Automation's Impact on Income and Wealth Inequality

Uneven Growth: Automation’s Impact on Income and Wealth Inequality Abstract Over the past forty years, economic growth in the United States has been unevenly distributed: income percentiles corresponding to the lower half of the distribution have stagnated while those at the top have sharply increased. At the same time, the aggregate labor share has fallen and wealth inequality has risen. We study technical change as a candidate cause of these trends. To this end, we develop a tractable theory that links technology to the personal income and wealth distributions, and not just the wage distribution as is commonly done in the existing literature. We use this theory to study the distributional effects of automation, defined as technical change that substitutes labor with capital. We isolate a new theoretical mechanism: automation may increase inequality via increasing returns to wealth. The flip side of this mechanism is that, relative to theories in which returns are unaffected, automation is more likely to lead to stagnant wages and therefore stagnant incomes at the bottom of the income distribution. We confront our model with the data and argue that automation can account for part of the observed trends in the distribution of wages, incomes and wealth as well as macroeconomic aggregates. Introduction Over the past forty years, economic growth in many advanced economies has been unevenly distributed. In the United States, while the aggregate economy has grown at roughly two percent per year, income percentiles corresponding to the lower half of the distribution have stagnated. At the same time, incomes at the 95th percentile have roughly doubled and top 1 percent incomes have roughly tripled.1 One potential driver of these trends that is often cited by pundits and policy makers alike is technical change, and in particular the automation of tasks performed by labor. A large literature in macro and labor economics has studied how technology and automation affect the distribution of labor incomes.2 But not all income is labor income and capital income is an important income source, particularly at the top of the distribution where incomes have increased the most. Existing theories therefore paint an incomplete picture of technology’s implications for overall income inequality.3 This shortcoming is particularly acute when it comes to automation, that is, technical change that substitutes labor with capital, thereby increasing the importance of capital in production. To understand how automation shapes inequality, we therefore need to study how the additional capital income is distributed and how it affects capital ownership – that is, wealth inequality. The objective of our paper is to develop a theory that links technology to the personal income and wealth distributions—and not just that of wages—and to use it to study the distributional effects of automation. Our framework provides a complete and integrated characterization of how technology and, more generally, changes in the economy’s production and market structure affect the personal distribution of income, wages, and capital ownership, as well as macroeconomic aggregates. As we explain in more detail below, we achieve this by adopting a perpetual youth structure with imperfect dynasties as in Blanchard (1985). The theory has two key features relative to the neoclassical growth model. First, the long-run supply of capital is less than perfectly elastic. Second, the theory generates a non-degenerate wealth distribution. It also features heterogeneity in skills and hence an interesting joint distribution of capital and labor income.4 Our main argument is that technology affects not only relative wages but also asset returns and this can have substantial distributional effects. This argument has two parts. 1 See for example Census Bureau (2015) and Piketty, Saez and Zucman (2018). See for example Katz and Murphy (1992), Krusell et al. (2000) and Autor, Katz and Kearney (2006) 3 Of course, many theories of technical change feature capital in production and therefore also feature capital income. However, these theories then typically assume that this capital is owned by a representative household, which implies either a degenerate or an indeterminate wealth distribution (with symmetric implications for the capital income distribution). 4 Importantly, ours is a theory of the personal income distribution and not just of the factor income distribution. The latter type of theory – for example “two class models” with capitalists and workers – cannot speak to a number of empirical regularities in developed countries, for example that individuals at the top of the labor income distribution typically also earn substantial capital incomes (and vice versa). 2 1 First, automation directly contributes to income inequality by increasing returns to wealth and the concentration of capital ownership. Second, relative to theories in which returns are unaffected, automation is also more likely to lead to stagnant wages and therefore stagnant incomes at the bottom of the income distribution (even in the long run). The key for understanding both parts of the argument is that automation increases the demand for capital relative to labor. Because the long-run supply of capital is upward-sloping, this demand shift permanently increases returns to wealth. Intuitively, equilibrium requires a higher return to wealth in order to incentivize individuals to accumulate and supply the increased amount of capital demanded in production. The first part, that automation directly increases wealth and income inequality, then follows because some individuals receive a higher return on their assets and grow their fortunes more rapidly. The second part, that wages are more likely to stagnate, follows because some of the productivity gains from automation accrue to owners of capital in the form of a higher return to their wealth. We deviate from a textbook neoclassical growth model in two simple ways. We let individuals differ in their skills so that technology affects relative wages. And we adopt a perpetual youth structure with imperfect dynasties, meaning that new cohorts are born with their labor income but inherit no assets. The perpetual youth structure gives us two features that are crucial to understand how automation affects inequality and macroeconomic aggregates. First, we obtain well-defined steady-state distributions for wealth and income. Second, unlike models that admit a representative household, our model generates a longrun supply of capital that is less than perfectly elastic. This second feature determines the equilibrium impact of automation on returns to wealth (prices) and the amount of capital used in production (quantities). Our model is deliberately simple and abstracts from labor income risk, a more realistic life-cycle and bequests, as well as return heterogeneity— important elements in quantitative theories of the wealth distribution.5 In exchange, we obtain analytical solutions for steady state aggregates and distributions of wages, incomes and wealth. Furthermore, the economy aggregates, and solving for its transitional dynamics is as easy as in the neoclassical growth model. We obtain two main analytical results that illuminate how automation affects inequality in our theory. First, the steady-state return to wealth exceeds the discount rate by a premium p × σ × αnet . Here p is the death rate, σ is the inverse of the intertemporal elasticity of substitution, and αnet is the net capital share—an object that rises with automation. Because this premium determines the equilibrium gap between the return to wealth and the discount rate it also determines the speed at which individuals accumulate wealth during their lives.6 Second, we show that conditional on wages, individuals’ total incomes follow 5 See for example Krusell and Smith (1998); Castañeda, Dı́az-Giménez and Rı́os-Rull (2003); Straub (2019); Hubmer, Krusell and Smith (2016). 6 The imperfect dynasties assumption ensures that, in steady state, individuals accumulate wealth during their lives even though the aggregate capital stock is constant. 2 an exact Pareto distribution in steady state. The scale of this distribution—a measure of its location—is determined by wages. The inverse of its tail parameter—a measure of its thickness—is given by the ratio of the individual accumulation rate and the death rate p. Our formula for the return premium implies that, in equilibrium, the inverse tail parameter exactly equals the net capital share αnet . Intuitively, automation affects income inequality via two channels. First, it affects wages, thereby shifting the scales of the conditional income distributions. Second, it increases the net capital share and returns to wealth, thereby resulting in a thicker top tail of these distributions. This second channel, that top tail inequality equals the net capital share, illustrates in a transparent fashion the new mechanism emphasized above that automation contributes to income inequality by permanently increasing returns to wealth and the concentration of capital ownership. It is important to note that this new channel differs from the common argument that a rise in the capital share leads to higher inequality because capital income is more unequally distribution than labor income (Meade, 1964; Piketty, 2014). In fact, we show that such compositional effects are small both relative to the data and to the changes in capital ownership generated by our model. The new channel also operates for other changes in the economy’s market structure that reduce the labor share, such as rising markups, but is absent for changes that leave it unaffected, such as skill-biased technical change affecting only relative wages. As just discussed, automation also affects inequality through wages. Our theory features the standard mechanism from a large existing literature that automation affects the relative wages of different skill types. Its more interesting predictions instead concern wage levels. In our theory, automation generates productivity gains but some of these productivity gains accrue to capital owners in the form of a higher return to their wealth. The higher cost of capital permanently limits the expansion of investment and output in response to this technological improvement. As a result, automation can lead to stagnant or falling real wages even in the long run, especially of workers whose skills are more susceptible to automation. This is in contrast to models that admit a representative household.7 In those models, the supply of capital is perfectly elastic, and automation leads to a substantial increase in capital accumulation and higher average wages in the long run (Acemoglu and Restrepo, 2018b; Caselli and Manning, 2018). Although our model is intentionally stylized, and some of the assumptions used are stark, we find the mechanisms underlying our results to be quite general. The two key ingredients behind our results are an upward-sloping long-run supply of capital—so that technology persistently affects asset returns—and a nexus between returns to wealth and 7 To be clear, our notion of “models that admit a representative household” allows for skill and wage heterogeneity. In particular, it includes models that make the common assumption that there are different skill types but these are all members of the same representative household, and models where Gorman aggregation theorems hold (see Theorem 5.2 in Acemoglu, 2009) 3 inequality. An upward-sloping capital supply seems to us a more natural and less extreme starting point than the perfectly-elastic capital supply of the neoclassical growth model and its relatives. It is also a feature of overlapping generations (OLG) models, models with a life-cycle component, and models with labor income risk and precautionary savings (as in Aiyagari-Bewley-Huggett models). A return-inequality nexus emerges naturally in models where the stochastic accumulation of wealth over time generates a skewed distribution of wealth. This includes models with stochastic bequest motives (Benhabib and Bisin, 2007), models with stochastic returns or discount rates (Krusell and Smith, 1998; Benhabib, Bisin and Zhu, 2011), or models with explosive growth and some stabilizing force, like the birth and death process with imperfect dynasties in our model (Wold and Whittle, 1957; Steindl, 1965; Jones, 2015).8 Also the argument of Piketty (2014, 2015) that top wealth inequality depends on “r − g” highlights precisely this return-inequality nexus. Summarizing, we view our model as the core of a more elaborate framework that alters some or all of the specific assumptions we made, but retains the two key features described above. After presenting the model and our main analytical results, we turn to a numerical evaluation of our model. Our objective here is not to conduct a quantitative exercise designed to judge the match of the model to the data. Instead, we view this exercise as a first step in exploring the range of implications of this class of models for aggregates and distributions under plausible parameterizations. In our numerical exercise, we study how the automation of routine jobs contributed to overall income inequality. To do so, we infer changes in automation by percentile of the wage distribution by computing its exposure to routine jobs, which the literature singles out as jobs that can be easily automated using computer software or other equipment (see Autor, Levy and Murnane, 2003; Autor, Katz and Kearney, 2006). The implicit assumption behind this particular application of our theory is that the automation of routine jobs since 1980 explains the declining share of labor in national income observed during this period. In line with the literature on wage polarization, we find that the automation of routine jobs explains about sixty percent of the observed changes in relative wages. More novel, we also find that the automation of routine jobs is able to generate a pattern of uneven growth reminiscent of that observed in the US over the last forty years. Two features combine to produce this pattern. First, our model generates a decline in real wages at the bottom and middle of the income distribution, which accounts for part of the income stagnation observed 8 See Benhabib and Bisin (2018) for a review of models capable of generating skewed wealth distributions. Among the models surveyed, only models where the tail of the wealth distribution is induced by the tail of the distribution of labor earnings (“models with skewed earnings”) lack a return inequality nexus. These include models with finite lives and no inheritances, and the simple versions of Aiyagari-Bewley-Huggett models (Stachurski and Toda, 2018). Furthermore, in Aiyagari-Bewley-Huggett models, automation may also affect incentives to engage in precautionary savings (by decreasing the importance of labor income relative to capital income, or by increasing the risk of unemployment as in Rachel, 2019), with implications for the rate at which individuals accumulate wealth. 4 at these percentiles. Second, our model generates a rising concentration of capital income at the top of the distribution, which accounts for the sharp rise of incomes at the top. Of course, we do not match the observed changes in income because we only focus on the role of automation—one of many technological and secular changes affecting the US economy. We also use our numerical exercise to contrast the predictions of our theory to those of a representative-household model which, for example, predicts that automation results in substantially larger expansions in investment and average wages. We conclude the paper by contrasting some of the predictions from our model with the data and concurrent trends. First, we discuss how to square our theory with the fact that risk-free interest rates are on a secular decline. Second, we discuss how markups—another factor reducing the share of labor in national income—fits into our story. Third, we use our framework to explore whether concurrent changes in capital taxation, demographics, and the falling price of investment goods—all of which contributed to the rising importance of capital in the economy—also increased inequality. Fourth, we discuss trends in capital accumulation and investment. Fifth, we discuss the historical and contemporary evidence linking the share of capital in national income to inequality and explain why one cannot make sense of this evidence by appealing to a compositional effect alone. Finally, we discuss whether our framework can account for the fast rise in inequality observed in the data and the importance of capital income in this trend. We view our paper as making contributions to two branches of literature. The first is the literature on routine jobs and automation. Like most theoretical papers in this literature, we use a task-based framework to model automation (Zeira, 1998; Acemoglu and Autor, 2011; Acemoglu and Restrepo, 2018b). Most of the papers in this literature focus on wage inequality (Autor, Levy and Murnane, 2003; Autor, Katz and Kearney, 2006; Acemoglu and Autor, 2011; Hémous and Olsen, 2018), or study the effect of automation on aggregates and wages using a representative household framework (Acemoglu and Restrepo, 2018b; Caselli and Manning, 2018). One exception is Sachs and Kotlikoff (2012), who study the possibility of immiserizing growth in an OLG model. We contribute to this literature by moving beyond representative-household models and exploring the implications of automation for inequality of total incomes across individuals. By doing so, we show that automation might contribute to rising incomes at the top and stagnant or declining wages at the bottom of the income distribution. Second, we contribute to the literature exploring the determinants of income inequality. Several papers explore this question quantitatively in general equilibrium models, including Castañeda, Dı́az-Giménez and Rı́os-Rull (2003); Kaymak and Poschke (2016); Hubmer, Krusell and Smith (2016); Straub (2019). Relative to these papers, our contribution is conceptual and lies in emphasizing how technical change might contribute to inequality by permanently increasing returns to wealth. In contrast, most of these papers explore the 5 effect of technology through wages. The mechanism generating a Pareto tail in our model is a common feature of random growth processes (see Gabaix, 2009, for a review). In fact, the birth-death process in our model is a simple and tractable example of a random growth process, and the idea that it leads to a Pareto distribution has been used before by many authors. Our main contribution to this literature is in providing an analytical characterization of income distributions in perpetual youth models, taking into account the effect of technology in general equilibrium. Methodologically, our model is close in spirit to models of the Gorman class (see Chatterjee, 1994; Caselli and Ventura, 2000). In these theories as well as ours, policy functions are linear, and aggregates do not depend on the wealth distribution (“macro matters for inequality, but inequality does not matter for macro”).9 Unlike in these models, our imperfect dynasties assumption ensures that we have a determinate wealth distribution in steady state and that our model does not admit a representative household. Section 1 lays out our theory of uneven growth. In Section 2 we take this model to the data with a simple calibration of changes in automation across the wage distribution, and in Section 3 we confront its predictions with observed trends in key macroeconomic and distributional variables. Section 4 concludes. 1 A Model of Uneven Growth The model is cast in continuous time. For expositional clarity, in the main text we outline the model in stationary form. Appendix B provides the exposition of the full model along the transition path. 1.1 Economic Environment Households. The economy is populated by individuals with different skills indexed by z, and we denote by `z the share of skill z. Individuals are born, age over time, die at Poisson rate p, and are then replaced by an offspring with the same skills. Individuals with a given skill z maximize the expected discounted stream of utility over 9 The linear policy functions in our theory deliver unrealistic consumption and saving behavior, for example marginal propensities to consume (MPCs) that are independent of household balance sheets. A large recent literature instead argues that models incorporating empirically realistic heterogeneity often deliver strikingly different aggregate implications than do representative agent models, precisely because aggregates depend on distribution. We view linear policy functions and other abstractions with unrealistic implications as costs worth paying in return for our theory’s analytical tractability. 6 their lifetime subject to a flow budget constraint and the natural debt limit: ˆ {cz (s),az (s)}s≥0 ∞ e−(%+p)s max 0 cz (s)1−σ ds 1−σ (1) s.t. ȧz (s) = wz + raz (s) − cz (s), and az (s) ≥ −wz /r where s is their age, az (s) denotes their asset holdings and cz (s) their consumption at age s, r is the interest rate and wz their wage income. Also, % is the pure rate of time preference, so that the effective discount rate is given by ρ := % + p. Our key assumption is that dynasties are imperfect; that is, not all wealth owned by individuals who die is passed to their offspring. In the main text, we operationalize this assumption in the simplest possible way by having newborns start their life with zero assets, az (0) = 0 and individuals consume all of their wealth upon death. This is a crude way of capturing the life cycle: people start their lives with little financial assets, accumulate assets over time, and towards the end of their lives, de-accumulate assets to finance their consumption. This is also the case in our model, but here individuals are born with zero financial assets and consume their remaining wealth instantaneously before death. Appendix B shows that this assumption can be derived from individual optimization in combination with an alternative assumption that we term “one day left to live”: at rate p individuals learn that they will die T time units from then on so that they optimally spend down their remaining wealth; we then consider the limiting case T → 0. The Appendix shows that also with this “one day left to live” assumption, saving and consumption decisions while alive solve (1).10 The assumption that dynasties are imperfect allows us to deviate in a tractable way from representative-household models, and it also ensures that the distribution of wealth is determinate. In contrast, when p = 0 or dynasties are perfect, the model admits a representative household and the wealth distribution is indeterminate (see Caselli and Ventura, 2000). Appendix D discusses other mechanisms that generate imperfect dynasties, including population growth (wealth dissipates as cohorts become larger), annuities (as in Blanchard, 1985), estate taxation, differences in altruism (as in Benhabib and Bisin, 2007), and retirement periods during which individuals consume some of their wealth (as in Castañeda, Dı́az-Giménez and Rı́os-Rull, 2003). Technology. Our description of the production process emphasizes the role of tasks as the fundamental unit of production. It also emphasizes that different tasks are completed by 10 As we show in Appendix B, the possibility of death has the standard effect of increasing the effective discount rate as in (1), but end-of-life consumption does not alter saving decisions in any other way. Intuitively, under the “one day left to live” assumption, individuals enjoy an infinite end-of-life consumption flow over an infinitesimal time interval T → 0. Because utility is strictly concave, the value of such end-of-life consumption converges to zero as T → 0. 7 workers of different skills, and that over time, some of the tasks supplied by workers of a given skill might become automated. Each skill type z works in a different sector that produces output Yz . The economy produces a final good Y using these sectoral outputs according to a Cobb-Douglas production function X Y γz = 1. Y =A Yzγz with z z Here, γz denotes the importance of the sectoral output produced by skill type z in production. The productivity shifter A captures the role of factor-neutral technological improvements. The production of sectoral output Yz involves the completion of a unit continuum of tasks u with a Cobb-Douglas aggregator: ˆ 1 ln Yz (u)du. ln Yz = 0 These tasks can be produced using capital and skill-z labor as follows: ( ψz `z (u) + kz (u) if u ∈ [0, αz ] Yz (u) = ψz `z (u) if u ∈ (αz , 1] The threshold αz summarizes the possibilities for the automation of tasks performed by workers of skill z. Tasks u ∈ [0, αz ] are technologically automated and can be produced by capital kz (u) or labor `z (u) at a unit cost of R and wz /ψz , respectively. The remaining tasks are not technologically automated and must be produced by labor. Total capital used in production is given by K, and its rental rate is given by R. Capital depreciates at a rate δ ≥ 0 and can be reallocated across tasks at no cost. We normalize total employment in the economy to 1, so that `z denotes the available supply of labor of skill z, with a wage given by wz . An increase in αz captures the development of automation technologies that expand the range of tasks in which capital is now able to substitute workers of type z. For example, workers engaged in white-collar office work devote their time to tasks such as accounting, keeping and locating records, and customer support. Workers engaged in blue-collar work devote their time to tasks such as welding, painting, assembling, machining, and supervision. Over time, technological improvements have allowed the automation of some of these tasks, while others, like customer support or supervision, remain the domain of workers. In the rest of the paper, we view A, γz , and αz as describing the state of technology, and explore the implications of exogenous shifts in αz for aggregates and inequality. 8 Equilibrium. Throughout, we assume competitive input and final good markets and define a steady state equilibrium of the economy as follows. Definition 1 The steady state equilibrium is given by aggregate output and capital, factor prices, a set of factor allocations, and consumption and saving policy functions such that: • capital and labor, {kz (u), `z (u)}z,u , are allocated in a cost minimizing way to produce output Y given factor prices {wz }z , R = r + δ. • policy functions az (s), cz (s) maximize individual utility given az (0) = 0, wz and r. • labor markets clear: ˆ 1 `z (u)du = `z for all z. 0 • the aggregate capital supplied by individuals and used by firms equals K: X ˆ ∞ X ˆ αz `z az (s)pe−ps ds. kz (u)du = K= z 1.2 0 z 0 Macroeconomic Aggregates To characterize the steady state equilibrium, we first study individuals’ savings and consumption decisions. Optimal asset accumulation by individuals determines the aggregate supply of capital in the economy. We then consider the firms’ problem of choosing the costminimizing mix of factor inputs, which determines the equilibrium demand for capital. We are able to derive a closed-form solution for both the supply of and demand for capital, and for the interest rate that equilibrates the capital market. Lemma 1 (Households policy functions) Suppose that r > (r − ρ)/σ. The solution to the individual problem is given by policy functions that are linear in total wealth, az (s)+wz /r: r−ρ wz  ȧz (s) = az (s) + , σ r   r−ρ  wz  cz (s) = r − az (s) + σ r (2) with az (0) = 0. Proof. Lemma A3 in Appendix B derives individual policy functions outside of the steady state. Lemma 1 is a special case of that more general formulation. The lemma shows that individuals consume a constant share of their effective wealth. As individuals age, their effective wealth grows at a rate equal to (r − ρ)/σ. This rate is identical for all individuals, irrespective of their skill and age. This makes the model 9 particularly tractable. But it also means that the model abstracts from realistic features, like differences in the propensity to consume and save out of total wealth. We now turn to the production side of the economy. To simplify the exposition, we make the following assumption for the rest of the paper: Assumption 1 (Full adoption of available automation technologies) wz >R ψz for all z. Assumption 1 ensures that in all tasks for which automation technologies are available, the cost of producing them with labor, wz /ψz , exceeds the cost of producing them with capital, R. As a result, all automation technologies will be adopted. Assumption 1 involves endogenous factor prices, so it needs to be verified in equilibrium. Lemma A1 in Appendix A shows that a sufficient condition for this assumption is that A is high, ensuring a high level of wages. The following lemma characterizes output and factor prices as functions of the stock of capital and technology. Lemma 2 (Equilibrium output and factor prices) Suppose Assumption 1 is satisfied. Then equilibrium output is Y = AK P z γz αz Y (ψz `z )γz (1−αz ) , (3) z where A is a constant that depends on parameters {A, αz , γz }. Factor prices are given by wz =(1 − αz ) γz Y, `z R =α where α is the average degree of automation in the economy: α := Y , K P z (4) αz γz . Proof. See Appendix A. Lemma 2 shows that aggregate output is given by a Cobb-Douglas production function. Factor shares are linked to the range of tasks performed by each factor, and the importance of these tasks in final output (the γz ’s). Automation changes the importance of labor in production. As tasks that were the domain of skill z get automated, this skill becomes less important in output (1 − αz declines) and capital gains importance (α rises).11 11 Our model abstracts from firms and industries, and so the decline of the labor share following improvements in automation manifests at the aggregate level. In practice, larger and growing firms might be more likely to deploy automation technologies, or firms already operating more automated production processes 10 Automation not only changes the distribution of income between capital and workers with certain skills, but also increases productivity by allowing the substitution of expensive labor for cheaper capital in more tasks. The contribution of increase in the αz ’s to changes in total factor productivity is given by   X wz d ln TFPα = γz ln dαz > 0, (5) ψz R z where ln  wz ψz R  > 0 denotes the percent reduction in costs obtained when a task performed by skill type z is automated.12 This formula states that the contribution of automation to TFP is a (weighted) sum of the cost-saving associated with automation at the micro-level. Lemma 1 showed how the return to wealth determines the rate at which individuals accumulate wealth and supply capital, and Lemma 2 characterized how technology shapes the demand for capital. We now characterize the equilibrium of the return to wealth and aggregates. We denote variables in the equilibrium steady state with an asterisk. Proposition 1 (Steady state characterization) The steady state is unique. The return to wealth, r∗ , is given by the solution to the equation 1 − ρ/r∗ α 1 = . ∗ ∗ pσ + ρ − r 1−αr +δ (6) In turn, r∗ determines the steady state level of the capital-output ratio (K/Y )∗ , output (Y ∗ ), and wages (wz∗ ) as the unique solution to equations (3) and (4). ∗ In steady state, the return to wealth and the net share of capital income, αnet , satisfy: ∗ r∗ = ρ + pσαnet . (7) Proof. See Appendix A for the full proof. Equation 6 shows how technology and demographics determine the equilibrium return to wealth in steady state. We can think of the return to wealth as determined by the supply of capital and the demand for capital. To elaborate this argument it is convenient to think in terms of units of capital relative to the total wage bill, so that we are working with the demand and supply of capital relative to labor. may expand at the expense of others as automation technologies improve. These dynamics are consistent with the evidence in Autor et al. (2017); Kehrig and Vincent (2018) on the role played by firms in driving the decline in the labor share, and with the theoretical model in Martinez (2019). Changes in αz in our model are meant to capture all these potential channels by which improvements in automation affect the aggregate production function. 12 The contribution of a technology to TFP is given by the resulting increase in output holding factor utilization constant. That is, d ln TFPα = d ln Y |K,`z . Given this definition, the total effect on output of automation is given by d ln Y = d ln TFPα + αd ln K, since the `z are fixed but K may adjust over time. 11 To derive the supply of capital, we use the characterization in Lemma 1. Let X denote the total effective wealth in the economy, where recall that effective wealth is the sum of financial wealth, az (s), and human wealth, wz /r. Integrating equation (2), we obtain r−ρ X − Effective wealth of individuals dying + Effective wealth of newborns. σ P Because of market clearing, X = K + w̄/r, where w̄ := z wz `z is the wage bill. Also, the effective wealth of individuals dying and newborns differ by pK, which is the amount of wealth consumed by individuals upon death and not passed as bequests. The difference of pK reflects the fact that we have imperfect dynasties: new generations are born with the same human capital as their elders, but with no financial wealth. These observations imply 0 = Ẋ = r−ρ w̄  0= K+ − pK, σ r which after rearrangement gives the (relative) supply of capital  K w̄ s = 1 − ρ/r . pσ + ρ − r (8) This curves gives an upward-sloping relationship between the return to wealth and the capital supplied by individuals. As the return to wealth rises, individuals increase their accumulation rate and supply more capital relative to their labor income. The upward-sloping line in Figure 1 illustrates this curve. Figure 1: Equilibrium determination of the return to wealth. On the other hand, from the production side, equation (4) shows that the demand for 12 capital (normalized by labor income) is given by  K w̄ d = 1 α . 1−αr+δ (9) This curves gives a downward-sloping relationship between the return to wealth and the capital demanded by firms (normalized by the wage bill). As the return that firms need to pay capital owners rises, they will use less capital relative to labor. Equation (6) gives the return r∗ at which the supply and demand for capital meet. The main implication of the supply and demand diagram in Figure 1 is that r∗ lies between ρ and ρ + pσ, which implies that the return to wealth has a premium above ρ. ∗ determines exactly where in this range r∗ lies. According to Equation (7) shows that αnet ∗ this equation, the premium of r∗ above ρ is given by pσαnet , which implies that in steady ∗ state, individuals accumulate wealth at a rate pαnet > 0.13 Note that aggregate wealth X is constant, but individuals accumulate wealth at a positive rate. This distinction arises due to the imperfect dynasties formulation, which requires individuals to start their lives with zero assets but eventually hold the capital stock of the economy as they age. Thus, in our model the individual rate of wealth accumulation will change with technology and fundamentals; whereas the aggregate rate in steady state will remain at zero. Figure 1 also illustrates the difference between the equilibrium in our model compared to a model with a representative household (our model with p = 0 or perfect dynasties). With a representative household, the supply of capital is perfectly elastic and the return to wealth is fixed at r = ρ. In such models, only the quantity of capital adjusts in the long run, ensuring that technology can have at most a short lived impact on asset returns. We view this as a knife edge prediction of representative-household models that our framework is able to relax. Despite the simplicity of our model, we find the logic behind the upward-sloping supply of capital and the result that the return to wealth has a premium above ρ linked to the net capital share to be quite general. The driving force behind this result is that the wealth of new cohorts is tightly linked to their human wealth—the net present discounted value of their wages— since younger cohorts hold less financial assets than older ones. A high return to wealth exceeding ρ is required to get entering cohorts to accumulate and supply a high amount of capital relative to their endowment of labor income. How high above ρ must The exact formula for r∗ in this equation follows from rearranging equation (8) and holds independently of the production structure of the economy. The reason why the return to wealth exceeds ρ is different from the known formula r = ρ + σg which holds along a balanced-growth path where technology grows at a rate g > 0. In that formula, the growth rate of the economy determines the return to wealth; whereas in ours, changes in factor shares affect the return to wealth even when the economy does not exhibit any growth. By using the fact that Ẋ = gX(t), we can generalize equation (7) to a setting with growth: r = ρ + σg + pσαnet . Here, it is still the case that the return to wealth would exhibit a premium above the return in a representative household economy, ρ + σg. 13 13 the return to wealth be? This will depend on the importance of capital relative to human wealth. In an economy where human wealth is less important than financial wealth (i.e., the net capital share is high), new-born individuals must accumulate assets rapidly and this requires a large premium of the interest rate above ρ. In an economy where human wealth is more important than financial wealth (i.e., the net capital share is low), new generations already own most of the wealth in the economy in the form of human wealth, and a slow individual accumulation rate is enough to get them to hold all financial assets. The results in Proposition 1 can be simplified when δ = 0. In this case, we obtain simple formulas for aggregates that we will use below to illustrate some of our findings. ∗ = α and Example 1 When δ = 0, we have that αnet r∗ = ρ + pσα, (K/Y )∗ = α . ρ + pσα (10) The following propositions explore the effect of automation on aggregates. Proposition 2 (The effects of automation on macroeconomic aggregates) The re∗ turn to wealth, r∗ , the individual accumulation rate (r∗ − ρ)/σ, the net capital share αnet , ∗ and the capital-output ratio, (K/Y ) , are all increasing functions of α. Output increases in all αz . Proof. See Appendix A. The proposition shows that the average extent of automation in the economy, α, increases the return to wealth, the individual accumulation rate, and the net share of capital. Moreover, these aggregates depend only on the average extent of automation, not the distribution of αz . This finding can be seen directly from Figure 2. An increase in α raises the demand for capital relative to labor. This demand shift increases the return to wealth and the individual accumulation rate, (r∗ − ρ)/σ, the capital-wage ratio, (K/w̄)∗ , and the net share of capital, which can be written in terms of the objects in Figure 1 as ∗ αnet = r∗ × (K/w̄)∗ /(r∗ × (K/w̄)∗ + 1). Though it cannot be seen from the figure, the proposition also shows that the capital-output ratio, (K/Y )∗ , expands as automation increases the relative demand for capital. These findings are intuitive. At a fundamental level, automation makes human wealth less important than financial wealth. As a result, new cohorts—who start life with their less valuable human wealth—need to accumulate assets at a higher rate so as to supply the capital required for production. A higher return to wealth is required to induce the rapid individual accumulation of assets—even though at the aggregate level, the expansion of capital could be small and will be limited by the higher user cost of capital, as shown in Figure 2. The result that returns to wealth and the individual accumulation rate rise with 14 Figure 2: Effect of automation—an increase in α—on the equilibrium return to wealth, r∗ . The red arrows illustrate how the equilibrium changes in our model and in a model that admits a representative household. automation is not unique to this model. As we discussed in the introduction, the same result applies in a broader class of models and in other models with a life cycle component and imperfect dynasties.14 For comparison, Figure 2 also depicts the effects of automation on returns and capital accumulation in the representative-household version of our model, that is, the special case with a death rate of p = 0. As in the neoclassical growth model, long-run capital supply is now infinitely elastic at r∗ = ρ. Compared to our model with an upward-sloping long-run capital supply, automation now results in a much larger expansion of investment and has no effect on the return to wealth. Considering again our model, does the higher demand for capital following an increase in automation primarily result in a higher return to capital or in an expansion of the capitalto-output ratio? Not surprisingly, given the comparison to the representative-household case in the preceding paragraph, the answer depends on the capital-supply elasticity, which is in turn linked to p. This can be illustrated in the simple case with δ = 0, so that the return to wealth and the capital-output ratio are given by the expressions in (10), both of which are increasing in average automation α. As these equations show, p determines the extent to which the higher demand for capital results in a higher return to capital (a high p generates a more inelastic capital supply) or in an expansion of the capital-to-output ratio (a low p generates a more elastic capital supply). 14 In models with uninsured labor income risk a-la Aiyagari, a related result applies. In these models, automation reduces the importance of labor income and hence of the precautionary motive for saving. As a result, the interest rate rises (see Proposition 9 in Auclert and Rognlie, 2018). However, the effect on individual accumulation rates is ambiguous, as returns increase but precautionary motives weaken. 15 Finally, the steady state effect of automation on output is given by d ln Y ∗ = 1 α d ln TFPα + d ln(K/Y )∗ > 0. 1−α 1−α (11) Output increases via two channels: because of automation increasing productivity (the first term) and the endogenous capital deepening resulting from the increase in capital accumulation by individuals.15 For high values of p, the expansion of output will be more modest as automation will lead to a small expansion of the capital-output ratio. The main takeaway of Proposition 2 is that, so long as p > 0 and dynasties are imperfect, automation will increase asset returns and the individual accumulation rate permanently. This has significant implications for output and the extent to which capital expands due to automation. As we show next, the same is true for wages and inequality. Proposition 3 (The long-run effects of automation on wages) • An increase in αz reduces the wage wz∗ relative to other wages wv∗ for v 6= z. • For a given change in the αz0 s, there exists a threshold p̄ > 0 such that, for p > p̄, the average wage w̄∗ falls; and for p < p̄, w̄∗ increases. Proof. See Appendix A. The effect of automation on relative wages is unambiguous and follows from the fact that wz = (1 − αz ) γ`zz Y (see also Hémous and Olsen, 2018; Acemoglu and Restrepo, 2018b). A more novel implication of the proposition is the possibility that automation may lead to stagnant wages for the average worker. Whether this is the case or not depends on p, which determines how inelastic the supply of capital is in steady state. Two complementary intuitions illustrate the importance of the capital supply in determining the behavior of the wage level. First, following a technological improvement that 15 This result contrast the findings by Kotlikoff and Sachs (2014) who study the effects of automation in a two period OLG model. In their setting, only young workers are able to save. By reducing young workers’ wages, automation may reduce the capital-output ratio in the long run, potentially causing a decline in output. 16 raises total factor productivity by d ln TFPα > 0, we have16 d ln TFPα = (1 − α)d ln w̄ + αd ln R, R=r+δ That is, productivity improvements accrue to capital owners in the form of a higher return to their wealth or to workers in the form of higher average wages. While this expression is very general and also holds outside of steady state and in a much broader class of models, consider now the steady state of our economy. When p > p̄, the supply of capital becomes so inelastic that all productivity gains from automation accrue to capital in the form of a higher return to wealth. When p < p̄, the supply of capital becomes so elastic that d ln R∗ ≈ 0 and most of the productivity gains from automation will accrue to labor—the inelastic factor.17 An alternative intuition comes from studying directly the behavior of the average wage. From Lemma 2, we have that w̄ = (1−α)Y . From (11) we know that automation results in an output expansion, with the magnitude depending on the productivity increase d ln TFPα and the expansion in capital supply d ln(K/Y )∗ . The effect of automation on the average wage is therefore determined by the relative strength of this output expansion and the displacement effect captured by the term 1 − α. With sufficiently inelastic capital supply (i.e., p > p̄), the displacement effect dominates. With sufficiently elastic capital supply (i.e., p < p̄), the output expansion dominates. 1.3 Wealth and Income Inequality We now study the wealth and income distribution. We take advantage of the fact that our model is block recursive: as Proposition 1 shows, the behavior of aggregates is independent of the wealth distribution. In what follows, we take the aggregates in steady state as given and study the resulting wealth distribution. Proposition 4 (Automation and the wealth and income distribution) Denote individuals’ effective wealth by xz (s) := az (s) + wz∗ /r∗ . The stationary distribution of effective 16 This result holds in general whenever aggregate output exhibits constant returns to scale and markets are competitive. Under these assumptions, we have Y = w̄L + RK where we now allow for movements in labor supply L to underline the argument’s generality so that w̄ denotes the average wage. Differentiating both sides of this identity, we get that, following a technological improvement, we have d ln Y = d ln TFPα + αd ln K + (1 − α)d ln L = (1 − α)d ln w̄ + αd ln R + αd ln K + (1 − α)d ln L, (12) where α = RK/Y . The expression in the main text follows by canceling the d ln K and d ln L terms. This derivation shows that the results in the proposition extend beyond our model: any type of technological change will raise wages provided that d ln R = 0. See Jaffe et al. (2019, ch.18/19) for a textbook treatment. 17 This part of the proposition is in line with papers that have studied the impact of automation in settings with a representative household or an infinitely elastic supply of capital, such as Simon (1965); Acemoglu and Restrepo (2018b); Caselli and Manning (2018). These settings correspond to the case with p = 0 in our model. 17 wealth for skill type z is  gz (x) = wz∗ r∗ ζ ζx −ζ−1 1 r∗ − ρ 1 ∗ = = αnet . ζ p σ , The conditional and unconditional wealth distributions satisfy  Pr(wealth ≥ a|z) = a + wz∗ /r∗ wz∗ /r∗ −1/α∗net , Pr(wealth ≥ a) = X  `z z a + wz∗ /r∗ wz∗ /r∗ −1/α∗net ; and the conditional and unconditional income distributions satisfy  Pr(income ≥ y|z) = max{y, wz∗ } wz∗ −1/α∗net , Pr(income ≥ y) = X  `z z max{y, wz∗ } wz∗ −1/α∗net ∗ In the special case with δ = 0, we have 1/ζ = α. In the general case with δ > 0, 1/ζ = αnet increases with the average extent of automation in the economy, α. Proof. See Appendix A. The proposition shows that the distribution of effective wealth for skill type z is Pareto ∗ . The driving force behind this result is with scale wz∗ /r∗ and tail parameter 1/ζ = αnet the nature of the process for the accumulation of effective wealth. People start life with effective wealth xz (0) = wz∗ /r∗ and scale it over time by growing it at a rate (r∗ − ρ)/σ. The distribution of wealth is stabilized by deaths that arrive at a rate p. Figure 3 describes the process of accumulation graphically. This special type of random growth process gives rise to a Pareto distribution (see Wold and Whittle, 1957; Steindl, 1965; Jones, 2015) with tail parameter 1 individual accumulation rate (r∗ − ρ)/σ = = . ζ death rate p As the formula shows, what matters for inequality is the ratio of the rate at which individuals accumulate wealth and the probability with which they die and consume all of it, p. The formula in the proposition follows from the observation that the steady state return to wealth ∗ is given by r∗ = ρ + pσαnet , which implies an individual accumulation rate of (r∗ − ρ)/σ = ∗ pαnet . The reason why we get a Pareto tail is that some individuals are lucky to live very long lives during which they manage to accumulate wealth exponentially. Instead of interpreting this mechanism literally, we see it as a metaphor for the fact that wealthy people tend to belong to dynasties that have accumulated wealth with no interruption for several generations. For this metaphor to apply, p should be re-interpreted as the probability that dynastic wealth 18 . Figure 3: Dynamics of effective wealth accumulation as individuals age. accumulation stops (say because of changes in the altruism of its members or other shocks) and the dynasty has to start accumulating wealth from scratch. This reinterpretation underscores the importance of bequests and dynastic wealth accumulation as drivers of wealth inequality at the very top. As recognized by several authors (see Castañeda, Dı́az-Giménez and Rı́os-Rull, 2003; Benhabib and Bisin, 2018), models with finite lives and no bequests have a hard time generating wealth distributions that are more skewed than earnings. A complementary view is that our model provides a tractable but unrealistic way of micro-founding a nexus between returns to wealth and inequality present in a broader class of models where the process of individual wealth accumulation results in skewed wealth distributions. This broader class of models includes models with stochastic bequest motives (see Benhabib and Bisin, 2007), models with stochastic returns or discount rates (see Krusell and Smith, 1998; Benhabib, Bisin and Zhu, 2011), or models with explosive growth and some stabilizing force, like the birth and death process with imperfect dynasties in our model (see Wold and Whittle, 1957; Jones, 2015). Despite differences in their details, in all these models a random growth process underlies the dynamics of wealth accumulation at the top of the wealth distribution, creating a natural nexus between return rates and inequality (see Gabaix et al., 2016). In particular, higher returns shift the rate at which people accumulate wealth, resulting in a larger mass of people populating the upper tail of the wealth distribution over time—those with uninterrupted accumulation at a higher than average rate. It follows that in this broader class of models, technological changes that lead to a higher return to wealth and more rapid individual wealth accumulation will generate a fatter tail in the wealth distribution, as is the case in our model. The distribution of effective wealth is important in an on itself because it tells us about 19 inequality in consumption and welfare (a corollary of Lemma 1). But our model also allows us to characterize the conditional and unconditional distributions of wealth and income, which is what we typically measure in the data. The remaining formulas in Proposition 4 provide an exact characterization of the wealth and income distributions. The formulas show that technology affects both distributions via wages—which determine the scale parameters— , but more novel via the net capital share—which determines the thickness of the tail of the wealth and income distribution. Intuitively, technologies like automation that increase the net capital share raise returns to wealth permanently. This allows some individuals to accumulate assets at a consistently higher return during long periods of time, generating more wealth inequality. All the models with a nexus between returns to wealth and inequality mentioned above share the feature that, by increasing returns to wealth, automation will increase wealth inequality. We now characterize the composition, sources of income, and income shares held by the top q income earners—the top q for short, so that the top 0.1 refers to individuals with the 10% highest incomes. We focus on the tail of the income distribution, where we are able to obtain a clear characterization. Proposition 5 (Composition and sources of income at top of income distribution) Let q̄ := Pr(income ≥ maxz wz ). For q < q̄, we have: • the probability that someone with a wage wz is in the top q is 1/α∗ `z wz net Pr(skill = z|top q) = P ; 1/α∗net ` w v v v • the share of labor income relative to total income held by the top q is P 1+1/α∗net E[labor income|top q] ∗ α∗net z `z wz = (1 − αnet )q P 1+α∗net ; E[income|top q] 1/α∗net z `z wz • the share of national income held by the top q is ∗ S(q) = Λq 1−αnet , (13) where Λ is a constant that depends on the wage distribution. Proof. See Appendix A. The proposition characterizes the skill composition of top income earners and their sources of income. The first part of the proposition shows that the share of individuals 20 of skill z among top earners depends on their wage wz relative to other wages wv . Technology might increase the share of high skill workers among top earners if it is skill biased (that is, if it increases the relative wage of high wage earners). In particular, the automation of tasks performed by workers in the middle and bottom of the skill distribution will increase the share of high skill workers among top earners. On the other hand, an increase in au∗ tomation (captured here by an increase in the net share of capital αnet ) makes relative wages less important in determining the composition of top income earners. Intuitively, many more individuals that managed to accumulate assets for long periods at the higher rate brought by automation will be among top earners independently of their wage. The second part of the proposition shows that, as we move up in the income distribution, individuals derive more of their income from capital ownership. This reflects the fact that the tail of the income distribution is increasingly made of successful investors for whom labor income represents a small part of their earnings. The final part of the proposition shows that the share of national income held by the top ∗ q is increasing in αnet , and therefore rises with automation. The formula here follows from the fact that for all levels of income above maxv {wv∗ }, all conditional income distributions have an exact Pareto tail whose thickness depends on the net capital share. The constant Λ adjusts for the different scales of the Pareto distributions below maxv {wv∗ }. When there is no heterogeneity in wages, Λ = 1 and we obtain the usual formula for the top q percent share in a Pareto distribution (see Jones, 2015). One implication of this formula is that technology might affect S(q) through wages (via the Λ term) but this would cause a proportional increase ∗ in S(q) for all q < q̄. Instead, by raising αnet , technology will increase the share of income held by higher percentiles disproportionately. 1.4 The Link Between Net Capital Share and Inequality Propositions 4 and 5 establish a link from the net capital share to income inequality. In this subsection we distinguish our mechanism from previous arguments emphasizing the importance of net capital shares for inequality. Starting with Meade (1964) and more recently with Piketty (2014), several authors have emphasized that a rise in the net capital share might generate inequality via a compositional effect. The argument is that because capital is more unequally distributed than labor income, a rise in the relative importance of capital would generate more income inequality. This argument differs from ours, since we emphasize how technology might increase wealth inequality and wage inequality directly, with major implications for income inequality. We can use our model to illustrate the differences between compositional effects and our mechanism. As above, denote by S(q) the top q percent income share. Also denote by S̃ k (q) and S̃ ` (q) the share of aggregate capital income and labor income earned by the top 21 q percent of the distribution of total income. It follows that18 S(q) = αnet × S̃ k (q) + (1 − αnet ) × S̃ ` (q). This simple formula is precisely the one derived by Meade (1964, p.34) and we can use it to decompose changes over time in S(q) as dS(q) = (S̃ k (q) − S̃ ` (q)) × dαnet + αnet × dS̃ k (q) + (1 − αnet ) × dS̃ ` (q) . | {z Compositional effect at q } | {z Changes in within factor distribution at q (14) } This decomposition highlights two shortcomings of theories emphasizing compositional effects. First, the difference between the share of capital income and labor income held by the top q, S̃ k (q) − S̃ ` (q) above, is not large enough to generate a substantial rise in income inequality. In the US in 1980, for the top 1%, roughly S(q) = 10%, S̃ ` (q) = 5% and S̃ k (q) = 20% so that a large increase in the net capital share of ten percentage points would yield an increase in the top 1% income share of only (S̃ k (q)− S̃ ` (q))×dαnet = 0.15×0.1 = 1.5 percentage points, or a proportional increase of 15%.19 As we discuss below, this number is small when compared to the effects in our model and in the data. Second, the emphasis on compositional effects misses the possibility that technology might have sizable effects on wage inequality—the term dS̃ ` (q)—and contribute to a more concentrated ownership of capital—the term dS̃ k (q). Our mechanism amends these shortcomings. First, in our model technology will have sizable effects on inequality, especially at the very top of the income distribution. Equation ∗ (13) implies a log-linear relation between S(q) and αnet of the form ∗ ln S(q) = ln Λ − ln(1/q) + ln(1/q) × αnet , (15) where ln(1/q) > 0 (q ∈ (0, 1] is in percent terms). Our model predicts that a 1 percentage point increase in the net capital share should raise the share of income earned by the top 0.1 percent by about 6.9%(= ln(1000); the share of income earned by the top 1 percent by about 4.6%(= ln(100); and the share of income earned by the top 10 percent by about 2.3%(= 18 Denote by y(q) the income of individuals at the top q percentile (the top q quantile), by y` (q) their labor income and by yk (q) their capital income. Further denote the corresponding aggregates by Y := ´1 ´1 ´1 y(q)dq, Yk = 0 yk (q)dq and Y` = 0 y` (q)dq. We have y(q) = yk (q) + y` (q) and Y = Yk + Y` and hence 0 y(q)/Y ´= αnet × yk (q)/Yk + (1 − αnet ) × y` (q)/Y` with αnet := Yk /Y . Hence the top q percent income share q S(q) = 0 y(v)dv/Y satisfies the equation above. 19 These are only rough magnitudes to illustrate the quantitative power of this composition effect. We conduct a precise calculations of this kind in Section 3. Meade (1964, Table 2.2) performs similar calculations but obtains much larger effects because he assumes that, for the top 1%, S̃ ` (q) = 6% and S̃ ( q) = 47%, which he defends as appropriate numbers for the United Kingdom in 1959. With these numbers, the compositional effect is given by (S̃ k (p) − S̃ ` (p)) × dαnet = 0.41 × 0.1 = 4.1 percentage points. 22 ln(10)). The effects for the top 1 percent share are three times larger than compositional effects. Second, our mechanism fully accounts for the effect of technology on the distribution of capital income and labor income. It is precisely because our model predicts that automation will generate a thicker tail of wealth that we get a sizable effect of changes in the capital share on income inequality. The following proposition shows that, in line with this discussion, at the top of the income distribution, changes in wealth inequality—the changes in the S̃ k (p) term—dominate compositional effects. Proposition 6 (Decomposing changes in inequality) Consider a change in the net cap∗ > 0 holding relative wages constant. As q → 0, the share of the total change ital share dαnet in S(q) explained by the composition effect converges to zero. Proof. See Appendix A The proposition implies that, following an increase in automation, income inequality rises due to a more concentrated ownership of capital at the top (and the usual changes in relative wages); not so much due to a compositional effect. This result can be illustrated when wz = w̄. Because there is no wage inequality, we have ∗ S(q) =q 1−αnet , S̃ ` (q) =q, S̃ k (q) =
1 ∗ ∗ q 1−αnet − (1 − αnet )q . ∗ αnet The compositional effect is then given by Compositional effect at q =
∗ 1 1−α∗net q − q dαnet > 0, ∗ αnet whereas the overall change in the share of income held by the top q percent is ∗ ∗ Total change at q = ln(1/q) × q 1−αnet dαnet > 0. The share of the total change in S(q) explained by the compositional effect is then given by ∗ 1 − q αnet , ∗ αnet ln(1/q) which converges to zero as q → 0. 23