Capital Taxation with Entrepreneurial Risk

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Capital Taxation with Entrepreneurial Risk Vasia Panousi 2010-56 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Capital Taxation with Entrepreneurial Risk Vasia Panousi Federal Reserve Board∗ October 22, 2010 Abstract Thispaperstudiesthee(cid:27)ectsofcapitaltaxationinadynamicheterogeneous-agenteconomy withuninsurableentrepreneurialrisk. Althoughitallowsforrichgeneral-equilibriume(cid:27)ectsand a stationary distribution of wealth, the model is highly tractable. This permits a clear analysis, not only of the steady state, but also of the entire transitional dynamics following any change in tax policies. Unlike either the complete-markets paradigm or Bewley-type models where idiosyncraticriskimpactsonlylaborincome,hereitisshownthatcapitaltaxationmayactually stimulate capital accumulation. This possibility emerges because of the general-equilibrium e(cid:27)ects of the insurance aspect of capital taxation. In particular, for the preferred calibrated version of the model, when the tax on capital is 25%, output per work-hour is 2.2% higher than it would have been had the tax rate been zero. Turning to the welfare e(cid:27)ects of a reform in capital taxation, it is examined how these e(cid:27)ects depend on whether one focuses on the steady state or also takes into account transitional dynamics, as well as how they vary in the cross-section of the population (rich versus poor, entrepreneurs versus non-entrepreneurs). ∗Email address: vasia.panousi@frb.gov. IamdeeplyindebtedtomyprimaryadvisorGeorge-MariosAngeletosfor hisconstantsupportandguidance. IamverygratefultomyadvisorsMikeGolosovandIvanWerningforextremely constructive feedback and discussions. I would like to thank Daron Acemoglu, Olivier Blanchard, V. V. Chari, Sylvain Chassang, Peter Diamond, Simon Gilchrist, Narayana Kocherlakota, Jiro Kondo, Dimitris Papanikolaou, James Poterba, JosØ-V(cid:237)ctor R(cid:237)os-Rull, Catarina Reis, Robert Townsend, Harald Uhlig and seminar participants at MIT,theFederalReserveBoard,theBankofPortugal,BernUniversity,GeorgetownUniversity,IndianaUniversity, the Kellogg School of Management, the New York Fed, the University of Notre Dame, Tufts University, the 2008 SED annual meeting, and the 2008 NSF/NBER Conference on General Equilibrium and Mathematical Economics at Brown University for useful comments. The views presented are solely those of the author and do not necessarily represent those of the Board of Governors of the Federal Reserve System or its sta(cid:27) members. 1

1 Introduction This paper studies the macroeconomic and welfare e(cid:27)ects of capital-income taxation in an environmentwhereagentsfaceuninsurableidiosyncraticinvestmentrisk. Suchriskisempiricallyimportant for entrepreneurs and wealthy agents, who, even though they represent a small fraction of the population, hold most of an economy’s wealth. In this context, capital taxation raises an interesting tradeo(cid:27) between the distortion of investment versus the provision of insurance against idiosyncratic capital-income risk. On the one hand, capital taxation comes at a cost, since it distorts agents’ saving decisions. On the other hand, it has bene(cid:28)ts, since it provides agents with partial insurance against idiosyncratic investment risk. This suggests that a positive tax on capital income could be welfare-improving, even if it reduced capital accumulation. Mostsurprisinglythough,itisshownthatapositivetaxoncapitalincomemayactuallystimulate capital accumulation. Indeed, the steady-state levels of the capital stock, output and employment mayallbemaximizedatapositivevalueofthecapital-incometax. Thispossibilityemergesbecause of the general-equilibrium e(cid:27)ects of the insurance aspect of capital taxation. This result stands in stark contrast to the e(cid:27)ect of capital taxation both under complete-markets models, and under incomplete-markets models with uninsurable labor-income risk alone. In these models, capitalincome taxation, irrespectively of whether it is welfare-improving or not, necessarily discourages capital accumulation. Model. This paper represents a (cid:28)rst attempt to study the e(cid:27)ects of capital-income taxation in a general-equilibrium incomplete-markets economy, where agents are exposed to uninsurable idiosyncratic investment risk. The framework builds on Angeletos (2007), who develops a variant of the neoclassical growth model that allows for idiosyncratic investment risk, and studies the e(cid:27)ects of such risk on macroeconomic aggregates. Agents own privately held businesses that operate under constant returns to scale. Agents are not exposed to labor-income risk, and they can freely borrow and lend in a riskless bond, but they cannot diversify the idiosyncratic risk in their private business investments. Abstracting from labor-income risk, borrowing constraints, and other market frictions, isolatestheimpactoftheidiosyncraticinvestmentriskandpreservesthetractabilityofthe general-equilibrium dynamics. The present model extends Angeletos’s model in the following ways. First, a government is introduced, imposing proportional taxes on capital and labor income, along with a non-contingent lump-sum tax or transfer. Second, agents have (cid:28)nite lives, which ensures the existence of a stationary wealth distribution. Third, there is stochastic, though exogenous, transition in and out of entrepreneurship, which helps capture the observed heterogeneity between entrepreneurs and non-entrepreneurs without the complexity of endogenizing occupational choice. Fourth, labor supply is endogenous. Clearly the (cid:28)rst element is essential for the novel contribution of the paper. The other three improve the quantitative performance of the model and demonstrate the robustness of the main result. 2

Preview of results. The main result of the paper is that an increase in capital-income taxation may actually stimulate capital accumulation. The intuition behind this result comes from recognizing that the overall e(cid:27)ect of the capital-income tax on capital accumulation can be decomposed in two parts. The (cid:28)rst part captures the response of capital to the tax in a setting with endogenous saving but exogenously (cid:28)xed interest rate. This is isomorphic to examining the e(cid:27)ects of the capital tax in a (cid:16)small open economy(cid:17). In this context, it is shown that an increase in the capital-income tax unambiguously decreases the steady-state capital stock. The second part, which is the core result of this paper, captures the importance of the general-equilibrium adjustment of the interest rate for wealth and capital accumulation. Here, an increase in the tax reduces the e(cid:27)ective variance of the risk agents are exposed to. This reduces the demand for precautionary saving, and therefore increases the interest rate, which in turn increases steady-state wealth. With decreasing absolute risk aversion, wealthier agents are willing to undertake more risk, and hence they will increase their investment in capital. In other words, the general-equilibrium e(cid:27)ect of the interest rate adjustment is a force that tends to increase investment and the steady-state capital stock. For plausible parameterizations of the closed economy, the general equilibrium e(cid:27)ect dominates for low levels of the capital-income tax, so that steady-state capital at (cid:28)rst increases with the tax. In particular, for the preferred calibrated version of the model, the steady-state capital stock is maximized when the tax on capital is 40%. So, for example, when the tax on capital is 25%, output per work-hour is 2.2% higher than what it would have been had the tax rate been zero. The result that the steady-state capital stock is inversely U-shaped with respect to the capital-income tax is robust for a wide range of empirically plausible parameter values. Furthermore, the tax that maximizes steady-state capital is increasing in risk aversion and/or the volatility of idiosyncratic risk. This (cid:28)nding reinforces the insurance interpretation of the tax system. Subsequently, the paper examines the aggregate and welfare e(cid:27)ects of eliminating the capitalincome tax. This is because an extensive discussion has been conducted within the context of the complete-marketsneoclassicalgrowthmodelaboutthewelfarebene(cid:28)tsofsettingthecapital-income tax to zero. In light of the main result here, revisiting this discussion is worthwhile. In particular, the aggregate and welfare e(cid:27)ects are presented from two di(cid:27)erent perspectives. On the one hand, one might be interested in examining the welfare of the current generation immediately after the policy reform, taking into account the entire transitional dynamics of the economy towards the new steady state with the zero tax. On the other hand, one might be interested in examining the welfare of the generations that will be alive in the distant future, i.e. at the new steady state with the zero tax. First, consider the macroeconomic e(cid:27)ects of eliminating the capital-income tax. When markets are complete, investment increases in the short run, and it is also higher at the new long-run steady state with the zero tax, compared to the old steady state with the positive tax. By contrast, in the present model of incomplete markets, investment falls in the short run, as well as in the long run. 3

Second, consider the welfare e(cid:27)ects of eliminating the capital-income tax. These vary across the di(cid:27)erenttypesofagents, thedi(cid:27)erentlevelsofwealth, andthecurrentversusthefuturegenerations. In the current generation, poor agents, whether entrepreneurs or non-entrepreneurs, prefer the zero tax. This is because most of their wealth comes from wage income, and, with capital (cid:28)xed, the present value of wages increases due to a fall in the interest rate. Rich agents, on the other hand, prefer a positive tax, since they bene(cid:28)t more from insurance provision. In the long run, all types of agents, and at all levels of wealth, prefer a positive tax on capital income. However, the cost of switching to a zero-tax regime is much higher for poorer than for wealthier agents of all types. This is because, in the long run, the elimination of the tax decreases the steady-state capital stock, thereby decreasing the present value of wages. Therefore poorer agents will su(cid:27)er the most, since human wealth constitutes a big part of their total wealth. Literature review. This paper focuses on entrepreneurial risk, because such risk is in fact empirically relevant, even in a (cid:28)nancially developed country like the United States. For example, Moskowitz and Vissing-Jłrgensen (2002) (cid:28)nd that 75% of all private equity is owned by agents for whomsuchinvestmentconstitutesatleasthalfoftheirtotalnetworth. Furthermore,85%ofprivate equity is held by owners who are actively involved in the management of their own (cid:28)rm.1 Given this evidence about the US, one expects that entrepreneurial risk must be even more prevalent in less developed economies, where a large part of production takes place in small unincorporated businesses and where risk-sharing arrangements are much more limited. Furthermore, idiosyncratic investment risk need not be interpreted as a(cid:27)ecting solely the owners of privately held businesses. In recent work, Panousi and Papanikolaou (2008) (cid:28)nd a signi(cid:28)cant and robustnegativerelationshipbetweenidiosyncraticriskandtheinvestmentofpubliclytraded(cid:28)rmsin the US. In addition, they show that this relationship is stronger in (cid:28)rms where the insider mangers hold a larger fraction of the (cid:28)rm’s shares, and they provide evidence for a possible explanation that has to do with managerial risk aversion. Combined with the work of Moskowitz and Vissing- Jłrgensen (2002), this demonstrates that a large fraction of total investment in the US, whether by publicly traded or privately held businesses, is sensitive to idiosyncratic risk, and therefore strengthens the empirical applicability of the present model setup. Thispaperrelatestothestrandofthemacroeconomicliteraturediscussingoptimaltaxationand the e(cid:27)ects of taxation. However, most of this literature has focused on labor income risk. Chamley (1986) and Judd (1985) (cid:28)rst established the result of zero optimal capital taxation in the long run when markets are complete. Atkeson, Chari and Kehoe (1999) generalized this result to most of the short run for an interesting class of preferences, and to the case of (cid:28)nitely lived agents. Aiyagari (1995) extended the complete-markets framework to include uninsurable labor income risk and borrowing constraints. In this context, when only a limited set of policy instruments are available, 1FurtherevidencefortheseobservationsisalsoprovidedbyQuadrini(00),Carroll(02),GentryandHubbard(00), and Cagetti and DeNardi (06). 4

it becomes optimal to tax capital in the long run: a positive capital tax increases welfare, but it unambiguously lowers the level of the capital stock.2 A related but di(cid:27)erent normative exercise is conducted by Davila et al. (2007). They examine constrained e(cid:30)ciency, in the spirit of Geanakoplos-Polemarchakis, within a version of Aiyagari’s model. This exercise does not allow for risk-sharing through taxes or any other instruments, and insteadconsidersane(cid:30)ciencyconceptwheretheplannerdirectlydictatestotheagentshowmuchto investandtotrade. AngeletosandWerning(2006)examineasimilarconstrainede(cid:30)ciencyproblem in a two-period version of a model with idiosyncratic investment risk. Albanesi (2006) considers optimaltaxationinatwo-periodmodelofentrepreneurialactivity,inaconstrainede(cid:30)ciencysetting, and following the Mirrlees optimal policy tradition. The bene(cid:28)t of her approach is that the source of incomplete risk-sharing is endogenously speci(cid:28)ed as the result of a private information (moral hazard) problem, and that there are no ad hoc restrictions placed on the tax instruments. However, hermodeldoesnotallowfordynamics,forlong-runconsiderations,orforgeneral-equilibriume(cid:27)ects like those studied in the present paper. In general, the extensive theoretical work on taxation originating from the Mirrlees and the new dynamic public (cid:28)nance tradition focuses on labor-income risk, as does the literature that examines the optimal progressivity of the tax code.3 The growing literature on the e(cid:27)ects of borrowing constraints on entrepreneurial choices has examined policy questions, and especially the implications of replacing a progressive with a proportionalincome-taxschedule, inanAiyagari-typeenvironment, i.e. withdecreasingreturnstoscaleat the individual level, borrowing constraints, and undiversi(cid:28)able labor income risk. Some examples in thisareaincludeLi(2002), DomeijandHeathcote(2003), Meh(2005), CagettiandDeNardi(2007), and Kitao (2007). Benabou (2002) develops a tractable dynamic general-equilibrium model of humancapitalaccumulationwithendogenouse(cid:27)ortandmissingcreditandinsurancemarkets. Within this framework he examines the long-run tradeo(cid:27)s of progressive taxation and education (cid:28)nance. Finally, Erosa and Koreshkova (2007) examine the long-run e(cid:27)ects of switching from progressive to proportional income taxation in a quantitative dynastic model of human capital. This paper also relates to the branch of the public (cid:28)nance literature that considers the e(cid:27)ects of capital taxation on portfolio allocation and risk-taking. Domar and Musgrave (1944) (cid:28)rst proposed the idea that proportional income taxation may increase risk-taking, by having the government 2Alvarez et al. (1992), Erosa and Gervais (2002), and Garriga (2003), show that in life-cycle models the optimal capital-income tax is in general di(cid:27)erent from zero, at least if the tax code cannot explicitly be conditioned on the age of the household. Conesa et al. (2008) quantitatively characterize the optimal capital- and labor- income tax inanoverlapping-generationsmodelwithidiosyncraticuninsurablelabor-incomeshocksandpermanentproductivity di(cid:27)erences across households, and (cid:28)nd for an optimal capital-income tax of 36%. Uhlig and Yanagawa (1995) show that,undermildconditions,highercapital-incometaxesleadtofastergrowthinanoverlapping-generationseconomy with endogenous growth. It should be noted, however, that the results of the present paper do not depend on a life-cycle or overlapping-generations setup. Instead, they arise in the context of the standard neoclassical framework of in(cid:28)nitely-lived agents. 3Some examples here include Golosov et al. (2003), Albanesi and Sleet (2005), Conesa and Krueger (2006), Werning (2007), and Reiter (2004). 5

bear part of the risk facing the agents.4 This idea was formalized by Stiglitz (1969), within a twoperiod single-agent model, where asset returns and the level of saving are exogenously given, but where the agent optimally chooses the allocation of his (cid:28)xed amount of saving between a risky and a riskless asset. Ahsan (1974) extended Stiglitz by endogenizing the intertemporal consumptionsaving decision in a two-period model. He showed that the partial-equilibrium e(cid:27)ect of capitalincome taxation on risk-taking is in general ambiguous. By contrast, in the (cid:16)small open economy(cid:17) version of the present model, which di(cid:27)ers from Ahsan’s in that the horizon is in(cid:28)nite and the return to capital is endogenous, it is shown that the steady-state capital stock is decreasing in the capital-income tax. This (cid:28)nding highlights that the results here are driven by, novel to the literature, general-equilibrium e(cid:27)ects. Asalreadymentioned,thepresentmodelbuildsonAngeletos(2007),whoabstractedfrompolicy questions and considered instead the e(cid:27)ect of investment risk on macroeconomic aggregates. The contributionofthepresentpaperistostudythee(cid:27)ectsofcapital-incometaxationonaggregatesand welfare. AngeletosandPanousi(2009),inaframeworkliketheoneinAngeletos(2007),examinethe e(cid:27)ects of government spending on macroeconomic aggregates, but for the case where government spending is (cid:28)nanced exclusively through lump-sum taxation. The rest of the paper is organized as follows. Section 2 presents the model. Section 3 describes individual behavior and the aggregate equilibrium dynamics. Section 4 characterizes the steady state in terms of aggregates and distributions. Section 5 presents and discusses the main theoretical result. Section 6 presents the calibration methodology and the parameter choices, along with the implications of the preferred calibrated model for aggregates and distributions. Section 7 quanti(cid:28)es thesteady-statee(cid:27)ectsofcapitaltaxation,aswellastheshort-runandlong-rune(cid:27)ectsofeliminating the capital-income tax. Section 8 examines the robustness of the results to the availability of a safe asset in positive net supply. Section 9 concludes. All proofs are delegated to the appendix. 2 The Model Time is continuous and indexed by t ∈ [0,∞). There is a continuum of agents distributed uniformly over[0,1]. Ateachpointintimeanagentcanbeeitheranentrepreneur, denotedbyE, oralaborer, denoted by L. The probabilities of switching between these two types are exogenous. In particular, the probability that an agent will switch from being an entrepreneur to being a laborer is p dt, EL and the probability that he will switch from being a laborer to being an entrepreneur is p dt. The LE measure of entrepreneurs in the economy at time t is denoted by χ . t Inwhatfollows, andforexpositionalsimplicity, laboristakentobeexogenous. Alloftheproofs, which are delegated to the appendix, and all of the calibrations, will consider the general case of endogenous labor, where preferences are homothetic between consumption and leisure, i.e. they are 4Sandmo (1977) extended this idea to the case of multiple risky assets. 6

of the King-Plosser-Rebelo (1988) speci(cid:28)cation. 2.1 Preferences All agents are endowed with one unit of time. Preferences are Epstein-Zin over consumption, c, and they are de(cid:28)ned as the limit, for ∆t → 0, of 5: U t = { (1−e−β∆t) c t 1−1/θ + e−β∆t (E t [ U t 1 + − ∆ γ t ] ) 1 1 − − 1 γ /θ } 1− 1 1/θ , (1) where β > 0 is the discount rate, γ > 0 is the coe(cid:30)cient of relative risk aversion, and θ > 0 is the elasticity of intertemporal substitution. For θ = 1/γ, this reduces to the case of standard expected utility, U = E (cid:82)∞ e−βsU(c )ds, where U(c ) = c1 t −1/θ . t t t s t 1−1/θ 2.2 Entrepreneurs When an agent is an entrepreneur, he owns and runs a (cid:28)rm operating a constant-returns-to-scale neoclassical production function F(k,l), where k is capital input and l is labor input. An entrepreneur can only invest in his own (cid:28)rm’s capital, although he supplies and employs labor in the competitive labor market. Capital investment in his (cid:28)rm is subject to uninsurable risk. The idiosyncratic shocks are i.i.d., hence there is no aggregate uncertainty. An entrepreneur can also save in a riskless bond. The (cid:28)nancial wealth of an entrepreneur i, denoted by xi, is the sum of his holdings in private t capital, ki, and the riskless bond, bi: t t xi = ki+bi. (2) t t t The evolution of xi is given by: t dxi = (1−τK) dπi+[ (1−τK)R bi +(1−τL)ω +T −ci ]dt, (3) t t t t t t t t t t where dπi are (cid:28)rm pro(cid:28)ts (capital income), R is the interest rate on the riskless bond, τK is the t t t proportionalcapital-incometax,ω isthewagerateintheaggregateeconomy,τL istheproportional t t labor-income tax, T are non-contingent lump-sum transfers received from the government, and ci t t is consumption. Finally, a no-Ponzi game condition is imposed. Firm pro(cid:28)ts are given by: dπi = [ F(ki,li)−ω li−δki ]dt + σkidzi, (4) t t t t t t t t where F(k,l) = kαl1−α with α ∈ (0,1), and δ is the mean depreciation rate in the aggregate economy. Idiosyncratic risk is introduced through dzi, a standard Wiener process that is i.i.d. t 5Lemma 1 in the appendix gives the formal description of preferences. 7

across agents and across time6. The scalar σ measures the amount of undiversi(cid:28)ed idiosyncratic risk, and is an index of market incompleteness, with higher σ corresponding to a lower degree of risk-sharing, and σ = 0 corresponding to complete markets. 2.3 Laborers When an agent is a laborer, he cannot invest in capital, and he can only hold the riskless bond. He also supplies labor in the competitive labor market. Financial wealth for a laborer i is therefore: xi = bi, (5) t t and its evolution is given by: dxi = [ (1−τK)R bi +(1−τL)ω +T −ci ]dt. (6) t t t t t t t t 2.4 Government At each point in time the government taxes capital and bond income at the rate τK, and labor int comeattherateτL. Partofthetaxproceedsisusedbythegovernmentforownconsumptionatthe t deterministic rate G . Government spending does not a(cid:27)ect the utility from private consumption t or the production technology. The remaining tax proceeds are then distributed back to the households in the form of non-contingent lump-sum transfers, T . The government budget constraint is t therefore: (cid:90) (cid:90) (cid:90) 0 = [ τLF ( ki,1)+τK(F ( ki,1)−δ) ki−G −T ] dt, (7) t Lt t t Kt t t t t i i i where F ( (cid:82) ki,1) is the marginal product of capital in the aggregate economy, F ( (cid:82) ki,1) is the Kt i t Lt i t marginal product of labor, and (cid:82) li = 1. i t 2.5 Finite lives and annuities All households face a constant probability of death, with Poisson arrival rate vdt at every instant in time.7 There is no intergenerational altruism linking a household to its descendants, and utility 6Idiosyncratic risk is modeled here as coming from uninsurable i.i.d. depreciation shocks. However these shocks could also be modeled as or interpreted as i.i.d. productivity shocks. 7The(small)positiveprobabilityofdeathisintroducedinordertoguaranteetheexistenceofastationarywealth distribution. In general, with (cid:28)nite lives and no altruism, Ricardian equivalence might fail, since some of the tax burdenassociatedwiththecurrentissueofabondisbornebyagentswhoarenotalivewhenthebondisissued. Here, for v = 0, Ricardian equivalence holds, because all agents can freely borrow in the riskless bond. The theoretical steady-stateresultsfortheaggregatesarederivedforv=0,andtheycarrythroughforvsmallbutpositive. However, it might still be the case that the dynamic e(cid:27)ects of time-varying policy changes possibly depend on the validity of Ricardianequivalence. Nonetheless,forthepurposesofthispaper,thegovernmentbudgetconstraintwillbewritten as in (7) for v positive but small. 8

is zero after death. The discount rate in preferences can then be reinterpreted as β = β˜+v, where β˜ is the psychological or subjective discount rate and v is the probability of death8. In order to isolate the e(cid:27)ects of capital-income risk, it is assumed that there exist annuity (cid:28)rms permitting all agents to get insurance against mortality risk, by freely borrowing the entire net present value of their future labor income. As a result, all agents have (safe) human wealth, denoted by h , and de(cid:28)ned as the present discounted value of their net-of-taxes labor endowment t plus government transfers:9 (cid:90) ∞ h t = e− (cid:82) t s((1−τ j K)Rj+v)dj((1−τ s L)ω s +T s )ds. (8) t Then, thetotale(cid:27)ectivewealth, wi, foranagentisde(cid:28)nedasthesumofhis(cid:28)nancialandhuman t wealth, i.e. wi ≡ xi +h . Hence, e(cid:27)ective wealth for an entrepreneur is given by: t t t wi = ki+bi +h , (9) t t t t and e(cid:27)ective wealth for a laborer is given by:10 wi = bi +h . (10) t t t 3 Equilibrium This section characterizes individual behavior and the general equilibrium in the economy. The analysis will be in closed-form, since, as will be shown, the wealth distribution is not a relevant state variable for the characterization of aggregate equilibrium dynamics. 3.1 Individual Behavior Entrepreneurschooseemploymentaftertheircapitalstockhasbeeninstalledandtheiridiosyncratic shock has been observed. Hence, since their production function, F, exhibits constant returns to 8Sinceutilityiszeroafterdeath,thisisavalidinterpretationthatdoesnotviolatetheaxiomsofexpectedutility. 9Let h˙ t =(R t +v)h t −ω t, and b t =−h t. Then, b˙ t =R t b t −vh t +ω t. These equations are consistent with each otherandwithmarketclearing,andtheyhavetwoalternativebutisomorphicinterpretations. First,inthebeginning of time, every agent borrows from annuity (cid:28)rms an amount equal to his entire human wealth. From then on, he repays this debt every period by giving up his wage plus interest to the annuity (cid:28)rms, and this only stops when he dies. Second, the annuity (cid:28)rms borrow from the agent his entire human wealth, and every period from then on they repay the agent by giving him wage plus interest, until the agent dies. Either of these interpretations is consistent with the analysis here. 10It is assumed that capital is fully fungible upon exit from entrepreneurship. The assumption of exogenous transitionprobabilitiesismaintainedherefortractability,inordertoensureaclosed-formsolution. Thisassumption could have the interpretation that, at some random point in time, the agent is given the chance to operate a highreturn,high-risktechnology,whileatsomeotherrandompointintimetheoptiontosaveinthisalternativetechnology is taken away (for example, the agent has an idea which depreciates at some exogenous rate). 9

scale, optimal (cid:28)rm employment and optimal pro(cid:28)ts are linear in own capital: li = l(ω )ki and dπi = r(ω )kidt+σkidzi, (11) t t t t t t t t where l(ω ) ≡ argmax [F(1,l)−ω l] and r(ω ) ≡ max [F(1,l)−ω l]−δ. Here, r ≡ r(ω ) is an t l t t l t t t entrepreneur’s expectation of the return to his capital prior to the realization of his idiosyncratic shock, as well as the mean of the realized returns in the cross-section of (cid:28)rms. The key result here is that entrepreneurs face risky, but linear, returns to their investment. The evolution of e(cid:27)ective wealth for an entrepreneur is described by: dwi = [ (1−τK)r ki+(1−τK)R (bi +h )−ci ]dt+σ(1−τK)kidzi. (12) t t t t t t t t t t t t The (cid:28)rst term captures the expected rate of growth of e(cid:27)ective wealth, and it shows that wealth grows when the total return to saving for an entrepreneur exceeds consumption expenditures. The secondtermcapturestheimpactofidiosyncraticrisk. Theevolutionofe(cid:27)ectivewealthforalaborer is described by: dwi = [ (1−τK)R (bi +h )−ci ]dt. (13) t t t t t t Let the fraction of e(cid:27)ective wealth an agent saves in the risky asset be: ki φi ≡ t . (14) t wi t Let an agent’s marginal propensity to consume out of e(cid:27)ective wealth be: ci mi ≡ t . (15) t wi t Let µ = (1−τK)r −(1−τK)R denote the risk premium. Since investment in capital is risky, t t t t t it has to be the case that r > R , otherwise no one would invest in capital. In other words, t t agents require a positive risk premium as compensation for undertaking capital investment. Let ρ ≡ φ (1 − τK)r + (1 − φ )(1 − τK)R denote the net-of-tax mean return to saving for an t t t t t t t entrepreneur, and let ρˆ ≡ ρ −1/2γφ2σ2(1−τK)2 denote the net-of-tax risk-adjusted return to t t t t saving for an entrepreneur. The net-of-tax return to saving for a laborer is simply (1 − τK)R . t t Then, since R < r , it has to be that (1−τK)R < ρˆ < ρ < (1−τK)r . t t t t t t t t Because of the linearity in assets of the budget constraints (12) and (13), and the homotheticity ofthepreferences,theoptimalindividualpolicyruleswillbelinearintotale(cid:27)ectivewealth,forgiven prices and government policies. Hence, for given prices and policies, an agent’s consumption-saving problem reduces to a tractable homothetic problem as in Samuelson’s and Merton’s classic portfolio analysis. Optimal individual behavior is then characterized by the following proposition. 10

Proposition 1. Let {ω ,R ,r } and {τK,τL,T ,G } be equilibrium price and policy t t t t∈[0,∞) t t t t t∈[0,∞) sequences. If an agent i is an entrepreneur, his optimal consumption, investment, portfolio, and bond holding choices, respectively, are given by: (1−τK)r −(1−τK)R ci = mEwi, ki = φ wi, φ = t t t t , bi = (1−φ )wi−h . (16) t t t t t t t γσ2(1−τK)2 t t t t t If an agent i is a laborer, his optimal consumption, investment, and bond holding choices, respectively, are given by: ci = mLwi, ki = 0, bi = wi−h . (17) t t t t t t t The marginal propensities to consume satisfy the following system of ordinary di(cid:27)erential equations: m m ˙ t E E = mE t −θβ+(θ−1)ρˆ t + 1 θ − − γ 1 p EL [ ( m m E L ) 1 1 − − γ θ −1 ] (18) t m m ˙ t L L = mL t −θβ+(θ−1)(1−τ t K)R t + 1 θ − − γ 1 p LE [ ( m m E L ) 1 1 − − γ θ −1 ]. (19) t From (16) and (17) it is clear that optimal consumption is a linear function of total e(cid:27)ective wealth, where the marginal propensity to consume depends only on the type of the agent, and not on the level of wealth. In other words, all entrepreneurs share a common marginal propensity to consume, mE, and all laborers share a common marginal propensity to consume, mL. The fraction t t φ of wealth invested in the risky asset by an agent who happens to be an entrepreneur is increasing t in the risk premium, decreasing in risk aversion, and decreasing in the e(cid:27)ective variance of risk, σ(1 − τK). Because of homotheticity and linearity, φ is the same across all entrepreneurs, and t t independent of the level of wealth. The policy for optimal bond holdings follows from (9) or (10), and(14). Thesystemof(18)and(19)isasystemoftwoEulerequations. Itshowsthatthemarginal propensities to consume, conditional on being an entrepreneur or a laborer, depend on two factors. First, on the process of the corresponding net-of-tax anticipated (risk-adjusted) returns to saving, in accordance with whether the elasticity of intertemporal substitution, θ, is higher or lower than 1. Second, on the probability that the agent might switch between being an entrepreneur and being a laborer. 3.2 General equilibrium Theinitialpositionoftheeconomyisgivenbythedistributionof(ki,bi)acrosshouseholds. Anequi- 0 0 librium is a deterministic sequence of prices {ω ,R ,r } , a deterministic sequence of policies t t t t∈[0,∞) {τK,τL,T ,G } ,adeterministicmacroeconomicpath{C ,K ,Y ,L ,W ,WE,WL} ,and t t t t t∈[0,∞) t t t t t t t t∈[0,∞) a collection of individual contingent plans ({ci,li,ki,bi,wi} ) for i ∈ [0,1], such that the folt t t t t t∈[0,∞) lowing conditions hold: (i) given the sequences of prices and policies, the plans are optimal for the 11

households; (ii) the labor market clears, (cid:82) li = 1, in all t; (iii) the bond market clears, (cid:82) bi = 0, in i t t t all t; (iv) the government budget constraint (7) is satis(cid:28)ed in all t; and (v) the aggregates are consistent with individual behavior, C = (cid:82) ci, L = (cid:82) li = 1, K = (cid:82) ki, Y = (cid:82) F(ki,li) = F( (cid:82) ki,1), t i t t i t t i t t i t t i t W = (cid:82) wi, WE = (cid:82) wi, and WL = (cid:82) wi , in all t. t i t t i,E t t i,L t Because individual consumption and investment are linear in individual wealth, aggregates at any point in time do not depend on the extend of wealth inequality at that time. Therefore here, in contrast to other incomplete-markets models, it is not the case that the entire wealth distribution is a relevant state variable for aggregate dynamics. In fact, for the determination of aggregate dynamics, it su(cid:30)ces to keep track of the mean of aggregate wealth, and of the allocation of total wealth between the two groups of agents. To that end, let the fraction of total e(cid:27)ective wealth held by entrepreneurs in the economy be: WE λ ≡ t . (20) t W t The aggregate equilibrium dynamics can then be described by the following recursive system. Proposition 2. In equilibrium, the aggregate dynamics satisfy: W˙ /W = λ (ρ −mE)+(1−λ )((1−τK)R −mL) (21) t t t t t t t t t 1 λ˙ /λ = (1−λ )φ µ +(1−λ )(mL−mE)+p ( −1)−p (22) t t t t t t t t LE λ EL t H˙ = ((1−τK)R +v)H −(1−τL)ω −(τLω +τK(F −δ)K −G ) (23) t t t t t t t t t Kt t t φ λ t t K = H , (24) t t 1−φ λ t t along with (18) and (19). Equation (21) shows that the evolution of total e(cid:27)ective wealth is a weighted average of two terms. The (cid:28)rst term is positive when the mean net-of-tax return to saving for entrepreneurs exceeds their marginal propensity to consume, and is weighted by the fraction of total wealth the entrepreneurs hold in the economy. The second term is positive when the net-of-tax return to saving for laborers exceeds their marginal propensity to consume, and is weighted by the fraction of total wealth the laborers hold in the economy. Equation (22) shows the endogenous evolution of the relativedistributionofwealthbetweenthetwogroupsofagents. Theevolutionofλdependsonthree factors. First, on the di(cid:27)erential excess return the entrepreneurs face on their saving, which is given by φ µ , where φ is the fraction of wealth invested in the risky asset, and µ is the risk premium. t t t t Second, on the di(cid:27)erence in the level of saving between entrepreneurs and laborers, as captured by the di(cid:27)erence in the marginal propensities to consume, mL −mE. Third, on the adjustment t t made for the transition probabilities. Note here that the evolution of consumption can be recovered by aggregating across individual optimal policies, so that CE = mEWE and CL = mLWL, and t t t t t t using (18), (19), (21), and (22). Equation (23) shows the evolution of total human wealth, using the 12

governmentbudgetconstraintT = τLω +τK(F −δ)K −G , whereF isthemarginalproduct t t t t Kt t t Kt of capital in the aggregate production function F(K,1), and where ω = F (K ,1) from market t Lt t clearing. Since W˙ = K˙ + H˙ , the resource constraint of the economy is also satis(cid:28)ed. Equation (24) is the bond market clearing condition. It comes from aggregating across individual capital and bond choices as given in (16) and (17), adding up, using BE+BL = 0, and using (20). From (24) it t t follows that, for given prices and human wealth, a decrease in λ decreases K. A fall in λ indicates that the entrepreneurs on average now borrow more from the laborers, hence their wealth will on average be lower. With decreasing absolute risk aversion, this will negatively a(cid:27)ect their willingness to take risk, and therefore investment and the capital stock will fall for given prices. 3.3 Steady state: characterization of aggregates A steady state is a competitive equilibrium as de(cid:28)ned in section 3.2, where prices, policies, and aggregates are time-invariant. For expositional purposes, and to illustrate that the results about the e(cid:27)ects of capital-income taxation on the aggregates are not due to the presence of two types of agents or to the probability of death, section 3.3 (as well as section 4 later on) will consider the case with λ = 1 and v = 011. However, section 3.4 will characterize the invariant distributions for the general case. The steady state is the (cid:28)xed point of the dynamic system in Proposition 2. Let government spending,G,beparameterizedasafractiongoftaxrevenue. Thefollowingpropositioncharacterizes the steady state. Proposition 3. (i) The steady state always exists and is unique. (ii) In steady state, the capital stock, K, and the interest rate, R, are the solution to: (cid:114) 2θγσ2 F (K)−δ = R+ (β−(1−τK)R) (25) K θ+1 φ(K,R) (1−τL)ω(K)+(1−g)(τLω(K)+τK(F (K)−δ)K) K K = , (26) 1−φ(K,R) (1−τK)R where F (K) is the marginal product of capital and ω(K) is the wage rate in the aggregate economy. K From (18) or (19) and (21) in steady state, and using the fact that φµ = (F −δ −R)2/γσ2, K we get equation (25). This condition gives the combinations of K and R that are consistent with wealth and consumption stationarity. Using (24) and (23) in steady state yields equation (26). This condition gives the combinations of K and R that are consistent with stationarity of human wealth and bond market clearing. 11The more general case is left for the appendix. 13

Atthispointitisusefultobrie(cid:29)ycomparethesteadystatetoitscomplete-marketscounterpart. From (25) note that the di(cid:27)erence from complete markets, in which case it would be F (K)−δ = K R, is the presence of the square-root term, which captures the risk premium, i.e. here µ(R) = (cid:112) 2 θ γσ2(β−(1−τK)R) / (θ+1) (cid:54)= 0. In other words, agents here require a (private) risk premium in order to invest in capital. In addition, combining (18) or (19) with (21), and using the fact that C = mW, we get C˙/C = θ (ρˆ −β) + 1 γ φ2 σ2 (1−τK)2, from which, in steady state, t 2 t t t we conclude that: 1 γ ρˆ = β − φ2 σ2 (1−τK)2 . (27) 2 θ Inotherwords,therisk-adjustedreturntosavingmustbejustlowenoughtoo(cid:27)settheprecautionary saving motive,12 which is present here because agents face risk in their consumption stream. Since (1−τK)R < ρˆ, it follows that (1−τK)R < β, i.e. the net interest rate is lower than it would have been under complete markets. This result is also true in Aiyagari (1994) and in other Bewley-type models, with the di(cid:27)erence that in Bewley models it is labor-income that introduces the risk in the consumption stream. Furthermore here, because F −δ > R, it could be the case either that K F − δ > β or F − δ < β. Hence, capital can be either lower or higher than under complete K K markets.13 This is in contrast to the e(cid:27)ects of labor-income risk on steady-state capital, and it is due to the fact that idiosyncratic investment risk introduces a wedge (the risk premium) between the return to the risky asset and the return to the riskless asset. 3.4 Steady state: characterization of invariant distributions At each point in time, agents die and are replaced by newborn agents, and the assumption is that the newborn agents are endowed with the wealth of the exiting agents.14 This force generates mean reversion and guarantees the existence of an invariant wealth distribution. Let ξi ≡ wi/W denote t t t the distance between individual and aggregate e(cid:27)ective wealth. Let Φ and Φ be the condi- L E tional invariant distributions for laborers and entrepreneurs respectively. The following proposition characterizes the invariant distributions. Proposition 4. The conditional invariant distributions Φ and Φ are characterized by the follow- L E 12Iftherisk-adjustedreturnwerehigherthanthiscriticallevel,consumption(andwealth)wouldincreaseovertime without bound, which would be a contradiction of steady state. Conversely, if the risk-adjusted return were lower thanthislevel,consumption(andwealth)wouldshrinktozero,whichwouldonceagainbeacontradictionofsteady state. 13Angeletos (2007) gives a condition that determines whether steady-state capital is higher or lower than under complete markets, and quanti(cid:28)es the e(cid:27)ects of idiosyncratic capital-income risk on steady-state aggregates. 14Hence, from a law of large numbers, each agent starts life with the sum of human wealth plus the mean wealth in the economy. 14

ing second order linear di(cid:27)erential system: ∂Φ L 0 = κ ξ + κ Φ + p Φ , 1 2 L EL E ∂ξ ∂2Φ ∂Φ 0 = κ ξ2 E + κ ξ E + κ Φ + p Φ , 3 ∂ξ2 4 ∂ξ 5 E LE L where κ ,κ ,κ ,κ ,κ are constants determined by steady-state aggregates. 1 2 3 4 5 The point to note here is that the tractability of the model allows for a very detailed characterization of the invariant distributions. This is particularly useful for the case of entrepreneurs, since it is reasonable to expect that the distribution of wealth over entrepreneurs will be, to a large extent, determined by the realization of entrepreneurial returns.15 4 Steady-State E(cid:27)ects of Capital Taxation This section presents the core of the contribution of this paper, which is the study of the steadystate e(cid:27)ects of capital-income taxation. Again, for illustration purposes, the assumption is that λ = 1 and v = 0. The main result here is that an increase in the capital-income tax may actually increaseinvestmentandthesteady-statecapitalstock. Thispossibilityarisesbecauseofthegeneralequilibrium e(cid:27)ects of the insurance aspect of capital taxation, which operate mainly through the endogenous adjustment of the interest rate. In order to illustrate this, the analysis will proceed by making the distinction between the case where the interest rate is (cid:28)xed, and the case where the interest rate is allowed to adjust endogenously. Note then that equation (25) expresses capital, K, as a function of the tax, τK, and the interest rate, R. If the interest rate were (cid:28)xed,16 then the steady-state capital stock would be Ko(τK,R), as given by (25), and where both τK and R are exogenous. Next, by plugging Ko(τK,R) from (25) into(26), wecansolvefortheclosed-economysteady-stateinterestrate, asafunctionofthecapitalincome tax. Let Rc(τK) denote the closed-economy solution for the interest rate. It follows then, that the closed-economy steady-state capital stock will be given by Kc(τK) = Ko(τK, Rc(τK)). Hence, the impact of the capital-income tax on the closed economy steady-state capital stock can be decomposed in two parts. The (cid:28)rst part describes how steady-state capital changes with the tax when the interest rate is kept constant or exogenously (cid:28)xed. The second part describes the general-equilibrium adjustment of the interest rate in the closed economy, and the subsequent e(cid:27)ects of this adjustment on capital accumulation. Thus, the total e(cid:27)ect of the capital-income tax 15WhereasthetractabilityoftheaggregatesfollowsfromAngeletos(2007),theresultaboutthetractabilityofthe invariant distributions is novel to the present paper. 16This would be the case, for example, in a (small) open-economy version of the present model. This would be an economy with the same preferences, technologies, and risks, but which is open to an international market for the riskless bond, thus facing an exogenously (cid:28)xed interest rate. 15

on the closed-economy steady-state capital stock can be decomposed as follows: dKc ∂Ko ∂Ko dRc = + , (28) dτK ∂τK ∂R dτK where the (cid:28)rst term is the e(cid:27)ect when the interest rate is (cid:28)xed, and the second term is the e(cid:27)ect whentheinterestrateisallowedtoadjust, i.e. itistheclosed-economyorgeneral-equilibriume(cid:27)ect. Let’s (cid:28)rst turn to the (cid:28)xed-interest rate e(cid:27)ect. The following corollary characterizes the e(cid:27)ect of capital-income taxation on capital accumulation when the interest rate is held constant. Corollary 1. When the interest rate is exogenously (cid:28)xed, an increase in the capital-income tax unambiguously reduces the steady-state capital stock, i.e. ∂Ko/∂τK < 0. This result follows immediately from (25), for a given R. Hence, when the interest rate is kept constant, capital falls with the tax, despite a direct insurance aspect of the tax that is still present, namelythatthetaxreducesthevarianceofnetreturns, σ(1−τK). Clearlythen, foragiveninterest rate, this channel is not strong enough to outweigh the distortionary e(cid:27)ect of capital taxation on investment. This result stands in contrast to the (cid:28)ndings of Ahsan (1974). Ahsan considers the simultaneousdeterminationofthesizeandthecompositionoftheoptimalportfolio, inatwo-period model with exogenous returns. He shows that the e(cid:27)ect of an increase in capital-income taxation on risk-taking and capital is in general ambiguous.17 The result here indicates that, once Ahsan’s setting is extended to incorporate endogenous capital return and in(cid:28)nite horizon, the ambiguity disappears and capital taxation always leads to a fall in the steady-state capital stock. It is then clear that, in addition to the direct insurance role of the tax, the endogenous adjustment of the interest rate is also required for the e(cid:27)ect of capital taxation on capital to become ambiguous once again. Let’s now turn to the general-equilibrium e(cid:27)ect, which captures the fact that in the closed economy the interest rate endogenously adjusts to clear the bond market, according to equation (26). This e(cid:27)ect further consists of two parts. First,anincreaseinthecapital-incometaxreducesthee(cid:27)ectivevolatilityofriskforentrepreneurs, σ(1−τK), and this is the direct insurance e(cid:27)ect mentioned above. As a result, the interest rate, which is below the discount rate in steady state, increases, essentially because of a reduction in the demand for precautionary saving, i.e. dRc/dτK > 0.18 In fact, the increase in the interest rate is so high, that the net interest rate, R(1 − τK), ends up increasing, despite the increase in the capital-income tax. 17Ahsan’s result is, in turn, a generalization of Stiglitz (1969), who examines the e(cid:27)ects of proportional capitalincome taxation in a two-period model, taking not only returns, but also the level of saving as exogenously given. 18Thisintuitiveresulthasnotbeenproveninthecontextofthein(cid:28)nitehorizonmodel,althoughaproofisavailable for the two period version of the closed economy, for small τK. There, it can be shown in closed-form that steadystate capital is inversely U-shaped with respect to the capital-income tax. Nonetheless, simulations show that in the in(cid:28)nite-horizon closed-economy model the net interest rate is always increasing in the tax, as section 6.1 will demonstrate. 16

Second, this increase in the (net) interest rate will generate two opposing e(cid:27)ects on saving and wealth accumulation, as can be seen from (25). On the one hand, an increase in the interest rate increases the opportunity cost of capital, and thus it tends to lower the steady-state capital stock. On the other hand, an increase in the interest rate tends to increase the return to saving, and hence the steady-state wealth of entrepreneurs. With decreasing absolute risk aversion, this increases entrepreneurs’ willingness to take risk, and hence it is a force that tends to increase the steady-state capital stock. This second e(cid:27)ect is due to the fact that here investment is sensitive to wealth, a mechanism which is absent when markets are complete. In other words, agents require a (private) risk premium in order to invest in capital, but this premium is lower at higher levels of wealth.19 Therefore,theoveralle(cid:27)ectofanincreaseinRonK isambiguous,asissummarizedinthefollowing corollary. Corollary 2. When the interest rate is taken to be exogenous, ∂Ko/∂R ⇔ θ > φ/(1−φ). The proof for this corollary also follows from equation (25), and is left for the appendix. The intuition behind this result is a bit convoluted, so it is worth examining step-by-step. Combining equations (18) or (19) and (21) in steady state, we get: ρ + (θ−1) ρˆ = θβ , (29) where ρ is the mean return to saving, and ρˆ is the risk adjusted return, both evaluated at the steady-state K and for given R. Of course, this condition is equivalent to (25), but it is more useful for developing intuition. Note (cid:28)rst that an increase in K necessarily reduces ρ+(θ−1)ρˆ. This is because an increase in K reduces f(cid:48)(K), and, for given φ, this reduces ρ and ρˆ equally, thus also reducing ρ+(θ −1)ρˆ. Of course, the optimal φ must also fall, but this only reinforces the negative e(cid:27)ect on ρ (since the portfolio is shifted towards the low-return bond), while it does not a(cid:27)ect ρˆ(because of the envelope theorem and the fact that φ maximizes ρˆ). Note next that an increase in R has an ambiguous e(cid:27)ect on ρ+(θ −1)ρˆ. This is because, for given φ, both ρ and ρˆ increase with R, but now the decrease in φ works in the opposite direction, contributing to lower ρ. Intuitively, though, this e(cid:27)ect should be small if φ was small to begin with. Moreover, the impact of ρˆ is likely to dominate if θ is high enough. Therefore, ρ + (θ − 1)ρˆ is expected to increase with R if and only if either φ is low or θ is high. Combiningthesetwoobservationsleadstotheconclusionthatsteady-stateK increaseswithRif and only if θ > φ/(1−φ). As shown in section 3.3, this condition is more likely to be satis(cid:28)ed when 19To see this wealth e(cid:27)ect more clearly, note that we can use (23) and (25) to write steady-state human wealth as H(R) = H(K(R)). Then, by bond market clearing, steady-state aggregate wealth is W(R) = K(R)+H(R). The appendix20 shows that W(cid:48)(R) > 0 ⇔ µ(cid:48)(R) < µ < 0. But from (25) it is easy to show that µ(cid:48)(cid:48)(R) < 0. Hence, W(cid:48)(R) > 0 ⇔ µ(cid:48)(R) < µ ⇔ R > R. In other words, when the interest rate is above a certain threshold, then an increase in the interest rate increases aggregate steady-state wealth. 17

R is su(cid:30)ciently high, since the steady-state risk premium is a decreasing function of R. Intuitively, when R is close to β, a marginal increase in R has such a strong positive e(cid:27)ect on steady-state wealth, that the consequent reduction in the risk premium more than o(cid:27)sets the increase in the opportunity cost of investment, ensuring that K increases with R. The following proposition now summarizes the discussion above and the main result of this section. Proposition 5. If θ > φ/(1 − φ) and dRc/dτK su(cid:30)ciently high, then dKc/dτK > 0, i.e. the closed-economy steady-state capital stock is increasing in the capital-income tax. In order to assess the empirical relevance of the relationship θ > φ/(1 − φ), one can use a simple back-of-the-envelope calculation that does not require any reference to the degree of market incompleteness, σ. In particular, take labor income to be 65% of GDP, and take the safe rate to be 2%. Then, steady-state H is about 33 times GDP, or 11 times K, if the steady-state capital-output ratio is taken to be 3. Hence, φ/(1−φ) = K/H = 0.1, which, as section 5 will discuss, is lower than most of the empirical estimates of θ that use micro data for the United States. Hence, in all likelihood, the condition θ > φ/(1−φ) is satis(cid:28)ed in the data. At the same time, a high positive value for dRc/dτK is intuitive, considering the (insurance) e(cid:27)ect of the tax on the demand for precautionary saving. It is therefore possible that, although ∂Ko/∂τK < 0, it could still be that dKc/dτK > 0 over some region, since ∂Ko/∂R ·dRc/dτK > 0 is very likely positive. This means that the generalequilibrium e(cid:27)ect of insurance on the adjustment of the interest rate, and the subsequent e(cid:27)ect of this adjustment on wealth accumulation, is crucial for overthrowing the negative e(cid:27)ect of the capital-income tax on capital when the interest rate is (cid:28)xed. The next sections will demonstrate how, for empirically plausible parameter values, this general-equilibrium e(cid:27)ect will produce the counter-intuitive result that increases in the capital-income tax will at (cid:28)rst increase steady-state capital, even with the (cid:28)xed interest-rate e(cid:27)ect working in the opposite direction. 5 Calibration and Steady-State Implications For the quantitative part of the paper, the benchmark model analyzed so far is extended to include endogenous labor. Preferences are assumed homothetic between consumption, c and leisure, n, according to the King-Plosser-Rebelo (1988) speci(cid:28)cation, and they are de(cid:28)ned as the limit, for ∆t → 0, of: U t = { (1−e−β∆t)(c1 t −ψnψ t )1−1/θ +e−β∆t( E t [U t 1 + − ∆ γ t ] ) 1 1 − − 1 γ /θ } 1− 1 1/θ . (30) The appendix presents all proofs for the general case of endogenous labor. Note also that the calibration will treat the case with two types of agents, i.e. there will be both entrepreneurs and 18

laborers in the model. This section will present the benchmark calibration, and will examine its implication for the steady-state aggregates and wealth distributions. The next section will then focus on the e(cid:27)ects of capital-income taxation for aggregates and welfare. 5.1 Simulations The dynamic system described in Proposition 2, and generalized to the case of endogenous labor, is highlytractablecomparedtootherincomplete-marketsmodels, wheretheentirewealthdistribution is a relevant state variable for aggregate equilibrium dynamics. The steady state of the system is found by setting the dynamics of all equations in Proposition 2 to zero. The algorithm (cid:28)rst solves for the steady-state aggregates, which are deterministic and characterized by Proposition 3. Subsequently,foranyhistoricallygiven(K ,χ ,XE),whereχ istheinitialmeasureofentrepreneurs 0 0 0 0 in the economy, and XE is the historically given (cid:28)nancial wealth of the entrepreneur group, and 0 using as boundary conditions the steady state values of (H,mE,mL), it integrates backward until the path of (K ,λ ,H ,mE,mL) is close enough to its steady-state value. t t t t t The method of (cid:28)nite di(cid:27)erences is used on the general version of the system in Proposition 4. The (cid:28)rst and second derivatives of the invariant distributions are replaced by their discrete time approximations. Theonlyconditionsimposedarethattheprobabilitydensityfunctionsintegrateto one, andthattheydonotexplodetotheright. TheemergingfunctionsΦ andΦ arewell-behaved L E and stable. Subsequently,Monte-Carlosimulationsareperformed. Theprocessesofdying,oftype-switching, and of the idiosyncratic capital-income shocks, are simulated using random number generators for series of 200,000 households and 100,000 years. The wealth distributions generated converge to those produced by the (cid:28)nite-di(cid:27)erences method, and their variances are stable as time increases. Finally, using these distributions, welfare calculations are performed. 5.2 Parameter choice The economy is parameterized by (α, β, γ, δ, θ, σ, ψ, v, p , p , τK, τL, G). Table 1 presents EL LE the parameter choices for the preferred benchmark model calibration. The parameter values chosen refer to annual data from the United States. The discount rate is β = 0.024. The preference parameter is ψ = 0.75, which is standard in the macro literature21. The income share of capital is α = 0.40. The depreciation rate is δ = 0.06. The probability of death is chosen to be v = 1/150, a compromise between having an empirically relevant probability of death and allowing for some altruism across generations. The probability of exiting entrepreneurship is p = 0.18. The probability of entering entrepreneurship is p = 0.025. These two values were EL LE estimatedfromthePSIDandSCFdata, andsubsequentlyusedforcalibrations, byQuadrini(2000). 21For example, King, Plosser, and Rebelo (1988), and Christiano and Eichenbaum (1992). 19

In Quadrini’s model, as well as here, they imply a fraction of entrepreneurs in the total population of 12%,22 which is in line with the data, as Quadrini and Cagetti and DeNardi (2006) document. The elasticity of intertemporal substitution is chosen to be θ = 1. The empirical estimates of the EIS vary a lot. Using aggregate British data and correcting for aggregation bias, Attanasio and Weber (1993) estimate θ to be about 0.7. Although the exact estimates from micro data vary across studies and speci(cid:28)cations, in most cases they are around 1, especially for agents at the top layers of wealth and asset holdings. For example, using data from the Consumer Expenditure Survey (CEX) and an Epstein-Zin speci(cid:28)cation, as in the present paper, Vissing-Jłrgensen and Attanasio (2003) report baseline estimates between 1 and 1.4 for stockholders. The proportional tax on capital income is τK = 0.25. The Congressional Budget O(cid:30)ce BackgroundPaper(December2006)reportsthattheaveragemarginalrateatwhichcorporatepro(cid:28)tsare taxed is 35%, whereas the average marginal rate at which non-corporate business income is taxed is around 26%−27%. The CRS Report for Congress (October 2003) details the capital income tax revisions and e(cid:27)ective tax rates due to provisions granted through bonus depreciations of 30% or 50%. If these provisions are taken into account, the average marginal capital income tax is between 20% − 25% for non-corporate businesses and between 25% − 30% for corporate businesses. The value of τK = 0.25 is chosen to be in the middle of these estimates.23 The proportional tax on labor income is τL = 0.35. The Congressional Budget O(cid:30)ce Background Paper (December 2006) reports that the median e(cid:27)ective marginal tax rate on labor income is 32%, inclusive of federal, state and payroll taxes.24 Incorporating the distortionary e(cid:27)ect of social security taxes would further increase this number, hence the choice made here. The level of government spending, G, is chosen so that the steady-state government-spending-to-GDP ratio is 20%. The coe(cid:30)cient of relative risk aversion is chosen to be γ = 8. The empirical estimation of γ is a complicated task, because, as Vissing-Jłrgensen and Attanasio (2003) detail, it requires making additional assumptions about the covariance of consumption growth with stock returns, the share of stocks in the (cid:28)nancial wealth portfolio, the properties of the expected returns to human capital, and the share of human capital in overall wealth. Using the Consumer Expenditure Survey (CEX), Vissing-JłrgensenandAttanasio(cid:28)ndestimatesofriskaversionforstockholdersintherangeof5−10, but with a broader sample and under di(cid:27)erent assumptions these estimates go up to 20−30. They alsocomparetheirresultstoCampbell(1996),whoestimatesγ intherangeof17−25,usingdataon monthly and annual returns, and assuming that the entire (cid:28)nancial portfolio is held in stocks. Alan and Browning (2008), use the PSID data to structurally estimate the joint distribution of discount factors and relative risk aversion coe(cid:30)cients. They (cid:28)nd that the lower educated households are less risk averse than the more educated households, and that the medians of the two relative risk 22The proof can be found in Lemma 3 of the appendix. 23Altig et al. (2001) report a proportional capital income tax of 20% at the federal level, but they also subject capital income to a 3.7% state tax. 24This number is also reported by Jokisch and Kotliko(cid:27) (2006). 20

aversion distributions are 6.2 and 8.4 respectively. Guiso and Paiella (2005), using data from the 1995 Bank of Italy Survey on Household Income and Wealth, estimate direct measures of risk aversion based on the maximum price a consumer is willing to pay to buy a risky asset. They (cid:28)nd that the median relative risk aversion is 6, if consumers have a one-year horizon, and it is 16, if they have a lifetime horizon. Dohmen et el. (2005) present evidence on the distribution of risk attitudes in the population, using survey questions and a representative sample of 22,000 individuals living in Germany. The behavioral relevance of their survey is tested by conducting a complementary (cid:28)eld experiment, based on a representative sample, and the conclusion is that the survey measure is a good predictor of actual risk-taking behavior. They (cid:28)nd that the bulk of the mass in the γ-distribution is located between 1−10. There is, however, a non-negligible mass of estimates in the range of higher values, up to 20. Barsky et al. (1997) measure risk aversion based on survey responses by participants in the Health and Retirement Study to hypothetical situations. They (cid:28)nd that most individuals fall in the category that has mean relative risk aversion of 15.8. Cohen and Einav(2005)useadatasetof100,000individuals’deductiblechoicesinautoinsurancecontracts, to estimate the distribution of risk preferences. They (cid:28)nd that the 82nd percentile in the distribution of the coe(cid:30)cient of relative risk aversion is about 13−15.25 The volatility coe(cid:30)cient is chosen to be σ = 0.15. The empirical estimation of the standard deviation of idiosyncratic entrepreneurial returns is a very di(cid:30)cult task, and has not as yet received much attention in the literature. So far, the most thorough attempt to measure idiosyncratic risk is by Moskowitz and Vissing-Jłrgensen (2002). They document poor diversi(cid:28)cation and extreme concentration of entrepreneurial investment, signi(cid:28)cant heterogeneity in individual investment choices, and high risk at the individual level due to high bankruptcy rates. However, because of the problemsarisingwhenimputinglaborincome, andbecauseofthelackofsu(cid:30)cienttimedimensioninthe Survey of Consumer Finances (SCF) data, they cannot provide an accurate estimate of the volatility of entrepreneurial returns for unincorporated businesses. In the end, they conjecture that the volatility of returns for private (cid:28)rms cannot be lower than the corresponding volatility of publicly traded (cid:28)rms, which the (cid:28)nd to be about 0.5 per annum.26 Davis et al. (2006) use the Longitudinal Business Database (LBD), which contains annual observations on employment and payroll for all establishments and (cid:28)rms in the private sector, to estimate the volatility of employment growth rates. They (cid:28)nd that, in 2001, the ratio of private to public volatility was in the range 1.43−1.75. 25Attanasio et al. (2002) also provide evidence of considerable heterogeneity in the point estimates of the relative risk aversion coe(cid:30)cient, using data from the UK Family Expenditure Survey over 1978-1995. Estimates of relative risk aversion in the range of 10 have also been reported by PÆlsson (1999), who uses Swedish cross-sectional data from tax returns in 1985. 26It is to be noted though, that their analysis focuses on the di(cid:27)erences in the cross-sectional volatility facing privateentrepreneurs. Butwhatreallymattersinthepresentmodelistheinvestmentvolatilityanentrepreneurfaces overtime,sincethetimedimensionistheonerelevantforcapitalaccumulation. Forthistimedimension,Panousiand Papanikolaou (2008) (cid:28)nd that the mean annual idiosyncratic investment volatility is approximately 0.4 for publicly traded (cid:28)rms in the US. 21

Given that the average annual standard deviation for public (cid:28)rms over 1990−1997 was 0.11,27 and that there is, at least in the context of the present model, a close relationship between volatility of pro(cid:28)ts and volatility of labor demand, the choice of σ = 0.15 could also be justi(cid:28)ed from this perspective. Finally, this choice generates an annual variance for steady-state consumption growth in the range indicated by the micro data, once consumer heterogeneity is taken into account.28 Parameters γ and σ are especially important for the calibrated model, for two reasons. First, theydirectlyin(cid:29)uenceλ, thefractionofwealthheldbyentrepreneursintheeconomy. Andthen, for example,asmentionedinsection3.2,whenλfalls,i.e. whentheentrepreneursborrowmorefromthe laborers, then, for given prices, K falls as well.29 Given this importance of λ, the calibrated model’s implicationsaboutλareagoodcriterionofmodelperformance. Aswillbeshowninsection5.3, the choices γ = 8 and σ = 0.15, which seem empirically relevant given the discussion above, produce, without an attempt to match it, a value for λ that is reasonably close to the values documented in the data. Second, parameters γ and σ relate to the interpretation of the capital-income tax as providing insurance. For this reason, comparative statics will also be performed, in section 6, to show how the tax that maximizes the steady-state capital stock varies with risk aversion and the volatility of risk. The main result, that steady-state capital is inversely U-shaped with respect to the capital-income tax, is preserved qualitatively for σ ∈ (0,1) and for γ ∈ (2,20]. 5.3 Implications for steady-state aggregates and distributions This section undertakes the examination of the quantitative performance of the model in terms of aggregates and wealth distributions, for three reasons. First, to show how wealth inequality is in(cid:29)uencedbytherandom-walkcomponentintroducedinwealthbytheidiosyncraticinvestmentrisk. Second, to demonstrate how wealth inequality depends on the excess returns to entrepreneurship, whichisanimportantquestioninitsownright,butalsoinviewoftheimpactofagentheterogeneity on capital accumulation. Third, to provide some additional con(cid:28)dence in the main quantitative results presented in the next section (section 6) about the e(cid:27)ects of capital-income taxation on capital accumulation, by showing that the model performs well in matching aspects of the US aggregate and welfare data. Table2presentstheimplicationsofthemodelforsteady-stateaggregates,andcomparesthemto the data from the US economy. The model’s capital-output ratio is 2.8. Investment is 17% of GDP. The safe rate is 2.5%. The steady-state fraction of entrepreneurs, χss, is 12%, and it matches the databychoiceofthetransitionprobabilities, asexplainedinsection5.2. Entrepreneurshold30%of 27As reported in Campbell et al. (2001). 28For example, A(cid:239)t-Sahalia et al. (2001), and Malloy et al. (2006). 29This result also carries over to the steady-state analysis. For example, when λ (cid:54)= 1, then, for θ = 1, equation (25)takestheformF K (K)−δ=R+ (cid:112) λ(R)−1 γσ2(β−(1−τK)R),whereλ=(β−(1−τK)R+p LE )/(β−(1− τK)R+p LE +p EL ). From this it is clear that, for given R, an increase in λ leads to a fall in steady-state K. 22

total wealth in the economy, where the equivalent of λ in the data is the ratio XE/X30. The share of total wealth held by entrepreneurs in the data ranges between 35%−55%. The model-generated value for λ is an indication that the model performs reasonably well, especially given the low value of σ used in the calibration, and also since the rest of the aggregates could have been matched by a standard neoclassical growth model. The fraction of entrepreneurs in the top 10% of the population is 18% in the model, whereas in the data this number ranges between 32%−54%.31 Next, Table 3 examines the wealth distribution generated by the model. The (cid:28)rst two rows present the percentiles for wealth computed by Quadrini (2000), using the PSID and SCF samples for1994and1992,respectively. Thelastrowistheconditionalwealthdistributionofthebenchmark calibrated model.32 Aiyagari’s(1994)benchmarkcalibrationpredictionsforthewealthholdingsofthetop5%andthe top 1% of the population are 13.1% and 3.2%, respectively. Hence, the present model demonstrates how the random-walk component introduced in wealth by entrepreneurial risk helps generate a fatter right tail in the wealth distribution.33 Next, Figure 1 plots the Lorenz curves for the model’s aggregate wealth and consumption distributions. The model produces results in the right direction, in that the distribution of wealth over the population is much more unequal than the distribution of consumption. The model’s Gini coe(cid:30)cient for wealth, conditional on wealth being positive, is 0.62. The model’s Gini coe(cid:30)cient for consumption is 0.15.34 In the data, the Gini coe(cid:30)cient for total net worth is 0.8, and the Gini coe(cid:30)cient for consumption is 0.32. Finally, Figure 2 presents the model’s conditional wealth distributions over entrepreneurs and laborers. Onthehorizontalaxisiswealthnormalizedbymeanannualincomeintheeconomy. Onthe vertical axis are frequencies. The solid line represents entrepreneurs, and the dashed line laborers. Consistent with the data, the distribution of wealth for the population of entrepreneurs displays a fattertailthantheoneforlaborers. Thisisduetotherandom-walkcomponentthattheuninsurable investment risk introduces into entrepreneurial wealth. Furthermore, the entrepreneurial wealth distribution is shifted to the right, and it has lower frequencies at lower levels of wealth. This is due to the higher mean return of the total entrepreneurial portfolio. Finally, the distributions of 30This is because, in the data, wealth is de(cid:28)ned as total net worth, i.e. it is (cid:28)nancial wealth, X, as de(cid:28)ned in the present model, plus housing. 31The data on entrepreneurs and wealth concentration is as reported in Cagetti and De Nardi (2006). 32Compared to the data, the model’s unconditional wealth distribution has a larger fraction of agents at negative levels of wealth, most likely because of the absence of borrowing constraints. 33Atractableextensionthatcouldimprovethemodel’spredictionaboutwealthconcentrationatthetopwouldbe tointroduceathirdstate,inwhichanagentgetstobeanentrepreneuroperatingaveryhighreturnorverylowrisk productionfunction. Then,thetransitionprobabilitiesbetweenthethreestatescanbefreelychosentomatchdesired momentsofthewealthdistribution. Inparticular,makingthegoodentrepreneurialstatetheleastpersistentandthe most likely to transition to the state of being a laborer would increase the precautionary saving, and therefore the wealth concentration, of the very rich agents. 34Thedi(cid:27)erencesintheGinicoe(cid:30)cientsareduetothepresenceofhumanwealth: sincepooreragentshavehigher human-to-(cid:28)nancial-wealth ratios, they can sustain relatively high consumption. This would not be the case in the presence of borrowing constraints. 23

wealth for both groups have signi(cid:28)cant mass of people with wealth higher than (cid:28)fty times mean income. In the model, the laborers at the right tail of the wealth distribution are former successful entrepreneurs. 6 E(cid:27)ects of Capital-Income Taxation Havinggainedsomecon(cid:28)denceabouttheoverallquantitativeperformanceofthemodel,thissection now proceeds to study the e(cid:27)ects of capital-income taxation in the benchmark calibration, where again the relevant parameters are those of Table 1. Subsection 6.1 examines the steady-state e(cid:27)ects of capital-income taxation, while subsection 6.2 examines the aggregate and welfare implications of abolishing the capital-income tax. 6.1 Steady state This section quanti(cid:28)es the main theoretical result of the paper, which is that an increase in the capital-income tax increases the steady-state capital stock, when the tax is low enough. As already explained, this result is due to the general-equilibrium e(cid:27)ect of the insurance aspect of the capitalincome tax, and it operates mainly through the endogenous adjustment of the interest rate. Figure 3 shows the behavior of the steady-state aggregates and welfare with respect to the capital-income tax. Capital (panel (a)) and output (panel (b)) are inversely-U shaped with respect to the capital-income tax, and they reach a maximum when τK = 0.4. The same is true for employment, the capital-labor (capital per work-hour) ratio, and output per work-hour. At τK = 0.4,steady-statecapitalperwork-houris6.75%higherthanwhenτ = 0,andoutputperwork-hour K is 2.65% higher. As shown in Figure 3(f), aggregate welfare is maximized at τK = 0.7, whether for entrepreneurs (solid line), laborers (dashed line), or the economy as a whole (dotted line).35 This is because of the combined direct insurance e(cid:27)ect of the tax, through the reduction in σ(1−τK), and the e(cid:27)ect of the tax on aggregates. Figure 3(c) shows that the net interest rate increases with the tax, and that it tends to the discount rate, β = 0.024, as τK → 1. This is the demonstration of the precautionary saving motive mentioned in section 4: when the capital-income tax increases, the e(cid:27)ective volatility of risk facing an entrepreneur decreases, which reduces the demand for precautionary saving, and therefore increases the interest rate. Figure 3(d) reinforces this interpretation of the capital-income tax as providing insurance: when the tax increases, the precautionary saving motive becomes weaker, and therefore entrepreneurs are satis(cid:28)ed with a lower risk premium. Figure 3(e) shows that the fraction of wealth held by entrepreneurs in the economy is decreasing in the capital-income tax. This results from the combination of the weaker precautionary saving motive, and the fall in the risk premium. 35Naturally,entrepreneurwelfareishigherthanlaborerwelfareforalltaxlevels. Inaddition,entrepreneurwelfare ishigherthanlaborerwelfareforalllevelsofwealth,sinceentrepreneursareunconstrainedintheirinvestmentchoices. 24

At this point, it is useful to compare the e(cid:27)ects of capital-income taxation in the present model to those under complete markets, where there is no scope for insurance (assuming that agents are homogeneous), as well as to those in the open economy version of the model, where only the direct insurance aspect of the tax is present. As already mentioned, here, at τK = 0.4, steadystate capital per work-hour and output per work-hour are 6.75% and 2.65%, respectively, higher than when τ = 0. By contrast, under complete markets,36 at τK = 0.4, steady-state capital per K work-hour and output per work-hour are 25% and 11% lower than when τK = 0. And (cid:28)nally, in the open-economy version of the model the aggregates fall all the way with the tax, but less so than under complete markets. In particular, at τK = 0.4, steady-state capital per work-hour and output per work-hour are 14% and 6% lower than when τK = 0. In addition, steady-state welfare is maximized when τK = 0.4, which re(cid:29)ects solely the direct insurance aspect of the tax. Finally, in order to reinforce the insurance interpretation of the tax system, Figure 4 presents robustness checks with respect to volatility, σ, and risk aversion, γ. On the vertical axis is the tax that maximizes the steady-state capital stock. When either the volatility of risk increases or risk aversion increases, the tax that maximizes the steady-state capital stock increases. These comparative statics also indicate that the main result of the paper is robust to the wide range of empirically plausible values of σ ∈ (0,1) and of γ ∈ (2,20]. In particular, for the low value of σ = 0.15, the capital-income tax that maximizes the steady-state capital stock is positive for all γ > 2, and it is actually zero when γ = 2. 6.2 Dynamics of eliminating the capital-income tax This section proceeds to examine the aggregate and welfare implications of eliminating the capitalincome tax. In the standard representative-agent complete-markets neoclassical model, the optimal capital-income tax is zero in the long run, as well as in most of the short run for an interesting class of preferences. Steady-state welfare is also decreasing in the level of the capital-income tax. These (cid:28)ndings have initiated an extensive debate as to the possible bene(cid:28)ts of eliminating the tax on capital income. By contrast, the main result of the present paper is that an increase in the capital-income tax may actually increase the steady-state capital stock. In light of this result, it is worthwhile to revisit the discussion on the implications of setting the capital-income tax to zero. Thee(cid:27)ectsonaggregatesandwelfarewhenthecapital-incometaxiseliminatedwillbeexamined from two perspectives. On the one hand, one might be interested in examining the welfare of the current generation immediately after the policy reform, taking into account the entire transitional dynamics of the economy towards the new steady state with the zero tax. On the other hand, one might be interested in examining the welfare of the generations that will be alive in the distant future, i.e. at the new steady state in the long run. 36The complete-markets calibration uses the relevant parameter values from the benchmark Table 1. 25

The present model can in fact examine the current-generation implications of policy reforms, because it is very tractable, compared to other incomplete-markets models, where the entire wealth distribution is a relevant state variable. Here, only the mean of the wealth distribution is relevant for aggregate dynamics, which constitutes a signi(cid:28)cant gain in tractability, and allows for the entire dynamic response of the economy, after a policy change, to be considered. This is important, since it has long been recognized that the immediate e(cid:27)ects of policy may well be very di(cid:27)erent from the long-run e(cid:27)ects. Here, the economy starts from the steady state described by the benchmark calibration parameters in Table 1, where the capital-income tax is τK = 0.25. Subsequently, the tax is set to zero, ceteris paribus. 6.2.1 Aggregate e(cid:27)ects This section presents the short-run and long-run responses of the aggregate variables to the policy reform that eliminates the capital-income tax. Table 4 shows the response of the aggregates on the impact of the policy reform (denoted by Current), as well as at the new steady state (denoted by Long Run), under both complete markets and the present model of incomplete markets. The e(cid:27)ects on the interest rate, R, the risk premium, µ, and the investment-output ratio, I/Y, are in percentage units. The rest of the numbers denote percentage changes. Under complete markets, a permanent (unanticipated) tax cut leads to an immediate negative jump in consumption and an immediate positive jump in investment. Capital slowly increases and convergestoahighersteady-statevalue,whileconsumptionisinitiallylowerandincreasesovertime. In other words, the long-run increase in investment requires an initial period of lower consumption, which inturnallowsforan immediateincreaseininvestmentaswell. Bycontrast, underincomplete markets, the exact opposite is the case. In light of the main mechanism of the paper, investment decreases in the long run. This allows for an immediate increase in consumption, and therefore necessitates a fall in current investment. In particular, the investment-output ratio falls by more than 3 percentage units. These e(cid:27)ects are also illustrated in Figure 5, which plots the impulse responses of the variables when the capital-income tax is eliminated. 6.2.2 Welfare e(cid:27)ects This section studies the welfare implications of eliminating the capital-income tax. These implicationsarerepresentedintermsofacompensatingdi(cid:27)erentialforeachlevelofwealthandeachtypeof agent, whether entrepreneur or laborer. In particular, starting from the old regime with τK = 0.25, the question is what fraction of his (cid:28)nancial wealth would an agent be willing to give up in order to avoid the impact of the new regime initiated by the policy change, either immediately or in the long run. 26

Figure 6 presents the welfare implications of abolishing the capital-income tax for entrepreneurs (solid line) and laborers (dashed line). Panel (a) shows the welfare implications for the current generation, taking into account the entire transitional dynamics of the economy towards the new steady-state, and panel (b) shows the welfare implications for the generations alive at the new steady state. Financial wealth normalized by annual mean income is on the horizontal axis, and the compensating di(cid:27)erentials are on the vertical axis. A negative number on the vertical axis indicates an agent who bene(cid:28)ts from the reform: the agent would have to be paid to be indi(cid:27)erent between the old regime and the regime initiated by the impact of the policy change, hence the agent prefers the new regime with the zero capital-income tax. Figure6(a)showsthatcurrent-generationpooragents, whetherentrepreneursorlaborers, prefer the zero capital-income tax regime. As wealth increases, both entrepreneurs and laborers prefer the positive capital-income tax regime. Finally, the mean cost of eliminating the tax is higher for the middle-class agents than for the very rich. Figure 6(b) shows that, in the long run, both types of agents and at all wealth levels prefer the steady state with the positive tax, the rich less so than the poor, and the entrepreneurs less so than the laborers. These cross-sectional di(cid:27)erences can be explained by referring to Figure 7, which plots, in the top row for the current generation, and in the bottom row for the long-run generation, the response of human wealth and of the (risk-adjusted) returns to saving for laborers and entrepreneurs, against the tax rate of the policy reform. Let’s (cid:28)rst turn to the immediate implications. The decrease in the capital-income tax from τK = 0.25 to τK = 0 increases the demand for precautionary saving, and therefore leads to a fall in the interest rate.37 Roughly speaking, since the capital stock is historically given and cannot change, the fall in the interest rate increases human wealth. For poor agents, whether entrepreneurs or laborers, human wealth constitutes a signi(cid:28)cant part of total wealth, and hence they bene(cid:28)t from the elimination of the tax. Furthermore, poor agents do not bene(cid:28)t much from insurance directly, since they invest little or nothing in the risky asset. Therefore, in the short run, poor agents prefer the zero capital-income tax regime, mainly because the elimination of the tax increases their safe income, and safe income is a big part of their total wealth. Turning to the long run, the elimination of the capital-income tax increases the demand for precautionary saving, and it therefore leads to a fall in the interest rate. But now, the generalequilibrium implications of the interest rate adjustment for capital accumulation become relevant. In particular, the fall in the interest rate reduces steady-state wealth and capital accumulation. It turns out that the fall in the steady-state capital stock dominates the fall in the interest rate, so that in the end steady-state human wealth falls. This adversely a(cid:27)ects poor agents of all types, since human wealth represents a big part of their total wealth. Because the risk-adjusted return for 37Immediatelyafterareformthatreducesthecapitaltaxfromaveryhighlevel,thenetinterestratemayactually increase. This possibility, which does not emerge in the long run, is due to the usual distortionary e(cid:27)ect of big tax increases on investment. 27

entrepreneurs, ρˆ, increases when the capital-income tax is eliminated, the cost of the policy change is not as high for an entrepreneur as it is for a laborer at any given level of wealth. Inconclusion,theeliminationofthecapital-incometaxhaswelfareimplicationsthatdi(cid:27)eracross time and in the cross-section of the population. These di(cid:27)erences are due to the general-equilibrium e(cid:27)ects of the interest rate adjustment on capital accumulation. In particular, they operate mainly through the di(cid:27)erent response of human wealth: immediately after the elimination of the tax, when the capital stock cannot adjust, human wealth increases, whereas in the long run, when capital accumulation changes endogenously, human wealth falls. Therefore, current-generation poor agents prefer a zero capital-income tax in the short run, whereas future-generation poor agents prefer a positive capital-income tax. Rich agents always prefer a positive tax, but less forcefully in the longrun, because in the long run the elimination of the tax increases the mean entrepreneurial portfolio return. 7 Extension: Introducing Publicly Traded Sector So far it has been assumed that all investment is subject to uninsurable idiosyncratic risk. This might not be an appropriate assumption for a country like the United States, where private equity actually accounts for about 40% of total (cid:28)nancial wealth. In order to capture this fact, and to demonstrate the robustness of the main mechanism, a second sector of production is here formally introduced. In this sector, all (cid:28)rms are publicly traded, and it is assumed that they can perfectly diversify away all idiosyncratic risks.38 Assume that both entrepreneurs and laborers can invest and work in the public sector. In other words, the public sector is an additional safe asset, but in positive net supply, as opposed to the bond, which is in zero net supply. Let the production function of the public sector be of the same form as the aggregate production function in the private sector, that is J(M,Λ) = ν−1MαΛ1−α, where M is the total capital and Λ is the total labor employed by public (cid:28)rms. Here ν is a scaling factor, ensuring that the mean return to capital in the public sector is lower than in the private sector.39 Let public sector capital be taxed at the rate τK, and public sector labor at the rate τL. t t By no arbitrage, in equilibrium, the risk-free rate will be equal to the marginal product of capital in the public sector, i.e. R = J (M,Λ), and the wage will be equal to the marginal product of labor t M in the public sector, i.e. ω = J (M,Λ). The rest of the equilibrium characterization proceeds as t Λ in the benchmark model, with bond holdings now replaced by the sum of bond and public equity holdings. So aggregate wealth in the economy will now be W = K +H +M . Hence, we obtain t t t t the following variant of Proposition 2. 38Thisisanextremeassumptionmadehereforanalyticalconvenience. Infact,thedataindicatesthatpublic(cid:28)rms do not have a perfectly diversi(cid:28)ed shareholder base. Himmelberg et al. (2002), using the Worldscope database for a panel of publicly traded (cid:28)rms across 38 countries, (cid:28)nd that the median insider equity ownership share is 40%. 39Otherwise no entrepreneur would invest in the private sector. 28

Proposition 6. In an equilibrium where both sectors are active, the aggregate dynamics satisfy: W˙ /W = λ (ρ −mE)+(1−λ )((1−τK)R −mL) (31) t t t t t t t t t 1 λ˙ /λ = (1−λ )φ µ +(1−λ )(mL−mE)+p ( −1)−p (32) t t t t t t t t LE λ EL t H˙ = ((1−τK)R +v)H −(1−τL)ω −(τLω +τK(F −δ)K −G ) (33) t t t t t t t t t Kt t t φ λ t t K = (H +M ) (34) t t t 1−φ λ t t R = J (M,Λ), ω = J (M,Λ), L +Λ = 1 (35) t M t Λ t t along with (18) and (19). The proof of Proposition 6 is straightforward. Following similar steps as in the model with only the private sector, we can show that R = R(ω ), and Λ = l(ω )M . t t t t t In the public sector, where there is no scope for insurance, an increase in the capital-income tax unambiguously reduces investment, so that public capital is a negative function of the tax. As a result,theoveralle(cid:27)ectofthetaxontheaggregatecapitalstockisingeneralambiguous. Inaddition though, the increase in the capital tax might now trigger a reallocation of resources, away from the low-risk and low-productivity public sector, towards the higher-risk and higher-productivity private sector, thus increasing total factor productivity. Hence, aggregate output may increase with the tax, even if aggregate capital falls. Coming now to the steady-state calibration,40 parameter ν is chosen so that the share of private capital in the aggregate capital stock is 40%. As in the one-sector economy, the calibration ensures that at τK = 0.25, the interest rate is about 2%, and that government spending as a fraction of GDP is 20%. At this level of the tax, the private equity premium is 2.6%, which is lower than its value of 4.2% in the one-sector economy, since there is now one additional (safe) asset.41 In the two-sector economy, the capital-income tax that maximizes aggregate capital and welfare is zero, and the per capita variables are all decreasing in the tax. However, at τK = 0.4, the capital-labor ratio in the public sector is 20% lower than it would have been if τK = 0, i.e. it falls by less than under complete markets, where it would have fallen by 25%. In fact, the capital-labor ratio in publicly traded (cid:28)rms is unambiguously always higher than under complete markets, since the interest rate under incomplete markets is lower than the discount rate. In addition, at τK = 0.4, the ratio of total output to total labor falls by 8.6%, whereas under complete markets it falls by 11%. To summarize, in steady state and for τK = 40%, output-per work-hour is approximately 15% higher than under complete markets when all production takes place in the private sector, and it 40Forexpositionalpurposes,thissectionwillonlydealwiththesteady-statee(cid:27)ectsofcapital-incometaxation. The dynamic implications of eliminating the tax are available upon request. 41Lowvaluesfortheprivatepremiumareconsistentwiththe(cid:28)ndingsofMoskowitzandVissing-Jłrgensen(2002). 29

is 3% higher than under complete markets when the private sector accounts for 40% of (cid:28)nancial assets. 8 Conclusions This paper studies the aggregate and welfare e(cid:27)ects of capital-income taxation in an environment whereagentsfaceuninsurableidiosyncraticentrepreneurialrisk. Thecounter-intuitiveresultemerging is that an increase in the capital-income tax may actually, due to its general-equilibrium insurance aspect, stimulate capital accumulation. This result stands in stark contrast to the e(cid:27)ects of capital-income taxation in either complete-markets models, or in Bewley-type incomplete-markets models, since in those models capital-income taxation necessarily discourages capital accumulation. Furthermore, the result is quantitatively signi(cid:28)cant: for the preferred calibration of the model, the steady-state levels of the capital stock, output, and employment are all maximized for a positive value of the capital-income tax, at which point output per work-hour is 2.65% higher than it would have been had the tax rate been zero.42 Although the present paper provides some useful guidance about the direction of optimal policy, itdoesnotsolveforthefullyoptimalpolicy. Aninterestingdirectionforfutureresearchistheformal study of optimal policy, either in the Ramsey tradition (though allowing for lump-sum taxes, as in the present model), or in the Mirrlees tradition of endogenizing the source of market incompleteness and having no ad hoc restrictions placed on the set of available instruments. This paper focuses on the e(cid:27)ects of uninsurable entrepreneurial risk, and abstracts from laborincomerisk,borrowingconstraints,anddecreasingreturnstoscaleattheindividuallevel. Extending the model to include these relevant aspects of the data and revisiting the e(cid:27)ects of capital taxation inthisrichersettingisimportant,notonlytogetabetterquantitativeevaluationoftheimplications ofcapitaltaxation, butalsotoexaminewhetherthegeneral-equilibriume(cid:27)ectsidenti(cid:28)edheremight interact with other sources of market incompleteness in an interesting way. For example, after an increase in the capital-income tax, the increase in steady-state wealth documented here could make borrowing constraints less binding. At the same time, the increase in the steady-state interest rate could also increase the cost of borrowing. Further investigating these rich general-equilibrium interactionswillgreatlyfacilitateabettertheoreticalandquantitativeassessmentoftheimplications of (cid:28)scal policy in dynamic heterogeneous-agent environments. 42Undercompletemarkets,andstartingfromasteadystatewithτK =0.25,thecapital-incometaxwouldhaveto be reduced to approximately τK =0.15, in order for output per work-hour to increase by 2.65%. 30

9 Appendix: Proofs Lemma 1. Let preferences be described by: J t = {(1−e−β∆t)(c1 t −ψnψ t )1−1/θ +e−β∆t(E t [J t 1 + − ∆ γ t ]) 1 1 − − 1 γ /θ }1− 1 1/θ , where c is consumption and n is leisure. Then, given the processes for c and n, the utility process is de(cid:28)ned as the solution to the following integral equation: (cid:90) ∞ U = E z(c ,U )ds, (36) t t s s t where   β (c1−ψnψ)1−1/θ z(c,U) ≡  t t −(1−γ)U. (37) 1−1/θ −1/θ+γ ((1−γ)U) 1−γ Proof of Lemma 1. De(cid:28)ne the functions: 1−1/θ ((1−γ)x) 1−γ g(x) = , 1−1/θ J1−γ U = t . t 1−γ Then: J 1−1/θ (c1−ψnψ)1−1/θ g(U ) = t = (1−e−β∆t) t t +e−β∆tg(E [U ]). t t t+∆t 1−1/θ 1−1/θ Take a (cid:28)rst order Taylor expansion in ∆t : (c1−ψnψ)1−1/θ g(U ) = g(U )+β t t ∆t−βg(U )∆t+g(cid:48)(U )E [∆U ]. t t t t t t 1−1/θ Then: (c1−ψnψ)1−1/θ β t t −βg(U ) 1−1/θ t E [∆U ] = − ∆t, t t g(cid:48)(U ) t where: g(U ) (1−γ)U t t = . g(cid:48)(U ) (1−1/θ) t Hence: E [∆U ] = −z(c ,n ,U )∆t, t t t t t where: β (c1−ψnψ)1−1/θ z(c ,n ,U ) ≡ [ t t −(1−γ)U ]. t t t t 1−1/θ −1/θ+γ ((1−γ)U t ) 1−γ 31

For a more general proof of the above and for a proof of existence and uniqueness of the solution to the integral equation (36) see Du(cid:30)e and Epstein (1992). Proof of Proposition 1. Because of the CRRA/CEIS speci(cid:28)cation of preferences, guess that the value function for an entrepreneur is: wE1−γ J(wE,t) = BE , t 1−γ where the term BE captures the time dimension. The Bellman equation for an entrepreneur is: t 0 = max z(cE,n E ,JE(wE,t)) + ∂JE (wE,t)[(φ(1−τK)r +(1−φ)(1−τK)R )wE−cE−(1−τL)ω nE] cE,nE,φ ∂wE t t t t t t ∂JE 1 ∂2JE + (wE,t) + (wE,t)σ2(1−τK)2φ2w2 + p [J(wL,t)−J(wE,t)], ∂t 2∂wE2 t EL where the function z is given by (37), and where the last term shows that the entrepreneur might switch into being a worker with probability p . Because of the homogeneity of JE in wE, the EL marginal propensity to consume and the portfolio choice will be the same for all entrepreneurs. The (cid:28)rstorderconditionfortheoptimalportfolioallocationgivestheconditionforφ in(16). Combining t the (cid:28)rst order conditions for consumption and leisure we get the optimal leisure choice: ψ 1 ni = ci. (38) t 1−ψ (1−τL)ω t t t From the envelope condition we get: mE ≡ B E 1 1 − − γ θ ( ψ 1 )−ψ(1−θ)(1−ψ)θβθ. 1−ψ(1−τL)ω Similarly, guess that the value function for a laborer is: wL1−γ J(wL,t) = BL , t 1−γ The Bellman equation for a laborer is: ∂JL 0 = max z(cL,nL,JL(wL,t)) + (wL,t)[R wL−cL−(1−τL)ω nL] cL,nL ∂wL t t t ∂JL + (wL,t) + p [J(wE,t)−J(wL,t)]. LE ∂t 32

Following similar steps, we get from the envelope condition that: mL ≡ B L 1 1 − − γ θ ( ψ 1 )−ψ(1−θ)(1−ψ)θβθ. 1−ψ(1−τL)ω It follows that: BE mE 1−γ = ( )1−θ . BL mL Using this, the (cid:28)rst order conditions, the envelope conditions, and plugging back into the Bellman equation we get (18) and (19). Proof of Proposition 2. Let R˜ be the e(cid:27)ective risk-free rate. The human wealth for each t individual i = E,L in the economy is hi t = (cid:82) t ∞ e− (cid:82) t sR˜ jdj((1−τ s L)ω s +T s )ds. The human wealth of the measure-χ t group of entrepreneurs is H t E = χ t (cid:82) t ∞ e− (cid:82) t sR˜ jdj((1−τ s L)ω s +T s )ds, and the human wealth of the measure-(1−χ t ) group of laborers is H t L = (1−χ t ) (cid:82) t ∞ e− (cid:82) t sR˜ jdj((1−τ s L)ω s +T s )ds. Hence total human wealth is H t = H t E + H t L = (cid:82) t ∞ e− (cid:82) t sR˜ jdj((1 − τ s L)ω s + T s )ds = hi t . Using the Leibniz rule, and substituting in from the government budget constraint (7), we get that the evolution of total human wealth is described by (23). Since only entrepreneurs invest in capital, the aggregate capital stock in the economy is given by K = φ WE. For an agent in the E and L t t t group respectively, bE +hE = (1−φ )wE and bL +hL = wL. Aggregating over each group, we t t t t t t t get BE +χ H = (1−φ )WE and BL +(1−χ )H = WL. Adding up and using the fact that t t t t t t t t BE +BL = 0, we get H = (1−φ )WE +WL. Now using W = WE +WL and K = φ WE, t t t t t t t t t t t t we get W = K +H . Combining H = (1−φ )WE +WL, K = φ WE, and λ = WE/W , we t t t t t t t t t t t t t get (24). Aggregating across leisure choices we get (ψ/(1−ψ))(1/((1−τL)ω ))C +L = 1, where t t t t C = mEWE +mLWL, WL = W −WE, and W = K +H . Aggregating across (12) and (13), t t t t t t t t t t t and adding up, using BE +BL = 0, H = HE +HL, and labor market clearing, we get: t t t t t 1 W = [(1−τK)r K +(1−τK)R H − C ]dt. t t t t t t t 1−ψ t UsingH = (1−φ )WE+WL,K = φ WE,µ = (1−τK)r −(1−τK)R ,andC = mEWE+mLWL, t t t t t t t t t t t t t t t t t and dividing through with W we get: t W˙ 1 t = (1−τK)r φ µ +(1−τK)R − (λ mE +(1−λ )mL), W t t t t t t 1−ψ t t t t t which gives (21) when we use ρ = φ µ +(1−τK)R . Aggregating across (12), and subtracting t t t t t from (21), we get (22). Proof of Proposition 3. Consider (cid:28)rst the case with λ = 1 and v = 0 (and labor is exogenous, soψ = 0). Combining(18)or(19)insteadystate,with(21)insteadystate,andusingthede(cid:28)nitions of ρ and ρˆ, we get equation (25). Combining (24) with (23) in steady state , we get equation (26). 33

Now, let µ(R) and φ(R) denote, respectively, the risk premium and the fraction of e(cid:27)ective wealth held in capital, when K is given by (25): (cid:114) (cid:115) 2θγσ2 2θ µ(R) ≡ (β−(1−τK)R) and φ(R) ≡ (β−(1−τK)R) . (39) 1+θ γσ2(1+θ) Note that µ(cid:48)(R) < 0 and φ(cid:48)(R) < 0. Next, let K(R) denote the solution to (25), or equivalently: (cid:20) (cid:21) 1 µ(R)+δ+R α−1 K(R) = . (40) α Finally, for τK (cid:39) 0, τL (cid:39) 0, δ (cid:39) 0, G = gY, and Y = f(K) = Kα, we can write equation (26) as: K(R)α−1 1−φ(R) D(R;g) ≡ (1−α−g) − , (41) R φ(R) where α+g < 1. To establish existence and uniqueness of the steady state , it su(cid:30)ces to show that there exists a unique R that solves D(R;g) = 0. For a given g, consider the limits of D as R → 0+ and R → β−. Note that µ(0) = (2θγσ2 β)1/2 is (cid:28)nite and hence both φ(0) and K(0) are (cid:28)nite. It 1+θ follows that: 1 1 lim D(R;g) = (1−α−g)K(0)α−1 lim − +1 = +∞ . R→0+ R→0+ R φ(0) Furthermore, µ(β) = 0, implying φ(β) = 0 and K(β) = K ≡ (f(cid:48))−1(β) is (cid:28)nite. Hence: compl 1 1 lim D(R;g) = (1−α−g)K(β)α−1 − lim +1 = −∞ . R→β− β R→β− φ(R) These properties, together with the continuity of D(R) in R, ensure the existence of an R ∈ (0,β) such that D(R) = 0. If D(R;g) is strictly decreasing in R, then we also have uniqueness. To show this, note that, from (41): ∂D K(R)α−1 (cid:20) K(cid:48)(R) (cid:21) φ(cid:48)(R) = (1−α−g) (α−1)R −1 + . (42) ∂R R2 K(R) φ(R)2 In addition: f(cid:48)(K) K(cid:48) 1 µ(cid:48)+1 φ(cid:48) γσ2µ(cid:48) Kα−1 = , = , and = , α K α−1f(cid:48)(K) φ2 µ2 where the dependence of K, µ, and φ on R has been dropped for notational simplicity. Hence: ∂D 1−α−gf(cid:48)(K) (cid:20) µ(cid:48)+1 (cid:21) γσ2µ(cid:48) = R −1 + = ∂R α R2 f(cid:48)(K) µ2 1−α−gRµ(cid:48)+R−f(cid:48)(K) γσ2µ(cid:48) = + . α R2 µ2 34

Since µ(cid:48)(R) < 0 and R < f(cid:48)(K(R)) for all R ∈ (0,β), it follows that ∂D/∂R < 0 for all R ∈ (0,β), which completes the argument. When v > 0, an extension of the proof above shows that there is a unique R solving D(R) = 0, where R ∈ (−v,β), and where: (1−gτL)(1−α)K(R)α−1+(1−g)τKf (R) 1 K D(R) ≡ − +1. ((1−τK)R+v) φ(R)λ(R) However,inthatcase,uniquenesshasnotbeenproved,althoughsimulationssuggestthatthesteady state is always unique. In order to see how the existence of two types of agents modi(cid:28)es the characterization of the steady state, consider next the case where λ (cid:54)= 1, and take θ = 1 for simplicity. Then, the marginal propensity to consume is always constant and equal to β, for both types of agents. Equation (21) in steady state yields λ = (β−(1−τK)R)/(φµ). Combining this with (22) in steady state gives: β−(1−τK)R+p LE λ = , (43) β −(1−τK)R+p +p LE EL which veri(cid:28)es that λ < 1. Plugging this back into (21) in steady state, we get: (β−(1−τK)R)(β−(1−τK)R+p +p ) LE EL φµ = , (β−(1−τK)R+p ) LE from which, if we use the de(cid:28)nition of µ, we get: (cid:115) 1 F (K)−δ = R+ γσ2(β−(1−τK)R), (44) K λ(R) as the relevant version of (25). Finally, combining (23) in steady state with (24), we get: φ(K,R) λ(R) (1−τL)ω(K)+(1−g)(τLω(K)+τK(F (K)−δ)K) K K = , (45) 1−φ(K,R) λ(R) (1−τK)R+v which is the relevant version of (26). Hence, the steady state is characterized by equations (43), (44), and (45). So, (43) expresses λ as a function of R, (44) expresses K as a function of R, and (45) solves for the equilibrium R, using K(R) and λ(R). When labor is endogenous and θ = 1, then mE = mL = (1−ψ)β, and the proofs above carry through the same way, only now with f (K/L) and ω(K/L). So for characterization of the steady K state we need to add the labor market clearing condition, and the steady state system will be in K,L,R. In particular, labor market clearing, combined with C = (1−ψ)βW, λ = WE/W, and WE = K/φ gives: ψβ 1 K/L L = ( +1)−1. (1−τL)ω(K,L)λ(R)φ(K,L,R) 35

Finally, when labor is endogenous and θ (cid:54)= 1 then: 1 mL−(1−τK)R+p 1−ψ LE λ = , 1 mL−(1−τK)R+p +p 1−ψ LE EL and (cid:115) γσ2(1−τ)2 1 µ = ( (mEλ+mL(1−λ))−(1−τK)R) . λ 1−ψ Hereweneedtoaddtwomoreequationstocharacterizethesteadystate,namelytheEulerconditions for the marginal propensities to consume. This will be a system of two equations in two unknowns to be solved as a function of steady state prices.43 Lemma 2. When the interest rate is exogenous, W(cid:48)(R) > 0 ⇔ R > R. Proof of Lemma 2. Let v = 0, λ = 1, τK (cid:39) 0, τL (cid:39) 0, G = gY, and Y = Kα. Then, from (23) in steady state, we have that H(R) = (1−α −g)K(R)α/R, while K as a function of R is given by (40). Hence, we can write W(R) = K(R)+H(R). Di(cid:27)erentiating with respect to R, we get that: α K2(α−1) φ W(cid:48)(R) > 0 ⇔ µ(cid:48)(R) < (α−1) −1 , R2 φ+α(1−φ) which means that the interest rate has to be higher than a given threshold, i.e. R > R, since from (39) it is easy to show that µ(cid:48)(cid:48)(R) < 0. ProofofProposition4. LetthenewbornhouseholdreceiveaweightedaverageaW +(1−a)wi, t t where 0 < a < 1, upon birth.44 Let d be the indicator function, where d = 1 for entrepreneurs t t and d = 0 for laborers. The dynamic system for the state vector (ξi,d ), where , ξi ≡ wi/W as t t t t t t in the text, is: ξ ˙i = µ(ξi,d )+σ(ξi,d )dzi−(ξi−1)dN1 t t t t t t t t d˙ = s(d )dN2, t t t wheredN1 isthePoissonprocessdenotingdeathwitharrivalratevdt,andwheredN2 isthePoisson t t switching process with arrival intensity p(I)dt: p(d) = p if d = 0 LE p(d) = p if d = 1, EL 43The conditions needed for establishing that λ>0 are satis(cid:28)ed in simulations. 44Onecouldrationalizethisthroughtheexistenceofanestatetaxon theagent: iftheagentdies, thegovernment takes away a(w t i−W t ) from his descendants. The idea is to keep the aggregate wealth una(cid:27)ected. Here, expected wealth for the agent at any point in time is v·(aW t +(1−a)w t i)+(1−v)·w t i, and aggregating across agents yields the desired result. The special case a = 1 implies that each newborn agent enters the economy endowed with the sum of the mean economy wealth plus his human wealth. 36

where: s(d) = 1 if d = 0 s(d) = −1 if d = 1, and: 1−λ µ(ξ ,1) = [ (mL−mE)+φ (1−λ )(1−τK)(r −R )]ξ t 1−ψ t t t t t t t t λ µ(ξ ,0) = [ (mE −mL)−φ λ (1−τK)(r −R )]ξ t 1−ψ t t t t t t t t σ(ξ ,1) = φ σ (1−τK)ξ t t t t t σ(ξ ,0) = 0. t Let Φ ≡ Φ(ξ,1) and Φ ≡ Φ(ξ,0) be the conditional distributions for entrepreneurs and labor- E L ers respectively. In steady state the conditional distribution Φ satis(cid:28)es the forward Kolmogorov L equation: ∂(µ(ξ,0)Φ ) v ξ−a L 0 = − −p(0)Φ +(pΦ )(ξ,0−η(0))−vΦ + Φ ( ), L L L L ∂ξ 1−a 1−a and the conditional distribution Φ satis(cid:28)es the forward Kolmogorov equation: E 1∂2(σ(ξ,d)2Φ ) ∂(µ(ξ,1)Φ ) v ξ−a E E 0 = − −p(1)Φ +(pΦ )(ξ,1−η(1))−vΦ + Φ ( ). 2 ∂ξ2 ∂ξ E E E 1−a E 1−a In the two equations above we need to calculate: (pΦ)(ξ,d−η(d)) = p(d−η(d))Φ(ξ,d−η(d)). To that end, let the old state be d, and the new state be d(cid:48). They are related through d(cid:48) = d+s(d), and we need to compute η(d(cid:48)) = s(d). For d = 0, we have d(cid:48) = 0 + s(0) = 0 + 1 = 1, and η(d(cid:48)) = η(1) = s(0) = 1, hence η(1) = 1. For d = 1, we have d(cid:48) = 1 + s(1) = 1 − 1 = 0, and η(d(cid:48)) = η(0) = s(1) = −1, hence η(0) = −1. Therefore: p(0−η(0))Φ(ξ,0−η(0)) = p(1)Φ(ξ,1) = p Φ , EL E and: p(1−η(1))Φ(ξ,1−η(1)) = p(0)Φ(ξ,0) = p Φ . LE L Substituting for µ(ξ ,0), µ(ξ ,1), σ(ξ ,0), σ(ξ ,1) and using the above, we can write the system of t t t t 37

the two Kolmogorov equations as: ∂2Φ ∂Φ v ξ−a 0 = c ξ2 E +c ξ E +c Φ +p Φ + Φ ( ) 1 ∂ξ2 2 ∂ξ 3 E LE L 1−α E 1−a ∂Φ v ξ−a L 0 = c ξ +c Φ +p Φ + Φ ( ), 4 5 L EL E L ∂ξ 1−a 1−a where: φ2σ2(1−τ)2 c = 1 2 c = 2φ2σ2(1−τ)2−[ 1 (m _ −mE)+φµ(1−λ)] 2 1−ψ c = φ2σ2(1−τ)2−[ 1 (m _ −mE)+φµ(1−λ)]−p −v 3 EL 1−ψ 1 c = λφµ− (m¯ −mL) 4 1−ψ 1 c = λφµ− (m¯ −mL)−p −v 5 LE 1−ψ Now, the Laplace transform for any variable y is de(cid:28)ned as: (cid:90) ∞ Y(s) = e−sty(t)dt, 0 and therefore: (cid:90) ∞ d Y(cid:48)(s) = − e−stty(t)dt = −L[ty(t)] ⇒ L[ty] = − Y(s), ds 0 and: (cid:90) ∞ (cid:90) ∞ Y(cid:48)(s) = − e−stty(t)dt ⇒ Y(cid:48)(cid:48)(s) = e−stt2y(t)dt = L[t2y(t)]. 0 0 Hence, we have that: (cid:90) ∞ L[ty(cid:48)] = e−stty(cid:48)dt = −sY(cid:48)(s)−Y(s), 0 and: (cid:90) ∞ L[t2y(cid:48)(cid:48)] = e−stt2y(cid:48)(cid:48)dt = s2Y(cid:48)(cid:48)(s)+4sY(cid:48)(s)+2Y(s). 0 Let c = 1 , k = a , and τ = ct−k, then dτ = cdt and t = τ+k. So we have: 1−a 1−a c t−a (cid:90) ∞ 1 L[y( 1−a )] = e−sty(ct−k)dt = c e−sk cL[y(t)] s→s/c = (1−a)e−k(1−a)L[y(t)] s→s(1−a) . k/c Therefore, when a = 1: t−a (cid:90) ∞ L[y( )] = (1−1)e−k(1−1) y(t)dt = 0·1·1, 1−a 0 38

if y is a probability density function. Hence, the last term in both Kolmogorov equations will drop out when a = 1. After changing variables to ξ = ex, and de(cid:28)ning ∂Φ /∂x ≡ Φ we get: E 2      Φ(cid:48) 0 c /c −1 p /c Φ L 5 4 EL 4 L       Φ(cid:48)  =  1 0 0  Φ   E    E  Φ(cid:48) c /c −3c −p /c −2+c /c −c /c Φ 2 2 1 1 LE 1 2 1 3 1 2 Since all coe(cid:30)cients are constant, and ξ is bounded, a Lipschitz condition is satis(cid:28)ed, hence the solutiontothesystemexistsandisunique.45 Theconditionaldensitiescanberecoveredbyinverting the Laplace transforms. Proof of Corollary 2. Consider the case with λ = 1 and v = 0. Then, using (18) or (19) and (21) in steady state, using the de(cid:28)nition of ρˆ, and taking the total di(cid:27)erential with respect to K and R, gives: ∂K φ−θ(1−φ) 1 = , ∂R φ(θ+1) F KK which proves that: ∂K φ > 0 ⇔ θ > . ∂R 1−φ Lemma 3. The steady state measure of entrepreneurs is given by p /(p +p ). LE LE EL Proof of Lemma 3. Callχthemeasureofentrepreneurstoday,andχ(cid:48) theirmeasuretomorrow. Then χ(cid:48) = χ(1−p )+(1−χ)p . But in steady state χ = χ(cid:48) hence χ = p /(p +p ). EL LE LE LE EL 45For proper (and non-restrictive) initial conditions, the solution is also stable. 39

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Parameters Values Preferences β 0.024 γ 8 θ 1 ψ 0.75 Technology α 0.40 δ 0.06 Probabilities v 0.0067 p 0.18 EL p 0.025 LE Government τK 0.25 τL 0.35 G/GDP 0.20 Risk σ 0.15 Table 1. Benchmark Calibration Values. K/Y I/Y G/Y R χss XE/X χss in top 10% US Data 2.7 17% 20% 2% 10−19% 35−55% 32−54% Model 2.8 17% 20% 2.5% 12% 30% 18% Table 2. Steady-State Aggregates. Top Percentiles 30% 20% 10% 5% 1% SCF 87.6 79.5 66.1 53.5 29.5 PSID 85.9 75.9 59.1 44.8 22.6 Model 76.25 63.68 44.44 29.80 10.53 Table 3. Distribution of Wealth in the US and in the Model. Current Long Run Incomplete Complete Incomplete Complete L −2.98 8.94 −0.0477 1.43 Y −1.80 5.27 −2.12 7.77 C 2.48 −5.52 −2.06 5.87 Y/L 1.22 −3.37 −2.08 6.25 I/Y −3.16 6.44 −0.52 2.48 R −1.21 1.29 −0.13 0 net Table 4. Dynamics of Eliminating the Capital-Income Tax. 44

Lorenz CuLrvoere fnozr WCueravleth for Wealth Lorenz CuLrovere fnozr CCuornvseu mfopr tCioonnsumption 1 1 1 1 0.9 0.9 0.9 0.9 0.8 0.8 0.8 0.8 0.7 0.7 0.7 0.7 0.6 0.6 0.6 0.6 0.5 0.5 0.5 0.5 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0 0 0 0 0 00.2 00..24 00..46 00..68 0.18 1 0 00.2 00..24 00..46 00..68 0.18 1 (a) Wealth (b) Consumption Figure 1: Lorenz Curves for Wealth and Consumption Wealth Distribution for Entrepreneurs and Laborers 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 10 20 30 40 50 60 wealth / mean income seicneuqerf Entrepreneurs Laborers Figure 2: Wealth Distribution for Entrepreneurs and Laborers 45

1.6 0.55 1.4 0.5 1.2 0.45 1 0.4 0.8 0.6 0.35 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 τK 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 τK (a) Aggregate Capital Stock (b) Aggregate Output 0.024 0.07 0.0235 0.06 0.023 0.05 0.0225 0.04 0.022 0.03 0.0215 0.02 0.021 0.0205 0.01 0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 τK 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 τK (c) Net Interest Rate (d) Risk Premium 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 τK 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (e) Fraction of Wealth Held by E (f) Welfare Figure 3: Steady State and Capital-Income Taxation 46

Risk Aversion 0.6 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 20 γ xaT gnizimixaM Volatility of Risk 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 σ xaT gnizimixaM Risk Aversion 0.6 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 20 γ (a) Risk Aversion xaT gnizimixaM Volatility of Risk 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 σ xaT gnizimixaM (b) Volatility of Risk Figure 4: Robustness Checks 31 0.025 0.035 30.5 0.03 0.02 30 0.025 29.5 0.015 0.02 29 0.015 0.01 28.5 0.01 28 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 τK 0.005 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 τK 0.005 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 τK HumanWealth: SR NetInterestRate: SR Risk-AdjustedReturn,ρˆ: SR 30 0.024 0.036 28 0.0235 0.034 26 0.023 0.032 24 0.0225 0.022 0.03 22 0.0215 20 0.028 0.021 18 0.026 0.0205 16 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 τK 0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 τK 0.024 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 τK HumanWealth: LR NetInterestRate: LR Risk-AdjustedReturn,ρˆ: LR Figure 7. Immediate (SR) vs Long Run (LR): Human Wealth and Saving Returns 47

10 8 6 4 2 0 −2 −4 0 5 10 15 20 25 30 35 40 45 segnahc tnecrep 8 Incomplete Markets Complete Markets 6 4 2 0 −2 t −4 0 5 10 15 20 25 30 35 40 45 (a) Aggregate Labor Supply segnahc tnecrep t (b) Aggregate Output 6 4 2 0 −2 −4 −6 0 5 10 15 20 25 30 35 40 45 segnahc tnecrep 8 6 4 2 0 −2 −4 t 0 5 10 15 20 25 30 35 40 45 (c) Aggregate Consumption segnahc tnecrep t (d) Labor Productivity 2 1.5 1 0.5 0 −0.5 −1 −1.5 0 5 10 15 20 25 30 35 40 45 segnahc tniop egatnecrep 1.5 1 0.5 0 −0.5 −1 t −1.5 0 5 10 15 20 25 30 35 40 45 (e) Investment-Output Ratio segnahc tniop egatnecrep t (f) Net Interest Rate Figure 5: Dynamics of Incomplete vs. Complete Markets: Eliminating the Capital-Income Tax 48

15 10 5 0 −5 −10 −15 −20 −50 0 50 100 150 200 250 300 350 400 450 wealth / mean income 0=Kτ diova ot pu nevig snoitcarf htlaew 45 40 35 30 25 20 15 10 Entrepreneurs 5 Laborers 0 −200 0 200 400 600 800 1000 1200 wealth / mean income (a) Immediate Welfare Implications 0=Kτ diova ot pu nevig snoitcarf htlaew Entrepreneurs Laborers (b) Long-Run Welfare Implications Figure 6: Welfare Implications of Eliminating the Capital-Income Tax 49