate, e.g. by increasing or decreasing the plateau level in a circumstance where output plateaus regardless of AI. --, ++, I, and II refer to cases in which AI allows for decreases to the long-run growth rate, increases to the long-run growth rate, Type I growth singularities, and Type II growth singularities. = refers to cases in which AI does not change the long-run output level or growth rate. 30 C means that AI pushes the human labor share to some positive constant, not necessarily lower or higher than the value it would take in the absence of AI. 31 L means that human wages are driven to some low but constant rate (typically the rental rate of effective capital). C means that they are pushed to some positive constant, not necessarily lower or higher than they would be in the absence of AI. All other symbols are defined as in the Growth column. The human labor share and wage are technically undefined in the models of endogenous technology production, since, as noted in §2.4, we cannot straightforwardly assume that the factors of technology production will tend to be paid their marginal products (or anything else in particular). As typically presented, however, human labor is the lone factor of final good production in these models, and the technology being produced is laboraugmenting. Barring any radical market interventions, therefore, the wage should therefore grow in line with technology and so with output. That is, it should exhibit growth rate decreases, increases, Type I singularities or Type II singularities as listed above.
 Conclusion The set of models discussed here cover a wide range of AI's possible longrun macroeconomic impacts. It can hopefully serve as a bridge between the tools of economics-whose use is typically restricted to shorter-term and smaller-scale possibilities (but need not be)-and the longer-term and largerscale questions posed by futurists, who typically do not draw on the tools of economics (but could, we believe, sometimes learn from doing so). Nevertheless, of course, many topics relevant to the economics of AI, and even of transformative AI, could not be covered here. Wage distribution is a-perhaps even the-central concern of the literature on the economics of AI, including much of the literature cited here. It is likewise a central concern of the less long-term-focused reviews of the economics of AI cited in §1. Indeed, wages and skill levels are of course empirically highly unequal. And this inequality has indeed increased in the recent past, a development many attribute to the rise of automation. Nevertheless, we have consistently referred to all wages and human abilities as homogeneous. The choice to focus on average wages and on the overall labor share is motivated in part by the supposition that, if we are truly considering the long run, the likeliest transformative possibility is that AI will outsmart us all. In this event, human talents will not save us; if we retain positive wages or a positive labor share, we will do so only because AI is put to use making us more productive, or because some tasks, like those performed by clergy or by hospice nurses, remain resistant to automation. Otherwise, as  Freeman (2015)  colorfully puts it, "[w]ithout ownership stakes, workers will become serfs working on behalf of robots' overlords". To be clear, however, this view may by all means be incorrect. We cannot rule out AI-induced scenarios in which, even in the long run, income is concentrated not entirely in the hands of the robot owners but also at least to some extent in the hands of the most skilled human individuals. The subject most conspicuously present, despite receiving relatively little direct attention in economics literature, is the possibility of lasting changes to the growth regime: growth rate shifts and Type I and II singularities. As this review perhaps illustrates, the lack of attention currently given to these possibilities is not a necessary consequence of all plausible economic modeling. Rather, it is the result of a widespread norm of focusing only on model scenarios in which long-run growth is constant. Even  Aghion et al. (2019) , who take the singularitarian growth potential of AI most seriously, focus less on scenarios in which labor and capital are highly substitutable in technology production on the grounds that, as long as φ > 0, "in this case researchers are not a necessary input and so standard capital accumulation is enough to generate explosive growth. This is one reason why the case of ρ < 0 . . . is the natural case to consider." Expressed motivations along these lines appear throughout the literature. Even outside discussions of AI, it is rare for economic growth literature to consider substantial and permanent growth rate increases, let alone growth explosions. This is presumably because such models would violate the "Kaldor fact" of constant exponential per-capita growth at 2-3% per year, which has roughly held in the industrialized world since roughly the Industrial Revolution. But on a longer timeframe, models of increasing growth would not be ahistorical; the growth rate was far lower before the Industrial Revolution, and before the Agricultural Revolution it was lower still. Empirical forecasts on the basis of these longer-run facts commonly predict radical future increases to growth, including substantial one-time rate increases and Type I and II singularities (see e.g.  Hanson (2000)  and  Roodman (2020) ). Some of the models underlying such forecasts lack economic foundations (e.g. that in  Hanson (2000) ), making it difficult to assess whether the forces driving historical superexponential growth are still at work today. Notably, however, some have economic foundations (e.g. that in  Kremer (1993) ) that imply that continued increases to the pool of effective workers or researchers, as advances in robotics and AI would permit, would continue the superexponential process. For a more thorough survey of the singularitarian literature, at least as of 2013, see  Sandberg (2013) . In short, if there were a sufficiently compelling theoretical reason to believe that the observed long-run trend of increasing growth will soon permanently halt, then we should of course dismiss transformative growth scenarios. But as many models of the economics of AI confirm, there is no shortage of mechanisms, once we allow ourselves to look for them, through which advances in automation could have transformative growth consequences. Futurists and economists interested in the long term might therefore do well to collaborate more on this point of common interest. Finally, the subject most conspicuously absent here that features most heavily in futurist discussion about AI is the most transformative macroeconomic possibility of all: the risk of an AI-induced existential catastrophe (see e.g.  Bostrom (2017) ). Unlike the possibility of transformative growth effects, AI risk appears to be absent from the economics literature not primarily "by choice" but because there is no particularly obvious mechanism through which accelerating automation or capital productivity, within existing models of production or growth, can pose a danger. It is possible, of course, to write down economic models in which production and/or technological development pose catastrophic risks in the abstract, as e.g.  Jones (2016)  and  Aschenbrenner (2020)  have done. As outlined above, however, the only growth-slowing AI possibilities economists have considered to date are those mediated by impacts on saving  (Sachs and Kotlikoff, 2012)  and on innovation incentives  (Aghion et al., 2019; Acemoglu and Restrepo, 2019) . These scenarios are very far from those that motivate most concern about AI risk. The latter typically feature superintelligent agents, with goals not fully aligned with ours, who take control of the world. The tools of economics can shed at least some light on these concerns as well. Most simply, to the extent that AI development poses such a risk, AI safety is a global and intergenerational public good. Through that lens, much of the analysis of public goods, and in particular many of the tools developed by environmental economists for the pricing and provision of climate risk mitigation, could apply to AI safety. More subtly, to the extent that AI risk arises from AIs' ability to control resources independently of human input, models in which the human labor share remains positive and significant should give us comfort. If human work remains a bottleneck to growth-say, if AI accelerates growth only by giving human workers instructions which they must physically perform-then humanity can in principle impoverish any robot overlords by going on strike. More worrying are models in which a unit of capital can grow, do research into capital-augmenting technology, and recursively self-improve all without human input. A thorough analysis of the links between the economics of AI and the issue of AI safety remains an important topic for further exploration. ,t + H/(DR t ). S K and S R are thus asymptotically constant and nonzero. Furthermore, sinceR t = f K,t S K,t K t + (H + S R,t DR t )/D, we must have H + S R,t DR t < DR t . It follows that S * and S  *  R are strictly less than 1. K 16  The model will approximate an AK model with A
 Aghion et al. (2019) ,Acemoglu and Restrepo (2018b)     §3.2 §4.4 Acemoglu and Restrepo (2018b) AI-diminished innovation incentives −− PS in production, capital-augmenting HS in production & task automation  §5.4 tech growth, & MS land constraint and creation ++ I → → 0 → 0 I §3.3 Hanson (2001)  §4.3 Hémous and Olsen (2014) PS in production, equipment-augmen-Learning by doing, w/intermed. ting tech gr., & MS land constraint feedback and/or automation ++ ++ → L §3.3  §5.1 Hanson (2001) PS in production & LS land constraint Learning by doing, with suffic. (regardless of tech) feedback and/or automation = II → → 0 §3.3 Korinek and Stiglitz (2019)  §5.1 Hanson (2001) HS in final good production, HS in LS in tech production, low research robotics production feedback, & asymptotic research task ++ ++ → L §3.4 Mookherjee and Ray (2017), Korinek and Stiglitz (2019) automation; or HS in tech production, negative research feedback, & research HS in final good production, LS in robotics production capital productivity growth  §5.2 Aghion et al. (2019) + C C §3.4 Mookherjee and Ray (2017), Korinek (2018) LS in tech production, intermed. PS in production & one-off capital-augmenting tech increase → zero research feedback, & research saving increase task automation; or HS in tech prod., research feedback & asymp. research ++ I → = §3.5 Korinek and Stiglitz (2019) capital prod. growth PS in production & capital-aug. tech  §5.2 Aghion et al. (2019) growth → saving decrease LS in tech production & high research − → → 0 §3.5 Sachs and Kotlikoff (2012), Sachs et al. (2015) feedback or HS in tech production & II MS in production & asymptotic or full positive research feedback task automation  §5.2 Aghion et al. (2019) I → I §4.2 Aghion et al. (2019) AI-assisted multiplication of LS in production & asymptotic task combinatorial idea discovery ++ automation  §5.3 Agrawal et al. (2019) ++ C ++ §4.2 Aghion et al. (2019) AI-assisted elasticity-change in idea LS in production & task automation discovery II and replacement  §5.3 Agrawal et al. (2019) ++ C ++
			 Sandberg (2013)  presented an "overview of models of technological singularity" before the past decade of economist engagement with AI and its transformative potential. Most of the models he summarizes therefore do not attempt to spell out how artificial intelligence, or indeed any particular transformative technology, would interact with standard economic models to produce the results in question. The models summarized here fill this gap.ECONOMIC GROWTH UNDER TRANSFORMATIVE AI
			 More formally, it is assumed that F LK > 0, with subscripts here denoting partial derivatives (and that F KL = F LK > 0, by Young's Theorem).
			 After accounting for capital depreciation, ρ may have to be significantly above 0 for capital accumulation to allow long-run growth. We will ignore capital depreciation throughout most of this document for simplicity.5 More generally, as should now be intuitive, when highly complementary production factors undergo different rates of accumulation or productivity growth, output's growth rate converges to that of the slowest-growing factor and its share goes converges to 1. Likewise with respect to complementary consumption goods, each requiring a different input. This is sometimes known as the "Baumol condition", afterBaumol's (e.g. Baumol  (1967)) seminal analyses of the increasing share of output spent on low-productivitygrowth sectors, such as live entertainment.6 The saving rate is in fact historically roughly constant, as famously observed by
			 Indeed, some reserve the term "endogenous" for growth models in which φ = 1, since
			 As Yudkowsky (2013)  points out, we might interpret this as a model in which AI comes in the form of " emulations"-a theoretical technology on which Hanson has written extensively-which are always technically feasible but which are, at first, prohibitively expensive, because effective capital is sufficiently scarce.13 Note that, were it not for the inclusion of the non-accumulable factor land, there would be no steady-state growth rate; in solving for it, we would have to divide by 0. Instead, the economy would be, asymptotically, an AK economy with exogenous capital productivity growth. As we saw in the previous section, we would have a Type I growth singularity.
			 Equivalently, we could say that output is produced by a single factor, labor, which can be supplied both by humans and by robots.
			 Technically, if flow utility is logarithmic in consumption (see footnote 18), lifetime utility falls to negative infinity as the consumption of the young falls to zero.
			 A = a -a (1-a) a-1 , which ranges from 1 (at a = 0 or 1) to 2 (at a = 1/2).ECONOMIC GROWTH UNDER TRANSFORMATIVE AI
			 Note that, in this case, effective capital growth g A + g Y will always equal gL ((1 − a)(1 − φ) + λ(1 − a))/((1 − a)(1 − φ) − λa) > g L .With effective capital growing more quickly than labor, the automation condition will always eventually be met for any fixed a.
			 The model requires φ > 0 such that the fishing-out effect does not predominate. As discussed in §2.4,Bloom et al. (2020)  estimate φ = −2.1.27  The formula for θ = 0 is the limiting case of the formula for θ > 0, as θ → 0.
			 Zeira, Joseph, "Workers, Machines, and Economic Growth," The Quarterly Journal ofEconomics, 1998, 113 (4), <PHONE>.