Optimal Capital Taxation with Idiosyncratic Investment Risk

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Optimal Capital Taxation with Idiosyncratic Investment Risk Vasia Panousi and Catarina Reis 2012-70 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.

Optimal Capital Taxation with Idiosyncratic Investment Risk Vasia Panousi and Catarina Reis Federal Reserve Board and Universidade Catolica Portuguesa∗ October 1, 2012 Abstract WeexaminetheoptimaltaxationofcapitalinaRamseysettingofageneral-equilibrium heterogeneous-agent economy with uninsurable idiosyncratic investment or capitalincome risk. We prove that the ex ante optimal tax, evaluated at steady state, maximizes human wealth, namely the present discounted value of agents’ income from sources that are not subject to capital risk. Furthermore, when the amount of idiosyncratic risk in the economy is higher than a minimum lower bound, the optimal tax is positive and it is precisely the tax that maximizes the economy-wide aggregates, such as the capital stock and output. By contrast, when the amount of risk is exogenously very low, the social planner (cid:28)nds it optimal to increase social risk taking by subsidizing investment in capital. ∗Email addresses: vasia.panousi@frb.gov, creis@ucp.pt. We would like to thank Peter Diamond, Glenn Follette, and Dimitris Papanikolaou for useful comments and discussions. We thank Felix Galbis-Reig for extremely insightful mathematical discussions, and Michael Barnett for excellent research assistance. The views presented in this paper are solely those of the authors 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 We study optimal capital taxation in a Ramsey framework and in an environment where agents face uninsurable idiosyncratic investment or capital-income risk. Such risk is empirically important for all investment decision makers, whether they are entrepreneurs and private business owners or managers of publicly traded (cid:28)rms. In this context, capital taxation raises an interesting trade-o(cid:27): On the one hand, it comes at the usual cost, as it distorts agents’ saving decisions. On the other hand, it has bene(cid:28)ts, as it provides agents with partial insurance against idiosyncratic capital-income risk. Moreover, this insurance aspect of capital taxation may even lead to higher capital accumulation in general equilibrium, contrary to what happens in models of complete markets or uninsurable labor income risk. Therefore, a positive tax on capital income could be welfare improving, and more so than in other Ramsey environments. Our modelling 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. These private businesses are subject to idiosyncratic risk that the agents cannot diversify away. However, agents are not exposed to labor-income risk, and they can also freely borrow and lend in a riskless bond. Abstracting from borrowing constraints, labor-income risk, and other market frictions, isolates the impact of the idiosyncraticinvestmentrisk, andpreservestractabilityofthemodel. Thereisagovernment, imposing a proportional tax on capital income, along with a non-contingent lump-sum tax or transfer. In a similar framework, Panousi (2012) performs comparative statics with respect to the capital income tax and (cid:28)nds that an increase in capital taxation may actually stimulate capital accumulation, due to a general equilibrium insurance aspect of the interest rate or safe rate adjustment. We fully characterize the optimal taxation problem, where the social planner maximizes ex ante welfare, taking into account the entire transitional dynamics of the economy toward the steady state with the optimal tax. We prove that the ex ante optimal tax, evaluated at steadystate, maximizeshumanwealth, namelythepresentdiscountedvalueofagents’future safe income, namely of income from wages or government transfers (or any other sources that are not subject to idiosyncratic risk). Furthermore, when the amount of idiosyncratic risk in the economy is higher than a minimum lower bound, the optimal tax is positive and it is precisely the tax that maximizes the economy-wide aggregates, such as the capital stock and output. By contrast, when the amount of risk is exogenously very low, then aggregate capital and output are falling with the tax at the optimum, and the social planner (cid:28)nds it optimal to increase social risk taking by subsidizing investment in capital. As a simple and illustrative example, we provide the case of an AK economy. There, if the interest rate is endogenous, the optimal tax is always constant, even o(cid:27) steady state, it maximizes the risk-adjusted growth rate of wealth (the di(cid:27)erence between the risk-adjusted return to saving and the marginal propensity to consume), and it is equal to 1 − A/σ2. Therefore, the optimal tax is positive when the mean return to the risky asset (capital), A, is lower than the variance of idiosyncratic capital-income risk, σ2. If the interest rate is exogenous, then the optimal tax is actually always exactly equal to zero. 2

Our paper belongs in the optimal taxation literature in the Ramsey tradition, though it also allows for lump-sum transfers, and it shows that the rationale for positive capital income taxation in the long run does not necessarily extend to the case where markets are incomplete due to the presence of idiosyncratic capital income risk. However, regardless of whether the ex ante optimal tax is positive or negative, the planner’s motivation for setting the optimal tax is always related to ensuring a su(cid:30)cient amount of social risk taking. When theexogenousriskintheeconomyissu(cid:30)cientlyhigh, thentheoptimalcapitaltaxispositive, sothattheensuinggeneralequilibriumadjustmentoftheinterestrateendogenouslyprovides agents with the insurance they need to undertake risky capital investment, despite the fact that the tax tends to reduce the mean return to saving. When the exogenous risk in the economy is su(cid:30)ciently low, then the optimal capital tax is negative, so that the capital subsidy encourages capital accumulation directly, by increasing the mean return to capital, despite the fact that the subsidy tends to reduce the interest rate and therefore wealth accumulation. This is because risk was low enough to begin with, and therefore the agents can tolerate the increase in the variance of risk resulting from the subsidy, both directly and indirectly through its general equilibrium e(cid:27)ects. 2 Related Literature We focus on an environment with idiosyncratic investment risk, because such risk is in fact empirically relevant for all investment decision makers, even in a (cid:28)nancially developed country like the United States. First, Moskowitz and Vissing-Jorgensen (2002), among others, (cid:28)nd that about 80 percent of all private equity in the US is owned by agents who are actively involved in the management of their own (cid:28)rm, and for whom such investment constitutes at least half of their total net worth. It seems plausible, then, that entrepreneurial risk must be even more prevalent in less developed economies. Second, Panousi and Papanikolaou (2012) show that the negative relationship between idiosyncratic risk and the investment of publicly-traded (cid:28)rms in the US is stronger in (cid:28)rms where the mangers hold a larger fraction of the (cid:28)rm’s shares. Combined, these (cid:28)ndings strengthen the empirical applicability of our model setup, because they demonstrate that a large fraction of total investment in the US is sensitive to idiosyncratic risk, essentially through the risk aversion of the agents making the investment decisions. This paper relates to the macroeconomic literature on optimal taxation. Most of this literature has focused either on complete markets or on incomplete markets with laborincome risk. Starting with the Ramsey literature of exogenously given market structure and exogenous policy instruments, Chamley (1986), Judd (1985), and Atkeson, Chari and Kehoe(1999)establishtheresultofzerooptimalcapitaltaxationwhenmarketsarecomplete. Correia (1996) shows that, in the neoclassical model, if there are restrictions on the taxation of production factors, then the tax rate on capital income in steady state is di(cid:27)erent from zero. Aiyagari (1995) extends the complete-markets framework to include uninsurable laborincome risk and borrowing constraints, and (cid:28)nds that the optimal capital tax is positive in 1 the long run. Chamley (2001) argues that it might be best to think of the reasoning behind 1A related but di(cid:27)erent normative exercise to that in Aiyagari (1995) is conducted by Davila, Hong, 3

Aiyagari’s positive optimal capital tax result as related to the ex-ante insurance or ex-post redistribution aspect of the tax: the planner taxes agents with high income realizations, and subsidizes agents with low income realizations, thereby equalizing consumption across di(cid:27)erent types of agents. Our paper contributes to the literature on optimal taxation in the Ramsey tradition, though also allowing for lump-sum transfers, and shows that, when markets are incomplete due to the presence of uninsurable capital-income risk, then the optimal capital tax will be di(cid:27)erent from zero, and will be positive or negative depending on general equilibrium insurance considerations. Moving to the Mirrlees literature of endogenous market incompleteness and endogenous policy instruments, Albanesi (2006) considers optimal taxation in a two-period model of entrepreneurial activity in a constrained-e(cid:30)ciency setting. Shourideh (2011) also studies the optimal taxation of entrepreneurial income. In his model, as in our paper, the intertemporal wedge determining the tax on wealth cannot be unambiguously signed. However, the incentive constraint seems to create a force towards a wealth subsidy, since increasing capital tends to loosen the incentive compatibility constraint in the future. In general, however, the extensive theoretical work on taxation originating from the Mirrlees tradition focuses on labor-income risk. This literature shows that, if insurance is limited due to the presence of asymmetricinformation, thenitmaybebesttorestrictfreeaccesstosavings. Thisresulthas in turn been interpreted as a justi(cid:28)cation for capital taxation. Some additional examples include Diamond and Mirrlees (1978), Golosov, Kocherlakota, and Tsyvinski (2003), Albanesi and Sleet (2006), and Golosov, Troshkin, Tsyvinsky and Weinzierl (2010). Farhi and Werning (2010) study optimal nonlinear taxation of labor and capital in a political economy model with heterogeneous agents, where policies are chosen sequentially over time, without commitment, as the outcome of democratic elections. They (cid:28)nd that credible policies show a concern for future inequality and that capital taxation emerges as an e(cid:30)cient redistributive tool for this purpose. The overlapping-generations literature has often found support for positive optimal capital taxation. Conesa, Kitao and Krueger (2009) quantitatively characterize the optimal capital and labor income taxes in an overlapping generations model with idiosyncratic uninsurable income shocks and permanent productivity di(cid:27)erences across households. They (cid:28)nd that the optimal capital-income tax rate is signi(cid:28)cantly positive at 36 percent, mainly 2 driven by the life-cycle structure of the model. Erosa and Gervais (2002), in an overlappinggenerations economy where agents’ productivity varies over time, (cid:28)nd that positive capital taxes may be optimal when labor taxes cannot be conditioned on age. Garriga (2003), and Peterman (2011) also (cid:28)nd similar results. A strand of the public (cid:28)nance literature has examined the e(cid:27)ects of capital taxation on risk taking, mostly in a partial equilibrium framework. Some examples include Domar and Musgrave (1944), Stiglitz (1969), Ahsan (1974), Sandmo (1977), and Kanbur (1981). These authors argued that, by e(cid:27)ectively reducing the variance of capital income, the capital tax allowed for increased social risk taking, leading to an increase in investment in the Krusell and Rios-Rull (2005), in the spirit of Geanakoplos-Polemarchakis. 2Domeij and Heathcote (2004) perform a similar exercise. Uhlig and Yanagawa (1996) show that higher capital income taxes lead to faster growth in an overlapping generations economy with endogeneous growth. 4

risky asset (capital). Our results are of similar (cid:29)avor, though re(cid:29)ecting general equilibrium considerations that this literature cannot capture, even in the case where the optimal tax turns out to be negative. As it turns out, a capital subsidy enhances risk taking in cases where the amount of exogenous risk in the economy was too low to begin with. Our paper is also related to Varian (1980), who assumes that di(cid:27)erences in observed income are due to exogenous di(cid:27)erences in luck. In a two-period model of endogenous saving, he (cid:28)nds that the optimal capital-income tax is positive, due to the trade-o(cid:27) it involves between the distortion in the saving decision and the provision of social insurance through redistribution. Finally, Angeletos and Panousi (2009) use a model similar to the one in the present paper to examine the e(cid:27)ects of government consumption on steady state aggregates, for the case where government spending is (cid:28)nanced solely through lump-sum taxes. 3 The basic model Time is continuous and indexed by t ∈ [0,∞). There is a continuum of in(cid:28)nitely lived households distributed uniformly over [0,1]. Each household consists of a worker and an entrepreneur. The worker is endowed with one unit of labor, supplied inelastically in a competitive labor market. The entrepreneur owns and runs a privately-held (cid:28)rm. Each (cid:28)rm employs labor in the competitive labor market, but can only use the capital stock invested by the particular household. Each (cid:28)rm is hit by idiosyncratic shocks, which the household can only partially diversify, as it cannot invest in other households’ (cid:28)rms. However, each household can freely save or borrow in a riskless bond (up to a natural borrowing constraint), which is in zero net supply. In terms of timing for the (cid:28)rm’s problem, (cid:28)rst capital is installed, then the idiosyncratic shock is realized, and lastly the labor choice is made. All uncertainty is purely idiosyncratic, and therefore aggregates are deterministic. Finally, the government imposes proportional taxes on savings and labor income, and balances the budget by giving back to agents, in the form of lump-sum transfers, the proceeds of taxation minus any government spending. Throughout the paper, for any variable y, the notation y is used as t short-hand notation for y(t), where t is time. 3.1 Households, (cid:28)rms, and idiosyncratic risk Preferences are logarithmic over consumption, c: (cid:90) ∞ U = E e−βsln(c ),ds (1) t t s t where β > 0 is the discount rate. The (cid:28)nancial wealth of a household i, denoted by ai, is the sum of its asset holdings in t private capital, ki, and in the riskless bond, bi, so that ai = ki +bi. The evolution of ai is t t t t t t given by the household budget: dai = (1−τ ) dπi +[ (1−τ )R bi + w +T −ci ]dt, (2) t t t t t t t t t 5

where dπi are the pro(cid:28)ts from the (cid:28)rm the household operates or the household’s capital t income, R is the the safe rate or the interest rate on the riskless bond, w is the wage rate in t t theaggregateeconomy,ci isconsumption,τ istheproportionalsavingsorcapital-incometax, t t applied to the income from the capital and the bond alike, and T are non-contingent lumpt sum transfers received from the government. A no-Ponzi-game condition is also imposed. Firm pro(cid:28)ts are subject to undiversi(cid:28)ed idiosyncratic risk: dπi = [ F(ki,li)−w li −δki ]dt + σkidzi. (3) t t t t t t t t Here, F is a constant-returns-to-scale neoclassical production function, assumed to be Cobb- Douglas for simplicity, namely F(k,l) = kαl1−α with α ∈ (0,1), where li is the amount of t labor the (cid:28)rms hires in the competitive labor market. In addition, δ is the mean depreciation rate in the aggregate economy. Idiosyncratic risk is introduced through dzi, a standard t Wiener process that is i.i.d. across agents and across time. Literally taken, dzi represents t a stochastic depreciation shock. However, these shocks can also be interpreted as stochastic productivity shocks. The scalar σ measures the amount of undiversi(cid:28)ed idiosyncratic risk, and it is an index of market incompleteness, with higher σ corresponding to a lower degree of risk-sharing, and with σ = 0 corresponding to complete markets. 3.2 Government At each point in time the government taxes capital income and bond income at the rate τ . The government also does some government spending at the rate G , where G does not t t t enter any production or utility functions. The proceeds of taxation, minus any government consumption, are then distributed back to the households in the form of non-contingent lump-sum transfers, T . The government budget constraint is therefore: t (cid:90) (cid:90) 0 = [ τ (F ( ki,1)−δ) ki −G −T ] dt, (4) t Kt t t t t i i (cid:82) (cid:82) where F ( ki,1) is the marginal product of capital in the aggregate economy, and li = 1. Kt i t i t For simplicity, we will henceforth set G = 0 for all t. 3 t 4 Equilibrium and steady state This section characterizes the equilibrium of the economy. First, it solves for households’ optimalplans,giventhesequencesofpricesandpolicies. Itthenaggregatesacrosshouseholds to derive the general equilibrium dynamics. 4.1 Individual behavior Entrepreneurs choose employment after their capital stock has been installed and their idiosyncratic shock has been observed. Hence, since their production function, F, exhibits 3None of our theoretical results hinge on this assumption. 6

constant returns to scale, optimal (cid:28)rm employment and pro(cid:28)ts are linear in own capital: li = l(w )ki and dπi = r(w )kidt+σ(1−τ )kidzi, (5) t t t t t t t t t wherel(w ) ≡ argmax [F(1,l)−w l]andr(w ) ≡ max [F(1,l)−w l]−δ. Here, r ≡ r(w ) t l t t l t t t is an 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, since there is no aggregate uncertainty. As in Angeletos (2007), the key result here is that entrepreneurs face linear, albeit risky, returns to their investment. To see how this translates to linearity of wealth in assets, let h denote a household’s human wealth, namely the present t discounted value of net-of-taxes labor endowment plus government transfers: (cid:90) ∞ h t = e− (cid:82) t s(1−τj)Rj dj(w s +T s )ds. (6) t Next, de(cid:28)ne total e(cid:27)ective wealth, xi, as the sum of (cid:28)nancial wealth and human wealth: t xi ≡ ai +h = ki +bi +h . (7) t t t t t t Total e(cid:27)ective wealth is then the only state variable relevant for the household’s optimization problem. The only constrained imposed is that consumption is non-negative, which implies non-negativity of total e(cid:27)ective wealth, so that: xi (cid:62) 0 ⇔ xi (cid:62) −h . t t t In other words, there is no ad hoc borrowing constraint, and agents can freely borrow and lend up to the natural borrowing limit. The evolution of total e(cid:27)ective wealth is then described by: dxi = [ (1−τ )r ki +(1−τ )R (bi +h )−ci ]dt+σ(1−τ )kidzi. (8) t t t t t t t t t t t t The (cid:28)rst term in 8 captures the expected rate of growth of e(cid:27)ective wealth, and it shows that wealth grows when entrepreneurial saving exceeds consumption expenditures. The second term captures the e(cid:27)ect of idiosyncratic risk. This linearity of wealth in assets, together with the homotheticity of preferences, ensures that the household’s consumption-saving problem reduces to a tractable homothetic optimization problem, as in Samuelson’s and Merton’s classic portfolio analysis. Therefore, the optimal individual policy rules will be linear in total e(cid:27)ective wealth, for given prices and government policies, as the next proposition shows. Proposition 1. Let {w ,R ,r } and {τ ,T } be equilibrium price and policy set t t t∈[0,∞) t t t∈[0,∞) quences. The household maximizes preferences as described in (1) subject to the total e(cid:27)ective wealth evolution constraint (8). The optimal consumption, investment, and bond holding choices, respectively, are given by: ci = m xi, ki = φ xi, bi = (1−φ )xi −h , (9) t t t t t t t t t t where the fraction of e(cid:27)ective wealth invested in capital, φ , is given by: t (1−τ )r −(1−τ )R t t t t φ = , (10) t σ2(1−τ )2 t and the marginal propensity to consume, m , is constant and equal to the discount rate in t preferences, i.e. m = β for all t. t 7

The fact that investment is subject to undiversi(cid:28)able idiosyncratic risk introduces a wage between the marginal product of capital and the risk-free rate, so that (1 − τ )R < t t (1 − τ )r . In other words, it has to be that, in equilibrium, the mean return to the risky t t asset (capital) exceeds the mean return to the safe asset (bond) by an amount equal to the positive (private) risk premium agents require as compensation for undertaking risky investment. The fraction of wealth invested in the risky asset, φ , is then increasing in this t risk premium, and decreasing in the e(cid:27)ective variance of risk, σ(1−τ ). Furthermore, it is t the same across all entrepreneurs and it does not depend on the level of wealth. In equation (9), optimal consumption is a linear function of total e(cid:27)ective wealth, where the marginal propensity to consume, m , is also independent of wealth. Moreover, because preferences t are logarithmic, the marginal propensity to consume is constant over time and equal to the discount rate in preferences, β. The wealth evolution constraint, incorporating bond market clearing and individual optimization, is: dxi = [(1−τ )r φ +(1−τ )R (1−φ )−β]xidt+σ(1−τ )φ xidzi. (11) t t t t t t t t t t t t Using (11) and Proposition 1, we get the following characterization for individual consumption dynamics. Lemma 1. The evolution of individual consumption, investment, and wealth is given by: dci dxi t = t = (ρ −β)dt+σ(1−τ )φ dzi, (12) ci xi t t t t t t where ρ = (1−τ )φ r +(1−τ )(1−φ )R is the total return to saving. Solving for ci, the t t t t t t t t evolution of individual consumption is: (cid:90) t (cid:90) t ci = ci · exp{ (ρˆ −β) ds + σ(1−τ )φ dzi}, (13) t 0 t s s s 0 0 where ρˆ = ρ − 1σ2(1−τ )2φ2 is the risk-adjusted return to saving. t t 2 t t The proof of equation (13) follows from Ito’s lemma. Here, ρ is the mean return to t saving, namely the total or overall portfolio return for the household. In other words, the totalreturntosavingisaweightedaverageofthe(net-of-tax)marginalproductofcapitaland the (net-of-tax) risk-free rate. Then, the risk-adjusted return to saving, ρˆ, is the certainty t equivalent of the overall portfolio return, and is lower than ρ because agents are risk averse t and face risk in their consumption stream. It follows that, in equilibrium, it will have to be (1−τ )R < ρˆ < ρ < (1−τ )r . t t t t t t 4.2 General equilibrium The initial position of the economy is given by the cross sectional distribution of (ki,bi) 0 0 across households. Households choose plans {ci,li,ki,bi,xi} for i ∈ [0,1], contingent t t t t t t∈[0,∞) on the history of their idiosyncratic shocks, and given the price sequence and the government 8

policy, so as to maximize their lifetime utility. Idiosyncratic risk washes out in the aggregate. Anequilibriumisthende(cid:28)nedasadeterministicsequenceofprices{w ,R ,r } , policies t t t t∈[0,∞) {τ ,T } , and macroeconomic variables {C ,K ,Y ,L ,X } , along with a collection t t t∈[0,∞) t t t t t t∈[0,∞) of individual contingent plans {ci,li,ki,bi,xi} for i ∈ [0,1], such that the following t t t t t t∈[0,∞) conditions hold: (i) given the sequences of prices and policies, the plans are optimal for (cid:82) the households; (ii) the labor market clears, li = 1, in all t; (iii) the bond market clears, (cid:82) i t bi = 0, in all t; (iv) the government budget constraint (4) is satis(cid:28)ed in all t; and (v) the t t (cid:82) (cid:82) (cid:82) aggregates are consistent with individual behavior, C = ci, L = li = 1, K = ki, (cid:82) (cid:82) (cid:82) t i t t i t t i t Y = F(ki,li) = F( ki,1), X = xi, in all t. Note that the aggregates do not depend t i t t i t t i t on the extend of wealth inequality, because individual policies are linear in wealth. De(cid:28)ne f(K) ≡ F(K,1) = Kα. From Proposition 1, the equilibrium ratio of capital to e(cid:27)ective wealth and the equilibrium mean return to savings are identical across agents and can be expressed as functions of the capital stock and risk-free rate: φ ≡ φ(K ,R ) and t t t ρ ≡ ρ(K ,R ). Similarly, the wage is w ≡ w(K ) = f(K ) − f(cid:48)(K )K = (1 − α)f(K ). t t t t t t t t t Using this, aggregating the policy rules across agents, and imposing bond market clearing, we arrive at the following characterization of the general equilibrium. Proposition 2. In equilibrium, the aggregate dynamics satisfy: ˙ X t = ρ −β (14) t X t ˙ H = (1−τ )R H −w −τ (F −δ)K (15) t t t t t t Kt t φ t K = H , (16) t t 1−φ t along with m = β. t Condition(14)followsfromaggregatingtheindividualwealthevolutionconstraintsacross agents, and using (9). It captures the evolution of total e(cid:27)ective wealth, and shows that wealth grows when the mean net-of-tax return to saving, ρ , exceeds the marginal propensity t toconsume,β. Condition(15)istheevolutionofhumanwealth,combinedwithfactormarket clearing and with the intertemporal government budget. Condition (16) represents clearing in the bond market and ensures that the bond is in zero net supply in the aggregate. From this point on, and for simplicity, we let δ = 0 without loss of generality. 4 The aggregate resource constraint of the economy is then: dK = [Kα −C ]dt. (17) t t t 4.3 Steady state The steady state is the (cid:28)xed point of the dynamic system described in Proposition 2. The following proposition characterizes the steady state. 4None of the theoretical results hinge on this assumption. 9

Proposition 3. The steady state always exists, and it is the unique solution to the following system of two equations in the capital stock, K, and the interest rate, R: (cid:112) r(K) = R+ σ2(β −(1−τ)R) (18) φ(K,R) w(K)+τr(K)K K = , (19) 1−φ(K,R) (1−τ)R where φ(K,R) ≡ (r(K)−R)/(σ2(1−τ)), r(K) ≡ f(cid:48)(K) = αKα−1, and w(K) ≡ (1−α)Kα. Equation (18) combines stationarity of wealth from (14) with the de(cid:28)nition of the mean return to saving, ρ = (1−τ)rφ+(1−τ)R(1−φ). Note from (14) that wealth stationarity requires ρ = β, namely that the mean return to saving is equal to the marginal propensity to consume. But since (1 − τ)R < ρ, it follows that (1 − τ)R < β in steady state. This is a manifestation of the precautionary saving motive, with a rationale similar to that in Aiyagari (1994). 5 Equation (19) follows from stationarity of human wealth in (15), bond market clearing in (16), and the government budget constraint. Lastly, we note that, in this model, as shown in Angeletos (2007), steady state capital is below complete markets, if and only if the elasticity of intertemporal substitution is higher than the ratio φ/(2−φ), where φ is the fraction of total e(cid:27)ective wealth invested in capital. Here, because the elasticity of intertemporal substitution is 1 with log preferences, and because φ < 1, the steady state aggregates will always be below the corresponding ones in complete markets. 5 The ex ante optimal tax The optimal tax in our framework is the one maximizing ex ante welfare. In particular, assume that at time t = 0 the economy rests at an arbitrary steady state, and that the social planner is considering implementing an unexpected policy change at some future time t = u. The question posed from an ex ante perspective is: What is the impact of that policy change on agents’ welfare at time t = 0, taking into account the entire transitional dynamics of the economy toward the new steady state resulting from the policy change? The tax maximizing agents’ utility at time t = 0 is then the (ex ante) optimal tax. We will (cid:28)rst characterize the planner’s problem of maximizing ex ante welfare, and then we will evaluate the solution of 6 the problem in steady state. 5In particular, agents have a precautionary saving motive, because the idiosyncratic investment risk generates risk in their consumption stream. Therefore, if the net interest rate were higher than the discount rate in preferences, savings and wealth would explode, which violates the notion of steady state. In fact, the netinterestratehastobelowerthanthediscountratebyexactlyasmuchasisneededforthecorresponding substitution e(cid:27)ect of a lower saving return to exactly o(cid:27)set the precautionary saving motive. 6Notehowever,thatwearetakingthetransitiontothenewsteadystateexplicitlyintoaccount,andthat we are not imposing a constant tax along the transition. 10

5.1 The planner’s problem From Proposition 1, it follows that the value function for an agent with initial wealth xi at 0 t = 0, given the tax sequence {τ }∞ , is the solution to the problem: t t=0 (cid:90) ∞ V(xi;{τ }∞ ) = max E e−βtln(ci)dt, (20) 0 t t=0 ci,φi −1 t 0 where the maximands are the optimal policy functions described in Proposition 1. Thesocialplanner’sobjectiveistochoosethetaxsequence{τ }∞ thatmaximizesexante t t=0 expectedutility, subjecttotheconditionsforindividualoptimizationandgeneralequilibrium ˜ in section 4. The planner’s objective function, W, is then weighted sum of agents’ value functions, Vi, where the weights, ψ(xi;τ ), depend on the initial wealth of each agent: 0 t (cid:90) W ˜ (xi;{τ }∞ ) = max V(xi;τ )ψ(xi;τ )dxi . (21) 0 t t=0 {τt}∞ t=0 0 t 0 t 0 i Without loss of generality, we will assume that at t = 0 the wealth distribution is concentrated at one point, so that all agents hold the same amount of capital, equal to the economy-wide aggregate capital stock, and therefore receive the same weight in the planner’s objective, so that ψ(xi;τ ) = 1. The following proposition characterizes the planner’s 0 t problem of maximizing ex ante welfare. Proposition 4. The planner chooses the sequence of taxes {τ }∞ to maximize the objective t t=0 function: (cid:90) ∞ (cid:90) t 1 W ˜ (xi;{τ }∞ ) = e−βt{ln(βxi)+ [(1−τ )r φ +(1−τ )R (1−φ )−β− σ2(1−τ )2φ2]ds}dt , 0 t t=0 0 s s s s s s 2 s s 0 0 (22) subject to the following constraints: r −R t t φ = , (23) t σ2(1−τ ) t dK = [Kα −C ]dt , (24) t t t dX /X = [(1−τ )r φ +(1−τ )R (1−φ )−β]dt , (25) t t t t t t t t r = αKα−1, w = (1−α)Kα , (26) t t t t T = τ r K . (27) t t t t Hence, at time t = 0, the (cid:28)rst order condition with respect to a change in the capital tax in period t = u is: ˜ ˜ ˜ ˜ ˜ ˜ dW ∂W dx ∂W dφ ∂W dr ∂W dR ∂W 0 = + + + + = 0 . (28) dτ ∂x dτ ∂φ dτ ∂r dτ ∂R dτ ∂τ u 0 u u u u u u u u 11

The planner’s objective follows from equations (9) and (13), and the fact that all uncertainty is idiosyncratic, namely that E (cid:82)t σ(1 − τ )φ dzi = 0. Equation (23) describes i 0 s s s the optimal portfolio allocation from Proposition 1, equation (24) is the aggregate resource constraint in the economy, equation (25) is the aggregate wealth evolution constraint using the de(cid:28)nition of the mean return to saving ρ , equation (26) is market clearing for the factors t of production, and equation (27) is the government budget constraint. Therefore, the planner maximizes an objective consisting of two terms. The (cid:28)rst captures the e(cid:27)ect of the entire path of future prices and tax policies on e(cid:27)ective wealth at time zero, x . This is because e(cid:27)ective wealth is the sum of asset holdings, a = k + b , which are 0 0 0 0 historically given, and human wealth, h , which is the present discounted value of future 0 wages and transfers (i.e. of future safe income), and therefore depends on the entire future path for the tax, the wage, the return to capital, and the interest rate. The second term in the planner’s objective captures both the direct e(cid:27)ects of the capital tax on the mean return to saving and on the e(cid:27)ective volatility of risk, as well as the indirect e(cid:27)ects of the tax through the corresponding adjustment of the risky return, r , the risk-free rate, R , and t t portfolio allocation, φ . Note that the term (1−τ )r φ +(1−τ )R (1−φ )− 1σ2(1−τ )2φ2 t t t t t t t 2 t t in the planner’s objective is actually the risk-adjusted return to saving, ρˆ. t In other words, the planner has to weigh two considerations when choosing the optimal tax sequence. First, how the path of the taxes will a(cid:27)ect the paths of wages and prices, thereby possibly maximizing time-zero wealth, x , through h . Second, whether the path of 0 0 the taxes will maximize the di(cid:27)erence between the risk-adjusted return to saving, ρˆ, and t the marginal propensity to consume, β, i.e. the risk-adjusted rate of growth of household consumption and wealth (see equation (13)). We can also see from (28) that, if x did not include human wealth, h , then x = 0 0 0 k +b would be historically given and therefore not relevant for the planner’s maximization 0 0 problem. In turn, absence of human wealth means that there is only risky income in the economy, as in an AK version of our model. In that case, the planner chooses the path of taxes to maximize ρˆ −β, which is exactly the di(cid:27)erence between the risk-adjusted return to t savingandthemarginalpropensitytoconsume, ortherisk-adjustedgrowthrateofindividual consumption and wealth. The next step is now to characterize each one of the derivatives in equation (28) in turn. Simple di(cid:27)erentiation of the objective function yields: ∂W ˜ 1 (cid:90) ∞ = e−βsds (29) ∂x x 0 0 0 ˜ ˜ ˜ ˜ The calculation of the derivatives ∂W/∂φ , ∂W/∂r , ∂W/∂R , and ∂W/∂τ is more u u u u complicated. Intuitively, these derivatives indicate the e(cid:27)ect on the objective at time t = 0 of a change in the functions φ , r , and R at time t = u, due to a tax change at that point t t t in time. Hence, the calculations use the de(cid:28)nition of a functional derivative, with the Dirac delta function as the appropriate test function. In turn, the Dirac delta function allows for an impulse change in the functions at time t = u, due to a change in the tax at that time, 12

while the functions remain unchanged at all other points in time. The outcome is: ∂W ˜ (cid:90) ∞ = [(1−τ )r −(1−τ )R −σ2(1−τ )2φ ] e−βsds (30) u u u u u u ∂φ u u ∂W ˜ (cid:90) ∞ = (1−τ )φ e−βsds (31) u u ∂r u u ∂W ˜ (cid:90) ∞ = (1−τ )(1−φ ) e−βsds (32) u u ∂R u u ∂W ˜ (cid:90) ∞ = [−r φ −R (1−φ )+σ2(1−τ )φ2] e−βsds . (33) ∂τ u u u u u u u u ˜ From (30), and using portfolio allocation from (10), we can see that ∂W/∂φ = 0. This u is because the optimal choice of φ actually maximizes the risk-adjusted return to saving, ρˆ. Equations (31) and (32) capture indirect e(cid:27)ects of the tax on welfare operating through asset returns, for the risky (capital) and the riskless (bond) asset, respectively. The weights on these terms, which depend on φ , capture the importance of each asset on the overall u portfolio. Equation (33) captures the direct e(cid:27)ect of the tax on welfare and consists of two terms. The (cid:28)rst term is −[r φ +R (1−φ )], and is negative, re(cid:29)ecting the standard result u u u u that the tax lowers the mean return to saving. The second term is σ2(1 − τ )φu, and is u t positive, re(cid:29)ecting the fact that the tax directly provides some insurance by lowering the e(cid:27)ective variance of risk, σ(1 − τ ). Using portfolio allocation from (10), we get that the u term in brackets in (33) is equal to −R . u In order to calculate the derivative dx /dτ , note (cid:28)rst that we can write x as: 0 u 0 (cid:90) ∞ x 0 ({τ t }∞ t=0 ) = k 0 +b 0 + e− (cid:82) 0 sp(τj)djq(τ s )ds , (34) 0 where a ≡ k +b is historically given and where: 0 0 0 p(τ ) ≡ (1−τ )R(τ ), (35) t t t q(τ ) ≡ w +T = [(1−α)αα/(1−α) +τ α1/(1−α)]r(τ )α/(α−1) , (36) t t t t t are, respectively, the equilibrium after-tax interest rate and the equilibrium safe income from wages and transfers, both as a function of prices only. The latter is given by the sum of w = (1 − α)(r /α)α/(α−1) and T = τ rα/(α−1)α1/(1−α), using equations (26) and (27). t t t t t Then, using the de(cid:28)nition of a functional derivative, the Dirac delta function, and the series expansion for the exponential, we get: dx dh dq dp (cid:90) ∞ 0 = 0 = e− (cid:82) 0 up(τs)ds − q t e− (cid:82) 0 tp(τs)dsdt , (37) dτ dτ dτ dτ u u u u u where dp dR = −R(τ )+(1−τ ) , (38) u u dτ dτ u u 13

dq dr = α1/(1−α)r(τ )α/(α−1) +[(1−α)αα/(1−α) +τ α1/(1−α)]α(α−1)−1r(τ )1/(α−1) . (39) u u u dτ dτ u u Thisleavesuswiththederivativesdφ/dτ ,dr/dτ ,anddR/dτ . Unfortunately,wecannot u u u explicitly characterize these derivatives along the transition. Therefore, in the next section, we will proceed to evaluate them in steady state. 5.2 Evaluating the ex ante optimal tax in steady state 7 In this section, we will evaluate the ex ante optimal policy in steady state. In steady state, andusingthefactthatrK = αKα, C = βX, andK = φX, theaggregateresourceconstraint (17) yields: rφ = αβ . (40) Substituting this as well as portfolio allocation from (10) into the aggregate wealth evolution constraint (25) in steady state yields: αβ (1−τ)(1−φ)( −φσ2(1−τ))−β[1−(1−τ)α] = 0 . φ De(cid:28)ne the left-hand-side of this equation as: αβ G(φ,τ) ≡ (1−τ)( −φσ2(1−τ))(1−φ)−β[1−(1−τ)α] (41) φ From the implicity function theorem it follows that: dφ ∂G/∂τ = − (42) dτ ∂G/∂φ where ∂G αβ = −( −φσ2(1−τ))(1−φ)+(1−τ)(1−φ)φσ2 −αβ, (43) ∂τ φ ∂G αβ αβ = (1−τ)(1−φ)(− −σ2(1−τ))−(1−τ)( −φσ2(1−τ)) . (44) ∂φ φ2 φ Hence, dφ/dτ could be either positive or negative. Using portfolio allocation (10) and (40), we then get respectively: dR dr dφ = − σ2(1−τ)+φσ2 , (45) dτ dτ dτ dr αβ dφ = − . (46) dτ φ2 dτ At this point, note also that, since r = αKα−1, it follows: dr dK = α(α−1)Kα−2 (47) dτ dτ 7In steady state, the variables are not a function of time, as time goes to in(cid:28)nity. 14

Therefore, when dK/dτ > 0, then dr/dτ < 0 and also dφ/dτ > 0 from (46). In other words, K and φ always have the same monotonicity with respect to the tax. The possibility that the capital stock might be increasing with the capital tax over some range of taxes has been studied in Panousi (2012). Here note that K and φ will be maximized for the same value of the capital tax. However, this need not be the case for human wealth, which in steady state is H = q/p. Combining 16, (46), and (47), we get that: dH dφ K α−φ = . (48) dτ dτ φ2 1−α We can show that φ < (1−τ)α in steady state. If the tax is positive, then φ < α always, and therefore H has the same monotonicity as K and φ. If the tax is negative, though, then α−φ might be either positive or negative, and it may change sign endogenously as the tax, and therefore φ, changes. Hence, for positive taxes, H gets maximized at the same point as K and φ. However, for negative taxes, if φ > α, it is possible that H increases, even while K falls. We will return to this point in what follows. This completes the derivation of all the equations needed to evaluate the planner’s (cid:28)rst order condition (28) in steady state. Plugging everything we have derived so far into (28), imposingsteadystate, doingsomealgebra, andusingthenotationy todenotethederivative τ of a variable y with respect to the tax in steady state, we get that the optimal tax in steady state solves: q e−βu lim{(q −p )+x [(1−τ)φr +(1−τ)(1−φ)R −R] } = 0 (49) u→∞ τ τ p 0 τ τ e−pu where the second term is zero as u → ∞, because p = (1−τ)R < β. Therefore, the following proposition characterizes the optimal tax in steady state. Proposition 5. The ex ante optimal tax, evaluated at steady state, solves: dq dp p(τ)− q(τ) = 0 , (50) dτ dτ where p(τ) = (1−τ)R(τ) is the steady state after-tax interest rate, q(τ) = [(1−α)αα/(1−α)+ τα1/(1−α)]r(τ)α/(α−1) is the steady state sum of wages and transfers (safe income), and dp/dτ and dq/dτ are given by (38) and (39) evaluated at steady state. Since H = q/p is the steady state value of human wealth, (50) can be equivalently written as: dH = 0 . (51) dτ In other words, conditions (50) or (51) show that, evaluated at steady state, the ex ante optimal tax maximizes human wealth, H, i.e. the present discounted value of wage and government transfer income. However, the optimal tax cannot be unambiguously signed, and could therefore be either positive or negative. If the optimal tax turns out to be positive, then it also maximizes aggregate capital, K, and aggregate consumption, C, in addition to also maximizing human wealth, H. This case 15

is illustrated in Figure 1. The top left panel plots the planner’s (cid:28)rst order condition (28), evaluated atsteady state, against the capital tax. The optimal tax is determined at the point where the (cid:28)rst order condition curve intersects the horizontal axis, which in this example occurs at τ = 0.33. At that point, dφ/dτ = 0, as shown in the top right panel, which means that the share of wealth invested in risky capital is at a maximum. Furthermore, the bottom left and rights panels show that aggregate capital, K, and human wealth, H, also attain their maximum at the same tax. If the optimal tax is negative, then it will not maximize capital and, for example, capital may be falling at the optimum. This case is illustrated in Figure 2. The top left panel again plots the planner’s (cid:28)rst order condition (28), evaluated at steady state, and shows that the optimal tax is actually a subsidy at τ = −0.13. The top right panel show that, at the optimal tax, dφ/dτ < 0, which means that the share of capital in wealth is falling at the optimum. The bottom left panel captures the fact that, at the optimum, human wealth is at a maximum, as shown in Proposition 5. The bottom right panel shows that, as is the case with φ, capital is also falling at the optimum. Note that this is a case where σ = 0.1, whereas the rest of the parameters remain as in Figure 1. We could also obtain results of similar (cid:29)avor if we set α = 0.1, while keeping the rest of the parameters as in Figure 1. In other words, when risk in the economy is below a minimum lower bound, as captured either by low volatility of risk or low returns to the risky asset in the production, then the planner (cid:28)nds it optimal to subsidize capital. This increases the e(cid:27)ective variance of risk, but it also increases capital accumulation, as capital is a decreasing function of the tax. The agents can in fact undertake this increase in risk endogenously, because the exogenous risk in the economy was too low to begin with. Hence, as already discussed, the optimal tax is always the one that maximizes the steady state value of human wealth, H. However, at the optimum, capital and the rest of the aggregates will either be at a maximum or they will be falling with the tax. Calibrations show that the latter will be the case when the variance of idiosyncratic risk, σ2 is very low, see Figure 3, panel (b). This is a case where basically there is not enough risk taking in the economy. By subsidizing capital, the planner increases the e(cid:27)ective variance of risk, (1−τ)2σ2, and also levies a lump sum tax on agents, thereby reducing their human wealth. In addition, because capital is falling all the way with the tax, the subsidy increases capital accumulation, which translates into endogenously higher risk taking in the economy. Agents are actually able to undertake this endogenous increase in risk, as the exogenous variance of risk, σ2, was very low to begin with. 5.3 A simple example: the AK model TheAK versionofthepresentmodelistheonewherethereisnosafeincomeintheeconomy, such as income from wages or government transfers. In this case, human wealth is zero, h = 0. Hence, as already mentioned in section 5 and indicated in Proposition 4, the planner 0 thenmaximizesthedi(cid:27)erencebetweentherisk-adjustedreturntosaving, ρˆ, andthemarginal propensity to consume, β, i.e. the risk-adjusted rate of growth of consumption and wealth. 16

In that case, the planner’s (cid:28)rst order condition becomes: ˜ ˜ ˜ ˜ ˜ dW ∂W dφ ∂W dr ∂W dR ∂W t t t = + + + = 0 . (52) dτ ∂φ dτ ∂r dτ ∂R dτ ∂τ t t t t t t t t Note that in the AK model, r = A for all t and hence dr /dτ = 0. In addition, since all t t t income is risky, φ = 1 for all t, which from (10) implies that R = A−σ2(1−τ) and also t that dR/dτ = σ2. In addition, ∂W ˜ /∂R = 0, and again ∂W ˜ /∂φ = 0. Putting all of this t t together, the planner’s (cid:28)rst order condition yields: A −R = 0 ⇔ τ = 1− (53) σ2 Note that this is the case even o(cid:27) the steady state (we have not imposed any steady state restrictions here), and therefore the optimal tax in the AK model is always constant, even along the transition. The optimal tax is positive when A < σ2, i.e. when the mean return to the risky asset is lower than the variance of idiosyncratic risk, a condition likely to be satis(cid:28)ed in the data. By contrast, if the variance of risk is very low, then the optimal tax will actually be negative, re(cid:29)ecting the planner’s desire for increased risk taking in the economy. 6 Conclusions We study the optimal taxation of capital in a Ramsey setting of a general-equilibrium heterogeneous-agent economy with uninsurable idiosyncratic investment or capital-income risk. We prove that the ex ante optimal tax, evaluated at steady state, maximizes human wealth, namely the present discounted value of agents’ income from sources that are not subject to capital risk. Furthermore, when the amount of idiosyncratic risk in the economy is higher than a minimum lower bound, the optimal tax is positive and it is precisely the tax that maximizes the economy-wide aggregates, such as the capital stock and output. By contrast, when the amount of risk is exogenously very low, the social planner (cid:28)nds it optimal to increase social risk taking by subsidizing investment in capital. Our paper contributes to the optimal taxation literature in the Ramsey tradition, though also allowing for lump-sum transfers, and it shows that the rationale for positive capital income taxation in the long run does not necessarily extend to the case where markets are incomplete due to the presence of idiosyncratic capital income risk. However, regardless of whether the ex ante optimal tax is positive or negative, the planner’s motivation for setting the optimal tax is always related to ensuring a su(cid:30)cient amount of social risk taking. When theexogenousriskintheeconomyissu(cid:30)cientlyhigh, thentheoptimalcapitaltaxispositive, sothattheensuinggeneralequilibriumadjustmentoftheinterestrateendogenouslyprovides agents with the insurance they need to undertake risky capital investment, despite the fact that the tax tends to reduce the mean return to saving. When the exogenous risk in the economy is su(cid:30)ciently low, then the optimal capital tax is negative, so that the capital subsidy encourages capital accumulation directly, by increasing the mean return to capital, despite the fact that the subsidy tends to reduce the interest rate and therefore wealth accumulation. This is because risk was low enough to begin with, and therefore the agents 17

can tolerate the increase in the variance of risk resulting from the subsidy, both directly and indirectly through its general equilibrium e(cid:27)ects. In both cases, the outcome of the optimal tax is therefore higher investment in the risky asset, higher capital accumulation, and a higher amount of social risk taking in the economy. 18

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Figure 1: Ex ante welfare 1: optimal tax maximizes capital 4000 3000 2000 1000 0 −0.5 0 0.5 1 tax K ,latipac 12000 10000 8000 6000 4000 2000 0 −0.5 0 0.5 1 tax p/q=H ,htlaew namuh 0.2 0 −0.2 −0.4 −0.6 −0.8 −0.5 0 0.5 1 tax τd/φd 5 0 −5 −10 −15 −0.5 0 0.5 1 tax cof Figure 1 uses parameter values α=0.7, σ =0.5, β =0.98. The top left panel plots the (cid:28)rst order condition (28) against the capital tax. The line cuts the horizontal axis at the ex ante optimal tax, which is positive at τ = 0.33. The top right panel plots the derivative of φ against the tax. This derivative is zero at the optimal tax. The bottom left panel plots human wealth, H, against the tax. Human wealth is maximized at the optimal tax, τ =0.33. The bottom right panel plots aggregate capital, K, against the tax. Capital is maximizes at the optimal tax, τ =0.33. 21

Figure 2: Ex ante welfare 2: optimal tax does not maximize capital 5 x 10 4 3 2 1 0 −0.5 0 0.5 1 tax K ,latipac 4 x 10 6 5 4 3 2 1 0 −0.5 0 0.5 1 tax p/q=H ,htlaew namuh 0 −0.5 −1 −1.5 −2 −2.5 −3 −0.5 0 0.5 1 tax τd/φd 10 0 −10 −20 −30 −40 −0.5 0 0.5 1 tax cof Figure 2 uses parameter values α=0.7, σ =0.1, β =0.98. The top left panel plots the (cid:28)rst order condition (28)againstthecapitaltax. Thelinecutsthehorizontalaxisattheexanteoptimaltax,whichisnegativeat τ =−0.13. The top right panel plots the derivative of φ against the tax. This derivative is always negative, indicating that φ is falling at the optimum. The bottom left panel plots human wealth, H, against the tax. Human wealth is maximized at the optimal tax, τ =−0.13. The bottom right panel plots aggregate capital, K, against the tax. Capital is falling at the optimum. 22

Figure 3: Comparative statics 0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 0.2 0.3 0.4 0.5 0.6 0.7 share of capital in production, α xat lamitpo 0.6 0.5 0.4 0.3 0.2 ex ante steady state 0.1 0 −0.1 −0.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 volatility of risk, σ (a) Share of Capital xat lamitpo ex ante steady state (b) Volatility of Risk 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 0.89 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 discount factor xat lamitpo ex ante steady state (c) Discount Factor Figure 3 performs robustness tests with respect to the main model parameters. On the horizontal axis are the various values of each parameter. On the vertical axis is the optimal capital tax. The blue line shows the ex ante optimal tax for each parameter value. The red line shows the steady state optimal tax for each parameter value. Panel (a) uses σ = 0.5, β = 0.98, and varies the values of α. Panel (b) uses α = 0.7, β =0.98, and varies the values of σ. Panel (c) uses α=0.7, σ =0.5, and varies the values of β. 23