Keywords: Capitalincome taxation, uninsurable idiosyncratic investment risk, general equilibrium, heterogeneous agents
Abstract:
This paper studies the effects of capital taxation in a dynamic heterogeneousagent economy with uninsurable entrepreneurial risk. Although it allows for rich generalequilibrium effects and 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 completemarkets paradigm or Bewleytype models where idiosyncratic risk impacts only labor income, here it is shown that capital taxation may actually stimulate capital accumulation. This possibility emerges because of the generalequilibrium effects of the insurance aspect of capital taxation. In particular, for the preferred calibrated version of the model, when the tax on capital is 25 percent, output per workhour is 2.2 percent higher than it would have been had the tax rate been zero. Turning to the welfare effects of a reform in capital taxation, it is examined how these effects depend on whether one focuses on the steady state or also takes into account transitional dynamics, as well as how they vary in the crosssection of the population (rich versus poor, entrepreneurs versus nonentrepreneurs).
This paper studies the macroeconomic and welfare effects of capitalincome taxation in an environment where agents face uninsurable idiosyncratic investment risk. Such risk is empirically important 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 tradeoff between the distortion of investment versus the provision of insurance against idiosyncratic capitalincome risk. On the one hand, capital taxation comes at a cost, since it distorts agents' saving decisions. On the other hand, it has benefits, since it provides agents with partial insurance against idiosyncratic investment risk. This suggests that a positive tax on capital income could be welfareimproving, even if it reduced capital accumulation.
Most surprisingly though, it is shown that a positive tax on capital income may actually stimulate capital accumulation. Indeed, the steadystate levels of the capital stock, output and employment may all be maximized at a positive value of the capitalincome tax. This possibility emerges because of the generalequilibrium effects of the insurance aspect of capital taxation. This result stands in stark contrast to the effect of capital taxation both under completemarkets models, and under incompletemarkets models with uninsurable laborincome risk alone. In these models, capitalincome taxation, irrespectively of whether it is welfareimproving or not, necessarily discourages capital accumulation.
Model. This paper represents a first attempt to study the effects of capitalincome taxation in a generalequilibrium incompletemarkets 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 effects of such risk on macroeconomic aggregates. Agents own privately held businesses that operate under constant returns to scale. Agents are not exposed to laborincome 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 laborincome risk, borrowing constraints, and other market frictions, isolates the impact of the idiosyncratic investment risk and preserves the tractability of the generalequilibrium 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 noncontingent lumpsum tax or transfer. Second, agents have finite 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 nonentrepreneurs without the complexity of endogenizing occupational choice. Fourth, labor supply is endogenous. Clearly the first 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.
Preview of results. The main result of the paper is that an increase in capitalincome taxation may actually stimulate capital accumulation. The intuition behind this result comes from recognizing that the overall effect of the capitalincome tax on capital accumulation can be decomposed in two parts. The first part captures the response of capital to the tax in a setting with endogenous saving but exogenously fixed interest rate. This is isomorphic to examining the effects of the capital tax in a "small open economy". In this context, it is shown that an increase in the capitalincome tax unambiguously decreases the steadystate capital stock. The second part, which is the core result of this paper, captures the importance of the generalequilibrium adjustment of the interest rate for wealth and capital accumulation. Here, an increase in the tax reduces the effective 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 steadystate 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 generalequilibrium effect of the interest rate adjustment is a force that tends to increase investment and the steadystate capital stock.
For plausible parameterizations of the closed economy, the general equilibrium effect dominates for low levels of the capitalincome tax, so that steadystate capital at first increases with the tax. In particular, for the preferred calibrated version of the model, the steadystate capital stock is maximized when the tax on capital is . So, for example, when the tax on capital is , output per workhour is higher than what it would have been had the tax rate been zero. The result that the steadystate capital stock is inversely Ushaped with respect to the capitalincome tax is robust for a wide range of empirically plausible parameter values. Furthermore, the tax that maximizes steadystate capital is increasing in risk aversion and/or the volatility of idiosyncratic risk. This finding reinforces the insurance interpretation of the tax system.
Subsequently, the paper examines the aggregate and welfare effects of eliminating the capitalincome tax. This is because an extensive discussion has been conducted within the context of the completemarkets neoclassical growth model about the welfare benefits of setting the capitalincome tax to zero. In light of the main result here, revisiting this discussion is worthwhile. In particular, the aggregate and welfare effects are presented from two different 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 effects of eliminating the capitalincome tax. When markets are complete, investment increases in the short run, and it is also higher at the new longrun 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.
Second, consider the welfare effects of eliminating the capitalincome tax. These vary across the different types of agents, the different levels of wealth, and the current versus the future generations. In the current generation, poor agents, whether entrepreneurs or nonentrepreneurs, prefer the zero tax. This is because most of their wealth comes from wage income, and, with capital fixed, 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 benefit 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 zerotax 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 steadystate capital stock, thereby decreasing the present value of wages. Therefore poorer agents will suffer 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 financially developed country like the United States. For example, Moskowitz and VissingJørgensen (2002) find that of all private equity is owned by agents for whom such investment constitutes at least half of their total net worth. Furthermore, of private equity is held by owners who are actively involved in the management of their own firm.^{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 risksharing arrangements are much more limited.
Furthermore, idiosyncratic investment risk need not be interpreted as affecting solely the owners of privately held businesses. In recent work, Panousi and Papanikolaou (2008) find a significant and robust negative relationship between idiosyncratic risk and the investment of publicly traded firms in the US. In addition, they show that this relationship is stronger in firms where the insider mangers hold a larger fraction of the firm'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 VissingJørgensen (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.
This paper relates to the strand of the macroeconomic literature discussing optimal taxation and the effects of taxation. However, most of this literature has focused on labor income risk. Chamley (1986) and Judd (1985) first 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 finitely lived agents. Aiyagari (1995) extended the completemarkets framework to include uninsurable labor income risk and borrowing constraints. In this context, when only a limited set of policy instruments are available, 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 different normative exercise is conducted by Davila et al. (2007). They examine constrained efficiency, in the spirit of GeanakoplosPolemarchakis, within a version of Aiyagari's model. This exercise does not allow for risksharing through taxes or any other instruments, and instead considers an efficiency concept where the planner directly dictates to the agents how much to invest and to trade. Angeletos and Werning (2006) examine a similar constrained efficiency problem in a twoperiod version of a model with idiosyncratic investment risk. Albanesi (2006) considers optimal taxation in a twoperiod model of entrepreneurial activity, in a constrained efficiency setting, and following the Mirrlees optimal policy tradition. The benefit of her approach is that the source of incomplete risksharing is endogenously specified as the result of a private information (moral hazard) problem, and that there are no ad hoc restrictions placed on the tax instruments. However, her model does not allow for dynamics, for longrun considerations, or for generalequilibrium effects like those studied in the present paper. In general, the extensive theoretical work on taxation originating from the Mirrlees and the new dynamic public finance tradition focuses on laborincome risk, as does the literature that examines the optimal progressivity of the tax code.^{3}
The growing literature on the effects of borrowing constraints on entrepreneurial choices has examined policy questions, and especially the implications of replacing a progressive with a proportional incometax schedule, in an Aiyagaritype environment, i.e. with decreasing returns to scale at the individual level, borrowing constraints, and undiversifiable labor income risk. Some examples in this area include Li (2002), Domeij and Heathcote (2003), Meh (2005), Cagetti and DeNardi (2007), and Kitao (2007). Benabou (2002) develops a tractable dynamic generalequilibrium model of human capital accumulation with endogenous effort and missing credit and insurance markets. Within this framework he examines the longrun tradeoffs of progressive taxation and education finance. Finally, Erosa and Koreshkova (2007) examine the longrun effects 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 finance literature that considers the effects of capital taxation on portfolio allocation and risktaking. Domar and Musgrave (1944) first proposed the idea that proportional income taxation may increase risktaking, by having the government bear part of the risk facing the agents.^{4} This idea was formalized by Stiglitz (1969), within a twoperiod singleagent model, where asset returns and the level of saving are exogenously given, but where the agent optimally chooses the allocation of his fixed amount of saving between a risky and a riskless asset. Ahsan (1974) extended Stiglitz by endogenizing the intertemporal consumptionsaving decision in a twoperiod model. He showed that the partialequilibrium effect of capitalincome taxation on risktaking is in general ambiguous. By contrast, in the "small open economy" version of the present model, which differs from Ahsan's in that the horizon is infinite and the return to capital is endogenous, it is shown that the steadystate capital stock is decreasing in the capitalincome tax. This finding highlights that the results here are driven by, novel to the literature, generalequilibrium effects.
As already mentioned, the present model builds on Angeletos (2007), who abstracted from policy questions and considered instead the effect of investment risk on macroeconomic aggregates. The contribution of the present paper is to study the effects of capitalincome taxation on aggregates and welfare. Angeletos and Panousi (2009), in a framework like the one in Angeletos (2007), examine the effects of government spending on macroeconomic aggregates, but for the case where government spending is financed exclusively through lumpsum 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 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 quantifies the steadystate effects of capital taxation, as well as the shortrun and longrun effects of eliminating the capitalincome 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.
Time is continuous and indexed by . There is a continuum of agents distributed uniformly over . At each point in time an agent can be either an entrepreneur, denoted by , or a laborer, denoted by . 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 , and the probability that he will switch from being a laborer to being an entrepreneur is . The measure of entrepreneurs in the economy at time is denoted by .
In what follows, and for expositional simplicity, labor is taken to be exogenous. All of the proofs, 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 of the KingPlosserRebelo (1988) specification.
All agents are endowed with one unit of time. Preferences are EpsteinZin over consumption, , and they are defined as the limit, for , of ^{5}:
(1) 
When an agent is an entrepreneur, he owns and runs a firm operating a constantreturnstoscale neoclassical production function , where is capital input and is labor input. An entrepreneur can only invest in his own firm's capital, although he supplies and employs labor in the competitive labor market. Capital investment in his firm 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 financial wealth of an entrepreneur , denoted by , is the sum of his holdings in private capital, , and the riskless bond, :
Firm profits are given by:
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 is therefore:
At each point in time the government taxes capital and bond income at the rate , and labor income at the rate . Part of the tax proceeds is used by the government for own consumption at the deterministic rate . Government spending does not affect the utility from private consumption or the production technology. The remaining tax proceeds are then distributed back to the households in the form of noncontingent lumpsum transfers, . The government budget constraint is therefore:
All households face a constant probability of death, with Poisson arrival rate at every instant in time.^{7} There is no intergenerational altruism linking a household to its descendants, and utility is zero after death. The discount rate in preferences can then be reinterpreted as , where is the psychological or subjective discount rate and is the probability of death^{8}.
In order to isolate the effects of capitalincome risk, it is assumed that there exist annuity firms 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 , and defined as the present discounted value of their netoftaxes labor endowment plus government transfers:^{9}
Then, the total effective wealth, , for an agent is defined as the sum of his financial and human wealth, i.e. . Hence, effective wealth for an entrepreneur is given by:
This section characterizes individual behavior and the general equilibrium in the economy. The analysis will be in closedform, since, as will be shown, the wealth distribution is not a relevant state variable for the characterization of aggregate equilibrium dynamics.
Entrepreneurs choose employment after their capital stock has been installed and their idiosyncratic shock has been observed. Hence, since their production function, , exhibits constant returns to scale, optimal firm employment and optimal profits are linear in own capital:
and  (11) 
The evolution of effective wealth for an entrepreneur is described by:
Let the fraction of effective wealth an agent saves in the risky asset be:
(15) 
Because of the linearity in assets of the budget constraints and , and the homotheticity of the preferences, the optimal individual policy rules will be linear in total effective wealth, for given prices and government policies. Hence, for given prices and policies, an agent's consumptionsaving 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.
From and it is clear that optimal consumption is a linear function of total effective 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, , and all laborers share a common marginal propensity to consume, . The fraction of wealth invested in the risky asset by an agent who happens to be an entrepreneur is increasing in the risk premium, decreasing in risk aversion, and decreasing in the effective variance of risk, . Because of homotheticity and linearity, is the same across all entrepreneurs, and independent of the level of wealth. The policy for optimal bond holdings follows from or , and . The system of and is a system of two Euler equations. It shows that the marginal propensities to consume, conditional on being an entrepreneur or a laborer, depend on two factors. First, on the process of the corresponding netoftax anticipated (riskadjusted) returns to saving, in accordance with whether the elasticity of intertemporal substitution, , is higher or lower than 1. Second, on the probability that the agent might switch between being an entrepreneur and being a laborer.
The initial position of the economy is given by the distribution of across households. An equilibrium is a deterministic sequence of prices , a deterministic sequence of policies , a deterministic macroeconomic path , and a collection of individual contingent plans for , such that the following conditions hold: (i) given the sequences of prices and policies, the plans are optimal for the households; (ii) the labor market clears, , in all ; (iii) the bond market clears, , in all ; (iv) the government budget constraint is satisfied in all ; and (v) the aggregates are consistent with individual behavior, , , , , , , and , in all .
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 incompletemarkets 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 suffices 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 effective wealth held by entrepreneurs in the economy be:
Equation shows that the evolution of total effective wealth is a weighted average of two terms. The first term is positive when the mean netoftax 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 netoftax 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 shows the endogenous evolution of the relative distribution of wealth between the two groups of agents. The evolution of depends on three factors. First, on the differential excess return the entrepreneurs face on their saving, which is given by , where is the fraction of wealth invested in the risky asset, and is the risk premium. Second, on the difference in the level of saving between entrepreneurs and laborers, as captured by the difference in the marginal propensities to consume, . Third, on the adjustment made for the transition probabilities. Note here that the evolution of consumption can be recovered by aggregating across individual optimal policies, so that and , and using , , , and . Equation shows the evolution of total human wealth, using the government budget constraint , where is the marginal product of capital in the aggregate production function , and where from market clearing. Since , the resource constraint of the economy is also satisfied. Equation is the bond market clearing condition. It comes from aggregating across individual capital and bond choices as given in and , adding up, using , and using . From it follows that, for given prices and human wealth, a decrease in decreases . A fall in 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 affect their willingness to take risk, and therefore investment and the capital stock will fall for given prices.
A steady state is a competitive equilibrium as defined in section , where prices, policies, and aggregates are timeinvariant. For expositional purposes, and to illustrate that the results about the effects of capitalincome taxation on the aggregates are not due to the presence of two types of agents or to the probability of death, section (as well as section later on) will consider the case with and ^{11}. However, section will characterize the invariant distributions for the general case.
The steady state is the fixed point of the dynamic system in Proposition . Let government spending, , be parameterized as a fraction of tax revenue. The following proposition characterizes the steady state.
From or and in steady state, and using the fact that , we get equation . This condition gives the combinations of and that are consistent with wealth and consumption stationarity. Using and in steady state yields equation . This condition gives the combinations of and that are consistent with stationarity of human wealth and bond market clearing.
At this point it is useful to briefly compare the steady state to its completemarkets counterpart. From note that the difference from complete markets, in which case it would be , is the presence of the squareroot term, which captures the risk premium, i.e. here . In other words, agents here require a (private) risk premium in order to invest in capital. In addition, combining or with , and using the fact that , we get , from which, in steady state, we conclude that:
(27) 
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 denote the distance between individual and aggregate effective wealth. Let and be the conditional invariant distributions for laborers and entrepreneurs respectively. The following proposition characterizes the invariant distributions.
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}
This section presents the core of the contribution of this paper, which is the study of the steadystate effects of capitalincome taxation. Again, for illustration purposes, the assumption is that and . The main result here is that an increase in the capitalincome tax may actually increase investment and the steadystate capital stock. This possibility arises because of the generalequilibrium effects 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 fixed, and the case where the interest rate is allowed to adjust endogenously.
Note then that equation expresses capital, , as a function of the tax, , and the interest rate, . If the interest rate were fixed,^{16} then the steadystate capital stock would be , as given by , and where both and are exogenous. Next, by plugging from into , we can solve for the closedeconomy steadystate interest rate, as a function of the capitalincome tax. Let denote the closedeconomy solution for the interest rate. It follows then, that the closedeconomy steadystate capital stock will be given by .
Hence, the impact of the capitalincome tax on the closed economy steadystate capital stock can be decomposed in two parts. The first part describes how steadystate capital changes with the tax when the interest rate is kept constant or exogenously fixed. The second part describes the generalequilibrium adjustment of the interest rate in the closed economy, and the subsequent effects of this adjustment on capital accumulation. Thus, the total effect of the capitalincome tax on the closedeconomy steadystate capital stock can be decomposed as follows:
Let's first turn to the fixedinterest rate effect. The following corollary characterizes the effect of capitalincome taxation on capital accumulation when the interest rate is held constant.
Corollary #.:
This result follows immediately from , for a given . 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, namely that the tax reduces the variance of net returns, . Clearly then, for a given interest rate, this channel is not strong enough to outweigh the distortionary effect of capital taxation on investment. This result stands in contrast to the findings of Ahsan (1974). Ahsan considers the simultaneous determination of the size and the composition of the optimal portfolio, in a twoperiod model with exogenous returns. He shows that the effect of an increase in capitalincome taxation on risktaking and capital is in general ambiguous.^{17} The result here indicates that, once Ahsan's setting is extended to incorporate endogenous capital return and infinite horizon, the ambiguity disappears and capital taxation always leads to a fall in the steadystate 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 effect of capital taxation on capital to become ambiguous once again.
Let's now turn to the generalequilibrium effect, which captures the fact that in the closed economy the interest rate endogenously adjusts to clear the bond market, according to equation . This effect further consists of two parts.
First, an increase in the capitalincome tax reduces the effective volatility of risk for entrepreneurs, , and this is the direct insurance effect 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. .^{18} In fact, the increase in the interest rate is so high, that the net interest rate, , ends up increasing, despite the increase in the capitalincome tax.
Second, this increase in the (net) interest rate will generate two opposing effects on saving and wealth accumulation, as can be seen from . On the one hand, an increase in the interest rate increases the opportunity cost of capital, and thus it tends to lower the steadystate capital stock. On the other hand, an increase in the interest rate tends to increase the return to saving, and hence the steadystate 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 steadystate capital stock. This second effect 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, the overall effect of an increase in on is ambiguous, as is summarized in the following corollary.
Corollary #.:
The proof for this corollary also follows from equation , and is left for the appendix. The intuition behind this result is a bit convoluted, so it is worth examining stepbystep. Combining equations or and in steady state, we get:
Note first that an increase in necessarily reduces This is because an increase in reduces , and, for given this reduces and equally, thus also reducing Of course, the optimal must also fall, but this only reinforces the negative effect on (since the portfolio is shifted towards the lowreturn bond), while it does not affect (because of the envelope theorem and the fact that maximizes .
Note next that an increase in has an ambiguous effect on This is because, for given both and increase with , but now the decrease in works in the opposite direction, contributing to lower Intuitively, though, this effect should be small if was small to begin with. Moreover, the impact of is likely to dominate if is high enough. Therefore, is expected to increase with if and only if either is low or is high.
Combining these two observations leads to the conclusion that steadystate increases with if and only if . As shown in section , this condition is more likely to be satisfied when is sufficiently high, since the steadystate risk premium is a decreasing function of . Intuitively, when is close to a marginal increase in has such a strong positive effect on steadystate wealth, that the consequent reduction in the risk premium more than offsets the increase in the opportunity cost of investment, ensuring that increases with .
The following proposition now summarizes the discussion above and the main result of this section.
In order to assess the empirical relevance of the relationship , one can use a simple backoftheenvelope calculation that does not require any reference to the degree of market incompleteness, . In particular, take labor income to be of GDP, and take the safe rate to be . Then, steadystate is about 33 times GDP, or 11 times K, if the steadystate capitaloutput ratio is taken to be 3. Hence, , which, as section will discuss, is lower than most of the empirical estimates of that use micro data for the United States. Hence, in all likelihood, the condition is satisfied in the data. At the same time, a high positive value for is intuitive, considering the (insurance) effect of the tax on the demand for precautionary saving.
It is therefore possible that, although , it could still be that over some region, since is very likely positive. This means that the generalequilibrium effect of insurance on the adjustment of the interest rate, and the subsequent effect of this adjustment on wealth accumulation, is crucial for overthrowing the negative effect of the capitalincome tax on capital when the interest rate is fixed. The next sections will demonstrate how, for empirically plausible parameter values, this generalequilibrium effect will produce the counterintuitive result that increases in the capitalincome tax will at first increase steadystate capital, even with the fixed interestrate effect working in the opposite direction.
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, and leisure, , according to the KingPlosserRebelo (1988) specification, and they are defined as the limit, for , of:
(30) 
The dynamic system described in Proposition , and generalized to the case of endogenous labor, is highly tractable compared to other incompletemarkets models, where the entire wealth distribution 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 to zero. The algorithm first solves for the steadystate aggregates, which are deterministic and characterized by Proposition . Subsequently, for any historically given , where is the initial measure of entrepreneurs in the economy, and is the historically given financial wealth of the entrepreneur group, and using as boundary conditions the steady state values of , it integrates backward until the path of is close enough to its steadystate value.
The method of finite differences is used on the general version of the system in Proposition . The first and second derivatives of the invariant distributions are replaced by their discrete time approximations. The only conditions imposed are that the probability density functions integrate to one, and that they do not explode to the right. The emerging functions and are wellbehaved and stable.
Subsequently, MonteCarlo simulations are performed. The processes of dying, of typeswitching, and of the idiosyncratic capitalincome 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 finitedifferences method, and their variances are stable as time increases. Finally, using these distributions, welfare calculations are performed.
The economy is parameterized by . Table 1 presents 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 . The preference parameter is , which is standard in the macro literature^{21}. The income share of capital is . The depreciation rate is . The probability of death is chosen to be , a compromise between having an empirically relevant probability of death and allowing for some altruism across generations. The probability of exiting entrepreneurship is . The probability of entering entrepreneurship is . These two values were estimated from the PSID and SCF data, and subsequently used for calibrations, by Quadrini (2000). In Quadrini's model, as well as here, they imply a fraction of entrepreneurs in the total population of ,^{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 . The empirical estimates of the EIS vary a lot. Using aggregate British data and correcting for aggregation bias, Attanasio and Weber (1993) estimate to be about 0.7. Although the exact estimates from micro data vary across studies and specifications, 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 EpsteinZin specification, as in the present paper, VissingJørgensen and Attanasio (2003) report baseline estimates between 1 and 1.4 for stockholders.
The proportional tax on capital income is . The Congressional Budget Office Background Paper (December 2006) reports that the average marginal rate at which corporate profits are taxed is , whereas the average marginal rate at which noncorporate business income is taxed is around . The CRS Report for Congress (October 2003) details the capital income tax revisions and effective tax rates due to provisions granted through bonus depreciations of or . If these provisions are taken into account, the average marginal capital income tax is between for noncorporate businesses and between for corporate businesses. The value of is chosen to be in the middle of these estimates.^{23} The proportional tax on labor income is . The Congressional Budget Office Background Paper (December 2006) reports that the median effective marginal tax rate on labor income is , inclusive of federal, state and payroll taxes.^{24} Incorporating the distortionary effect of social security taxes would further increase this number, hence the choice made here. The level of government spending, , is chosen so that the steadystate governmentspendingtoGDP ratio is .
The coefficient of relative risk aversion is chosen to be . The empirical estimation of is a complicated task, because, as VissingJørgensen and Attanasio (2003) detail, it requires making additional assumptions about the covariance of consumption growth with stock returns, the share of stocks in the financial 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), VissingJørgensen and Attanasio find estimates of risk aversion for stockholders in the range of 510, but with a broader sample and under different assumptions these estimates go up to 2030. They also compare their results to Campbell (1996), who estimates in the range of 1725, using data on monthly and annual returns, and assuming that the entire financial 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 coefficients. They find that the lower educated households are less risk averse than the more educated households, and that the medians of the two relative risk 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 find that the median relative risk aversion is 6, if consumers have a oneyear 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 individuals living in Germany. The behavioral relevance of their survey is tested by conducting a complementary field experiment, based on a representative sample, and the conclusion is that the survey measure is a good predictor of actual risktaking behavior. They find that the bulk of the mass in the distribution is located between 110. There is, however, a nonnegligible 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 find that most individuals fall in the category that has mean relative risk aversion of 15.8. Cohen and Einav (2005) use a data set of 100,000 individuals' deductible choices in auto insurance contracts, to estimate the distribution of risk preferences. They find that the 82nd percentile in the distribution of the coefficient of relative risk aversion is about 1315.^{25}
The volatility coefficient is chosen to be . The empirical estimation of the standard deviation of idiosyncratic entrepreneurial returns is a very difficult 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 VissingJørgensen (2002). They document poor diversification and extreme concentration of entrepreneurial investment, significant heterogeneity in individual investment choices, and high risk at the individual level due to high bankruptcy rates. However, because of the problems arising when imputing labor income, and because of the lack of sufficient time dimension in the 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 firms cannot be lower than the corresponding volatility of publicly traded firms, which the find 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 firms in the private sector, to estimate the volatility of employment growth rates. They find that, in 2001, the ratio of private to public volatility was in the range 1.431.75. Given that the average annual standard deviation for public firms over 19901997 was 0.11,^{27} and that there is, at least in the context of the present model, a close relationship between volatility of profits and volatility of labor demand, the choice of could also be justified from this perspective. Finally, this choice generates an annual variance for steadystate consumption growth in the range indicated by the micro data, once consumer heterogeneity is taken into account.^{28}
Parameters and are especially important for the calibrated model, for two reasons. First, they directly influence , the fraction of wealth held by entrepreneurs in the economy. And then, for example, as mentioned in section , when falls, i.e. when the entrepreneurs borrow more from the laborers, then, for given prices, falls as well.^{29} Given this importance of , the calibrated model's implications about are a good criterion of model performance. As will be shown in section , the choices and , which seem empirically relevant given the discussion above, produce, without an attempt to match it, a value for that is reasonably close to the values documented in the data. Second, parameters and relate to the interpretation of the capitalincome tax as providing insurance. For this reason, comparative statics will also be performed, in section , to show how the tax that maximizes the steadystate capital stock varies with risk aversion and the volatility of risk. The main result, that steadystate capital is inversely Ushaped with respect to the capitalincome tax, is preserved qualitatively for and for .
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 influenced by the randomwalk component introduced in wealth by the idiosyncratic investment risk. Second, to demonstrate how wealth inequality depends on the excess returns to entrepreneurship, which is an important question in its own right, but also in view of the impact of agent heterogeneity on capital accumulation. Third, to provide some additional confidence in the main quantitative results presented in the next section (section ) about the effects of capitalincome taxation on capital accumulation, by showing that the model performs well in matching aspects of the US aggregate and welfare data.
Table 2 presents the implications of the model for steadystate aggregates, and compares them to the data from the US economy. The model's capitaloutput ratio is . Investment is of GDP. The safe rate is . The steadystate fraction of entrepreneurs, , is , and it matches the data by choice of the transition probabilities, as explained in section
. Entrepreneurs hold of
total wealth in the economy, where the equivalent of in the data is the ratio ^{30}. The share of total wealth held by entrepreneurs in the data ranges between . The modelgenerated value for is an indication that the model performs reasonably well, especially given the low
value of 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
of the population is in the model, whereas in the data this number
ranges between .^{31}
Next, Table 3 examines the wealth distribution generated by the model. The first two rows present the percentiles for wealth computed by Quadrini (2000), using the PSID and SCF samples for
1994 and 1992, respectively. The last row is the conditional wealth distribution of the benchmark calibrated model.^{32}

Aiyagari's (1994) benchmark calibration predictions for the wealth holdings of the top and the top of the population are and , respectively. Hence, the present model demonstrates how the randomwalk 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 coefficient for wealth, conditional on wealth being positive, is 0.62. The model's Gini coefficient for consumption is 0.15.^{34} In the data, the Gini coefficient for total net worth is 0.8, and the Gini coefficient for consumption is 0.32.
Finally, Figure 2 presents the model's conditional wealth distributions over entrepreneurs and laborers. On the horizontal axis is wealth normalized by mean annual income in the economy. On the 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 fatter tail than the one for laborers. This is due to the randomwalk component that the uninsurable 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 wealth for both groups have significant mass of people with wealth higher than fifty times mean income. In the model, the laborers at the right tail of the wealth distribution are former successful entrepreneurs.
Having gained some confidence about the overall quantitative performance of the model, this section now proceeds to study the effects of capitalincome taxation in the benchmark calibration, where again the relevant parameters are those of Table 1. Subsection examines the steadystate effects of capitalincome taxation, while subsection examines the aggregate and welfare implications of abolishing the capitalincome tax.
This section quantifies the main theoretical result of the paper, which is that an increase in the capitalincome tax increases the steadystate capital stock, when the tax is low enough. As already explained, this result is due to the generalequilibrium effect 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 steadystate aggregates and welfare with respect to the capitalincome tax. Capital (panel ) and output (panel ) are inverselyU shaped with respect to the capitalincome tax, and they reach a maximum when . The same is true for employment, the capitallabor (capital per workhour) ratio, and output per workhour. At , steadystate capital per workhour is higher than when , and output per workhour is higher. As shown in Figure , aggregate welfare is maximized at , 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 effect of the tax, through the reduction in , and the effect of the tax on aggregates.
Figure shows that the net interest rate increases with the tax, and that it tends to the discount rate, , as . This is the demonstration of the precautionary saving motive mentioned in section : when the capitalincome tax increases, the effective volatility of risk facing an entrepreneur decreases, which reduces the demand for precautionary saving, and therefore increases the interest rate. Figure reinforces this interpretation of the capitalincome tax as providing insurance: when the tax increases, the precautionary saving motive becomes weaker, and therefore entrepreneurs are satisfied with a lower risk premium. Figure shows that the fraction of wealth held by entrepreneurs in the economy is decreasing in the capitalincome tax. This results from the combination of the weaker precautionary saving motive, and the fall in the risk premium.
At this point, it is useful to compare the effects of capitalincome 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 , steadystate capital per workhour and output per workhour are and , respectively, higher than when . By contrast, under complete markets,^{36} at , steadystate capital per workhour and output per workhour are and lower than when . And finally, in the openeconomy version of the model the aggregates fall all the way with the tax, but less so than under complete markets. In particular, at , steadystate capital per workhour and output per workhour are and lower than when . In addition, steadystate welfare is maximized when , which reflects 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, , and risk aversion, . On the vertical axis is the tax that maximizes the steadystate capital stock. When either the volatility of risk increases or risk aversion increases, the tax that maximizes the steadystate 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 and of . In particular, for the low value of , the capitalincome tax that maximizes the steadystate capital stock is positive for all , and it is actually zero when .
This section proceeds to examine the aggregate and welfare implications of eliminating the capitalincome tax. In the standard representativeagent completemarkets neoclassical model, the optimal capitalincome tax is zero in the long run, as well as in most of the short run for an interesting class of preferences. Steadystate welfare is also decreasing in the level of the capitalincome tax. These findings have initiated an extensive debate as to the possible benefits of eliminating the tax on capital income. By contrast, the main result of the present paper is that an increase in the capitalincome tax may actually increase the steadystate capital stock. In light of this result, it is worthwhile to revisit the discussion on the implications of setting the capitalincome tax to zero.
The effects on aggregates and welfare when the capitalincome tax is eliminated will be examined 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.
The present model can in fact examine the currentgeneration implications of policy reforms, because it is very tractable, compared to other incompletemarkets 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 significant 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 effects of policy may well be very different from the longrun effects.
Here, the economy starts from the steady state described by the benchmark calibration parameters in Table 1, where the capitalincome tax is . Subsequently, the tax is set to zero, ceteris paribus.
This section presents the shortrun and longrun responses of the aggregate variables to the policy reform that eliminates the capitalincome 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 effects on the interest rate, , the risk premium, , and the investmentoutput ratio, , 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 converges to a higher steadystate value, while consumption is initially lower and increases over time. In other words, the longrun increase in investment requires an initial period of lower consumption, which in turn allows for an immediate increase in investment as well. By contrast, under incomplete 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 investmentoutput ratio falls by more than 3 percentage units. These effects are also illustrated in Figure 5, which plots the impulse responses of the variables when the capitalincome tax is eliminated.
This section studies the welfare implications of eliminating the capitalincome tax. These implications are represented in terms of a compensating differential for each level of wealth and each type of agent, whether entrepreneur or laborer. In particular, starting from the old regime with , the question is what fraction of his financial 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.
Figure 6 presents the welfare implications of abolishing the capitalincome tax for entrepreneurs (solid line) and laborers (dashed line). Panel shows the welfare implications for the current generation, taking into account the entire transitional dynamics of the economy towards the new steadystate, and panel 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 differentials are on the vertical axis. A negative number on the vertical axis indicates an agent who benefits from the reform: the agent would have to be paid to be indifferent 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 capitalincome tax.
Figure shows that currentgeneration poor agents, whether entrepreneurs or laborers, prefer the zero capitalincome tax regime. As wealth increases, both entrepreneurs and laborers prefer the positive capitalincome tax regime. Finally, the mean cost of eliminating the tax is higher for the middleclass agents than for the very rich. Figure 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 crosssectional differences 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 longrun generation, the response of human wealth and of the (riskadjusted) returns to saving for laborers and entrepreneurs, against the tax rate of the policy reform. Let's first turn to the immediate implications. The decrease in the capitalincome tax from to 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 significant part of total wealth, and hence they benefit from the elimination of the tax. Furthermore, poor agents do not benefit much from insurance directly, since they invest little or nothing in the risky asset. Therefore, in the short run, poor agents prefer the zero capitalincome 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 capitalincome 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 steadystate wealth and capital accumulation. It turns out that the fall in the steadystate capital stock dominates the fall in the interest rate, so that in the end steadystate human wealth falls. This adversely affects poor agents of all types, since human wealth represents a big part of their total wealth. Because the riskadjusted return for entrepreneurs, , increases when the capitalincome 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.
In conclusion, the elimination of the capitalincome tax has welfare implications that differ across time and in the crosssection of the population. These differences are due to the generalequilibrium effects of the interest rate adjustment on capital accumulation. In particular, they operate mainly through the different 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, currentgeneration poor agents prefer a zero capitalincome tax in the short run, whereas futuregeneration poor agents prefer a positive capitalincome 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.
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 of total financial 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 firms 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 , where is the total capital and is the total labor employed by public firms. Here 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 , and public sector labor at the rate . By no arbitrage, in equilibrium, the riskfree rate will be equal to the marginal product of capital in the public sector, i.e. , and the wage will be equal to the marginal product of labor in the public sector, i.e. . The rest of the equilibrium characterization proceeds as 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 . Hence, we obtain the following variant of Proposition .
(31)  
(32)  
(33)  
(34)  
(35) 
The proof of Proposition is straightforward. Following similar steps as in the model with only the private sector, we can show that , and .
In the public sector, where there is no scope for insurance, an increase in the capitalincome tax unambiguously reduces investment, so that public capital is a negative function of the tax. As a result, the overall effect of the tax on the aggregate capital stock is in general ambiguous. In addition though, the increase in the capital tax might now trigger a reallocation of resources, away from the lowrisk and lowproductivity public sector, towards the higherrisk and higherproductivity private sector, thus increasing total factor productivity. Hence, aggregate output may increase with the tax, even if aggregate capital falls.
Coming now to the steadystate calibration,^{40} parameter is chosen so that the share of private capital in the aggregate capital stock is . As in the onesector economy, the calibration ensures that at , the interest rate is about , and that government spending as a fraction of GDP is . At this level of the tax, the private equity premium is , which is lower than its value of in the onesector economy, since there is now one additional (safe) asset.^{41} In the twosector economy, the capitalincome tax that maximizes aggregate capital and welfare is zero, and the per capita variables are all decreasing in the tax. However, at , the capitallabor ratio in the public sector is lower than it would have been if , i.e. it falls by less than under complete markets, where it would have fallen by . In fact, the capitallabor ratio in publicly traded firms is unambiguously always higher than under complete markets, since the interest rate under incomplete markets is lower than the discount rate. In addition, at , the ratio of total output to total labor falls by , whereas under complete markets it falls by .
To summarize, in steady state and for , outputper workhour is approximately higher than under complete markets when all production takes place in the private sector, and it is higher than under complete markets when the private sector accounts for of financial assets.
This paper studies the aggregate and welfare effects of capitalincome taxation in an environment where agents face uninsurable idiosyncratic entrepreneurial risk. The counterintuitive result emerging is that an increase in the capitalincome tax may actually, due to its generalequilibrium insurance aspect, stimulate capital accumulation. This result stands in stark contrast to the effects of capitalincome taxation in either completemarkets models, or in Bewleytype incompletemarkets models, since in those models capitalincome taxation necessarily discourages capital accumulation. Furthermore, the result is quantitatively significant: for the preferred calibration of the model, the steadystate levels of the capital stock, output, and employment are all maximized for a positive value of the capitalincome tax, at which point output per workhour is 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, it does not solve for the fully optimal policy. An interesting direction for future research is the formal study of optimal policy, either in the Ramsey tradition (though allowing for lumpsum 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 effects of uninsurable entrepreneurial risk, and abstracts from laborincome risk, borrowing constraints, and decreasing returns to scale at the individual level. Extending the model to include these relevant aspects of the data and revisiting the effects of capital taxation in this richer setting is important, not only to get a better quantitative evaluation of the implications of capital taxation, but also to examine whether the generalequilibrium effects identified here might interact with other sources of market incompleteness in an interesting way. For example, after an increase in the capitalincome tax, the increase in steadystate wealth documented here could make borrowing constraints less binding. At the same time, the increase in the steadystate interest rate could also increase the cost of borrowing. Further investigating these rich generalequilibrium interactions will greatly facilitate a better theoretical and quantitative assessment of the implications of fiscal policy in dynamic heterogeneousagent environments.
Proof of Proposition 3.1. Because of the CRRA/CEIS specification of preferences, guess that the value function for an entrepreneur is:
(38) 
Proof of Proposition 3.2. Let
be the effective riskfree rate. The human wealth for each individual in the economy is
. The human wealth of the measure group of entrepreneurs is
, and the human wealth of the measure
group of laborers is
. Hence total human wealth is
. Using the Leibniz rule, and substituting in from the government
budget constraint
, we get that the evolution of total human wealth is described by
. Since only entrepreneurs invest in capital, the aggregate capital stock in the economy is given by
. For an agent in the E and L group respectively,
and . Aggregating over each
group, we get
and . Adding up and
using the fact that
, we get
. Now using
and
, we get
. Combining
,
, and
, we get
. Aggregating across leisure choices we get
, where
,
, and
. Aggregating across
and
, and adding up, using
,
, and labor market clearing, we get:
Proof of Proposition 3.3. Consider first the case with and (and labor is exogenous, so ). Combining
or
in steady state, with
in steady state, and using the definitions of and
, we get equation
. Combining
with
in steady state , we get equation
.
Now, let and denote, respectively, the risk premium and the fraction of effective wealth held in capital, when is given by :
In order to see how the existence of two types of agents modifies the characterization of the steady state, consider next the case where , and take for simplicity. Then, the marginal propensity to consume is always constant and equal to , for both types of agents. Equation in steady state yields . Combining this with in steady state gives:
When labor is endogenous and , then , and the proofs above carry through the same way, only now with and . So for characterization of the steady state we need to add the labor market clearing condition, and the steady state system will be in . In particular, labor market clearing, combined with , , and gives:
Finally, when labor is endogenous and then:
Proof of Proposition 3.4. Let the newborn household receive a weighted average
, where , upon birth.^{44} Let be the indicator function, where for entrepreneurs and for laborers. The dynamic system for the state vector
, where ,
as in the text, is:
if  
if 
if  
if 
Let and be the conditional distributions for entrepreneurs and laborers respectively. In steady state the conditional distribution satisfies the forward Kolmogorov equation:
and the conditional distribution satisfies the forward Kolmogorov equation: In the two equations above we need to calculate:
Let
,
, and
, then
and
. So we have:
After changing variables to , and defining
we get:
Proof of Corollary 2. Consider the case with and . Then, using
or
and
in steady state, using the definition of
, and taking the total differential with respect to and , gives:
References