Sufficient Statistics for Top End Wealth Inequality

Transcription

1 Sufficient Statistics for Top End Wealth Inequality Dan Cao Georgetown University Wenlan Luo Georgetown University July 2016 Abstract Are there simple and observable aggregate statistics that might help determine and forecast the degree of wealth inequality? Do they explain the differences in wealth inequality across economies and over time? To answer these questions, we build a general equilibrium, neoclassical growth model, in which the stationary wealth distribution has heavy right tail with the Pareto tail index, θ. In the simplest cases of the model, the tail index depends on interest rate r, growth rate g, aggregate labor income share EY, and aggregate capital to output ratio, KY, summarized in a formula θ = ˆθ r, g, EY, KY rather than the simple gap r g, put forth in Thomas Piketty s Capital in the 21st century. In addition, financial development, production technology, corporate and wealth taxes affect the tail index through their effects on these sufficient statistics. When we calibrate the model to the U.S. economy, we find that the model requires significant persistent heterogeneous returns to investment in order to generate the tail index of U.S. wealth distribution. In this calibration, earnings inequality and, to a lesser extent, initial wealth distribution have negligible effects on top end wealth inequality. The transition of the model economy after a uniform wealth destruction shock or changes in corporate tax but not after a financial deregulation shock produces the joint dynamics of EY, KY, and wealth inequality experienced in major developed economies after World War II. Lastly, we find empirical evidence for persistent heterogeneous returns in the PSID surveys. Keywords: Top End Wealth Inequality; Pareto Wealth Distribution; Sufficient Statistics; Heterogeneous Returns; r-g; Corporate Tax; Financial Development For useful comments and discussions we thank Daron Acemoglu, George Akerlof, Jinhui Bai, Alberto Bisin, Mary-Ann Bronson, Paco Buera, Martin Evans, Garance Genicot, Pedro Gete, Mark Huggett, Roger Lagunoff, Ben Moll, Thomas Piketty, Vincenzo Quadrini, Martin Ravallion, Richard Rogerson, John Rust, and participants at Georgetown macroeconomic seminar. We also thank Son Le for excellent research assistance and his initial involvement in the project. Corresponding author. address: dc448@georgetown.edu 1

2 1 Introduction The recent empirical work by Piketty 2014, Saez and Zucman 2016, and others has found that the wealth distribution in major developed countries is highly unequal, especially in the U.S. The distribution has thick right tails, which are well approximated by Pareto Law. Are there simple and observable aggregate statistics that might help determine and forecast the degree of wealth inequality, and relatedly the Pareto tail index? Do they explain the differences in wealth inequality across economies and over time? Piketty 2014 proposes a theory, formalized in Piketty and Zucman 2015, that the Pareto tail index, θ, is determined by a simple sufficient statistics: the gap between interest rate, r, and the growth rate of the economy, i.e., θ = ˆθ 0 r g, for some function ˆθ 0.. While this simple theory is very appealing and makes important predictions about the future of wealth inequality in major developed countries, 1 it has come under fierce criticism including Acemoglu and Robinson 2015, Jones 2015, Mankiw 2015, Krusell and Smith 2015, Ray 2015, and others. One criticism that stands out is that, by focusing on r g, Piketty 2014 implicitly assumes that the saving rate out of capital income is equal to 1. 2 In this paper, we show that when we endogenize this saving rate in a fully optimizing, general equilibrium model, the tail index is a function of the model s parameters and observable equilibrium variables other than, and in addition to, the simple gap r g. 3 In the simplest cases, we have the following formula: θ = ˆθ 1 r, g, EY, KY, where EY and KY are the aggregate labor income share and capital to output ratio, respectively. In general equilibrium, the observables, EY and KY, provide information about the endogenous saving rate, as well as the implied aggregate capital accumulation and equilibrium rate of return to capital or rates of return to capital if the rates differ across agents. Returning to the question asked at the beginning of this paper, our formula for θ sug- 1 For example, recent low growth in OECD countries should imply high wealth inequality in the near future. 2 Admittedly, the formula in Piketty and Zucman 2015 for top end inequality involves the exogenous average saving rate their equation for ω between equations 15.3 and 15.5, despite lack of emphasis. However, this exogenous saving rate might be unobservable, unlike r and g. 3 Our results also apply under the assumption that the saving rate is exogenously given but is constant over time. What turns out to be more important is the general equilibrium implications of this saving rate. 2

3 gests that, although there might be useful sufficient aggregate statistics, in general, one would need more information about the economy than r g to determine and forecast top end wealth inequality. Relatedly, in general equilibrium, observable aggregates provide information about unobservable parameters that are valuable for such an enterprise. Moreover, in the class of models that we consider, under the restrictive assumption that the agents in the model face the same rate of returns to capital, i.e., interest rate r, as assumed by Piketty 2014 and Piketty and Zucman 2015, the model, when calibrated to match the historical values of EY, KY, r, g in the U.S., has a tail index of the stationary wealth distribution that is too high. That is top end wealth inequality is too low compared to the estimates from U.S. data. Heterogeneous rates of return to capital are indispensable in helping the model match the historical tail index. Our model builds on a general equilibrium framework in continuous time and features idiosyncratic investment risk as in Quadrini 2000 and Angeletos In the model, the agents share the same utility function but might differ in their investment returns, which are determined by their investment productivity. 4 For tractability, we assume that at any instant, there are at most two levels of investment returns, high or low. In the special case of homogenous returns that we also consider, the two returns are the same. The agents are subject to idiosyncratic shocks, as in Huggett 1993 and Aiyagari 1994, with Poisson arrival rates. If an agent is hit by a shock, her return switches from high to low, or low to high. In addition, to capture earnings inequality, we allow for permanent heterogeneity in labor productivity across agents. There is a bond market that allows the agents to save or borrow, subject to a borrowing constraint. Given the idiosyncratic investment returns and the financial market structure, the agents make the consumption, saving, and investment decisions to maximize their inter-temporal expected utility, taking interest rate and wage rate as given. We study the properties of the recursive competitive equilibrium of the model, in which prices are determined such that relevant markets, bond and labor markets, clear at all time. The model delivers several sets of results. First, in the stationary balanced growth path of the model, the stationary wealth distribution has a right Pareto tail, with the tail index determined by a simple quadratic equation. Following Piketty 2014, we define top end wealth inequality as the tail index, θ a lower tail index corresponds to more wealth 4 Heterogenous investment returns in our model can have several interpretations. They might arise from different degrees of financial sophistication while investing in publicly traded financial instruments, as documented in Calvet, Campbell, and Sodini 2007, differential access to high return investments in private businesses, or from entrepreneurship as documented and modeled in Quadrini 2000 and Cagett and De Nardi While we do not take a stand on the precise channel leading to heterogeneous returns, our model has a direct entrepreneurship interpretation. 3

4 inequality. Piketty 2014 and Piketty and Zucman 2015 argue that top end inequality is a function of the difference between interest rate and the growth rate of the economy. In our second set of results, we generalize this result or rather intuition to our model economy with heterogenous investment returns. We show that θ = ˆθ 2 g H g, where g H is the endogenous growth rate of wealth of the agent with weakly higher investment returns, relative to the endogenous growth rate of the economy. In addition, ˆθ 2 is strictly decreasing, i.e., the higher the gap g H g, the higher top end wealth inequality. The aforementioned formula contains endogenous and potentially unobservable variables. In two special cases, we obtain sharper characterizations and more user-friendly formulae. First, in the special case of homogenous returns, we show that θ = ˆθ 1a r, g, n, EY, KY, where n is population growth rate. Second, under strictly heterogenous investment returns but log utility and without population growth, we have θ = ˆθ 1b r, g, EY, KY. Therefore, at least in these cases, top end inequality is determined by factors other than the simple difference between interest rate and growth rate of the economy. In addition, we show that, everything else being equal, lowering labor income share, or aggregate growth rate, is associated with an increase in top end wealth inequality ˆθ 1b ˆθ 1b g EY > 0 and > 0, as discussed in Piketty However, an increase in capital to output ratio corresponds to a decrease in top end wealth inequality ˆθ 1b KY suggested by Piketty > 0, unlike the outcome We then calibrate the model to the U.S. economy and show that the parameters can be chosen to match important moments in the data, including the aggregate capital to output ratio, labor income share, and top end wealth inequality. In the calibration, in order to generate a Pareto tail index of 2 for top end wealth inequality, we need strictly heterogeneous returns with a difference of around 4% in un-levered returns 6% in levered returns that lasts on average 10 years. In this calibrated model, in which top end wealth inequality is determined by heterogenous investment returns, we also look at the importance of earnings inequality and initial wealth distribution at the beginning of the agents lifetime - initial wealth inequality - on top end wealth inequality. We find that varying 4

5 earnings inequality and initial wealth inequality, while keeping the aggregate variables constant, do not alter top end wealth inequality as measured by the Pareto tail index, or to a lesser extent, the top wealth share statistics top 0.1%, 1%, or 5% wealth shares. Outside the stationary balanced growth path of the calibrated model, we examine the joint dynamics of top end wealth inequality and the aggregate variables, capital to output ratio and labor income share, over transitional paths after a significant destruction of physical capital which was experienced in major European countries during WWI and WWII and after changes in corporate tax which were experienced in the U.S. during and after WWII and in the 1970s. We find that the dynamics of capital to output ratio and labor income share in the model have similar patterns as in the data increasing capital to output ratio and decreasing earning to output ratio over time. In addition, top end inequality follows a U-shape over the transitional paths, as also observed in the data by Piketty 2014, however in much smaller magnitudes and at lower speeds. We also consider the transitional path after a relaxation of borrowing constraints, which corresponds to financial deregulation in the 1980s in the U.S. Although relaxing borrowing constraints increases top end wealth inequality, it leads to lower capital to output ratio and higher labor income share, neither of which is consistent with the evidence in U.S. data documented in Piketty and Zucman Lastly, as the model calls for persistent heterogenous returns to match top end wealth inequality in the data, we find some evidence for persistent differences in returns to wealth across households from the PSID surveys. For each household in the survey, we define returns as the ratio between the sum of financial income - asset income from farm and business, rent, interest, dividends, income from royalty and trust funds excluding capital gains because of data quality - and core asset the sum of net worth, excluding other debts, home equity, and vehicles. The resulting returns to wealth display great heterogeneity and persistence. We also find that persistent heterogeneous returns can be inferred from wealth mobility. For example, households which have experienced episodes of high returns are also more likely to show upward wealth mobility in the future. The current paper belongs to the theoretical literature on Pareto distribution for wealth. Early work started with Stiglitz 1969; more recently, Benhabib, Bisin, and Zhu 2011, 2015, 2016, Moll 2012, Toda 2014, Achdou et al. 2015, Nirei and Aoki 2016, and others have provided interesting and important mechanisms to generate stationary wealth distribution with Pareto tails. 5 The Pareto tail index in these papers are given by formulae involving unobservable parameters and variables. For example, Benhabib, Bisin, and Zhu 2011, 2015, 2016 provide formulae for the tail index that require knowledge of the 5 Benhabib and Bisin 2016 provide an excellent survey of the literature. 5

6 entire distribution of stochastic investment returns and the entire demographic structure of the economy. In this context, Piketty 2014 s and Piketty and Zucman 2015 s r g theory is compelling since one needs to observe only simple aggregate statistics from an economy to gauge the degree of wealth inequality in that economy. In this paper, following the intuition in Piketty 2014 and Piketty and Zucman 2015, we not only show that wealth distribution has a right Pareto tail as found in the earlier literature, but we also attempt to go a step further by characterizing how the tail parameters vary with the underlying parameters and observable aggregate variables in the economy. In general equilibrium, observable aggregates provide information about unobservable parameters that help determine and forecast wealth inequality. This mechanism is absent in partial equilibrium papers, such as Benhabib, Bisin, and Zhu 2011, 2015, Our paper is also related to the growing literature studying idiosyncratic investment risks. In this literature, most papers, including Angeletos 2007, Toda 2014, Benhabib, Bisin, and Zhu 2015, Piketty and Zucman 2015, and Nirei and Aoki 2016, assume I.I.D. investment risks, or, more precisely, investment returns. 6 In this paper, we extend their analysis to allow for persistent idiosyncratic investment risks. From this perspective, our paper is closely related to Benhabib, Bisin, and Zhu 2011, 2016, Buera and Shin 2011, Moll 2012, Moll 2014, Achdou et al. 2015, Buera and Moll 2015, who consider persistent idiosyncratic investment risks. Our paper can be seen as a simplification of these papers. We use a simple two-level idiosyncratic investment risk Poisson process, which allows us to obtain sharp analytical characterizations of top end wealth inequality and transitional paths in general equilibrium with or without production. 7 Heterogenous returns in our paper can be interpreted as returns to entrepreneurs versus workers, as in the literature on entrepreneurship, including Quadrini 2000, Cagett and De Nardi 2006, and Buera In these papers, entrepreneurs have access to more productive production technologies than non-entrepreneurs. This assumption can be mapped into heterogeneous returns to investment in our model. However, unlike these papers, we do not assume fixed-costs of starting up a business or decreasing returns to scale. This assumption allows us to preserve the homogeneity of the optimization problem of the agents, which simplifies the characterization of wealth distribution 6 Piketty and Zucman 2015 assume I.I.D. saving rates, but this assumption is isomorphic to I.I.D. investment risks assumption. The analytical results in Nirei and Aoki 2016 rely on I.I.D. investment shocks, but the authors relax this assumption in their quantitative investigation. 7 Benhabib et al and Gabaix et al. 2015, Online Appendix provide analytical characterizations of the Pareto tail index for wealth distribution under persistent investment shocks as we do, but only in a partial equilibrium setting in which the rates of returns are exogenous. General equilibrium might lead to interesting comparative statics, such as a non-monotone effect of financial development on the tail index, as shown in Moll 2012 and in our Proposition 2. 6

7 and transitional paths. Another strand of literature starting with Huggett 1996 attempts to explain the high degree of wealth inequality using models with idiosyncratic earnings shocks over the lifecycle of households. However, this class of models cannot match wealth inequality at the very top unless one assumes very large temporary earnings shocks as in Castaneda et al Using a similar model, Kaymak and Poschke 2016 show that increases in earnings inequality have accounted for most of the increase in wealth inequality in the U.S. since the 1970s. In our model, in which wealth inequality is driven by persistent heterogeneous investment returns, changing earnings inequality has negligible effects on wealth inequality. Benhabib, Bisin, and Zhu 2011 and Benhabib, Bisin, and Luo 2015 also emphasize this point in a partial equilibrium context. Over transitional paths, Gabaix et al. 2015, Online Appendix have recently argued that heterogenous returns give rise to fast transitional dynamics of top end wealth inequality. However, in their analysis, the authors take the returns on wealth as exogenously given. In our paper, the returns are endogenously determined by the total supply of capital and labor along transitional paths. We find that, despite significant return heterogeneity, the transition of top end wealth inequality is rather slow. Empirically, we find some evidence for heterogeneous and persistent returns to wealth in the data from PSID surveys. Fagereng et al have found similar evidence in Norwegian household level data. Saez and Zucman 2016 have also found that, for foundations, total rates of returns - including unrealized capital gains - rise sharply with foundation wealth. Using Swedish household level data, Calvet, Campbell, and Sodini 2007 documented that more financially sophisticated households earn higher mean returns from investments in publicly traded financial instruments. The rest of this paper is organized as follow. Section 2 presents a general model with capital accumulation, production, and labor. Section 3 presents a special case, the AK model, and results on top end inequality and transitional dynamics. In Section 4, we calibrate the main model to the U.S. economy and investigate the quantitative implications of the model. Lastly, Section 5 presents empirical evidence for persistent heterogeneity in idiosyncratic investment returns and their implications for wealth mobility. Section 6 concludes. 2 An Neoclassical Economy in Continuous Time In this section, we develop a neoclassical growth model in continuous time as presented in Acemoglu 2009 but with heterogenous agents facing idiosyncratic death shocks and 7

8 investment shocks, which determine the productivity of their capital investment. The agents can finance their projects only by issuing risk-free debt, collateralized by their capital. 2.1 The Environment Time - t - is continuous and runs from 0 to. 8 The economy is populated by a continuum of agents that are indexed by h N t = [0, N t ] where N t is population size at time t. Investment Productivity Let i h t denote the individual state of agent h at time t. We assume that i h t follows a two-state Markov chain, i h t I = {L, H}, which capture low i h t = L and high investment productivity i h t = H, with Poisson switching rates λ LH and λ HL from one state to the other. By the law of large numbers, the fraction of agents with high investment returns and low investment returns are respectively λ LH M H = λ HL + λ LH λ M L = HL = 1 M λ HL + H. λ LH In each instant, the agents can produce output using a constant-return-to-scale technology, which depends on the idiosyncratic state, using capital and labor 9 Y t = F i h t k h t, l h t. We assume that the high type is more productive than the low type. Assumption 1. For any k, l > 0, F H k, l > F L k, l. While in this paper we focus on the case in which there are only two-level of investment productivities, we can extend easily the model to allow for more heterogeneity in this dimension. In addition, when agents with lower productivity actively produce in equilibrium, this model is equivalent to one in which a competitive representative firm 8 We have also studied and solved the model in discrete time. However, the solution for the equilibrium wealth distribution is much simpler in continuous time. 9 The production function includes depreciation, i.e. Y t is net output. The CES production function in Section 4 is an example. 8

9 uses the less productive production function and entrepreneurs that have access to the more productive production. The macroeconomic literature on entrepreneurship such as Quadrini 2000 and Cagett and De Nardi 2006 often embraces this interpretation. Labor Productivity Each agent is also endowed with x h t efficiency units of labor. We assume that x h t grows at the rate g x common across households: x h t+t = x h t eg xt. 10 We assume that the initial labor productivity of an agent is determined when she is born. Normalized by a constant trend G x,t = e gxt, defined over R + with c.d.f Ψx and mean x h t G x,t is initially drawn from a distribution Death Shocks, Population Growth, and Redistribution Population N t grows at a constant rate n 0 and the initial population is normalized to 1. We also assume that agents are hit by death shock, arriving at Poisson rate λ > 0.Therefore, there are n + λn t t new borns in each infinitesimal [t, t + t] time interval. The investment productivity of the new born can be H or L with the fraction M H and M L respectively. The labor productivity of the new borns, relative to the constant trend G x,t, are drawn from the distribution Ψ.. The total wealth of the dying agent is redistributed to the new borns according to a redistribution function Γ i,t ψ. That is, conditional on investment productivity type i, each agent receives a draw of fraction ψ from the distribution Γ i,t Pr ψ ψ = Γ i,t ψ. The agent then obtains a fraction ψ of the total wealth of the dying agents per new born. We also assume that the support of Γ i,t belongs to R +. Since in each instant, there is a mass λ tn t of dying agents and a mass λ + n N t of new borns, and the total wealth of the dying agents is redistributed fully, we must have M H ψdγ H,t ψ + M L ψdγ L,t ψ = 1. In most cases, we assume that the redistribution function preserves the total wealth of agents with the same investment productivity. 10 In Online Appendix E, we extend the model to allow for heterogenous but deterministic growth rate of labor productivity across agents. As shown in Gabaix et al. 2015, heterogenous growth rates of labor productivity can potentially help explain high labor income inequality and a rapid rise in labor income inequality. However, we cannot allow for idiosyncratic productivity shocks over the agents lifetime since the optimization problem of the agents would no longer be homogenous in total wealth. 11 The model does not exhibit the scale effect, so normalizing the mean of Ψ to 1 is without loss of generality. 9

10 Assumption 2. The distribution function is type preserving i.e. M i ψdγ i,t ψw t = W i,t ψ for i = H, L, where W t is aggregate financial wealth and W i,t is the aggregate financial wealth of all agents with investment productivity i conditional aggregate. Death shocks and redistribution are crucial in preventing the wealth distribution from ever expanding and give rise to a stationary distribution. 12 The redistribution functions Γ i,t also correspond to wealth inequality at the beginning of the agents lifetime - initial wealth inequality. In the calibration of the model in Section 4, we use the estimates by Huggett et al of wealth inequality at the ages in PSID for Γ i. Utility, Constraints, and Optimization function The agents share the same instantaneous utility c 1 σ 1 σ if σ = 1 uc = logc if σ=1. Let {r t, e t } t=0 denote the sequence of interest rates and wages. Agent h chooses a sequence of capital holding k h t, bond holding bh t, and consumption ch t to maximize subject to dw h t dt [ max E exp ρ + λ t ] u c ct h t h dt,kh t,bh 0 t } = max {F i k h lt h t, lt h e t lt h + r t bt h + e t xt h ct h 2 ht w h t = k h t + b h t, 3 where wt h denote the financial wealth of agent h at time t. The budget constraint, 2, and portfolio constraint, 3, are standard. At time t, change in financial wealth, dwh t dt, consists of return from capital investment, interest from bond holding, r t b t, labor earning, e t xt h, minus consumption, ct h. Financial wealth, wh t, is allocated to capital holding kh t and bond holding bt h. The agent is also subject to the borrowing constraint 5 defined below. Let Q t denote the present discounted value of future labor income for each agent, i.e. 12 See Gabaix 2009 for other types of assumptions that guarantee the existence of a stationary wealth distribution with Pareto tail such as a reflecting barrier. 1 10

11 human wealth, Q h t = 0 t exp r t+t dt e t+t xt+t h dt. Since x h t grows at the rate g x, we also have Q h t = q tx h t, where q t = The dynamics of Q h t and q t are then given by 0 t exp r t+t g 0 1 x dt 1 e t+t dt. dq h t dt = r t Q h t e t x h t and dq t dt = r t g x q t e t. 4 Given Qt h, we assume that the borrowing constraint takes the form 0 k h t and 0 mk h t + bt h + Qt h, 5 where m 0, 1. It is important that the borrowing constraint is in terms of borrowing plus the present discounted value of labor income so that the optimization problem of the agent is homogeneous, as we will see below. 13 The definition of an equilibrium is standard. Definition 1. For any initial distribution of wealth { w0 h } h N, a competitive equilibrium is 0 described by stochastic processes: the processes for interest rates and wage rates {r t, e t } t=0, wealth distribution { w h t } t=0,h N t, capital and bond holdings { k h t, bh t } t=0,h N t and consumption { c h t } t=0,h N t ; such that i The allocation { w h t, kh t, bh t, ch t } solves agent h s maximization problem 1 given the processes for interest rates and wage rates. and ii Bond and labor markets clear in each instant h N t b h t dh = 0, lt h dh = xt h dh. h N t h N t Notice that by Walras Law, the market clearing condition in the bond market implies 13 Angeletos 2007 makes a similar assumption on borrowing constraint, except for m = 0. 11

12 that market for consumption goods also clear, i.e total output equals to total consumption plus total investment: F i k ht h Nt h t, lt h dh = c h dwt h t dh + h N t h N t dt dh. Together with the portfolio choice constraint 3, the market clearing condition in the bond market implies market clearing in capital market: k h t dh = wt h dh. h N t h N t Given the homogeneous structure of the model, in the following subsection, we show that a competitive equilibrium has a simpler representation. Indeed, the definition of competitive equilibrium suggests that we need to keep track of the whole wealth distribution to solve for the equilibrium. However, taking advantage of the homogeneity of the agents optimization problem, we show that only some aggregates are sufficient to determine the equilibrium in economy. Having solved for the equilibrium prices and policy functions, we can then go back to solve for the equilibrium wealth distribution. In this sense, solving for an equilibrium is decoupled into solving for the equilibrium aggregates and solving for the equilibrium wealth distribution. 2.2 Markov Equilibrium To obtain sharper characterization of a competitive equilibrium, we first characterize the agents optimal dynamic strategies using the Hamilton-Jacobi-Bellman HJB equation for their value functions in a competitive equilibrium. To simplify the notation, we use the following auxiliary functions, R i e and S i e: R i ek = max F i k, l el lt h and S i ek denote the maximizer. Because F i has constant returns to scale, we have F i k, S i ek = R i ek + es i ek. R i e and S i e are basically the payments to capital and labor per unit of capital used in the production function F i. Let Vt, it h, xh t, wh t denote the value function from the maximization 1 of agent h. The following lemma provides the PDE that characterizes V. 12

13 Lemma 1. The HJB equation for V is ρ + λ V V t = max c,k,b uc + V x g xx t + V w R i h t e tk + r t b c + e t x t + λ i h t, i h t Vt, ih t, x h t, w h t Vt, i h t, x h t, w h t 6 where the maximization problem is subject to the constraints wt h = k + b and 0 k and 0 mk + b + Qt h we adopt the standard notation that H = L and L = H for i h t and it h. Proof. Online Appendix C. We conjecture and verify that the value function V has the form: V h t, it h, xt h, wt h vt, it h = wt h + xh t q 1 σ t if σ = 1 vt, it h + 1 ρ+ λ log wt h + xh t q t if σ=1. In addition, we work with b t = b t + Qt h. Given the functional form for V, the HJB equation for the agents, 6, becomes ρ + λvt, i h t vt, ih t t = max c,k,b H i h t c, k, b; vt, i h t + λ i h t, i h t vt, ih t vt, i h t 7 where uc + 1 σvr i e t k + r t b c when σ = 1 H i c, k, b; t, v = logc + 1 ρ+ λ R ie t k + r t b c when σ = 1 and the maximization problem is subject to 8 1 = k + b and 0 k and 0 mk + b. 9 This implies that the policy functions are linear in total wealth, i.e. financial wealth plus human wealth: k h t = k i h t b t h = b t it h ct h = c t it h t wt h + Qt h wt h + Qt h wt h + Qt h. 13

14 Let W i,t = h:it h=i wh t dh denote the total wealth conditional on investment productivity type i. The market clearing condition in the bond market becomes: b H t W H,t + M H G t N t q t + b L t W L,t + M L G t N t q t G t N t q t = 0, 10 and in the labor market becomes: S H e t k H,t W H,t + M H G t N t q t + S L e t k L,t W L,t + M L G t N t q t = G t N t 11 which depends only on W H,t, W L,t, q t. The following lemma provides the equations that determine the dynamics of these three variables. Lemma 2. In a competitive equilibrium, the aggregate dynamics of aggregate conditional wealth satisfy, for i {H, L} dw i,t dt = k i,t R ie t + bi,t r t ci,t W i,t + q t M i G t N t λ i, i W i,t + λ i,i W i,t + λ M i ψdγ i,t ψw i,t M i ψdγ i,t ψw i,t dq t dt M ig t N t g x q t M i G t N t ψ ψ 12 and q t satisfies 4. Proof. Online Appendix C. The analysis above suggests that W H,t, W L,t, q t are sufficient endogenous state variables to determine interest rate, wage rate and allocations at time t. The following definition formalizes the intuition. Definition 2. A Markov equilibrium is a competitive equilibrium in which the equilibrium prices r t, e t and policy functions, ci,t, k i,t, b i,t depend only on the aggregate state variables W H,t, W L,t, q t. It is also important to characterize the dynamics of the aggregate growth rate of the economy and of the wealth shares of agents with the same investment productivity. Indeed, let g t denote the growth of the aggregate total wealth, which is given by g t = 1 d W H,t + W L,t + Q t, W H,t + W L,t + Q t dt 14

15 and X t is the wealth share of agents with high investment productivity: X t = W H,t + M H N t q t W t + Q t. The following result characterizes the dynamics of g t and X t as functions of the agents policy functions. Lemma 3. In a competitive equilibrium, the dynamics of the growth rate of total wealth is given by In addition, X t satisfies g t = R H e t k H,t + r t b H,t H,t c Xt + R L e t k L,t + r t b L,t L,t c 1 Xt Q t + n. 13 W t + Q t dx t dt = g H,t g L,t X t1 X t λ HL X t + λ LH 1 X t + λ M H ψdγ H,t ψ1 X t M L ψdγ L,t ψx t ψ ψ Q t λm H M L ψdγ H,t ψ ψdγ L,t ψ ψ ψ W t + Q t Q t + n M W t + H X t, 14 Q t where gi,t denote the relative growth rate of type i agents: g i,t R ie t k i,t + r tb i,t c i,t g t. Proof. Online Appendix C. From the expression for g t in 13 and the definition of gi,t above, we obtain 0 = g H,t X t + g L,t 1 X t + n. 15 W t + Q t Q t 15

16 When Assumption 2 holds, equation 14 simplifies to dx t dt = g H,t g L,t X t1 X t λ HL X t + λ LH 1 X t Q t + n M W t + H X t. 16 Q t The results above fully characterize the aggregate dynamics of the economy without the knowledge of the entire wealth distribution among the agents in the economy. Given the aggregate dynamics, the following lemma provides the PDEs that characterize the evolution of the wealth distribution over time. Let ω h t denote the total wealth financial plus human wealth of agent h relative to the average total wealth in the population at time t: and let p i t, ω denote: ωt h = wh t + q txt h, W t +Q t N t p i t, ω = Prω h t ω, i h t = i. By the law of large numbers, p H t, ω p L t, ω, is also the fraction of agents with high investment return low investment return and with relative total wealth exceeding ω. p H t, ω and p L t, ω satisfy the following Kolmogorov forward equation. Lemma 4. p H and p L satisfy the following PDEs: p i t, ω t = p it, ω ω ω gi,t + n λ i, i + λ + n p i t, ω + λ i, i p i t, ω + λ + n ω W t + Q t J i,t, q tg t N t W t λ λ+n W t λ λ+n 17 where J i,t y 1, y 2 = Γ i,t y 1 y 2 x dφx and p i also satisfies the boundary conditions: Proof. Online Appendix C. lim p it, ω = 1 and lim p i t, ω = 0. ω 0 ω 16

17 2.3 Stationary Balanced Growth Path In this subsection, we look for a stationary equilibrium as in Huggett 1993 and Aiyagari 1994, in which interest rate, wage rate, rates of return on investment, wealth redistribution functions, and relative wealth distributions remain unchanged over time. However, the economy as a whole grows at a constant rate. Definition 3. A stationary stationary balanced growth path, or balanced growth path BGP for short, is a Markov equilibrium in which interest rate, wage rate, rates of return on investment, growth rate, and relative wealth distribution are constant over time. The concept of stationary balanced growth path is standard see for example Huggett et al and Acemoglu and Cao With strictly concave production functions, we can easily show that in a BGP the economy grows that the rate n + g x and W i,t = W i e g x+nt. Since interest rate and wage rate are constant over time, q t q, and by 4, q is determined by: 0 = r g x q ē. 18 The value functions in 7 become vt, i v i, where v i satisfies ρ + λ v i = max c,k,b H ic, k, b; v i + λ i, i v i v i 19 where uc + 1 σvr i ēk + r b c when σ = 1 H i c, k, b; v = logc + 1 ρ+ λ R iēk + r b c when σ = 1 and the maximization is subject to the constraints 9. The dynamics of aggregate wealth, 12, simplifies to 20 g x + n W i = k i R iē + b i r c i W i + qm i λ i, i W i + λ i,i W i + λ M i ψdγ i ψ W i M i ψdγ i ψ W i g x qm i. 21 The markets clearing conditions 10 and 11 become ψ ψ b H X + b L 1 X = q W + q, 22 17

18 and ēs H ē k H X + ēs L ē k L 1 X = ē W + q, 23 where X = W H +M H q W H + W L + q. The following theorem characterizes the BGP in the economy. Theorem 1 Stationary Balanced Growth Path. Assume Assumption 1 holds. Depending on the parameters of the model, a stationary BGP is characterized by six equations: two HJB equations 19 for H- and L-agents, the two equations determining W H and W L, 21, the two market clearing conditions, 22 and 23. However, the unknowns differ in three different cases: Case 1: r = R L ē. Then k H, b H = 1 1 m, 1 m m and b L = 1 k L. The six unknowns are v H, v L, k L, ē, W H, W L. Case 2: R L ē < r < R H ē. Then k H, b H = 1 1 m, 1 m m and k L, b L = 0, 1.The six unknowns are v H, v L, r, ē, W H, W L. Case 3: r = R H ē. Then k L, b L = 0, 1 and b H = 1 k H. The six unknowns are v H, v L, k H, ē, W H, W L. Lastly, in Case 1 and Case 2: R H ē k H + r b H c H > R Lē k L + r b L c L. Proof. Appendix A. In Case 1, given that r = R L ē, L-agents are indifferent between producing using their production function and lending to the H-agents at interest rate r. In equilibrium, they do both, i.e. 0 k L 1 and k L, b L are determined by the equilibrium conditions. The same logic applies to Case 3. One important implication of this theorem is that in Case 1 and Case 2, the agents with the higher rate of returns on investment H-agents save more. While consumption can be higher or lower due to the opposite income and substitution effects, the effect to higher rates of returns on saving is unambiguous. 14 This result is consistent with Saez and Zucman 2016 who find that saving rates tend to rise with wealth The proof of this property is not straightforward due to the switching probability between the two rates of returns. When σ < 1, we can show that the substitution effect on consumption dominates the income effects, therefore c H < c L so the result on saving rate is immediate. It is more difficult to prove the result when σ > 1, so that c H > c L. 15 We also find some empirical evidence for higher saving rates conditional on higher returns to wealth in PSID. 18

19 Theorem 1 does not tell us the conditions under which the equilibrium belongs to Case 1, 2, or 3. For a special case of this model, the AK model in Section 3 and with log utility, Proposition 1 provides a complete characterization. We show that Case 1 happens when m is low, Case 2 when m is intermediate, and Case 3 when m is high. We have verified numerically that the same pattern holds for the current model with production. In the BGP characterized by Theorem 1, the PDEs for stationary wealth distribution, 17, become 0 = p iω ω ω ḡ i + n λ i, i + λ + n p i ω + λ i, i p i ω + λ + n ω W H + W L + q q J i, W H + W L λ λ+n where J i is defined in Lemma 4 with Γ i,t Γ i. W H + W L λ λ+n The following theorem characterizes the stationary wealth distribution., 24 Theorem 2 Pareto Tail. Assume that the supports of Φ and Γ i,t are bounded above, then in a stationary BGP, we obtain the following properties of the stationary distributions. 1 The stationary distribution of total wealth financial plus human wealth has right Pareto tail. More specifically, p H ω = Ψ Hω θ p L ω = Ψ Lω θ for all ω ω for some Ψ H, Ψ L > 0 and ω > 0, and θ is the negative root of the following quadratic equation: λhl + λ + n ḡ H + n λlh + λ + n λ + η ḡ L + n + η HL λ LH ḡ H + n ḡ = L + n 2 The distribution of financial wealth follows an approximate power law with the same tail index as the one for the distribution of total wealth, i.e. there exist d > d > 0 such that dw θ Pr when w, where θ is determined in Part 1. Proof. Appendix A. w h t W t /N t w dw θ, 19

20 Similar results still hold if the supports of Φ and Γ i are unbounded but the distributions have thin tails, or Pareto tail with the tail index strictly higher than θ. However, we need to relax the definition of Pareto tail to asymptotic Pareto tail as in Acemoglu and Cao Following Piketty 2014, we define top end wealth inequality as the upper tail index, θ in Theorem 2. Lower θ corresponds to fatter tail of the wealth distribution, i.e. higher degree of wealth inequality. The results in Theorem 2 deserve some discussion. Homogenous Rates of Returns First of all, in the special case without heterogeneous investment risk, ḡ H = ḡ L = n q W+ q and λ HL = λ LH = 0, by 25, the tail index is given by θ = 1 + λ 1 + q W. 26 n In the AK model presented in Section 3, q = 0, we recover the formula for the tail index given in Jones 2015 θ = 1 + λ n. As noted by Jones 2015, this formula contradicts the intuition put forth in Piketty 2014: in general equilibrium, lower population growth leads to lower top end wealth inequality. With concave production functions, using 18 and the observation that ē W = KY EY, 26 implies that 17 θ = ˆθ 1a EY, KY, r; gx, λ, n := 1 + λ 1 + EY n KY 1 r g x. 27 In Online Appendix F, we present a direct derivation of 27 in the standard neoclassical growth model a la Ramsey-Cass-Koopman with labor productivity growth. 18 We show that θ = λ + n r c g x. After solving out for consumption rate c using equilibrium conditions, we arrive at 27. Using the standard values for parameters λ, n, g x and the historical values of EY and 16 A distribution density ϕ has asymptotic Pareto tail χ if for any ξ > 0 there exists B, B and x such that Bx χ 1+ξx < ϕx < Bx χ 1 ξx for all x x. 17 Notice that the tail index depends on r g x and n separately, unlike what is conjectured by Piketty 2014, i.e. θ = ˆθ 0 r g where g = g x + n. 18 In a companion note Online Appendix II, we also present an equivalent formulation with exogenous labor augmenting technological progress. 20

21 KY in the U.S. data see Table 1, 27 implies a tail index of which is too high compared to those observed in the U.S. data which are between 1.4 and 2 over the years. We will show below that allowing for heterogenous returns in the model can help match the empirical tail index. Heterogeneous Rates of Returns Returning to the general model with heterogenous returns, equation 25 also tells us how θ depends on ḡ H, ḡ L and the transitional probabilities λ HL, λ LH and population growth. We obtain sharper characterizations of how θ varies if we assume that there is no population growth and that wealth redistribution is type preserving. Theorem 3 Sufficient Statistics I. Assume n = 0 and Assumption 2 holds. In a stationary BGP with k L > 0, i.e. Case 1 in Theorem 1, top end wealth inequality is a function of the relative growth rate of the high type and the persistent parameters: θ = ˆθ 2 ḡ H ; λ HL, λ LH, λ. In addition, top end wealth inequality increases in ḡ H, i.e. ˆθ 2 ḡ H Proof. Appendix A. This is a generalization of the now famous result from Piketty 2014 that top end wealth inequality depends on r g, which corresponds to formula 27 above. With heterogenous investment returns, top end inequality depends on the relative growth rate of wealth of the high type agents, which in turn depends on their investment returns, equilibrium interest rate and endogenous consumption and portfolio choice decisions: < 0. ḡ H = R Hē k H + r b H c H ḡ. The first two terms in the right hand side capture the rate of returns on the optimal portfolio of the H-type. The third term captures the fact that a fraction of the returns is consumed. This term is implicitly ignored by Piketty 2014 when he focuses on the gap r g as the sole determinant of wealth inequality. This implicit assumption is criticized by Mankiw 2015 and Ray In a BGP, the aggregate labor income share and capital to output ratio are given by: EY ē F H 1, S H ē + δ k H W H + M H Q + F L 1, S L ē + δ k L W L + M L Q, 21

22 and KY W H + W L F H 1, S H ē + δ k H W H + M H Q + F L 1, S L ē + δ k L W L + M L Q. How are KY and EY related to each other? By looking at the distribution of output between wages and returns to capital, we obtain k H X k H X + k L 1 X R k L Hē + δ + 1 X k H X + k L 1 X R Lē + δ KY = 1 } {{ EY }. }{{} captital income share average rate of return to capital 28 Notice that this identity is a generalization of the Piketty 2014 s First Fundamental Law of Capitalism: rβ = α, in his model with homogeneous returns. In our model with heterogenous returns, capital income share is equal to the product of the weighted average of the rates of return to capital of the two types of agents and the capital to output ratio. 19 The following theorem links the top end inequality to the two aggregate statistics. Theorem 4 Sufficient Statistics II. Assume n = 0 and Assumption 2 holds and uc = log c. In a stationary BGP with k L > 0, top end inequality θ is a function of the aggregate labor income share, capital to output ratio, interest rate, together with the primitive parameters g x, δ, λ HL, λ LH, λ: In addition, ˆθ 1b KY > 0, ˆθ 1b EY > 0 and ˆθ 1b g x > 0. Proof. Appendix A. θ = ˆθ 1b EY, KY, r; gx, δ, λ HL, λ LH, λ. Because k L > 0, we are in Case 1 of Theorem 1 which implies that R H > R L, i.e. returns are strictly heterogenous. Therefore, formula 27 for homogenous returns does not apply. Consequently, the comparative statics with respect to KY also differ in the two cases: ˆθ 1a KY < 0, while ˆθ 1b KY > 0. The last two comparative statics in this theorem are consistent with Piketty 2014 s discussion: higher labor income share, or higher growth rate of the economy since n = 0, 19 The production functions F i implicitly incorporate depreciation. So to measure total output - gross output - we add back depreciation to the net output given by F i. This practice is more consistent with the modern formulation of the neoclassical growth models such as Cass Krusell and Smith 2015 show that the distinctions between gross output versus net output and gross saving rate versus net saving rate are important for the predictions of the neoclassical growth models. 22

23 g = g x, corresponds to higher Pareto tail index and thus lowers top end wealth inequality. However, the first comparative statics differ from Piketty 2014 s prediction that higher capital to output ratio is associated with higher wealth inequality. 20 Theorem 4 makes restrictive assumptions such as no population growth and log utility, however it is still fairly general since it allows for any production function as well as any degree of financial friction. For example, changing m certainly changes the equilibrium stationary BGP, as shown in Proposition 1 below. However, top end inequality only changes to the extent that EY, KY, and r change because of the change in m. Theorem 4 shows that in general top end inequality depends on factors other than the simple gap r g suggested by Piketty If the assumptions of Theorem 4 are not satisfied, one should expect that the determinants of the top end wealth inequality are fairly complex, and might not be summarized in a simple formula with simple aggregate statistics as inputs. 3 AK Growth Model In this section we assume that uc log c and F i k, l = A i k where A H > A L > 0. We further assume type-preserving wealth redistribution, i.e. Assumption 2 holds, and no population growth. These assumptions allow us to sharply characterize the balance growth paths as well as transitional dynamics. In Online Appendix G, we also characterize the dynamics of the economy under both idiosyncratic and aggregate shocks to investment returns. We show that the aggregate dynamics depend crucially on the process of idiosyncratic shocks. Although this section helps grasp some qualitative properties of the general model, impatient readers can skip directly to Section 4 in the first reading of the paper. 3.1 Stationary Equilibrium Theorem 1 does not tell us which case happens depending on the exogenous parameters of the model. Under log utility, the following proposition completely characterizes the 20 A casual observation of cross-country data on capital to income ratio and top end wealth inequality from Piketty 2014, Piketty and Zucman 2014, Alvaredo and Saez 2009, and Saez and Zucman 2016 suggests that countries with higher top end wealth inequality might have lower capital to income ratio. For example the U.S. has the highest level of top end wealth inequality, but has lower capital to income ratio, compared to France, Spain, and the U.K. Our preliminary cross-country regression of top 1% wealth share on KY, controlling for other factors, shows that the coefficient on KY is negative and significant. 23

24 equilibrium. 21 Proposition 1. In a stationary BGP, one of the following three cases happens: Case 1: Low m If then r = A L and X < 1 m. Case 2: Intermediate m If M L 1 + A H A L λ HL +λ LH > m > 0, 29 M L > m > M L 1 + A H A L λ HL +λ LH, 30 then and X = 1 m. Case 3: High m If r = A H 1 mλ HL mλ LH m A L, A H 1 > m > M L, 31 then r = A H and X > 1 m Proof. Appendix B.1. The case of log utility also allows us to determine in closed forms the Pareto tail index of the stationary wealth distribution which is a solution to a quadratic equation similar to 25. Therefore, we can also characterize how the degree of financial friction affects the tail index, or equivalently top end wealth inequality. Proposition 2 Financial Kutznet s Curve. In stationary BGPs, top end wealth inequality varies in m differently depending on which case in Proposition 1 the equilibrium belongs to: Case 1 Low m: Top end wealth inequality is increasing in m. Case 2 Intermediate m: Top end wealth inequality is decreasing in m. Case 3 High m: Top end wealth inequality is independent of m. Proof. Appendix B.1. Numerically, Moll 2012 finds a similar result in a production economy with a continuum of investment productivity types and we borrow the term Financial Kutznet s Curve from his analysis. As he explains, in Case 1, top end inequality is increasing in 21 One minor difference relative to Theorem 1 is that the growth rate of the economy is now totally endogenous, instead of being determined exogenously by g x and n. 24

25 m because of the leverage effect, i.e. higher m allows the more productive agents to borrow more at an interest rate determined by the rate of returns to the less productive agents, this magnifies the differences in returns and increases wealth inequality. However, at higher m, i.e., in Case 2, top end wealth inequality is decreasing because of the return equalization effect, i.e. higher m increases the demand to borrow by the more productive agents which pushes up the interest rate earned by the less productive agents which is strictly higher than their own rate of return. This increase in interest rate reduces the differences in returns and decreases wealth inequality. If m is even higher, one reaches Case 3 in which the interest rate earned by the less productive agents is pushed up to the rate of return of the more productive agents. There is essentially no wealth inequality in a stationary BGP in this case. As Piketty 2014 suggests, a powerful tool to reduce wealth inequality is wealth tax. In the following proposition, we consider a proportional wealth tax, τ > 0. We assume that the proceed from taxing wealth is distributed among the new borns. To simplify the analysis, we assume that the redistribution of wealth tax proceeds is distributed within the same type, i.e. tax proceeds from the high-type agents is distributed to the new borns of high type and similarly for low type. 22 The advantage of this assumption is that the aggregate dynamics do not change. However, the following result should not depend on the precise redistribution scheme. Proposition 3 Wealth Tax. Assume uc = log c. There exists τ > 0, such that the Pareto tail index is decreasing in τ for τ 0, τ. Proof. Appendix B.1. As also shown in the appendix, the effectiveness of the wealth tax in reducing wealth inequality depends on the fundamental of the model, such as m, λ HL, λ LH and λ. At first sight, this appears to be a trivial result. However, as shown in Jones 2015, under homogenous returns, wealth taxes do not affect top end wealth inequality because of the general equilibrium effect. Therefore, this proposition shows that heterogenous returns are important for the effectiveness of wealth taxes in reducing top end wealth inequality. 3.2 Transitional Path In this subsection, we investigate the dynamics of the economy over transitional paths. The initial conditions are such that the economy does not start out at a stationary equilib- 22 The formulation of the model with wealth tax is similar to the one with corporate tax presented in Online Appendix D. 25