Scenarios for the Transition to AGI

Scenarios for the Transition to AGI* Anton Korinek and Donghyun Suh University of Virginia, Brookings, and GovAI March 2024

Abstract

We analyze how output and wages behave under dierent scenarios for technological progress that may culminate in Articial General Intelligence (AGI), dened as the ability of AI systems to perform all tasks that humans can perform. We assume that human work can be decomposed into atomistic tasks that dier in their complexity. Advances in technology make ever more complex tasks amenable to automation. The eects on wages depend on a race between automation and capital accumulation. If the distribution of task complexity exhibits a suciently thick innite tail, then there is always enough work for humans, and wages may rise forever. By contrast, if the complexity of tasks that humans can perform is bounded and full automation is reached, then wages collapse. But declines may occur even before if large-scale automation outpaces capital accumulation and makes labor too abundant. Automating productivity growth may lead to broad-based gains in the returns to all factors. By contrast, bottlenecks to growth from irreproducible scarce factors may exacerbate the decline in wages. Keywords: articial general intelligence, tasks in compute space, automation, capital accumulation JEL Codes: O33, E24, J23, O41 *We would like to thank Ajay Agrawal, Tamay Besiroglu, Erik Brynjolfsson, Tom Davidson, Joe Stiglitz and seminar participants at Anthropic, Google, GovAI, and OpenAI for helpful conversations and comments. 1 Introduction Recent advances in AI promise signicant productivity gains, but have also renewed fears about the displacement of labor. A growing number of both AI researchers and industry leaders suggest that it is time for humanity to prepare for the possibility that we may soon reach Articial General Intelligence (AGI) AI that can perform all cognitive tasks at human levels and thus automate them. This raises a number of fundamental economic questions. What would the transition to AGI look like? What would AGI imply for output, wages, and ultimately human welfare? Would wages rise or collapse? Our paper introduces an economic framework to think about these questions and evaluate alternative scenarios of technological progress that may culminate in AGI. Our starting assumption is that human work can be decomposed into unchanging atomistic tasks that dier in how complex they are. Advances in technology make ever more complex tasks amenable to automation. We capture this by assuming that there is a threshold of task complexity that can be automated at a given time, captured by an automation index. This index grows exogenously over time, in line with regularities such as Moore's Law. Although our results hold more broadly, we suggest that in the Age of AI, a natural measure of task complexity is the amount of compute (shorthand for computational resources) required for the execution of a task by machines. Some tasks, such as adding up numbers in a spreadsheet, can be performed with minimal computation. In contrast, others require a substantial amount of computation for machines, despite seeming natural and eortless for humans, such as navigating a bipedal body over an uneven surface. We describe how tasks dier in computational complexity using a distribution function that captures tasks in complexity space or, referring to our preferred interpretation, tasks in compute space. Throughout the paper, we analyze two opposing cases for the distribution of tasks in complexity space, which result in sharply dierent economic outcomes. First, we consider the possibility that human tasks are of unbounded complexity, illustrated in the left-hand panel of Figure 1. In this case, advances in the automation index, illustrated by the right-ward movement of the vertical frontier of automation, imply that more and more tasks are automated over time, but that there always remain tasks and by extension jobs that cannot be automated. Second, we consider a bounded distribution of task complexity, which reects that the computational capabilities of the human brain are nite, as discussed, e.g., in Carlsmith (2020). Bounded distributions result in full automation within nite time when the frontier of automation crosses the 1This includes, for example, the CEOs of the three leading AGI labs, OpenAI's Sam Altman, Google Deepmind's Demis Hassabis, and Anthropic's Dario Amodei (Kruppa, 2023; Time, 2024). It also includes the world's most renowned AI researchers, for example two of the godfathers of deep learning, Georey Hinton and Yoshua Bengio. 2Although the computational complexity of tasks is most evident for cognitive tasks, the automation of physical tasks is also greatly constrained by the computational complexity involved, as captured, e.g., by Moravec's paradox (Moravec, 1988). We analyze an extension that explicitly accounts for cognitive and physical tasks in Section 4. Complexity of tasks Fraction of tasks Frontier of automation 2024 Maximum complexity Bounded distribution Complexity of tasks Fraction of tasks Frontier of automation 2024 Unbounded distributionFigure 1: Unbounded and bounded distributions of tasks in complexity space maximum complexity of tasks performed by humans. An alternative interpretation for tasks being to complex to automate is that society may choose not to automate certain tasks even when it is feasible to do so. This may apply, for example, to some of the tasks performed by priests, judges, or lawmakers. We lay out an economic model in which atomistic tasks are gross complements that are combined to produce nal goods. In the spirit of Zeira (1998) and Acemoglu and Restrepo (2018, 2022), all tasks can be performed by labor, and automated tasks can be performed by either labor or capital. However, unlike in the described works, our main focus is on the edge cases that arise as we come close to full automation. Our analysis begins by examining the equilibrium under xed supplies of capital and labor. We show that automation can have dramatic impacts on wages and output even before it reaches all tasks. There exists a threshold level of the automation index that separates two distinct regions. As long as the index remains below the threshold, labor remains scarce relative to capital, and wages remain high. However, once the automation index surpasses the threshold, the economy enters a second region, where the scarcity of labor is alleviated, despite the presence of some tasks that still need to be performed by humans. In this region, labor and capital become perfect substitutes at the margin so wages decline starkly to equal the marginal product of capital. The economy exhibits behavior akin to an AK model. Next, we characterize the eects of automation on the economy's factor price frontier (FPF), which reects all possible combinations of factor prices that may result from a given level of technology under dierent capital/labor ratios. The FPF provides general insights into the eects of automation that do not depend on specic assumptions on capital accumulation. We nd that for a given level of automation, wages lie within a bounded interval that expands as the automation index rises but only as long as automation is incomplete. Once all tasks are automated, the factor price frontier discontinuously collapses to a single point at which the eective returns to labor and capital are equalized. For given factor endowments, the eects of automation on wages are hump-shaped: for low levels of automation, advances in automation increase wages as the economy becomes more productive, but for higher levels of automation, wages decline due to the displacement of labor. We analyze dynamic settings and show that the eects on wages are determined by a race between automation and capital accumulation. In addition to the previous two opposing eects on wages from rising productivity and labor displacement, automation also triggers capital accumulation that moves the economy up on the factor price frontier, increasing wages. We characterize an upper bound on output and wages that is reached in the limit case that the capital stock can instantaneously adjust to its optimal level whenever automation advances. We show a powerful analytic result: For any optimizing representative agent with linearly separable intertemporal preferences, the eects of automation on output and wages will lie between a lower bound captured by the constant-capital case and the described upper bound. When the complexity distribution of tasks is bounded, full automation is reached in nite time and leads to a collapse in wages, no matter what savings behavior the representative agent pursues. For unbounded complexity distributions of tasks, we show that if the tail of remaining tasks is suciently thick, wages will rise forever. By contrast, if the tail of unautomated tasks is too thin, wages will eventually collapse. Next, we simulate a range of scenarios to illustrate our ndings numerically. (Figure 8 on page 24 shows the main results.) We start with a business-as-usual scenario, which captures the traditional notion that a constant fraction of tasks is automated each period, similar to Aghion et al. (2019). This corresponds to a Pareto distribution for task complexity together with exponential growth in the automation index. Since the maximum complexity of tasks in this scenario is unbounded, true AGI will not be reached in nite time. In our calibration, both output and wages rise forever in this scenario, at a pace similar to what advanced countries have experienced over the past century. Next we consider two AGI scenarios that span the range of estimates provided by Georey Hinton, one of the godfathers of deep learning, who estimated in May 2023 that AGI may be reached within 5 to 20 yearsafter declaring that he had suddenly switched [his] views on whether these things are going to be more intelligent than us. In our baseline AGI scenario we assume a bounded task distribution such that full automation is obtained within 20 year. In an aggressive AGI scenario we assume a shorter-tailed distribution that implies full automation within ve years. Our simulation results imply ten times faster growth than in the business-as-usual scenario, especially in the aggressive AGI scenario. However, wages collapse as the economy approaches full automation. In a fourth scenario, we consider the possibility that there is a large bout of automation in the near termfor example because AI rapidly automates cognitive jobsbut that there remains a long tail of tasks that are harder to automate. As a result of the initial bout of automation, the economy enters the 3As economists, we hope that the computational complexity of at least some atomistic tasks that go into writing economics papers is far into that right tail. Alas, this might be wishful thinking. region in which labor loses its relative scarcity value, and wages in our simulation collapse. However, after capital accumulation has caught up suciently, labor becomes suciently scarce again so that the economy returns to region 1 and wages rise in line with output growth. We extend our baseline model to analyze several additional important considerations. First, we consider the role of xed factors (such as minerals or matter) and show that they may pose a bottleneck that holds back economic growth and worsens the outlook for wages, ultimately leading to stagnation accompanied by a wage collapse. Next, we add an innovation sector to analyze the potential for automating technological progress and show that this lifts the returns of all factors including wages. We illustrate that sucient automation may give rise to a growth singularity whereby output takes o. Furthermore, we analyze societal choices to retain certain jobs as exclusively human even when they can be automated (e.g., priests and judges), and show that a sucient volume of such nostalgic jobs may help to keep labor sufciently scarce so that wages continue to grow even when full automation is technically possible. We analyze the wage-maximizing rate of automation and show that slowing down automation in an AGI scenario may deliver signicant gains to workers albeit at the cost of forgoing a growing fraction of output. Next, we evaluate the impact of automation on workers with heterogeneous skills and susceptibility to being automated. We nd that automation in such a scenario may give rise to an ever-declining fraction of superstar workers earnings ever-growing wages, whereas the majority of the labor force is starkly devalued by automation. Finally, we explore the role of compute as an example of specic capital that is tailored to automating specic tasks. We observe that in the short term, compute may earn very high returns, but after an adjustment period during which sucient compute has been accumulated (and which may last long), compute may become just another form of capital that earns the same return as all other types of capital. Related Literature The foundational work of Aghion et al. (2019) explores the impact of articial intelligence on economic growth, oering valuable insights into how technological advances in AI, including AGI, might inuence future economic trajectories. Jones (2023) underscores the risk of technological progress, emphasizing existential riska concept crucial in AGI discussions. Davidson (2023) analyzes a model of the factors that may lead to a take-o in economic growth if technology advances near AGI but does not focus on the wage implications. Besiroglu et al. (2022) show how advances in AI may accelerate growth by speeding up R&D, and Erdil and Besiroglu (2023) review the factors by which AGI may give rise to exponential growth. Trammell and Korinek (2023) provide a useful survey on the broader implications of advanced articial intelligence on economic growth. A critical body of literature explores the dynamics between labor and automation. Seminal works by Acemoglu and Restrepo (2018, 2022) and Autor (2019) provide insights into how automation reshapes labor markets, focusing on technology as a substitute for individual worker tasks or how workers and tasks can complement or substitute for technology. Eloundou et al. (2023) and Felten et al. (2023) provide excellent empirical analyses of which tasks are amenable to automation by the current wave of foundation models. These studies oer a useful lens for understanding the economic implications of AI before AGI is reached. Our contribution to these strands of literature is to look at the limit case of what happens if either all work tasks are automated or we asymptote towards a world in which all tasks are automated. Our paper is also related to a broader literature on AGI and superintelligence literature. Good (1965) was the rst to articulate the potential of an intelligence explosion if AGI is reached. Bostrom (2014) provides a comprehensive exploration of superintelligence, highlighting the potential capabilities of AGI and the profound implications these might have for society. Yudkowsky (2013) discusses several of the economic implications of the transition to AGI. 2 A Compute-Centric Model of Automation

2.1 Tasks in Compute Space

compute /k@m-"pyüt/ verb: to determine by calculation The system computed the length of the shortest path. noun: the combined computational resources available for information processing tasks Modern AI relies on vast amounts of compute. Etymology: Derived from the Latin verb "computare," meaning "to count, sum up, or reckon together," the word "compute" entered the English language as a verb in the 16th century. The noun form "compute" gained prominence more recently with the advent of high-performance digital computers and the increasing need to describe the resources required for computation. Atomistic Job Tasks A central assumption of our analysis is that the work performed by humans is composed of tasks and sub-tasks or unchanging atomistic task that dier in how easily they are automated. In our baseline model, we focus on cognitive tasks and their potential for automation. In this setting, an atomistic task is a well-dened computational assignment that contributes to the accomplishment of a larger job task. These atomistic tasks are fundamental and are signicantly smaller than the tasks that are listed in O*Net. Table 1 lists, for example, the top-5 O*Net tasks of economists: to study data; conduct and disseminate research; compile, analyze and report data; supervise research; and teach. Each of these O*Net job tasks involves a wide variety of dierent atomistic tasks. For example, the O*Net task teach theories of economics may require rst planning the overall Study economic and statistical data in area of specialization, such as nance, labor, or agriculture. Conduct research on economic issues, and disseminate research ndings through technical reports or scientic articles in journals. Compile, analyze, and report data to explain economic phenomena and forecast market trends, applying mathematical models and statistical techniques. Supervise research projects and students' study projects. Teach theories, principles, and methods of economics. Table 1: Top-5 Tasks performed by economists (O*Net database) task, recalling dierent economic theories, synthesizing a structure, preparing slides, formulating lectures, synthesizing speech and aect, decoding and responding to student questions, preparing problem sets, distributing problem sets, grading problem sets, and so onall while keeping track of the plan. It may also require tasks such as recognizing emotional expressions on students' faces, using theory of mind to evaluate student progress and dynamically adjust the structure, etc. All of these tasks involve a set of basic human brain functions, which constitute a form of computation. Some of these functions are easily performed by machines and therefore highly susceptible to cognitive automation (Korinek, 2023), whereas others are more dicult. What matters for our purposes here is how computation-intensive they are using machines. Recent literature on technology on labor markets observes that innovation typically gives rise to new job tasks (e.g., Acemoglu and Restrepo, 2018; Autor, 2019). This holds true when viewed from the perspective of high-level job tasks such as those captured by O*Net. However, when viewed from an atomistic level that reects basic brain functions, innovation merely recombines atomistic tasks in novel ways to produce novel high-level tasks and jobs. For example, the novel task of prompt engineering may require atomistic tasks such as dening a desired output, crafting an initial prompt, entering it, reading the output, evaluating it, deciding whether to iterate, and nally sharing the outputall functions that existed long before the invention of generative AI systems that triggered prompt engineering. Task Complexity and Compute Intensity Our baseline model emphasizes dierences in complexity as a key dimension when studying the automation potential of tasks. Our preferred interpretation for what makes tasks dicult to automate is their compute intensity, which refers to the amount of computational resources required to perform a specic task. Compute intensity can easily be measured by the amount of oating point operations (FLOP) that Figure 2: Training compute of frontier AI systems over time (Copyright © 2024 by Epoch under a CC-BY-4.0 license; Sevilla et al., 2022.) need to be executed to perform a given task. The computational complexity for machines to execute a task often diers starkly from how easy or dicult it is for humans. Still, it is the computational complexity for machines that determines whether a task can be automated. Advances in Computing One of the main drivers of recent advances in AI has been the increased availability of computing power. Moore's Law, rst described by Gordon Moore (1965), describes that the performance of cuttingedge computer chips doubles approximately every two years. The regularity has held for the past sixty years. Additionally, the amount of compute deployed in cutting-edge AI systems has grown even faster over the past decade, doubling roughly every six months, as shown in Sevilla et al. (2022) and depicted in Figure 2. Improvements in algorithms have further accelerated the growth in capabilities of cutting-edge AI systems (Besiroglu et al., 2022). For our analysis below, we assume that there is an automation index that captures the maximum complexity of tasks that can be automated. This index grows exponentially at an exogenous rate, mirroring the type of advances captured by Moore's Law and Figure 2. As the automation index increases, a growing mass of tasks can be automated. 4For example, Moravec (1988) observed that some tasks that feel easy to execute for humans, such as vision processing, employ a large amount of dedicated grey matter and are, in fact, computationally quite intensive. There are also certain tasks that require very little compute in dedicated machines but that are dicult for the human brain since it has not evolved for them: for example, arithmetic operations take just one FLOP on a basic computer but require signicant amounts of grey matter for human brains to perform, likely involving the equivalent of billions of FLOP.

2.2 Baseline Model

Consider a representative household in a static economy who is endowed with L = 1 units of labor and K > 0 units of capital. There is a continuum of tasks that dier in their computational complexity i. The distribution function Φ(i) reects the cumulative mass of tasks with complexity ≤ i and satises Φ(0) = 0 and limi→∞ Φ(i) = 1. If the distribution function is dierentiable, we call its derivative ϕ(i) the density of tasks of complexity i. Examples are shown in Figure 1. To produce aggregate output Y , we combine all the tasks of dierent complexity using a CES aggregator with elasticity of substitution σ. Y = A Z i y(i) σ−1 σ dΦ(i) σ σ−1 (1) where y(i) is the amount of type i tasks employed in the production of output. We generally assume σ < 1, reecting that the atomistic tasks are gross complements. Each task is performed using capital k(i) and labor ℓ(i) according to the production function y(i) = aK (i)k(i) + aL(i)ℓ(i) (2) where the coecients aK (i) and aL(i) reect the eciency of capital and labor. We assume that the exogenous index I reects the state of automation and denes a complexity threshold such that all tasks below the threshold can be performed with either capital or labor but all the tasks above the threshold require labor. We normalize the technological parameters aK (i) = aL (i) = 1 except that aK (i) = 0 if i ≥ I. In other words, y (i) = ( k (i) + ℓ (i) for i < I ℓ (i) for i ≥ I Strategies The representative agent supplies her endowments of labor and capital every period at the prevailing factor prices w and R and makes no interesting economic decisions. The representative rm in the economy maximizes prots by hiring capital k (i) and labor ℓ (i) for each task at the prevailing factor prices w and R to produce y (i), which is then combined to produce nal output. The rm's maximization problem is max k(i),ℓ(i) Y − R Z i k(i)dΦ(i) − w Z i ℓ(i)dΦ(i) s.t. (1), (2) Equilibrium An equilibrium in the baseline model consists of a set of {k (i) , ℓ (i) , y (i)}i≥0 and factor prices w and R such that the representative rm solves its maximization problem and markets for capital and labor clear, i.e., Z i k(i)dΦ(i) = K Z i ℓ(i)dΦ(i) = 1 Since there are no market imperfections, the described equilibrium also constitutes the rst-best of the economy. 2.3 Equilibrium: Characterizing Two Regions Scarcity of Labor For given factor endowments (K, L), there are two possible regimes for the scarcity of labor, depending on the level of the automation index I: If the index is low enough so that labor is relatively scarce, then the return on labor is greater than the return on capital, w > R, and the scarce labor is employed solely in those tasks that cannot be automated. Conversely, if the state of automation is suciently advanced that only a small fraction of tasks are exclusive to human labor, then w = R holds, and labor is employed not only in the remaining unautomated tasks but also in some of the automated tasks. At the margin, capital and labor are perfect substitutes. Lemma 1 (Scarcity of labor). For given (K, L), there is a threshold value for the state of automation ˆI that is dened by Φ ˆI = K/L 1 + K/L (3) and increasing in the K/L-ratio such that there are two regions: Region 1: If I < ˆI, then labor is scarce compared to capital. In this regime, labor is employed only for tasks with i > I. Output is Y = F (K, L; I) = A h K σ−1 σ Φ(I) 1 σ + L σ−1 σ (1 − Φ(I)) 1 σ i σ σ−1 (4) and wages satisfy w = A σ−1 σ (Y /L) 1 σ · (1 − Φ(I)) 1 σ > R Region 2: If I ≥ ˆI, then the relative scarcity of labor is relieved, and labor earns the same return as capital w = R = A; if the inequality is strict, some labor is deployed alongside capital for tasks with i < I, and labor and capital are perfect substitutes for the marginal task. Output is given by the linear function Y = F (K, L) = A (K + L) (5) Conversely, for given I, there is a threshold κ (I) = Φ(I)/ [1 − Φ (I)] such that the economy is in region 1 if K/L > κ (I) and in region 2 if K/L ≤ κ(I). The threshold κ(I) is increasing in I, i.e., if K/L is marginally above the threshold, further automation pushes the economy from region 1 into region 2 where the scarcity of labor is relieved. Proof. Assume rst that all labor is employed in tasks with i ≥ I and observe that the symmetry of the production function across all tasks implies that an identical amount of capital k = K/Φ(I) will be employed in each task below the threshold and an identical amount of labor ℓ = L/(1 − Φ(I)) for each task above the threshold for given aggregate K and L. The production function can then be written as Y = F (K, L; I) = A h k σ−1 σ Φ(I) + ℓ σ−1 σ (1 − Φ(I)) i σ σ−1 = A h K σ−1 σ Φ(I) 1 σ + L σ−1 σ (1 − Φ(I)) 1 σ i σ σ−1 proving equation (4). The rm's optimization problem implies the rst-order conditions FK = A σ−1 σ Y 1 σ · K− 1 σ Φ(I) 1 σ = R (6) FL = A σ−1 σ Y 1 σ · L− 1 σ (1 − Φ(I)) 1 σ = w (7) By comparing these two expressions, we can see that the return on capital R is less than the return on labor w as long as K− 1 σ Φ(I) 1 σ < L− 1 σ (1 − Φ(I)) 1 σ or, equivalently, k > ℓ, i.e., the capital assigned to each automated task is greater than the labor assigned to unautomated tasks. Expressing this in terms of aggregate supplies of factors, the condition is K L > κ(I) := Φ(I) 1 − Φ(I) (8) The right-hand side is an increasing function of I that goes from 0 to ∞. By implication, there is a value of I such that the inequality is violated for all I > ˆI. When that threshold is crossed, it is more ecient to allocate some labor to tasks with i < I, and the marginal unit of labor is perfectly substituable with capital. By implication, k (i) = ℓ (i) = K +L, and the CES aggregator simplies to equation (5). Alternatively, the threshold for Φ(I) can be expressed explicitly as an increasing function of the K/L-ratio by solving for Φ ˆI = K/L 1 + K/L The remaining results stated in the lemma follow immediately. Intuitively, region 1 reects the world as we have experienced it over the past 200 years, in which capital and labor are complementary in production, and labor is comparatively scarce. Figure 3 shows that for given factor supplies, a higher automation index I increases the mass of tasks that can be accomplished with capital, implying that the available capital is spread over a greater number of tasks and becomes scarcer. Conversely, automation reduces the mass of tasks that is exclusive to labor, implying that the available labor can be concentrated on fewer tasks and becomes less scarce. As the automation index reaches the threshold ˆI, there are so few tasks left that are exclusive to labor that labor no longer enjoys a scarcity advantage over capital, and the returns on the two factors are equated. i y(i) I k ℓ i y(i) I I′Figure 3: Automation and the scarcity of labor Note that the threshold ˆI depends solely on relative factor supplies, not on the elasticity of substitution σ between capital and labor. As soon as labor is no longer scarce, it will be used interchangably with capital in the marginal task, and this holds even when individual tasks are highly complementary as reected by low values of the elasticity of substitution (as long as σ > 0).

2.4 Factor Price Frontier (FPF)

For an analysis of the eects of advances in automation I on factor returns, let us characterize the factor price frontier associated with the rm's technology. The factor price frontier depicts all possible combinations of factor prices R and w that will result from dierent proportions of factor supplies K and L in a competitive economy with prot-maximizing rms under a given technology. Lemma 2 (Factor Price Frontier (FPF)). For a given automation index I, the factor price frontier slopes downwards, starting from a limiting point w∗(I) = A(1 − Φ(I)) 1 σ−1 and R = 0 as K/L → ∞ to the point w = R = A when K/L ≤ κ(I). Increases in A move the FPF proportionately outwards. Increases in I raise w∗(I) and swivel the factor price frontier clock-wise. Proof. We obtain the factor price frontier from the aggregate cost function, which is the dual of the aggregate production function. The associated unit cost function represents the minimum cost at which a competitive optimizing rm can produce one unit of nal output, given factor prices w and R. In the region of I < ˆI, the unit cost function associated with equation (4) is C(w, R; I) = 1 A R1−σ Φ(I) + w1−σ (1 − Φ(I)) 1 1−σ R w w∗ 45◦ A AR w ( K L ) 1 A A A′ A′Figure 4: Factor price frontier and its dependence on A Since we employed the nal good as the numeraire good, this cost function needs to equal 1 in a competitive economy. The factor price frontier when I < ˆI is thus given by all pairs of (w, R) that satisfy the equation C (w, R; I) = 1, or equivalently, w = A1−σ − R1−σ Φ(I) 1 − Φ(I) 1 1−σ (9) Asymptotically, as K/L goes to innity, we can see from equations (6) and (7) that the return to capital R goes to zero, whereas the wage converges to w∗(I) = lim K/L→∞ w = A h 0 · Φ(I) 1 σ + (1 − Φ(I)) 1 σ i 1 σ−1 (1−Φ(I)) 1 σ = A(1−Φ(I)) 1 σ−1 (10) Conversely, when I ≥ ˆI, the cost function is simply C (w, R; I) = min {w, R} /A, and the factor price frontier is degenerate and consists of a single point w = R = A. The factor price frontier is illustrated in Figure 4. The area above the 45 degree line corresponds to Region 1 of Lemma 1, reecting a high capital-labor ratio K/L > κ(I) and w > R. Higher capital intensity K/L moves factor returns up and to the left along the frontier, i.e., it increases w and reduce R. Conversely, when K/L ≤ κ(I), we enter Region 2 of the lemma, and the factor price frontier corresponds to a single dot on the 45 degree line at which w = R = A. The right panel of Figure 4 shows how an increase in the level of technology A pushes out the factor price frontier for any ratio of K/L, it scales the returns of all factors proportionately. This exemplies how the factor price frontier serves as a convenient tool to describe how factor returns are impacted by technological changes across any levels of factors supplies. The Automation Path on the Factor Price Frontier We next turn to the eects of automation for a given capital stock K or equivalently, capital intensity k = K/L. Then it is easy to see that: Lemma 3 (Automation and Output). An increase in automation dΦ(I) raises output as long as I < ˆI, and leaves output unaected otherwise. Proof. For I < ˆI, the result follows by dierentiating expression (4), dY dΦ(I) = 1 σ − 1 A σ−1 σ Y 1 σ · k σ−1 σ − ℓ σ−1 σ Given σ < 1, the derivative is positive as long as k > ℓ, which is the condition for being in region 1 in Lemma 1 in which the production function is relevant. For I ≥ ˆI, the relevant production function is (5), which is independent of I. Intuitively, for output to rise, capital must be suciently abundant, delivering a productivity gain from deploying the amply available capital to a greater number of tasks. This is frequently termed the productivity eect of automation. Let us look at factor returns next. Lemma 4 (Automation and Factor Returns). (i) An increase in automation dΦ(I) always raises R as long as I < ˆI. The eect on w is hump-shaped: there is a threshold I∗(K/L) with Φ(I∗(·)) ∈ (0, 1) such that wages w rise in Φ(I) as long as I < I∗(K/L) or, equivalently, as long as K/L > κ∗(I), but decline in Φ(I) for I > I∗(K/L) or, equivalently, K/L < κ∗(I). (ii) For Φ(I) = 0, the return on capital is R = 0, and wages equal w = A. For Φ(I) ≥ κ/(1 + κ), both equal R = w = A. The latter condition always holds if Φ(I) = 1. Proof. The limit results follow readily from equations (6) and (7) and from the second part of Lemma 1. By dierentiating equation (6), with respect to Φ(I), we can see that automation always raises the return on capital. To see how automation aects wages, consider the derivative of log w with respect to Φ from the rm's optimality condition (7): d log w dΦ(I) = 1 σ − 1 1 σ k σ−1 σ − ℓ σ−1 σ (Y /A) 1−σ σ − 1 σ 1 1 − Φ(I) . (11) The rst term reects the productivity eect of automation, which is positive under condition (8), reecting that producing the marginal task using a relatively more abundant k units of capital rather than a scarce ℓ units of labor increases output. The second term captures the displacement eect of automation and reduces labor income. It reects that the labor used in each unautomated task ℓ = L/(1 − Φ) increases, as captured by the term in the denominator, thereby pulling down the marginal product of labor. As I rises, wages rise at rstat Φ(I) = 0 we nd d log w dΦ(I) = 1 1−σ > 0. As I becomes larger, the rst term in (4) declines, reaching zero for I = ˆI, and R w 45◦Figure 5: Factor price frontier and automation the absolute value of the second term grows and eventually dominates the rst termin the limit of Φ → 1, the second term becomes innitely large. Thus, there exists an intermediate value I∗(K/L) after which further automation reduces wages. Notice that the rst term is increasing in K/L whereas the second term is independent of K/L. The threshold can alternatively be expressed as K/L < κ∗(I). Figure 5 illustrates that an increase in automation I rotates the factor price frontier clockwise, for example, from the dotted to the dashed and solid lines. If the economy is in the labor-scarce region 1 (above the 45 degree line), automation raises wages for a given return of capital and also the maximum wage level w∗(I). For given K/L, the path of factor prices that results from rising automation I is illustrated by the hump-shaped bold line with arrows in the gure. Along the path, R rises continually whereas w at rst rises but eventually falls. When automation reaches ˆI(K/L), the economy ends up in the degenerate equilibrium with w = R = A on the 45-degree line.

2.5 Automation and Factor Earnings

Figure 6 shows the eects of automation on total output for given factor supplies as well as its split into the wage bill and the total returns to capital. The horizontal axis depicts the fraction Φ (I) of automated tasks, which goes from zero to one. The left-hand panel illustrates the case of equal capital and labor endowments, K = L = 1, and modest complementarity with an elasticity of substitution σ = 0.5 between the two. As long as the economy is in the scarcelabor region (Region 1), output is a strictly monotonic function of automation. At rst, automation almost exclusively benets labor, and the returns to capital (a) Equal factor endowments (b) Abundant eective capital Figure 6: Static equilibria under rising automation are minuscule. But as automation increases and we come closer to Region 2, the wage bill reaches a ceiling and starts to decline. Further automation still raises output, but the returns to capital grow faster than output, at the expense of the wage bill. When Region 2 is reached at Φ = 0.5, both factors earn equal returns. Given equal endowments, this translates into capital and labor shares of one-half each. The right panel of the gure shows an alternative scenario in which the eective supply of capital is ten times higher than labor, i.e., L = 1 and K = 10, and in which the two are strong complements with σ = 0.2. The abundance of capital and the strong complementarity imply that the region in which most of the benets go to labor is much larger, but so is the drop in the wage bill once a critical threshold is surpassed: whereas wages seem to be growing exponentially in Φ (I) up until Φ ≈ 0.80, they experience a precipitous decline by about 85% starting around Φ ≈ 0.83, accompanied by a meteoric rise in the returns to capital. When Region 2 is reached at Φ = 10/11, factor returns are equalized, and given the relative factor endowments, the capital share of the economy is ten times the labor share. This example highlights that the fate of labor can change rapidly when certain thresholds are crossed. Crucially, the eect of automation on outputand per-capita income depends on the capital available. In the illustration in the left panel, full automation merely doubles output; in the right panel, output grows eleven-fold. This observation naturally leads us to the next step of our analysisto analyze how automation interacts with capital accumulation in a dynamic setting. 3 Dynamics: The Race between Automation and Capital Accumulation The dynamics of output and wages depend not only on technological advances captured by the automation index Ibut also on capital accumulation and by extension on the savings behavior of the agents in the economy. This section analyzes these forces in a dynamic setting.

3.1 Automation Scenarios

Progress in automation We assume that the automation index I grows exponentially over time at an exogenous rate of g, reecting Moore's law and similar regularities. For an initial I0, the time path of I (omitting the time index t for conciseness) is given by I = I0egt We can equivalently write that log I = log I0 + gt grows linearly at the rate g. We consider dierent distributions Φ (i) of task complexity or tasks in compute space to capture alternative scenarios for the advent of AGI: Business-As-Usual Scenario (Unbounded Distribution) We model unbounded complexity distributions of tasks Φ (i) as Pareto, implying that log i is described by an exponential distribution, log i ∼ Exp(λ) with decay parameter λ. The resulting cumulative distribution function is Φ(i) = 1 − e−λ log i. If the automation index I grows exponentially at rate g, the fraction of non-automated tasks declines at rate λ · g. This distribution has an innite right tail, meaning that there will always be tasks that cannot be automated. Baseline and Aggressive AGI Scenarios (Bounded Distributions) For our AGI scenarios, we assume a bounded complexity distribution of tasks to capture the scenario that the tasks that can be performed by human brains is limited by an upper bound so automation crosses the threshold ˆI within nite time. We assume that Φ(i) follows a power function Φ(i) = 1 − (1 − log i/ log Imax)β with β = 1 and with normalization Imax = I0egT such that all tasks are automated after T years. Following Hinton's predictions, we set T = 20 in the baseline AGI scenario and T = 5 in the aggressive AGI scenario. For I > Imax, we keep Φ(i) = 1 capturing full automation. Bout of Automation (Mixed Distribution) We consider a fourth scenario in which rapid advances in AI automate a large fraction of tasks within a short time span, but in which we assume that there remains an unbounded tail of tasks that cannot be automated, for example, because of legal or cultural reasons. Analytically, we assume a mixture of the two scenarios above. Specically, Φ(i) is dened as Φ(i) = ω 1 − (1 − log i/ log Imax)β + (1 − ω) 1 − e−λ log i where ω ∈ [0, 1] is a weight parameter. We assume the same values for the parameters of the Pareto and power function distributions as in the previous two cases. 5Power function distributions are a special case of beta distributions with a beta parameter α = 1.

3.2 Consumer Problem

The representative household seeks to maximize its lifetime utility by choosing consumption Ct over time: max {Ct} U = Z ∞ 0 e−ρtu(Ct)dt (12) subject to the law of motion for capital: ˙Kt = F (Kt, Lt; It) − δKt − Ct (13) for given K0. The current-value Hamiltonian for this problem is: Hc = u(Ct) + μt [F (Kt, Lt) − δKt − Ct] The rst-order conditions with respect to consumption and capital are: ∂Hc ∂Ct = u′(Ct) − μt = 0 ∂Hc ∂Kt = μt [FK − δ] = − ˙μt + ρμt Dierentiating the rst optimality condition with respect to time yields u′′(Ct) ˙Ct = ˙μt, and substituting into the second optimality condition gives ˙Ct Ct = 1 η(Ct) [FK (Kt, Lt) − ρ − δ] (14) where η(Ct) = − u′′(Ct)Ct u′(Ct) is the elasticity of intertemporal substitution. Limit Behavior in Region 2 When the economy is in region 2, then FK = A. If the agent's utility function exhibits constant elasticity of substitution η, then the Euler equation implies a constant growth rate of consumption gC = ˙Ct Ct = A − ρ − δ η (15) Let us assume that A > ρ + δ so consumption growth is positive and consider the case that the economy remains in region 2 foreverfor example, because full automation Φ(I) = 1 has been reached. Then the economy will converge towards a balanced growth path in which gY = gK = gC as in (15) and the savings rate s∞ = 1 − C/Y is constant. From (13), we obtain that gK = ˙Kt Kt = sA(Kt + L) Kt − δ As limt→∞ L/Kt = 0, we can equate gC = gK and solve for the long-run savings rate s∞ = A − ρ − δ + ηδ Aη = 1 η − ρ + (1 − η)δ Aη Bounds Assume an initial I0 and K0 that satisfy FK (K0, L; I0) ≥ ρ + δ, i.e., there was no excessive capital accumulation in the past. Then the following proposition holds for any intertemporal utility function that is linearly separable as specied in (12) with a twice continuously dierentiable, increasing, and strictly concave period utility function u(C): Proposition 5 (Bounds for Output and Wages). For any distribution Φ (i) of tasks in compute space and exogenous growth in the automation index It, the paths of capital, output, and wages lie between lower and upper bounds K− ≤ Kt ≤ K+ t , Y − t ≤ Yt ≤ Y + t and w− t ≤ wt ≤ w+ t . The lower bounds are dened by the xed-capital case with K− = K0 ∀t and Y − t = F (K−, L, It), w− t = FL(K−, L, It). The lower bound on wages rst rises in It and then declines in It. It declines to A in nite time if full automation is reached asymptotically, i.e., if limI→∞ Φ(I) = 1. If Φ(It) < 1, an upper bound K+ t for capital is dened by FK K+ t , L, It = R = ρ + δ ∀t as long as a solution exists; otherwise we set K+ t = ∞. The upper bounds for output and wages are Y + t = F (K+ t , L, It) and w+ t = FL(K+ t , L, It). All three upper bounds are increasing in the automation index It. If automation is full, Φ(It) = 1, the upper bounds are K+ t = ∞ and Y + t = ∞, and the upper bound on wages discontinuously collapses to w+ t = A. Proof. Observe that for any twice continuously dierentiable period utility function that is increasing and strictly concave, the elasticity in the Euler equation

(14) satises η(Ct) ∈ (0, ∞). Consumption on the optimal path is increasing as long as FK > ρ + δ and constant when FK = ρ + δ. Our characterization of the factor price frontier delivers most of the remaining results. For the lower bound, observe that increases in I and Φ(I) raise the marginal product FK for given K, triggering additional capital accumulation, which raises output and wages above the lower bound. For the upper bound, observe that by the Euler equation, capital accumulation will never exceed the upper threshold K+ t , which is given by K+ t = Aσ L(1 − Φ(It)) 1 σ−1 Φ(It) (Rσ−1 − Aσ−1Φ(It)) σ σ−1 . (16) For given It, output and wages are increasing in Kt, implying that they must lie between the lower and upper bounds dened by K− and K+ t . If Φ(It) = 1, the production function is AK-style, and Lemma 4 implies that wt = A. On the factor price frontier, the lower bound on wages w− is pinned down by the automation path in Figure 5; it collapses to A in nite time if the economy asymptotically converges to full automation. As long as Φ(I) < 1, the upper bound on wages w+ t is pinned down by the intersection of the corresponding factor price frontier with a vertical line at R = ρ + δ and rises without bounds in I. However, when full automation Φ(I) = 1 is reached, the upper bound on wages w+ t discontinuously collapses to A, which equals the lower bound and must therefore equal the equilibrium wage. This result is independent of intertemporal preferences and savings behavior and occurs in nite time if the distribution of task complexity Φ(I) is bounded, as in our two AGI scenarios. The Balancing Savings Rate To further investigate the race between automation and capital accumulation, we analyze the threshold at which the wage eects of automation and capital accumulation precisely oset each other. For this, we take the total dierential of the equilibrium wage, wt = FL(Kt, L; It), and set dwt = 0 to nd FKL(Kt, L; It) dKt dt + FLI (Kt, L; It) dIt dt = 0 (17) Suppose, for simplicity, that δ = 0 so we can denote the savings rate at t by st = ˙Kt/Yt. Also, note that FLI (Kt, L; It) dIt dt = FLΦ ˙Φt. Then stYt · FKL = −FLΦ ˙Φt The left-hand side is the increase in wages due to capital accumulation. The right-hand side is the change in wages due to automation. As we observed above in Lemma 4, the term FLΦ encompasses the productivity eect and the displacement eect of automation on wages. Dividing by the cross-derivative FKL, the fraction FLΦ FKL captures the wage eects of automation relative to capital accumulation. After some algebra, we obtain FLΦ FKL = σ 1 − σ (k/ℓ) 1−σ σ − κ + 1 1 − σ k The rst term in the brackets is the productivity eect that is increasing in the relative abundance of capital k/ℓ if capital is very abundant compared to labor, then using capital for newly automated tasks signicantly raises output. The second term is the displacement eect that is increasing in κ and thus in the automation index I. (Recall that for the given level of automation I, κ (I) = Φ(I)/ [1 − Φ (I)] reects the threshold of the capital/labor ratio below which the economy is in region 2 such that the scarcity of labor is lifted.) Intuitively, if a large fraction of tasks has already been automated, then further automation of marginal tasks will result in a large fall in labor demand since the automated labor has to be reallocated to an ever smaller set of human-only tasks. By rearranging terms, we obtain the following expression for the savings rate that, given Φ and g, perfectly osets the eect of automation on wages ˜st = κ + 1 1 − σ − σ 1 − σ (k/ℓ) 1−σ σ · Kt Yt · ϕtIt Φt · g (18) The condition tells us that, to oset the eect of automation on wages, the savings rate must be increasing with (i) the displacement eect net of the productivity eect, (ii) the capital-output ratio, (iii) the relative mass of automated tasks at the current compute threshold for task automation, and (iv) the growth of compute g. Intuitively, a large fraction of output must be invested if (i) the displacement eect reduces wages signicantly, or (ii) there is already a large amount of capital stock in the economy, or (iii) a large amount of tasks are being automated, or (iv) automation is fast. The expression in (18) tells us about the threshold level for the savings rate at time t above which wages rise and below which wages fall, given the extent of automation occurring at time t. In other words, it characterizes the short-run behavior of wages as an outcome of the race between automation and capital accumulation. Long-Run Dynamics for Unbounded Task Distributions To further illuminate the trade-o in (18), we turn to the long-run dynamics of wages. To do so, we start by characterizing the conditions for the existence of a balanced growth path (BGP). We dene a BPG as an equilibrium path on which output and capital stock grow at a constant rate and factor shares remain constant. Lemma 6. Suppose that as t increases, Φ(It) → 1, and focus on the limit case. Then the return to capital converges to A. Moreover, output and capital stock grow at the rate (A−ρ−δ)/η and the savings rate converges to (A−ρ−δ+ηδ)/Aη. Proof. If the economy is in region 1 in the limit, then the production function converges to lim Φ→1 A h K σ−1 σ Φ 1 σ + L σ−1 σ (1 − Φ) 1 σ i σ σ−1 = AK If the economy is in region 2 in the limit, then F (K, L) = A(K + L). In both cases, lim Φ→1 FK = A As a result, the Euler equation implies ˙Ct Ct = 1 η [FK − ρ − δ] → 1 η [A − ρ − δ] as Φ → 1. Output and capital must grow at the same rate, which implies that the savings rate must satisfy ˙Ct Ct = ˙Kt Kt = s∞Yt Kt − δ Since limΦ→1 Yt Kt = A, this requires that ˙Kt/Kt → s∞A − δ. Therefore, we have s∞A − δ = 1 η [A − ρ − δ] s∞ = A − ρ − δ + ηδ Aη λ · g gw (1 − σ) · A−ρ−δ η A−ρ−δ η A−ρ−δ η Case 3 Case 2 Case 1 0Figure 7: Wage growth rate (gw) as a function of the rate of automation (λg) Since we are interested in the long-run dynamics of wages, we make a simplifying assumption that the savings rate is given exogenously at a constant value s∞, which can be interpreted as the long-run savings rate. Under the assumption, s is a key parameter determining the rate of capital accumulation. Depending on the value of s∞ relative to the rate of automation g, the race between automation and capital accumlulation can result in three possible outcomes. The following proposition summarizes the results. Proposition 7 (Race between automation and capital accumulation). Suppose the complexity distribution of tasks is Pareto and that the economy starts in region 1, i.e., I0 < ˆI0. Then the growth of wages and long-run labor shares are characterized by two thresholds on the rate of automation λg:

1. If λg > A−ρ−δ η then limt→∞ wt = A and the labor share converges to zero.

2. If A−ρ−δ η · (1 − σ) < λg ≤ A−ρ−δ η then wages grow exponentially at an asymptotic rate 1 σ A−ρ−δ η − λg and the labor share converges to one.

3. Lastly, if λg ≤ A−ρ−δ η ·(1−σ) then wages grow exponentially at an asymptotic rate λg 1−σ and the labor share converges to 1 − (A−ρ−δ+ηδ)/η λg 1−σ +δ σ−1 σ . Proof. See Appendix A.1. Intuitively, the proposition illustrates how wages evolve as the result of a race between automation and capital accumulation. As observed above, the fraction Parameter Value Description ρ 0.04 Discount rate η 2 Risk aversion parameter δ 0.1 Depreciation rate σ 0.5 Elasticity of substitution A 0.5 Total factor productivity L 1 Labor endowment Φ0 0.608 Initial fraction of automated tasks K0 4.6 Initial capital stock Table 2: Parameter values for the numerical illustration A−ρ−δ η is proportional to the long-run savings rate of the economy. In the rst case, if the rate of task automation λg is too high compared the savings rate, then the automation index I crosses the threshold ˆI in nite time and the economy transitions into region 2, where wages collapse to A and remain stagnant. If the rate of task automation λg is at an intermediate value, then wage growth is constrained by capital accumulation. Wages grow perpetually at rate 1 σ ( A−ρ−δ η − λg), which is proportional to the savings rate minus the rate of automation. Finally, if λg is low enough, then the rate of automation rather than capital accumulation constrains wage growth. In other words, wage growth depends on how fast automation increases the eciency of factor allocation and allows the utilization of abundant capital. Indeed, the growth rate of wages (and of the entire economy) in this regime is increasing in the rate of automation. Figure 7 illustrates the three cases in Proposition 7. The gure plots the long-run growth rate of wages as a function of the rate of automation. If λg is suciently low as in case 1, then the wage growth rate is increasing in the rate of automation as the upward-sloping part of the curve indicates. Once λg surpasses the rst threshold value, the growth rate of wages starts to decline as λg increases further. Lastly, if λg surpasses the second threshold value, then wages do not grow in the long run and stay at A.

3.3 Numerical Illustration

To provide an illustration of the theoretical results, we present simulations of the four automation scenarios described in Section 3.1. Table 2 summarizes the parameter values that were common to all the simulations. The rst ve parameters are standard in the literature, and L = 1 is a normalization. We chose Φ0 and K0 to match a 66% initial labor share with capital at its steady state for that level of technology. Figure 8 presents the results. Panel (a) shows the traditional automation (a) Business-as-usual scenario (b) Baseline AGI scenario (c) Aggressive AGI scenario (d) Mixed scenario Figure 8: Simulations of the four scenarios scenario business-as-usual, in which Φ(i) reects a rate of task automation of λg = 0.01 per year. The upper part of the panel shows the output, split into the returns to capital (red, upper area) and the wage bill (green, lower area), on a logarithmic scale. The lower part of the panel shows the fraction of unautomated tasks 1 − Φ on a logarithmic scalefor panel (a), this is a straight line, capturing exponential decay. We observe that in the business-as-usual scenario, output grows at approximately 2% per year, and both the returns to capital and the wage bill rise approximately in tandem (with a small decline in the labor share due to the eects of automation). Note that this scenario corresponds to case 3 in Proposition 7, i.e., capital accumulation is suciently fast so that growth is constrained by the speed of automation. Panels (b) and (c) show the AGI scenarios, in which the fraction of unautomated tasks collapses to zero in 20 or 5 years, respectively. In the baseline AGI scenario, wages grow slightly during the initial periods but then collapse before full automation is reached. After the collapse, wages are equal to the returns to capital, and the economy remains in region 2 where labor and capital are perfectly substitutable, with steady-state growth of 18% per year. In the aggressive AGI scenario, the wage collapse happens after about 3 years. Since the scarcity of labor is relieved earlier than in the baseline AGI scenario, the growth take-o occurs earlier. Panel (d) shows the bout-of-automation scenario. During the initial periods, a large fraction of tasks are automated, leading to wage collapse similar to the aggressive AGI scenario as the economy enters region 2labor is abundant because of the rapid automation and comparatively low capital stock. However, over time, the economy accumulates more capital, making labor scarcer again. Around year 9, the economy has accumulated sucient capital so that it returns to region 1. Wages rise above A and start growing again in line with further (slower) advances in automation and further capital accumulation. This scenario illustrates the possibility that labor demand may collapse due to rapid automation but recover later because of a long tail of tasks that cannot be automated. 4 Extensions

4.1 Fixed Factors and the Return of Scarcity

If labor is dethroned as the most important factor of production, it becomes useful to disentangle the remaining factors, which have traditionally been lumped together into capital in the economic models of the Industrial Age. Let us distinguish between factors that are in xed supply and factors that are reproducible and can therefore be accumulated. We continue to call all reproducible factors capital, including compute, robots, power plants, and factories. By contrast, factors in xed supply include land, space, minerals, or solar radia- tion. It is dicult to predict which scarce factors will matter the most in an AGI-powered future in the short term, it is likely that microchips and the semiconductor fabrication equipment (fabs) used for producing these chips will be bottlenecks, but these are clearly reproducible. By contrast, the raw materials going into the production of chips, for example certain rare earth minerals, are irreproducible. In the longer-term, matter or, equivalently, energy (E = mc2) may be the ultimately source of scarcity. For the purposes of our analysis, we incorporate a xed factor in our analysis that we label M for minerals or matter. We assume that the aggregate production function is a Cobb-Douglas aggregator of the task composite and M , Y = A Z i y(i) σ−1 σ dΦ(i) σ σ−1 ·α M 1−α (19) where α ∈ [0, 1] is the share of the composite among total output. Then, a version of Lemma 1 applies, separating two regimes: Lemma 8. For given (K, L), the automation threshold ˆI is dened by (3) as in the original lemma and is independent of M . It denes two regions: Region 1: If I < ˆI, then labor is scarce compared to capital and employed only for unautomated tasks. Output is given by Y = F (K, L, M ; I) = A h K σ−1 σ Φ(I) 1 σ + L σ−1 σ (1 − Φ(I)) 1 σ i σ σ−1 ·α M 1−α (20) Wages and the returns to M satisfy w = αA h K σ−1 σ Φ(I) 1 σ + L σ−1 σ (1 − Φ(I)) 1 σ i σ σ−1 ·α−1 L− 1 σ (1 − Φ(I)) 1 σ M 1−α > R Q = (1 − α)Y /M Region 2: If I ≥ ˆI, then the relative scarcity of labor is relieved; if the inequality is strict, labor and capital are perfect substitutes for the marginal task. Output is given by Y = F (K, L, M ) = A(K + L)αM 1−α (21) Wages and the return to M satisfy w = R = αA(K + L)α−1M 1−α (22) Q = (1 − α)A(K + L)αM −α Proof. The proof follows along the same lines as the proof of Lemma 1. 6During the Industrial Age, labor was considered in xed supply at the relevant times scale raising humans took so long that their supply could be approximated as exogenous whereas human capital was a reproducible factor. The presence of the xed factor M does not aect the key characteristics of the production function described in Lemma 1 such as the threshold for the automation index beyond which labor is no longer scarce compared to capital. Similar results apply for the eects of automation on wages: Lemma 9 (Automation and Wages with M ). For given capital intensity K/L, an increase in automation dΦ(I) always raises R for I < ˆI. The eects on w is hump-shaped: there is a threshold I∗(K/L) with Φ(I∗(·)) ∈ (0, 1) such that wages w rise in Φ(I) as long as I < I∗(K/L) but decline in Φ(I) for I > I∗(K/L). The threshold I∗ with M is lower than in Lemma 4. In the limit cases of Φ(I) = 0 and Φ(I) ≥ κ/(1 + κ), wages are given by (22). The limit is reached for any K/L ratio if Φ(I) = 1. Proof. The eect of automation on wages for a given K/L-ratio is similar to Lemma 4: d log w dΦ(I) = σ σ − 1 α − 1 1 σ k σ−1 σ − ℓ σ−1 σ (Y /A) 1−σ σ − 1 σ 1 1 − Φ(I) (23) The only dierence is the multiplicative term σ σ−1 α − 1, which is smaller than 1 σ−1 for α < 1. Thus, the productivity eect is smaller with the xed factor M , meaning that wages start to decline at lower levels of I. Although the presence of a xed factor preserves the key results on the automation threshold and the wage eects of automation, the long-run dynamics of the economy changefor the worse. In particular, we nd that wages will always decline to the return on capital as the economy will always enter region 2 in nite time. Proposition 10. If lim Φ(I) = 1, then the economy enters region 2 in nite time, and wages equal the returns on capital w = R = ρ + δ. The labor share equals αL/(K∗ + L), where K∗ is dened by w = R = αA(K∗ + L)α−1M 1−α = ρ + δ Proof. If the economy is in region 2 after some nite time T , it will converge towards a steady state in which (K∗ +L) are pinned down by the Euler equation (14), resulting in the expression above. We observe that K∗ is the maximum capital level that an optimizing agent will accumulate in this economy since FK < ρ + δ for any region 1 allocation, as can be seen from the economy's factor price frontier. This implies that the economy will enter region 2 no later than when the automation threshold reaches the scarcity of labor threshold ˆI s.t. Φ( ˆI) = K∗/L/(1 + K∗/L), as dened in Lemma 1. Intuitively, Proposition 10 tells us that if there is a xed factor then automation eventually outpaces capital accumulation regardless of the distribution of tasks. This contrasts with Proposition 7, which shows that wages may grow indenitely if a sucient amount of tasks is always left to labor. Figure 9: Factor shares with xed factor M in traditional scenario Figure 9 illustrates the implications under the assumption that the CobbDouglas for M is (1 − α) = .10 in the traditional scenario, in which a constant fraction of tasks is automated every period. As can be seen, wages peak after about 10 years, and the economy enters region 2 after 25 years, slowly converging to the steady-state level of capital K∗. In stark contrast to our simulation results in Section 3, this illustrates that even though there is an innite tail of unautomated tasks, the presence of a xed factor bottlenecks capital accumulation and implies that labor loses its scarcity status in nite time.

4.2 Automating Technological Progress

Our analysis so far has focused on automation as the only form of technological advancement and has taken as given the technology parameter A, which is considered as the main driver of productivity gains in the neoclassical growth model. This has allowed us to derive a number of powerful results on the eects of automation in goods production on output and wages. However, there are widespread predictions that advances in AI not only will make output production more ecient but also will speed up technological progress (Aghion et al., 2019; Agrawal et al., 2023; Davidson, 2023). At the most basic level, the production of R&D that drives technological progress consists of atomistic computational taskslike any other production process described earlier in the paper. For example, Agrawal et al. (2023) suggest that scientic hypothesis generation can be viewed as the making of predictions over a vast combinatorial space. We denote the complexity distribution of tasks involved in R&D by the distribution function Gamma Γ(i), which may dier from the complexity distribution of tasks Φ(i) involved in producing outputperhaps R&D involves on average more complex computational tasks. W.l.o.g., we assume that our ability to automate both R&D and production are captured by the same automation index I. Building on our earlier task production function and on the endogenous growth setup of Jones (1995), we assume that advances in the technology parameter A are driven by an ideas production function combining atomistic computational tasks {x(i)} that involve computational complexity as reected in the distribution function Γ(i), log ˙A = log Aθ + Z log x(i)dΓ(i), where the parameter θ captures the potential for knowledge spillovers or for decreasing returns to knowledge accumulation. Similarly with the production of nal goods, we assume that automated tasks can be performed by either capital or labor whereas unautomated tasks require labor, x (i) = ( kA (i) + ℓA (i) for i < I, ℓA (i) for i ≥ I. To keep our analysis tractable, we assume that there is an exogenous supply of knowledge workers LA = 1 that can only work in ideas production in addition to the unit supply of workers LY = 1 who are solely engaged in nal output production. Moreover, we assume unitary elasticity of substitution between tasks in the output production function so σ = 1. Analogs of lemma 1 hold for both production functions. As long as labor is scarce (region 1), we observe that the production functions of nal output and knowledge satisfy F (K, L) ≃ AKΦ(I) Y L1−Φ(I) Y and ˙A ≃ Aθ KΓ(I) A L1−Γ(I) A where KY and KA are the aggregate amounts of capital employed in nal output or knowledge production. Following the hypothesis of Aghion et al. (2019) and the proof of Trammell and Korinek (2023), it can then be shown that once automation in the two production functions has proceeded suciently, growth in the described economy will experience what Aghion et al. (2019) term a type II singularity. The intuition is that a rapidly growing capital stock generates an explosion in R&D output and technological progress that feeds on itself, resulting innite output in nite time. The following proposition states this result formally under the assumptions of a constant savings rate s ∈ (0, 1), a constant allocation of capital across nal output and ideas production, and no depreciation for tractability. Proposition 11. The economy enters a path of super-exponential growth in technology A and output Y that diverges to innity in nite time once the automation index I reaches the level I♡such that Γ I♡ (1 − Φ (I♡)) (1 − θ) > 1 The marginal product of labor in the production of nal output diverges to innity alongside output. Proof. Observe that once the described threshold has been passed, the production functions F (K, L) ≃ AKΦ(I♡) Y L1−Φ(I♡) Y and ˙A ≃ Aθ KΓ(I♡) A L1−Γ(I♡) A for Figure 10: Output and wage growth under technological progress constant fractions of capital KY = cK and KA = (1 − c)K are lower bounds for the actual production functions using the (still increasing) automation level Φ(I) and Γ(I). The result is then a direct application of the proof in Trammell and Korinek (2023) (Appendix B.2). Note that wt ≥ At no matter if output production is in Region 1 or Region 2 as dened in Lemma 1. Therefore the marginal product of labor in the production of nal output also diverges to innity. The condition in the proposition depends on three parameters sucient automation in the production of ideas Γ(·), sucient automation in the production of nal output Φ(·), and sucient returns to the accumulation of knowledge. Remarkably, as long as the production of nal output has been suciently automated (suciently high Φ), the remaining two parameters can take on any nite levels. This highlights that sucient automation in the production of nal output and thus capital accumulation will always lead to an explosion in growth. Figure 10 illustrates the path of wages under a CES production function for output and optimal savings, numerically illustrating the explosive path of output growth and type-II singularity that we identied in the proposition. The convexity of the curve on a log-scale indicates that the growth rate of wages is ever-increasing due to the acceleration of technological progress. (Note that the log scale hides that the labor share of output is declining.) In the gure, we have cut o the simulation at t = 10. The singularity occurs shortly thereafter. In summary, even if automation induces wages to collapse to the returns to capital at A, rapid technological progress from the automation of R&D allows workers to benet from the advancement of AI once sucient automation has taken place. More generally, the force described in this subsection is plausible, and sucient progress in AI will likely indeed lead to rapid technological advances and increases in living standards. At the same time, it is also likely that both the production of output and of ideas will eventually be bottlenecked by xed factors, as we emphasized in Section 4.1. A model that comprehensively incorporates both eects is beyond the scope of this paper.

4.3 Nostalgic Jobs or Limits on Automation

Our baseline model assumed that the automation of work was driven solely by technological factors, occurring as soon as the compute requirements of performing specic tasks were reached. However, even if it is technologically feasible to perform certain tasks, our society may decide that it is preferably for those tasks to remain exclusively human. For example, Korinek and Juelfs (2023) observe that jobs such as priests, judges, or lawmakers may remain exclusively human long after the time when they can be performed at equal or superior levels by machines, labelling such jobs nostalgic jobs. For the purposes of our analysis, we assume that there is a separate distribution function Ψ(I) that captures how far the automation index I must advance for society to choose to automate task I in addition to the distribution Φ (I) capturing the technological possibility of automation. The inequality Ψ (I) ≤ Φ(I) reects that society can only choose to automate tasks that are feasible to automate. The inequality is strict if there are tasks that could be automated from a technical perspective but aren't for societal reasons. If limI→∞ Ψ(I) < Φ(I) ≤ 1, then this captures that there are tasks that humans choose to never automate even though they could be. The described setup can also capture situations in which tasks are delegated to machines with a delay, i.e., for higher levels of the automation index I than what is technologically feasible. Korinek and Juelfs (2023) describe two reasons for why this may occur: First, as the capabilities of machines to perform certain tasks become better and better than human abilities, it may become increasingly untenable for the tasks to be left to humans. For example, if AI systems demonstrably become much fairer judges with fewer biases and noise than human judges, it may become untenable to leave many judicial deicisons to error-prone humans. Second, with sucient advances in robotics, it may become more and more dicult to distinguish humans and AI-powered robots performing human services. They observe that a robot priest with greater emotional intelligence than humans and a more comprehensive theory of human minds than a human priest may be able to perform the tasks typically performed by human priests quite perfectly, or intentionally somewhat imperfectly so as to not give away that it is a robot. Both of these categories require that the performance of AI systems is suciently above human levels, corresponding to a suciently high level of the automation index I. Maximizing Wage Growth Consider the problem of a government with the objective to maximize wage growth by imposing limits on automation and choosing an optimal path Ψ(I) ≤ Ψ(I). The following result characterizes the optimal Ψ(I) among all Pareto distributionsgiven exponential advances in the automation index I, this amounts to the government choosing an optimal constant rate of automation per time period. Proposition 12 (Maximizing Wage Growth). Suppose Ψ is a Pareto distribution dened as Ψ(It) = 1 − I−λ 0 e−λgt where I0 is the initial automation index and λg is the rate of task automation. Then the long-run growth rate of wages is maximized for λg = (1 − σ) · A−ρ−δ η , assuming that Ψ(I) ≤ Ψ(I)∀I for this distribution. As a result, wages grow at rate A−ρ−δ η . Proof. The proof follows from Proposition 7. The rate of automation is lowest in case 3. And the wage growth rate is increasing in λg. Thus, the wage growth rate increases until λg = (1 − σ) · A−ρ−δ η . Once λg surpasses (1 − σ) · A−ρ−δ η , the growth rate of wages decreases in λg until λg = A−ρ−δ η at which the growth rate equals zero. Therefore, the maxmum growth rate of wages is A−ρ−δ η at λg = (1 − σ) · A−ρ−δ η . Figure 7 on page 22 provides a graphical illustration of this ndingthe peak of the wage growth rate as a function of λg is A−ρ−δ η . Figure 11 shows what happens if we slow down progress in the baseline AGI scenario from Section 3 so that wages growth is maximized. Up until period 14, a wage-maximizing planner is constrained by the natural pace of automation and sets Ψ(I) = Φ(I) over that stretch. After that point, the baseline AGI scenario implies rapid declines in the labor share, but the planner sets Ψ(I) < Φ(I) to slow down eective automation. The left panel of the gure shows the paths of output and wages, and the right panel depicts the two variables in relative terms for the two scenarios. Up until period 14, the paths in the two scenarios roughly coincide (with a minor gap opening since the AGI scenario triggers rapid capital accumulation in advance of the economy achieving full automation). Thereafter, the wage-maximizing planner obtains a path of exponentially growing wages, as predicted by the proposition, whereas wages in the AGI scenario collapse. Notably, the right-hand panel also illustrates the output cost of foregoing the possibility of full automation. As can be seen, the output cost of holding back automation is low at rst, but eventually, almost 100% of the output potential of the economy is lost by holding back automation. Our nding illustrates that slowing down automation may be a powerful tool to increase wages, albeit it comes at the cost of reducing output growth. The described policy is feasible under both of the AGI scenarios simulated in the previous section and always results in exponentially growing wages instead of the collapse wihin a matter of years that would otherwise occur when AGI automates human tasks too quickly.

4.4 Heterogeneous Worker Skills

When labor is heterogeneous, individuals are hit by the eects of automation at dierent times, depending on the extent to which their skills are automated. In practice, workers dier along many dierent dimensions, and each worker's labor may be complemented or substituted for in dierent ways by technological Figure 11: Comparison of output and wages under Φ and Ψ advances. One of the classical ways of accounting for heterogeneity in the labor market, going back to Katz and Murphy (1992), is to split workers into skilled and unskilled based on a threshold level of educational attainment. An additional distinction, introduced by Autor et al. (2003), was to categorize workers according to whether they hold cognitive or manual jobs performing routine or non-routine activities. Under the described paradigm, we could capture the distribution of tasks in compute space separately for each of the resulting buckets (e.g., routine cognitive workers), and analyze how advances in computing capabilities will aect that type of workers. Recent advances in AI have raised the possibility that many cognitive tasks, including non-routing tasks, may be automated relatively soon (e.g. Korinek, 2023). However, ongoing advances in robotics make it likely that non-routine manual jobs will be similarly aected to cognitive tasks by the recent wave of progress in foundation models (Ahn et al., 2022). For our purposes here, we found it useful to consider labor that diers in uni-dimensional but continuous manner. We assume that workers dier in an exogenous parameter that we label skill J, which reects the maximum level of task complexity that the worker can perform. Workers' skill levels are described by the distribution function Υ(J). For analytical simplicity, we assume that Φ(I) ≥ Υ(I).7 For a level of the automation index I, a fraction Υ(I) of workers are perfectly substitutable by machines and earn wage wj = A. A fraction 1 − Υ(I) is not substitutable, but given that the remaining workers are suciently skilled, they are all eective substitutes for each other and earn wage wj = FL (K + Υ(I), 1 − Υ(I)). In contrast to our baseline model, this captures the concern that automation may make workers on the lower rungs of the skill distribution redundant, whereas workers who are able to perform at higher levels of skill may benet from automation. In the long run, assuming less than full automation (Φ(I) < 1 for any 7This assumption implies that for any level of automation I, there are suciently many skilled workers at all unautomated complexity levels left so that we can treat unautomated workers as perfect substitutes for each other. nite I), the share of workers who are gainfully employed will decline over time and will asymptote to 1 − limJ→∞ Υ(J), i.e., only workers who can perform unautomated tasks with arbitrary computational complexity will earn higher returns than capital. If Υ(J) = 1 for nite J, then the role of all human labor will lose its scarcity value in the same manner as in the AGI scenarios in our baseline model. Conversely, if Υ(J) asymptotes to 1, then there may be evergrowing inequality among workers: an ever-declining fraction of workers at the top may see incomes rise without bounds, whereas a fraction of the population that asymptotes towards one will see wages collapse to the level that equates the return on capital A. Heterogeneity in both skill and productivity The described setup could easily be extended to include heterogeneity in individual worker productivity in addition to heterogeneity in skill. Assume that workers not only have dierent skill levels Jj but are also endowed with dierent eciency units of labor Lj per time period. This may capture, for example, that there may be two economists who can both write papers up to complexity J, but one of them is twice as fast at it than the other. This could explain the empirial observation that workers in the same occupation sometimes earn signicantly dierent wages. Complementary human capital An alternative lens that may be relevant in the current era of cognitive automation is that workers possess dierent levels of human capital that is aected by automation. To keep our discussion simple, assume again that each worker j is characterized by a skill level Jj as well as an exogenous amount of human capital Hj > 0, which enables them to supply Lj = Hj eciency units of labor per time period. As the automation index I surpasses a given worker with skill level Jj , the human capital that they possessed is fully devalued. The loss is greater and more painful for workers with more human capital.

4.5 Compute as Specic Capital

An important feature of the ongoing AI take-o is the scarcity of compute. In our baseline model, we followed the standard neoclassical practice of modeling capital as uniform, capable of being deployed in the production of any task. As illustrated in Figure 3, automation of new tasks then implies that the existing capital stock can be smoothly allocated to a larger number of tasks, unlocking immediate productivity gains. In practice, however, many types of capital are specic to the task for which they were created and dicult or impossible to reallocate, corresponding to what the literature has traditionally called putty-clay capital. In the current context, the most salient type of specic capital on which AI systems rely is compute, which is in very limited supply, slowing down the deployment of AI systems 8The notion was rst introduced by Leif Johansen (1959) once putty has been turned into clay, it cannot be turned into another shape and expanded by Solow (1962) and others. Figure 12: Specic capital and factor prices for new tasks. Another example of specic capital is organizational capital, including the capital derived from investments into developing new processes for deploying new technologies in rms. We expand our framework by assuming that each unit of capital investment is specic to a task i and can only be invested once the task is automated, i.e., once I ≥ i. This leaves the task production function (2) unaected but modies the capital accumulation constraint: instead of a single law of motion for capital (13), the consumer needs to separately keep track of each type of capital k (i) since capital that is deployed for one task cannot be redeployed later. In an economy in which automation is proceeding slowly and steadily, the consumer problem is unchanged as the resulting constraints on capital redeployment are slackevery instant of time, a density ϕ(It) of new tasks is automated, and sucient capital for those tasks is instantaneously accumulated. By contrast, if the economy experiences a bout of progress that leads to a discrete mass of tasks suddenly being amenable to automation, the accumulation of the relevant specic capital may lag behind. The rapid rise of LLMs at the time of writing may be an example of such a bout. To illustrate this analytically, assume that a discrete mass of tasks ∆t = Φ(It) − Φ(It− ) > 0 is automated at time t, and let us interpret the specic capital k(It) required for these tasks as compute. At time t, no compute has been accumulated yet, k(It) = 0, so all type-It tasks are performed by humans at wage w, even though they could technically be automated. Figure 12 illustrates the factor returns as a function of the accumulation of compute k(It) while holding the inputs of labor and other capital constant: at rst, k(It) = 0, and the economy starts out at the left side of the gure where labor and compute are, at the margin, perfect substitutes so the returns on the two are equated. The rental rate on traditional capital is comparatively low. Over time, compute capital k(It) is accumulated and progressively substitutes for labor. As long as compute remains below the rst threshold k(It) < k1, illustrated by the rst vertical line in the gure, labor and compute remain perfect substitutes at the margin, but wages w decline with the addition of more compute, whereas the return on traditional capital rises. Within this region, all capital investment goes into compute. Once sucient compute (k1) is accumulated so that all humans are replaced from type-It tasks, all labor is allocated to the remaining unautomated tasks with i > It, and the marginal product of compute decouples from wages. All capital investment continues to be devoted to compute; wages wt and the returns to traditional capital rise whereas the return on compute k(It) declines sharply until it reaches the marginal product of all other types of capital. This is the middle region between k1 and k2 where only the return on compute, captured by the dotted curve, is decreasing. Once the second threshold is passed, the marginal product of compute and other capital is equated. Any additional capital investment is spread proportionately across all types of specic capital k(i), i ≤ I, and leads to a decline in the return on capital. In summary, the race between automation and capital accumulation leads to a non-monotonic response of wages, depicted by the blue curve marked with dots, and the returns to traditional capital, depicted by the red curve marked with squares. The following proposition characterizes the thresholds for the amount of specic capital and summarizes the non-monotonic response of factor prices to the accumulation of this specic capital analytically. Proposition 13 (Specic capital and factor returns). Suppose that the current amount of the specic capital is given by k(It). There are threshold values k1 and k2 > k1 such that (i) if k(It) < k1 then the wage decreases and the rental rate of the traditional capital increases with k(It), (ii) if k1 ≤ k(It) < k2 then both the wage and the rental rate of traditional capital increase with k(It), and (iii) if k(It) ≥ k2 then specic capital k(It) is only accumulated alongside traditional capital, and the wage increases with capital accumulation. Proof. See appendix. In summary, rapid advances in automation may lead to episodes in which certain types of specic capital (like compute) may exhibit very high returns, but since capital is reproducible, the resulting accumulation of specic capital will ultimately dissipate the excess returns. The implication is that after an adjustment period, specic capital for newly automated processes will be just another form of capital earning the market rate of return. 5 Conclusions This paper models the economic impact of the transition torwards articial general intelligence on output and wages. We develop a compute-centric framework that represents work as consisting of tasks that vary in their computational complexity and study how exponential growth in computing power will aect automation and the advent of articial general intelligence (AGI). The paper illuminates how dierent plausible assumptions about the complexity distribution of tasks across "compute space" translate into dramatically dierent scenarios for economic outcomes. If the task distribution has an innite Pareto tail, reecting unlimited complexity of human work, then the we show that wages can rise indenitely if the tail is suciently thick, as capital accumulation automates ever more complex tasks but there always remains enough for human labor. However, if the Pareto tail is too thin, then automation ultimately outpaces capital accumulation and causes a collapse in wages. Moreover, if the complexity of tasks humans can perform is bounded, mirroring computational limits on human cognition, then we demonstrate that wages would at rst surge as machines displace more and more human labor, but would eventually collapse, even before full AGI is reached. Beyond these scenarios, the paper provides several powerful general insights. Using the economy's factor price frontier, we show that the eects of automation follow an inverse U-shape, rst increasing wages by utilizing abundant capital but eventually decreasing wages due to labor displacement. We show that sufcient capital accumulation is essential to prevent automation from depressing wages. Adding xed factors like land causes wages to eventually decline. Yet automating innovation itself can restart wage growth after an initial automationdriven collapse. The novel compute-centric approach opens up a new perspective for analyzing the economic impact of articial intelligence. Interesting next steps include incorporating labor and capital adjustment costs, modeling endogenous innovation, analyzing distributional impacts more fully, studying macroeconomic dynamics and policies, and evaluating the possibility of an intelligence explosion with AGI. By presenting several rigorous scenarios for how the transition to AGI may unfold, we hope that this paper will make an important contribution to enabling economists, policymakers and the public to examine alternative futures and to prepare for the technological transformations on the horizon.

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After some algebra, the growth rate of capital is ˙Kt Kt = stA K σ−1 σ t Φ 1 σ t + L σ−1 σ (1 − Φt) 1 σ σ σ−1 Kt − δ = stA Φ 1 σ t + Kt− σ−1 σ L σ−1 σ (1 − Φt) 1 σ σ σ−1 − δ = stA 1 + L σ−1 σ Kt σ−1 σ Φt 1 − Φt 1 σ σ σ−1 · Φ 1 σ−1 t − δ Note that the growth rate of capital stock depends on the behavior of Ωt ≡ Kt σ−1 σ Φt 1−Φt 1 σ . In particular, Φt 1−Φt grows approximately at an exponential rate under the Pareto assumption because Φt 1 − Φt = 1 − I−λ t I−λ t Iλ t − 1 = Iλ 0 eλgt − 1 ≈ Iλ 0 eλgt We consider three cases and see whether each case is consistent with derivation above. First, suppose Ωt → ∞. Then capital stock must grow at a suciently low rate at least for large t. That is, the long-run growth rate of capital gK must satisfy gK < λg 1 − σ In this case, the growth rate of capital is ˙Kt Kt → sA 1 + 0 σ σ−1 · 1 − δ = s∞A − δ where s∞ is the long-run savings rate dened in Lemma 6. Then we have gK = s∞A − δ < λg 1−σ and thus the following upper bound on the long-run savings rate s∞ < 1 A λg 1 − σ + δ Secondly, consider the case where Ωt → 0. Then capital stock must grow at a rate such that gK > λg 1 − σ In this case, the growth rate of capital converges to a negative value ˙Kt Kt → sA · 0 · 1 − δ = −δ But this contradicts the lower bound on gK and puts an upper bound on the growth rate of capital stock. Lastly, suppose Ωt converges to a nonzero constant. Then it must be the case that gK = λg 1 − σ which requires Ωt → Ω where Ω is some constant satisfying ˙Kt Kt → s∞A 1 + L σ−1 σ Ω σ σ−1 −δ = λg 1−σ . In the three cases, capital stock grows asymptotically at either s∞A − δ or λg/(1 − σ). To characterize the long-run labor income share, note that the labor share is LSt = wtL Yt = A σ−1 σ (Yt/L) 1 σ (1 − Φt) 1 σ L Yt = A σ−1 σ Y 1 σ −1 t L1− 1 σ (1 − Φt) 1 σ = A σ−1 σ Y − σ−1 σ t L σ−1 σ (1 − Φt) 1 σ = A σ−1 σ A− σ−1 σ K σ−1 σ t Φ 1 σ t + L σ−1 σ (1 − Φt) 1 σ −1 L σ−1 σ (1 − Φt) 1 σ = K σ−1 σ t Φ 1 σ t + L σ−1 σ (1 − Φt) 1 σ −1 L σ−1 σ (1 − Φt) 1 σ = L σ−1 σ (1 − Φt) 1 σ K σ−1 σ t Φ 1 σ t + L σ−1 σ (1 − Φt) 1 σ = L σ−1 σ K σ−1 σ t Φt 1−Φt 1 σ + L σ−1 σ = L σ−1 σ Ωt + L σ−1 σ In the rst case, Ωt → 1 and so LSt → 0. In the second case, Ωt → 0 and so LSt → 1. Lastly, in the third case, Ωt → ¯K σ−1 σ ¯I 1 σ . Since s∞A 1 + L σ−1 σ Ω σ σ−1 − δ = λg 1−σ , it follows that s∞A 1 + L σ−1 σ Ω σ σ−1 − δ = λg 1 − σ 1 + L σ−1 σ Ω σ σ−1 = 1 s∞A λg 1 − σ + δ Ω + L σ−1 σ Ω σ σ−1 = 1 s∞A λg 1 − σ + δ Ω Ω + L σ−1 σ σ σ−1 = s∞A λg 1−σ + δ 1 − L σ−1 σ Ω + L σ−1 σ σ σ−1 = s∞A λg 1−σ + δ 1 − L σ−1 σ Ω + L σ−1 σ = s∞A λg 1−σ + δ σ−1 σ L σ−1 σ Ω + L σ−1 σ = 1 − s∞A λg 1−σ + δ σ−1 σ Therefore, LSt → 1 − s∞A λg 1−σ +δ σ−1 σ = 1 − (A−ρ−δ+ηδ)/η λg 1−σ +δ σ−1 σ . If the economy starts in region 1 then the economy stays in region 1 as long as ˙ˆIt ˆIt ≥ ˙It It That is, the threshold grows faster than the automation index. Under the Pareto assumption, the inequality is equivalent to ˙Kt Kt ≥ 1 + Kt/L Kt/L · λg where 1+Kt/L Kt/L converges to one from above. Thus, the above inequality delivers a lower bound on the savings rate for the economy to asymptotically stay in region 1: s∞A − δ > λg (24) The inequality ensures that capital accumulation is suciently fast compared to automation. If it is violated then It crosses ˆIt eventually and wages collapse to A. To further examine how capital accumulation and automation shape the asymptotic behavior of wages, consider the growth rate of wages ˙wt wt = 1 σ ˙Yt Yt − 1 σ ˙Φt 1 − Φt derived from wt = FL and log FL = log A + 1 σ log(Y /A) − 1 σ log L + 1 σ log(1 − Φ(I)) d log FL dt = 1 σ d log Y dt + 1 σ d log(1 − Φ(I)) dt = 1 σ d log Y dt + 1 σ 1 1 − Φ (− ˙Φ) . The above equation shows that the growth rate consists of output growth and the displacement eect of automation. Note that the growth rate of output is ˙Yt Yt = SK ˙Kt Kt + 1 1 − σ ˙Φt 1 − Φt SL − SK 1 − Φt Φt where SK ≡ K σ−1 σ t Φ 1 σ t K σ−1 σ t Φ 1 σ t +L σ−1 σ (1−Φt) 1 σ and SL ≡ L σ−1 σ (1−Φt) 1 σ K σ−1 σ t Φ 1 σ t +L σ−1 σ (1−Φt) 1 σ , omitting time subscripts for notational simplicity. The rst term is growth due to capital accumulation and the second term is growth due to the productivity eect of automation. The wage growth rate is then ˙wt wt = 1 σ SK ˙Kt Kt + 1 1 − σ ˙Φt 1 − Φt SL − SK 1 − Φt Φt − ˙Φt 1 − Φt (25) That is, wages rise as long as capital accumulation and the productivity eect dominate the displacement eect. In fact, this is another version of (18), which can be seen by setting ˙wt = 0, and tells us what determines the growth rate of wages. Under the Pareto assumption, we have ˙wt wt = 1 σ SK ˙Kt Kt + 1 1 − σ · λg · SL − SK 1 − Φt Φt − λg where λg is the rate of automation adjusted by the decay rate of the fraction of tasks for labor. Notice that SK = (∗) (∗) + L σ−1 σ If λg > (1 − σ)(s∞A − δ) (i.e. the rst case in the beginning of the proof) then SK → 1 since (∗) → ∞. Note that ˙wt wt = 1 σ SK ˙Kt Kt + 1 1 − σ · λg · SL − SK 1 − Φt Φt − λg As t → ∞, the growth rate of wages converges as follows ˙wt wt → 1 σ 1 · (s∞A − δ) + 1 1 − σ · λg · 0 − 1 · 0 − λg = 1 σ s∞A − δ − λg Thus, if s∞A−δ > λg (but s∞A−δ < λg/(1−σ)) then wages grow exponentially at an asymptotic rate 1 σ s∞A − δ − λg = 1 σ A−ρ−δ η − λg . In the case where capital stock asymptotically grows at λg/(1 − σ), SK converges to one. As a result, the growth rate of wages converges as follows ˙wt wt → 1 σ SK λg 1 − σ + 1 1 − σ · λg · 1 − SK − SK · 0 − λg = 1 σ SK λg 1 − σ + λg 1 − σ · (1 − SK ) − λg = 1 σ λg 1 − σ − λg = λg 1 − σ ∴ ˙wt wt → λg 1 − σ If s∞A − δ ≤ λg then wages decline until the automation index crosses the threshold and collapse to A, as (24) indicates. A.2 Proof of Proposition 13 Proof of Proposition 13. The production function can be written as Yt = A K σ−1 σ t Φ(It−) 1 σ + K(I) σ−1 σ t ∆ 1 σ t + L σ−1 σ (1 − Φ(It)) 1 σ σ σ−1 FK = A K σ−1 σ t Φ(It−) 1 σ + K(I) σ−1 σ t ∆ 1 σ t + L σ−1 σ (1 − Φ(It)) 1 σ 1 σ−1 K− 1 σ t Φ(It−) 1 σ FK(It) = A K σ−1 σ t Φ(It−) 1 σ + K(I) σ−1 σ t ∆ 1 σ t + L σ−1 σ (1 − Φ(It)) 1 σ 1 σ−1 K(I)− 1 σ t ∆ 1 σ t FL = A K σ−1 σ t Φ(It−) 1 σ + K(I) σ−1 σ t ∆ 1 σ t + L σ−1 σ (1 − Φ(It)) 1 σ 1 σ−1 L− 1 σ (1 − Φ(It)) 1 σ If FL < FK(It), then the specic capital and labor are perfectly substitutable. That is, K(It) L < ∆t 1 − Φ(It) k(It) = K(It) ∆t < L 1 − Φ(It) in which case, the production function can be written as Yt = A K σ−1 σ t Φ(It−) 1 σ + (K(It) + L) σ−1 σ (1 − Φ(It−)) 1 σ σ σ−1 If FK > FK(It), then the specic capital and the traditional capital are perfectly substitutable. That is, K(It) Kt > ∆t Φ(It−) k(It) = K(It) ∆t > Kt Φ(It−) in which case, the production function can be written as Yt = A (Kt + K(It)) σ−1 σ Φ(It) 1 σ + L σ−1 σ (1 − Φ(It)) 1 σ σ σ−1