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\begin{document}

\title{Uneven Growth:\\ Automation's Impact on Income and Wealth Inequality\thanks{We thank Daron Acemoglu, Fabrizio Perri, Matt Rognlie, Richard Rogerson, Stefanie Stantcheva, Ivan Werning and seminar participants at various institutions for useful comments and Bob Allen, Daniel Waldenstram and Gabriel Zucman for sharing data with us. We also thank Benny Kleinman, Nils Lehr, and Max Vogler for outstanding research assistance. The views presented here are solely those of the authors and do not represent the views of the Bank of England or its policy committees.  Moll received funding for this project from the Leverhulme Trust and the European Union's Horizon 2020 research and innovation programme under grant number No. GA: 865227. Restrepo thanks the National Science Foundation for its support under award No. 2049427.}}
\author{ Benjamin Moll  \\[-7pt]
%EndAName
{\small LSE}
\and  Lukasz Rachel  \\[-7pt]
%EndAName
{\small Princeton University }\\[-7pt]
\and  Pascual Restrepo  \\[-7pt]
%EndAName
{\small Boston University  }\\[-7pt]
}
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\begin{abstract}
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The benefits of new technologies accrue not only to high-skilled labor but also to owners of capital in the form of higher capital incomes. This increases inequality. To make this argument, we develop a tractable theory that links technology to the personal income and wealth distributions -- and not just that of wages -- and use it to study the distributional effects of automation. We isolate a new theoretical mechanism: automation increases inequality via returns to wealth. The flip side of such return movements is that automation is more likely to lead to stagnant wages and therefore stagnant incomes at the bottom of the distribution. We use a multi-asset model extension to confront differing empirical trends in returns to productive and safe assets and show that the relevant return measures have increased over time. Automation accounts for part of the observed trends in income and wealth inequality and macroeconomic aggregates.

\textbf{Keywords:} inequality, wealth, capital, returns, wages, labor share, technology, automation 
\end{abstract}

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\section*{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 doubled and top 1 percent incomes tripled.\footnote{See for example \citet{census-facts} and \citet{piketty-saez-zucman}.}

One potential driver of these trends that is often cited by pundits and policy makers alike is technical change, and in particular automation. A large literature in macro and labor economics has studied how technology and automation affect the distribution of labor incomes.\footnote{See for example \citet{katz-murphy}, \citet{krusell-capital} and \citet{autor-polarization}} But not all income is labor income and capital 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. This shortcoming is particularly acute when it comes to automation, technical change that substitutes labor with capital, and which therefore increases the importance of capital in the economy. 

We argue that the benefits of new technologies---and in particular automation technologies---accrue not only to high-skilled labor but also to owners of capital in the form of higher capital incomes. This increases inequality. To make this argument, we develop a tractable framework that allows us to study the impact of technology on factor prices and the personal income and wealth distributions---and not just that of wages---and use it to study the distributional effects of automation.

Our framework provides a complete 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.\footnote{Importantly, ours is a theory of the \emph{personal} income distribution and not just of the \emph{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).} We achieve this by assuming that households differ in their skills and household wealth accumulation is subject to \emph{dissipation shocks} which leave them with zero assets and only their labor income, thereby capturing the hazards of accumulating and maintaining a fortune. Dissipation shocks allow us to deviate in a tractable fashion from models that admit a representative household (like variants of the neoclassical growth model) and introduce two key features relative to such models that are crucial for understanding how automation affects inequality and aggregates.\footnote{Our notion of ``models that admit a representative household" allows for skill heterogeneity and includes models that assume there are different skill types but these are all members of the same representative household, and models where Gorman aggregation holds \citep[see Theorem 5.2 in][]{acemoglu-book}.} First, our theory generates well-defined and tractable steady-state distributions for wealth and income. Second, the long-run supply of capital is less than perfectly elastic and therefore determines the long-run effect of technology on returns to wealth (prices) and the amount of capital used in production (quantities).

The framework underscores two key and novel channels through which technology affects income inequality and contributes to the pattern of uneven growth described above. First, automation increases wealth and capital income inequality by raising returns to wealth. Second, 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. The key to understanding both results is that the long-run capital supply in our model is upward-sloping. Automation increases the demand for capital relative to labor and, because supply is upward-sloping, this demand shift \emph{permanently} increases returns to wealth. The first part, that automation directly increases wealth and income inequality, then follows because households receive a higher return to 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 do not accrue to workers but rather to owners of capital in the form of a higher return to their wealth. Neither of these mechanisms would be operational in  textbook representative household models, in which the long-run capital supply is perfectly elastic and therefore returns are unaffected by technology \citep[see, for example][]{acemoglu-restrepo,caselli-manning}.

We first illustrate these mechanisms using a simple baseline model and later extend this model in a number of directions. In the baseline model, capital is the only asset in the economy and all households earn the same return to their wealth. We obtain two main analytical results that illustrate how automation affects inequality.

First, the steady-state return $r$ exceeds the discount rate $\rho$ by a \emph{return gap} $r-\rho=p  \sigma  \alpha_{net}$, where $p$ is the arrival rate of dissipation shocks, $\sigma$ is the inverse of the intertemporal elasticity of substitution, and $\alpha_{net}$ is the net capital share -- an object that rises with automation. Therefore, in contrast to textbook representative household models in which this return equals the (fixed) discount rate, the return in our model increases with automation.

Second, the wealth accumulation process of individual households generates an exact Pareto distribution for both effective wealth (the sum of financial and human wealth) and income conditional on wages. Tail inequality -- as measured by the inverse of the Pareto shape parameter -- equals the net capital share $\alpha_{net}$, meaning that both top income and wealth inequality increase with automation. Intuitively, the gap between the return to wealth and the discount rate determines the speed at which \emph{individual households} accumulate wealth in the absence of a dissipation shock, which in turn determines the thickness of the wealth distribution. In equilibrium, this gap equals $p  \sigma  \alpha_{net}$ and therefore the net capital share $\alpha_{net}$ pins down the tail index of these distributions. This result illustrates how automation contributes to income inequality by permanently increasing returns to wealth and the concentration of capital ownership.\footnote{ 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 distributed than labor income \citep{meade,piketty-capital}. As we discuss in detail, such compositional effects are small relative to the data and to the changes in capital ownership generated by our model.} 

The flip side of the finding that the return to wealth rises with automation is that wages of displaced workers are more likely to fall, not only \emph{relative} to other skill types (as commonly emphasized in the literature), but also in \emph{levels}. In our theory, the upward-sloping supply of capital limits the expansion of investment and output in response to technological improvements. As a result, automation can lead to stagnant or falling real wages even in the long run, especially for workers whose skills are more susceptible to automation. 

Although our baseline model is intentionally stylized, the mechanisms underlying our results are more 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 inequality. The first ingredient, an upward-sloping capital supply, seems to us a more natural and less extreme starting point than the perfectly-elastic capital supply of the textbook 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). The second ingredient, a return-inequality nexus, emerges naturally in models in which stochastic wealth accumulation at the individual level generates a fat-tailed wealth distribution. This includes models with stochastic returns or discount rates \citep{krusell-smith, benhabib-bisin-zhu,benhabib-bisin-zhu-JET} and models with explosive growth coupled with a birth and death process \citep{wold-whittle,steindl,benhabib-bisin-altruism,jones,sargent-wang-yang}.\footnote{See \citet{benhabib-bisin-jel} for a review of models capable of generating fat-tailed 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 simple versions of Aiyagari-Bewley-Huggett models \citep{stachurski-toda}.
} Also the argument of \citet{piketty-capital,piketty-clarification} that top wealth inequality depends on ``$r-g$'' highlights precisely this nexus. 

In our baseline model, automation has important distributional consequences because it raises \emph{the} return to wealth. One of the main challenges in applying this theory to the data is that returns on different assets have displayed divergent empirical trends: treasury rates have declined over time \citep[e.g.][]{rachel-summers} while the return to other assets, including US business capital and equity have increased \citep[e.g.][]{caballero-farhi-gourinchas,gomme-ravikumar-rupert,reis}. What is then the relevant return to evaluate our theory? And how should we account for changes in the growth rate of the economy and other forces affecting returns?  

To answer these questions, we extend our model to feature multiple assets with different returns, richer capital income risk, markups, taxation and long-run growth. Despite the added complexity, the model remains tractable. In our extended model, the return to risky capital exceeds the return to safe bonds because of compensation for risk or financial frictions that prevent arbitrage.\footnote{The observed divergence between real risk-free interest rates and the returns  to capital  has been previously emphasized by \citet{caballero-farhi-gourinchas} and \citet{farhi-gourio} who propose various candidate explanations ranging from rising market power to rising discounts on safe assets. }  We show that the key return measure determining wealth inequality is the overall return to investors' wealth, $r_W$, which depends on investors' portfolio allocation and leverage. The safe rate is not necessarily informative about this return. In fact, when investors are leveraged, lower safe rates \emph{increase} this return and hence wealth inequality. Moreover, automation leads to a higher return to investors' wealth and a widening spread between risky and safe returns, with ambiguous implications for the level of the safe rate. Finally, the model also shows that, in a growing economy, wealth inequality depends on the \emph{return gap} $r_W-\rho-\sigma  g$, and that this gap always rises with automation.  

These results show that, when assessing the validity of our mechanism, one should focus on the return to investors' wealth and the return gap. Building on this insight, we provide a series of estimates for the return to investors' wealth $r_W$ and the return gap. Contrary to the observed trend in safe rates and in line with our theory, the return to wealth rose by 0.5--2 percentage points and the return gap rose by 1.5--3.5 percentage points since 1980.

The addition of risky capital in the extended model generates higher levels of top-tail inequality. In our baseline model, top income inequality as measured by the inverse of the Pareto shape parameter equals $\alpha^*_{net}\approx 0.2$ which is much lower than the corresponding empirical values of around 0.7. In contrast, the extended model with capital income risk can generate the higher tail inequality that we see in the data. 

After presenting our main analytical results, we turn to a numerical evaluation of the model. 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 using exposure to routine jobs, which the literature singles out as jobs that can be easily automated using computer software or other equipment \citep[see][]{autor-levy-murnane,autor-polarization}. To benchmark the extent to which these jobs have been automated, we calibrate the automation of routine jobs since 1980 to match the declining share of labor in GDP during this period. 

The automation of routine jobs generates 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 at these percentiles. Second, it generates a rising concentration of capital income at the top of the distribution, which accounts for the sharp rise of incomes at the top. In particular, automation generates an increase in tail income inequality (measured by the inverse of the Pareto index) from 0.54 to 0.65, about 75\% of the increase estimated by \citet{piketty-saez-zucman}.

The first contribution of our paper is to the literature on technological change and automation. Like most theory papers in this area, we use a task-based framework to model automation \citep{zeira,acemoglu-autor,acemoglu-restrepo}. Papers in this literature focus on wage inequality 
\citep[][]{autor-levy-murnane, autor-polarization,acemoglu-autor,hemous-olsen}, or study the effect of automation on aggregates and wages using a representative household framework \citep[][]{acemoglu-restrepo,caselli-manning}. One exception is \citet{sachs-kotlikoff}, 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 and wealth.  

Our second contribution is to the literature exploring the determinants of wealth inequality. 
Several papers explore this question quantitatively in general equilibrium models, including \citet{krusell-smith, castaneda-et-al,kaymak-poschke,straub} and \citet{hubmer-krusell-smith}. Relative to this literature, we isolate a new theoretical mechanism: technical change affects income and wealth distribution via returns to wealth. In contrast, other papers studying how technology affects wealth distribution focus on its effects through wage dispersion. Our model is deliberately simple and abstracts from labor income risk and a realistic treatment of life-cycle and bequests---important elements in quantitative theories of the wealth distribution. In exchange, we obtain analytical solutions for the steady-state distributions of wages, incomes and wealth.\footnote{The mechanism generating a Pareto tail in our model is a common feature of random growth processes \citep[see][for a review]{gabaix-AR}. In fact, the process with dissipation shocks 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 by many authors before us. Among those, the closest to our work are \citet{cao-2017}, who also derive formulas for the tail parameter of the wealth distribution in terms of aggregate statistics and provide comparative statics, but do not explore the role of automation as we do.} Furthermore, the economy aggregates, and solving for its transition dynamics is as easy as in the neoclassical growth model. Thus, our model is closer in spirit to models of the Gorman class \citep[see][]{chatterjee,caselli-ventura} in which policy functions are linear and aggregates do not depend on the wealth distribution.\footnote{Because policy functions are linear ``macro matters for inequality, but inequality does not matter for macro."  A large recent literature argues that models incorporating empirically realistic heterogeneity in household balance sheets and marginal propensities to consume often deliver different aggregate implications than do representative agent models, precisely because aggregates depend on distributions. We view linear policy functions and other abstractions with unrealistic implications as costs worth paying in return for our theory's analytical tractability.} However, in contrast to these models, dissipation shocks generate a determinate steady-state wealth distribution and imply that our model does not admit a representative household.

Section \ref{sec:model} lays out our baseline theory of uneven growth. Section \ref{sec:extension} presents the extended model and a measurement exercise showing that the relevant return measures have increased in line with our theory. In Section \ref{sec:calibration} we take the extended model to the data with a calibration of changes in automation across the wage distribution, and in Section \ref{sec:confronting} we show that this generates a pattern of uneven growth like that observed in the data. Section \ref{sec:conclusion} concludes. 


\section{Benchmark Model of Uneven Growth}\label{sec:model}
\vspace{-3mm} 
 
The model is cast in continuous time. For expositional clarity, we outline the model in its stationary form.  Appendix \ref{appendix proofs} provides proofs and derivations. Appendix \ref{appendix transition} describes the model along the transition path.  

\subsection{Economic Environment}

\paragraph{Households:} There is a unit continuum of households that differ in their skills $z$ and supply their labor inelastically, with $\ell_z$ denoting the population share of each skill. Households maximize standard preferences over utility flows from consumption subject to a flow budget constraint and a natural debt limit:
\begin{align}\max_{\{c_{z,t}, a_{z,t}\}_{t \geq 0}} \ & \int_0^\infty e^{-\rho t} \frac{c_{z,t}^{1-\sigma}}{1-\sigma}dt \label{equation: max problem} \\
\mbox{subject to: }&\dot{a}_{z,t} = w_z + r a_{z,t} - c_{z,t}, \text{ and } a_{z,t} \geq -w_z/r, \notag \end{align} 
where $a_{z,t}$ denotes assets, $c_{z,t}$ consumption, $r$ the return to wealth and $w_z$ wage income. 

Without any modification, Gorman's aggregation theorem applies \citep[see theorem 5.2 in][]{acemoglu-book} and this model would admit a representative household. Independently of the production side, the steady state would involve a constant interest rate equal to the discount rate $\rho$ (or $\rho+\sigma g$ in a growing economy). Moreover, the wealth distribution would be indeterminate \citep[see][]{caselli-ventura}, and so the theory would be ill-suited to study how technology affects income and wealth inequality.

We break Gorman aggregation by assuming that the accumulation of wealth is subject to \emph{dissipation shocks}, which arrive at a rate $p>0$. We operationalize this assumption in the simplest possible way by assuming that after receiving a dissipation shock, households consume all of their wealth and are left with zero assets, so that following a dissipation shock we have $a_{z,0}=0$.\footnote{For households with debt, we assume that dissipation shocks act as a cancellation of their debt. Alternatively, one could assume that households cannot borrow, or that dissipation shocks only hit households with a positive level of assets. These assumptions are not important for our results. As we will show, the steady state of our model involves all households holding positive assets, i.e. noone holds debt.} In our formulation and in all this section, $t$ denotes the time elapsed since the last dissipation shock and not calendar time. Appendix \ref{appendix dissipation} shows formally that in the presence of dissipation shocks, households solve  \eqref{equation: max problem} and discount the future at a rate $\rho := \varrho + p$, where $\varrho$ is the pure rate of time preference.  In what follows, we will refer to the case with $p=0$ as the representative-household benchmark.  
 
Dissipation shocks provide a reduced-form way of capturing the risks and hazards involved in raising and maintaining a fortune, both over time and across generations. These risks are a defining feature of wealth accumulation, as emphasized by recent work that documents substantial churn at the top of the wealth distribution \citep{gomez,zheng}. Appendix \ref{appendix: foundations} provides several  micro-foundations for dissipation shocks, many of which are key ingredients in existing theories of wealth inequality:

\begin{itemize}
	\item \emph{perpetual youth:} individuals die with a constant probability $p$ and are not altruistic toward their offspring. If there are no accidental bequests \citep[either because of an annuity market, as in][or because people consume their wealth right before they die]{blanchard}, newborns start life with zero assets and only their labor income, mimicking the effect of a dissipation shock.\footnote{When there is an annuity market, the return obtained by households is $r+p$, where the addition of $p$ accounts for the income from annuities. See Appendix \ref{appendix:annuities} for a more detailed discussion. \par 
This setup can be generalized to allow for some bequests (e.g. due to partial altruism or a ``joy of giving" bequest motive) as in \citet{benhabib-bisin-altruism}, so long as individuals do not pass on their entire wealth to their offspring thereby forming a perfect infinitely-lived dynasty.} 
	
	\item \emph{finite lives and stochastic altruism:} finitely-lived individuals are part of a dynasty and pass on their wealth over time. At rate $p$, the current member of the dynasty stops being altruistic and consumes all of her wealth so that the dynasty is interrupted.
	
	\item \emph{population growth:} $p$ captures net increases in population. Being born is isomorphic to receiving a dissipation shock because newborns start life with  zero assets. 	
	
	\item \emph{discount rate shocks:} at rate $p$, households become infinitely impatient and consume all of their wealth. This is a simple and tractable version of the more general process for discount rates assumed in \cite{krusell-smith}. 
	
	\item \emph{uninsured capital income risk:} there are multiple investment projects, households invest in a single project, and at rate $p$ the capital produced in this project becomes obsolete and looses all its value, i.e. a return of minus one hundred percent.\footnote{This micro-foundation requires two reasonable assumptions. First, households cannot be perfectly diversified across product lines. Second, risk-sharing is limited. Otherwise, an investment company could collect all investments and offer households a risk free return of $r-p$, which they would prefer to a return of $r$ coupled with the risk of loosing all their capital with probability $p$.   
	} This is a simple version of more general processes for idiosyncratic returns to wealth \citep[e.g.][]{benhabib-bisin-zhu,benhabib-bisin-zhu-JET}.
\end{itemize} 


\paragraph{Technology:} Our description of the production process follows  \citet{acemoglu-restrepo} and emphasizes the role of tasks, which are either completed by workers or automated. Each skills $z$ can be used to produce a different unit continuum of tasks, a fraction $\alpha_z$ of which are technologically automated and can be produced by capital. In addition, tasks are combined via a Cobb-Douglas production function. Appendix \ref{appendix proofs} shows that, in this setup, the economy's production side reduces to an aggregate production function in terms of capital and labor 
\begin{equation}\label{equation: aggregate output}
Y(K):=\mathcal{A}K^{\sum_{z}\alpha_{z}\eta_z}\prod_{z}\left(\psi_{z}\ell_{z}\right)^{(1-\alpha_{z})\eta_z},
\end{equation}
where $\eta_z$ denotes the importance in value added of the tasks that can be performed by skill $z$, $\psi_z$ denotes the productivity of labor at these tasks, and $K$ denotes the aggregate stock of capital in the economy. The productivity term $\mathcal{A}$ 	is a constant that depends on parameters $\{A,\alpha_z , \eta_z\}$, where $A$ captures the role of factor-neutral technological improvements.  

Labor of skill $z$ is rented at a wage $w_z$. Capital is produced one-for-one from the final good and depreciates at a rate $\delta\geq 0$, which implies a rental rate of $R=r+\delta$. Therefore the unit cost of producing a task with capital is $R$ and that of producing it with labor is $w_z/\psi_z$ where $\psi_z$ is the productivity of skill $z$. The representation in equation \eqref{equation: aggregate output} assumes that all tasks that can be produced by capital are allocated to capital in equilibrium, which imposes the following assumption:\footnote{Assumption \ref{assumption: adoption} involves endogenous factor prices, so it needs to be verified in equilibrium. Lemma \ref{lemma: adoption} in Appendix \ref{appendix proofs} shows that it holds when productivity $A$ is high.}  

\begin{assumption}[Full adoption of automation technologies]\label{assumption: adoption} The cost of producing tasks with labor, $w_z/\psi_z$, exceeds the cost of producing them with capital, $R$:
$$ \frac{w_z}{\psi_z } > R \quad \mbox{\emph{for all}} \ z.$$ 
\end{assumption}

Equation \eqref{equation: aggregate output} represents aggregate output as a Cobb-Douglas production function with factor shares linked to the range of tasks performed by each factor. The share of capital is
$$\alpha:=\sum_{z}\alpha_z\eta_z,$$
which gives the average degree of automation in the economy. Our main exercise studies the effects of improvements in automation technologies parameterized as exogenous increases in the $\alpha_z$'s. An increase in $\alpha_z$ captures the automation of tasks previously performed by workers of skill $z$ in the sense that it expands the range of tasks in which capital can now substitute for these workers. 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. As tasks that were the domain of skill $z$ get automated, this skill loses importance in production ($1-\alpha_z$ declines) and capital gains importance ($\alpha$ rises). Assumption \ref{assumption: adoption}  ensures that an increase in $\alpha_z$ raises TFP and that these technologies are adopted. Changes in the $\alpha_z$'s thus provide a flexible way of capturing technological improvements that involve an expansion in the range of tasks that capital and machines can do at the expense of some workers. 

%\footnote{Our model abstracts from firms. See \citet{martinez} and \citet{hubmer-restrepo} for models that account for the diffusion of automation technologies across firms, and \citet{autor-etal} and \citet{kehrig-vincent} for evidence on the role of firms in driving the decline in the aggregate labor share.}

\paragraph{Equilibrium:} We assume competitive input and final good markets and define  a steady state equilibrium of the economy as follows. 

\begin{definition} \label{def: st st}
	The steady state equilibrium is given by aggregate output and capital, factor prices, and consumption and saving policy functions such that: 
	\begin{itemize}
		\item factor prices equal marginal products, $w_z=(1-\alpha_z)\frac{\eta_z}{\ell_z}Y(K)$, $R=\alpha\frac{Y(K)}{K}$,
		and $r=R-\delta$;
		\item policy functions $a_{z,t}, c_{z,t}$ maximize utility given $a_{z,0}=0$, $w_z$ and $r$, where $t$ denotes time since the last dissipation shock experienced by a household;
		\item capital markets clear: 
		$K=\sum_z\ell_z\int_{0}^\infty a_{z,t}pe^{-pt}dt$,
		where $pe^{-pt}$ is the share of households that have accumulated wealth without experiencing a dissipation shock for $t$ periods. 
	\end{itemize}
\end{definition}

We first study households' consumption and saving decisions, which determine the supply of capital in the economy.

\begin{lemma}[Policy functions]\label{lemma: households policy st st} Suppose that $r>(r-\rho)/\sigma$. Starting from $a_{z,0}=0$, the solution to \eqref{equation: max problem} is given by policy functions that are linear in effective wealth, $a_{z,t}+w_z/r$:
\begin{equation}
\dot{a}_{z,t} =\frac{r-\rho}{\sigma}\left(a_{z,t}+\frac{w_z}{r}\right), \qquad  c_{z,t}=\left(r-\frac{r-\rho}{\sigma}\right)\left(a_{z,t}+\frac{w_z}{r}\right). \label{equation: x dynamics} 
\end{equation}
\end{lemma}
\noindent The lemma shows that households accumulate assets at a rate $(r-\rho)/\sigma$ and consume a constant share of their effective wealth. The linear policy functions imply that, in our model, the behavior of aggregates will be independent of the wealth and income distribution. In what follows, we will  explore the behavior of aggregates in response to automation first, and then study how changes in factor prices and returns shape the income and wealth distribution. 

\subsection{Behavior of Aggregates and Wages}

We characterize the equilibrium of the return to wealth and aggregates in steady state. 
We denote variables in the equilibrium steady state with an asterisk. 

\begin{proposition}[Return to wealth and aggregate effects of automation]\label{proposition: steady state} The steady state is unique. The return to wealth, $r^*$, is given by the solution to the equation
	\begin{equation}
\frac{1-\rho/r^*}{p \sigma + \rho -r^*}=\frac{\alpha}{1-\alpha} \frac{1}{r^*+\delta}.\label{equation: implicit r}
\end{equation}
Equivalently, denoting by $\alpha^*_{net}:=r^*K^*/(r^*K^* + \sum_z w_z^* \ell_z)$ the net share of capital income, the return to wealth satisfies
\begin{equation}
r^*=\rho+p \sigma \alpha^*_{net}. \label{equation: return premium}
\end{equation} 
The return to wealth, $r^*$, the net capital share $\alpha_{net}^*$ , and the capital-output ratio, $(K/Y)^*$, are all increasing functions of $\alpha$. Moreover, output increases in all $\alpha_z$. 
\end{proposition}

\noindent We can think of the return to wealth as determined by the supply of and demand for capital. It is convenient to think in terms of capital relative to labor income
$$
k:=\frac{K}{\bar{w}},
$$
where $\bar{w}:=\sum_z w_z \ell_z$ is the wage bill in the economy. To derive the supply of capital, we use the characterization in Lemma \ref{lemma: households policy st st}. Integrating equation \eqref{equation: x dynamics} yields
$
\dot{K}=\frac{r-\rho}{\sigma}\left(K+\frac{\bar{w}}{r}\right) - p K,
$
where the term $-pK$ accounts for the wealth loss due to dissipation shocks. In steady state $\dot{K}=0$ and hence
$$
\frac{r-\rho}{\sigma}\left(K+\frac{\bar{w}}{r}\right)=p K.
$$
Rearranging gives the (relative) supply of capital for $r\geq \rho$:
\begin{equation}
k^s=\frac{1-\rho/r}{p \sigma + \rho -r}\label{equation:supply capital}.
\end{equation}
This equation gives the upward-sloping long-run capital supply curve depicted in Figure \ref{figure: supply demand}. Intuitively, as the return to wealth rises, households accumulate and supply more capital (relative to their labor income). 

\begin{figure}[ht]
	\centering\includegraphics[width=.7\linewidth]{graphs/equilibrium_figures_main_new.pdf}
	\vspace{-1cm}
	\caption{The supply and demand for capital and the determination of the return to wealth.\label{figure: supply demand}}
\end{figure}

Figure \ref{figure: supply demand} also plots a capital demand curve which we obtain from the economy's production side: combining $\bar{w}=(1-\alpha)Y(K)$ and $RK=\alpha Y(K)$ where $R=r+\delta$ yields
\begin{equation}
k^d=\frac{\alpha}{1-\alpha}\frac{1}{r+\delta}.\label{equation: demand capital}
\end{equation}
and so (normalized) capital demand is a decreasing function of the return to wealth.

Equation \eqref{equation: implicit r} gives the return $r^*$ implicitly as the unique point at which demand equals supply. Figure \ref{figure: supply demand} shows that $r^*$ lies between $\rho$ and $\rho+p \sigma$, which implies that the return to wealth exceeds $\rho$. Equation \eqref{equation: return premium} shows that $\alpha^*_{net}$ determines exactly where in this range $r^*$ lies, with the gap $r^*-\rho$ given by $\sigma   p \alpha_{net}^*$.\footnote{
The formula for $r^*$ in equation \eqref{equation: return premium} holds for \emph{any} constant returns to scale production function. In particular, $0=\dot{K}=\frac{r-\rho}{\sigma}\left(K+\frac{\bar{w}}{r^*}\right) - p K$ implies $r=\rho+p \sigma \frac{rK}{rK+\bar{w}}=\rho+p \sigma \alpha_{net}$. Thus, the exact modeling of automation and our assumed production function is not required for this formula. Likewise, any technology that increases the net capital share will raise the return to wealth and contribute to higher wealth inequality. } 
  
Figure \ref{figure: supply demand} also illustrates the difference between our model and the representative-household benchmark. With a representative household, the supply of capital is perfectly elastic and the return to wealth is fixed at $r^*=\rho$. In such models, technology can have at most a short-lived impact on asset returns.\footnote{\label{footnote: growth}
	Recall that our simple model does not feature sustained growth. The reason why the return to wealth exceeds $\rho$ in our model is therefore different from the one embedded in Ramsey's formula $r=\rho+\sigma g$, which holds along a balanced-growth path of a \emph{growing} economy. Equation \eqref{eq: returns with g} in Section \ref{sec:extension} generalizes equation \eqref{equation: return premium} to an environment with sustained growth and shows that the equilibrium return is given by $\rho+\sigma g + p \sigma \alpha^*_{net}$, where $\alpha_{net}^*$ is now the capital share in income net of depreciation and growth.}  

The logic behind the upward-sloping supply of capital and the result that the return gap $r-\rho$  is linked to $\alpha^*_{net}$ in equilibrium is intuitive and general.
The driving force behind this result is that all the financial wealth and capital in the economy derives from labor income that households save and invest over time. Because everyone at some point in time had nothing but their human wealth, the equilibrium return to wealth $r$ must exceed the discount rate $\rho$ so as to incentivize households to accumulate and supply capital.	How high above $\rho$ must the return to wealth be? This depends on the importance of capital relative to human labor. If the net capital share is high, and hence capital is important relative to human labor, households must accumulate capital rapidly and this requires that the return to wealth exceeds $\rho$ by a large amount. This explanation also shows that all that is required for our theory is that households experience \emph{some} wealth dissipation. The assumption that they loose all their wealth following a dissipation shock is made only for simplicity.

Proposition \ref{proposition: steady state} shows that automation increases the net capital share and the return to wealth. This finding can be seen directly from Figure \ref{figure: supply demand shift}, where an increase in $\alpha$ raises the demand for capital relative to labor. This demand shift increases the return to wealth, the ratio of capital to human wealth, $k^*$, and the net share of capital. 

\begin{figure}[ht]
	\centering\includegraphics[width=.7\linewidth]{graphs/equilibrium_figures_shift_new.pdf}
	\vspace{-1cm}
	\caption{Effect of automation---an increase in $\alpha$---on the equilibrium return to wealth, $r^*$. The green arrows illustrate how the equilibrium changes in our model and in the representative-household benchmark. \label{figure: supply demand shift}}
\end{figure} 

Does the higher demand for capital following an increase in automation result primarily in a higher return to capital or in an expansion of the capital-to-output ratio? The answer depends on the capital-supply elasticity, which is inversely linked to $p$. To illustrate this, consider the special case $\delta=0$ in which we obtain simple formulas for aggregates: 
\begin{example}\label{example:delta=0}
	When $\delta=0$, we have that $\alpha^*_{net}=\alpha$ and
	\begin{equation}
	r^*=\rho+p \sigma \alpha, \qquad (K/Y)^*=\frac{\alpha}{\rho+p \sigma \alpha}. \label{eq:return_simple}
	\end{equation}
\end{example} 
As these equations show, for high values of $p$, automation leads to a sizable increase in the return to capital and a mild expansion in the capital-to-output ratio. Instead, in the representative-household benchmark with $p=0$, automation results in a larger expansion of capital and has no effect on the return to wealth.

Finally, the Proposition shows that output rises with automation. Appendix \ref{appendix proofs} shows that, to a first-order approximation, the effect of automation on output is 
\begin{equation}\label{eq:dlnY}
d\ln Y^*=\frac{1}{1-\alpha}\underbrace{\sum_z \eta_z \ln\left(\frac{w_z}{\psi_z R}\right)d\alpha_z}_{:= d\ln \textrm{TFP}_{\alpha}}+\frac{\alpha}{1-\alpha} d\ln (K/Y)^*>0.
\end{equation}
Output increases via two channels. First, by allowing the substitution of labor for cheaper capital, automation increases TFP (the term $d\ln \textrm{TFP}_{\alpha}>0$). The contribution of automation to TFP is a (weighted) sum of the cost-saving gains at the task-level $\ln\left(w_z/(\psi_z R)\right)$ (which are positive due to Assumption \ref{assumption: adoption}). Second, output increases due to endogenous capital accumulation. For high values of $p$, this second force is modest, and automation generates a more limited output expansion. 

The following proposition summarizes the effects of automation on real wages. 

\begin{proposition}[The long-run effects of automation on wages]\label{proposition: wages}
	\hspace{3cm}
	
	\vspace{-0.4cm}
	
	\begin{itemize}		
	\item An increase in $\alpha_z$ reduces the wage $w_z^*$ relative to other wages $w_v^*$ for $v\neq z$.
	
	\item For a given increase in the $\alpha_z's$, there exists a threshold $\bar{p}>0$ such that, for $p>\bar{p}$, the average wage $\bar{w}^*$ falls; and for $p<\bar{p}$, $\bar{w}^*$ increases.
	\end{itemize}
\end{proposition}

The effect of automation on relative wages is unambiguous and follows from the fact that $w_z=(1-\alpha_z)\frac{\eta_z}{\ell_z} Y$ \citep[see also][]{hemous-olsen,acemoglu-restrepo-tasks}.

A more novel implication of the proposition is the possibility that automation may lead to stagnant wages for the average worker, which necessarily implies a more pronounced \emph{real decline} in the wages of workers displaced by automation. Whether this is the case or not depends again on $p$, which determines how inelastic the supply of capital is in steady state. 

To understand this, consider an improvement in automation technologies that raises TFP by $d\ln \textrm{TFP}_{\alpha}>0$. The dual version of the Solow residual implies that\footnote{This result holds in general whenever aggregate output exhibits constant returns to scale and markets are competitive. Under these assumptions, we have $Y=\bar{w}L+RK$ where we now allow for movements in labor supply $L$ to underline the argument's generality so that $\bar{w}$ denotes the average wage. Differentiating both sides of this identity, we get that, following a technological improvement, we have
\begin{align*}
d\ln Y =d\ln \textrm{TFP}_{\alpha}+ \alpha d\ln K + (1-\alpha)d\ln L = (1-\alpha) d\ln \bar{w} +\alpha d\ln R + \alpha d\ln K + (1-\alpha)d\ln L,  
\end{align*}
where $\alpha=RK/Y$. 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 \citet[][ch.18/19]{chicago-price-theory} for a textbook treatment.}
\begin{align}\label{equation: tfp and prices}
d\ln \textrm{TFP}_{\alpha}=(1-\alpha) d\ln \bar{w} +\alpha d\ln R, \qquad R=r+\delta
\end{align}
That is, productivity improvements accrue either to workers in the form of higher average wages or to capital owners in the form of a higher return to wealth. In the representative-household benchmark $p=0$, supply is perfectly elastic and hence $d\ln R = 0$ in the long run so that all productivity gains from automation eventually accrue to labor  -- the inelastic factor.\footnote{This part of the proposition is in line with papers that studied the impact of automation in settings with a representative household or an infinitely elastic supply of capital, such as \citet{simon,caselli-manning, acemoglu-restrepo-tasks}. In these papers, any improvement in technology must necessarily increase average wages, even if some groups see a reduction in real wages.
} However, as $p$ increases and capital supply becomes more inelastic, an increasingly larger share of these productivity gains accrues to capital in the form of a higher return. When $p>\bar{p}$ so that capital supply is sufficiently inelastic, most of these gains accrue to capital so that average wages fall permanently, $d\ln\bar{w}<0$.\footnote{A complementary intuition comes from the fact that $\bar{w}=(1-\alpha) Y$. From \eqref{eq:dlnY} we know that automation results in an output expansion, with the magnitude depending on the productivity increase $d\ln \textrm{TFP}_{\alpha}$ 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 $1-\alpha$. With sufficiently inelastic capital supply (i.e., $p>\bar{p}$), the displacement effect dominates. With sufficiently elastic capital supply (i.e., $p<\bar{p}$), the output expansion dominates.}

\paragraph{Transitional dynamics and a numerical example:} To illustrate the results from Propositions \ref{proposition: steady state} and \ref{proposition: wages} and how the transition dynamics deviate from the representative-household benchmark, we present a numerical example that shows the response of aggregates to an improvement in automation technologies---a permanent increase in the $\alpha_z$'s.

Appendix \ref{appendix transition} shows that the transition dynamics for the aggregate variables are governed by the stable arm of the system of differential equations
\begin{align*}
\dot{C}-p \dot{K}=&\frac{1}{\sigma}\left(Y'(K)-\delta-\rho\right)(C-p K) -\mu p K\\
\dot{K}=&Y(K)-\delta K-C\\
\frac{\dot{\mu}}{\mu}=&\mu-Y'(K)+\delta+\frac{1}{\sigma}\left(Y'(K)-\delta-\rho\right),
\end{align*}
where $\mu$ is the (common) marginal propensity to consume out of effective wealth. This is similar to the usual system of equations determining the equilibrium in the neoclassical growth model, but is now coupled with an additional forward-looking equation for $\mu$. Moreover, $C-pK$---the consumption not triggered by dissipation shocks---follows a standard Euler equation, with an adjustment $-\mu p K$ which accounts for the share $\mu$ of dissipated wealth $p K$ that would have been consumed.

%\footnote{As for the steady state of the model, we can solve the transition dynamics without knowing the evolution of the wealth distribution. This block-recursive property implies that one can first solve for aggregates and then use the behavior of aggregates to compute the evolution of distributions. This also implies that, in this model, ``macro matters for inequality'' but ``inequality does not matter for macro.''}  

For this example, we use a subset of the parameters from the calibration of our extended model in Section \ref{sec:calibration}. In particular, we feed into the model a gradual increase in the $\alpha_z$'s that starts in 1980 and matches the observed increase in the capital share between 1980  and 2014 from 34.5\% to 43\%. We also assume that automated tasks see a cost reduction of 30\% (so that aggregate TFP increases by 2.5\% due to automation), which makes this a relatively small shock in terms of productivity gains. Finally, we set a value of $p=4.5\%$, which implies a fairly high capital supply semi-elasticity, $d\ln K/dr$, of 50. 

\begin{figure}[!ht]
	\centering 
	\includegraphics[width=.9\linewidth]{numerical_analysis/output/Figure_paper_transition.pdf}
	\caption{Transitional dynamics following an improvement in automation technologies taking place from 1980 to 2014. The solid line presents the transition dynamics in our model. The dashed line presents the transition dynamics in the representative-household benchmark. \label{figure: aggregates}}
\end{figure}

Figure \ref{figure: aggregates} presents the transition dynamics for the labor share, output per worker, the net investment rate, the capital-output ratio, the return to wealth, and the average wage per hour. For comparison, we also plotted the transition dynamics for the representative-household benchmark. 

In our model, automation leads to a modest expansion of output of 11\%. Automation generates a modest expansion in the capital-output and investment-output ratios of 15\% and permanently increases the return to wealth from 6.5\% to 7.2\%. Thus, our model with an upward-sloping long-run 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." Intuitively, while automation leads to an increase in the gross capital share $\alpha= R K/Y$, a substantial fraction of this increase may show up in returns, $R$, rather than quantities, $K/Y$.\footnote{To see this in more detail, recall from above that the 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 can be seen from manipulating the definition of the capital share $\alpha= R K/Y$:
	\begin{equation*}
		\frac{\left(K/Y\right)_{2014}}{\left(K/Y\right)_{1980}} = \frac{\alpha_{2014}/\alpha_{1980}}{R_{2014}/R_{1980}} 
	\end{equation*}
	Now consider two scenarios, and assume in both that $R_{1980}=11.5\%$ as in our calibration. If the rental rate $R$ is unchanged so that $R_{2014}/R_{1980}=1$, as in the representative household model, the capital-output ratio must increase by 25 percent. But suppose instead that the rental rate $R$ rises by one percentage  as in our model. Then $\frac{\left(K/Y\right)_{2014}}{\left(K/Y\right)_{1980}}=\frac{0.43/0.345}{12.5\%/11.5\%} = 1.15$. Rather than increasing by 25\%, the capital-output ratio increases by only 15\%. As we discuss in Appendix \ref{appendix investment}, this is well within the range observed in the data.}

Despite the increase in the average output per worker, and in line with Proposition \ref{proposition: wages}, mean wages go down by 3\% in 2020 and by 1\% in the long run.\footnote{Equation \eqref{equation: tfp and prices} 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 $\alpha=0.345$---the base point of our calibration---shows that a small increase in $R^*$ of 10\% (or 1 percentage point) is enough to ensure that the 2.5\% increase in TFP driven by automation results in a decline in real wages, $\bar{w}$, of roughly 1\%.} This is in contrast to what would happen in the representative-household benchmark, where automation leads to a more pronounced economic expansion
propelled by a boom in investment, a \emph{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 numerical example is fairly elastic (with a semi-elasticity $d\ln K/dr = 50$), the response of macroeconomic aggregates to automation differs significantly from what one would get in a representative household model. 

\subsection{Wealth and Income Inequality\label{sec:ineq}}

We now study the wealth and income distributions. Recall that the steady-state return to wealth satisfies $r^*>\rho$. Therefore, while aggregate capital $K$ is constant in steady state, individual households accumulate wealth at a positive rate, with some experiencing dissipation shocks and losing all their wealth at rate $p$. The combination of accumulation and dissipation gives rise to a unique, non-degenerate distribution of effective wealth, which in turn determines the income distribution. 

\begin{proposition}[Automation and the wealth and income distribution]\label{proposition: wealth inequality} Denote households' effective wealth by  $x_{z,t}:=a_{z,t}+w_z^*/r^*$. The stationary distribution of effective wealth for skill type $z$ is Pareto, with a PDF
	\begin{align*}
	f_z(x)=&\left(\frac{w_z^*}{r^*}\right)^{\zeta}\zeta x^{-\zeta-1}, & \frac{1}{\zeta}:=\frac{1}{p}\frac{r^*-\rho}{\sigma}=\alpha_{net}^*.
	\end{align*} 
	The conditional and unconditional wealth distributions satisfy 
	\begin{align*}
	\Pr(\textrm{wealth}\geq a| z)&=\left(\frac{a+w^*_z/r^*}{w^*_z/r^*}\right)^{-\zeta}, &\Pr(\textrm{wealth}\geq a)=\sum_z \ell_z\left(\frac{a+w^*_z/r^*}{w^*_z/r^*}\right)^{-\zeta};
	\end{align*}
	and the conditional and unconditional income distributions satisfy
	\begin{align*}
	\Pr(\textrm{income}\geq y|z)&= \left(\frac{\max\{y,w^*_z\}}{w^*_z}\right)^{-\zeta}, &\Pr(\textrm{income}\geq y)=\sum_z \ell_z \left(\frac{\max\{y,w^*_z\}}{w^*_z}\right)^{-\zeta}.
	\end{align*}
	Moreover, the tail index $1/\zeta=\alpha^*_{net}$ increases with automation, $\alpha$.
\end{proposition}


The proposition shows that the distribution of effective wealth for skill type $z$ is Pareto with scale $w_z^*/r^*$ and tail index $1/\zeta=\alpha_{net}^*$. 
The driving force behind this result is the random growth process governing the accumulation of wealth: starting from their human wealth $x_z(0)=w_z^*/r^*$, households accumulate capital over time at a rate $(r^*-\rho)/\sigma$. The distribution of wealth is stabilized by the dissipation shocks, which arrive at a rate $p$. Figure \ref{figure: jones} depicts the process of accumulation. This random growth process gives rise to a Pareto distribution \citep[see][]{wold-whittle,steindl,jones} with tail index
\begin{align*}
\frac{1}{\zeta}=\frac{\textrm{individual household accumulation rate}}{\textrm{dissipation rate}}=\frac{(r^*-\rho)/\sigma}{p}=\alpha_{net}^*. 
\end{align*} 
As the formula shows, inequality depends on the ratio of the rate at which households accumulate wealth and the probability with which their wealth dissipates, $p$, and in equilibrium this is exactly given by the net capital share $\alpha^*_{net}$.

\begin{figure}[!ht]
	\centering
	\includegraphics[width=.8\linewidth]{graphs/figure_steindl.pdf}
	\vspace{-2.3cm}\caption{Dynamics of effective wealth accumulation.\label{figure: jones}}
\end{figure}

The reason why we get a Pareto tail is that, while the average household retains a constant level of wealth due to the dissipation shocks, some fortunate households manage to accumulate wealth exponentially at a rate $(r^*-\rho)/\sigma>0$, populating the tail of the wealth distribution over time. A higher return means more rapid accumulation and a fatter tail. Although our model delivers this insight in a stylized way, we see dissipation shocks as a tractable way to capture the nexus between returns to wealth and inequality that is present in a broader class of more realistic models where the stochastic accumulation of wealth results in fat-tailed distributions. This broader class includes models with stochastic bequest motives \citep[see][]{benhabib-bisin-altruism}, models with stochastic returns or discount rates \citep[see][]{krusell-smith, benhabib-bisin-zhu,benhabib-bisin-zhu-JET}, or the extension of our model presented in Section \ref{sec:extension}. While the details differ, in all of these models wealth accumulation at the top of the distribution is governed by a random growth process and this creates a natural nexus between returns and inequality \citep[see][]{GLLM}. In these models, technological changes that lead to a higher return to wealth increase the rate at which successful households accumulate wealth and hence lead to a more fat-tailed wealth distribution.  

The distribution of effective wealth is important in and of itself because it tells us about inequality in consumption and welfare (a corollary of Lemma \ref{lemma: households policy st st}). 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 \ref{proposition: wealth inequality} characterize the wealth and income distributions. The formulas show that technology affects both distributions through two channels: first, via wages which determine the distributions' scale parameters; second -- and this is more novel -- via the net capital share which determines the return to wealth and the distributions' tail index, $1/\zeta$. 

%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 return to wealth and the tail index of the income distribution, $1/\zeta$). 

The formulas in Proposition \ref{proposition: wealth inequality} illustrate why automation might generate uneven growth. By depressing the real wages of displaced workers, automation reduces incomes at the bottom of the distribution, especially for displaced workers with no assets. By raising the return to wealth, automation generates rising income inequality at the top driven by the rapid and uninterrupted capital accumulation of some households. Both effects depend crucially on the less than perfectly elastic supply of capital, which implies that a smaller share of the productivity gains from automation accrues to workers and more accrues to wealthy capital owners in the form of higher returns to their wealth. 

 \vspace{-3mm}
\paragraph{Composition of top income earners:} Proposition \ref{proposition: wealth inequality} can be used to study the composition of top income earners and quantify their share of income. As we show in Appendix \ref{appendix:composition}, for small $q$,  the share of national income held by the top $q$ (e.g., the top 0.1\%) is
\begin{align}
S(q)=\Lambda q^{1-\alpha^*_{net}},\label{eq:topq share}
\end{align}
where $\Lambda$ is a constant that depends on the wage distribution. When there is no heterogeneity in wages, $\Lambda=1$ and we obtain the usual formula for the top $q$ percent share in a Pareto distribution \citep[see][]{jones}. One implication of this formula is that technology might affect $S(q)$ through wages (via the $\Lambda$ term) but this would cause a proportional increase in $S(q)$ for all $q<\bar{q}$. Instead, by raising $\alpha^*_{net}$, technology will increase the share of income held by higher percentiles disproportionately. 

Moreover, the probability that someone with a wage $w_z$ reaches the top $q$ percent is
\begin{align*}
\Pr(\textrm{skill}=z| \textrm{top } q)=\frac{\ell_z w_z^{1/\alpha_{net}^*}}{\sum_v \ell_v w_v^{1/\alpha_{net}^*}},
\end{align*}
which shows that high wage households are over-represented at the top of the income distribution as is the case in the data. Interestingly, wages become less relevant in determining who reaches the top in a more automated world with higher returns. The appendix also shows that, as we move up the income distribution, households derive more of their income from capital ownership, since the tail is increasingly made of successful investors for whom labor income represents a small part of their earnings. 

%As shown by these formulas, technologies that increase the relative wage of high wage earners, like the automation of middle and low-skill jobs and skill-biased technical change more generally, will increase the share of high skill workers and the importance of labor income among top earners. On the other hand, by raising the net capital share, automation also makes relative wages less important in determining the composition of top income earners, and reduces the importance of labor income at the top of the distribution. Intuitively, many more households that managed to accumulate assets for long periods at the higher rate brought by automation will be among top earners independently of their wage.
 \vspace{-3mm}
\paragraph{Other mechanisms:} Starting with \citet{meade} and more recently with \citet{piketty-capital}, several authors have emphasized that a rise in the net capital share will generate income 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 mechanically increases income inequality. This argument differs from ours, since we emphasize how technology might lead to a more concentrated ownership of wealth. 

To see the difference, denote by $S(q)$ the top $q$ percent income share. Also denote by $\tilde{S}_k(q)$ and $\tilde{S}_\ell(q)$ the share of aggregate capital income and labor income earned by the top $q$ percent of the distribution of \emph{total} income. It then follows that
\begin{equation}\label{eq:meade}
S(q)=\alpha_{net} \tilde{S}_k(q)+(1-\alpha_{net}) \tilde{S}_\ell(q).
\end{equation}
This formula was already used by \citet[][p.34]{meade} and we re-derive it in Appendix \ref{appendix:composition}. Next, we decompose changes over time in $S(q)$ as 
\begin{equation*}
dS(q)=\underbrace{\vphantom{\Bigg(}(\tilde{S}_k(q)-\tilde{S}_\ell(q)) d \alpha_{net}}_{\textrm{Compositional effect at }q} + \underbrace{\vphantom{\Bigg(}\alpha_{net} d \tilde{S}_k(q) +(1-\alpha_{net})  d \tilde{S}_\ell(q)}_{\textrm{Changes in within factor distribution at }q}.
\end{equation*}
This decomposition highlights the difference between compositional effects and our mechanism. In our mechanism, automation increases the concentration of capital ownership, leading to a rise in income inequality via changes in the within factor term. The decomposition also shows that composition effects are small in practice. For example, for the top 1\% in the US in 1980 we had $\tilde{S}_\ell(q)=8.3\%$ and  $\tilde{S}_k(q)=15.3\%$, so that an increase in the net capital share of one percentage point would yield an increase in the top 1\% income share of only $(\tilde{S}_k(q)-\tilde{S}_\ell(q))  d \alpha_{net} = 0.07 \times 0.01 = 0.07$ percentage points.\footnote{In 1980, the top 1\% earned 10\% of US income. 35\% of the income of that group was capital income and 65\% was labor income (as opposed to a 20\% and 80\% breakdown in the aggregate) \citep[see][]{piketty-saez}. Using these numbers, we have $\tilde{S}_k(q) = \frac{\alpha_{net}(q)}{\alpha_{net}} S(q)=15.3\%$ and 
$\tilde{S}_\ell(q) = \frac{1-\alpha_{net}(q)}{1-\alpha_{net}} S(q)=8.3\%$. } 
This effect is small when compared to the effects in our model and the observed increase of top tail inequality in the  data, where the top 1\% income share increased from 10\% to 20\% in 1980--2012. 

\section{Extended Model: Multiple Assets and Growth\label{sec:extension}}
\vspace{-3mm}

We now present an extended version of our model that clarifies
how automation and concurrent changes in capital markets affect the returns of different assets. This extension identifies the key returns that are linked to rising wealth inequality and explains how these returns can be measured. All proofs and derivations are in Appendix \ref{appendix extension}.


\subsection{An Extended Version of the Model}\label{sec:extended_model}

The extension modifies the baseline model in four ways:
\begin{enumerate}
	\item We assume that there are two assets, risky capital (or equity) and safe bonds, and that only a fraction of the population can invest in capital, but are constrained to do so because of risk or financial frictions. 
	
	\item We include markups in the production sector, which allows for the possibility that part of the decline in the labor share might be due to markups, and that the return to capital may include some of the profits generated by markups. 
	
	\item We model the taxation of capital, which will also affect capital markets.
	
	\item We allow for factor-neutral technological change causing sustained growth at a rate $g$.   
\end{enumerate}


\vspace{-3mm} 
\paragraph{Households and Investors:} Households maximize their expected lifetime utility. We work with Duffie-Epstein-Zin preferences
$$v_0=\mathbb{E}_0 \int_0^\infty f(c_t,v_t)dt \quad \mbox{with} \quad f(c,v)=\frac{\rho (1-\gamma) v}{1-\sigma}\left(\left(\frac{c}{((1-\gamma) v)^{1/(1-\gamma)}}\right)^{1-\sigma}-1\right),$$
where $\gamma>0$ measures risk aversion and $\sigma>0$ is the inverse of the IES. As in the baseline model, households face dissipation shocks at rate $p$ so that the discount rate $\rho$ equals $\varrho+p$.

A fraction $\chi\in (0,1)$ of  households can invest in both capital and safe bonds. We refer to these households as investors, and assume that investors come with equal probability from all skill groups. The remaining $1-\chi$ households can only invest in bonds.

The bond pays a safe return $r_B$. The return to capital instead features idiosyncratic risk: the after-tax return over a time interval of length $dt$ equals $dR_t = r_K dt + \nu dW_t$, where $r_K$ is the average after-tax return per dollar of capital, $\nu$ is the standard deviation of returns, and $W_{t}$ is an idiosyncratic standard Brownian motion. This return risk might arise from shocks to the productivity of businesses or might also capture idiosyncratic capital gains.

The budget constraint of an investor with wage $w_z$ is therefore given by
\begin{equation*}
	da_{z,t}+db_{z,t} = (r_K a_{z,t} + r_B  b_{z,t} +w_z +T- c_{z,t} )dt +  a_{z,t} \nu dW_t.
\end{equation*}

\vspace{-3mm}
\paragraph{Taxation:} Capital income from bond holdings and equity is taxed at a rate $\tau$ and the revenue is returned to households via a common lump-sum transfer $T$.\footnote{To simplify the exposition, we assume that idiosyncratic capital returns---the term $ a_{z,t} \nu dW_t$ in investors' budget constraint---are not taxed. If these were taxed, the after-tax income risk would be $(1-\tau)  a_{z,t} \nu dW_t$, and taxes would reduce capital income risk. Instead, here we are only interested in the effect of taxes on the expected after-tax returns $r_K$ and $r_B$.} 

\vspace{-3mm}
\paragraph{Technology and Market Structure:} Firms operate the same technology as in our baseline model but now charge a constant markup $\varphi\geq 1$ with all profits accruing to equity owners. In addition, to generate balanced growth, we assume that $\psi_z$ grows at a constant rate of $g>0$ for all skill groups $z$. 

\vspace{-3mm}
\paragraph{Balanced-growth Equilibrium (BGE):} In a balanced-growth equilibrium, output, capital, bond holdings, and wages grow at a rate $g>0$, and the returns $r_K$ and $r_B$ are constant, with $r_B>g$ so that human wealth $(w_z+T)/(r_B-g)$ is finite. Moreover, in the main text, we will focus on characterizing the BGE of a closed economy in which bonds are in zero net supply. We assume that $\rho+(\sigma-1)g>0$, which is a sufficient condition to ensure that the equilibrium exists and features finite wealth.

Define the return to investors' wealth as
\begin{equation}\label{eq:r_W_def}
	r_W:= \underbrace{\kappa r_K+(1-\kappa) r_B\phantom{\bigg|}}_{\text{portfolio return}}+\underbrace{\frac{1}{2}(\sigma-1)\gamma\nu^2{\kappa}^2\phantom{\bigg|}}_{\text{risk adjustment}}
\end{equation}
where $\kappa:=\min\left\{\frac{1}{1-\theta}, \frac{r_K-r_B}{\gamma  \nu^2}\right\}$ denotes the share of investors' effective wealth invested in risky capital. The formula for $r_W$ takes the return to investors' portfolio, $\kappa r_K + (1-\kappa) r_B$, and adjusts it by the extra term $\frac{1}{2}(\sigma-1)\gamma \nu^2\kappa^2$, which accounts for the effect of capital income risk on investors' saving decisions.\footnote{In the presence of capital income risk,  investors recognize that a given investment will bring them a lower future consumption equivalent. This generates an income effect (the lower value of future consumption pushes current consumption down and savings up) and a substitution effect (the lower consumption equivalent pushes current consumption up and savings down). For $\sigma=1$ these effects cancel, but for $\sigma>1$, as in most calibrations of the (inverse of the) intertemporal elasticity of substitution, the income effect dominates, and capital income risk leads to higher savings, which shows up in our formula as if investors faced a higher return. See also the discussion in \citet{Obstfeld94} and \citet{Angeletos07}.} The return $r_W$ determines investors' optimal wealth accumulation. In particular, their optimally chosen effective wealth (defined now as $x_{z,t}=a_{z,t}+b_{z,t}+\frac{w_z+T}{r_B-g}$)  follows a random growth process of the form
\begin{align*}
	d x_{z,t} = \frac{r_W-\rho}{\sigma} x_{z,t} dt +\kappa \nu x_{z,t} dW_t,
\end{align*}
where, in addition, $x_{z,t}$ is reset to $x_{z,0}=\frac{w_z+T}{r_B-g}$ with probability $p$. Non-investors solve the same problem as in the baseline model so that Lemma \ref{lemma: households policy st st} applies and their effective wealth (given by $x_{z,t}=b_{z,t}+\frac{w_z+T}{r_B-g}$) also resets with probability $p$ and otherwise evolves as
\begin{align*}
	\dot{x}_{z,t}=\frac{r_B-\rho}{\sigma} x_{z,t}.
\end{align*}

In what follows, define the \emph{after-tax share of capital inclusive of profits} in GDP
\begin{equation}\label{eq:alpha_pi}
	\widetilde{\alpha}:=(1-\tau)\left(\frac{\varphi-1}{\varphi}+\frac{\alpha}{\varphi}\right),
\end{equation}
which increases in $\alpha$, $\varphi$ and the keep rate $1-\tau$. As we will see, all these primitives have the same effects on returns and wealth inequality operating through $\widetilde{\alpha}$. 

\begin{proposition}\label{proposition:bg}
	The model admits a BGE where aggregates grow at a constant rate $g$.
	\begin{itemize}
		\vspace{-3mm}
		\item  If investors are risk neutral, $r_W^*=r_K^*=r_B^*=r^*>\rho+\sigma g$, and $r^*$ is increasing in $\widetilde{\alpha}$. 
		
		\item If investors are risk averse,  
		\begin{align*}
			r_W^*,r_K^*>r_B^*, \rho+\sigma g.
		\end{align*}
		
		\item In both cases, the distributions of wealth and income have Pareto tails with tail index
		\begin{align}
			\frac{1}{\zeta}:=\frac{r^*_W-\rho-\sigma g-\frac{\sigma {\kappa^*}^2\nu^2}{2}+\sqrt{\left(r^*_W-\rho-\sigma g-\frac{\sigma{\kappa^*}^2\nu^2}{2}\right)^2+2\sigma^2{\kappa^*}^2\nu^2p}}{2p\sigma}.
			\label{eq:tail index}
		\end{align}
	\end{itemize}
\end{proposition}

The first part of the Proposition shows that, when investors are risk neutral, the return to capital equals that on bonds. This common return satisfies a generalization of \eqref{equation: return premium}:
\begin{align}
	r^*=\rho+\sigma g + p \sigma \alpha^*_{net}, \quad\quad \alpha^*_{net}:=\frac{(r^*-g)K^*}{(r^*-g)K^*+\sum_{z}w^*_z\ell_z} \label{eq: returns with g},
\end{align}
with the difference that now, $\alpha^*_{net}$ is the capital share net of depreciation, capital taxes, and growth (but including all profits). Appendix \ref{appendix extension} derives demand and supply curves that uniquely define $r^*$ and shows that both $r^*$ and $\alpha^*_{net}$ increase in  $\widetilde{\alpha}$.

The second part of the proposition shows that, if investors are risk averse, the return to capital and investors' wealth exceeds the return on bonds, and also $\rho+\sigma g$ -- the level of returns in a BGE of the representative-household benchmark.\footnote{Appendix \ref{appendix extension} provides a more general version of the model where investors face borrowing constraints. This is another force that could drive a wedge between bond rates and the return to capital. } 

The third part of the proposition shows that top-tail inequality depends on the return to investors' wealth $r_W^*$. More precisely, top tail inequality depend on the \emph{return gap:} 
\begin{align}
	\textrm{return gap}:=r_W^*-\rho-\sigma g.\label{equation:adjusted return}
\end{align} 
To see why, note that the wealth and income of an average household grow at a rate of $g$. Thus, the tail of these distributions will be populated by successful investors who manage to accumulate wealth uninterruptedly at an average rate of $(r_W^*-\rho)/\sigma$ that exceeds $g$. The return gap measures precisely how fast successful investors accumulate wealth relative to the rest of the economy, determining the thickness of the income and wealth distribution. In a way, our formula for the return gap and its link to top-tail inequality generalizes Piketty's ``$r-g$'' insight. In fact, the return gap is equal to $r-g$ in the special case when preferences are logarithmic and there is no discounting (so that investors reinvest all their wealth).

The expression for the return gap \eqref{equation:adjusted return} and its link to top inequality in equation \eqref{eq:tail index} clarify how wealth inequality will respond to a change in equilibrium safe returns $r_B^*$ or returns to capital $r_K^*$. The answer depends on investors' portfolio shares $\kappa^*$. In particular, note that if investors are primarily invested in business capital ($\kappa^*$ close to one), inequality will be largely unaffected by movements in the safe return $r_B^*$. In fact, \eqref{equation:adjusted return} shows that if investors are leveraged, $\kappa^*>1$, it is a \emph{reduction} in safe rates that increases inequality. Instead, the return to business capital $r_K^*$ will be tightly connected to inequality.

Equation \eqref{eq:tail index} shows that tail inequality will also depend on investors' exposure to risky capital, $\kappa$. This is because returns are stochastic, and so some investors will obtain even larger returns in a sustained way, generating a thicker tail for income. This second effect implies that, relative to the formula for the tail index in Proposition \ref{proposition: wealth inequality}, the extended  model with additional capital income risk generates higher income inequality at par with what we see in the data.\footnote{Our extended model provides a more realistic description of the process by which investors reach and leave the top of the distribution. Because of capital income risk, some investors will quickly reach the top after obtaining a series of high returns. By the same token, there is now much more churn among this group, with investors dropping out of the top after a dissipation shock or obtaining a series of low returns. Recent work documents the importance of churn at the top of the wealth distribution \citep{gomez,zheng}. }

The implications of our extended model for returns and inequality and their connection to technology and other factors affecting the demand for capital can be illustrated by approximating the equilibrium around $\widetilde{\alpha}=0$.\footnote{Proposition \ref{proposition:bg and changes in alpha} in Appendix \ref{appendix extension} provides a complementary result and shows that, for $\widetilde{\alpha}<\bar{\alpha}$,   $r^*_W$,  $r^*_W - \rho - \sigma g$,  $r_K^*-r_B^*$, and $1/\zeta^*$  increase  in $\widetilde{\alpha}$ in the full non-linear model.   } 

\begin{proposition}\label{proposition:approximation}
	For small values of $\widetilde{\alpha}$, the economy admits a unique BGE. In this equilibrium, aggregates grow at a constant rate $g$ and, to a first-order approximation, we have
	\begin{align*}
		r_K^*=&\rho+\sigma g+ \left(p\sigma + \frac{\gamma\nu^2}{\chi}\right)\alpha^*_{net} &
		r_B^*=&\rho+\sigma g+ p\sigma \alpha^*_{net}&
		\kappa^*=&\frac{1}{\chi}\alpha^*_{net},
	\end{align*}
	where $\alpha^*_{net}=\frac{\rho+(\sigma-1)g}{\rho+\sigma g+\delta} \widetilde{\alpha}$ gives the capital share net of depreciation and growth.
	Along this BGE, inequality satisfies (to a first-order approximation)
	\begin{align*}
		\frac{1}{\zeta^*}=\alpha^*_{net}+\alpha^*_{net} \frac{\frac{\nu^2}{p\chi^2}}{1+\sqrt{1+2\frac{\nu^2}{p\chi^2}}},
	\end{align*}
	and, to a second-order approximation,  the return gap is given by 
	\begin{align*}
		r_W^*-\rho-\sigma g=p\sigma\alpha^*_{net}+\frac{1}{2}(\sigma+1)\frac{\gamma\nu^2}{\chi^2}\left(\alpha^*_{net}\right)^2.
	\end{align*}
\end{proposition}

The proposition shows that the return gap and top-tail inequality depend on $\widetilde{\alpha}$ and its effect via the net capital share $\alpha^*_{net}$. Qualitatively, this is the same insight as in our baseline model. But now, our model explicitly distinguishes between the return to capital and the return to bonds, and provides us with an explicit formula for the return gap in this extended model. Importantly, our model now shows that automation widens the spread between $r_K-r_B=\frac{\gamma\nu^2}{\chi}\alpha^*_{net}$. The extended model can therefore make sense of a situation where a decline in $\rho$ or $g$ reduce $r_B$ (with no effect on inequality) while at the same time automation raises $r_K$, the return gap, and top-tail inequality. 

The formula for $1/\zeta^*$ highlights the fact that top-tail inequality is higher than in our baseline model. The first term in the formula captures the fact that a high value of $\widetilde{\alpha}$ requires a higher drift in the process of capital accumulation among investors---the same force as in our baseline model. In particular, when $\nu=0$, the formula in \eqref{eq:tail index} boils down to $(r_W^*-\rho-\sigma g)/(p \sigma)=\alpha^*_{net}$, which is the natural extension of 
the formula in our baseline model to a growing economy. The second term captures the fact that returns are stochastic, and investors' exposure to this extra risk is also determined by $\widetilde{\alpha}$ and given by $\kappa^*=\alpha^*_{net}/\chi$. This extra source of capital income risk generates a fatter tail, as some investors are lucky to obtain high returns consistently.\footnote{This mechanism continues to operate and becomes stronger when labor income is risky. In this case, automation reduces the importance of labor income risk (since wages became a smaller share of income) and increases the importance of capital income risk for investors. This shift in the nature of risk still contributes to higher inequality at the top because tail inequality depends on the multiplicative capital income risk and not on additive labor income risk. Moreover, risk averse investors will respond to the decline in background labor income risk by increasing their exposure to risky capital \citep[see, for example, section 6.1.1 in][]{campbell_viceira}, which leads to a further expansion of $\kappa$ and a more pronounced increase in wealth inequality in response to improvements in automation technologies.} For example, taking $\nu=7.7\%, p=4.5\%$ and $\chi=6.6\%$ (our calibrated parameters below), and for $\alpha^*_{net}=15\%$, we get $1/\zeta^*=0.66$. The extended model is thus capable of producing levels of wealth inequality that match  the data.  

\subsection{Other Drivers of an Increasing Net Capital Share}

Our extended model implies that automation is not the only driver of rising wealth inequality. Higher markups ($\varphi$) or lower capital taxation ($\tau$) have similar effects. Higher markups and lower capital taxes also increase the \emph{after-tax share of capital inclusive of profits} $\widetilde{\alpha}$ defined in equation \eqref{eq:alpha_pi} and Proposition \ref{proposition:approximation} showed that it is this object that determines asset returns and top wealth inequality. Intuitively, all of automation, higher markups and lower capital taxes increase the demand for capital and hence the return to wealth, the return gap, and the spread between the return to capital and safe assets. 

While our paper's main focus is on understanding the consequences of automation on income and wealth inequality, we view our framework's ability to speak to these alternative trends as one of its key advantages.\footnote{For example, \citet{cao-2017} use a similar framework to study the role of financial deregulation and lower capital taxes on wealth inequality.} In our quantitative exercise below and the discussion that follows, we study the effects of automation on inequality. But readers should keep in mind that the model-implied effects of automation on capital markets, returns, and wealth inequality at the top of the distribution are identical to those generated by an increase in markups or a reduction in capital taxes causing the same reduction in the after-tax share of capital inclusive of profits. That being said, these other forces would have different implications for wage inequality and aggregates such as investment.

\subsection{Has the Return Gap Risen?\label{sec:returns}}

The extended model shows that top-tail inequality depends on the return gap obtained by investors, which is in turn determined in equilibrium by the extent of automation, markups, and capital taxation in the economy. We turn to a measurement exercise that shows that the return gap has risen since 1980 despite a secular decline in safe rates. We highlight the main takeaways and relegate a detailed discussion of data and measurement to Appendix \ref{appendix returns}.

Figure \ref{figure:grr} plots estimates of returns on bonds---our proxy for $r_B$---
and three measures of returns to business capital---our proxy for $r_K$---in the US. The blue line depicts the long-run return to safe assets, measured using \citeauthor{holston-laubach-williams}' (\citeyear{holston-laubach-williams}) estimate of the real natural interest rate for the United States. The safe rate declined by 3 percentage points in 1960--2020. Most of the decline can be explained by demographic factors or developments in international markets, like the ``savings glut" (captured by a lower $\rho$ in our model) and slower trend productivity growth \citep[see][]{rachel-summers,auclert-malmberg-martenet-rognlie}. 

\begin{figure}[ht]
	\centering
	\includegraphics[width=.7\linewidth]{measuring_returns/output/Figure_paper_returns.pdf}
	\caption{Estimates of the return to US capital, $r_K$, and the safe rate, $r_B$, \footnotesize \newline \textit{Notes---}see Appendix \ref{appendix returns} for data sources and measurement details.\label{figure:grr}}
\end{figure}

In contrast with what we found for the safe rate, all three measures of returns to capital in Figure \ref{figure:grr} exhibit an increasing trend, rising by about 1--2 percentage point from 1970--2020. The first series is from \citet{gomme-ravikumar-rupert}, who measure the return to US business capital as $r_K:=Y_k/K+\Pi_k$. Here $Y_k$ is net capital income (net output generated by business capital minus labor compensation and taxes), $K$ is the stock of capital measured at replacement cost, and $\Pi_K$ captures expected capital gains arising from changes in the price of capital over time.\footnote{Appendix \ref{appendix returns} Figure \ref{figure:grr components} shows that, relative to the income component, capital gains are more volatile but exhibit no visible trend. Thus, the upward trend in returns to business capital is driven by the income component of returns. Also, the series from \citet{gomme-ravikumar-rupert} excludes housing and focuses only on productive assets.} We plot a version of their measure that sets expected capital gains to their average realization over the 1960--2020 period. We complement this series with two additional measures of the return to capital: the return to productive (i.e. non-financial) assets in the non-financial corporate sector, and the return to business capital in the non-corporate sector. These returns are also measured as $r_K:=Y_k/K+\Pi_k$, with productive capital valued at replacement cost using data from NIPA and the Integrated Macroeconomic Accounts---IMAs, as described in Appendix \ref{appendix returns}.\footnote{Appendix \ref{appendix returns} provides different proxies for returns to equity, including the dividend yield, realized 10-year returns, and realized yearly returns. Depending on the measure, returns to equity have either remained stable or have increased over time. The exception is a decline in the dividend yield in the recent period since the 1980s; however the dividend yield does not display a clear trend when viewed over a longer horizon since the 1960s. These findings align with \citet{greenwald-et-al} who document an increase in realized stock market returns over the postwar period and estimate that 2.1 pp of this increase are accounted for by persistent factor-share shocks, which in our model correspond to increases in $\tilde{\alpha}$.} All three series paint a consistent picture of a gradual increase in the return to capital since 1980. 
 
We now turn to estimates of the return to investors' wealth, $r_W$, and the return gap $r_W - \rho - \sigma g$. We compute the return to investors' wealth  as
$r_W=\kappa r_K+(1-\kappa) r_B$,
where $r_K$ is the return to productive capital from \citet{gomme-ravikumar-rupert}
and $r_B$ is given by \citet{holston-laubach-williams} estimate of the US natural interest rate.\footnote{Our calculations exclude housing and mortgages from investors' portfolios. We also focus on investors' financial portfolios. Thus our calculations are particularly relevant for the tail of the wealth distribution, where the importance of human wealth diminishes. We also ignore the risk-adjustment term in equation \eqref{eq:r_W_def}, which generates conservative estimates of the increase in $r_W$ (if $\sigma\geq1$ as in our calibration).} 

We provide several estimates for the return to investors' wealth and the return gap based on different measures of $\kappa$. In our model, $\kappa$ equals the productive capital held by investors divided by their net worth with values exceeding 1 meaning that investors are leveraged. However, when computing $\kappa$, we have to adjust for the fact that: (i) not just investors but also the businesses they invest in may be leveraged (so that these businesses assets exceed their equity); (ii) not all equity is held by investors; and (iii) investors also hold government and corporate bonds. This implies
$$\kappa=\frac{\textrm{business capital}}{\textrm{business equity}+\textrm{bonds}-\textrm{personal debt}},$$ 
where the business capital held by investors is given by
$$
\textrm{business capital}=\textrm{equity share held by investors}\times\textrm{firms' total assets}.
$$ 
This approach accounts for (i) because part of firms' assets are financed through liabilities, and so firms' productive capital will typically exceed their equity. Thus, our measure for $\kappa$ will exceed 1 when investors are leveraged themselves or invest in highly leveraged businesses.

We compute $\kappa$ for households at the top of the income and wealth distribution, which we identify with investors in our model. We measure all the components using data from Distributional National Accounts (DINAs) of \citet{piketty-saez-zucman} and the Federal Reserve's Distributional Financial Accounts (DFAs). We find that $\kappa$ lies between 1 and 1.5 depending on the data source, and on whether we include government bonds in our calculations. The measures that exclude government bonds, and can therefore be more closely mapped to our model, are all close to 1.3 in 1980 and exhibit a mild upward trend since. These values suggest that investors at the top of the distribution top up every dollar of their own resources with 30 cents of borrowing from households. These findings are consistent with the evidence of substantial leverage at the top, including on the balance sheets of businesses held by those at the top \citep[see, for example, ][who estimate values of $\kappa$ of 1.5  for entrepreneurs]{gomez-gouin-bonenfant}. 
%\footnote{Note that if we compute an aggregate $\kappa$ across all households in the economy using the above definition we would obtain a value of $\kappa=K/(K+B)<1,$ where $B$ denotes government bonds. This aggregate $\kappa$ is therefore uninformative of the portfolios and leverage held by investors, which is what determines their returns and contributes to wealth inequality.}


Figure \ref{figure:returns to wealth} provides our estimates for the return to investors' wealth based on our estimates for $r_K, r_B$ and $\kappa$. 
We provide estimates for the top 1\% of earners (from the DINAs) and the wealthiest 1\% (from the DFAs). Since our model does not feature government bonds, we also report estimates that exclude government bond holdings from investors' portfolios (the dashed lines). Finally, we also show the return that accrues to a  fictional investor who holds no corporate bonds or personal debt (the solid black line). All of the resulting measures of returns to wealth have increased over time, by 0.5--2 percentage points since the 1980s. This increase is driven both by the rise in the return to productive capital $r_K$ and by the decline in the return to bonds $r_B$, which raises the return to investors' wealth when they take leverage positions with $\kappa>1$, as in the data.

\begin{figure}[ht]
	\includegraphics[width=\linewidth]{measuring_returns/output/Figure_returns_and_adj_returns_sigma2.pdf}
	\caption{\label{figure:returns to wealth} Estimated returns to wealth (left panel) and the return gap for $\sigma=2$ (right panel). \textit{Notes---}see Appendix \ref{appendix returns} for details. }
\end{figure}

Our theory's precise implication is that it is the return gap $r_W - \rho -\sigma g$, rather than the return to wealth itself, that drives wealth and income inequality. The right panel in Figure \ref{figure:returns to wealth} combines the data on returns shown in the left panel with the estimates of the trend growth rate of the economy from the Congressional Budget Office, to compute the return gap over time as $r_W-\sigma g$.\footnote{In our calculations we use a 20-year moving average of the growth rate of per-capita real potential GDP, together with the CBO latest (2021) forecast, as the proxy for the expected trend growth rate of the economy. Our measure of $g$ declines by around 1.2 percentage point relative to 1980. We also ignore the discounting term $\rho$. This simplification is conservative because the effective discount rate has arguably declined over time, e.g. due to demographic trends, so that it would have pushed up the return gap.} The decline in trend growth rate means that the return gap has increased by more than the return to investor's wealth since the 1980s.


These estimates support the idea that, despite the large decline in the safe rate of return, returns to wealth and in particular the return gap rose by 1.5--3.5 percentage points since the 1980s. This divergence is driven by (i) the increase in the returns to US productive capital, $r_K$, (ii) the fact that households at the top of the income and wealth distribution are leveraged, and so the decline in safe rates actually increased the return to their portfolios, and (iii) the lower trend growth during this period. Our theory highlights the fact that this growing return gap is a potential driver of the observed rise in wealth and income inequality.

\section{Calibration and Implications for Aggregates}\label{sec:calibration}

To study the effects of automation, we feed exogenous changes in $\alpha_z$ into the extended model and explore the consequences for aggregates and inequality. We will focus on changes in automation between 1980 and 2014, a period with a marked shift in technology towards automation \citep{acemoglu-restrepo-jep}, especially of routine tasks  \citep{autor-levy-murnane,acemoglu-autor}.\footnote{Our focus on this period does not imply that there was no automation before. As discussed in \citet{acemoglu-restrepo-jep}, before 1980 jobs were automated in some specific industries and tasks, but this was counteracted by other technological improvements that raised labor shares in other industries or introduced new labor-intensive roles for labor in production. As a result, the aggregate labor share---the key object determining how technology affects wealth inequality---remained stable before 1980. }  This exercise isolates the effects of automation while holding other drivers of distributional and macroeconomic trends at their 1980 levels.

\subsection{Calibration}

We interpret $z$ as indexing workers in a given percentile of the wage distribution, so that we have 100 skill groups. We think of increases in $\alpha_{z}$ from 1980 to 2014 as driven by the automation of routine tasks in which group $z$ specializes. To capture this, we adopt the shift-share specification
\begin{align}
	\frac{1}{1-\alpha_{z,2014}}-\frac{1}{1-\alpha_{z,1980}}=\omega^R_{z} \left(\frac{1}{1-\alpha_{2014}}-\frac{1}{1-\alpha_{1980}}\right),\label{equation: alpha z}
\end{align}
which assumes that the observed decline in the labor share from 1980 to 2014 was entirely driven by automation, and then apportions the increase in automation across skill groups based on their relative exposure to routine jobs. The change in $\alpha_{z}$ depends on two factors: First, we have the revealed comparative advantage of workers in each percentile in routine jobs, $\omega^R_{z}$, which adjusts for the incidence of automation.\footnote{This is given by 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. We measure this term using data from the 2000 Census---a point in the middle of our study period---and based on the ONET definition of routine jobs in \citet{acemoglu-autor}. Appendix \ref{appendix calibration} provides the details and a derivation of \eqref{equation: alpha z}.} Second, we have a common shift term that captures the overall extent of automation taking place in this period. We benchmark this shift by assuming that the observed decline in the labor share during 1980--2014 is due to automation, which implies that the shift term matches the increase in the capital share during this period (from $\alpha_{1980}=34.5\%$ to $\alpha_{2014}=42.8\%$ in the BLS series for the non-financial corporate sector).\footnote{In particular, we view this calibration as asking: if the observed decline in the labor share were explained by the automation of routine jobs, what would the distributional and aggregate consequences be?	
We recognize that there are other forces behind the decline in the labor share, including rising markups \citep{autor-etal,de-loecker-eeckhout}. Although our calibration assumes no markups, one could also use our model and feed a combination of rising markups and automation that matches the observed decline in the labor share. The extended model in Section \ref{sec:extension} shows that this would create similar implications for wealth inequality at the top, since both shocks affect top inequality via $\tilde{\alpha}$.  
}  Finally, we normalize $\alpha_{z,1980}$ in 1980 to be equal across all $z$, which requires $\alpha_{z,1980}=\alpha_{1980}=34.5\%$---the gross capital share in 1980.

This procedure results in the 1980 and 2014 values for $\alpha_{z}$ plotted in Figure \ref{figure: paths alpha}, which shows greater exposure to the automation of routine jobs among workers at the middle and bottom of the wage distribution. The average change in $\alpha_{z}$  is of 8.4 percentage points, which roughly matches the observed decline in the labor share during this period.  Finally, in terms of timing, we feed a smooth increase in $\alpha_{z}$ from its value in 1980 to its final value in 2014, though this only affects the transitional dynamics presented in Figure \ref{figure: aggregates}. 

\begin{figure}[!ht]
	\centering 
	\includegraphics[width=.65\linewidth]{numerical_analysis/output/figure_paper_shock.pdf}
	\caption{Calibrated $\alpha_z$ by wage percentile in 1980 and the new steady state.\label{figure: paths alpha}}
\end{figure}

For the remaining labor parameters, we calibrate $\eta_z$ to match the wage distribution in 1980 (obtained from the 1980 Census). The $\eta_z$'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 $\psi_z$ to ensure that human labor is 30\% more costly than capital in automated tasks. This number is in line with studies exploring the cost-saving gains from industrial robots \citep[see][]{acemoglu-restrepo-robots}.  As already discussed, this implies that automating a task reduces its cost by 30\%, and that, to a first-order approximation, the productivity gains from the calibrated increase in $\alpha_{z,t}$ between 1980 and 2014 are only 2.4\%---close to 10\% of the total increase in TFP during this period \citep{fernald-tfp}.

\begin{table}[!ht]
	\caption{List of calibrated parameter values}\label{table: calibration}
	\centering
	\resizebox{.97\linewidth}{!}{\begin{tabular}{clcl}
			\toprule
			\multicolumn{2}{l}{\textsc{Description}} & \textsc{Value} & \textsc{Target / Source} \\
			\midrule \\[-1.5ex]
			\multicolumn{2}{l}{\textsc{Preferences}} & &  \\
			$\sigma$ & Inverse IES& 2 & Standard calibration \\
			$\gamma$ & Risk aversion & 2 & Standard calibration \\
			$p$ & Dissipation rate (yearly) & 4.5\% & Target capital-supply elasticity $d\ln K/dr \approx 50$ \\ 
			$\rho$ & Discount rate (yearly) & 1\% &Target $r=6.5\%$ \\ 
			$\chi$ & Share investors & 6.6\% & Target $\kappa_{1980}=1.3$ and $\frac{1}{\zeta_{1980}}=0.54$ \\
			$\nu$ & Capital risk (yearly) & 7.7\% & Target $\kappa_{1980}=1.3$ and $\frac{1}{\zeta_{1980}}=0.54$\\
			%$\theta$ & Borrowing constraint & 0.5 & Standard calibration\\  	[1ex]
		\multicolumn{2}{l}{\textsc{Technology}} & &  \\
			$g$ & Growth rate of $\psi_z$ (yearly)      & 1.5\%& Standard calibration\\
			$\delta$ & Depreciation rate (yearly) & 5\% & Standard calibration\\
			$A$ & Hicks-neutral productivity term & 0.14 & $Y/L$ in 1980\\
			$\eta_z$ & Skill demand shifters in 1980 & vector & Wage levels in 1980 Census/ACS \\
			$\psi_{z,1980}$ & Productivity of labor relative to capital & vector & Automation reduces costs by 30\% $(= \frac{w_z}{\psi_z R})$\\ [1ex]
			\multicolumn{2}{l}{\textsc{Automation Shock}} & &  \\ 
			$ \alpha_{1980}$ & Capital share in 1980 & 0.345 &  BLS labor share in 1980 \\
			$ \alpha_{2014}$ & Capital share in 2014 & 0.428 &  BLS labor share in 2014\\
			$\omega^R_{z}$ & Exposure to routine jobs by pctile & vector & 2000 Census/ACS\\
			\bottomrule
	\end{tabular}}
	\begin{minipage}{\linewidth}
		\vspace{0.2cm}
		\footnotesize\textsl{Notes:} The table provides the parameters used in our baseline calibration of the model. Our calibration assumes no markups ($\varphi=0$) and no capital taxes ($\tau=0$). For details, see the main text and Appendix \ref{appendix calibration}. 
	\end{minipage}
	
\end{table}

Turning to investors, we focus on a calibration with risk averse investors with $\gamma=2$. We jointly calibrate the share of investors in the economy $\chi$ and the extent of capital income risk $\nu$ to match the observed tail index of the income distribution in 1980 ($1/\zeta_{1980}=0.54$) and our estimates of investors' exposure to business capital $\kappa  = 1.3$.\footnote{\label{footnote: income inequality data}The share of income held by the top 0.1\% rose from 4\% in 1980 to a peak of 12\% in 2007 \citep{jones, piketty-saez}, which implies a tail index of 0.54 in 1980 and 0.7 in 2007. In this model, the tail index of the wealth distribution and the income distribution coincides, and so by matching the tail of the income distribution we cannot match the thicker tail of wealth inequality observed in the data.}  This procedure yields $\nu=7.7\%$ and $\chi=6.6\%$.\footnote{
The calibrated capital income risk $\nu$ is reasonable and in line with estimates in the literature. For example \citet{FGMP} estimate a standard deviation of value-weighted before-tax real returns to net worth of 8.6\% and an unweighted standard deviation of 22.1\% (see their Table 3); They also estimate that 74\% of return differences are due to idiosyncratic risk as opposed to individual fixed effects (see their Table 8) which would imply a value-weighted risk-only standard deviation of $74\% \times 8.6\% = 6.4\%$. We show in Appendix \ref{appendix calibration} that our results are quantitatively robust to targeting different values of $\kappa$, including $\kappa = 1$ and $\kappa = 1.5$. We also present results for a low risk aversion calibration in this appendix, with $\gamma=0.1$, and results where investors are borrowing constrained.} 

The remaining parameters are chosen to match aggregates in 1980 and estimates of the elasticity of capital supply. 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 return to capital equals $r_K=6.5\%$. We pick $\rho$ to ensure a 6.5\% return to capital and $p$ to target a long-run elasticity of capital supply $d\ln K/dr$ of about 50. This choice of $p$ is conservative, in the sense that much of the evidence suggests a more inelastic supply of capital, as discussed in Appendix \ref{appendix calibration}.  Table \ref{table: calibration} summarizes the parameters used in our exercise.  

\vspace{-2mm}

\subsection{Implications for Aggregates and Returns}

We now explore the effects of changes in the $\alpha_z$'s on macroeconomic aggregates. Figure \ref{figure: aggregates} already plotted the transition dynamics of our baseline model following a gradual increase in automation. In what follows, we will therefore focus on steady-state comparisons. 

\begin{figure}[!ht]	
	\centering 
	\includegraphics[width=.85\linewidth]{numerical_analysis/output/Figure_paper_alpha_aggregates_notbinding.pdf}
	\caption{\label{figure:alpha aggregates}Steady-state effects of changes in $\alpha=\sum_z\alpha_z\eta_z$.}
\end{figure}

Figure \ref{figure:alpha aggregates} summarizes the steady-state effects of automation, which we plot against $\alpha=\sum_z \alpha_z\eta_z$ -- the average extent of automation in the economy which also equals the capital share. The figure thus shows how different levels of automation, moving the capital share from 34.5\% (its 1980 level) and up to 42.8\% (its 2014 level) would affect the steady-state behavior of aggregates. In line with the discussion in Section \ref{sec:extension}, automation leads to a sharp rise in the return to capital; while the return on bonds and safe assets increases slightly. As a result, we see a widening gap between $r_K$ and $r_B$. Because some of the productivity gains from automation accrue to investors, mean wages are stagnant and median wages decline, and we see a modest expansion in output and the capital-output ratio. Finally, despite the large distributional implications that we will document, automation generates an increase in TFP of 2\% -- smaller but of a similar magnitude to our first-order approximation. In Appendix \ref{appendix investment} we further show that the modest increase in the capital-output ratio of 15\% implied by our model is in line with that observed in the data.

\section{Uneven Growth: Inequality in Model and Data}\label{sec:confronting}

We now turn to the implications of our model for wage and income inequality. Section \ref{sec:wage_inequality} briefly describes our model's implications for changes in wages and wage inequality. Section \ref{sec:num findings} then presents our model's implications for inequality of overall income, including not only wages but also capital income, and Section \ref{sec:uneven_data} confronts these model implications with the data. Section \ref{sec:transition_speed} briefly discusses a discrepancy between model and data regarding the speed at which these changes occur at the top of the income distribution.

\subsection{Wage Inequality}\label{sec:wage_inequality}

As we already saw in Figure \ref{figure:alpha aggregates}, average wages in our model are roughly stagnant when the capital share increases. However, the constant average wage masks substantial heterogeneity. This can be seen in Figure \ref{figure: relative wages} which plots the change in steady-state wages by wage percentile. 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 10\%. 
In contrast, the real wages of workers at the 95th percentile of the wage distribution rise by 7\%. For comparison, the figure also plots on a different vertical axis the observed change in wages by percentile between 1980 and 2012--2016 (using data from the US Census and the American Community Survey (ACS)). 

%\footnote{Because our model's wage distribution is very skewed (just like in the data), the rising wages for the top 25 percent of the wage distribution in Figure \ref{figure: relative wages} are enough to offset the declining wages for the bottom 75 percent of the distribution so that the average wage in Figure \ref{figure:alpha aggregates} is roughly constant.}


\begin{figure}[!ht]	
	\centering 
	\includegraphics[width=.8\linewidth]{numerical_analysis/output/Figure_paper_wages_notbinding.pdf}
	\caption{Predicted change in wages by wage percentile (left axis) and observed change in wages by wage percentile (right axis). \footnotesize
		\textit{Notes---} Observed wage changes computed using the 1980 Census and 2012--2016 ACS. See Appendix \ref{appendix calibration} for details. \label{figure: relative wages}}
\end{figure}


\subsection{Uneven Growth in the Model}\label{sec:num findings}

We now turn to our model's implications for \emph{income} inequality, including also capital income. Figure \ref{figure: quantiles labor capital} presents the change in total income for different percentiles of the income distribution across steady states. The figure reveals a pattern of uneven growth. Below the 50th percentile, households experienced declines in total income of 5--10\%. Between the 50th and 80th percentile, households experienced stagnant incomes. This is in contrast to the top income quantiles, which experienced an increase in income ranging from 20\% (for households in the top 1\%--0.5\%) to 55\% (for the top 0.1\%). The figure shows that automation is capable of generating substantial income gains at the very top of the distribution and stagnant or declining incomes at the bottom. 

\begin{figure}[!ht]	
	\centering 
	\includegraphics[width=0.9\linewidth]{numerical_analysis/output/Figure_paper_quantiles_nobinding_onepanel_sameaxis_cut5pct.pdf}
	\caption{Predicted change in income by income percentile decomposed into the contribution of capital and labor income. The dotted red line plots the results from the representative-household benchmark.{\footnotesize  \textit{Notes---}see Appendix \ref{app:inc_dist_extended} for a detailed explanation of the figure's construction.}\label{figure: quantiles labor capital}}
\end{figure}

Both wage and wealth inequality combine to produce the pattern observed in Figure \ref{figure: quantiles labor capital}. The blue-shaded area plots the contribution of changes in labor income. The fall in real wages for households at the bottom of the wage distribution contributed to declining incomes for households 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 are highly dispersed. People at the bottom of the distribution have few assets and only invest in bonds and safe assets, and so do not benefit as much from an increase in the return to capital. In contrast, a higher return to capital allows investors 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 (see Figure \ref{figure: relative wages}), the contribution of wages at the top of the overall income distribution is small or even negative. As discussed in section \ref{sec:ineq}, there are two effects at play here. On the one hand, the skill-biased nature of automation raises the wages of households 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 investors 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 low or negative contribution of labor income.\footnote{Formally,  the expected wage of households at the top of the income distribution is $\sum_z \ell_z w_z^{1+1/\alpha^*_{net}}/\sum_z \ell_z w_z^{1/\alpha^*_{net}}$. Skill-biased changes in wages raise this expected wage, but increases in $\alpha^*_{net}$ reduce it.  Note that as we keep moving up the tail, the contribution of labor income converges to zero since households' own mostly capital income.}
 
This finding can be contrasted with the representative-household benchmark. In this model the steady-state wealth distribution is indeterminate, but one can still trace its evolution starting from a given initial distribution, and so we assume the same initial distribution of wages and wealth as in the extended model with dissipation shocks  (see Appendix \ref{appendix transition} for details). The dashed line in Figure \ref{figure: quantiles labor capital} shows the change in income for different percentiles of the income distribution in this model. The representative household model does not generate uneven growth: there is a fairly uniform increase in income between 10\% and 18\% for all quantiles. This reflects two differences. First, there is no wage stagnation in the representative household model, since the infinitely elastic supply of capital implies that all productivity gains from automation accrue to workers. Second, the temporary increase in returns to wealth benefits all households equally, as they are all able to scale their effective wealth by the same amount. This is in contrast to our model with dissipation shocks, where only a few households benefit from the higher return to capital. 

How strong is the new mechanism in our model linking changes in the capital share due to automation, returns, and the resulting increase in income inequality? To address this question, Figure \ref{figure:alpha inequality} plots the behavior of the return gap $r_W^* - \rho - \sigma g$ and the resulting tail index of the income distribution as functions of the capital share in our calibrated model. 

\begin{figure}[!ht]	
	\centering 	
	\includegraphics[width=.75\linewidth]{numerical_analysis/output/Figure_paper_alpha_inequality_notbinding.pdf}
	\caption{\label{figure:alpha inequality}Steady-state inequality effects of changes in $\alpha=\sum_z \alpha_z \eta_z$.}
\end{figure}

The return gap in our calibrated model rises by 1.2 percentage points from 4 to 5.2 percentage points per year. Despite this relatively modest rise, the tail index of the income distribution rises from 0.54 (the targeted level in 1980) to 0.65, which is roughly 70\% of the observed increase in the data. A large part of this increase in tail inequality is due to the effect described in our baseline model, namely that investors' return to wealth $r_W$ increases and that they therefore accumulate wealth at a faster rate. However, in our extended model with idiosyncratic capital income risk, there is also an additional effect: when the return spread $r_K-r_B$ increases, investors increase the share of equity in their portfolios and become more exposed to this risky capital. The right panel of the figure shows that the higher return to wealth (the ``drift effect") generates an increase in the tail index from 0.54 to 0.63; while greater exposure to risk explains the increase from 0.63 to 0.65.

%Of course, this steady-state comparison must be taken with a grain of salt, since as we discuss in Section \ref{sec:transition_speed}, the transition dynamics in our model are slow.

\subsection{Uneven Growth in the Data}\label{sec:uneven_data}

The main implication of our model is that technological change involving the automation of tasks performed by low and middle-wage workers generates the pattern of uneven growth documented in Figure \ref{figure: quantiles labor capital}: stagnant or decreasing labor incomes at the bottom and rising incomes at the top, with an increasing role for capital income at the top of the distribution. 

We explore whether the evolution of capital and labor income in the US economy has followed a similar pattern of uneven growth using two data sources: the NBER IRS public use sample from 1980 to 2012 and the synthetic micro-files from the Distributional National Accounts (DINAs) of \citet{piketty-saez-zucman} for the same period.\footnote{The NBER IRS public use sample is available through the NBER. See \url{https://users.nber.org/~taxsim/gdb/} and \url{https://www.nber.org/taxsim-notes.html}. The DINAs merge semi-public and private IRS data with national accounts and the US Census/ ACS to construct consistent measures of wealth and income across the income and wealth distribution. The synthetic micro-files for the Distributional National Accounts can be accessed via Gabriel Zucman's website \url{http://gabriel-zucman.eu/usdina/}. 
We present additional decompositions for 1980--2007 in Appendix \ref{appendix DINAs}.} The IRS dataset is based on administrative tax records and yields reliable information on incomes for the very top of the distribution. However, it only records \emph{fiscal income} and therefore omits pensions and other sources of income that are tax-exempt. Likewise, fiscal income misses income flows such as corporate taxes, retained earnings, or housing services for home owners. The DINAs account for this ``missing income" by imputing tax-exempt sources of income and measuring corporate income and other sources of capital income using a ``capitalization approach.'' This approach allocates aggregate measures of equity from the IMAs to households according to positive corporate income reported to the IRS, and then imputes corporate income based on these stock measures. The main advantage of the DINAs are the more comprehensive income concept being used that adds up to national income but this comes up at the cost of 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 missing income and, in particular, missing capital income. 

 
\begin{figure}[!ht]
	\centering
	
	Panel (a)---IRS data
	
	\includegraphics[width=.7\linewidth]{additional_evidence/psz_irs_data/output/Figure_paper_consistent_irs_share_1980_2012_onepanel.pdf}
	
	Panel (b)---DINAs data
	
	\includegraphics[width=.7\linewidth]{additional_evidence/psz_irs_data/output/Figure_paper_consistent_psz_share_1980_2012_onepanel.pdf}
	
	
	\caption{Income growth by percentile between 1980--2012 and decomposition into capital and labor income. \footnotesize \textit{Notes---} see Appendix \ref{appendix DINAs} for details.}\label{fig:IRS}
\end{figure}

Figure \ref{fig:IRS} provides our analysis of uneven growth in the United States. Panel (a) uses the IRS data and panel (b) the DINAs. 
The dash-dotted black lines labeled ``total income" plot the average annual income growth between 1980 and 2012 for different percentiles of the income distribution. 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 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 \ref{figure: quantiles labor capital}).\footnote{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) labeled ``total income" is the same as that in Figure II(a) of \citet{piketty-saez-zucman} 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 \citet{piketty-saez}" which is essentially our IRS series.}

What is the contribution of capital and labor income to the pattern of uneven growth in Figure \ref{fig:IRS}? Decomposing income growth into these components is difficult because a substantial share of income is derived from self-employment or ownership of passthrough entities, and these cannot be easily classified as labor or capital income. 
As a starting point, we follow \citet{piketty-saez-zucman} who propose measuring capital income as the sum of corporate income (including the income of both C- and S-corporations) plus 30\% of the mixed income from non-corporate passthrough businesses (including self-proprietorships and partnerships). However, \citet{smith-yagan-zidar-zwick} criticize this approach and argue that (in addition to the 70\% of non-corporate passthrough income already classified as wage income by Piketty, Saez and Zucman) also about 75\% of S-corporation income should be counted as labor income rather than capital income. The argument is that these 75\% of S-corporation income are, in fact, returns to the human capital of owners and are only reported as corporate income to minimize tax obligations. To address this criticism, we also present an alternative measure of capital income that relabels 75\% of S-corporation income as labor income.

The shaded areas in both panels of Figure \ref{fig:IRS} decompose the growth in total income into parts due labor income, capital income, and the ambiguous S-corporation income.\footnote{Denoting by $y_t(q)$ the $q$th income percentile at time $t$ and by $y_{\ell,t}(q)$ and $y_{k,t}(q)$ that percentile's labor and capital income, we decompose the annualized growth rate from a base year $t=0$, $\frac{1}{T}\frac{y_{T}(q)-y_0(q)}{y_0(q)}$, into a part due to labor income $\frac{1}{T}\frac{y_{\ell,T}(q)-y_{\ell,0}(q)}{y_0(q)}$ and a part due to capital income $\frac{1}{T}\frac{y_{k,T}(q)-y_{k,0}(q)}{y_0(q)}$.} The blue area is the contribution of what both \citet{piketty-saez-zucman} and \citet{smith-yagan-zidar-zwick} classify as labor income and the red area is the contribution of what they agree is capital income. The green area in the middle corresponds to the contribution of the 75\% of S-corporations income that \citet{piketty-saez-zucman} attribute to capital and \citet{smith-yagan-zidar-zwick} attribute to the human capital of owners. The sum of the green and red areas thus provides the contribution of capital to income growth under the classification favored by \citet{piketty-saez-zucman}, while the red area provides the contribution of capital to income growth under the classification favored by \citet{smith-yagan-zidar-zwick}.\footnote{In these figures, we measure S-corporation income using the reported \emph{fiscal income}. Appendix \ref{appendix DINAs} shows an alternative strategy using DINAs where we compute corporate income from S-corporations using the capitalization method. Both approaches yield very similar quantitative results.} 

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 declining or stagnant real wages. However, further up in the distribution and especially within the top 1 percent, capital income becomes 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. Capital income becomes important within the top 1\% and especially at the top 0.1\% where it accounts for around 60\% of the cumulative growth from 1980-2012, or 30\% if one focuses only on the red area. 
In contrast, the DINA 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. When using the DINA data, the relabeling of S-corporations income does not alter these conclusions.\footnote{ 
These discrepancies across datasets are to be expected, given the differences in coverage between the IRS and DINAs. In particular, the share of capital in fiscal income using the IRS data is of less than 10\% whereas this share rises to 25--30\% in the DINAs (a level that matches the net capital share in national income). Thus, the IRS data is likely to understate the contribution of capital to income growth, and relabeling part of S-corporations income as labor income is more consequential when using the IRS data. }
 
As a whole, the empirical patterns in both data sets are consistent with those generated by our model. Our analysis suggests that the exact contribution of capital to income growth at the very top of the distribution  depends on the dataset and assumptions used, but ranges from 30\% (in the IRS data) to 70\% (in the DINAs). Moreover, both sources of data suggest that incomes at the middle and bottom have stagnated due to slow labor-income growth. 

The main difference between the patterns in these figures and that in our model is that, in the data, wage income plays a more prominent and positive role at the top of the income distribution. This suggests that there are forms of skill-biased technical change other than automation affecting relative wages at the top which are not included in our model (see the discussion of Figure \ref{figure: quantiles labor capital}).  

Our discussion focused on US trends since the 1980s. But it is natural to ask whether our model's prediction that rises in the net capital share are accompanied by large increases in top income inequality receives support from other countries or time periods. Appendix \ref{appendix evidence} discuss the available evidence, which shows that this link is visible in a panel of countries \citep[see][]{bengtsson-waldenstrom}  and during historical periods of rapid mechanization, such as the industrial revolution in Britain \citep[see][]{allen,lindert}.


\subsection{Speed of Top Inequality Dynamics\label{sec:transition_speed}} 

Figure \ref{fig:IRS} showed that the data features not only very large but also very fast changes in top inequality from 1980 to 2012. While our theory can account for large changes in inequality between steady states, it cannot generate rapid transition dynamics of top income inequality. 
That theories like ours cannot generate fast transition dynamics is a known result:  \citet{GLLM} 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 \ref{proposition: wealth inequality} -- 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. \citet{GLLM} 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 \citep{FGMP,bach-calvet-sodini}. 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.

\vspace{-2mm}  
\section{Conclusion}\label{sec:conclusion}
\vspace{-2mm}

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 through which technology affects inequality. The benefits of new technologies accrue not only to high-skilled labor but also to owners of capital in the form of higher capital incomes and returns.

There are several fruitful avenues for future work. First, one could use our 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, 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. 

%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.

Second, one could devise 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.  Our model in Section \ref{sec:extension} is a first step in that direction. More elaborate versions of our model could include realistic life-cycle structures, a careful treatment of intergenerational transfers, and additional sources of 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 in the data. Alternatively, theories with changing asset prices---another salient feature of the data that we do not model---are promising for generating fast wealth inequality dynamics.

Finally, ours is a model where ``macro matters for inequality, but inequality does not matter for macro.'' An implication is that the supply curve of capital does not depend on the wage distribution. Therefore, in our Figure \ref{figure: supply demand shift}, automation shifts only capital demand but not capital supply (even though it shifts the wage distribution). In more general models, however, additional mechanisms shifting the capital supply curve can be at play. For example, \citet{straub} proposes a model in which high permanent-income types have a higher saving rate, a form of scale dependence. Skill-biased technical change like automation then reallocates permanent income from households with low saving rates to those with high saving rates and hence expands the supply of saving. Therefore, to the extent that high permanent-income types empirically have higher saving rates, automation would shift not only capital demand but also capital supply to the right and this would provide an offsetting effect on returns to wealth. Our theory clarifies that, even in the case where such additional mechanisms are operational, the net effect of technology on top-tail income and wealth inequality depends on whether the higher demand for capital brought by automation translates into a higher rate of wealth accumulation for rich households in equilibrium. Exploring this question in models where ``inequality matters for macro'' is another interesting area for future research.  ff


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\begin{appendices}


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\begin{center}
{\Large {\textbf{Online Appendix to ``Uneven Growth: Automation's Impact on Income and Wealth Inequality"}}}

Benjamin Moll, Lukasz Rachel and Pascual Restrepo \\

\today

\end{center}


\section{Derivations and Proofs for the Baseline Model\label{appendix proofs}}

\subsection{Task Model Micro-foundation and Details}\label{app:task_model}

This appendix provides a micro-foundation for the aggregate production function in equation \eqref{equation: aggregate output}. The appendix also provides a derivation of equation \eqref{eq:dlnY} and primitive conditions to ensure Assumption \ref{assumption: adoption} holds. 

Each skill type $z$ works in a different sector that produces output $Y_z$. The economy produces a final good $Y$ using these sectoral outputs according to a Cobb-Douglas aggregator
\begin{equation*}
Y=A\prod_{z} Y_z ^{\eta_z}  \quad 
\mbox{with} \quad \sum_z \eta_z = 1.
\end{equation*}
Here, $\eta_z$ 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 $Y_z$ involves the completion of a unit continuum of tasks $u$, which are then combined via a Cobb-Douglas aggregator:
\begin{equation*}
\ln Y_z=\int_0^1 \ln \mathcal{Y}_{z}(u) du.
\end{equation*}
These tasks can be produced using capital and skill-$z$ labor as follows:
\begin{align*}
\mathcal{Y}_{z}(u)=\left\{
\begin{array}{lc}
\psi_{z} \ell_{z}(u) + k_{z}(u) & \text{ if } u\in [0,\alpha_z]\\
\psi_{z} \ell_{z}(u) & \text{ if } u\in (\alpha_z,1].\\
\end{array}
\right.
\end{align*} 
The threshold $\alpha_z$ summarizes the possibilities for the automation of tasks performed by workers of skill $z$. Tasks $u\in [0,\alpha_z]$ are technologically automated and can be produced by capital $k_{z}(u)$ or labor $\ell_{z}(u)$. The remaining tasks are not technologically automated and must be produced by labor. 

The unit cost of producing a task with capital is $R$ and that of producing it with labor is $w_z/\psi_z$. Denote by $p_z(u)$ the price of task $\mathcal{Y}_z(u)$, and by $p_z$ the price of sector $z$ output $Y_z$. Cost minimization in the production of sector $z$ output implies that the quantity of task $u$ used is given by
	\begin{equation*}
	\mathcal{Y}_z(u)=\frac{p_z Y_z}{p_z(u)}
	\end{equation*}
	
	Assumption \ref{assumption: adoption} implies that all tasks $u\in[0,\alpha_z]$ are produced with capital. It follows that for those tasks, $p_z(u)=R$ and the quantity of capital required to produce $\mathcal{Y}_z(u)$ is $p_zY_z/R$. It follows that the total amount of capital used in sector $z$ is:
	\begin{equation}\label{eq:Kdemand}
	K_z=\frac{\alpha_z p_zY_z}{R}
	\end{equation}
	
	Assumption \ref{assumption: adoption} implies that all tasks $u\in(\alpha_z,1]$ are produced with labor. It follows that for those tasks, $p_z(u)=\frac{w_z}{\psi_z}$ and the quantity of labor required to produce $\mathcal{Y}_z(u)$ is $p_zY_z/w_z$. It follows that the total amount of labor of skill $z$ used in sector $z$ is:
	\begin{equation}\label{eq:Ldemand}
	\ell_z=\frac{(1-\alpha_z)p_zY_z}{w_z}
	\end{equation}
	
	With perfect competition, the price of sector $z$ output equals the marginal cost of production. Because tasks are combined via a Cobb-Douglas aggregator, the price is given by the dual $\ln(p_z)=\int_{0}^{1}\ln(p_z(u))du$. It follows that
	\begin{equation}\label{eq:priceCD}
	p_z=R^{\alpha_z}\left(\frac{w_z}{\psi_z}\right)^{1-\alpha_z}.
	\end{equation}
	
	Combining the formula for $p_z$ in \eqref{eq:priceCD}
	with capital and labor demand conditions \eqref{eq:Kdemand}
	and \eqref{eq:Ldemand} gives the production of sector $z$ as a function of the total capital and labor used in this sector, $K_z$ and $\ell_z$:
	\begin{equation}
	Y_z=\left( \frac{K_z}{\alpha_z}\right)^{\alpha_z}\left(\frac{\psi_z \ell_z}{1-\alpha_z}\right)^{1-\alpha_z}\label{eq: derived prof fn}
	\end{equation}
	
	We now turn to aggregate output. We normalize the price of the final good to 1, so that the demand for sector $z$ output satisfies $p_z Y_z = \eta_z Y$. 
	
	Using equations \eqref{eq:Kdemand}, we can compute the demand for capital in sector $z$ as
	\begin{equation*}
	K_z =\frac{\alpha_z p_z Y_z}{R}=\frac{\alpha_z\eta_z Y}{R}. 
	\end{equation*}
	Adding this formula across sectors, it follows that the total amount of capital used in the economy is
	\begin{equation}
	K =\alpha \frac{Y}{R}\label{eq:KdemandAggregate}, 
	\end{equation}
	where recall that $\alpha:=\sum_z \alpha_z\eta_z$. The share of capital allocated to sector $z$ is therefore equal to
	\begin{equation}
	K_z = K \frac{\alpha_z\eta_z }{\alpha}. 
	\end{equation}
	
	Substituting this formula into \eqref{eq: derived prof fn} we get:
	\begin{equation}
	Y_z=\left(K\frac{\eta_z}{\alpha}\right)^{\alpha_z}\left(\frac{\psi_z L_z}{1-\alpha_z}\right)^{1-\alpha_z}.
	\end{equation}
	Substituting sectoral outputs into the aggregate production function we obtain the formula in equation \eqref{equation: aggregate output}, with 
	\begin{equation}\label{eq:A_def}
	\mathcal{A} := A\alpha^{-\alpha}\prod_{z}\left(\eta_z(1-\alpha_{z})\right)^{-\eta_z(1-\alpha_{z})}\prod_{z} \eta_z^{\eta_z}
	\end{equation}
	
\begin{proof}[Proof of the expression for productivity gains from automation] The formula in equation \eqref{equation: aggregate output} can be written as
	\begin{align*}\label{appeq: aggregate output convenient}
	Y=A  \prod_z \eta_z^{\eta_z}   \left(\frac{K}{\alpha}\right)^{\alpha} \prod_{z}\left(\frac{\psi_{z}\ell_{z}}{\eta_z(1-\alpha_z)}\right)^{\eta_z(1-\alpha_{z})}.
	\end{align*}
	
	It follows that
	\begin{align*}
	d\ln Y =&\eta_z \ln \left(\frac{K}{\alpha}\right)-\ln \left(\frac{\psi_z\ell_z}{\eta_z(1-\alpha_z)}\right)+ \alpha d\ln K \\
	=&\eta_z \ln \left(\frac{Y}{R}\right)-\ln \left(\psi_z\frac{Y}{w_z}\right)+ \alpha d\ln K \\
	=&\eta_z \ln \left(\frac{w_z}{\psi_z R}\right)+ \alpha d\ln K >0.
	\end{align*}
	The third row substitutes factor prices for their marginal products.
	Subtracting $\alpha d\ln Y$ from both sides of this equation and dividing through by $1-\alpha$ yields the formula in \eqref{eq:dlnY}.
\end{proof}

\begin{lemma}[Lemma ensuring adoption of automation technologies]\label{lemma: adoption}
	Suppose that for all $z$, the following inequality holds:
	\begin{equation}
	(\rho+p \sigma +\delta)^{-\frac{1}{1-\alpha}}\mathcal{A}^{\frac{1}{1-\alpha}}\alpha^{\frac{\alpha}{1-\alpha}}  > \frac{1}{(1-\alpha_z)\eta_z}\frac{\ell_z\psi_z}{\prod_v (\ell_v\psi_v)^{\frac{\eta_v(1-\alpha_v)}{1-\alpha}}},\label{appeq: inequality}
	\end{equation}
	where $\mathcal{A}$ is defined in \eqref{eq:A_def}. The equilibrium will involve the adoption of all available automation technologies. The above inequality holds for values of $A$ above a threshold $
	\bar{A}$.
\end{lemma}

\begin{proof}
	We assume that all automation technologies are adopted and verify that in equilibrium, the condition above ensures that $w^*_z>\psi_z R^*$. 
	
	In steady state, we have that  $\rho+p \sigma >r^*$, which can be seen from the fact that $r^*=\rho+p \sigma \alpha^*_{net}$. Using the fact that $r^*+\delta=\alpha\frac{Y}{K}$, we can rewrite $\rho+p \sigma >r^*$ as
	\begin{align}
	(K/Y)^*>\frac{\alpha}{\rho+p \sigma +\delta}.\label{appeq inequality ky} 
	\end{align}
    Turning to wages, we have that
    \begin{align*}
    	w_z^*=&(1-\alpha_z)\frac{\eta_z}{\ell_z} Y^*\\
    	  =&(1-\alpha_z)\frac{\eta_z}{\ell_z} \mathcal{A}^{\frac{1}{1-\alpha}}{(K/Y)^*}^{\frac{\alpha}{1-\alpha}}\prod_{v} (\psi_v \ell_v)^{\frac{\eta_z(1-\alpha_z)}{1-\alpha}} \\
    	  >&\frac{(1-\alpha_z)\eta_z}{\ell_z}\mathcal{A}^{\frac{1}{1-\alpha}}\alpha^{\frac{\alpha}{1-\alpha}} \prod_{z} (\psi_z\ell_z)^{\frac{\eta_z(1-\alpha_z)}{1-\alpha}}  (\rho+p \sigma +\delta)^{-\frac{\alpha}{1-\alpha}},
    \end{align*}
    where the last line uses inequality \eqref{appeq inequality ky}.
	
	Finally, because $\psi_z (\rho+p \sigma +\delta)> \psi_z R^*$, a sufficient condition to ensure $w^*_z>\psi_z R^*$ is 
	\begin{align*}
	\frac{(1-\alpha_z)\eta_z}{\ell_z}\mathcal{A}^{\frac{1}{1-\alpha}}\alpha^{\frac{\alpha}{1-\alpha}} \prod_{z} (\psi_z\ell_z)^{\frac{\eta_z(1-\alpha_z)}{1-\alpha}}  (\rho+p \sigma +\delta)^{-\frac{\alpha}{1-\alpha}}>\psi_z (\rho+p \sigma +\delta).
	\end{align*}
	
	This inequality is equivalent to \eqref{appeq: inequality}. Finally, the definition of $\mathcal{A}$ in \eqref{eq:A_def} implies that \eqref{appeq: inequality} holds for large values of $A$, concluding the proof of the lemma.
\end{proof}

\subsection{Propositions for Baseline Model \ref{appendix proofs}}

\begin{proof}[Proof of Lemma \ref{lemma: households policy st st}] Let $x_{z,t}:=w_z/r+a_{z,t}$ denote effective wealth. Equation \eqref{equation: max problem} can be rewritten as:
	\begin{align}\max_{\{c_{z,t}, x_{z,t}\}_{t \geq 0}} \ & \int_0^\infty e^{-\rho t} \frac{c_{z,t}^{1-\sigma}}{1-\sigma}dt \label{appeq: max problem} \\
	\mbox{s.t. }&\dot{x}_{z,t} = r x_{z,t} - c_{z,t}, \text{ and } x_{z,t} \geq 0 \notag \end{align} 
	
	The Hamiltonian associated with this maximization problem is
	\begin{align}
	H(c_z, x_z, \lambda_z):=\frac{c_z^{1-\sigma}}{1-\sigma}+\lambda (r x_z - c_z), \label{appeq: max hamiltonean}
	\end{align}
	where $\lambda_z$ is the co-state for effective wealth. 
	
	We can write the candidate solution given in Lemma \ref{lemma: households policy st st} as (time arguments are ignored to save on notation)
	\begin{align}
	\dot{x}_z=&\frac{r-\rho}{\sigma} x_z &
	c_z=&\left(r-\frac{r-\rho}{\sigma}\right) x_z.\label{appeq: max path}
	\end{align}
	We will show that the unique solution to this system of differential equations starting from $x_{z,0}=w_z/r$ solves the maximization problem in \eqref{appeq: max problem}. 
	
	Theorem 7.14 in \citet{acemoglu-book} implies that this candidate path reaches an optimum if there exists a co-state variable $\lambda_z$ such that:
	\begin{enumerate}
		\item the path satisfies the restrictions $\dot{x}_{z} = r x_{z} - c_{z}$,  and $x_{z} \geq 0$;
		
		\item the following necessary conditions hold:  
		\begin{align*}
		c_z^{-\sigma}=&\lambda_z, \\
		\rho \lambda_z-\dot{\lambda_z}=& r \lambda_z;
		\end{align*}
		
		\item the maximized Hamiltonian $M(x_z, \lambda_z)=\max_{c} H(c, x_z, \lambda_z)$ is concave in $x_z$ along the candidate path;
		
		\item the transversality condition holds. That is, for the candidate path, we have
		\begin{align*}
		\lim_{s \rightarrow \infty} e^{-\rho s} x_z \lambda_z=&0. 
		\end{align*}
		and for all other feasible paths, $\hat{x}_z$, we have
		\begin{align*}
		\lim_{s \rightarrow \infty} e^{-\rho s} \hat{x}_z \lambda_z\geq&0. 
		\end{align*}
\end{enumerate}

To prove condition 1, note that starting from any $x_{z,0}\geq 0$, we will have $x_z\geq 0$. Moreover, for any path satisfying equations \eqref{appeq: max path} the flow budget constraint holds:
\begin{align*}
r x_z-c_z &= r x_z-\left(r-\frac{r-\rho}{\sigma}\right) x_z\\
          &= \frac{r-\rho}{\sigma} x_z \\
          &= \dot{x}_z.
\end{align*}

To prove condition 2, define $\lambda_z:=\left(r-(r-\rho)/\sigma\right)^{-\sigma} x_z^{-\sigma}>0$ (here we used the condition $r>(r-\rho)/\sigma$). By construction, $c_z^{-\sigma}=\lambda_z$. Moreover:
\begin{align*}
\rho \lambda_z -\dot{\lambda}_z=&\rho \left(r-\frac{r-\rho}{\sigma}\right)^{-\sigma} x_z^{-\sigma} 
+\left(r-\frac{r-\rho}{\sigma}\right)^{-\sigma}  \sigma x_{z,t}^{-\sigma-1} \dot{x}_z\\
=& \left(\rho+\sigma\frac{\dot{x}_z}{x_z} \right) \left(r-\frac{r-\rho}{\sigma}\right)^{-\sigma} x_{z}^{-\sigma} \\
=&  \left(\rho+\sigma\frac{\dot{x}_z}{x_z} \right) \lambda_z \\
=& \left(\rho+\sigma \frac{r-\rho}{\sigma} \right) \lambda_z \\
=& r\lambda_z.
\end{align*} 

To prove condition 3, note that
\begin{align*}
\max_c H(c, x_z, \lambda_z)= \frac{\lambda_z^{\frac{\sigma-1}{\sigma}}}{1-\sigma}+\lambda_z(r x_z -\lambda_z^{-\frac{1}{\sigma}}),
\end{align*}
which is concave (linear) in $x_z$.  

To prove the first part of condition 4, note that along the candidate path, $x_z$ grows at a rate $\frac{r-\rho}{\sigma}$, and $\lambda_z$ at a rate $\rho-r$. It follows that the first part of the transversality condition holds if
\begin{align*}
-\rho+\frac{r-\rho}{\sigma}+\rho-r <0,
\end{align*}
which is equivalent to the condition $r>(r-\rho)/\sigma$. 

The second part of the transversality condition follows from the fact that, along any feasible path, we have $\hat{x}_z\geq 0$. 

It follows that the candidate paths given in Lemma \ref{lemma: households policy st st} provide optimal paths for consumption and asset accumulation in a steady state. 
\end{proof}	


\begin{proof}[Proof of Proposition \ref{proposition: steady state}]

The main text presents the derivations of the supply curve (equation \ref{equation:supply capital}) and the demand curve (equation \ref{equation: demand capital}). 

The supply curve
$(K/\bar{w})^s$ increases from zero to infinity as $r$ increases from $\rho$ to $\rho+p \sigma $. For $r<\rho$ households supply no capital. For $r>\rho+p \sigma $, households amass a divergent amount of capital. 

The demand curve $(K/\bar{w})^s$ decreases from $(\alpha/(1-\alpha))/(\rho+\delta)>0$ to $(\alpha/(1-\alpha))/(\rho+p \sigma +\delta)>0$ as $r$ increases from $\rho$ to $\rho+p \sigma $. 

These observations imply that equation \eqref{equation: implicit r} has a unique solution $r^*$ and that this unique solution lies in $(\rho,\rho+p \sigma )$. In fact, $r^*$ can be computed analytically as
\begin{align}
r^*=\frac{- ((1-\alpha) \delta - \rho - \alpha p \sigma )  + \sqrt{((1-\alpha) \delta - \rho - \alpha p \sigma )^2 + 4 (1-\alpha)\rho \delta}}{2} \label{appeq: analytic r}
\end{align}

The equilibrium return $r^*$ determines the remaining aggregates as follows. First, the capital-output ratio is given by
\begin{align*}
(K/Y)^*=\frac{\alpha}{r^*+\delta}.
\end{align*}
The output level is given by
\begin{align*}
Y^*=\mathcal{A}^{\frac{1}{1-\alpha}}\left(\frac{\alpha}{r^*+\delta}\right)^{\frac{\alpha}{1-\alpha}} \prod_z (\ell_z\psi_z)^{\frac{\eta_z(1-\alpha_z)}{1-\alpha}}.
\end{align*}
These two equations combined imply
\begin{align}
K^*=\mathcal{A}^{\frac{1}{1-\alpha}}\left(\frac{\alpha}{r^*+\delta}\right)^{\frac{1}{1-\alpha}} \prod_z (\ell_z\psi_z)^{\frac{\eta_z(1-\alpha_z)}{1-\alpha}}.\label{appeq: capital}
\end{align}

Turning to wages, we have that $w_z^*=(1-\alpha_z)\frac{\eta_z}{\ell_z} Y^*$, which implies
\begin{align*}
w_z^*=(1-\alpha_z)\frac{\eta_z}{\ell_z}\mathcal{A}^{\frac{1}{1-\alpha}}\left(\frac{\alpha}{r^*+\delta}\right)^{\frac{\alpha}{1-\alpha}} \prod_z (\ell_z\psi_z)^{\frac{\eta_z(1-\alpha_z)}{1-\alpha}}.
\end{align*}

Finally, equation \eqref{equation: return premium} can be derived from the household side. As explained in the main text, in steady state we must have
\begin{align*}
0=\frac{r^*-\rho}{\sigma}\left(K^*+\frac{\bar{w}^*}{r^*}\right)-p K^*.
\end{align*}
This expression can be rearranged as
\begin{align*}
r^*=\rho+p \sigma \frac{ r^*K^*}{r^*K^*+\bar{w}^*}=\rho+p \sigma \alpha^*_{net}.
\end{align*}

Note that the condition $r^*>(r^*-\rho)/\sigma$, which is needed to ensure the households' policy functions are an optimum is equivalent to $\rho+p\alpha^*_{net}(\sigma-1)>0$, which we assume holds throughout.  

Turning to the comparative statics exercises, we can rearrange \eqref{equation: implicit r} as:
\begin{equation*}
\frac{\left(1-\frac{\rho}{r^*}\right) \left(r^* + \delta\right)}{p \sigma  + \rho -r^*}=\frac{\alpha}{1-\alpha}.
\end{equation*}
The right hand side of this equation is increasing in $r^*$, and the left is increasing in $\alpha$. It follows that $r^*$ is increasing in $\alpha$. 

The household accumulation rate is given by $(r^*-\rho)/\sigma$, and so it also increases with $\alpha$.

The net capital share satisfies the identity in equation \eqref{equation: return premium}, and so it increases with $\alpha$. 

Denote the capital-output ratio by $(K/Y)^*$. We have
\begin{align*}
\alpha^*_{net}=\frac{r^*(K/Y)^*}{1-\delta (K/Y)^*}.
\end{align*}
Rearranging this equation and using the fact that $r^*=\rho+p \sigma \alpha^*_{net}$, we obtain
\begin{align}
(K/Y)^*=\frac{\alpha^*_{net}}{\rho+(p \sigma +\delta)\alpha^*_{net}},\label{appeq: equation kappa}
\end{align}
which is an increasing function of $\alpha^*_{net}$, and hence $\alpha$. 

Finally, turning to output, from equation \eqref{equation: aggregate output} we have
\begin{align*}
d\ln Y^*=\frac{1}{1-\alpha}d\ln \textrm{TFP}_{\alpha}+\frac{\alpha}{1-\alpha} d\ln (K/Y)^*.
\end{align*}
Because $d\ln \textrm{TFP}_{\alpha}>0$ and $d\ln (K/Y)^*>0$ following any increase in the $\alpha_z$'s, we have that automation always increase output.\end{proof}


\begin{proof} [Proof of Proposition \ref{proposition: wages}] Equilibrium factor prices imply that relative wages satisfy
\begin{align*}
\frac{w_z}{w_v}=\frac{(1-\alpha_z)}{(1-\alpha_v)}\frac{\eta_z}{\eta_v}\frac{\ell_v}{\ell_z}. 
\end{align*}	
It follows that an increase in $\alpha_z$ reduces $w_z/w_v$ for $v\neq z$.

Turning to the wage bill, we can add up individual wages for all $z$ to obtain:
\begin{align*}
\bar{w}=(1-\alpha) Y.
\end{align*}

It follows that
\begin{align*}
d\ln \bar{w}=-\frac{1}{1-\alpha}\sum \eta_z d\alpha_z+\frac{1}{1-\alpha}d\ln \textrm{TFP}_{\alpha}+\frac{\alpha}{1-\alpha} d\ln (K/Y)^*.
\end{align*}
We now show that the terms $d\ln \textrm{TFP}_{\alpha}$ and $ d\ln (K/Y)^*$ are both decreasing in $p$ and converge to zero as $p$ increases. Because the term $-(1/(1-\alpha_z))\sum \eta_z d\alpha_z$ is negative, this establishes the existence of the threshold $\bar{p}$. 

We first analyze the term $d\ln \textrm{TFP}_{\alpha}$. This is given by
\begin{align*}
d\ln \textrm{TFP}_{\alpha}=\sum_z\eta_z \ln\left(\frac{w^*_z}{\psi_z R^*}\right)=\sum_z\eta_z \ln\left(\frac{K^*}{\alpha}\right)-\eta_z\ln\left(\frac{\psi_z\ell_z}{\eta_z(1-\alpha_z)}\right),
\end{align*}
where we used the formulas for equilibrium factor prices. It is enough to show that $K^*$ is decreasing in $p$ and that $K^*$ converges to zero as $p$ increases. Because $K^*$ is given by \eqref{appeq: capital}, it is enough to show that $r^*$ is increasing in $p$ and that $r^*$ converges to infinity as $p$ increases.

The fact that $r^*$ increases in $p$ follows from equation \eqref{equation: implicit r}. An increase in $p$ contracts the supply of capital, which results in a higher $r^*$. Moreover, equation \eqref{appeq: analytic r} shows that $r^*\rightarrow \infty$ as $p\rightarrow \infty$. Note that the formal limit of $d\ln \textrm{TFP}_{\alpha}$ as $p\rightarrow \infty$ is zero, since as $K^*$ declines, we eventually reach a point where Assumption \ref{assumption: adoption} starts failing and increases in $\alpha_z$ do not affect productivity. 

We now turn to the term $d\ln (K/Y)^*$. We have that
\begin{align*}
\alpha=(K/Y)^*(\rho+p \sigma \alpha^*_{net}+\delta).
\end{align*}
Differentiating this expression we obtain
\begin{align}
1=\frac{p \sigma \alpha^*_{net}}{\rho+p \sigma \alpha^*_{net}+\delta}\frac{\partial \ln \alpha^*_{net}}{\partial \ln \alpha}+\frac{\partial \ln (K/Y)^*}{\partial \ln \alpha}.
\label{appeq: system 1}
\end{align}
Moreover, equation \eqref{appeq: equation kappa} implies that
\begin{align}
\frac{\partial \ln (K/Y)^*}{\partial \ln \alpha}=\frac{\rho}{\rho+(p \sigma+\delta)\alpha^*_{net}} \frac{\partial \ln \alpha^*_{net}}{\partial \ln \alpha}.\label{appeq: system 2}
\end{align}
Solving equations \eqref{appeq: system 1} and \eqref{appeq: system 2}, we obtain:
\begin{align*}
\frac{\partial \ln (K/Y)^*}{\partial \ln \alpha}=\frac{1}{1+\displaystyle\frac{p \sigma \alpha^*_{net}}{\rho+p \sigma \alpha^*_{net}+\delta}\frac{\rho+(p \sigma+\delta)\alpha^*_{net}}{\rho}}.
\end{align*}

We now show that the elasticity $\frac{\partial \ln (K/Y)^*}{\partial \ln \alpha^*_{net}}$ converges to zero as $p$ rises. A sufficient condition for this to be the case is that $\alpha^*_{net}$ is nondecreasing in $p$, which holds when $\delta=0$ and $\alpha^*_{net}=\alpha$. To show this is the case more generally, start from the fact that $\alpha=R  (K/Y)$. Rewriting the right hand side in terms of $\alpha^*_{net}$, we obtain
\begin{align*}
\alpha=(\rho+p \sigma \alpha^*_{net}+\delta)  \frac{\alpha^*_{net}}{\rho+(p \sigma +\delta)\alpha^*_{net}}. 
\end{align*}
This equation can be rearranged as
\begin{align*}
\alpha\left(1-\frac{\delta(1-\alpha^*_{net})}{\rho+p \sigma \alpha^*_{net}+\delta}\right)=\alpha^*_{net}.
\end{align*}
This equation defines $\alpha^*_{net}$ implicitly as a function of $\alpha$ and $p$. The left hand side is increasing in $\alpha^*_{net}$ and intercepts the right-hand side (the 45-degree line) from above at a single equilibrium point. An increase in $p$ shifts the left-hand side upwards, which results in a higher equilibrium value for $\alpha^*_{net}$ as claimed. 

The above argument shows that there exists some $\bar{p}$ such that, for $p>\bar{p}$, $d\ln \bar{w}<0$. To conclude the proof, we show that $\bar{p}>0$. This follows from the fact that, for $p=0$, $d\ln \bar{w}>0$. 

To show this, note that  for $p=0$ we get
\begin{align*}
d\ln (K/Y)^*=\frac{d\alpha}{\alpha}=\frac{1}{\alpha}\sum_z \eta_z d\alpha_z,
\end{align*}
and therefore
\begin{align*}
d\ln \bar{w}=&-\frac{1}{1-\alpha}\sum \eta_z d\alpha_z+\frac{1}{1-\alpha}d\ln \textrm{TFP}_{\alpha}+\frac{\alpha}{1-\alpha} \frac{1}{\alpha}\sum_z \eta_z d\alpha_z=&\frac{1}{1-\alpha}d\ln \textrm{TFP}_{\alpha}>0.
\end{align*}\end{proof}


\begin{proof} [Proof of Proposition \ref{proposition: wealth inequality}] Below we derive the effective wealth distribution, the wealth distribution, and the income distribution. To save on notation, we do not include asterisks when denoting steady state objects. 

\emph{Effective wealth distribution:} Denote the stationary density of effective wealth conditional on a given skill type $z$ by $f_z(x)$. $f_z$ satisfies the Kolmogorov Forward Equation (KFE)
\begin{align*}
0 = -\partial_x\left(\frac{r-\rho}{\sigma} x f_z(x)\right) - p f_z(x)
\end{align*} 
on $(w_z/r,\infty)$. We guess and verify that $f$ is Pareto, i.e. $f_z(x) = c\zeta x^{-\zeta-1}$ for some constants $c$ and $\zeta$. Substituting in the guess
\begin{align*}
0 &= \zeta \frac{r-\rho}{\sigma} c\zeta x^{-\zeta-1} - p c\zeta x^{-\zeta-1} \\
0 &= \zeta \frac{r-\rho}{\sigma}  - p\\
\frac{1}{\zeta} &= \frac{r-\rho}{p \sigma } = \alpha_{net}
\end{align*}
Since $f_z(x)=c\zeta x^{-\zeta-1}$ must integrate to $1$ on $(w_z/r,\infty)$, we must have $c = (w_z/r)^{-\zeta}$. Hence this is a Pareto distribution with tail parameter $\zeta= \frac{1}{\alpha_{net}}$ and scale parameter $x_z(0) = w_z/r$. 

Because the distribution of effective wealth is Pareto, the conditional counter-CDF for effective wealth of each skill type $z$ is of the form:
\begin{equation}\label{eq:F_z}
\Pr(\text{effective wealth} \geq x | z) = \left(\frac{x}{w_z/r}\right)^{-\frac{1}{\alpha_{net}}}, \quad x \geq w_z/r. 
\end{equation}

\emph{Wealth distribution:} We now derive the counter-CDF for wealth. Recall that effective wealth $x$ is $x := a + w_z/r$. Therefore
\begin{align*}
\Pr( \textrm{wealth} \geq a | z) = \Pr( \textrm{effective wealth} \geq a +w_z/r| z)
=\left(\frac{a + w_z/r}{w_z/r}\right)^{-\frac{1}{\alpha_{net}}}, \quad a \geq 0
\end{align*} 
To find the unconditional distribution, we add across the different skill-types, which yields
\begin{align*}
\Pr( \textrm{assets} \geq a) = \sum_z \ell_z \left(\frac{a + w_z/r}{w_z/r}\right)^{-\frac{1}{\alpha_{net}}}.
\end{align*}

\emph{Income Distribution:} We now derive the counter-CDF for income. The income of a person with effective wealth $x$ is $rx$. Therefore
\begin{align*}
\Pr(\textrm{income}\geq y| z)=\Pr( \textrm{effective wealth} \geq y/r| z)
=\left(\frac{y/r}{w_z/r}\right)^{-\frac{1}{\alpha_{net}}}, \quad y \geq w_z.
\end{align*}
To find the unconditional distribution, we add across the different skill-types, which yields
\begin{align*}
\Pr(\textrm{income}\geq y) =  \sum_z \left(\frac{\max{\{y,w_z\}}}{w_z}\right)^{-\frac{1}{\alpha_{net}}}.
\end{align*}

Finally, when $\delta=0$, we have $\alpha_{net}=\alpha$ and $\frac{1}{\zeta} = \alpha$. When $\delta>0$, Proposition \ref{proposition: steady state}, implies that $\frac{1}{\zeta}$ is increasing in $\alpha$. \end{proof}


 




\section{Derivations and Proofs for the Extended Model\label{appendix extension}}

\subsection{Derivations and Lemmas}

Before presenting the proofs, we generalize the model in the text so that investors could also face a borrowing constraint of the form
\begin{align*}
-b_{z,t} \leq \theta  a_{z,t}+\frac{w_z+T}{r_B-g},
\end{align*}
where $\theta\in (0,1]$ parameterizes the extent to which investors can pledge their capital.\footnote{
	The usual formulation used in the literature is $-b_{z,t} \leq \theta  a_{z,t}$. Relative to this, our formulation assumes that human wealth is pledgeable, which makes the model more tractable.} The results in the main text follow in the special case with $\theta=1$. We also provide a lemma characterizing investors policy functions.

\begin{lemma}[\citet{AHLLM}]\label{lemma:savings}
	Let $r_I=\kappa r_K+(1-\kappa) r_B$. Investors' policy functions are given by
	\begin{align*}
	c_{z,t}=&\frac{\rho +(\sigma-1) r_I-\frac{1}{2}(\sigma-1)\gamma\nu^2\kappa^2}{\sigma} x_{z,t},\\
	\kappa=&\min\left\{\frac{1}{1-\theta},\frac{r_K - r_B}{\gamma  \nu^2}\right\}.
	\end{align*}
	which imply that effective wealth follows a random growth process:
	\begin{align*}
	dx_{z,t}=&\frac{r_I-\rho+\frac{1}{2}(\sigma-1) \gamma \nu^2 \kappa^2}{\sigma} x_{z,t} dt+ 
	\kappa \nu x_{z,t} dW_t.
	\end{align*}
\end{lemma}

\begin{proof} Using the definition of effective wealth and the fact that in a balanced growth equilibrium wages and tax revenue grow at a constant rate $g$, we have
	\begin{align*}
	dx_{z,t}=da_{z,t}+db_{z,t}+ g \frac{w_z+T}{r_B-g}dt.	
	\end{align*}
	Substituting the investors' budget constraint in place of 
	$da_{z,t}+db_{z,t}$ and rearranging terms we obtain
	\begin{align*}
	dx_{z,t}=&r_B\left(a_{z,t}+b_{z,t}+\frac{w_z +T}{r_B-g}\right)dt
	+(r_K-r_B) a_{z,t} dt + \nu a_{z,t} dW_t - c_{z,t} dt\\
	=&r_B x_{z,t} dt
	+(r_K-r_B) a_{z,t} dt + \nu a_{z,t} dW_t - c_{z,t} dt.
	\end{align*}
	
	Let $\kappa$ denote the share of effective wealth held in equity, so that $a_{z,t}=\kappa x_{z,t}$. We can rewrite the budget constraint as
	\begin{align}
	dx_{z,t}= (r_I x_{z,t}-c_{z,t}) dt+\nu  \kappa  x_{z,t} dW_t.\label{eq:budget x}
	\end{align} 
	
	In what follows, we drop the subscript $z$ and time $t$, and examine the savings problem in terms of the state variable $x$---effective wealth. The HJB equation for this problem is
	\begin{align*}
	0=\max_{c>0, \kappa \in [0, 1/(1-\theta)]} f(c,v(x))+(r_I x-c)v'(x)+\frac{1}{2}\nu^2 \kappa^2 x^2 v''(x).
	\end{align*}
	Using the Duffie--Lions aggregator and guessing $v(x)=\Lambda  x^{1-\gamma}/(1-\gamma)$, the HJB equation becomes
	\begin{align*}
	\frac{\rho  \Lambda  x^{1-\gamma}}{1-\sigma}=\max_{c>0, \kappa \in [0, 1/(1-\theta)]}& \frac{\rho  \Lambda  x^{1-\gamma}}{1-\sigma} \left(\frac{c}{\Lambda^{1/(1-\gamma) } x}\right)^{1-\sigma}  + (r_I x - c) \Lambda x^{-\gamma} - \frac{\gamma}{2} \nu^2 \kappa^2 \Lambda  x^{1-\gamma}.
	\end{align*}
	
	The optimal consumption and portfolio choice are given by
	\begin{align*}
	c=&\rho^{\frac{1}{\sigma}}  \Lambda^{-\frac{1}{\sigma}\frac{1-\sigma}{1-\gamma}}  x & 
	\kappa=&\min\left\{\frac{1}{1-\theta},\frac{r_K - r_B}{\gamma  \nu^2}\right\}.  
	\end{align*}
	Plugging into the HJB equation and canceling terms, we get
	\begin{align*}
	\frac{\rho}{1-\sigma}=\frac{\rho}{1-\sigma}\left(\frac{\rho^{\frac{1}{\sigma}}  \Lambda^{-\frac{1}{\sigma}\frac{1-\sigma}{1-\gamma}}}{\Lambda^{1/(1-\gamma) }}\right)^{1-\sigma}  + r_I - \rho^{\frac{1}{\sigma}} \Lambda^{-\frac{1}{\sigma}\frac{1-\sigma}{1-\gamma}} - \frac{1}{2}\gamma\nu^2 \kappa^2.
	\end{align*}
	
	This is an equation in $c/x=\rho^{\frac{1}{\sigma}}  \Lambda^{-\frac{1}{\sigma}\frac{1-\sigma}{1-\gamma}}$,
	which yields
	\begin{align*}
	c/x=\frac{\rho +(\sigma-1) r_I-\frac{1}{2}(\sigma-1)\gamma\nu^2\kappa^2}{\sigma}.
	\end{align*}
	The policy function for $c_{z,t}$ follows from this expression, and the behavior of $x_{z,t}$ follows after plugging this policy function in the the budget constraint in equation \eqref{eq:budget x}.
\end{proof} 


\subsection{Propositions for Extended Model}

\begin{proof}[Proof of Proposition \ref{proposition:bg}]
	Lemma \ref{lemma:savings} implies that investors accumulate wealth at a rate
	\begin{align}
	\mu_I=\frac{r_W-\rho}{\sigma}.\label{equation:growth rate wealth}
	\end{align}
	Lemma \ref{lemma: households policy st st} applies to the remaining households, whose wealth then grows at a rate
	\begin{align*}
	\mu_H=\frac{r_B-\rho}{\sigma}.
	\end{align*}
	
	In what follows, we will describe the BGE in terms of $r_W$ and $r_B$. In particular, we define $h(r_W-r_B)=r_K-r_B$ and $m(r_W-r_B)=\kappa$ implicitly as the solution to
	\begin{align*}
		r_W-r_B=&m h+\frac{1}{2}(\sigma-1)\gamma\nu^2m^2&
		m=&\min\left\{\frac{1}{1-\theta}, \frac{h}{\gamma\nu^2}\right\}.	
	\end{align*}
    We can then write the return to capital and portfolio choice implicitly as:
	\begin{align*}
	r_K=&h(r_W-r_B)+r_B &
	\kappa=&m(r_W-r_B),
	\end{align*}
	where $h$ is a continuous and increasing function and $m$ is continuous and nondecreasing. 
	
	Denote by $X_I$ the aggregate effective wealth of investors and by $X_H$ the aggregate effective wealth of the remaining households, and bond holdings by $B_I$ and $B_H$, respectively. Optimal household saving behavior combined with the dissipation shocks implies that
	\begin{align*}
	\dot{X}_I=&\mu_I  X_I - p (K+B_I), 
	&\dot{X}_H=&\mu_H  X_H - p  B_H. 
	\end{align*}
	Moreover, because the value of capital installed in firms must be equal to the total capital owned by investors, aggregate effective wealths are given by
	\begin{align*}
	X_I=&K+B_I+\chi  \frac{\bar{w}+T}{r_B-g}&
	X_H=&B_H+(1-\chi)  \frac{\bar{w}+T}{r_B-g}.
	\end{align*} 
	Human wealth now depends on the sum of wages and transfers, whose present discounted value is obtained by dividing them by $r_B-g$ to account for their growth over time. It is convenient to analyze the BGE in terms of capital and bonds normalized by the value of human wealth, which are constant along a BGE:
	\begin{align*}
	k_n=&\frac{K}{(\bar{w}+T)/(r_B-g)} &
	b_I=&\frac{B_I}{(\bar{w}+T)/(r_B-g)} &
	b_H=&\frac{B_H}{(\bar{w}+T)/(r_B-g)}.
	\end{align*}
	
	
	Along a BGE, the effective wealth of investors grows at a rate $g$, so that $\dot{X}_I= g X_I$ and $\dot{X}_H= g X_H$. The effective wealth of investors grows at a rate $g$ if and only if $r_W> \rho+\sigma g$ and the BGE values of $k_n$ and $b_I$ satisfy 
	\begin{align}
	\label{equation:wealth investors}\left(\frac{r_W-\rho}{\sigma}-g\right) \left(k_n+b_I+\chi\right)=&p (k_n+b_I).
	\end{align}
	In addition, because a fraction $\kappa=m(r_W-r_B)$ of investors wealth is held in equity,
	\begin{align}
	\label{equation:portfolio investors} m(r_W-r_B)  \left(k_n+b_I+\chi\right)=k_n.
	\end{align}
	Likewise, the effective wealth of households grows at a rate $g$ if and only if $r_B> \rho+\sigma g$ and
	\begin{align}
	\label{equation:savings households}\left(\frac{r_B-\rho}{\sigma}-g\right) \left(b_H+1-\chi\right)=&p  b_H,
	\end{align}
	or $r_B\leq \rho+\sigma g$ and $b_H=0$. 
	
	We now characterize the production side of the economy. We focus on a balanced-growth equilibrium in which Assumption \ref{assumption: adoption} holds, so that output is given by \ref{equation: aggregate output}.
	Because of markups, total wage payments are now given by
	\begin{align*}
	\bar{w}=\frac{1-\alpha}{\varphi} Y,
	\end{align*}
	which implies that firms pay a share $(1-\alpha)/\varphi$ of their revenue to labor; while the remaining share of revenue $1-(1-\alpha)/\varphi$ constitutes gross capital income which is taxed at a rate $1-\tau$ (recall that we assumed a gross tax on capital income) and must cover for the depreciation of capital. Thus, we can compute the after-tax gross income from capital as
	\begin{align*}
	(r_K+\delta) K =(1-\tau)\left(1-\frac{1-\alpha}{\varphi}\right)Y.
	\end{align*}
	Tax revenue from capital taxation, and thus the lump-sum transfers, are equal to
	\begin{align*}
	T=\tau \left(1-\frac{1-\alpha}{\varphi}\right) Y.
	\end{align*}
	
	We can combine the equations for after-tax gross capital income and labor income and transfers to obtain an expression of the demand for financing by firms:
	\begin{align}
	\label{eq:demand for equity}k_n=\frac{\widetilde{\alpha}}{1-\widetilde{\alpha}} \frac{r_B-g}{h(r_W-r_B)+r_B+\delta}.
	\end{align}
	
	A balanced-growth equilibrium is characterized by constant values for $r_W, r_B, k_n, b_I, b_H$ that solve equations \eqref{equation:wealth investors}, \eqref{equation:portfolio investors}, \eqref{equation:savings households} and \eqref{eq:demand for equity} and where $r_B$ is  endogenous and ensures market clearing in the bonds market
	\begin{align*}
	b_I+b_H=0.
	\end{align*}
	In this case, we will assume that $\rho+(\sigma-1)g>0$, which is a sufficient condition to ensure that the equilibrium exists and features finite wealth.
	
	We start with the case in  which investors are risk neutral ($\gamma=0$ in the Duffie--Lions aggregator) and their borrowing constraint does not bind. We will provide necessary and sufficient conditions for this to be the case below. 
	
	Because we assumed that the borrowing constraint does not bind, it must be the case that $r_W=r_K=r_B=r^*$. 
	
	Adding equations \eqref{equation:wealth investors} and \eqref{equation:savings households}, we obtain
	\begin{align}
	\left(\frac{r^*-\rho}{\sigma}-g\right) \left(k_n+1\right)=p k_n.
	\end{align}
	Solving for $k_n$ and substituting into equation \eqref{eq:demand for equity} shows that an equilibrium is fully determined by a level of returns that satisfies:
	\begin{align}
		\frac{1-(\rho+(\sigma-1) g)/(r^*-g)}{\sigma (p+g)+\rho- r^*}=\frac{\widetilde{\alpha}}{1-\widetilde{\alpha}} \frac{1}{ r^*+\delta}.\label{eq:general r*}
	\end{align} 
	The left hand side of this equation is increasing in $r^*$; while the right-hand side is decreasing in $r^*$. Moreover, at $r^*=\rho+\sigma g$, the left-hand side is lower than the right-hand side; and at $r^*=\rho+\sigma (p+g)$, the the left-hand side converges to infinity and exceeds the right-hand side. This implies a unique solution exists and satisfies  $r^*\in (\rho+\sigma g, \rho+\sigma (p+g))$.
	
	We now derive conditions for $\theta$ that ensure the borrowing constraint does not bind. Denote by $k_n^*$ the value of $k_n$ in the balanced growth equilibrium above, which is given by
	\begin{align*}
	k_n^*=\frac{\widetilde{\alpha}}{1-\widetilde{\alpha}} \frac{r^*-g}{r^*+\delta},
	\end{align*} 
	and is independent of $\theta$ by construction. 
	Equations \eqref{equation:wealth investors} and \eqref{equation:savings households} imply that investors must borrow and amount
	\begin{align*}
	b^*=(1-\chi) k_n^*
	\end{align*}
	from households. 
	and so we must have
	\begin{align*}
	\kappa^*=\frac{k_n^*}{k_n^*-b^*+\chi}=\frac{1}{\chi} \frac{k_n^*}{k_n^*+1}.
	\end{align*}
	It follows that the borrowing constraint will be slack if and only if
	\begin{align*}
	\frac{1}{1-\theta}\geq\frac{1}{\chi} \frac{k_n^*}{k_n^*+1} \Leftrightarrow \theta\geq 1-\chi \frac{k_n^*+1}{k_n^*}:=\bar{\theta}.
	\end{align*}
	Note that $\bar{\theta}\leq 1$, as claimed in the proposition. 
	
	We now turn to the case in which investors are risk averse and/or $\theta<\bar{\theta}$ and we have a closed economy. In what follows, we will assume that $r_W>\rho+\sigma g$, which must hold in any equilibrium. To see this, notice that for $r_W\leq\rho+\sigma g$, investors do not accumulate wealth and the supply of capital is zero, which cannot be the case in a BGE. 
	
	Combining equations \eqref{equation:wealth investors} and \eqref{equation:savings households}, we obtain a supply of (normalized) capital
	\begin{align}
	k_n=\frac{p \sigma m(r_W-r_B)}{\sigma (p+g)+\rho-r_W} \chi.
	\end{align}
	Combining this with the demand for firm financing in equation \eqref{eq:demand for equity} yields the market clearing condition in the capital market:
	\begin{align}
	D_K(r_W, r_B)=\frac{\widetilde{\alpha}}{1-\widetilde{\alpha}}
	 \frac{r_B-g}{h(r_W-r_B)+r_B+\delta}-
	\frac{p \sigma m(r_W-r_B)}{\sigma (p+g)+\rho-r_W} \chi=0\label{equation: capital market clearing}
	\end{align}
	where $D_K(r_W, r_B)$ is the excess demand for capital, and the curve  $D_K(r_W, r_B)=0$ defines the locus of points for which the capital market clears.
	Likewise, the market clearing condition in the bond market is given by
	\begin{align}
	D_B(r_W, r_B)=\frac{p \sigma (m(r_W-r_B)-1)}{\sigma (p+g)+\rho-r_W} \chi+\chi-\frac{r_B-\rho-\sigma g}{\sigma (p+g)+\rho-r_B} (1-\chi)=0,
	\label{equation: bond market clearing}
	\end{align}
	where $D_B(r_W, r_B)$ is the excess demand for bonds, and the curve  $D_B(r_W, r_B)=0$ defines the locus of points for which the bond market clears.
	
	The following Lemmas characterize the behavior of these loci.
	
	\begin{lemma}\label{lemma:capital market clearing}
		The curve $D_K(r_W,r_B)=0$ gives a continuous and upward sloping locus in the $(r_W, r_B)$ space defined for $r_W\in (g, \rho+\sigma (p+g))$  and $r_B\in (g, r_W)$.
		Moreover: 
		\begin{enumerate}
			\item as $r_W\downarrow g$, the locus converges to the point $(g,g)$
			\item as $r_W\uparrow \rho+\sigma (p+g)$, the locus converges to the point $(\rho+\sigma (p+g),\rho+\sigma (p+g))$.
		\end{enumerate}
	\end{lemma}
	
	\begin{lemma}\label{lemma:bond market clearing}
		The curve $D_B(r_W,r_B)=0$ gives a continuous and initially decreasing locus in the $(r_W, r_B)$ space defined for 
		$r_W\in (\rho+\sigma g, \rho+\sigma (p+g))$ and $r_B< r_W$. Moreover:
		\begin{enumerate}
			\item as $r_W\downarrow \rho+\sigma g$, the locus converges to the point $(\rho+\sigma g,\rho+\sigma g)$;
			\item as $r_W\uparrow \rho+\sigma (p+g)$, the locus converges to the point
			$(\rho+\sigma(p+g), \tilde{r}_B)$, where
			\begin{align*}
			\tilde{r}_B:=\rho+\sigma(p+g)-\frac{1}{2}(\sigma+1)\gamma\nu^2.
			\end{align*}
			\item let $\bar{\gamma}:=2 p \frac{\sigma}{1+\sigma}(1-\chi)$. 
			\begin{itemize}
				\item if $\gamma \nu^2>\bar{\gamma}$, then $r_W-r_B$ increases from zero to a maximum of $\frac{1}{2}(\sigma+1)\gamma\nu^2$ along the locus $D_B(r_W,r_B)=0$;
				\item if $\gamma \nu^2<\bar{\gamma}$, then $r_W-r_B$ increases from zero up to a maximum and then decreases and reaches $\frac{1}{2}(\sigma+1)\gamma\nu^2$ along the locus $D_B(r_W,r_B)=0$.
			\end{itemize}
		\end{enumerate}
	\end{lemma}
	
	The proof of these lemmas is technical and relegated to Appendix \ref{appendix extension supplement}. 
	
	\begin{figure}[ht]
		\centering\includegraphics[width=.75\linewidth]{graphs/equilibrium_bonds_capital.pdf}
		\caption{Typical configurations of the locus for $D_K(r_W,r_B)=0$ and $D_B(r_W,r_B)=0$.\label{figure:equilibrium loci} }
	\end{figure} 
	
	The two lemmas combined imply that the loci $D_K(r_W,r_B)=0$ and $D_B(r_W,r_B)=0$ are as depicted in Figure \ref{figure:equilibrium loci}. 
	The lemmas imply that at $r_W=\rho+\sigma  g$, the locus for $D_B(r_W, r_B)=0$ is above that for $D_K(r_W, r_B)=0$ (recall that, by assumption,  $\rho+\sigma g>g$). On the other hand, at $r_W=\rho+\sigma  (p+g)$, the locus for $D_K(r_W, r_B)=0$ is above that for $D_B(r_W, r_B)=0$. The intermediate value theorem then implies that these loci intercept at a point $r^*_W, r^*_B$ and an equilibrium exists.  Moreover, as the figure shows, $r^*_W>r^*_B>g$ and $r^*_W\in (\rho+\sigma g, \rho+\sigma (p+g))$. 
	
	Finally, the characterization of the tail properties of the income and wealth  distribution follows as a corollary of  Proposition \ref{app proposition: normalized wealth distribution}. 
	%For investors with effective wealth $x_{z,t}$, their income flow is $(\kappa r_K+(1-\kappa)r_B) x_{z,t}dt +\kappa\nu x_{z,t} dW_t$. Thus, income flows are the product of a random variable that follows a Pareto distribution with tail index $\zeta$ and a random variable that follows a normal distribution, which has an approximate Pareto tail with index $\zeta$. 
\end{proof}

\begin{proposition}\label{app proposition: normalized wealth distribution}
	Let $x$ denote effective wealth and define \emph{normalized wealth} by $$\widetilde{x}=x\Big/\frac{w_z}{r_B}.$$ Let $f_H(\widetilde{x})$ and $f_I(\widetilde{x})$ denote the PDFs of the distributions of normalized wealth for households and investors.
	The distribution of normalized wealth for households is given by
	\begin{align}
	f_H(\widetilde{x})=\zeta_H \widetilde{x}^{-\zeta_H-1} \text{ for } \widetilde{x}\geq 1, \label{app dist: households}
	\end{align}
	where 
	$$\frac{1}{\zeta_H}:=\frac{r_B-\rho-\sigma g}{p \sigma},$$
	and the distribution of normalized wealth for investors is given
	\begin{align}
	f_I(\widetilde{x})=\left\{\begin{array}{cl}
	\displaystyle\frac{1}{1/\zeta_P-1/\zeta_N} \widetilde{x}^{-\zeta_P-1} & \text{ for } \widetilde{x}\geq 1\\
	\displaystyle\frac{1}{1/\zeta_P-1/\zeta_N} \widetilde{x}^{-\zeta_N-1} & \text{ for } \widetilde{x}\in [0,1),
	\end{array}\right.  \label{app dist: investors}
	\end{align}
	where
	$$
	\frac{1}{\zeta_P}:=
	\frac{r_W-\rho-\sigma g-\frac{\sigma \kappa^2\nu^2}{2}+\sqrt{\left(r_W-\rho-\sigma g-\frac{\sigma\kappa^2\nu^2}{2}\right)^2+2\sigma^2\kappa^2\nu^2p}}{2p\sigma}
	>0$$
	and
	$$
	\frac{1}{\zeta_N}:=\frac{r_W-\rho-\sigma g-\frac{\sigma \kappa^2\nu^2}{2}-\sqrt{\left(r_W-\rho-\sigma g-\frac{\sigma\kappa^2\nu^2}{2}\right)^2+2\sigma^2\kappa^2\nu^2p}}{2p\sigma}<0.$$

	Moreover, the distribution for investors' income flows has a Pareto tail with index $1/\zeta_P$.
\end{proposition}

\begin{proof}
	The evolution of (normalized) effective wealth for households is given by
	\begin{align*}
	\dot{\widetilde{x}_t}=(\mu_H-g) \widetilde{x}_t,
	\end{align*} 
	where $$\mu_H:=\frac{r_B-\rho}{\sigma}$$ and $\widetilde{x}_t$ resets to 1 with probability $p$.
	
	The Kolmogorov-forward equation characterizing $f_H$ in steady state is then given by
	\begin{align}
	0=-\partial_x \left((\mu_H-g) x f_H(x)\right)-p f_H(x) + p \wp(x-1), \label{app eq: kfe household}
	\end{align}
	where $\wp(.)$ is the Dirac's delta function (a mass of 1 at 0). To solve this differential equation, we guess and verify that 
	$$f_H(x)=C_H x^{-\zeta_H-1}, \text{ for } x\geq 1.$$
	Plugging this guess in the KFE equation \eqref{app eq: kfe household}, we obtain
	\begin{align*}
	0=\zeta_H (\mu_H-g) C_H x^{-\zeta_H-1}- p C_H x^{-\zeta_H-1}, \text{ for } x\geq 1. 
	\end{align*}
	This implies
	\begin{align*}
	\frac{1}{\zeta_H}=\frac{\mu_H-g}{p}=\frac{r_B-\rho-\sigma g}{p \sigma},
	\end{align*}
	as claimed in the proposition. Moreover, because the density $f_H(x)$ must integrate to 1, we obtain 
	\begin{align*}
	1=\int_1^{\infty} C_H x^{-\zeta_H-1} dx \Rightarrow C_H=\zeta_H. 
	\end{align*}
	
	The evolution of (normalized) effective wealth for investors is given by
	\begin{align*}
	d\widetilde{x}_t=(\mu_I-g) \widetilde{x}_t dt + \kappa \nu \widetilde{x}_t dW_t,
	\end{align*} 
	where $$\mu_I:=\frac{r_W-\rho}{\sigma}$$ and $\widetilde{x}_t$ resets to 1 with probability $p$.
	
	The Kolmogorov-forward equation characterizing $f_I$ in steady state is then given by
	\begin{align}
	0=-\partial_x \left((\mu_I-g) x f_I(x)\right)
	+\frac{1}{2} \partial_{xx} \left(\nu^2\kappa^2 x^2 f_I(x)\right)
	-p f_I(x) + p \wp(x-1), \label{app eq: kfe investors}
	\end{align}
	where $\wp(.)$ is the Dirac's delta function (a mass of 1 at 0). To solve this differential equation, we guess and verify a piece-wise solution of the form 
	$$f_I(x)=C_P x^{-\zeta_P-1}, \text{ for } x\geq 1,$$
	and 
	$$f_I(x)=C_N x^{-\zeta_N-1}, \text{ for } x\in (0,1),$$ which allows for the possibility that the distribution might be different to the left and to the right of the reinjection point (note also that the process for effective wealth implies that $x>0$). 
	
	Plugging this guess in the KFE equation \eqref{app eq: kfe household}, we obtain
	\begin{align*}
	0=\zeta_P (\mu_I-g) C_P x^{-\zeta_P-1}
	+(\zeta_P-1)\zeta_P\frac{1}{2} \nu^2\kappa^2 C_P x^{-\zeta_P-1}
	- p C_P x^{-\zeta_P-1}, \text{ for } x\geq 1. 
	\end{align*}
	This implies a quadratic equation for $\zeta_P$ given by
	\begin{align*}
	0=\zeta_P (\mu_I-g)
	+(\zeta_P-1)\zeta_P\frac{1}{2} \nu^2\kappa^2 - p. 
	\end{align*}
	Because the integral of $f_I(x)$ must converge on $(1,\infty)$, $\zeta_P$ must be equal to the unique positive root of the above quadratic equation, which is given by
	\begin{align*}
	\zeta_P=\frac{\left(\frac{\kappa^2\nu^2}{2}-\mu_I+g\right)+\sqrt{\left(\frac{\kappa^2\nu^2}{2}-\mu_I+g\right)^2+2\kappa^2\nu^2p}}{\kappa^2\nu^2}. 
	\end{align*}
	Multiplying the numerator and denominator by $$\left(\frac{\kappa^2\nu^2}{2}-\mu_I+g\right)-\sqrt{\left(\frac{\kappa^2\nu^2}{2}-\mu_I+g\right)^2+2\kappa^2\nu^2p}$$ and rearranging, this formula yields 
	\begin{align*}
	\frac{1}{\zeta_P}=\frac{\mu_I-g-\frac{\kappa^2\nu^2}{2}+\sqrt{\left(\mu_I-g-\frac{\kappa^2\nu^2}{2}\right)^2+2\kappa^2\nu^2p}}{2p},
	\end{align*} 
	which is the same as the formula provided in the lemma.
	
	Likewise, plugging our guess for $x\in (0,1)$, we obtain
	the same quadratic equation for $\zeta_N$ given by
	\begin{align*}
	0=\zeta_N (\mu_I-g)
	+(\zeta_N-1)\zeta_N\frac{1}{2} \nu^2\kappa^2 - p. 
	\end{align*}
	Because the integral of $f_I(x)$ must converge on $(0,1)$, $\zeta_N$ must be equal to the unique negative root of the above quadratic equation, which is given by
	\begin{align*}
	\frac{1}{\zeta_N}=\frac{\mu_I-g-\frac{\kappa^2\nu^2}{2}-\sqrt{\left(\mu_I-g-\frac{\kappa^2\nu^2}{2}\right)^2+2\kappa^2\nu^2p}}{2p},
	\end{align*} 
	which is the same as the formula provided in the lemma.
	
	Finally, we turn to the constants $C_P$ and $C_N$. First, because $f_I(x)$ must be continuous at $x=1$, we obtain $C_P=C_N$. Second, because the density $f_I(x)$ must integrate to 1, we obtain 
	\begin{align*}
	1=\int_0^{1} C_N x^{-\zeta_N-1} dx +  \int_1^{\infty} C_P x^{-\zeta_P-1} dx \Rightarrow C_P=C_N=\frac{1}{1/\zeta_P-1/\zeta_N}. 
	\end{align*}

%%%Added on 2/20/22%%%
Turning to the income distribution, Lemma \ref{app lemma: approximation} shows that, over a short period of time $t$, the income received by an investor with effective wealth $x_{z}$ can be approximated as
\begin{align*}
	y_{z,t}=x_{z,0}(r_It + \kappa \nu \sqrt{t} u),
\end{align*}
where $u\sim N(0,1)$. For large $y$, we have that
\begin{align*}
	\Pr(y_{z,t}\geq y)\propto \int_{-\frac{r_I\sqrt{t}}{\kappa\nu}}^{\infty}  (r_It + \kappa \nu \sqrt{t} u)^{\zeta_P} y^{-\zeta_P} \phi(u) du\propto y^{-\zeta_P},
\end{align*}
where $\phi(u)$ is the pdf of a standard normal, and the second equality follows from the fact that $\int_{0}^{\infty} (r_It + \kappa \nu \sqrt{t} u)^{\zeta_P}\phi(u) du$ is a finite constant for any value of $\zeta_P\geq 0$ (which in turn follows from the fact that the normal distribution has finite moments). Thus, the income distribution also has a Pareto tail with tail index $1/\zeta_P$. 
\end{proof}


\begin{proof}[Proof of Proposition \ref{proposition:approximation}]
The  equilibrium equations for $r_K, r_B,\kappa,  k_n, b_I$ and $b_H$ are
\begin{align*}
	\left(\frac{r_B+\kappa\cdot (r_K-r_B)+\frac{1}{2}(\sigma-1)\gamma\nu^2\kappa^2-\rho}{\sigma}-g\right)\cdot (k_n+b_I+\chi)=&p\cdot (k_n+b_I)\\
	\left(\frac{r_B-\rho}{\sigma}-g\right)\cdot (b_H+1-\chi)=&p\cdot b_H\\
	\kappa\cdot (k_n+b_I+\chi)=&k_n\\
	\frac{1}{\gamma\nu^2}(r_K-r_B)=&\kappa\\
	b_I+b_H=&0\\
	\frac{\alpha}{1-\alpha}\cdot \frac{r_B-g}{r_K+\delta}=&k_n. 
\end{align*}

For $\alpha=0$, we get $r_K=r_B=\rho+\sigma g$, $\kappa=k_n=b_H=b_I=0$. 

Linearizing the system of equations around this equilibrium, we get
\begin{align*}
	\frac{dr_B}{\sigma}\cdot \chi=&p\cdot (dk_n+db_I)&
	\frac{dr_B}{\sigma}\cdot (1-\chi)=&p\cdot db_H\\
	d\kappa\cdot \chi =& dk_n&
	dr_K-dr_B=&\gamma\nu^2 d\kappa\\
	db_I+db_H=&0&
	dk_n=&\frac{\rho+(\sigma-1)g}{\rho+\sigma g+\delta} \alpha.	
\end{align*}
Which yields the solution
\begin{align*}
	dr_K=&\left[p\sigma + \frac{\gamma\nu^2}{\chi}\right] \frac{\rho+(\sigma-1)g}{\rho+\sigma g+\delta} \alpha \\
	dr_B=&p\sigma \cdot\frac{\rho+(\sigma-1)g}{\rho+\sigma g+\delta} \alpha\\
	d\kappa=&\frac{1}{\chi}\cdot \frac{\rho+(\sigma-1)g}{\rho+\sigma g+\delta} \alpha.
\end{align*}
It follows that for small values of $\alpha$ we can approximate all equilibrium objects as
\begin{align*}
	r_K=&\rho+\sigma g+\left[p\sigma + \frac{\gamma\nu^2}{\chi}\right] \frac{\rho+(\sigma-1)g}{\rho+\sigma g+\delta} \alpha \\
	r_B=&\rho+\sigma g+p\sigma \cdot\frac{\rho+(\sigma-1)g}{\rho+\sigma g+\delta} \alpha\\
	\kappa=&\frac{1}{\chi}\cdot \frac{\rho+(\sigma-1)g}{\rho+\sigma g+\delta} \alpha\\
	k_n=&\frac{\rho+(\sigma-1)g}{\rho+\sigma g+\delta} \alpha.
\end{align*}

Let $\alpha^*_{net}$ denote the capital share net of growth and depreciation in equation \eqref{eq: returns with g}. By definition, this is equal to $k_n/(k_n+1)$, and so for small values of $\alpha$ we get the approximation $\alpha^*_{net}=k_n$ which implies $\alpha^*_{net}=\frac{\rho+(\sigma-1)g}{\rho+\sigma g+\delta} \alpha$.

We conclude that
\begin{align*}
	r_K=&\rho+\sigma g+ \left[p\sigma + \frac{\gamma\nu^2}{\chi}\right]\alpha^*_{net} \\
	r_B=&\rho+\sigma g+ p\sigma \cdot\alpha^*_{net}\\
	\kappa=&\frac{1}{\chi}\alpha^*_{net},
\end{align*}
as claimed in the proposition. 

Finally, the formula for the return gap in equation \ref{eq:r_W_def} implies
\begin{align*}
	r_W-\rho-\sigma g=p\sigma\alpha^*_{net}+\frac{1}{\chi^2}{\alpha^*_{net}}^2\gamma\nu^2.
\end{align*}

Using this expression for the return gap, we can compute the tail index for inequality as
\begin{small}\begin{align*}
	\frac{1}{\zeta}=\frac{p\sigma\alpha^*_{net}+\frac{1}{\chi^2}{\alpha^*_{net}}^2\gamma\nu^2-\frac{1}{\chi^2}{\alpha^*_{net}}^2\frac{\sigma}{2}\nu^2
		+\sqrt{(p\sigma\alpha^*_{net}+\frac{1}{\chi^2}{\alpha^*_{net}}^2\gamma\nu^2-\frac{1}{\chi^2}{\alpha^*_{net}}^2\frac{\sigma}{2}\nu^2)^2+ \frac{2}{\chi^2}{\alpha^*_{net}}^2\sigma^2\nu^2p}	
	}{2p\sigma}.
\end{align*}\end{small}
For small values of $\alpha$ this can be linearized as
\begin{align*}
	\frac{1}{\zeta}=\alpha^*_{net}+\alpha^*_{net}\cdot \frac{\frac{\nu^2}{p\chi^2}}{1+\sqrt{1+2\frac{\nu^2}{p\chi^2}}}
\end{align*}

\end{proof}


Proposition \ref{proposition:approximation} provides explicit formulas that are valid for small values of $\tilde{\alpha}$. We now provide an additional proposition characterizing the comparative statics of our extended model away from $\tilde{\alpha}=0$.

\begin{proposition}\label{proposition:bg and changes in alpha} 
	Suppose that investors are risk averse and/or $\theta<\bar{\theta}$. There exists a threshold  $\bar{\alpha}\in (0,1]$ such that, for $\widetilde{\alpha}<\bar{\alpha}$, the balanced-growth equilibrium in the closed economy is unique, and following an increase in $\widetilde{\alpha}$, we have that:
	\begin{itemize}
		\item The return to wealth $r^*_W$, the return gap $r^*_W - \rho - \sigma g$, and the return spread $r_K^*-r_B^*$ all strictly increase, and the portfolio share of capital $\kappa^*$ weakly increases;
		
		\item Top tail inequality $1/\zeta^*$ in \eqref{eq:tail index} strictly increases.
	\end{itemize}
\end{proposition} 


\begin{proof}[Proof of Proposition \ref{proposition:bg and changes in alpha}]
	Let $\bar{r}_W$ denote the point at which $r_W-r_B$ is maximized along the locus $D_B(r_W, r_B)=0$. This definition implies that $\bar{r}_W=\rho+\sigma (p+g)$ if $\gamma\nu^2<\bar{\gamma}$, and $\bar{r}_W<\rho+\sigma (p+g)$ if $\gamma\nu^2>\bar{\gamma}$. 
	
	We first show that there exists a threshold $\bar{\alpha}$ such that, for $\widetilde{\alpha}<\bar{\alpha}$, there is a unique equilibrium, and this equilibrium satisfies $r_W^*<\bar{r}_W$.
	
	Suppose there are multiple equilibria, and let $r^M_W(\widetilde{\alpha})$ denote the value of $r^*_W$ in the equilibrium with the highest return to wealth. In this equilibrium, the locus for $D_K(r_W, r_B)$ cuts the locus for $D_B(r_W, r_B)$ from below, and so an increase in $\widetilde{\alpha}$ shifting the locus for $D_K(r_W, r_B)$ outwards results in a higher $r^M_W(\widetilde{\alpha})$. Moreover, as $\widetilde{\alpha}\rightarrow 0$, $r^M_W(\widetilde{\alpha})\rightarrow \rho+\sigma g$, which implies that for small values of all $\widetilde{\alpha}$, we have $r_W^*<\bar{r}_W$ in any equilibria. Finally, note that as $\widetilde{\alpha}\rightarrow 0$, the locus for $D_K(r_W, r_B)$ converges to the 45 degree line and so we will have a unique equilibrium. This follows from the fact that, as shown in Lemma \ref{lemma:bond market clearing}, the locus for $D_B(r_W, r_B)=0$ moves away from the 45 degree line for $r_W \in (\rho+\sigma g, \breve{r}_W)$, and retains a gap of at least $\frac{1}{2}(\sigma+1)\gamma\nu^2$ from there on. It follows that we can pick a $\bar{\alpha}$ such that, for $\widetilde{\alpha}<\bar{\alpha}$, the equilibrium is unique and satisfies $r_W^*<\bar{r}_W$.
	
	\begin{figure}[ht]
		\centering\includegraphics[width=.9\linewidth]{graphs/equilibrium_bonds_capital_shift.pdf}
		\caption{Effects of changes in the demand for capital on $r_W^*$ and $r_B^*$.\label{figure:equilibrium loci shift} }
	\end{figure} 
	
	For $\widetilde{\alpha}<\bar{\alpha}$, the equilibrium will look as in Figure \ref{figure:equilibrium loci shift}, and increases in $\widetilde{\alpha}$ will result in a higher $r^*_W$. Moreover, because $r^*_W<\bar{r}_W$, we have that the gap $r^*_K-r^*_B$ rises following an increase in automation (so long as $\widetilde{\alpha}$ remains below $\bar{\alpha}$). 
\end{proof}


\clearpage

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\end{appendices}


\begin{appendices}
	
	
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	\begin{center}
		{\Large {\textbf{Supplementary Material for ``Uneven Growth: Automation's Impact on Income and Wealth Inequality"}}}
		
		Benjamin Moll, Lukasz Rachel and Pascual Restrepo \\
		
		\today
		
	\end{center}


\section{Dissipation Shocks \label{appendix dissipation}}

This section provides a formal treatment of dissipation shocks, and shows that they only affect households' saving problem by making them discount the future at a higher rate. We work in a general non-stationary environment. To do so, in this section, $t$ will denote calendar time. 

Formally, we assume that with Poisson rate $p$, households become impatient and only value their consumption in the following $T$ periods. We work with the limit as $T \rightarrow 0$, i.e. households become infinitely impatient. 

Let $V_{z,t,a}^T$ denote the value of lifetime consumption for a household who enters the impatience state at time $t$ with assets $a\geq 0$ (recall that households who enter this state with negative assets see a cancellation of their debt). This value function is given by
\begin{align}\label{eq:VT2}
V_{z,t,a}^T =& \max_{\{c_{z,t+\tau,a}^T, a_{z,t+\tau,a}^T\}} \int_0^{T} e^{-\varrho \tau}\frac{(c_{z,t+\tau,a}^T)^{1-\sigma}}{1-\sigma}d\tau \\
\mbox{s.t.} \quad & \dot{a}^T_{z,t+\tau,a} = w_{z,t+\tau} + r_{t+\tau} a^T_{z,t+\tau,a}- c^T_{z,t+\tau,a}, \quad a^T_{z,t+T,a} \geq 0, \quad a^T_{z,t,a}=a.\notag 
\end{align}	
The following lemma characterizes the behavior of this value function and the consumption of households in the impatience state.  

\begin{lemma}[Characterization of the impatience state]\label{lemma: dying}
	For $T \rightarrow 0$,  the value function $V_{z,t,a}^T$ converges to zero
	$$\lim_{T \rightarrow 0} V_{z,t,a}^T \rightarrow 0 \quad \forall a.$$
	In this limit case, the flow consumption of 
	the impatient is $p K_t$, where $K_t$ are total asset holdings in the economy. 
\end{lemma}

Before proceeding with the Lemma's proof, we briefly explain the intuition for the result that $ V_{z,t,a}^T \rightarrow 0$ as $T \rightarrow 0$. Households enjoy an infinite consumption flow over an infinitesimal time interval $T \rightarrow 0$. But because utility is strictly concave, the value of such consumption converges to zero as $T \rightarrow 0$. In terms of the notation in the proof below, we have
$$V_{z,t,a}^T \approx \int_0^T u(a/T)dt = T\frac{(a/T)^{1-\sigma}}{1-\sigma} =  T^\sigma \frac{a^{1-\sigma}}{1-\sigma} \rightarrow 0 \quad \mbox{as} \quad T \rightarrow 0.$$

\begin{proof}
	The Euler equation for problem \eqref{eq:VT2} implies that consumption satisfies
	\begin{align}\label{appeq: cons path}
	c^T_{z,t+\tau,a}=c^T_{z,t,a} \exp\left(\int_0^\tau \frac{1}{\sigma}(r_{t+s}-\varrho)ds\right).
	\end{align}
	
	Because $a^T_{z,t+T,a}\geq0$, any optimal consumption path must involve $a^T_{z,t+T,a}=0$. Therefore, the consumption path satisfies the budget constraint
	\begin{align*}
	\int_0^T \exp\left(-\int_0^\tau r_{t+s} ds\right) c^T_{z,t+\tau,a}d\tau=a+\int_0^T \exp\left(-\int_0^\tau r_{t+s} ds\right) w_{z,t+\tau}d\tau
	\end{align*}
	
	Plugging equation \eqref{appeq: cons path} into the budget constraint, we obtain the following expression for $c^T_{z,t,a}$:
	\begin{align*}
	c^T_{z,t,a}=\frac{a+\int_0^T \exp\left(-\int_0^\tau r_{t+s} ds\right) w_{z,t+\tau}d\tau}{\int_0^T \exp\left(-\int_0^\tau (r_{t+s}(1-1/\sigma)+\varrho/\sigma) ds\right)d\tau}.
	\end{align*}
	Thus, the optimal consumption path satisfies
	\begin{align}\label{appeq: optimal consumption}
	c^T_{z,t+\tau,a}=\frac{a+\int_0^T \exp\left(-\int_0^\tau r_{t+s} ds\right) w_{z,t+\tau}d\tau}{\int_0^T \exp\left(-\int_0^\tau (r_{t+s}(1-1/\sigma)+\varrho/\sigma) ds\right)d\tau} \exp\left(\int_0^\tau \frac{1}{\sigma}(r_{t+s}-\varrho)ds\right).
	\end{align}
	
	Plugging equation \eqref{appeq: optimal consumption} into \eqref{eq:VT2}, we obtain
	\begin{align*}
	V_{z,t,a}^T=&\frac{1}{1-\sigma}\left(a+\int_0^T \exp\left(-\int_0^\tau r_{t+s} ds\right) w_{z,t+\tau}d\tau\right)^{1-\sigma} \\ &\left(\int_0^T \exp\left(-\int_0^\tau (r_{t+s}(1-1/\sigma)+\varrho/\sigma) ds\right) d\tau\right)^{\sigma}.
	\end{align*}
	As $T\rightarrow 0$, we have that $a+\int_0^T \exp\left(-\int_0^\tau r_{t+s} ds\right) w^T_{z,t+\tau}d\tau$ converges to $a$ and the term $\int_0^T \exp\left(-\int_0^\tau (r_{t+s}(1-1/\sigma)+\varrho/\sigma) ds\right) d\tau$ converges to zero. It follows that $V^T_{z,t,a}$ converges to zero, as claimed in the lemma. 
	
	We now turn to computing the flow consumption of households in the impatience state. For $\tau \in [0,T]$, let $h_{z,a, t-\tau}$ denote the mass of households of skill $z$ who entered the impatience state with assets $a$ at time $t-\tau$. 
	Their flow consumption in the impatience state equals
	\begin{align*}
	C^T_t=&\sum_z \int_a \int_0^T c^T_{z,t+\tau,a} h_{z,a, t-\tau} d\tau da\\
	=&\sum_z \int_a \int_0^T \frac{a+\int_0^T \exp\left(-\int_0^\tau r_{t+s} ds\right) w_{z,t+\tau}d\tau}{\int_0^T \exp\left(-\int_0^\tau (r_{t+s}(1-1/\sigma)+\varrho/\sigma) ds\right) d\tau}\times \\
	&\hspace{3cm} \exp\left(\int_0^\tau \frac{1}{\sigma}(r_{t+s}-\varrho)ds\right) h_z(a, t-\tau) d\tau da \\
	=& \sum_z \int_a \left(a+\int_0^T \exp\left(-\int_0^\tau r_{t+s} ds\right) w_{z,t+\tau}d\tau\right) \times \\&\hspace{3cm}\left( \frac{\int_0^T\exp\left(\int_0^\tau \frac{1}{\sigma}(r_{t+s}-\varrho)ds\right) h_{z,a,t-\tau}  d\tau }{\int_0^T \exp\left(-\int_0^\tau (r_{t+s}(1-1/\sigma)+\varrho/\sigma) ds\right)  d\tau}\right) da.
	\end{align*}
	
	To compute the limit of $C^T_t$ as $T\rightarrow 0$, note that
	\begin{align*}
	\lim_{T \rightarrow 0} a+\int_0^T \exp\left(-\int_0^\tau r_{t+s}ds\right) w_{z,t+\tau}d\tau=a,
	\end{align*}
	and, by an application of L'H\^opital's rule
	\begin{align*}
	\lim_{T \rightarrow 0} \frac{\int_0^T\exp\left(\int_0^\tau \frac{1}{\sigma}(r_{t+s}-\varrho)ds\right) h_{z,a,t-\tau} d\tau }{\int_0^T \exp\left(-\int_0^\tau (r_{t+s}(1-1/\sigma)+\varrho/\sigma) ds\right)  d\tau}=h_{z,a,t}.
	\end{align*}
	It follows that 
	\begin{align*}
	\lim_{T \rightarrow 0}C^T_t=\sum_z \int_a a h_{z,a,t} da=p K_t.
	\end{align*}
	The last step follows from the fact that the probability of entering the impatience state is $p$, independently of assets and skills. 
	
	Note that in the proof above, we assumed that all households entering the impatience state have positive assets. Recall that upon entering this state, households with negative assets see a cancellation of their debt, which results in a one time negative consumption equal to minus their debt generating no disutility. This ensures that the above proof goes through even when some households have negative assets. By construction, for these households we have $V_{z,t,a}^T=0$ 
\end{proof}

Lemma \ref{lemma: dying} and the remarks above imply that the consumption and saving decisions of households solve the maximization problem in \eqref{equation: max problem} in the main text.

\subsection{Other Foundations for Dissipation Shocks\label{appendix: foundations}}

This appendix provides micro-foundations for dissipation shocks. First, we consider a model with stochastic death, no altruism, and an annuity market as in \citet{blanchard}. Second, we consider a model with infinitely lived dynasties and population growth in the form of new households emerging outside of existing dynasties. Third, we consider a model with finite lives and bequests where altruism evolves stochastically within a dynasty. Finally, we present a model with multiple capital lines subject to obsolescence shocks. In all of these models, an equation similar to \eqref{equation: return premium} determines long-run returns and the distribution of wealth also follows a Pareto distribution. 

\subsubsection{Stochastic deaths and annuities}\label{appendix:annuities}

Death is stochastic, with people dying with a Poisson probability $p$. As in \citet{blanchard}, there is an annuity market such that, when people die, they give their wealth to an insurance company. In exchange, they receive a flow income of $p a_{z,t}$ when alive. 

As in the main text, we focus on a steady state of this economy and use $t$ to denote the age of a person.
Consumption decisions solve the problem
\begin{align}\max_{\{c_{z,t}, a_{z,t}\}_{t \geq 0}} \ & \int_0^\infty e^{-(\varrho+p) t} \frac{c_{z,t}^{1-\sigma}}{1-\sigma}dt \label{equation: max problem extension 2} \\
\mbox{s.t. }&\dot{a}_{z,t} = w_z + (r+p) a_{z,t} - c_{z,t}, \text{ and } a_{z,t} \geq -w_z/(r+p) \notag \end{align} 

Analogous to Lemma \ref{lemma: households policy st st}, the optimal saving and consumption policy functions are
\begin{align*}
\dot{a}_{z,t} =\frac{r-\varrho}{\sigma}\left(a_{z,t}+\frac{w_z}{r+p}\right), \qquad  c_{z,t}=\left(r+p-\frac{r-\varrho}{\sigma}\right)\left(a_{z,t}+\frac{w_z}{r+p}\right) 
\end{align*}
with $a_{z,0}=0$. 

Let $X$ denote effective wealth of the economy in steady state. We have that $X=H+K$, where $H=\bar{w}/(r+p)$ denotes human wealth, and $K$ denotes the value of the capital stock. 

Following the derivation in the main text, it follows that the aggregate behavior of $X$ is given by:
\begin{equation}\label{eq:blanchard}
0=\dot{X}=\frac{r-\varrho}{\sigma} X - p K.
\end{equation}
Relative to our baseline model, the difference is that in this equation, the rate of accumulation depends on $r-\varrho$, rather than $r-\rho$. This difference is driven by the incentives to accumulate assets introduced by the annuity market: wealth now pays return $r+p$ so that the individual wealth accumulation rate is $(r+p-\varrho-p)/\sigma = (r-\varrho)/\sigma$. As in our baseline model, we still have the term $- p K$ on the right-hand side of equation \eqref{eq:blanchard}. 
This term now captures the wealth paid by dying individuals to the insurance company upon death. The insurance company redistributes this wealth to all living households via a higher return to their wealth $r+p$. The term $(r-\varrho)/\sigma$ already accounts for the redistributed wealth, so the $- p K$ term ensures there is no double-counting. 

Note that equation \eqref{eq:blanchard} is equivalent to the equations for aggregates in \citet{blanchard} who analyzes the special case of logarithmic utility, $\sigma=1$. In particular, using our notation, equations (5), (6) and (7) in the published version of his paper are: $C = (\varrho+p)(H+K),\dot{H}=(r+p)H - \bar{w}$ and $\dot{K} = rK + \bar{w} - C$. Using our definition of effective wealth $X:=H+K$, we have $\dot{X} = (r+p)H + rK - (\varrho+p)(H+K) = (r-\varrho)X - p K$ which is the special case of \eqref{eq:blanchard} with $\sigma=1$.

Net capital income now includes annuity payments and so it is given by $(r+p)K$. Therefore, rearranging \eqref{eq:blanchard} and using the fact that $X=K+\bar{w}/(r+p)$, the steady state return to wealth satisfies
\begin{align*}
r=\varrho + p\sigma \frac{(r+p)K}{(r+p)K+ \bar{w}}=\varrho + p\sigma \alpha^*_{net}.
\end{align*}
Alternatively, the analogue to \eqref{equation:supply capital} for long-run capital supply is $\left(K/\bar{w}\right)^s = \frac{r-\varrho}{(p \sigma + \varrho  - r)(r+p)}$. 

Following the same steps as in the baseline model, it follows that effective wealth follows a Pareto distribution. Wages give the scale parameters, and the common tail parameter is given by
\begin{align*}
\frac{1}{\zeta}= \frac{1}{p}\frac{r-\varrho}{\sigma} =\alpha^*_{net},
\end{align*}
where we have used that the rate of individual wealth accumulation depends on $r-\varrho$ as discussed below equation \eqref{eq:blanchard}.

The model with annuities therefore yields exactly the same expressions for the steady state return to wealth and the tail parameter of the wealth distribution as in our baseline model except for one difference: the effective discount rate $\rho=\varrho+p$ is replaced by the unadjusted discount rate $\varrho$.

\subsubsection{Population growth}

We now assume that dynasties are infinitely lived and experience no dissipation shocks. However, at a rate $p$, new dynasties emerge. These new dynasties start with zero assets. 

As in the main text, we focus on a steady state of this economy and use $t$ to denote the age of a dynasty.
Consumption decisions solve the problem
\begin{align}\max_{\{c_{z,t}, a_{z,t}\}_{t \geq 0}} \ & \int_0^\infty e^{-\varrho t} \frac{c_{z,t}^{1-\sigma}}{1-\sigma}dt \label{equation: max problem extension 3} \\
\mbox{s.t. }&\dot{a}_{z,t} = w_z + r a_{z,t} - c_{z,t}, \text{ and } a_{z,t} \geq -w_z/r. \notag \end{align} 

Analogous to Lemma \ref{lemma: households policy st st}, the optimal saving and consumption policy functions are
\begin{align*}
\dot{a}_{z,t} =\frac{r-\varrho}{\sigma}\left(a_{z,t}+\frac{w_z}{r}\right), \qquad  c_{z,t}=\left(r-\frac{r-\varrho}{\sigma}\right)\left(a_{z,t}+\frac{w_z}{r}\right) 
\end{align*}
with $a_{z,0}=0$. 

Let $X$ denote effective wealth of the economy in steady state. We have that $X=H+K$, where $H=\bar{w}/r$ denotes human wealth, and $K$ denotes the value of the capital stock. Because of population growth, we are interested in a steady state where aggregates grow at a rate $p$. 

Following the derivation in the main text, it follows that the aggregate behavior of $X$ is given by:
\begin{align*}
p X=\dot{X}=\frac{r-\varrho}{\sigma} X + p H.
\end{align*}
Relative to our baseline case, there are two differences in this equation. First, total wealth grows at a rate $p$ since population increases over time. Second, the term $+pH$ captures the wealth brought by new dynasties, who start life with nothing but their labor income. 

Rearranging this equation, and using the fact that $X=H+K$, we obtain:
\begin{align*}
0=\frac{r-\varrho}{\sigma} X -p K,
\end{align*}
which implies that, in steady state 
\begin{align*}
r=\varrho + p\sigma \alpha^*_{net}.
\end{align*}

Let us now turn to the wealth distribution. Let $f_z(x)$ denote the distribution of effective wealth $x_z$ for households with wage $w_z$. Following the same steps as in the baseline model, we obtain the Kolgomorov Forward Equation:
\begin{align*}
0 = -\left(\frac{r-\varrho}{\sigma} x f_z(x)\right)' -p f_z(x)
\end{align*} 
on $(w_z/r,\infty)$. The term $-p f_z(x)$ now captures the loss of probability mass as new individuals are born and start their lives with effective wealth $w_z/r$. Following the same steps as in the baseline model, it follows that effective wealth follows a Pareto distribution. 

\subsubsection{Finite lives and stochastic altruism}

We now consider a model with finite lives and stochastic altruism. Individuals live for a period of length $T$. They are born as one of two types: altruistic, which happens with probability $e^{-pT}$, or non-altruistic (selfish), which happens with probability $1-e^{-pT}$. Individuals decide how much to consume and save during their lifetimes and how much wealth to pass on to their offspring when they die at age $T$: if they are altruistic they leave their children some wealth; instead, if they are selfish, they consume all their wealth before they die. Therefore, the dynasty continues with probability $e^{-pT}$ but is interrupted with probability $1-e^{-pT}$.

As in the main text, we focus on a steady state of this economy and use $t$ to denote age. Let $v^T_A(a)$ denote the value function for an altruistic individual who starts life with assets $a$ and $v^T_N(a)$ that for a non-altruistic individual. We have
\begin{align}
v_A^T(a)=\max_{\{c_{z,t}, a_{z,t}\}_{t\in [0,T]}} \ & \int_0^T e^{-\varrho t} \frac{c_{z,t}^{1-\sigma}}{1-\sigma}dt + e^{-(\varrho+p) T} v^T_A(a_{z,T})+ e^{-\varrho T}\left(1-e^{-p T}\right) v^T_N(a_{z,T}), \label{equation: max problem extension 4a} \\
\mbox{s.t. }&a_{z,0}=a, \quad \dot{a}_{z,t} = w_z + r a_{z,t} - c_{z,t}, \text{ and } a_{z,t} \geq -w_z/r. \notag 
\end{align}
and
\begin{align}v^T_N(a)=\max_{\{c_{z,t}, a_{z,t}\}_{t\in [0,T]}} \ & \int_0^T e^{-\varrho t} \frac{c_{z,t}^{1-\sigma}}{1-\sigma}dt, \label{equation: max problem extension 4b} \\
\mbox{s.t. }&a_{z,0}=a, \quad\dot{a}_{z,t} = w_z + r a_{z,t} - c_{z,t}, \text{ and } a_{z,t} \geq -w_z/r. \notag \end{align} 

We now show that, as $T\rightarrow 0$, this model collapses to our baseline model with dissipation shocks.

At any point in time, a fraction $\frac{1}{T}(1-e^{-pT})$ of households starts life with no assets. Taking the limit as $T\rightarrow0$, this fraction converges to $p$. 

Lemma \ref{lemma: dying} then implies that $\lim_{T \rightarrow 0}v^T_N(a_{z,T})\rightarrow 0$ and the consumption of people in the non-altruistic state is $pK$. 

Using this result, and expanding equation \eqref{equation: max problem extension 4a} around $T=0$, we obtain
\begin{align*}
0=\max_{c_{z,0}}\frac{c_{z,0}^{1-\sigma}}{1-\sigma}
-(\varrho + p) v_A(a_{z,0}) 
+v_A'(a_{z,0})\dot{a}_{z,0}, 
\end{align*}
where $v_A(a_{z,0})=\lim_{T \rightarrow 0}v^T_A(a_{z,T})$. It follows that $v_A(a_{z,0})$ satisfies the same Bellman equation that characterizes optimal consumption and saving decisions in our baseline model. Thus, both models generate the same value functions, consumption and saving decisions. 

\subsubsection{Uninsured capital income risk (capital obsolescence shocks)}

Finally, we consider a model of uninsured capital income risk. There are multiple types of capital indexed by $i$ representing different investment projects and $k_i$ denotes the stock of capital of type $i$ households have already accumulated. Capital of type $i$ is either useful in production or obsolete which we denote by $\zeta_i\in \{0,1\}$, with $\zeta_i=0$ denoting obsolescence. Total capital services in the economy are 
\begin{align*}
K=\int_{i} \zeta_ik_{i} di.
\end{align*}
Obsolescence occurs at Poisson rate $p$. For example, obsolescence might be due to the fact that a whole technology class or an intermediate product line is suddenly displaced by a new one via creative destruction.

Each household is assigned a unique investment project $i$. While capital of type $i$ is useful in production it pays a return $r$. When capital becomes obsolete (which happens at rate $p$), the household draws a new investment project and start with zero capital in this project, i.e. when an obsolescence shock hits, the affected capital has a return of minus one hundred percent (or an instantaneous continuous-time return of $-\infty$). This formulation implies that a household $z$ with an active project $i(z)$ solves the problem
\begin{align*}
\max_{\{c_{z,t}, k_{i(z),t}\}_{t \geq 0}} \ & \int_0^\infty e^{-\varrho t} \frac{c_{z,t}^{1-\sigma}}{1-\sigma}dt  \\
\mbox{subject to: }&\dot{k}_{i(z),t}=r k_{i(z),t} + w_z - c_{z,t}, \text{ and } c_{z,t}, k_{i(z),t}\geq 0, \end{align*} 
where in addition, $k_{i(z),t}$ resets to zero with probability $p$---capturing the fact that the household needs to start a new investment project.

As in Lemma 1, household policy functions (assuming $r>\rho$) are given by
\begin{equation*}
\dot{k}_{i(z),t} =\frac{r-\rho}{\sigma}\left(k_{i(z),t}+\frac{w_z}{r}\right), \qquad  c_{z,t}=\left(r-\frac{r-\rho}{\sigma}\right)\left(k_{z,t}+\frac{w_z}{r}\right),
\end{equation*}
with $k_{i(z),t}$ reseting to zero with probability $p$. 

As in the main text, it follows that the aggregate behavior of $K$ satisfies
\begin{align*}
\dot{K}=\frac{r-\rho}{\sigma}\left(K+\frac{\bar{w}}{r}\right)-pK \quad \Rightarrow \quad  
r=\rho+p \sigma \alpha^*_{net}.
\end{align*}
This implies the same dynamics for wealth and wealth inequality that we obtained in the model with dissipation shocks. The only difference relative to our baseline model is that, in this case, capital that becomes obsolete cannot be consumed or re-invested. This implies a different Euler equation and resource constraint given by 
\begin{align*}
\dot{C}=&\frac{1}{\sigma}(Y'(K)-\delta-\rho)C-\mu p K,\\
\dot{K}=&Y(K)-(\delta+p)K-C.
\end{align*}

Finally, note that this formulation assumes limited risk sharing and diversification. If households were to decide how much to invest in each active capital line, they would invest a constant fraction in each and obtain a risk free return of $r-p$. Equivalently, if an investment company were to pool all investments, it would offer all households a common and risk-free return of $r-p$. In both cases we would recover Gorman aggregation; the steady state would involve $r=\rho$ and the wealth distribution would remain indeterminate. 

\section{Results Regarding the Composition of Income\label{appendix:composition}}

\subsection{Income composition at top of income distribution}\label{app:composition_prop}

This appendix provides a proposition exploring the implications of our formulas in the baseline model for the composition of the income distribution. 

\begin{proposition}[Composition and sources of income at top of income distribution]\label{proposition: composition top wealth} 
	Let $\bar{q}:=\Pr(\textrm{income}\geq \max_{z}w_z)$. For $q<\bar{q}$, we have:
	\begin{itemize}
		\item the probability that someone with a wage $w_z$ is in the top $q$ is
		\begin{align*}
		\Pr(\textrm{skill}=z| \textrm{top } q)=\frac{\ell_z w_z^{1/\alpha_{net}^*}}{\sum_v \ell_v w_v^{1/\alpha_{net}^*}};
		\end{align*}
		\item the share of labor income relative to total income held by the top $q$ is
		\begin{align*}
		\frac{\mathbb{E}[\textrm{labor income} | \textrm{top } q] }{\mathbb{E}[\textrm{income} |\textrm{top } q ]}= (1-\alpha^*_{net}) q^{\alpha^*_{net}}\frac{\sum_{z}\ell_z w_z^{1+1/\alpha^*_{net}}}{\left(\sum_{z}\ell_z w_z^{1/\alpha^*_{net}}\right)^{1+\alpha^*_{net}}};
		\end{align*}
		\item the share of national income held by the top $q$ is
		\begin{align*}
		S(q)=\Lambda q^{1-\alpha^*_{net}},
		\end{align*}
		where $\Lambda$ is a constant that depends on the wage distribution.
	\end{itemize}
	
\end{proposition}

\begin{proof}[Proof of Proposition \ref{proposition: composition top wealth}] We start by deriving the probability that households with skill $z$ are among the top $q$ income earners. To save on notation, we do not include asterisks when denoting steady state objects. 
	
	Let $y(q)$ denote the income of the $q$th higher earner. That is:
	\begin{align*}
	\Pr(\textrm{income}\geq y(q))=q.
	\end{align*}
	By definition, for $q<\bar{q}$ we have $y(q)>\max_z w_z$. We can thus compute $y(q)$ explicitly as
	\begin{align*}
	q=\Pr(\textrm{income}\geq y(q))=\sum_z \ell_z \left(\frac{y(q)}{w_z}\right)^{-1/\alpha_{net}},
	\end{align*}
	which implies
	\begin{align}
	y(q)=q^{-\alpha_{net}}\left(\sum_z \ell_z w_z^{1/\alpha_{net}}\right)^{\alpha_{net}}. \label{appeq: top quantile}
	\end{align}
	
	An application of Bayes' rule implies
	\begin{align*}
	\Pr(\textrm{skill}=z|\textrm{top }q)=\frac{\ell_z\Pr(\textrm{income}\geq y(q)|z)}{\sum_v \ell_v\Pr(\textrm{income}\geq y(q)|v)}
	=\frac{\ell_zy(q)^{-1/\alpha_{net}}w_z^{1/\alpha_{net}}}{\sum_v \ell_vy(q)^{-1/\alpha_{net}}w_v^{1/\alpha_{net}}}
	=\frac{\ell_zw_z^{1/\alpha_{net}}}{\sum_v \ell_v w_v^{1/\alpha_{net}}}.
	\end{align*}	
	
	We now turn to the share of labor income at the top of the income distribution. The expected labor income for households at the top $q$ is given by
	\begin{align*}
	\mathbb{E}[\textrm{labor income}|\textrm{top }q] = \sum_z w_z \Pr(\textrm{skill} = z|\textrm{top }q) = \sum_z \frac{\ell_z w_z^{1+1/\alpha_{net}} }{\sum_{v} \ell_v w_{v}^{1/\alpha_{net}} } =\frac{\sum_z \ell_z w_z^{1+1/\alpha_{net}} }{\sum_{z} \ell_z w_{z}^{1/\alpha_{net}} }.
	\end{align*}
	The expected income for households at the top $q$ is given by
	\begin{align*}
	\mathbb{E}[\textrm{income}|\textrm{top }q] = \sum_z \mathbb{E}[\textrm{income}|\textrm{income}\geq y(q)] \Pr(\textrm{skill} = z|\textrm{top }q)=\frac{y(q)}{1-\alpha_{net}}.
	\end{align*}
	Here, we used the fact that $\mathbb{E}[\textrm{income}|\textrm{income}\geq y(q)]=\frac{y(q)}{1-\alpha_{net}}$, a well-known property of Pareto distributions. 
	
	It follows that
	\begin{align*}
	\frac{\mathbb{E}[\textrm{labor income}|\textrm{top }q]}{\mathbb{E}[\textrm{income}|\textrm{top }q]}=\frac{1-\alpha_{net}}{y(q)}\frac{\sum_z \ell_z w_z^{1+1/\alpha_{net}} }{\sum_{z} \ell_z w_{z}^{1/\alpha_{net}} }=(1-\alpha_{net}) q^{\alpha_{net}}\frac{\sum_{z}\ell_z w_z^{1+1/\alpha_{net}}}{\left(\sum_{z}\ell_z w_z^{1/\alpha_{net}}\right)^{1+\alpha_{net}}}
	\end{align*}
	
	Finally, we compute the share of national income earned by the top $q$. For $q\leq \bar{q}$, the top $q$ earn an income
	\begin{align*}
	T(q)=q \mathbb{E}[\textrm{income}|\textrm{top }q] = \frac{1}{1-\alpha_{net}}q^{1-\alpha_{net}} \left(\sum_z \ell_z w_z^{1/\alpha_{net}}\right)^{\alpha_{net}}. 
	\end{align*}
	It follows that the top $q$ earn a share of national income equal to:
	\begin{align*}
	S(q)=\frac{S(q)}{S(\bar{q})}S(\bar{q})=\frac{T(q)}{T(\bar{q})}S(\bar{q})=\frac{q^{1-\alpha_{net}}}{\bar{q}^{1-\alpha_{net}}}S(\bar{q}).
	\end{align*}
	The result in the proposition follows by letting $\Lambda=\frac{1}{\bar{q}^{1-\alpha_{net}}}S(\bar{q})$. \end{proof}


\subsection{Meade's Formula about Compositional Effects}\label{app:meade}
\citet[][p.34]{meade} states equation \eqref{eq:meade} which links changes in the  top $q$ percent income share to changes in the net capital share $\alpha_{net}$. We here provide a quick derivation. As in Appendix \ref{app:composition_prop} denote by $y(q)$ the income of the $q$th highest earner. This income is composed of capital income $y_k(q)$ and labor income $y_\ell(q)$
\begin{equation}\label{eq:meade1}
y(q) = y_k(q) + y_\ell(q)
\end{equation}
Next denote the corresponding aggregates by $Y:=\int_0^1 y(q)dq,Y_k:=\int_0^1 y_k(q)dq$ and $Y_\ell:=\int_0^1 y_\ell(q)dq$ and the net capital share by $\alpha_{net} = Y_k/Y$. Dividing \eqref{eq:meade1} by $Y$ we have
$$\frac{y(q)}{Y} = \alpha_{net} \frac{y_k(q)}{Y_k} + (1-\alpha_{net})\frac{y_\ell(q)}{Y_\ell}$$ 
Therefore the top $q$ percent income share $S(q)=y(q)/Y$ satisfies \eqref{eq:meade} where $\tilde{S}_k(q):=y_k(q)/Y_k$ and $\tilde{S}_\ell(q):=y_\ell(q)/Y_\ell$ are the shares of aggregate capital income and labor income earned by the top $q$ percent of the distribution of \emph{total} income.


\section{Transitional Dynamics in the Baseline Model\label{appendix transition}}

This section describes the full model in a non-stationary environment. Because firms rent capital from households, the production structure remains unchanged. We therefore focus on the household problem in a non-stationary environment. The section concludes with the proof of Proposition \ref{proposition: transition dynamics}, which characterizes the transitional dynamics of our model.   

\subsection{Savings Problem in a Non-stationary Environment}

Unlike in the main text, in what follows it will be convenient to keep track of time and cohorts, rather than time since last dissipation shock. Thus, we use $t$ to denote calendar time, and $b$ to denote the time at which the wealth of a household was last reset. Also, define human wealth at time $t$
$h_{z,t}$ for households with skill $z$ as
\begin{align*}
h_{z,t}:=\int_{t}^{\infty}e^{-\int_{t}^{s}r_{\tau}d\tau}w_{z,s}ds.
\end{align*}

Lemma \ref{lemma: dying} implies that the consumption and saving decisions of households solve a variant of \eqref{equation: max problem} in the main text generalized to a non-stationary environment:
\begin{align}\max_{\{c_{z,t,b}, a_{z,t,b}\}_{t \geq b}} \ & \int_b^\infty e^{-(\varrho+p) (t-b)} \frac{c_{z,t,b}^{1-\sigma}}{1-\sigma}dt \label{equation: max problem full} \\
\mbox{s.t. }&\dot{a}_{z,t,b} = w_{z,t} + r_t a_{z,t,b} - c_{z,t,b}, \text{ and } a_{z,b,b} = 0, a_{z,t,b}\geq -h_{z,t}.\notag 
\end{align} 
Here, $c_{z,t,b}$ and  $a_{z,t,b}$ denote consumption and assets at time $t$ of a household from the cohort who experienced their last dissipation shock at time $b$. 

To characterize the solution to the problem in equation \eqref{equation: max problem full}, we generalize the definition of effective wealth to
\begin{align*}
x_{z,t,b}:=a_{z,t,b}+h_{z,t}.
\end{align*}


\begin{lemma}[Households policy functions outside the steady state]\label{lemma: household policy transition} Suppose that $r_t$ converges to $r^*$ and  $r^*>(r^*-\rho)/\sigma$. 
	The unique interior solution to the household problem in equation \eqref{equation: max problem full} is given by policy functions that are linear in effective wealth
	\begin{align}
	\dot{x}_{z,t,b}&=(r_t-\mu_t)x_{z,t,b}, \label{eq: x dynamics transition} \\
	c_{z,t,b}&=\mu_tx_{z,t,b}, \notag
	\end{align}
	for $t\geq b$, 	with $x_{z,b,b}=h_{z,b}$. 
	
	Here $\mu_t$ denotes the marginal propensity to consume out of wealth, and satisfies the differential equation:
	\begin{align}\label{equation: mu}
	\frac{\dot{\mu}_t}{\mu_t}=\mu_t-r_t+\frac{1}{\sigma}(r_t-\rho)
	\end{align}
\end{lemma}

\begin{proof}
	The maximization problem can be rewritten using effective wealth as
	\begin{align}\max_{\{c_{z,t,b}, x_{z,t,b}\}_{t \geq b}} \ & \int_t^\infty e^{-(\varrho+p) (t-b)} \frac{c_{z,t,b}^{1-\sigma}}{1-\sigma}dt \label{equation: max problem full x} \\
	\mbox{s.t. }&\dot{x}_{z,t,b} = r_t x_{z,t,b} - c_{z,t,b}, \text{ and } x_{z,t,b} \geq 0 \notag 
	\end{align} 
	
	The Hamiltonian associated with this maximization problem is
	\begin{align}
	H(c_z, x_z, \lambda_z):=\frac{c_z^{1-\sigma}}{1-\sigma}+\lambda (r x_z - c_z), \label{appeq: max hamiltonean full}
	\end{align}
	where $\lambda_z$ is the co-state for effective wealth. 
	
	We show that the unique solution to equation \eqref{eq: x dynamics transition} starting from $x_{z,b,b}=h_{z,b}$ solves the maximization problem in \eqref{appeq: max problem}. 
	
	Theorem 7.14 in \citet{acemoglu-book} implies that this candidate path reaches an optimum if there exists a co-state variable $\lambda_{z,t,b}$ such that:
	\begin{enumerate}
		\item the path satisfies the restrictions $\dot{x}_{z,t,b} = r_t x_{z,t,b} - c_{z,t,b}$,  and $x_{z,t,b} \geq 0$;
		
		\item the following necessary conditions hold:  
		\begin{align*}
		c_{z,t,b}^{-\sigma}=&\lambda_{z,t,b}, \\
		\rho \lambda_{z,t,b}-\dot{\lambda}_{z,t,b}=& r \lambda_{z,t,b};
		\end{align*}
		
		\item the maximized Hamiltonian $M(x_z, \lambda_z)=\max_{c} H(c, x_z, \lambda_z)$ is concave in $x_z$ along the candidate path;
		
		\item the transversality condition holds. That is, for the candidate path, we have
		\begin{align*}
		\lim_{t \rightarrow \infty} e^{-\rho t} x_{z,t,b} \lambda_{z,t,b}=&0. 
		\end{align*}
		and for all other feasible paths, $\hat{x}_{z,t,b}$, we have
		\begin{align*}
		\lim_{t \rightarrow \infty} e^{-\rho t} \hat{x}_{z,t,b} \lambda_{z,t,b}\geq&0. 
		\end{align*}
	\end{enumerate}
	
	To prove condition 1, note that starting from $x_{z,b,b}=h_{z,b}$, we will have $x_{z,t,b}\geq 0$ for all $t\geq b$. Moreover, for any path satisfying equations \eqref{appeq: max path} the flow budget constraint holds:
	\begin{align*}
	r_t x_{z,t,b}-c_{z,t,b}=&r_t x_{z,t,b}-\mu_t x_{z,t,b}\\
	=&(r_t-\mu_t) x_{z,t,b} \\
	=&\dot{x}_z.
	\end{align*}
	
	To prove condition 2, define $\lambda_{z,t,b}:=\mu_t^{-\sigma} x_{z,t,b}^{-\sigma}>0$. By construction, $c_{z,t,b}^{-\sigma}=\lambda_{z,t,b}$. Moreover:
	\begin{align*}
	\rho \lambda_{z,t,b} -\dot{\lambda}_{z,t,b}=&\rho \mu_t^{-\sigma} x_{z,t,b}^{-\sigma} 
	+\mu_t^{-\sigma}  \sigma x_z(s)^{-\sigma-1} \dot{x}_z
	-\sigma \mu_t^{-\sigma-1}\dot{\mu}_tx_{z,t,b}^{-\sigma} \\
	=&\lambda_{z,t,b} \left(\rho 
	+ \sigma \frac{\dot{x}_{z,t,b}}{x_{z,t,b}}
	+\sigma \frac{\dot{\mu}_t}{\mu_t}\right).
	\end{align*} 
	Using the equations for $\dot{x}_{z,t,b}$ (equation \eqref{appeq: max path}) and $\dot{\mu}_{z,t,b}$ (equation \eqref{equation: mu}), we obtain:
	\begin{align*}
	\rho \lambda_{z,t,b} -\dot{\lambda}_{z,t,b}=\lambda_{z,t,b} \left(\rho + \sigma (r_t-\mu_t) + \sigma \left(\mu_t-r_t+\frac{1}{\sigma}(r_t-\rho)\right) \right)=r_t\lambda_{z,t,b}
	\end{align*}
	
	
	To prove condition 3, note that
	\begin{align*}
	\max_c H(c, x_z, \lambda_z)= \frac{\lambda_z^{\frac{\sigma-1}{\sigma}}}{1-\sigma}+\lambda_z(r x_z -\lambda_z^{-\frac{1}{\sigma}}),
	\end{align*}
	which is concave (linear) in $x_z$.  
	
	To prove the first part of condition 4, note that along the candidate path, $x_z$ grows asymptotically at a rate $\frac{r^*-\rho}{\sigma}$, and $\lambda_{z,t,b}$ at a rate $\rho-r^*$ ($\mu_t$ converges along the candidate path). It follows that the first part of the transversality condition holds if
	\begin{align*}
	-\rho+\frac{r^*-\rho}{\sigma}+\rho-r^* <0,
	\end{align*}
	which is equivalent to the condition $r^*>(r^*-\rho)/\sigma$. 
	
	The second part of the transversality condition follows from the fact that, along any feasible path, we have $\hat{x}_{z,t,b}\geq 0$. 
	
	It follows that the candidate path provides optimal paths for consumption and asset accumulation outside of the steady state.
\end{proof}

\subsection{The Transition of Aggregates and Distributions}

The following proposition characterizes the transition dynamics for the macroeconomic aggregates and the distribution of effective wealth. As for the steady state equilibrium, the transition 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.

\begin{proposition}[Transitional dynamics]\label{proposition: transition dynamics} The behavior of the macroeconomic aggregates, $C$ and $K$ is given by the unique stable solution to the system of differential equations 
	\begin{align*}
	\dot{C} =&\frac{1}{\sigma}(r -\rho)(C - p K)-\mu p K + p\dot{K} \\
	\dot{K} =&Y-\delta K-C, \\
	\frac{\dot{\mu}}{\mu}=&\mu-r+\frac{1}{\sigma}(r-\rho)
	\end{align*}
	where $\mu$ denotes the rate at which households consume their effective wealth (to simplify notation, we removed the time dependence of  aggregates). Also, recall that $Y$ is given by $Y(K)$ in equation \eqref{equation: aggregate output} and $r$ is given by $Y'(K)-\delta$
	
	Along the transition path, households accumulate effective wealth at a rate $r_t-\mu_t$, which implies that the distribution of effective wealth for households with skill $z$, $f_z(x,t)$ evolves according to the Kolmogorov Forward Equation
	\begin{align}	
	\frac{\partial f_z(x,t)}{\partial t} = - \frac{\partial}{\partial x}\left[(r_t-\mu_t) x f_z(x,t)\right] - p f_z(x,t) + p \wp(x - h_{z,t}) \label{eq:kfe}  
	\end{align}
	where $\wp(.)$ is the Dirac delta function, and $h_{z,t}$ is a time-varying reinjection point.
\end{proposition}

\begin{proof} We start by deriving the equation for $\dot{C}$. Aggregate consumption is given by
	\begin{align*}
	C_t=\sum_z\ell_z \int_{-\infty}^t c_{z,t,b} pe^{-p(t-b)}\ell_z db+p K_t.	
	\end{align*}
	Here, $pe^{-p(t-b)}$ is the mass of households with skill $z$ at time $t$ who received their last dissipation shock at time $b$. The term $p K_t$ is the flow consumption of households in the impatience state, which was derived in Lemma \ref{lemma: dying}. 
	
	Differentiating this equation, we obtain
	\begin{align*}
	\dot{C}_t=&\sum_z\ell_z \int_{-\infty}^t \dot{c}_{z,t,b} pe^{-p(t-b)} db +p\dot{K}_t\\
	&+p\sum_z\ell_z c_z(t,t) -p\sum_z\ell_z \int_{-\infty}^t c_{z,t,b} pe^{-p(t-b)} db.
	\end{align*}
	These terms can be simplified as follows. The first term captures consumption growth over time. Using households' Euler equation, we can rewrite this term as
	\begin{align*}
	\sum_z\ell_z \int_{-\infty}^t \dot{c}_{z,t,b} pe^{-p(t-b)} db=&\sum_z\ell_z \int_{-\infty}^t \frac{r_t-\rho}{\sigma}c_{z,t,b} pe^{-p(t-b)} db \\
	=&\frac{r_t-\rho}{\sigma}(C_t-p K_t). 
	\end{align*}
	
	The third and fourth term capture the permanent decline in consumption due to the dissipation of wealth at time $t$. This is equal to $p\mu_tK_t$, which gives the product of the rate at which households consume their wealth and $p K_t$, which denotes the total wealth dispersed.
	
	Plugging these simplified values in the expression for $\dot{C}$, we obtain
	\begin{align*}
	\dot{C}_t=\frac{r_t-\rho}{\sigma}(C_t-p K_t)+p\dot{K}_t-p\mu_t K_t.
	\end{align*}
	
	The equation for $\dot{K}$ is the usual resource constraint, and the equation for $\dot{\mu}$ was derived above in Lemma \ref{lemma: household policy transition}.
	
	Turning to the distribution of effective wealth, the Kolgomorov Forward Equation \eqref{eq:kfe} follows from the fact that households accumulate effective wealth at a rate $r_t-\mu_t$ (an implication of Lemma \ref{lemma: households policy st st}) but experience a dissipation shock with probability $p$. After these shocks, households' effective wealth jumps to $h_{z,t}$. 
\end{proof}

The proposition shows that the transition 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 $\mu_t$, which controls the common marginal propensity to consume out of effective wealth. Also, 
the Euler equation has some extra terms to account for the changes in consumption due to dissipation shocks. 

Turning to the evolution of the wealth distribution, suppose the initial distribution of effective wealth conditional on skills is given by
\begin{align}
\Pr(x_{z, t_0}>x)=\left(\frac{x}{w^*_{z, t_0}/r^*_{t_0}}\right)^{-\zeta}, \label{eqq:initial dist}
\end{align}
as in Proposition \ref{proposition: wealth inequality}. Here, $x_{z, t_0}$ is a random variable denoting the effective wealth of households with skill $z$.

Following an increase in automation at time $t=t_0$, households with skills $z$ and effective wealth $x_{z, t_0}$ see a revaluation of their human wealth of $\Delta_z$ (this could be negative for households experiencing a real decline in wages over time). This implies that the distribution of $x_z$, denoted by $f_z(x,t)$, starts from
\begin{align*}
f_z(x, t_0)=\left(\frac{w_{z, t_0}^*}{r^*_{t_0}}\right)^{\zeta}\zeta (x-\Delta_z)^{-\zeta-1} \quad \text{for } x\geq \frac{w_{z, t_0}^*}{r^*_{t_0}}+\Delta_z.  
\end{align*}
and from there on evolves according to the Kolgomorov Forward Equation \eqref{eq:kfe}.

\subsection{Wealth Distribution in the Representative Household Benchmark}

This subsection characterizes the dynamics of the wealth and income distribution in the representative-household benchmark. We will use a superscript $h$ to denote the corresponding values of aggregates and to distinguish them from the aggregate paths for wages and interest rates in our model with dissipation shocks.

In the representative-household benchmark, 
the wealth distribution is 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 transition dynamics of the wealth distribution are uniquely defined. 

To make things comparable, assume as in the main text that the initial distribution of effective wealth is given by \eqref{eqq:initial dist}, and coincides with that in our model with dissipation shocks. Following an increase in automation at time $t=t_0$, households with skills $z$ and effective wealth $x_{z, t_0}$ see a revaluation of their human wealth of $\Delta^{h}_z$ (this will differ from the revaluation in our model since wages behave differently in the representative-household benchmark---see Proposition \ref{proposition: wages}). People then accumulate assets starting from $x_{z, t_0}+\Delta^h_z$ at a common rate $r^{h}_t-\mu^{h}_t$, which is temporarily above zero but converges to zero over time (recall that in the representative-household benchmark, $r^{h}_t$ and $\mu^{h}_t$ converge to $\rho$, reflecting the fact that the supply of capital is fully elastic). This temporary period of accumulation scales everyone's effective wealth by the same amount, $M$, but does not contribute to thicker tails in effective wealth. The resulting distribution of effective wealth is thus given by
\begin{align*}
\Pr(x_z>x)=\left(\frac{x/M-\Delta^{h}_z}{w_{z, t_0}^*/r_{t_0}^*}\right)^{-\zeta_0}, \quad \text{ for } x\geq M  (\Delta^{h}_z+w_{z, t_0}^*/r_{t_0}^*).
\end{align*}
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. As usual, the distribution of income then inherits all of the tail properties of the effective wealth distribution.  

\section{Additional Derivations and Proofs for the Extended Model\label{appendix extension supplement}}

\subsection{Supply and Demand in Extended Model}


\textbf{Note 1:} We can describe the balanced-growth equilibrium in Proposition \eqref{proposition:bg}
using supply and demand diagrams for capital normalized by labor income, $k$, and bonds, $b_I$ and $b_H$. We start with the special case where investors are risk neutral and $\theta\geq \bar{\theta}$ so that financial frictions do not bind. This implies that all assets yield the same return $r$ and this is also the return obtained by investors, $r^*_W=r^*_K=r^*_B=r$. Adding equations \eqref{equation:wealth investors} and \eqref{equation:savings households}, and using the fact that $b_I+b_H=0$, we obtain an expression for the supply of capital (normalized by labor income):
\begin{align}
k^s=\frac{1-(\rho+(\sigma-1) g)/(r-g)}{\sigma (p+g)+\rho-r}.\label{equation:supply risk neutral}
\end{align}
On the other hand, equation \eqref{eq:demand for equity} gives the demand for capital:
\begin{align}
k^d=\frac{\widetilde{\alpha}}{1-\widetilde{\alpha}} \frac{1}{r+\delta}.\label{equation:demand risk neutral}
\end{align}
The common return that equalizes supply and demand is given by the solution to \eqref{eq:general r*}, which generalizes equation \eqref{equation: implicit r} in the main text. This return now lies in $(\rho+\sigma g, \rho+\sigma(p+g))$.

We now turn to the case where investors are risk averse and/or financial frictions bind. 
Combining equations \eqref{equation:wealth investors} and \eqref{equation:portfolio investors}, we obtain a curve that describes the capital supplied by investors for a given level of the return to their wealth $r_W$ and the bond rate $r_B$:  
\begin{align*}
k^s=\frac{ p  \sigma   m(r_W-r_B)}{\sigma (p+g)+\rho-r_W} \frac{1}{r_B-g} \chi.
\end{align*}
Using equation \eqref{eq:demand for equity}, we can write the demand for capital as
\begin{align*}
k^d=\frac{\widetilde{\alpha}}{1-\widetilde{\alpha}} \frac{1}{h(r_W-r_B)+r_B+\delta}
\end{align*}
As shown in Figure \ref{figure:supply--demand--capital},
the supply curve gives an upward-sloping locus between $k^s$ and $r_W$; while the demand for capital in \eqref{eq:demand for equity} gives a downward-sloping locus between $k^d$ and $r_W$.

\begin{figure}[ht]
	\centering
	\includegraphics[width=.8\linewidth]{graphs/capital_market_new.pdf}
	\vspace{-5mm}
	
	\caption{\label{figure:supply--demand--capital} Depiction of the equilibrium in the capital market. }
\end{figure}

Turning to the bond market, equation \eqref{equation:portfolio investors} implies that net borrowing by investors is 
\begin{align*}
-b_I=\frac{ p  \sigma   (m(r_w-r_B)-1)}{\sigma (p+g)+\rho-r_W} \chi+\chi,
\end{align*}
which implies that investors borrow more from households as $r_B$ declines (reflecting the fact that investors would like to borrow at the lower rate $r_B$ to invest at the higher rate $r_W$). On the other hand, equation \eqref{equation:savings households} implies that net savings by households are
\begin{align*}
b_H=\frac{r_B-\rho-\sigma g}{\sigma (p+g)+\rho-r_B} (1-\chi).
\end{align*}
which increase in $r_B$, reflecting the fact that a higher safe rate leads to more savings by households. 
Figure \ref{figure:supply--demand--bonds} depicts the equilibrium in the bond market. In this panel, a movement to the right along the horizontal axis indicates higher households savings and more borrowing by investors. As the diagram shows, the curve describing investors' borrowing  rotates clockwise as they become more risk averse, and the interception of these curves yields an equilibrium bond rate $r_B^*$. 

\begin{figure}[ht]
	\centering
	\includegraphics[width=.8\linewidth]{graphs/bond_market_close.pdf}
	
	\caption{\label{figure:supply--demand--bonds} Depiction of the equilibrium in the bond market. }
\end{figure}

The proposition ensures the existence of $r^*_W, r^*_B$ such that the capital market clears ($k^d=k^s$) and the bond market clears ($b_I+b_H=0$).\footnote{
	Proposition \ref{proposition:bg} claims the existence of an equilibrium in the closed economy, but does not show it is unique. Proving uniqueness here is challenging because the excess demand functions for capital and bonds do not satisfy the gross substitutes property. This reflects the fact that changes in $r_B$ could lead investors to supply more capital as they accumulate higher levels of wealth; and changes in $r_W$ could also increase the demand for bonds when investors are risk averse.  
} Moreover, as apparent from the figures (and as was the case in our baseline model), we always have that $r_W^* \in (\rho+\sigma g, \rho+\sigma (g+p))$, which shows that the equilibrium return to wealth exceeds its level in a representative household economy. Moreover, we always have $r_W^*\geq r^*_B$, with equality if and only if investors are risk averse and $\theta\geq \bar{\theta}$, so that the max-leverage line is sufficiently to the right. 

\textbf{Note 2:} Alternatively, we can assume that bond markets are open, so that $r_B$ is fixed in international markets at a level $\bar{r}_B$, with
\begin{align*}
g<\bar{r}_B<\rho+\sigma(p+g).
\end{align*}
In this case, a balanced-growth equilibrium is characterized by constant values for $r_W, k, b_I, b_H$ that solve equations \eqref{equation:wealth investors}, \eqref{equation:portfolio investors}, \eqref{equation:savings households} and \eqref{eq:demand for equity}, and where $b_I+b_H$ determines whether the economy lends to or borrows from the rest of the world.

In this case, we have that the equilibrium is given by 
\begin{align*}
D_K(r_W^*, \bar{r}_B)=0.
\end{align*}
Lemma \ref{lemma:capital market clearing} implies that, for any $\bar{r}_B\in (g, \rho+\sigma (p+g))$, there is a unique solution $r_W^*$ to this equation. This shows that the equilibrium exists and is unique.  Moreover, as the figure shows, $r^*_W>\bar{r}_B$ and $r^*_W\in (\rho+\sigma g, \rho+\sigma (p+g))$.

In the open economy, the demand and supply of capital remain unchanged, with the only difference that now $r_B=\bar{r}_B$ is exogenous and determined in international markets. For values of $\bar{r}_B$ that are above the closed economy level of $r_B^*$, households will lend money to the rest of the world. Instead, for values of $\bar{r}_B$ that are below the closed economy level of $r_B^*$, both households and investors will borrow from foreigners at the low international rate.

\subsection{Proof of Lemmas \ref{lemma:capital market clearing} and \ref{lemma:bond market clearing}}


\begin{proof}[Proof of Lemma \ref{lemma:capital market clearing}] 
	For $r_W\in (g, \rho+\sigma(p+g))$ and $r_B\in (g, r_W)$, the function $D_K(r_W, r_B)$ is continuous and decreasing in $r_W$ and continuous and increasing in $r_B$. To prove the last claim, note that it is sufficient to show that
	\begin{align*}
	\frac{r_B-g}{h(r_W-r_B)+r_B+\delta}
	\end{align*}
	is increasing in $r_B$. We can rewrite this fraction as
	\begin{align*}
	\frac{1}{\displaystyle \frac{h(r_W-r_B)}{r_B-g}+1+\frac{g+\delta}{r_B-g}},
	\end{align*}
	which is clearly increasing in $r_B$ since the function $h$ is non-negative and increasing and $r_B-g>0$. 
	
	Moreover, for any $r_W\in (g, \rho+\sigma(p+g))$, there is a unique $r_B\in (g, r_W)$ that satisfies $D_K(r_W, r_B)=0$. To see this, note that 
	\begin{align*}
	D_K(r_W, g)<0 \text{ and } D_K(r_W, r_W)>0.
	\end{align*}
	The intermediate value theorem then implies the existence of a unique point $r_B\in (g, r_W)$  that satisfies $D_K(r_W, r_B)=0$.
	
	Taking these observations together, the implicit function theorem implies that the equation $D_K(r_W, r_B)=0$ defines a continuous and increasing locus for $r_W\in (g, \rho+\sigma(p+g))$ and $r_B\in (g, r_W)$, as claimed in the lemma. 
	
	To prove property 1, note that for $r_B=g$, we have
	\begin{align*}
	D_K(r_W, g)=-
	\frac{p \sigma m(r_W-g)}{\sigma (p+g)+\rho-r_W} \chi,
	\end{align*}
	and $D_K(r_W, g)=0$ then requires $r_W=g$.
	
	To prove property 2, take limits as $r_W\uparrow \rho+\sigma (p+g)$. If $r_B\nrightarrow \rho+\sigma (p+g)$, we have  
	\begin{align*}
	\lim_{r_w\uparrow \rho+\sigma (p+g)} D_K(r_K, r_W)=-\infty. 
	\end{align*}
	Thus, we must have $r_B\rightarrow \rho+\sigma (p+g)$ as claimed. 
\end{proof}

\begin{proof}[Proof of Lemma \ref{lemma:bond market clearing}] For $r_B\leq \rho+\sigma(p+g)$, the function $D_B(r_W, r_B)$ is continuous and decreasing in $r_B$. On the other hand, for $r_W\in [r_B, \rho+\sigma(p+g))$, the function $D_B(r_W, r_B)$ is continuous and u-shaped: it first decreases in $r_W$ and could become increasing afterwards.  
	
	Moreover, for any $r_W\in (\rho+\sigma g, \rho+\sigma(p+g))$, there is a unique $r_B< r_W$ that satisfies $D_B(r_W, r_B)=0$. To see this, note that 
	\begin{align*}
	\lim_{r_B\rightarrow -\infty} D_B(r_W, r_B)=\frac{p \sigma \theta/(1-\theta)}{\sigma (p+g)+\rho-r_W} \chi+\chi>0,
	\end{align*}
	and 
	\begin{align*}
	\lim_{r_B\rightarrow \infty} D_B(r_W, r_W)=\frac{\sigma g+\rho-r_W}{\sigma (p+g)+\rho-r_W}<0.
	\end{align*}
	The intermediate value theorem then implies the existence of a unique point $r_B<r_W$  that satisfies $D_B(r_W, r_B)=0$.
	
	Taking these observations together, the implicit function theorem implies that the equation $D_B(r_W, r_B)=0$ defines a continuous and initially decreasing locus for $r_W\in (\rho+\sigma g, \rho+\sigma(p+g))$ and $r_B<r_W$, as claimed in the lemma.
	
	We now turn to properties 1 and 2. Property 1 follows from the fact that 
	\begin{align*}
	D_B(\rho+\sigma g, \rho+\sigma g)=0.
	\end{align*}
	For property 2, take limits as $r_W\uparrow \rho+\sigma (p+g)$. If $m(r_w-r_B)\nrightarrow 1$, we have that
	\begin{align*}
	D_B(\rho+\sigma g, r_B)=\pm\infty.
	\end{align*} 
	Thus, as $r_W\uparrow \rho+\sigma (p+g)$, we must have that $m(r_w-r_B)\rightarrow 1$. In particular, this requires that $r_B\rightarrow \tilde{r}_B$. To see this, note that
	\begin{align*}
	m(r_w-r_B)\rightarrow 1 \Leftrightarrow r_K-r_B=\gamma\nu^2.
	\end{align*}
	The definition of $r_W$ implies that
	\begin{align*}
	r_W-\tilde{r}_B=\frac{1}{2}(\sigma+1)\gamma\nu^2.
	\end{align*}
	Substituting $r_W= \rho+\sigma (p+g)$ and solving for $\tilde{r}_B$, yields the formula for $\tilde{r}_B$ in the lemma. 
	
	We conclude with property 3. First, define 
	\begin{align*}
	\breve{r}_B=\rho+\sigma g + p \sigma \chi.
	\end{align*}
	we now show that along the locus $D_B(r_W, r_K)$, 	$m(r_w-r_B)>1$ for $r_B>\breve{r}_B$ and $m(r_w-r_B)<1$ for $r_B<\breve{r}_B$, with equality if and only if $r_B=\breve{r}_B$. To show this, note that 
	\begin{align*}
	D_B(r_W, \breve{r}_B)=\frac{p \sigma (m(r_W-r_B)-1)}{\sigma (p+g)+\rho-r_W} \chi. 
	\end{align*}
	Because the function $D_B(r_W, r_B)$ is decreasing in $r_B$, we have that for all points $r_B>\breve{r}_B$ along the locus $D_B(r_W, r_B)=0$,
	\begin{align*} 
	0=D_B(r_W, r_B)<\frac{p \sigma (m(r_W-r_B)-1)}{\sigma (p+g)+\rho-r_W} \chi,
	\end{align*}
	which implies $m(r_W-r_B)>1$. 
	Likewise, for all points $r_B<\breve{r}_B$ along the locus $D_B(r_W, r_B)=0$,
	\begin{align*} 
	0=D_B(r_W, r_B)>\frac{p \sigma (m(r_W-r_B)-1)}{\sigma (p+g)+\rho-r_W} \chi,
	\end{align*}
	which implies  $m(r_W-r_B)<1$. Clearly, equality holds if and only if  $r_B=\breve{r}_B$. 
	
	Moreover, suppose that the locus $D_B(r_W, r_B)=0$ contains $\breve{r}_B$. For $r_B>\breve{r}_B$, we have that $m(r_W-r_B)>1$, which implies that $D_B(r_W, r_B)$ is increasing in $r_W$ and the locus $D_B(r_W, r_B)=0$ is upward-sloping from $\breve{r}_B$ onwards. 
	
	We now consider the two cases identified in the lemma. Suppose that $\gamma\nu^2>\bar{\gamma}$. This necessarily implies that $\breve{r}_B>\tilde{r}_B$.
	Because the locus $\Delta_B(r_W, r_B)=0$ is u-shaped and both extremes are below $\breve{r}_B$, the entire locus lies below $\breve{r}_B$ and for all points along this locus we have $m(r_W-r_B)<1$. 
	
	To show that $r_W-r_B$ increases along this locus as we raise $r_W$, we proceed by contradiction. Suppose that this is not the case, then for two points $(r'_W, r'_B)$ and $(r''_W, r''_B)$ with  $r''_W>r'_W$ along this locus, we have $m(r'_W-r'_B)=m(r''_W-r''_B)=m_0\in (0,1)$, which also requires $r'_W-r'_B= r''_W-r''_B=d_0$. It follows that 
	both of these points satisfy the equation $D_B(r_W,r_B)=0$, which can be written as
	\begin{align*}
	\frac{p \sigma (m_0-1)}{\sigma (p+g)+\rho-r_W} \chi+\chi-\frac{r_W-d_0-\rho-\sigma g}{\sigma (p+g)+\rho-r_W+d_0} (1-\chi)=0.
	\end{align*}
	This equation is strictly decreasing in $r_W$ for $r_W\in (\rho+\sigma g, \rho+\sigma (p+g))$, and so it cannot hold for two different values of $r_W$. This contradiction establishes the claim that, for $\gamma\nu^2>\bar{\gamma}$, $r_W-r_B$ rises along the locus for $D_B(r_W, r_B)$.
	
	
	Finally, suppose that $\gamma\nu^2<\bar{\gamma}$,  which implies that $\tilde{r}_B> \breve{r}_B> \rho+\sigma g$. In this case $\tilde{r}_B$ lies in the locus $D_B(r_W, r_B)$. The exact same argument presented above implies that, for $r_W \in (\rho+\sigma g, \breve{r}_W)$, the gap $r_W-r_B$ widens as we increase $r_W$ along the locus $\Delta_B(r_W, r_B)=0$. 
	
	We now show that the gap $r_W-r_B$ is single peaked for $r_W \in (\breve{r}_W, \rho+\sigma (p+g))$. As shown above, for $r_W \in (\breve{r}_W, \rho+\sigma (p+g))$, we have that $m(r_W-r_B)>1$ (and hence $r_W-r_B>\frac{1}{2}(\sigma+1)\gamma\nu^2$) along the locus $\Delta_B(r_W, r_B)=0$. Because 
	$m(r_W-r_B)>1$, we must have that $r_B>\rho+\sigma g$
	for $r_W \in (\breve{r}_W, \rho+\sigma (p+g))$; otherwise, $D_B(r_W, r_B)>0$. 
	
	Recall that as $r_W\uparrow \rho+\sigma (p+g)$, we have that $r_B\rightarrow \tilde{r}_B$ and $r_W-r_B\rightarrow\frac{1}{2}(\sigma+1)\gamma\nu^2$ along the locus $\Delta_B(r_W, r_B)=0$. Thus, the gap $r_W-r_B$ reaches at least one peak at some point  $r_W\in (\breve{r}_W, \rho+\sigma (p+g))$. Suppose by way of contradiction, that the gap $r_W-r_B$ is not single peaked in this interval, as claimed in the lemma. Then it must have at least two peaks. This implies the existence of three points $(r'_W, r'_B)$, $(r''_W, r''_B)$ and $(r'''_W, r'''_B)$ along this locus and with $m(r'_W-r'_B)=m(r''_W-r''_B)=m(r'''_W-r'''_B)=m_0>1$, and $r'_W-r'_B= r''_W-r''_B= r'''_W-r'''_B=d_0$. However, in each of this points, we must have $D_B(r_W, r_B)=0$, which requires
	\begin{align*}
	\frac{p \sigma (m_0-1)}{\sigma (p+g)+\rho-r_W} \chi+\chi-\frac{r_W-d_0-\rho-\sigma g}{\sigma (p+g)+\rho-r_W+d_0} (1-\chi)=0. 
	\end{align*}
	This is a quadratic equation in $r_W$ and so it has at most two solutions, contradicting our initial assertion, and showing that $r_K-r_B$ is single peaked for $r_W \in (\breve{r}_W, \rho+\sigma (p+g))$. 
\end{proof}

\subsection{Wealth and Income Distribution in the Extended Model}\label{app:inc_dist_extended}

Proposition \ref{proposition:bg} and \ref{app proposition: normalized wealth distribution} characterized the distribution of effective wealth and the \emph{tail} of the income distribution. A complete analytic characterization of the income distribution is no longer feasible in the extended model.  This appendix shows how to partially characterize and numerically compute the income distributions, e.g. so as to construct Figure \ref{figure: quantiles labor capital}.

The flow of income over a small time interval of length $dt$ received by an investor with wealth $x_{z,t}$  is given by
\begin{equation}\label{eq:income_flow}
	dy_{z,t}=r_I x_{z,t} dt + \sigma_y x_{z,t} d W_t, 
\end{equation}
where $r_I=\kappa r_K + (1-\kappa) r_B$ and $\sigma_y=\kappa \nu$. Likewise, the flow of income received by a household with normalized wealth $x_{z,t}$ is given by
\begin{equation}\label{eq:income_flow_HH}
	\dot{y}_{z,t}=r_B x_{z,t}.
\end{equation}

The objective is to approximate the income received over a small  time period. 

We say that a function $X_t$ is of the order of $f(t)$ as $t\rightarrow 0$ if there exists a constant $M$ such that, for $t\rightarrow 0$, $|X_t|< M f(t)$. Likewise, we say that a random variable $X_t$ is of the order of $f(t)$  as $t\rightarrow 0$ if $\mathbb{E}[X_t^2]$ is of the order of $f(t)^2$  as $t\rightarrow 0$.
We use the notation $A_t=B_t+\mathcal{O}(f(t)) \text{ as } t\rightarrow 0$ to indicate that $A_t-B_t$ is of the order of $f(t)$   as $t\rightarrow 0$. 

The definition for $X_t$ being of the order of $f(t)$ as $t\rightarrow 0$  implies that: i. the mean and standard deviation of $X_t$ are also of the order of $f(t)$; and ii. for any $\epsilon>0$ there exists an $M_{\epsilon}$ such that $\Pr(X_t>M_{\epsilon} f(t))<\epsilon$ as $t\rightarrow 0$. That is,  $X_t$ tends to be below $M_{\epsilon}f(t)$ with probability converging to 1.

To show this, define $\mu_t=\mathbb{E}[X_t]$ and let $\sigma_t^2=\mathbb{E}[(X_t-\mu_t)^2]$ denote the variance of $X_t$. For (i), note that $\mathbb{E}[X_t^2]=\mu_t^2+\sigma_t^2\geq \mu_t^2$. Because $\mathbb{E}[X_t^2]$ is of the order of $f(t)$   as $t\rightarrow 0$, there exist a constant $M$ such that $Mf(t)^2>\mathbb{E}[X_t^2]=\mu_t^2+\sigma^2_t\geq \mu_t^2, \sigma^2_t$ as $t\rightarrow 0$. These inequalities imply that $\mu_t$ and $\sigma_t$ are also of the order of $f(t)$  as $t\rightarrow 0$.

For (ii), we use an application of Chebyshev's inequality:
\begin{align*}
	\Pr(X_t>\mu_t+N\sigma_t)<\frac{1}{N^2}.
\end{align*}
In particular, because $\mu_t$ and $\sigma_t$ are of the order of $f(t)$ as $t\rightarrow0$, for any $\epsilon>0$, there exists a constant $M_{\epsilon}$ such that $M_{\epsilon}f(t)>\mu_t+\epsilon^{-1/2}\sigma_t$ as  $t\rightarrow 0$. Chebyshev's inequality (applied with $N=\epsilon^{-1/2}$) then implies
\begin{align*}
	\Pr(X_t>M_{\epsilon}f(t))<\Pr\left(X_t>\mu_t+\epsilon^{-1/2}\sigma_t\right)<\epsilon \text{ as } t\rightarrow 0.
\end{align*}

The derivation of our approximation result also uses the following technical lemma:

\begin{lemma}\label{bound on ito integral} 
	Let $X_t$ be an stochastic process adapted to the natural filtration of te Brownian motion $W_t$. If $X_t$ is of the order of $t^k$ as $t\rightarrow0$, then:
	\begin{itemize}
		\item[i.] $\int_0^t X_s ds$ is of the order of $t^{k+1}$ as $t\rightarrow0$;
		\item[ii.]  $\int_0^t X_s dW_s$ is of the order of $t^{k+1/2}$ as $t\rightarrow0$.
	\end{itemize}
\end{lemma}

Intuitively, the lemma says that integrating with respect to time over $[0,t]$ is ``as if'' we multiplied the order of the variable by $t$; whereas integrating with respect to $dW_s$ over $[0,t]$ is ``as if'' we multiplied the order of the variable by $\sqrt{t}$---the order of $W_t$. 

\begin{proof}
	For (i), we have 
	\begin{align*}
		\mathbb{E}\left[\left(\int_0^t X_s ds\right)^2\right]\leq t \mathbb{E}\left[\int_0^t X_s^2 ds\right]= t \int_0^t \mathbb{E}\left[X_s^2\right] ds\leq M t \int_0^t s^{2k} ds = \frac{M}{2k+1} t^{2k+2} \text{ as } t\rightarrow 0
	\end{align*}
	The first inequality follows from Cauchy-Schwarz. The second one from the fact that $X_s$ is of the order of $t^{k}$. This sequence of inequalities  
	shows that $\int_0^t X_s ds$ is of the order of $t^{k+1}$ as $t\rightarrow0$.
	
	For (ii) we have that $\int_0^t X_s dW_s$ is a random variable with mean 0 (since $X_s$ is adapted and thus orthogonal to $dW_s$) and variance $\mathbb{E}\left[\int_0^t X_s^2 ds\right]$ (a result of Ito' Isometry). 
	Moreover:
	\begin{align*}
		\mathbb{E}\left[\int_0^t X_s^2 ds\right]=\int_0^t \mathbb{E}\left[X_s^2\right] ds\leq M  \int_0^t s^{2k} ds = \frac{M}{2k+1} t^{2k+1} \text{ as } t\rightarrow 0.
	\end{align*}
	It follows that the variance of the random variable $\int_0^t X_s dW_s$ is of the order of $t^{2k+1}$ as $t\rightarrow0$ and so  the random variable $\int_0^t X_s dW_s$ is of the order of $t^{k+1/2}$ as $t\rightarrow0$.
\end{proof}


The following lemma provides our main approximation result for the income received by households over one unit of time.  

\begin{lemma}\label{app lemma: approximation}
	Let $y_{z,t}$ denote the total income received by an investor between time $0$ and $t$. We have that
	\begin{align}
		y_{z,t}\approx x_{z,0} \left( r_I t + \sigma_y W_t\right).\label{app dist: income approximation investor} 
	\end{align}
	More precisely, 
	\begin{align*}
		y_{z,t}= x_{z,0} \left( r_I t + \sigma_y W_t\right)+\mathcal{O}(\max\{\sigma_yt^{3/2},r_It^{2},\sigma^2_yt\}) \text{ as } t\rightarrow 0.
	\end{align*}
	
	Likewise, let $y_{z,t}$ denote the total income received by a household between time 0 and $t$. We have that
	\begin{align}
		y_{z,t}\approx x_{z,0} r_B t.\label{app dist: income approximation household} 
	\end{align}
\end{lemma}
The interpretation of the Lemma is that we can obtain a good approximation to the income generated by an asset between $s=0$ and $s=t$ simply from the asset return (e.g. $r_I t + \sigma_y W_t$ in the case of investors) and the initial asset balance $x_{z,0}$ while one can ignore changes in the asset balance $x_{z,s}$ between $s=0$ and $s=t$. The following example helps clarify: an asset pays a return of $r=5\%$ that accrues continuously over the year. With an initial asset balance of \$100 the total capital income generated throughout the year will not just be $5\% \times \$100 = \$5$ but instead it will be larger than \$5. This is because the asset balance which is the basis of the return itself increases throughout the year. However, the Lemma shows that $5\% \times \$100 = \$5$ provides a good approximation to overall capital income, i.e. the effect we just described has only a small effect. In particular,  the following upper bound on the approximation error is easy to see: the asset balance could increase from \$100 to \$105. But then capital income would still be only $5\% \times \$105 = 5.25\%$ i.e. only $0.25\% = 5\% \times 5\%$ larger than the approximate capital income of $5\% \times \$100 = \$5$.



\begin{proof} We provide the proof for the case of investors. The approximation used for households then follows as a corollary. Throughout the proof we will ignore the subscripts $z$ and $I$ to simplify notation. 
	
	Ito's lemma applied to the log of investors' wealth implies that
	\begin{align*}
		d\ln x_{t}=\left(\mu-\frac{\sigma_y^2}{2}\right) dt + \sigma_y dW_t.
	\end{align*}	
	Integrating this expression in $[0,t]$ implies
	\begin{align*}
		\ln x_{t}-\ln x_{0}=\left(\mu-\frac{\sigma_y^2}{2}\right) t + \sigma_y W_t,
	\end{align*}	
	and so investor's wealth satisfies 
	\begin{align}
		x_{t}=x_{0}\exp\left(\left(\mu-\frac{\sigma_y^2}{2}\right)t + \sigma_y W_t\right)\label{appeq exponential eq}
	\end{align}
	and income over a period $[0,t]$ is given by
	\begin{align}
		y_{t}=\int_0^t r x_{0}\exp\left(\left(\mu-\frac{\sigma_y^2}{2}\right)s+\sigma_y W_s\right) ds+
		\int_0^t \sigma_y x_{0}\exp\left(\left(\mu-\frac{\sigma_y^2}{2}\right)s+\sigma_y W_s\right) dW_s\label{appeq exponential eq for income}
	\end{align}
	
	
	We can expand $\exp\left(\left(\mu-\frac{\sigma_y^2}{2}\right)t+\sigma_y W_t\right)$ using a Taylor expansion as
	\begin{align*}
		\exp\left(\left(\mu-\frac{\sigma_y^2}{2}\right)t+\sigma_y W_t\right)=1+\sigma_y W_t +\mathcal{O}(t) \text{ as } t\rightarrow 0.
	\end{align*}
	Note that $W_t$ is of the order of $\sqrt{t}$ since its variance is $t$. Note also that the residual term $\mathcal{O}(t)$ is adapted to the natural filtration of $W_t$ (it is a function of $W_t$ in this case). 
	
	Plugging this approximation in equation \eqref{appeq exponential eq for income}, we obtain
	\begin{align*}
		y_{t}=&\int_0^t r x_0\left(1+\sigma_y W_s +\mathcal{O}(s)\right) ds+
		\int_0^t \sigma_y x_{0}\left(1+\sigma_y W_s +\mathcal{O}(s)\right) dW_s\\
		=&x_{0}\left(r t + \sigma_y W_t\right)+\\
		&
		\int_0^t r x_0 \sigma_y W_s ds +\int_0^t r x_0 \mathcal{O}(s) ds
		+\int_0^t \sigma^2_y x_{0} W_s dW_s +\int_0^t \sigma_y x_{0}\mathcal{O}(s) dW_s
		\text{ as } t\rightarrow 0.
	\end{align*}
	Using Lemma \ref{bound on ito integral}, we can bound these integrals as
	\begin{align*}
		\int_0^t r x_0 \sigma_y W_s ds=&\mathcal{O}(\sigma_yt^{3/2})
		&\text{ as } t&\rightarrow 0\\
		\int_0^t r x_0 \mathcal{O}(s) ds=&\mathcal{O}(rt^{2})
		&\text{ as } t&\rightarrow 0\\
		\int_0^t \sigma^2_y x_{0} W_s dW_s=&\mathcal{O}(\sigma^2_yt)
		&\text{ as } t&\rightarrow 0\\
		\int_0^t \sigma_y x_{0}\mathcal{O}(s) dW_s=&\mathcal{O}(\sigma_y t^{3/2})
		&\text{ as } t&\rightarrow 0.
	\end{align*}
	These bounds imply that $y_{t}=x_{0}\left(r t + \sigma_y W_t\right)
	+\mathcal{O}(\max\{\sigma_yt^{3/2},rt^{2},\sigma^2_yt\})$
	as $t\rightarrow 0$.\end{proof}

We now use Proposition \ref{app proposition: normalized wealth distribution} and \ref{app lemma: approximation} to characterize an approximate income distribution. This proposition provides all of the formulas used in the construction of Figure \ref{figure: quantiles labor capital}. 

\begin{proposition}\label{app proposition: extended model distributions}
	The CDF of income received over one unit of time for households with wage $w_z$ can be approximated as
	\begin{align}
	G_{z,H}(y):=
	\left\{\begin{array}{cl} 1-\left(\frac{y}{w_z}\right)^{-\zeta_H} & \text{ for } y \geq w_z\\
	0 & \text{ for } y < w_z
	\end{array}\right.
	, \label{app dist: households income}
	\end{align} 
	and its PDF is
	\begin{align}
	g_{z,H}(y):=
	\left\{\begin{array}{cl} \frac{\zeta_H}{w_z} \left(\frac{y}{w_z}\right)^{-\zeta_H-1} & \text{ for } y \geq w_z\\
	0 & \text{ for } y < w_z
	\end{array}\right.
	, \label{app dist: households income pdf}
	\end{align} 
	
	The CDF of income received over one unit for investors with wage $w_z$ can be approximated as
	\begin{align}
	G_{z,I}(y):=1-\int_0^{\infty} f_{H}(x) \left(1-\Phi\left( \frac{1}{\sigma_y}\left(\frac{y/x}{w_z/r_B}- r_I\right)\right)\right) dx, \label{app dist: investors income}
	\end{align}  
	and its PDF is 
	\begin{align}
	g_{z,I}(y):=\int_0^{\infty} f_{H}(x) \phi\left( \frac{1}{\sigma_y}\left(\frac{y/x}{w_z/r_B}- r_I\right)\right)\frac{1}{\sigma_y}\frac{1/x}{w_z/r_B} dx, \label{app dist: investors income pdf}
	\end{align}  
	
	Moreover, the unconditional CDF of income satisfies 
	\begin{align}
	G(y)=\chi \sum_z \ell_z G_{z,I}(y) + (1-\chi) \sum_z \ell_z G_{z,H}(y), \label{app dist: unconditional income}
	\end{align}
	and its PDF is
	\begin{align}
	g(y)=\chi \sum_z \ell_z g_{z,I}(y) + (1-\chi) \sum_z \ell_z g_{z,H}(y), \label{app dist: unconditional income pdf}
	\end{align}
	
	Finally, the labor income of people with an income $y$ is given by
	\begin{align}
	w(y)=\frac{\chi \sum_z w_z \ell_z g_{z,I}(y) + (1-\chi) \sum_z w_z \ell_z g_{z,H}(y)}{g(y)}. \label{app dist: wage income}
	\end{align} 
\end{proposition}

\begin{proof}
	Lemma \ref{app lemma: approximation} implies that we can approximate the income received by households over one unit of time as $y_{z,1}\approx x_{z,0} r_B$. Thus, we can compute the probability that a household with wage $w_z$ has an income above $y$ as
	\begin{align*}
	\Pr(\textrm{income}\geq y | \textrm{household}, w_z)& =\Pr(x_{z,0} r_B \geq y | \textrm{household}, w_z) \\
	&=\Pr(x_{z,0} \geq \frac{y}{r_B} | \textrm{household}, w_z) \\
	& =\Pr(\widetilde{x}_{z,0} \geq \frac{y}{w_z} | \textrm{household}, w_z) \\
	& =\left(\frac{y}{w_z}\right)^{-\zeta_H} \text{ for } y \geq w_z.
	\end{align*} 
	This implies that $G_{z,H}(y)$ is given by the expression in equation \eqref{app dist: households income}, and its associated PDF is given by \eqref{app dist: households income pdf}.
	
	Likewise, Lemma \ref{app lemma: approximation} implies that we can approximate the income received by investors over one unit of time as $$y_{z,1}\approx x_{z,0} \left( r_I + \sigma_y W_1\right).$$ Thus, we can compute the probability that an investor with wage $w_z$ has an income above $y$ as
	\begin{align*}
	\Pr(\textrm{income}\geq y | \textrm{investor}, w_z)& =\Pr\big(x_{z,0} \left( r_I + \sigma_y W_1\right) \geq y \big| \textrm{investor}, w_z\big) \\
	& =\Pr\Bigg(\widetilde{x}_{z,0} \left( r_I + \sigma_y W_1\right) \geq \frac{y}{w_z/r_B} \Bigg| \textrm{investor}, w_z\Bigg) \\
	&= \int_0^{\infty} f_{H}(x) \Pr\Bigg(W_1 \geq \frac{1}{\sigma_y}\left(\frac{y/x}{w_z/r_B}- r_I\right) \Bigg| \widetilde{x}_{z,0}=x, \textrm{investor}, w_z\Bigg)      dx \\
	&= \int_0^{\infty} f_{H}(x) \left(1-\Phi\left( \frac{1}{\sigma_y}\left(\frac{y/x}{w_z/r_B}- r_I\right)\right)\right) dx. 
	\end{align*} 
	This implies that $G_{z,I}(y)$ is given by the expression in equation \eqref{app dist: investors income}, and its associated PDF is given by \eqref{app dist: investors income pdf}. 
	
	Finally, the formulas in \eqref{app dist: unconditional income} and \eqref{app dist: unconditional income pdf} follow from adding up the conditional distribution of income over skills and investors and households. The formula for the wage income for people with total income $y$ follows from an application of Bayes' rule. 
\end{proof}


\paragraph{Construction of Figure \ref{figure: quantiles labor capital}.} To construct Figure \ref{figure: quantiles labor capital}, we proceed as follows:
\begin{itemize}
	
	\item First, we use equation \eqref{app dist: unconditional income} to compute income quantiles $y_t(q)$ as
	\begin{align*}
	G_t(y_t(q))=q
	\end{align*}
	both for the initial steady state (denoted by $t=0$) and the final steady state (denoted by $t=T$). 
	
	
	\item Second, we compute the labor income and capital income earned in each quantile, $y_{\ell,t}(q)$ and $y_{k,t}(q)$, respectively, as
	\begin{align*}
	y_{\ell,t}(q)=&w_t(y_t(q))& y_{k,t}(q)=&y_t(q)-w_t(y_{t}(q))&
	\end{align*}
	where $w_t(y_t(q))$ was defined in \eqref{app dist: wage income}.
	
	\item Third, we compute the change in income by quantile as
	\begin{align*}
	\frac{y_T(q)-y_0(q)}{y_0(q)},
	\end{align*} 
	which can be decomposed into a part driven by labor income
	\begin{align*}
	\frac{y_{\ell,T}(q)-y_{\ell,0}(q)}{y_0(q)},
	\end{align*} 
	and a part driven by capital income
	\begin{align*}
	\frac{y_{k,T}(q)-y_{k,0}(q)}{y_0(q)}.
	\end{align*} 
	Figure \ref{figure: quantiles labor capital} plots the total change in income by quantile and the contribution of labor and capital income. 
	
	\item Finally, the figure also plots the change in income by quantile in the representative-household benchmark. Appendix \ref{appendix transition} explains how we computed the evolution of the income distribution in this case. 
\end{itemize}

\section{Calibration Details\label{appendix calibration}}

This appendix presents details of the calibration exercise.

\subsection{Wage data} 

We compute hourly wages for 1990 using the 1990 Census and for 2014 using the 2012--2016 American Community Survey. We keep the sample of salaried workers between 25 and 54 years of age living in continental US. Following common practice in the literature, we replace top coded wage income by 1.5 times the top code. We compute hourly wages by dividing wage income by hours per week times weeks worked. We then converted hourly wages to 2007 dollars using the personal consumption expenditure index, from the BEA. Finally, we winsorized hourly wages between 2 and 180 dollars. 

We then computed average wages for workers in each of the 100 wage percentiles. When computing these averages, we weight observations by the Census or ACS weight times total hours of labor supplied, so that we obtain the average hourly wage for workers in each percentile. 

Due to changes in the amount of wage data top coded, the data for the top 1 percentile exhibits visible discontinuities over time. We address this issue by fitting a log linear model for log wages as a function of the rank, using the fitted regression to impute the mean wage for the top 1 wage earners. We estimate this model for the percentiles above the 90th percent, exploiting the fact that the top tail of wages has an approximate Pareto shape. 

Finally, in Figure \ref{figure: relative wages} (and only for this figure) we smooth the observed change in wages by taking a moving average over consecutive bins over 10 percentiles. 

\subsection{Calibration of $\alpha_{z,t}$} 

As explained in the main text, we make three assumptions:
\begin{enumerate}
	\item[A1] the patterns of relative specialization $\omega^R_{z}$ vary little over time and can be approximated by their value in the year 2000;
	
	\item[A2] in 1980, a common share $\alpha_0$ of routine and non-routine tasks were automated;
	
	\item[A3] over time, the share of routine tasks that is automated is common across skill groups and given by $\alpha^R_{t}$, with $\alpha^R_{1980}=\alpha_0$. 
\end{enumerate}

Equation \eqref{equation: alpha z} can be derived as follows. Let $\bar{\omega}^R_{z,t}$ denote the share of wage income earned by workers with skill $z$ in routine jobs at time t. 

Let $w^R_{z,t}$ denote the wage income earned by workers with skill $z$ in routine jobs and let $y^R_{z,t}$ denote the value added generated in those jobs. Let $w^N_{z,t}$ denote the wage income earned by workers with skill $z$ in non-routine jobs and let $y^N_{z,t}$ denote the value added of those jobs. 

Assumptions A2 and A3 imply that we can write
\begin{align*}
\frac{1}{1-\alpha_{z,t}}=\frac{y^R_{z,t}+y^N_{z,t}}{w^R_{z,t}+w^N_{z,t}}=\bar{\omega}^R_{z,t}\frac{1}{1-\alpha^R_{t}}+(1-\bar{\omega}^R_{z,t})\frac{1}{1-\alpha_0}.
\end{align*}

This equation can be rewritten as
\begin{align}\label{equation: alpha derivation 1}
\frac{1}{1-\alpha_{z,t}}=\bar{\omega}^R_{z,t}\left(\frac{1}{1-\alpha^R_{t}}-\frac{1}{1-\alpha_0}\right)+\frac{1}{1-\alpha_0}. 
\end{align}

Let $\bar{\omega}_{R,t}$ denote the share of wage income derived from routine jobs across all workers. We also have
\begin{align*}
\frac{1}{1-\alpha_t}=\frac{\sum_z y^R_{z,t}+\sum_z y^N_{z,t}}{\sum_z w^R_{z,t}+\sum_z w^N_{z,t}}=\bar{\omega}^R_{t}\frac{1}{1-\alpha^R_{t}}+(1-\bar{\omega}^R_{t})\frac{1}{1-\alpha_0}.
\end{align*}
This equation can be rewritten as
\begin{align}\label{equation: alpha derivation 2}
\frac{1}{1-\alpha_t}-\frac{1}{1-\alpha_0}=\bar{\omega}^R_{t}\left(\frac{1}{1-\alpha^R_{t}}-\frac{1}{1-\alpha_0}\right). 
\end{align}

Combining equations \eqref{equation: alpha derivation 1} and \eqref{equation: alpha derivation 2}, we obtain 
\begin{align*}
\frac{1}{1-\alpha_{z,t}}=\frac{\bar{\omega}^R_{z,t}}{\bar{\omega}^R_{t}}\left(\frac{1}{1-\alpha_t}-\frac{1}{1-\alpha_0}\right)+\frac{1}{1-\alpha_0}. 
\end{align*}
Equation \eqref{equation: alpha z} in the main text follows from the fact that $\omega^R_{z}=\frac{\bar{\omega}^R_{z,t}}{\bar{\omega}^R_{t}}$, and we assumed that relative specialization does not vary over time. 

To operationalize the measurement of $\alpha_{z,t}$, we compute $\omega^R_{z}$ using Census data for 2000.\footnote{We measure $\omega^R_{z}$ using the 2000 Census---a point in the middle of the period we study. In our model, the composition and specialization patterns of a skill group are 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 $\omega^R_{z}$ as describing the level of specialization of different groups in routine jobs. We also experimented with measuring  $\omega^R_{z}$ using the 1980 Census and obtained similar results. The reason is that $\omega^R_{z}$ is highly correlated over time (the correlation between the 1980 and 2000 measures is of 0.9714).} 
We keep the sample of salaried workers between 25 and 54 years of age living in continental US, and we clean wage data in the exact same way as above. 
Following the literature, we code an occupation as routine if it is in the top tercile of jobs with the highest routine content according to $O*NET$. We define the routine content of an occupation as total routine inputs minus the average of routine inputs, cognitive inputs, and manual inputs involved in this job. The construction of these inputs is explained in \citet{acemoglu-autor} and available for download from their websites. We also experimented and obtained similar findings using the classification of occupations as routine and non-routine used in \citet{autor-dorn}. This classification is based on the Dictionary of Occupational Titles, which preceded $O*NET$. 

\subsection{Empirical estimates of the capital-supply elasticity}

As mentioned in the main text, an elasticity of capital to rental rates of 50 is large relative to the available empirical estimates. The following table summarizes our review of the empirical literature estimating this elasticity. 
For each paper, we report the implied elasticity $d\ln K/dr$, which can be directly compared to our calibration target of 50. As is apparent from our survey, all these estimates put the elasticity below 35, which is much more inelastic than what our model predicts. 

\begin{table}[htbp]\caption{Summary of available estimates of the capital-supply elasticity.}
	\resizebox{1.0\textwidth}{!}{
		\begin{tabular}{lp{11em}ccccp{15em}}
			\toprule
			&  &  &  & \textsc{Implied} &  & \\
			\multicolumn{1}{c}{\textsc{Study}} & \multicolumn{1}{c}{\textsc{What Elasticity?}} & \multicolumn{1}{c}{\textsc{Formula}} & \textsc{Estimate} &  \textbf{$d\ln K/dr$} & \textsc{Duration} & \multicolumn{1}{c}{\textsc{Methodology}} \\\midrule
			\textsl{Wealth Taxation:} \\
			Zoutman (2015) & Stock of housing and financial assets w.r.t tax on their sum	 & $\frac{d\log(K)}{d\log(\tau)}$ & -0.045 & \textbf{9}     & 5 Years & Quasi experiment - 2001 Dutch capital tax reform. Response of households is tracked using panel data.\\[1ex]
			Jakobsen et al. (2018) & Taxable wealth w.r.t. net-of-tax rate for the very rich / moderately rich & $\frac{d\log(K)}{d\tau}$ & [-8,-25] & \textbf{[8,25]} & 8 years & Quasi experiment - 1989 wealth tax reform in Denmark. Diff-in-diff using variation in tax ceilings and the tax exemption level. \\[1ex]
			Br\"{u}lhart et al. (2017) & Semi-elasticity - taxable wealth (excluding pensions) w.r.t. wealth tax rate & $\frac{d\log(K)}{d\tau}$ & [-23,-35] & \textbf{[23,35]} & -     & Cross-regional and time variation in the Swiss wealth-tax system.\\[1ex]
			\textsl{Income Taxation:}\\
			Kleven and Schultz (2014) & Positive taxable capital income w.r.t. net-of-tax rate & $\frac{d\log(rK)}{d\log(1-\tau)}$ & [0.1,0.3] & \textbf{[1.25,7.5]} & 3-7 years & Danish tax reforms and full-population administrative data since 1980.\\[1ex]
			\bottomrule
	\end{tabular}}	
	\label{tab:addlabel}
\end{table}


\subsection{Alternative calibrations\label{app:low_risk_aversion}}

Table \ref{table: altcal} summarizes the parameters used in alternative calibrations (top panel), and shows that our results, such as changes in returns and in top inequality, are robust to different assumptions about parameter values (bottom panel). Column 1 shows results for a calibration that matches an exposure to risky capital of $\kappa=1$ in 1980. Column 2 shows our baseline calibration, which matches a value of $\kappa=1.3$ in 1980. Column 3 shows results for a calibration that matches an exposure to risky capital of $\kappa=1.5$ in 1980. Finally, Column 4 returns to our baseline value for $\kappa$ but now assumes that investors have low levels of risk aversion and are instead borrowing constrained. In particular, we assume that investors can pledge up to a third of their wealth when borrowing ($\theta=1/3$), which implies $\kappa$ binds at 1.5 All calibrations deliver a similar increase in the top-tail inequality index $1/\zeta$ from 0.54 to 0.65--0.66.

\begin{table}[ht]
	\caption{Alternative calibrations of the extended model}\label{table: altcal}
	\resizebox{\linewidth}{!}{\begin{tabular}{ccccc}
			\toprule 
			& (1) & (2) & (3) & (4)\\
			& \phantom{\textsc{Constrained}}&\phantom{\textsc{Constrained}} &\phantom{\textsc{Constrained}} &\textsc{Constrained}\\
			& \textsc{Target $\kappa=1$} & \textsc{Target $\kappa=1.3$} & \textsc{Target $\kappa=1.5$} &  \textsc{investors}\tabularnewline
			\midrule
			Risk aversion $\gamma$ & 2 & 2 & 2 & 0.1\tabularnewline
			%\midrule 
			Capital risk $\nu$ & 0.09 & 0.077 & 0.067 & 0.06\tabularnewline
			%\midrule 
			Share investors $\chi$ & 0.063 & 0.066 & 0.061 & 0.045\tabularnewline
			\midrule 
			\multicolumn{5}{c}{\emph{initial steady state $\rightarrow$ final steady state }}\tabularnewline
			\midrule 
			$\kappa$ & $1\rightarrow1.1$ & $1.3\rightarrow1.5$ & $1.5\rightarrow1.7$ & $1.5$\tabularnewline
			%\midrule 
			$r_{B}$ & $4.1\%\rightarrow4.2\%$ & $4.9\%\rightarrow5.2\%$ & $5.2\%\rightarrow5.5\%$ & $4.2\%\rightarrow4.4\%$\tabularnewline
			%\midrule 
			$r_{K}$ & $6.5\%\rightarrow7.0\%$ & $6.5\%\rightarrow7.0\%$ & $6.5\%\rightarrow7.0\%$ & $6.5\%\rightarrow7.4\%$\tabularnewline
			%\midrule 
			$r_{W}$ & $7.7\%\rightarrow8.8\%$ & $8.0\%\rightarrow9.2\%$ & $8.2\%\rightarrow9.4\%$ & $7.7\%\rightarrow9.0\%$\tabularnewline
			%\midrule 
			$1/\zeta$ & $0.54\rightarrow0.65$ & $0.54\rightarrow0.65$ & $0.54\rightarrow0.66$ & $0.54\rightarrow0.66$\tabularnewline
			\bottomrule
	\end{tabular}}
\end{table}


\section{Measuring Returns to Wealth\label{appendix returns}}


This section provides the details behind our measurement of the returns to productive capital held by corporations and and private (noncorparate) businesses in the US. We compute these series using annual data from the Integrated Macroeconomic Accounts (IMA) and the National Income and Product Accounts (NIPA) for 1960--2020. We compute returns to capital as $r_K := Y_k/K + \Pi_k$ where $Y_k$ is capital income, $K$ is the value of the capital stock and $\Pi_K$ is the expected revaluation component (or capital income gains).

\paragraph{Return to capital in non-financial corporate sector:} We measure capital income as net operating surplus (IMA Table S5, Line 10) minus taxes paid by corporations (IMA Table S5, Line 21) and other transfers (IMA Table S5, Line 22). We measure the stock of capital held by US corporations by the value of productive assets (IMA Table S5, Line 109). Note that these values are end-of-period. We adjust for this by using the value of capital from the previous period adjusted for PCE inflation. Finally, we compute the realized revaluation component as the ratio of 
the change in productive capital prices during the year (IMA Table S5, Line 88) and the lagged stock of productive assets. We compute the expected revaluation component as the average of the realized term over the 1960--2020 period. 

\paragraph{Return to capital in non-corporate sector:} To measure capital income, we start from proprietors' income (NIPA Table 2.1, Line 9) and adjust this by multiplying it by the capital share in the corporate sector in NIPA. This is computed as one minus the share of net of tax corporate income (NIPA Table 1.13, Lines 3 minus Line 9) used for compensating employees (NIPA Table 1.13, Line 4). This adjustment accounts for the fact that proprietors' income mixes labor and capital income, and uses the income split from the corporate sector to apportion the income in the non-corporate private sector. We measure the stock of capital held by US private businesses by the value of their productive assets (IMA Table S3, Line 121). Finally, we compute the realized revaluation component as the ratio of 
the change in productive capital prices during the year (IMA Table S3, Line 92) and the lagged stock of productive assets. We compute the expected revaluation component as the average of the realized term over the 1960--2020 period. 

\paragraph{Adjusting for taxes:} We present after-tax returns, which we obtained by applying the average tax rate paid by households on their income to the returns described above (these return already account for taxes paid by corporations, but not for taxes paid by equity owners on their dividends and capital income gains). The average tax rate on income is computed as personal taxes (NIPA Table 2.1, Line 26) divided by: wages and salaries (NIPA Table 2.1, Line 26) plus business income (NIPA Table 2.1, Line 9) plus rental and dividend income (NIPA Table 2.1, Lines 12 and 13). 

\subsection{Behavior of the income and revaluation component}

Our baseline series of returns assume that expected revaluation is equal to the average realized revaluation over 1960--2020.  This choice is motivated by the fact that, relative to the income component, the revaluation component is highly volatile, with large fluctuations at high frequencies but no visible trend. Figure \ref{figure:grr components} illustrates this by plotting both the income and revaluation components constructed as described above. The different behavior of the income and revaluation components across these series is due to the fact that the return to capital computed by \citet{gomme-ravikumar-rupert} excludes housing, which has a more sizable and volatile revaluation component.    


\begin{figure}[ht]
	\includegraphics[width=\linewidth]{measuring_returns/output/Figure_paper_returns_composition.pdf}
	\caption{Decomposition of the returns to US business capital into an income component (left panel) and a revaluation component (right panel)  \footnotesize \newline \textit{Notes---}see Appendix \ref{appendix returns} for data sources and measurement details.\label{figure:grr components}}
\end{figure}

\subsection{Measuring $\kappa$}

Suppose that there are two types of households in the economy, investors and (the remaining) households. Let $K$ denote the total capital stock, which  is financed through equity $E$ ($E_I$ owned by investors and $E_H$ by households) and debt issuance $D$ ($D_I$ held by investors and $D_H$ by households). Suppose there is also a government that supplies $B$ of safe bonds to the market ($B_I$ held by investors and $B_H$ by households). Furthermore, households and investors can lend directly to each other, and we denote the loans from households to investors by $L_I$ and the loans from investors to households by $L_H$.  

Investors' own a total amount of business capital given by: 
$$
K_{I}=\text{share of equity held by investors} \times \text{firms' assets} = \frac{E_{I}}{E_{I}+E_{H}}\left(E_{I}+E_{H}+D_{I}+D_{H}\right).
$$ 
It follows that investors' ownership of business capital relative to their net worth is: 
$$\kappa=\frac{ \frac{E_{I}}{E_{I}+E_{H}}\left(E_{I}+E_{H}+D_{I}+D_{H}\right)}{E_I+D_{I}+B_{I}+L_H-L_{I}},$$ which corresponds to the formula given in the main text. 

In the DINA, we use various definitions of investors, including households in the top 1\% (10\%, 0.1\%) of the income distribution. We then measure the components for the two groups $I,H$ as follows (original variable names in italics): $E$: equity and business assets (\emph{hwequ + hwbus}); $D$: currency, deposits, bonds and loans minus municipal bonds, currency and money market and bond funds (\emph{hwfix - muni - currency - mmbondfund}); $B$: municipal bonds, currency and money market and bond funds (\emph{muni + currency + mmbondfund}); $L$: non-mortgage debt (\emph{nonmort}). 

In the DFA, we define the investors as households in the top 1\% of the net worth and of the income distributions. We then measure the components for the two groups $I,H$ as follows: $E$: corporate equities and mutual fund shares plus equity in noncorporate business; $D$: corporate and foreign bonds plus time deposits and short-term investments; $B$: US government and municipal bonds, and money market funds shares; $L$: loans liabilities minus home mortgages minus consumer credit.

Figure \ref{figure:shares} shows the resulting series for $\kappa$ for various definitions of the investors group and across sources. The right hand panel shows the respective estimates for $\kappa$ excluding the holdings of government bonds in investors' portfolios.  

\begin{figure}[ht]
	\begin{centering}
		\includegraphics[width=\textwidth]{measuring_returns/output/Figure_kappa_I_shares}
	\end{centering}
	\caption{\label{figure:shares}Estimates of $\kappa$---business capital controlled by investors divided by their net worth. The right panel shows series for $\kappa$ that excludes government bonds from our calculation.}
\end{figure}

\subsection{Return to US Equity}

We compute returns to equity as $r_E := d + \Pi_v$ where $d$ is the dividend yield and $\Pi_v$ are expected capital income gains due to changes in market valuation. We measure the return to equity on US corporations (which accrues to US households investing at home but also to non-US households investing in the US) and the return to equity obtained by US households from investing in the stock market and mutual funds (which might include equity abroad as well as bonds held by mutual funds).  

\paragraph{Return to equity for US non-financial corporate sector:} The dividend yield is computed as dividends (IMA Table S5, line 17) over the lagged market value of equity (IMA Table S5, line 156). Realized capital gains are given by changes in the market value of equity due to prices (IMA Table S5, lines 102 and 103) over the lagged market value of equity. 

\paragraph{Return to corporate equity for US households:} The dividend yield is computed as dividend income (NIPA Table 2.1, line 15) over the lagged market value of corporate equity held by US households directly or via mutual funds (IMA Table S3, line 118 and 119). Realized capital gains are given by changes in the market value of equity due to prices (IMA Table S3, lines 90 and 91) over the lagged market value of corporate equity held by US households. 

\paragraph{Average return to equity for US households:} We also computed the return to both private and corporate equity obtained by US households. The income component is computed as dividend income plus the adjusted proprietors' income described above
(NIPA Table 2.1, line 15 and an adjusted version of Line 9) over the lagged market value of equity held by US households directly or via mutual funds (IMA Table S3, line 118, 119, and 121). Realized capital gains are given by changes in the market value of equity due to prices (IMA Table S3, lines 90, 91, and 92) over the lagged market value of equity held by US households. 

\paragraph{Results:} Figure \ref{figure:equity return} plots different estimates for returns to equity. The panels differ in the treatment of the expected capital gains term. The left panel plots the dividend yield only.  The middle panel adds to it the expected capital gain assuming that it is equal to the average realization of the change in stock prices for the next 10 years. Effectively, this is the realized return of holding equity for 10 years. The right panel plots a measure of actual yearly realized returns, using actual revaluations observed in the data. Overall, the dividend yield declines since the highs recorded in the 80s, but is stable over the longer period since 1960s. The Figure also documents the significant volatility of the actual revaluations.

\begin{figure}[ht]
	\begin{centering}
		\includegraphics[width=\textwidth]{measuring_returns/output/Figure_returns_equity.pdf}
	\end{centering}
	\caption{\label{figure:equity return}Estimates of $r_E$---return to equity. }
\end{figure}



\section{Decomposing Percentile-Specific Income Growth \label{appendix DINAs}}

\subsection{Measuring capital and labor income using the DINAs}

The Distributed National Accounts (DINAs) combine individual level tax and survey data with national accounts in an effort to create a set of distributional accounts that fully account for national income and wealth (PSZ, 2018). PSZ have released annual public use micro-files of the DINAs, which build heavily on the IRS PUFs. A core advantage of the former over the latter is that the income and wealth items reported in the DINAs aggregate up to national accounts as opposed to taxable personal income only. This is especially important given the increasing importance of non-pecuniary benefits such as employer based healthcare and non-taxable income such as pensions. We choose 2012 as the final year of our analysis as the underlying tax records end in 2012 as well.\\ 

Our primary measure of individual income in the DINA is ``pre-tax national income'' (peinc) , which is composed of factor income from labor and capital as well as the primary surplus of the pension system and non-profit sector. Note that the surplus might be negative. We focus on this measure instead of narrower income definitions as it aggregates to national income. \\

The DINA measure capital and labor income by applying a range of adjustments to the underlying ``personal factor income''. Firstly, PSZ add the revenues from sales and exercise taxes and income from social insurance to labor and capital according to their factor shares. Secondly, social contributions are subtracted from labor income as is payable investment income to pension funds from capital income. Finally, PSZ distribute mixed income from pass-through entities and partnerships using a constant capital share of 30\%. Note that this does not apply to S-corporation profits, which PSZ entirely attribute to capital income. We discuss this below.\\

The adjusted labor income is composed of measured labor compensation, which includes wages as well as other employment based benefits such as employer based health insurance, mixed income from pass-through entities and the above mentioned adjustments:

\begin{equation*}
\textrm{labor income}_{PSZ} = \textrm{labor comp.} + 0.7 \times \textrm{mixed income} + \textrm{adjustments}
\end{equation*}

Capital income in turn is the sum of income from equity, fixed income assets, housing assets, pass-through businesses and pensions minus debt payments:

\begin{equation*}
\begin{split}
\textrm{capital income}_{PSZ} = &\textrm{equity inc.} + \textrm{fixed income inc.} + \textrm{housing inc.} +0.3 \times\textrm{mixed inc} \\
&+ \textrm{pension inc.} - \textrm{debt payments} + \textrm{adjustments}.
\end{split}
\end{equation*}

A couple of things are worth pointing out at this point. Firstly, housing income includes imputed rental payments for owner occupants as in the NIPA tables. Secondly, equity income is composed of the dividends as well as the entirety of S-corporation profits.\\

The treatment of S-corporation income has been subject so some controversy. In particular, SYZZ argue that the capital share of mixed income including S-corporation should be 25\% instead of 30\% for pass-through entities and 100\% for S-corporations. To address this potential concern, we investigate two potential adjustments.  Firstly, we simply reallocate 70\% of the reported fiscal S-corporation income:

\begin{align*}
&\textrm{labor income}_{SYZZ.A} = \textrm{labor income}_{PSZ} + 0.7\times \textrm{S-corp. fiscal inc.} \\
&\textrm{capital income}_{SYZZ.A} = \textrm{capital income}_{PSZ} - 0.7\times \textrm{S-corp. fiscal inc.}\\
\end{align*}

Alternatively, we follow the adjustment described in the Appendix to PSZ, which takes into account portfolio composition along the wealth distribution as well as the imputation method used for capital income in PSZ. In particular, PSZ define \textrm{a(p) = S-corp. equity of percentile p $\times$ aggregate S-corporation income to equity ratio} and \textrm{b(p) = C-corp. equity of percentile p $\times$ aggregate C-corporation income to equity ratio}. Importantly, C-corporation income includes dividends as well as retained earnings. The adjusted S-corporation income for percentile $p$ is then calculated as \textrm{a/(a+b) $\times$ equity income of percentile p}. We apply the same formula as above using this adjusted S-corporation income measure to calculate $\textrm{labor income}_{SYZZ.B}$ and $\textrm{capital income}_{SYZZ.B}$ respectively.

\subsection{Additional figures}

Figure \ref{fig:IRS appendix} provides a decomposition of income growth for the 1980--2007 period. On the other hand, Figure \ref{fig:IRS appendix 2} provides a decomposition for 1980--2012 where we measure S-corporation income using the DINAs and the capitalization method described above. 

\begin{figure}[!ht]
	\centering
	
	Panel (a)---IRS data
	
	
	\includegraphics[width=.6\linewidth]{additional_evidence/psz_irs_data/output/Figure_paper_consistent_irs_share_1980_2007_onepanel.pdf}
	
	Panel (b)---DINAs data
	
	\includegraphics[width=.6\linewidth]{additional_evidence/psz_irs_data/output/Figure_paper_consistent_psz_share_1980_2007_onepanel.pdf}
	\caption{\footnotesize Capital income and the rise in top inequality 1980--2007 (IRS and DINAs data)}\label{fig:IRS appendix}
\end{figure}

\begin{figure}[!ht]
	\centering
	\includegraphics[width=.6\linewidth]{additional_evidence/psz_irs_data/output/Figure_paper_consistent_psz_share_1980_2012_alternative_onepanel.pdf}
	\caption{\footnotesize Capital income and the rise in top inequality 1980--2007 (DINAs data), alternative measurement of S-corporations income.}\label{fig:IRS appendix 2}
\end{figure}



\section{Trends of Other Variables in Model and Data}\label{appendix trends}

\subsection{Investment and Capital Deepening\label{appendix investment}}
Our model predicts that automation should lead to an expansion in investment and the capital-output ratio, though less so than in a representative household model (Figure \ref{figure: supply demand shift}). Figure \ref{fig:K/Y} 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 \citep[see][]{gourio-klier}. In line with our findings for the capital-output ratio, 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 and since the 1970s.\footnote{
	An alternative measure of investment that is sometimes used in other papers is the net investment rate (net investment divided by existing capital). However, along a balanced-growth path, the net investment rate equals the growth rate of the economy. Long-run trends in the net investment rate are thus uninformative of the extent of automation in the economy, which is one of the many types of technological progress determining the growth rate of the economy in the long run.} 

\begin{figure}[tph]
	\centering
	\includegraphics[width=\textwidth]{additional_evidence/investment_data/output/Figure_paper_investment.pdf}
	\caption{Percent change since 1950 in capital and investment to GDP ratios in the US. \newline
		\footnotesize \textit{Notes---} 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).\label{fig:K/Y}}
\end{figure}


In a similar vein, our model predicts a moderate increase in TFP of 2.2\% between 1980 and 2014, which accounts for less than 10\% of the overall increase in productivity during this period. We view this as plausible given that automation is one of many technological improvements determining productivity during this period. More importantly, these numbers underscore the point that, in our model, improvements in automation can bring modest gains in productivity, output, and investment that are well within the bounds of what we see in the data, while at the same time having sizable distributional consequences.   



\subsection{The Capital Share and Inequality Across Countries and Longer Time Periods}\label{appendix evidence}
Does the prediction that rises in the net capital share are accompanied by large increases in top income inequality receive support from the data? 
\citet{bengtsson-waldenstrom} explore this link in a long panel with data for 15 countries going back to 1891. Their data shows that a 1 percentage point increase in the net capital share is associated with: a  6.68\% increase in the top 0.1 percent share of income; a 3.85\% increase in the top 1 percent share of income; and a 1.56\% increase in the top 10 percent  share of income.\footnote{In the published paper, \citet{bengtsson-waldenstrom} use a log-log specification, whereas our theory calls for the log-linear specification in equation \eqref{eq:topq share}. 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. }


\begin{figure}[!ht]
	\centering 
	\includegraphics[width=\linewidth]{additional_evidence/capshares_us_data/output/Figure_paper_us.pdf}
	\caption{Net capital share and top 1\% income share in the US. \newline
		\footnotesize\textit{Notes---} Data for capital shares from \citet{piketty-saez-zucman} and NIPA. The series for net capital shares are smoothed using a 5-year moving average. Data for the top 1\% and top 0.1\% income shares in the US from the World Inequality Database, which excludes capital gains.\label{figure: US}}
\end{figure}

Figure \ref{figure: US} illustrates this link for the US during the postwar period  using data on the net capital share from \citet{piketty-saez-zucman} (available on the World Inequality Database), as well as data from NIPA on the net capital share of the US corporate sector. Besides the net capital share, we also report \emph{capital shares net of growth}, which are the relevant statistic determining the importance of capital in the version of our model with growth (see equation \eqref{eq: returns with g}).\footnote{To construct capital shares net of growth we follow the approach in Section \ref{sec:returns} and measure $g$ using a 10-year moving average of the CBO's estimation of the potential growth rate of the economy. In addition, we measure the capital output ratios involved in the formula for the capital share net of growth from NIPA.} All of these measures reveal rising capital shares since the 1970s by 6--8 percentage points. The right panel shows that the increase in the net capital share was accompanied by a significant rise in the top 1\% and top 0.1\% income share starting also during the 1970s (also from the World Inequality Database). 



As already discussed, our mechanism is capable of generating this type of sizable changes in top income inequality. This is in contrast to the standard compositional effect emphasized in the literature and defined more precisely in Section \ref{sec:ineq}. As discussed in that section, an increase in the net capital share of 6 percentage points would generate an increase in the top 1\% income share of only 0.42 ($=0.07\times 6$) percentage points via compositional effects, which is small compared to what we see in the data and what our model is capable of generating.\footnote{ \citet[][Table 2.2]{meade} performs similar calculations for the United Kingdom circa 1959 but obtains much larger compositional effects because he assumes that, for the top $1\%$, $\tilde{S}_{\ell}(q)=6\%$ and $\tilde{S}_k(q)=47\%$, which he defends as appropriate numbers. With these numbers, the compositional effect is given by $(\tilde{S}_k(q)-\tilde{S}_\ell(q)) \times d \alpha_{net} = 0.41\times 0.01 = 0.41$ percentage points.  }

\begin{figure}[!ht]
	\centering 
	\includegraphics[width=\linewidth]{additional_evidence/capshares_uk_data/output/Figure_paper_engel.pdf}
	\caption{Gross capital share, the return to capital, and the top 5\% income share during the British industrial revolution. \footnotesize \textit{Notes---}Data from \citet{allen} and \citet{lindert}.\label{figure: historical episodes}}
\end{figure}

The link between capital shares and inequality can be also seen in other historical periods during which automation (or mechanization) was a dominant force, like the onset of the industrial revolution in Britain. As documented in \citet{allen} and reproduced here in Figure \ref{figure: historical episodes}, 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 \citet{lindert} 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.



\clearpage


\end{appendices}


\end{document} 