The Transition of Aggregates and Distributions

1.5 The Transition of Aggregates and Distributions Appendix B presents the full description of the model outside of steady state. The following proposition characterizes the transitional dynamics for the macroeconomic aggregates and the distribution of effective wealth. As for the steady state equilibrium, the transitional dynamics are block recursive: we can first characterize the behavior of macroeconomic aggregates and then use them to trace the evolution of the wealth distribution. Proposition 7 (Transitional dynamics) The behavior of the macroeconomic aggregates, C and K is given by the unique stable solution to the system of differential equations 1 Ċ = (r − ρ)(C − pK) − µpK + pK̇ σ K̇ =Y − δK − C, µ̇ 1 =µ − r + (r − ρ) µ σ where µ denotes the rate at which individuals consume their effective wealth (to simplify notation, we removed the time dependence of aggregates). Also, recall that Y is given by equation (3) and r is given by equation (4). Along the transition path, individuals accumulate effective wealth at a rate r(t) − µ(t), which implies that the distribution of effective wealth for individuals with skill z, gz (x, t) evolves according to the Kolgomorov Forward Equation ∂ ∂gz (x, t) = − [(r(t) − µ(t))xgz (x, t)] − pgz (x, t) + pδ̃(x − hz (t)) (16) ∂t ∂x ´ ∞ ´s where δ̃(·) is the Dirac delta function, and hz (t) = t e− t r(τ )dτ wz (s)ds is a time-varying reinjection point. Proof. See Appendix B. The proposition shows that the transitional dynamics for aggregates are no more complicated than those in the usual representative-household model. The main difference is that we need to keep track of the extra variable µ(t), which controls the common marginal propensity to consume out of effective wealth. Also, the Euler equation has some extra terms to account for the difference in consumption between new cohorts and the cohorts they replace (the term µ(t)pK(t)), as well as the consumption of the dying (the terms pK(t) and pK̇(t)). The fact that technology contributes to wealth inequality by permanently raising the return to wealth is the main result of our model. We can use the result in Proposition 7 to explain how this mechanism plays out over time and contrast our findings to an economy that admits a representative household. Suppose the initial distribution of effective wealth conditional on skills is given by  Pr(xz0 > x) = x ∗ wz0 /r0∗ −ζ0 , (17) ∗ as in Proposition 4. Here, xz0 is a random variable denoting the effective where 1/ζ0 = αnet0 wealth of individuals with skill z. First consider our model with an upward-sloping long-run supply of capital. Following an increase in automation at time t = t0 , individuals with skills z and effective wealth xz0 see a revaluation of their human wealth of ∆z (this could be negative for individuals experiencing a real decline in wages over time). This implies that the distribution of xz , denoted by gz (x, t), starts from  gz (x, t0 ) = ∗ wz0 r0∗  ζ0 ζ0 (x − ∆z )−ζ0 −1 for x ≥ ∗ wz0 + ∆z . r0∗ and from there on evolves according to the Kolgomorov Forward Equation (16). The first term in equation (16) captures the rate at which individuals accumulate assets. This rate equals r(t) − µ(t) and converges to (r∗ − ρ)/σ, where r∗ > r0∗ since automation increases the return to wealth permanently. The remaining terms capture the birth and death process. People die with probability p and are replaced by their offspring, who start life with an effective wealth equal to x0z (t). The birth and death process ensures that the distribution gz (x, t) converges to a new Pareto distribution which is independent of the starting one. This discussion implies that technology affects the effective wealth distribution and its evolution in two ways. First, technology will influence the effective wealth distribution via wages, which determine the revaluation effects in the short run, and the reinjection points in the long run. Second, technologies that make capital more important in production will influence the effective wealth distribution by permanently increasing the return to wealth, which causes individuals to accumulate wealth more rapidly during their lives, and generates a Pareto distribution with a thicker tail.20 These dynamics can be contrasted to what would happen in a model with individuals who differ in their skills but that admits a representative household, as in Caselli and Ventura (2000). The dynamics of aggregates and the distribution of effective wealth in this class of models corresponds to the special case with p = 0 in Proposition 7 (we will use a superscript h to distinguish the value of aggregates in this case). The wealth distribution is now 20 Other forms of skill-biased technical change that do not involve changes in the share of capital will have a different effect on the income distribution. Such changes will generate a revaluation effect in the short run and affect the steady-state distribution of income only through the reinjection points. As a result, these other forms of skill-biased technical change will affect the scale parameters of the income distribution, but wont generate a thicker tail by raising the returns to wealth. indeterminate, in the sense that any distribution is consistent with equilibrium in steady state. Despite the indeterminacy, starting from a given initial distribution of wealth and wages, the transitional dynamics of the wealth distribution are uniquely defined. To make things comparable, assume that the initial distribution of effective wealth is given by (17), and coincides with that in our model. Following an increase in automation at time t = t0 , individuals with skills z and effective wealth xz0 see a revaluation of their human wealth of ∆hz (this will differ from the revaluation in our model since wages behave differently in an economy that admits a representative household—see Proposition 3). People then accumulate assets starting from xz0 + ∆z at a common rate rh (t) − µh (t), which is temporarily above zero but converges to zero over time (recall that in the representative-household model, rh (t) and µh (t) converge to ρ, reflecting the fact that the supply of capital is fully elastic). This temporary period of accumulation scales everyones’ effective wealth by the same amount, M , but does not contribute to thicker tails in effective wealth. The resulting distribution of effective wealth is given by  Pr(xz > x) = x/M − ∆hz ∗ wz0 /r0∗ −ζ0 , ∗ for x ≥ M · (∆hz + wz0 /r0∗ ). This is a shifted Pareto distribution, with the shifts explained by the changes in wages. Unlike in our model, the new steady state distribution has the same tail parameter as the initial distribution. To summarize, in an economy that admits a representative household, technology only affects inequality through wages, which determine the revaluation effects ∆hz . The temporary increase in return rates scales everyone’s wealth, but does not contribute to inequality. 2 2.1 Model Meets Data Calibration As discussed in the introduction, we feed changes in αz to the model and explore the consequences of this particular type of technological change for aggregates and inequality in wages and wealth. We will focus on changes in automation between 1980 and 2014, a period with a marked shift in technology towards automation (Autor and Salomons, 2018; Acemoglu and Restrepo, 2019), especially of tasks performed by workers in routine jobs, both in manufacturing and in services (Autor, Levy and Murnane, 2003; Acemoglu and Autor, 2011).21 21 Our focus on this period does not imply that there was no automation before then. As discussed in Acemoglu and Restrepo (2019), before 1980 jobs were automated in some specific industries and tasks, but automation was counteracted by other technological improvements that raised labor shares in other industries To bring the model to the data, we interpret z as indexing the group of workers in a given percentile of the wage distribution, so that we have 100 skill groups. The main ingredient in our calibration is a measure of how automated the tasks being performed by workers in each percentile of the wage distribution have been over time, αz (t). In what follows, we will use a time argument to indicate which variables change over time. We assume that changes in αz (t) are driven by the automation of routine jobs, and that all routine jobs have been automated at the same rate over time. To operationalize this assumption, we posit the rule of motion for αz (t) (see Appendix C for a derivation of this equation): 1 1 − = ωzR 1 − αz (t) 1 − α0  1 1 − 1 − α(t) 1 − α0  . (18) Here, α0 denotes the extent to which non-routine tasks are automated (assumed invariant over time), and ωzR denotes the share of labor income derived by workers in percentile z from routine jobs relative to the labor income derived by all workers from routine jobs—a measure of the comparative advantage held by these workers in routine jobs. Equation 18 implies that groups of workers who specialize in routine jobs have a bigger share of the tasks they performed being automated over time. The implicit assumption here is that the observed decline in the labor share since 1980 is driven by the automation of routine jobs. Figure 4: Calibrated αz by wage percentile in 1980 and the new steady state (left panel), and the implied behavior of the aggregate labor share compared to the data (right panel). Using equation (18), we measure αz (t) for all percentiles of the wage distribution by computing their ωzR using the 2000 Census—a point in the middle of the period we study.22 or introduced new labor-intensive roles for labor in production. As a result, the labor share—the key object determining how technology affects wealth inequality—remained stable during this period. Technological change might have contributed to rising wage inequality before 1980, but our mechanism did not contribute to rising wealth inequality back then. 22 In our model, the composition of a skill group is assumed invariant. However, in the data, the composition of workers in a given wage percentile might change over time, as the relative ranking of groups of workers with different characteristics changes. In our baseline calibration, we used the 2000 values for ωzR as describing We normalize αz (1980) to be equal across all z (this pins down α0 = α(1980)), and pick the level of α(1980) and α(2014) to match the (gross) capital share in these years (0.345 and 0.42, respectively, in the BLS series for the non-financial corporate sector). Finally, we assume a linear increase in αz (t) from its value in 1980 to its final value in 2014. This procedure results in the change between 1980 and 2014 in αz (t) plotted in the left panel of Figure 4. The average change in αz (t) (weighted by γz ) is of 8.4 percentage points, which roughly matches the observed decline in the labor share during our period of analysis (8.3 p.p decline).23 The right panel of Figure 4 plots the implied behavior of the labor share P given the change in αz (t) over time ( 1 − α(t) = γz (1 − αz (t)) in our model) and the BLS series for the labor share in the corporate non-financial sector. Turning to the remaining parameters, we calibrate γz to match the wage distribution in 1980 (obtained from the 1980 Census). Note that the γz ’s might have changed over time as a result of other forms of skill biased technical change not modeled here, but we do not explore this possibility. We pick ψz to ensure that human labor is 30% more costly than using capital in the production of automated tasks. This number is in line with studies exploring the costsaving gains from using industrial robots in manufacturing (see Acemoglu and Restrepo, 2019). Because it is not clear that one can extrapolate from these studies, in Appendix C we present results assuming cost saving gains of 15% and 45%. For our baseline value of ψz , the automation of routine jobs explains 15% of the gains in productivity experienced by the US economy during the 1980-2014 period. We view this estimate as plausible given that automation is one of many technological improvements determining productivity during this period. The remaining parameters are standard and chosen to match aggregates in 1980. We take a capital output ratio of K/Y = 3, which implies a rental rate of capital R = 11.5%. We take a depreciation rate of 5% so that the net capital share equals 23% and the return to wealth equals r = 6.5%. We pick ρ = p = 3.85% to ensure a 6.5% return to wealth and target a long-run elasticity of capital supply d ln K/dr of about 50 (we get an elasticity of 51.28).24 We view this choice of p as conservative, in the sense that much of the evidence suggests a more inelastic supply of capital (see Appendix C). The factor-neutral technology term, A, is chosen to match output per hour in 1980. We do not feed changes in A to the model, which would capture other sources of technological improvement different from the level of specialization of different groups in routine jobs. We also experimented with measuring ωzR using the 1980 Census and obtained similar results. The reason is that ωzR is highly correlated over time (the correlation between the 1980 and 2000 measures is of 0.9714). 23 The average change in αz (t) in the model is slightly larger than the one observed in the data. The reason is that over time, we also have changes in γz that are not fed to the model. In particular, the observed changes in γz imply that other technological changes resulted in a reallocation of value added from highly automated skills to less automated ones at the top of the wage distribution. 24 As Appendix D shows, in a more general model, the role of p is replaced by p + n + δH . here n is the growth rate of population and δH the rate at which human capital depreciates as individuals age past their prime. Many plausible combinations of p, n, and δH yield a combined value of p + n + δH = 3.85%. Table 1: List of calibrated parameter values Description Value Target / Source Preferences σ Inverse IES % Pure rate of time preference (p.a.) p Death rate (p.a.) ρ Effective discount rate (p.a.) 3 0% 3.85% 3.85% Standard calibration Target r = 6.5% Target capital-supply elasticity d ln K/dr ≈ 50 p+% Technology A Hicks-neutral productivity term δ Depreciation rate (p.a.) Z Number of skill types `z Labor supply of each type α(1980) Capital share in 1980 α(2014) Capital share in 2014 α0 Capital share in non-routine tasks ωzR Routine jobs share in each pctile in 2000 γz Skill demand shifters in 1980 ψz Productivity of labor relative to capital 0.14 5% 100 1% 0.345 0.428 0.345 vector vector vector Y /L in 1980 Standard calibration Each type = percentile of wage distribution Each type = percentile of wage distribution BLS labor share in 1980 BLS labor share in 2014 Constant αz (1980) across z Acemoglu-Autor + 2000 Census/ACS Wage levels in 1980 Census/ACS Automation reduces costs by 30% (= ψwzzR ) Notes: The table provides the parameters used in our baseline calibration of the model. For details, see the main text and Appendix C. automation. Table 1 summarizes the parameters used in our exercise. 2.2 Numerical findings We now present the transitional dynamics and steady state implications of a change in αz over time. Because we are not feeding any other shocks to the model, the results here must be interpreted as counter-factual changes relative to the trend that the economy would have experienced in a world with not automation. Figure 5 presents the transitional dynamics for the labor share (summarizing the technological change fed to the model), output per worker, the net investment rate, the capitaloutput ratio, the return to wealth, and the average wage per hour. For comparison, we also plotted the transitional dynamics for the representative household case following the same shift in technology. In our model, automation leads to a modest expansion of output and the capital-output ratio of 11% and 15%, respectively. Despite the increase in the average output per hour of labor, mean wages go down by 4.5% in 2020 and by 2% in the long run. The productivity gains from automation accrue to capital owners in the form of a higher return to their wealth, which goes up by 0.9 percentage points in the long run and 1.3 p.p. by 2020.25 25 Equation (12) shows why a small increase in the return to wealth of roughly 1 percentage point can have a large effect on wages. Using this equation with α = 0.4—close to the midpoint of our calibration—shows that a small increase in R∗ of 1 percentage point or 10% in percent terms is enough to ensure that the 3% increase in TFP driven by automation results in a decline in real wages, w̄, of roughly 2%. Figure 5: Transitional dynamics for the main aggregates following an improvement in automation technologies in 1980. The solid line presents the transitional dynamics in the baseline calibration of our model. The dashed line presents the transitional dynamics when p = 0 and the model admits a representative household. This is in contrast to what would happen in the representative-household model, where automation leads to a more pronounced economic expansion propelled by a boom in investment, a temporary increase in the return to wealth, and higher average wages in the long run. The differences between these models underscore the importance of the capital-supply elasticity. Even though the supply of capital in our model is still fairly elastic, the response of macroeconomic aggregates to automation differs significantly from what one would get in a representative-household model. The average decline in wages of 0.5 dollars masks substantial heterogeneity. Figure 6 plots the change in wages by percentile from 1980 onwards. In line with the literature emphasizing how the automation of routine jobs affected workers in the middle of the wage distribution, our model generates a polarization of wages over time. The real wage of workers below the 80th percentile of the wage distribution declines over time, but the most pronounced effects are for workers at the 25th percentile of the wage distribution, whose real wages fall by Figure 6: Predicted change in wages by wage percentile (left axis) and observed change in wages by wage percentile (right axis). Observed wage changes computed using the 1980 Census and 2012-2016 ACS. See Appendix C for details. 10%. In contrast, the real wage of workers at the 95th percentile of the wage distribution rises by 5%. For comparison, the figure also plots on a different vertical axis the observed change in wages by percentile between 1980 and 2014 (using data from the US Census and the ACS). The model explains 60% of the observed variation in relative wages. Although other technologies not considered here also affected the level of wages and their relative change during this period, our model suggests that automation might have been a significant contributor to wage inequality and to anemic wage growth at the low end of the wage distribution. We now turn to the implications of our model for income inequality. Figure 7 presents the change in total income at each quantile of the income distribution. The right panel zooms to the top of the income distribution. The figure reveals substantial uneven growth. Below the 50th quantile, individuals experienced declines in total income of 5%. Between the 50th and 80th percentile, individuals experienced a modest income growth of 5%. This is in contrast to the top income quantile, which experienced an increase in income ranging from 30% (for the top 1%) to 50% (for the top 0.1%). Although other technologies not modeled here also shifted incomes, the figure shows that automation is capable of generating substantial rises of income at the very top of the distribution and declines or stagnant incomes at the bottom. Both wage and wealth inequality combine to produce the pattern observed in Figure 7. The blue-shaded area plots the contribution of changes in labor income. The fall in real wages for individuals at the bottom of the wage distribution contributed to declining incomes for individuals at the middle and bottom of the income distribution. The red-shaded area plots the contribution of changes in capital income. This is uniformly positive, as everyone benefits from a higher return to wealth. But the benefits from a higher return to wealth Change in income by percentile of the income distribution 1 60 60 Total income growth Part due to wage income Part due to capital income Rep. household model 0.9 50 0.8 percent change 40 50 40 0.7 0.6 30 30 0.5 20 20 0.4 10 10 0.3 0 0 0.2 0.1 -10 0 -10 0 0 200.1 0.2 40 0.3 60 0.4 income percentile 0.5 800.6 99 0.7 100 0.8 0.999.5 1 100 top tail Figure 7: Predicted change in income by income percentile decomposed into the contribution of capital and labor income. The right panel zooms to the top tail of the income distribution. The line with circle dots plots the change in a model that admits a representative household. are highly dispersed. People at the bottom of the distribution have few assets, and so do not benefit as much from an increase in the return to wealth. In contrast, a higher return to wealth allows some individuals at the top of the income distribution to accumulate large swaths of wealth and earn a high capital income. Interestingly, although automation is skill biased and raises wages at the top of the wage distribution relative to the middle and bottom (see Figure 6), changes in wage income have a negative contribution at the top of the income distribution. Intuitively, there are two effects. On the one hand, Proposition 5 shows that the top of the income distribution is populated by more skilled workers. The skill-biased nature of automation tends to raise the wages of individuals at the top of the distribution more than at other percentiles. However, the permanent increase in returns also means that the top of the income distribution becomes increasingly populated by low wage individuals with very high capital incomes. This shift in the composition of top earners dominates at the top of the income distribution and generates the observed negative contribution of labor income. This negative effect would not be present for other forms of skill biased technical change that do not results in higher returns to wealth.26 26 Formally, the expected wage of individuals at the top of the income distribution is P 1+1/α∗ 1/α∗ net / z `z wz net . Skill-biased changes in wages raise this expected wage, but increases in z `z wz P To illustrate the relevance of having a model where the return to wealth increases permanently, we study the behavior of income inequality in a model with individuals with different skills that admits a representative household (see Caselli and Ventura, 2000). In this model, the return to wealth does not depend on technology, and technology only affects income inequality through wages. The green dashed line in Figure 7 shows the change in income by quantile in this model. Both the representative-household model and ours start from the same conditional distributions of wages and wealth and are subject to the same changes in technology. However, the representative-household model does not generate uneven growth: there is a fairly uniform increase in income between 10% and 18% for all income quantiles. This reflects two differences. First, there is no wage stagnation in the representative-household model, which implies a less pronounced decline in labor income at the middle and bottom of the distribution. Second, the temporary increase in returns to wealth benefits all individuals equally, as they are all able to scale their effective wealth by the same amount. This is in contrast to our model with imperfect dynasties, where individuals who manage to accumulate wealth for longer periods will benefit disproportionally from the higher return to wealth. Figure 8: Predicted behavior of income at different percentiles of the income distribution over time. For each percentile, we plot its behavior relative to its steady-state value in 1980. Figure 8 shows the evolution of several income quantiles over time. In line with our findings above, we see a fanning out of the income distribution, with incomes below the 50th percentile declining over time and incomes above the 50th percentile rising. The fanning ∗ αnet reduce it. See the related discussion of Proposition 5. Note that as we keep moving up the tail, the contribution of labor income converges to zero since individuals’ own mostly capital income. out of top incomes continues over time and takes place slowly, but even by 2020 the model predicts substantial uneven growth. 3 Confronting the Model with Past and Current Trends Our theory has predictions for the evolution of a number of key macroeconomic aggregates as well as the dynamics of the distributions of income and wealth. One purpose of this section is to confront these predictions with the data and to discuss to what extent our model is consistent with observed trends in key variables, both quantitatively and qualitatively. We also use our model as a laboratory to examine a number of other trends that we have not discussed so far but that have been emphasized elsewhere, such as rising markups, falling capital taxes or a rising efficiency of producing investment goods. We do this to showcase the theory’s breadth and its applicability as a framework for thinking through many of the most important trends in modern, advanced economies. Secular trends in returns to wealth. Ceteris paribus, our theory predicts that automation increases returns to wealth. This can be most transparently seen in the expression for the steady-state rate of return in Example 1 with δ = 0, namely r∗ = ρ + pσα which is increasing in average automation α. To what extent observed trends in measured returns to wealth are consistent with this prediction? Before proceeding to discuss these empirical trends, it is important to clarify one key point. In our model, what matters for the evolution of wealth inequality is not the level of the return to wealth r but rather its deviation from the rate of time preference r−ρ. This gap determines the rate of wealth accumulation (r − ρ)/σ and top tail inequality (see Lemma 1 and Proposition 4). Therefore, for understanding trends in capital income and wealth inequality, secular trends in returns to wealth are, in a sense, not directly relevant and it would be preferable to contrast the model’s prediction for the rate of wealth accumulation with its empirical counterpart. We return to this point below. With this caveat in mind, Figure 9 plots various measures of returns to capital. To begin, the solid line with diamonds in panel A plots the real return to holding 5-year US treasury bills, that is a risk-free interest rate. As is well-known the last thirty years have seen a strong secular decline in this series: between 1980 and 2014, the treasury rate declined from around eight percentage points per year to zero. Along the lines of the discussion in the preceding paragraph, one possible interpretation of this trend is that the secular decline in safe real interest rates is due to other secular changes captured in a reduced form by a decline in the rate of time preference, ρ, for example demographics or the “saving glut.” As equation (10) shows, forces that are captured by a decline in ρ will result in a one-to-one decline in the return to wealth r∗ . These changes would have no effects on individuals’ accumulation rate Figure 9: Secular trends in various measures of returns to wealth. The top panel presents after-tax series from Gomme, Ravikumar and Rupert (2011) and the return to a 5-year treasury bill. The bottom panel presents pre-tax series of returns to private equity, computed as in Moskowitz and Vissing-Jørgensen (2002). All series for returns exclude capital gains. nor wealth inequality, both of which depend on α but not ρ. Automation, i.e. a concurrent ∗ increase in α would keep r∗ = ρ + pσαnet from declining by the same amount as ρ, thereby generating an increase in r∗ − ρ and wealth inequality. Risk-free interest rates or rates of return on other safe assets like AAA corporate bonds (rsafe from now on) may provide little information on the importance of capital income and the overall return to wealth obtained by US households, especially those at the top of the wealth distribution. Treasuries and AAA bonds represent a small fraction of the assets in the economy and using risk-free interest rates to infer movements in the average return of all productive assets in the US requires strong assumptions. Also, using risk-free interest rates to impute net capital income as rsafe × K leads to a large and volatile residual category of “factorless income” whose behavior is hard to justify (see Karabarbounis and Neiman, 2018) Panel A of Figure 9 therefore plots two alternative return series. Both of these aim to measure the return to the overall stock of capital in the US economy using national accounts data. The approach defines capital income Yk as net output minus wage payments and taxes and computes the return to capital as r := Yk /K where K is the stock of capital measured at replacement cost. That is, the approach computes r so that rK matches aggregate capital income from the national accounts. This strategy recognizes that all profits eventually accrue to renters of capital and equity owners, and as such, must be taken into account when computing capital income and the return to capital. This is the approach followed by Poterba (1998); Mulligan (2002); Gomme, Ravikumar and Rupert (2011); Karabarbounis and Neiman (2018) and the BLS multifactor productivity program. Figure 9 plots the after-tax return to US business capital (excluding housing) and to all US capital (including housing) using the data from Gomme, Ravikumar and Rupert (2011). Both measures exhibit an increasing trend between 1980 and 2017, with this measure of the return to wealth rising by 1 to 2 percentage points.27,28 Are individuals benefiting from the increasing return to US business capital and overall capital as in our model? Though we do not have direct evidence on the return that people (especially those with high incomes) obtain on their assets, we can use the return to private equity to get a sense of this. The bottom panel of Figure 9 plots a measure of returns from private proprietor and partnerships. Following Moskowitz and Vissing-Jørgensen (2002), we compute these returns using data on private equity from the US Flow of Funds Accounts (FFA) and profits generated by these business from NIPA. The return to private equity shows a marked increase from 13 percentage points per year in 1980 to an average level of 17 percentage points between 1990-2015. Analogous measures constructed from the Survey of Consumer Finances (SCF) confirm that the returns to proprietorships and partnerships (P&P), and to S and C corporations have retained their levels since 1990 and have not followed the sharp decline observed for treasury bills.29 Summarizing, since the 1980s the US has seen an upward trend in various measures of returns to wealth, with the exception of risk-free interest rates. To the extent that the increasing return measures are most informative about returns earned by individuals at the 27 We plot the after-tax returns because this is what is relevant for inequality—an issue we discuss in more detail below. The pre-tax series behaves in the same way after 1980 but exhibits a declining trend from 1950 to 1980. This declining trend is partly explained by the sharp decline in capital taxation during this period (see Poterba, 1998). Finally, we plot a series that excludes capital gains, since capital gains are volatile during this period but do not exhibit any clear trends. 28 The observed divergence between real risk-free interest rates and the returns to capital shown in Figure 9 has been previously emphasized by Caballero, Farhi and Gourinchas (2017) and Farhi and Gourio (2018) who propose various candidate explanations ranging from rising market power to rising discounts on safe assets. Our baseline model does not feature any elements that drive a wedge between different rates of returns and is therefore silent on this divergence. We briefly discuss an extension of our model with markups below. 29 As before, we plot a series that excludes capital gains, since capital gains are volatile during this period but do not exhibit any clear trends. Unlike the series for the overall return to capital, this measures of return to private equity are pre-tax, since information on tax rates faced by individuals who own private equity is not available. top of the income and wealth distributions, these secular trends can qualitatively explain the concurrent increase in income and wealth inequality. An alternative perspective is that the returns might be on a declining tend due to other forces such as demographic changes (captured by a lower ρ in our model), which as we discuss below, do not affect top tail inequality. Our mechanism could still increase the rate of wealth accumulation (r − ρ)/σ and top tail inequality, even as these other secular forces operate to reduce r∗ .30 Markups and profits. The approaches described above to measure r and returns to private equity assume that all non-labor income is capital income. One potential issue with this approach is the observation that markups might have risen over time (see De Loecker and Eeckhout, 2017), which implies that a portion of capital income represents economic profits (payments to capital that are above its “normal” return rate). Despite the ongoing debate regarding the rise of markups, the distinction between “normal” returns and economic profits is not consequential for our model. Appendix D provides a generalization of our model including markups and economic profits. The formulas for the return to wealth and the tail parameter of wealth inequality (in the simpler case with δ = 0) generalize to r∗ =ρ + pσ((1 − π)α + π) = ρ + pσ(1 − labor share) 1 =(1 − π)α + π = 1 − labor share, ζ where π is the profit share of output. The extension shows that the return to wealth and the behavior of wealth inequality depends on the importance of capital income inclusive of economic profits. Intuitively, claims on economic profits are also part of the total stock of financial assets that individuals must own, and as such they play the same role as capital. The above generalization shows that the implications of our model for the return to wealth and the behavior of wealth inequality depend only on the labor share in the economy. Both automation and rising markups will reduce the labor share and increase the return to wealth and top-tail wealth inequality. To the extent that markups are rising, they will also contribute to rising wealth inequality and higher returns to wealth, but would have different implications than automation for aggregates and wage inequality.31 Another secular force behind the potential decline in r∗ is the deceleration of economic growth (see Grossman et al., 2017). Suppose that δ = 0 and ψz grows over time at a rate g. We have r∗ = ρ + pσα + σg (see also footnote 13). As shown in Appendix D, what matters for inequality in this environment is the individual accumulation rate relative to the growth rate of the economy: r−ρ σ − g = pα. It follows that a deceleration in growth will lower r∗ but wont generate an increase in top tail inequality (with δ > 0, it will actually reduce top tail inequality but not by much). 31 Using a quantitative model, Boar and Midrigan (2019) identify a number of additional channels through which concentration and markups may affect inequality and show that policies that reduce firm concentration may actually have the unintended effect of increasing income and wealth inequality. 30 Other forces affecting capital markets. Going back to the diagram in Figure 1, we can think of our theory as explaining how changes in the relative demand for capital lead to a simultaneous increase in the return to wealth, the ratio of capital to wages, and inequality. We can use our model to study the role of other forces affecting capital markets. On the one hand, we have changes in the supply of capital, driven by demographic trends and by technological changes in the investment sector. On the other hand, we have changes in capital taxation. The main point we wish to make is that all these forces shape the equilibrium of capital markets and contributed to a rise in capital-output or capital-wages ratios since 1950. But so long as capital and labor are gross complements (as they are in our model), such changes cannot account for recent trends in wealth inequality and the rising after-tax return to wealth. To discuss these forces, we generalize our model slightly so that capital is produced using qK units of the final good and capital income is taxed at a rate τK ≥ 0 (revenue is used to finance government expenditure). As in our baseline model, wealth inequality depends on the after-tax return to wealth, which we denote by r∗ . Appendix D shows that the formulas for the steady-state return to wealth and the tail parameter of wealth inequality (in the simpler case with δ = 0) generalize to r∗ =ρ + pσ α(1 − τK ) , α(1 − τK ) + (1 − α) 1 α(1 − τK ) = ζ α(1 − τK ) + (1 − α) (19) These formulas show that increases in the supply of capital do not contribute to inequality. Demographic changes increasing savings (a reduction in ρ) cause an equal decline in r∗ , affect neither the individual accumulation rate nor inequality. Improvements in the investment sector (a rise in qK ) reduce the price of a unit of capital, but do not affect the return to wealth nor inequality. The logic behind these results is the same: although expansions in the supply of capital increase the the quantity of capital, they cause a fully offsetting reduction in the return rate per unit of capital, r∗ /qK . In our model, this is driven by the fact that the demand for capital has an elasticity of −1 (recall that technology is Cobb-Douglas). As a result, the importance of capital income relative to wages (r∗ /qK ) × K/w̄—the key object determining the rate of accumulation and inequality—remains constant and equal to α(1 − τK )/(1 − α). This line of reasoning shows that, so long as capital and labor are gross complements with an elasticity close to 1, as some of the empirical evidence suggests, the demand for capital will be inelastic and expansions in the supply of capital will bring a small reduction in inequality. Only if capital and labor are highly substitutable, expansions in the supply of capital will increase the net share of capital and result in more inequality.32 32 In particular, one would need capital and labor to be net substitutes. More generally, let Fnet (K, {`z }) denote net output and F (K, {`z }) denote gross output. A decline in ρ results in a higher net share of Regarding taxes, equation (19) shows that, as in a typical market, a reduction in capital taxes will lead to an increase in capital deepening, a lower interest rate faced by firms, r∗ /(1 − τK ), but a higher after-tax return to wealth r∗ . Capital taxes in the US declined sharply between 1950 and 1980 but stabilized since 1980. Our model predicts that the sharp decline in capital taxation contributed to a declining pre-tax return, an increasing after-tax return to wealth, rising wealth inequality, and an expansion of capital, but mostly during 1950-1980. Investment and capital. Our model predicts that automation should lead to an expansion in investment and the capital-output ratio, though less so than in a representativehousehold model (Figure 2). Figure 10 plots various measures of the empirical counterparts of these variables. The left panel plots two series from the BLS capturing how the value of capital services used by the private sector has evolved relative to GDP, and a series from the BEA giving the value of the US stock of private non-residential fixed assets relative to GDP. Since the 1970s the US capital-to-GDP ratio has increased somewhat according to these measures, with the increase being more pronounced for the BLS measure of capital services (see Gourio and Klier, 2015). The right panel shows that since the 1970s, the ratio of private non-residential fixed investment to GDP (from the BEA) also increased somewhat. Because the price of investment goods declined dramatically during this period, we also find it useful to look at the behavior of the quantity of private non-residential fixed investment relative to the quantity of GDP. The right panel shows that in terms of quantities, investment grew faster than GDP through the postwar period. The prediction in our model of an expansion in the capital-output and investment-output ratios of 15% (see Figure 5) is at the low end of what these series show. Our point here is not that the effects of automation predicted by our model quantitatively account for the observed trends in investment and capital accumulation (clearly, many other factors affect these variables). Rather, it is that our model with an upward-sloping longrun capital supply provides a partial answer to the question “if the decline in the US labor share was driven by automation, shouldn’t investment and capital have increased?” The answer is “yes, but only slightly, precisely because capital supply is upward-sloping.” To see this in more detail, recall from above that the (gross) capital share increased from 0.345 to 0.43 percent between 1980 and 2014, a 25 percent increase. How much would we expect the capital-output ratio to increase? The answer is simple and can be gauged from manipulating capital if and only if capital is a net substitute for labor (that is ∂ ln Fnet /∂ ln K is increasing in K). A decline in ρ results in a higher net share of capital if and only if capital is a net substitute for labor (that is ∂ ln Fnet /∂ ln K is increasing in K). A increase in qK results in a higher net share of capital if and only if capital is a gross substitute for labor (that is ∂ ln F/∂ ln K is increasing in K). See Rognlie (2015) for a related discussion of models capable of explaining the recent rise in the net capital share. . Figure 10: Percent change since 1950 in capital and investment to GDP ratios in the US. Panel A: ratio of capital services to value added for private and nonfarm sectors (BLS Multifactor Productivity series); and ratio of private nonresidential fixed assets (BEA fixed-assets Table 4.1 Line 1 plus Table 4.4 Line 1) to GDP (NIPA Table 1.1.5 Line 1). Panel B: ratio of private nonresidential fixed investment (NIPA Table 1.1.5 Line 9) to GDP (NIPA Table 1.1.5 Line 1); and ratio of quantity of private nonresidential fixed investment (NIPA Table 1.1.3 Line 9) to real GDP (NIPA Table 1.1.3 Line 1). the definition of the capital share α = RK/Y : (K/Y )2014 α2014 /α1980 = (K/Y )1980 R2014 /R1980 (20) Now consider two scenarios, and assume in both that R1980 = 11.5% as in our calibration. Consider first the case where the rental rate R is unchanged so that R2014 /R1980 = 1, as in the representative-household model. In that case, the capital-output ratio must increase by 25 percent. But suppose instead that the rental rate R rises by one percentage point, (K/Y ) 0.43/0.345 as it is the case in our model. Then (K/Y )2014 = 12.5%/11.5% = 1.15. Rather than increasing 1980 by 25%, the capital-output ratio increases by only 15%. Summarizing, it does not follow that automation predicts massive expansions in investment and capital accumulation. While automation leads to an increase in the gross capital share α = RK/Y , a substantial fraction of this increase may show up in R rather than K/Y . The link between capital shares and inequality in the data. Does the prediction that rises in the net capital share are accompanied by large increases in top income inequality receive support from the data? Bengtsson and Waldenström (2018) explore this link in a long panel with data for 21 countries going back to at least the 1930s. Their analysis shows that a 1 percentage point increase in the net capital share is associated with: a 6.67% increase in the top 0.1 percent share of income; a 3.84% increase in the top 1 percent share of income; and a 1.55% increase in the top 10 percent share of income.33 Figure 11 illustrates this link 33 In the published paper, Bengtsson and Waldenström (2018) use a log-log specification, whereas our theory suggests estimating a log-linear specification. We used their data and the same methodology behind their estimates in Table 2 of their paper to estimate the log-linear specifications reported here. Interestingly, for the US during the postwar period using the data in Bengtsson and Waldenström (2018) as well as the most recent update of Piketty and Zucman (2014) data. Since 1980, the 8 percentage point increase in the net capital share was accompanied by a 60% increase (in log points) in the top 1% share (a 7.5% increase for every one percentage point increase in the net capital share). All these estimates are of the same order of magnitude to what the mechanism in our model would predict—recall from (15) that in our model, a 1 percentage point increase in the capital share is associated with a ln(1/q) increase in the top q share. The 8 percentage point increase in the US net capital share should result in a 37% increase in the top 1% share of income. Figure 11: Net capital share and top 1% income share in the US. Data from Bengtsson and Waldenström (2018) and Piketty and Zucman (2014). The figure superimposes the contribution of compositional effects to the rise in top 1% income share. In contrast, the standard compositional effect emphasized in the literature and defined more precisely in Section 1.4 cannot account for a quantitatively meaningful increase in top income inequality. This is easy to see from a back-of-the-envelope calculation based on equation (14) using US data. In 1980, the top one percent income share was 10% and about 35% of the income of that group was capital income (see Piketty and Saez, 2003). The fact that αnet = 23% implies that the shares of capital and labor incomes going to the top 1% of the total income distribution were S̃ k (q) = 15.3% and S̃ ` (q) = 8.3%.34 Therefore, if there had only been a composition effect, the 8 percentage point increase in the US net capital share would have increased the top 1% income share by dS(q) = (S̃k (q) − S̃` (q)) × dαnet = (0.153−0.083)×0.08 = 0.56 percentage points. This is equivalent to a 5.6% increase, a tenth their data show an increase in the share of income held by the top 0.1% relative to the top 1% and the top 10%, as well as an increase in the share of income held by the top 1% relative to the top 10%, which is consistent with capital shares being associated with a thicker Pareto tail of the income distribution. 34 As in footnote 18, denote by y(q), yk (q) and y` (q) the income, capital income and labor income of the top q percent, the corresponding aggregates by Y, Yk , Y` and S(q) = y(q)/Y, S̃k (q) = yk (q)/Yk . Then, using that yk (q) = αnet (q)y(q) where αnet (q) = 35% is that quantile’s share of capital income, and yk (q) = S̃k (q)Yk = S̃k (q)αnet Y , we have S̃k (q) = (αnet (q)/αnet )S(q) = (35%/23%) × 10% = 15.3%. of what we observe in the data and a fifth of what our mechanism generates. The dashed line in Figure 11 shows the small contribution of compositional effects to the evolution of the top 1% income share.35 The link between capital shares and inequality can be also seen at other historical periods during which automation (or mechanization) was a dominant force, like the onset of the industrial revolution in Britain. As documented in Allen (2009) and reproduced here in Figure 12, from 1760 to 1840, the capital share (excluding land) rose from 20% to 40% in Britain and the labor share declined from 60% to 50%. In line with our model, the return to wealth (what Allen terms the profit rate) doubled from 10 percentage points to 20 percentage points at the same time as average wages stagnated. Data from Lindert (2000) show a sharp rise in income inequality starting exactly at this period, as can be seen from the evolution of the top 5% income share in Britain, plotted in the right panel of the figure. Figure 12: Gross capital share, the return to capital, and the top 5% income share during the British industrial revolution. Data from Allen (2009) and Lindert (2000). Capital income and the rise in top inequality. One of this paper’s main premises is that not all income is labor income and that, at any given point in time, capital income is important, particularly at the top of the income distribution. Along the same lines, our theory predicts that capital income is important for generating a large increase in top 35 Writing fifty years ago, Meade (1964) painted a bleak picture of automation’s impact, explicitly linking it to capital income and wealth inequality: “But what of the future? Suppose that automation should drastically reduce [the labor share...] There would be a limited number of exceedingly wealthy property owners; the proportion of the working population required to man the extremely profitable automated industries would be small; wage-rates would thus be depressed; [...] we would be back in a super-world of an immiserised proletariat and of butlers, footmen, kitchen maids, and other hangers-on.” He then goes on to justify the connection between the capital share and income inequality with the composition-effect logic explained above. Our analysis suggests that Meade was right to emphasize a connection between the capital share and inequality but that he had in mind the wrong mechanism (see Figure 11). inequality – see Figure 7. A natural question is therefore to what extent empirical trends are consistent with this prediction. annual growth 1980-2012 (in %) 6 5 0.8 Panel A. Change in income by percentile of the income distribution, IRS data 6 Top 0.1% Total income Part due to wages Part due to capital & entrepreneurial income 4 3 2 0 6 0.6 3 2 0.4 1 0.2 0 10 0 1 5 0.8 0 20 0.1 30 0.2 40 50 0.3 60 0.4 700.5 80 0.6 90 0.7100 2 -1 99 0.8 0.999.5 1 100 income percentile top tail Panel B. Change in income by percentile of the income distribution, PSZ data 6 Total income Part due to wages Part due to capital & entrepreneurial income Top 0.1% 4 3 Top 0.1% 5 4 1 -1 annual growth 1980-2018 (in %) 1 Top 0.1% 5 4 0.6 3 2 0.4 1 1 0.2 0 -1 0 0 10 0 20 0.1 30 0.2 40 50 0.3 60 0.4 income percentile 700.5 80 0.6 90 0.7100 -1 99 0.8 0.999.5 1 100 top tail Figure 13: Capital income and the rise in top inequality (IRS and PSZ data) To this end, we use US data to examine to what extent uneven growth in the US economy is accounted for by capital income and wage income. We use two different data sources: the NBER IRS public use sample from 1980 to 2012 and the data of Piketty, Saez and Zucman (2018) from 1980 to 2018 (PSZ, hereafter).36 The IRS dataset is based on administrative tax records and therefore yields reliable information for the very top of the distribution. However, it only records fiscal income and therefore omits tax-exempt income such as pension income. To remedy this, PSZ try to impute this “missing income” by making various adjustments to 36 The NBER IRS public use sample is available through the NBER. See https://users.nber.org/∼taxsim/ gdb/ and https://www.nber.org/taxsim-notes.html. This IRS data is also used in Piketty and Saez (2003). For the PSZ data, we do not have micro data at the individual or household level (in contrast to the IRS data). The figure instead uses tabulations by income percentiles from the Piketty, Saez and Zucman (2018) dataset that were kindly computed by Gabriel Zucman. the fiscal income series and bringing to the table other data sets as well as making various assumptions. Given this, the main advantage of the PSZ data are the more comprehensive income concept being used that also adds up to national income but this comes up at the cost of strong assumptions and numerous imputations. Conversely, the main advantage of the IRS data is the transparency and simplicity of the data construction but this comes up at the cost of substantial amounts of “missing income.” Figure 13 provides our analysis of the sources of uneven growth in the United States. Panel (a) uses the IRS data and panel (b) the PSZ data. The dash-dotted black line labelled “total income” in both panels plots the average annual income growth between 1980 and 2012 for the 100 percentiles of the income distribution. The right-hand side of each panel provides a “zoom” of the top tail of the distribution where much of the action is. As this figure shows, the income percentiles corresponding to the lower half of the distribution have stagnated or declined between 1980 and 2012. In contrast, top income percentiles have grown rapidly. For example, in both data sets, the top 0.1 percentile increased at a yearly rate of more than 5 percent. The hockey stick shape of income growth at different percentiles in the figure is qualitatively similar to that predicted by our model (see Figure 7).37 The blue and red shaded areas in both panels of Figure 13 decompose the growth in total income into a part due to labor income and a part due to capital and entrepreneurial income.38 In both datasets, capital income is unimportant for income growth at the bottom of the distribution. As in our model, the stagnant incomes at the bottom of the distribution are driven by declines in real wages. However, further up in the distribution and especially within the top 1 percent, capital and entrepreneurial income becomes more and more important. The extent to which capital income accounts for the observed uneven growth and at which percentiles it starts to be important differs between the two datasets. The IRS data in panel (a) suggest that up to the 90th percentile, essentially all growth in total income is accounted for by labor income and that capital and entrepreneurial income are mostly important within the top 1% and especially at the top 0.1% where capital and entrepreneurial income account 37 The change in total incomes in panels (a) and (b) is qualitatively similar but there are some quantitative differences in terms of both the average growth and the relative growth across different income percentiles. The blue line in panel (b) labelled “total income” is the same as that in Figure II(a) of Piketty, Saez and Zucman (2018) and they discuss this discrepancy. In particular, they note that their series yields “more growth for the bottom 90% since 1980 than suggested by the fiscal data studied by Piketty and Saez (2003)” which is essentially our IRS series. They go on to say that “the main reason for this discrepancy is that the tax-exempt income of bottom 90% earners – which fiscal data miss – has grown since 1980” and that “all of this increase derives from the rise of imputed capital income earned on tax-exempt pension plans.” Piketty, Saez and Zucman’s Appendix C.2 discusses in more detail the importance of capital income for the rise of the top 1% income share and compares their results to those of Smith et al. (2019). 38 Denoting by yt (q) the qth income percentile at time t and by y`,t (q) and yk,t (q) that percentile’s labor 0 (q) and capital income, we decompose the T -year growth rate from a base year t = 0, yT (q)−y , into a part due y0 (q) to labor income y`,T (q)−y`,0 (q) y0 (q) and a part due to capital income yk,T (q)−yk,0 (q) . y0 (q) Figure 13 plots an annualized version of this decomposition in which we scale the annualized T -year growth rate, (yT (q)/y0 (q))1/T − 1 by the fractions of T -year growth (yT (q)/y0 (q)) − 1 due to each income type. See Appendix A.2 for more detail. for around 60% of the cumulative growth from 1980-2012. In contrast, the PSZ data in panel (b) suggest that capital income plays an important role already around the median of the income distribution, accounts for around half of cumulative growth of the top 1% and is twice as important as labor income for the rise in top 0.01% incomes. The empirical pattern in both data sets is qualitatively consistent with that generated by our model – see Figure 7. The fact that wage income plays a more prominent and positive role at the tail of the income distribution in the data than in our model suggest there are forms of skill-biased technical change other than automation affecting relative wages (see the discussion of Figure 7 in Section 2.2). Dynamics of top income and wealth inequality. The past forty years have seen a rapid rise in top income inequality in the United States (see e.g. Piketty and Saez, 2003; Piketty, Saez and Zucman, 2018). There is also an ongoing debate about the dynamics of top wealth inequality,39 with some observers arguing that top wealth inequality has also increased rapidly. While our theory can account for large changes in inequality between steady states – see for example Figure 7 – it cannot generate the rapid transition dynamics of top income inequality observed in the data. Figure 8 shows that the model-implied transition dynamics at the top of the distribution are very slow and, for example, the 99.9 percentile of the income distribution still has not converged 80 years after the initial shock. That theories like ours cannot generate fast transition dynamics is a known result: Gabaix et al. (2016) have shown that standard theories of the Pareto tails of the income and wealth distributions, which build on a random growth mechanism, generate transition dynamics that are too slow relative to those observed in the data. Our theory is exactly a special case of such a theory – see Proposition 4 – and is therefore subject to the same criticism. The good news is that we know how to “fix” random growth theories to deliver fast transition dynamics like those observed in the data. Gabaix et al. (2016) show that what is needed are particular deviations from Gibrat’s law, what they call “type- and scale-dependence.” For the case of wealth dynamics, heterogeneous and persistent rates of return to wealth are one candidate for generating such type- and scale-dependence and seem to be a prevalent feature of the data (Fagereng et al., 2016). Future work should build more quantitatively serious theories of the general-equilibrium interaction between technology and income and wealth distribution that feature these model elements.40 39 See for example Kopczuk (2015), Saez and Zucman (2016), and Bricker et al. (2016). Alternatively, theories with changing asset prices – another model feature of the data that we do not model – are promising for generating fast wealth inequality dynamics. 40 4 Conclusion In this paper, we developed a tractable framework to study the effects of technology on income inequality. Our theory allowed us to go beyond wages and to explore how technology affects wealth inequality and overall income inequality. We used our framework to study the effects of automation and identified a new channel by which technology may affect inequality. Technology affects not only wages but also asset returns to wealth and this can have substantial distributional effects. Because the productivity gains from automation accrue to owners of capital in the form of higher returns, automation does not necessarily lead to a large expansion of investment, output, and wages. Instead, the higher returns raise the cost of capital and moderate the response of investment. Automation may therefore result in stagnant or declining real wages, especially at the bottom and middle of the wage distribution. There are two fruitful avenues for future work. First, one could use our tractable framework to study the distributional consequences of other types of technical change, changes in market structure and markups, and government policies, like the taxation of capital or estates, or redistributive policies. For example, it could be worthwhile to integrate our model with a theory of international trade so as to examine the effects of globalization on income and wealth distribution, and not just that of wages as is common in the trade literature. Similarly, our analytically tractable theory featuring a less-than-perfectly-elastic capital supply and non-degenerate wealth distribution may serve as a useful laboratory for exploring the optimal taxation of capital income and wealth. Second, one could think of building more elaborate quantitative models to study the effect of technologies on inequality. As explained in the introduction, these more elaborate models should retain the two key features underscored by our analysis: an upward-sloping supply of capital and a return inequality nexus. More elaborate versions of our model could include realistic life-cycle structures, a careful treatment of bequests and intergenerational transfers, and heterogeneity in portfolio and return rates. Successful quantitative extensions should also include some form of scale-dependence and type-dependence to account for the rapid rise of inequality observed in the data. In connecting our model to the data, one crucial questions is how important is the rise of capital income at the top of the distribution. This is still an open and interesting question. Some researchers assert that capital income plays a more prominent role in European countries. If so, perhaps our theory applies more forcefully to the European context. After all, Western Europe is ahead of the US when it comes to industrial automation (see Acemoglu and Restrepo, 2018a), and the decline in the labor share appears common to all these countries (Karabarbounis and Neiman, 2013). In the US, wages pay a more prominent role, pointing to the importance of other forms of skill-biased technical change different from the automation of routine jobs. Besides wages, entrepreneurial income seems to play an important role at the top of the income distribution (see Smith et al., 2019). One key question going forward is to understand how much of the rise in entrepreneurial income is the result of technologies that allow entrepreneurs to grow their firms without paying a high share of the generated income in wages, or if rising entrepreneurial incomes at the top reflect other changes in regulation, market structure, and the way we reward talent. That is, we need to understand whether rising entrepreneurial incomes reflect the higher returns brought by automation or these other factors.