Capital Taxation in a Chamley-Judd Economy with a Collateral Constraint

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1 Capital Taxation in a Chamley-Judd Economy with a Collateral Constraint Nina Biljanovska International Monetary Fund Alexandros Vardoulakis Federal Reserve Board January 2017 Abstract We study optimal long-run capital taxation in a closed economy with heterogeneous agents where borrowing is restricted by collateral constraints. The tax system in this environment serves a dual role: first, it is used to finance government consumption; second, it serves to alleviate the distortions arising from binding collateral constraints. Financial frictions distort intertemporal optimization margins and a positive tax on capital income is chosen to affect these distortions. The tax on capital income crucially depends on the attitudes towards risk. If collateral constrained capitalists are risk-averse, the tax on capital income is always positive in the long-run. On the other hand, when they are risk-neutral, the tax on capital income converges to zero. JEL classification: E60, E61, E62, H21 Keywords: Ramsey taxation, collateral constraint We are grateful to David Arseneau and seminar participants at the Federal Reserve Board for fruitful comments and suggestions. All remaining errors are ours. The views expressed in this paper are those of the authors and do not necessarily represent those of the International Monetary Fund, its Executive Board or its Management, nor of the Federal Reserve Board of Governors or anyone in the Federal Reserve System. International Monetary Fund. nbiljanovska@imf.org. Website: Board of Governors of the Federal Reserve System. alexandros.vardoulakis@frb.gov. Website:

2 1 Introduction What should the tax on capital income be in the long-run? Chamley (1986 and Judd (1985, working in somewhat different settings, found that in a dynamic Ramsey model with infinitely lived agents and no distortions in the economy capital should be untaxed given that the economy converges to a steady state (Straub and Werning, The result is based upon the intuition that capital income taxation induces differentiated consumption tax on present and future consumption. In other words, taxing capital income distorts individuals intertemporal consumption behavior as they substitute the more heavily taxed future consumption with current consumption (see also Chari et al., 1994, Atkeson et al., This paper recasts the optimal taxation problem in an economy with a collateral constraint. In particular, we examine whether the Chamley-Judd result of a zero tax on capital income in the long-run survives in an economy where agents face borrowing constraints akin to those present in Kiyotaki and Moore (1997 and Iacoviello (2005. We retain the environment in Judd (1985, which consists of two-classes of agents, workers and capitalists, but we allow them to discount the future differently. Capitalists are relatively more impatient and want to borrow from patient workers. Moreover, we modify the bond market structure by having capitalists borrowing be limited by a collateral requirement. The consideration of a collateral constraint is important since one of the key assumptions in Chamley (1986 and Judd (1985 is the ability of private agents to freely shift consumption inter-temporally, whereas the presence of a collateral constraint precludes it. Given that collateral constrained economies have been extensively used in macroeconomic analysis, it is of interest to understand how the presence of the collateral constraint influences the key results on long-run capital taxation. In an economy with endogenous collateral constraints of the class considered here, agents exhibit a tendency to over-utilize capital (compared to the unconstrained level since accumulating an additional unit of capital does not only result in an additional unit of capital in the future, but it also relaxes the collateral constraint in the present. Therefore, as argued by Fostel and Geanakoplos (2005 and Geanakoplos and Zame (2013, capital embeds a collateral premium, which pushes the marginal product of capital down. The Ramsey planner takes into consideration how capital accumulation matters for the implementable path of past and future consumption, given that capitalists cannot freely shift consumption inter-temporally. A Ramsey planner who is endowed only with linear 1

3 taxation tools cannot undo the financial friction resulting in binding collateral constraints, but could use these tools to affect the distorted inter-temporal margins. 1 A positive tax on capital income reduces the demand for capital and brings its marginal product closer to parity with the planner s evaluation of inter-temporal consumption. 2 Consequently, the government plays a dual role. The first goal is well-known in the Ramsey literature, which is to minimize distortions resulting from linear taxation used to finance exogenous government expenditure. The second goal, which has only recently been recognized in the Ramsey literature by Park (2014, is to internalize the externalities of capital in order to improve welfare in the economy. We similarly show, albeit in a different framework, that a tax on labor income is used to finance the expenditures of the government, whereas a tax on capital income is non-zero in the long-run and it serves to correct for the ineffi ciencies induced by the binding collateral constraint. Importantly, the magnitude of the tax on capital income in the long-run depends on the risk-aversion parameter, or equivalently the elasticity of intertemporal substitution, of capitalists. We show that when capitalists are riskneutral, the planner sets a zero tax on capital income in the steady state. However, when capitalists are risk-averse, the tax of capital income is positive in the long-run and increases as the elasticity of inter-temporal substitution decreases. Hence, our positive capital tax result in the presence of collateral constraints does not obtain as a special case for certain parameters governing the motive of capitalists to smooth consumption inter-temporally. For example, Lansing (1999 shows that the zero tax result in Judd (1985 can be invalidated, but only when capitalists have logarithmic utilities. The paper is organized as follows. Section 2 describes the economic environment. Section 3 derives the Ramsey problem and computes the optimal tax policy. Section 4 provides further intuition and results about the role of taxes in the economy. Finally, section 5 concludes. 1 See Biljanovska (forthcoming for an evaluation of a variety of corrective policy tools in the presence of collateral constraints and their ability to replicate first- and second-best allocations. Itskhoki and Moll (2015 also study optimal dynamic Ramsey policies when borrowing is constrained, but tax proceeds are rebated back to the same agents and, thus, policy is Pigouvian. Our analysis differs, importantly, because we do not allow for corrective Pigouvian taxation or other transfers, but rather study distortionary taxation used to fund government expenditure as is the norm in the Ramsey literature. Positive capital taxation would arguably be more diffi cult to obtain in our framework, since it would remove resources for resource-constrainted capitalists. 2 Given that the two types of agents exhibit different discounting of future consumption, we should note that we do not assume a specific discount factor for the planner, but rather endogenously derive the rate at which the planner evaluates intertemporal consumption. 2

4 Related Literature. Park (2014 studies an optimal Ramsey taxation problem in an environment where agents face a limited commitment problem as in Alvarez and Jermann (2000. She finds that the Ramsey government faces two conflicting objectives: first, to finance government expenditure; second, to internalize the externality of labor and capital to improve risk sharing. The steady state tax on capital income is levied to remove the external costs of capital; whereas the tax on labor income is used to finance the remaining budgetary needs of the government. Our paper similarly finds that the tax on capital income serves to correct for the suboptimal inter-temporal consumption-capital investment decision, which are not necessarily due to the presence of pecuniary externalities. Another paper that is related to ours is Reis (2012. She finds that the tax on capital income is positive at the deterministic steady state as long as the benevolent government is more impatient than the private agents, accumulates debt and is not able to commit to future policies. She finds that these conditions need to hold for a positive tax on capital income to emerge in the long-run. In our paper, contrary to hers, there are three agents: workers, capitalists and a benevolent government which does not face a commitment problem. The positive capital taxation in the long-run is due to the collateral premium and the different discount factors between workers and capitalists rather due to government impatience and its inability to commit. Aguiar and Amador (2016 model a small open economy with impatient agents compared to the rest of the world and a government with limited commitment. They show that labor income taxes can go to zero in the long-run, while capital income taxes may not be zero. Our paper is different because we model a closed economy with heterogeneous agents where the role of capital as collateral distorts inter-temporal margins and calls for positive capital taxation. Our paper is more broadly related to the literature studying optimal capital income taxation. Chamley (1986 and Judd (1985 established the result of zero capital taxation in the long-run, which rests critically on the possibility of shifting consumption across periods through perfect capital markets (see for e.g. Chamley, More recently, Chari et al. (2016 revisit the Chamley-Judd result in an environment that allows for a richer tax system (i.e. without caps on linear taxation, unlike Straub and Werning, who impose limits on taxation, and find that the tax on capital income is still zero in the long-run. However, the environment they consider does not involve any (financial market imperfections. In our setup, a collateral constraint yields inter-temporal wedges that, as we will see, call for a positive tax on capital income in the long-run. 3

5 A positive tax on capital income has also been found in models with uninsurable idiosyncratic risk and/or borrowing limits (see Aiyagari, 1995, and subsequent literature, as well as in life-cycle model frameworks (for example, Conesa et al., In the current paper, we find that tax on capital income is positive in the long-run. It is required to decrease the capital accumulation and bring the rental rate of capital closer to the borrowing rate, which is in line with the finding by Aiyagari (1995, but accrues from the presence of a collateral premium rather than from a precautionary savings motive. 2 The economy This section presents the economy and characterizes the set of attainable allocations. The economy is populated by two types of infinitely-lived agents that behave competitively, workers and capitalists. Workers provide labor hours, but they also lend to capitalists. Capitalists invest in capital and own a production technology that employs both capital and labor to produce a homogeneous good. The investment and production activity is financed by earnings and by issuing bonds sold to workers. Since capitalists cannot commit to repay their debt, their borrowing capacity is limited by a collateral constraint as in Kiyotaki and Moore (1997 and Iacoviello ( Workers There is a continuum of identical workers, whose objective is to maximize the sum of future utilities β t u (c t, 1 l t, where c t is consumption, 1 l t is leisure with l t denoting labor hours and β is the subjective discount factor taking values 0 < β < 1. Assumption 1. The utility function u (c t, 1 l t exhibits constant relative riskaversion (CRRA and is separable in consumption and labor. It is continuous, twice differentiable, increasing in consumption and decreasing in labor, and is globally concave. The Inada conditions are also met: lim c 0 u c (c, 1 l, lim l L u l (c, 1 l and lim c u c (c, 1 l, lim l 0 u l (c, 1 l 0. The function is defined for positive values of consumption and for values of l with 0 < l < L. 4

6 For every unit of work supplied, workers get wages, net of tax, ( 1 τ t l wt l t. They can also shift consumption across periods by investing in one period corporate bonds, b w t+1, issued by capitalists, for which they receive a return of r t. Taking market prices (the price of the consumption good, set to be a numeraire, the wage rate and the lending rate as given, workers make optimal consumption, labor, and investment in corporate bonds decisions to maximize the present value of the utility subject to the following budget constraint c t + bw t+1 b w t + ( 1 τ l t wt l t. (1 1 + r t The equilibrium conditions that characterize the solution to the workers problem are given by the Euler condition u c,t = βu c,t+1 (1 + r t, (2 and the optimal labor supply decision w t (1 τ l t = u l,t u c,t. (3 The subscripts on the utility function in (2 and (3 denote the respective partial derivative at time period t. We will preserve the same notation throughout the paper. 2.2 Capitalists There is a continuum of identical capitalists, who maximize the following sum of discounted utilities γ t u c (c c t, where c c t denotes capitalists consumption and γ is the subjective discount factor, with 0 < γ < 1. The utility function has the same properties as those detailed in Assumption 1. Assumption 2. Capitalists discount the future more heavily than workers, γ < β. This assumption is introduced to ensure that capitalists are not completely self-financed and that the collateral constraint binds in equilibrium. 3 3 See for example Kiyotaki and Moore (1997, Iacoviello (

7 Capitalists invest in capital, which accumulates following the law of motion k t+1 = i t + (1 δ k t, (4 where i t denotes investment and δ denotes the depreciation rate. They employ capital and labor, n t, using Cobb-Douglas production technology to produce a homogeneous good F (k t, n t = k α t n 1 α t, where α denotes the share of capital in the production process. Assumption 3. The production function, F (k t, n t, has a constant returns to scale and is increasing and concave in both its arguments. The following conditions hold: i. F (0, n = F (k, 0 = 0; ii. lim k 0 F k (k, n, lim n 0 F n (k, n ; iii. lim n F n (k, n, lim k F k (k, n 0. The function is defined for positive values of k and n. Capitalists issue a one period corporate bond, b c t+1, which is repaid at rate r t. They also pay tax to the government on their capital gains, τ k t. Thus, capitalists must meet the following budget constraint c c t + b c t + i t ( 1 τ k t [F (kt, n t w t n t ] + bc t r t. (5 However, due to their inability to commit to repay the debt capitalists capacity to borrow is bounded by a collateral constraint of the form εk t+1 bc t r t, (6 where ε < 1 is the liquidity value of capital and is kept constant for simplicity. 4 Capitalists choose optimally consumption, labor, capital investment and borrowing. The equilibrium conditions that characterize the solution to capitalists problem are given by w t = F n,t, (7 u c c,t (1 µ t = γu c c,t+1 (1 + r t, (8 4 The collateral constraint can be derived from a renegotiation process between the borrowers and the lenders, as shown in Hart and Moore (

8 [( u c c,t (1 εµ t = γu c c,t+1 1 τ k t+1 Fk,t δ ], (9 where µ t denotes the Lagrange multiplier on the collateral constraint, and F n,t and F k,t denote the marginal products of labor and capital, respectively at time t. The complementarity slackness condition is given by ( µ t εk t+1 bc t+1 = 0, µ 1 + r t 0. t Lemma 4. If γ < β, then the collateral constraint always binds at the steady state, i.e. µ > 0. Proof. The steady state version of eq. (8 can be obtained by eliminating the time subscripts, yielding 1 µ = γ. Substituting in the definition for 1+r, obtained from 1+r the steady state version of eq. (2, yields 1 γ = µ. It follows that if γ < β, µ is β constant and positive at the steady state. The proof shows that the tightness of the collateral constraint at the steady state is determined by the difference in agents discount factors. The less patient the capitalists, the tighter the constraint; capitalists borrow as much as possible since the borrowing rate is lower than their discount rate. The presence of a binding collateral constraint affects capitalists inter-temporal optimal choices with respect to borrowing and capital investment. In particular, eq. (9 suggests that foregoing one unit of consumption at time period t does not only bring an additional unit of capital at t + 1, but it also relaxes the collateral constraint at t. Hence, capital entails a collateral premium. 2.3 Government Following Judd (1985, the government cannot issue bonds and runs a balanced budget. It receives revenues from linear taxes on labor, τ l t, and capital income, τ k t, each period, and consumes an exogenous amount of g t 0. Given this, the government s budget constraint reads g t = τ l tw t l t + τ k t [F (k t, n t w t n t ]. (10 Now we can define the competitive equilibrium of the outlined economy. 7

9 2.4 Competitive Equilibrium Definition Given initial values for capital, k 0, and borrowing b w 0 = b c 0 = b 0, and an exogenous value for government spending, g t, a competitive equilibrium with a collateral constraint is a sequence of allocations { } c t, l t, b w t+1, c c t, n t, b c t+1, k t+1, prices {r t, w t }, and government policies { τ k t, τ t} l such that (i given prices and policies, the allocations solve workers and capitalists maximization problems; (ii the government budget constraint (10 is satisfied; (iii labor, bonds, and goods markets clear: l t = n t, b w t+1 = b c t+1 = b t+1, c t + c e t + i t + g t = F (k t, l t, t. 3 Ramsey Optimal Policy This section derives the tax on capital and labor income as a solution to the problem of the Ramsey government. We formulate the problem following the socalled "primal approach," consisting of the planner choosing allocations directly instead of policy instruments among the set of allocations attainable by the competitive economy with a collateral constraint. 3.1 Implementable Allocations Following the approach developed in Lucas and Stokey (1983, the initial step of the primal approach consists of finding the set of allocations that can be supported as a competitive equilibrium with a collateral constraint for some sequence of prices and policy instruments. To this end, we derive the conditions that these allocations need to satisfy such that they can be decentralized as a competitive equilibrium with a collateral constraint. Then, introducing these conditions as constraints in the Ramsey planner s maximization problem makes sure that any allocations chosen by the planner can also be sustained in the competitive economy with a collateral constraint. Lucas and Stockey (1983 showed that for frictionless markets, competitive equilibrium imposes a single period-zero implementability constraint on allocations (for each agents. However, in presence of borrowing limits, competitive equilibrium allocations must satisfy the same restrictions as in Lucas and Stokey 8

10 (1983, as well as additional ones that impose that capitalists do not exceed their borrowing limit at any point in time. Lemma 5. Given initial values for capital, k 0, borrowing b w 0 = b c 0 = b 0, as well as an exogenous value for government spending, g t, a sequence of allocations { } ct, l t, b w t+1, c c t, n t, b c t+1, k t+1 can be supported as a competitive equilibrium with a collateral constraint if and only if all markets clear and the following conditions are met: (i Resource constraint, c t + c c t + i t + g = F (k t, l t, t. (ii Period-zero implementability constraint of workers β t m (c t, l t = u c,0 b 0, (11 where m (c t, l t u c,t c t + u l,t l t. (iii Period-zero implementability constraint of capitalists [ ( ] γ t u c c,tc c t = u c c,0 Fk,0 1 τ k 0 k0 + (1 δ k 0 b 0 (12 (iv (Per-period Implementability constraints of capitalists γ t+1 s u c c,t+1c c t+1 = u c c,s t=s where 1 + r (c s, c s+1 = (v Collateral constraint, uc,s βu c,s+1. ( k s+1 b s+1, s 0, ( r (c s, c s+1 εk t+1 b t+1, t. ( r (c t, c t+1 Proof. See Appendix. In a frictionless economy such as the one presented in Judd (1985, the allocations characterizing the competitive equilibrium can be summarized by the economy s resource constraint and the period-zero implementability constraints of workers and capitalists, i.e. (11 and (12, respectively. The presence of the collateral constraint, limiting agents ability to shift consumption inter-temporally, 9

11 leaves these restrictions intact, but adds two other sequences of constraints. Constraint (13 requires capitalists to adhere to a per-period implementability constraint in addition to their period-zero implementability constraint. This condition, relative to the frictionless market case, requires that the next period allocations lie in a subspace determined by the limit on borrowing. In particular, it requires that the allocations at each date must be such that the present discounted value of future consumption does not exceed the net asset value. 5 Finally, condition (14 requires that the debt limit is respected. As we show later, in an economy where workers and capitalists have the same discount factors such as the one in Judd (1985, the per-period implementability constraint does not bind due to non-binding (financial frictions in the economy. The reason is, in absence of (financial market frictions, capitalists consumption choices are not necessarily restricted to a fraction of the net wealth; instead agents can freely shift consumption inter-temporally. To the contrary, the per-period implementability constraint binds in the economy with a collateral constraint. This is the key difference in the formulation of the problem for the competitive economy with a collateral constraint and the frictionless economy in Judd (1985, which, as shown later, is the source of a positive capital income tax. The final remark regarding the formalization of the Ramsey problem is concerning τ k 0. While other papers have shown that the tax on capital in the initial period should not be a choice variable in order to avoid the uninteresting result of the tax rate at t = 0 having the role of a lump-sum tax, financing all government expenditure, this assumption is redundant here. Namely, in the current setup, letting τ k 0 to be a choice variable for the planner at time period t = 0 or imposing additional restrictions on it would not affect the tax on capital income in the long-run. 3.2 The Ramsey Problem Given Lemma 5 the problem of the Ramsey planner can be formulated as a problem of choosing allocations (among the implementable set, and not policy instruments and prices, in order to maximize a given social welfare function subject to the constraints constituting a competitive equilibrium. The social welfare in this economy is given by the utilitarian welfare function that depends on exogenously given Pareto-Negishi weights, ω for workers and (1 ω for capitalists. Then, the 5 For a more detailed study of this case, see for e.g. Aiyagari et al. (

12 problem of the Ramsey government is given by max {c t,c c t,lt,k t+1,b t+1 } { ωβ t u (c t, 1 l t + (1 ω γ t u c (c c t } subject to conditions (i (v in Lemma 5. We analyze optimal taxation by solving the Ramsey optimization problem. We start by composing a Lagrangian. We attach the following Lagrange multipliers,, λ w, λ c, λ cc t, µ RP t, to constraints (i (v in Lemma 5, respectively. Then, the Lagrangian of the Ramsey government can be written as 6 L = { ωβ t u (c t, 1 l t + (1 ω γ t u c (c c t } + [F (k t, l t + (1 δ k t c t c c t k t+1 g t ] +λ w β t {u c,t c t + u l,t l t u c,o b 0 } +λ c γ { [ ( ]} t u c c,tc c t u c c,0 Fk,0 1 τ k 0 k0 + (1 δ k 0 b s=1 λ cc s µ RP t γ t+1 s u c c,t+1c c t+1 t=s ( εk t+1 b t r (c t, c t+1 ( λ cc t u c b t+1 c,t k t r (c t, c t+1 We assume that the government has the ability to commit to the contingent policy rules it announces at date 0. To derive optimal policy, we exploit the first order necessary conditions of the Ramsey problem t 1: 7 c t : ωβ t u c,t + β t λ w m c,t =, (15 6 Note that we do not make any assumptions regarding the growth rate of any of the Lagrange multiplier in the planner s optimization problem. Instead, in section 3.3 we derive the converegence rates of all the multipliers following Reis ( In the interest of space, we omit the first order conditions of the Ramsey government at t = 0. However, as mentioned earlier, this does not matter for our results on the long-run tax rate on capital income. 11

13 These conditions are necessary for an optimal solution to the planner s problem. 8 c c t : (1 ω γ t u c c,t + γ t λ ( c u c c,t + u c cc,tc c t + ( u c c,t + u c cc,tc c t t 1 s=1 λ cc s γ t s λ cc t u c cc,t ( k t+1 b t+1 =, ( r (c t, c t+1 b t+1 : λ cc t u c c,t = µ RP t, (17 k t+1 : ( ε 1 µ RP t = +1 (F k,t δ, (18 l t : ωβ t u l,t β t λ w m l,t = F l,t. (19 The key difference between the Ramsey planner s problem subject to a collateral constraint and the one without arises from eq. (16, (17 and (18. In particular, eq. (16 that also feeds in eq. (17 and (18 incorporates some additional terms absent from its frictionless counterpart. Assuming a CRRA utility function for capitalists of the form u c = cc1 σ 1 σ c c with elasticity of inter-temporal substitution EIS 1/σ C, we can rewrite eq. (16 by substituting in eq. (17 (1 ω γ t u c c,t + γ t λ c ( u c c,t + u c cc,tc c t + Ψt =, where Ψ t ( 1 1 t 1 EIS s=1 µ RP s γ t s + µ RP t 1 EIS 1 c c t ( b t+1 k t+1. ( r (c t, c t+1 This equation suggests that the planner internalizes that the social benefits from consumption include the direct effect on utility, (1 ω γ t u c c,t; the cost of distortions due to the revenue burden with distortionary income taxes, γ t λ c (u c c,t + u c cc,tc c t; and the cost arising from the binding collateral constraint, Ψ t. The first two terms in (?? are the familiar ones from the Ramsey literature without financial frictions. The last term, Ψ t, is novel and depends on the shadow value of the collateral premium, µ RP s, the willingness of agents to substitute income inter-temporally, i.e. the inter-temporal elasticity of substitution EIS,, 1 σ c as well as on capitalists net wealth as a fraction of their consumption. The whole term shows that the presence of a binding collateral constraint affects the choice of consumption. On one hand, the planner takes into consideration that a mar- 8 It is well known that these conditions are necessary, but not suffi cient because the implementability constraints are not convex. This issue is generally ignored in the literature and the planner s first order optimality conditions are simply assumed to be both necessary and suffi cient for a global optimum. 12

14 ginal increase in current consumption, c c t, matters for the implementable profile of consumption up to t 1, because capitalists are restricted from freely smoothing consumption inter-temporally due to the financial frictions. Given that the collateral constraint can bind in any period, the Ramsey planner needs to assure that allocations are implementable at each point in time, hence the dependence of the current consumption decision on the full path of past consumption. This is reflected in the first term in (20 which is increasing in the EIS. When the EIS is high and the motive for consumption smoothing is low, sacrificing past consumption for current consumption is not very costly, and vice versa. On the other hand, because of the binding collateral constraint, the planner also accounts for the period t trade-off between investing own wealth in capital versus consuming it. This effect is captured in the second term in (20 and its magnitude depends on the relative level of own funds invested in capital to consumption as well as on the EIS; lower EIS implies higher costs associated with sacrificing a unit of consumption. As we will see later, these two elements play the key role in determining the tax on capital income since they both stem from the fact that financial frictions distort inter-temporal margins and impede inter-temporal smoothing. Finally, from equations (6 and (17, respectively, it can easily be shown that when µ t = 0 and µ RP t Judd (1985. = 0, the planner s problem collapses to the familiar one in 3.3 Optimal Taxes This section derives analytical formulas for the tax rates on capital and labor income at the steady state using the above derived optimality conditions. Let c, l, c c, b and k be the steady state values for workers consumption, labor, capitalists consumption, borrowing and capital, respectively, which are constant. Furthermore, the multipliers at the steady state need to converge to some constant, as well. To this end, before deriving the steady state tax levels of capital income, a couple of remarks regarding the Lagrange multipliers in the Ramsey problem are in order. First, it is worth noting from eq. (17 that if µ RP t = 0 then also λ cc t = 0, and the economy converges to the familiar Judd (1985 economy. On the other hand, as long as the collateral constraint binds in the Ramsey government problem, i.e. µ RP t > 0, then also λ cc t > 0. Therefore, as it will be shown later, a binding collateral constraint in the Ramsey planner s problem is a key driver for positive tax on capital income in the long-run. 13

15 Second, since agents in this economy entertain different discount factors, it is not straight forward at what rate, the Lagrange multiplier on the resource constraint, will be converging to in the steady state. This growth rate is important because it is needed in the derivation of the tax on capital income in the steady state. Lemma 6. In the steady state, the shadow value of agents consumption in the Ramsey problem,, grows at β, i.e. lim t +1 = β. Proof. See Appendix. Then, in the steady state the consumption capital investment decision of the Ramsey planner becomes 1 ( ε 1 lim µ RP t t = β (F k + 1 δ. (21 Moreover, to determine the tax on capital income, we need to find the value at µ which the ratio of the two multipliers converges, i.e. lim RP t t, where µ RP t denotes the Lagrange multiplier on the collateral constraint in the Ramsey planner s problem. Lemma 7. In the steady state, the ratio µrp t converges to lim t µ RP t = u c c γa U β γ ccb, (22 c where A (u c c + u c ccc c and B ( k Proof. See Appendix. 1+r b. Finally, since the Lagrange multipliers on the resource constraint,, and the Lagrange multiplier on the inter-temporal budget constraint of workers, λ w, as shown later, enter in the equations of the tax rates, it will be important at this point to highlight that they are finite and constant. This can be shown by using equations (15 and (19, i.e. the first order conditions in the planner s problem withe respect to workers consumption and labor respectively, and solving for and λ w as functions of steady state variables. 14

16 3.3.1 Tax on capital income Proposition 8. The tax on capital income in the steady state is always positive as long as capitalists are risk-averse and it is given by τ k = (1 ε µ (γ lim RP t t βµ, (23 F k βγ µ where lim RP t t and µ are derived in Lemmas 4 and 7. Proof First, we derive the proof when capitalists are risk-averse, and then when they are risk-neutral. The first step in the derivation is to combine the Euler conditions with respect to capital of the planner and of capitalists at the steady state, i.e. equation (21 and (9 respectively, 9, such that we get 1 ε (1 µ τ k γf k γ = 1 ( ε 1 lim t µrp t β. After isolating τ k, the expression for the tax rate can be derived, (23. Then, in order to show that the tax rate on capital income is positive at the steady state, the proof requires showing that the following inequality does not hold µ RP t t γ lim < βµ. (24 We prove this by contradiction. Risk-averse capitalists. Suppose that the utility function of capitalists takes the following CRRA form, u c (c c t = c c1 σc t / (1 σ c, with σ c > 0. Start from expression (23 and substitute the expression for lim t µ RP t / derived in the lemma above, e.q. (22. Then, exploiting the properties of the utility function in A (u c c + u c ccc c = (1 σ c u c c and using the collateral constraint to eliminate borrowing, b, from B (k b/(1 + r = (1 ε k, after a few algebraic manipulation, the inequality (24 can be rewritten as k c c > γ (1 ε (β γ. 9 The steady state version of eq. (9 is obtained by removing the time subscripts. 15

17 Using the budget constraint of capitalists and their consumption-capital investment Euler condition in the steady state, we can find the capital to consumption ratio, given by k γ =. Substituting this expression in the ratio above c c (1 ε(1 γ yields, 1 < β, which is clearly a contradiction. This result proves the part of the proof showing that when capitalists are risk-averse, the tax on capital income is positive. Risk-neutral capitalists. Suppose that the utility function of capitalists is linear in consumption, with CRRA parameter σ c = 0. In this case, the second derivative of the utility function is zero, and A (u c c + u c ccc c = u c c. Then, the limit of the ratio of the Lagrange multiplier becomes lim t µ RP t / = (β γ/γ. Substituting this in (23 yields tax on capital income equal to zero, τ k = 0, for risk-neutral capitalists. Q.E.D. Technically, as argued in Chari and Kehoe (1999, the presence of additional restrictions in the Ramsey problem may yield non-zero capital income taxation optimal in the long-run. More specifically, these additional constraints need to be both binding and to depend on the capital stock, as it is the case with the per-period implementability constraint (13, for the optimality of a non-zero tax on capital income. From the expression for the tax rate (23, we can see that the tax rate on capital income serves to close the gap that exists between the Lagrange multipliers on the collateral constraint in the Ramsey problem and the competitive economy, respectively, weighted by the discount factors. Moreover, from the proof outlined above, clearly the tax on capital income in the steady state depends on the riskaversion parameter. Observing the expression for the tax rate, (23, we can see that while the discrepancy in the discounting between households and capitalists, governing the tightness of the collateral constraint in the agents economy calls for a subsidy on capital, the presence of a binding collateral constraint in the Ramsey planner s economy calls for a positive tax on capital income. When capitalists are risk-averse the latter effect prevails, whereas when they are risk-neutral, the two effects cancel each other out. To find the reason why the planner has an incentive to impose a positive tax on capital income when capitalists are risk-averse versus a zero tax when they are risk-neutral, one needs to first understand the exact purpose for which the distortionary tax rates were raised in the first place. In an economy without any 16

18 market frictions or imperfections, the objective of the Ramsey planner is to raise taxes to finance government expenditure such that the distortions are minimized. In this economy, however, the optimal allocations are not only distorted because of the presence of distortionary linear taxes, but also because of the presence of financial friction, i.e. the collateral constraint. Therefore, the goal of the Ramsey planner in this environment is twofold: first, it minimizes the distortions from taxation; second, it alleviates the ineffi ciencies induced by the binding collateral constraint. Section 4 examines in more detail the double role faced by distortionary taxes Tax on labor income Proposition 9. In the steady state, the tax on labor income is given by τ l = 1 ω + λw m c,t u c,t ω + λ w m l,t u l,t. (25 Moreover, for σ w 1, where σ w denotes the inverse of the EIS parameter in the utility function of households for consumption, the tax on labor income is always positive in the long-run. Proof. To derive an expression for the tax on labor income, combine the optimal consumption and labor conditions of workers with the corresponding conditions of the planner. Isolating τ l, yields equation (25. Further, if the inverse of the EIS parameter σ w 1, the tax rate on labor is always positive in the steady state. To show this, rewrite the expression for the tax rate as τ l = λw ( ml,t u l,t mc,t u c,t ω + λ w m l,t (26 u l,t For logarithmic utility in labor, standard in the literature, and u c,t > 0 and m c,t u cc,t c t + u c,t 0 for σ w 1, the tax rate is always positive. Note that the tax on labor income in the steady state is unaffected by the presence of the binding collateral constraint since the intra-temporal margin is not affected by the presence of the collateral constraint If a working capital loan were also collateralized, as in Jermann and Quadrini (2012 for example, then the collateral constraint would also affect the intra-temporal margin and would make the tax rate on labor income sensitive with respect to the collateral constraint. See for e.g. Biljanovska (forthcoming. 17

19 4 The Role of Distortionary Taxes The main argument against capital income taxation in the long-run is that it disproportionally taxes present and future consumption and, hence, distorts originally undistorted inter-temporal optimization margins. The financial frictions, resulting in binding collateral constraints, introduce inter-temporal distortions in the competitive equilibrium. A planner may, then, want to use distortionary taxation to restore effi ciency in inter-temporal margins on top of funding government expenditure. To examine what portions of the tax rates is used to alleviate the frictions from the binding collateral constraint, consider the case when lump-sum taxes are available to the Ramsey planner. Then, the Ramsey government does not need to rely on distortionary taxes to finance its expenditures since it can always choose lump-sum taxes. 11 Corollary 10 below summarizes the properties of the tax system in the steady state when lump-sum taxes are available. Corollary 10. When lump-sum taxes are available, the optimal tax system in the steady state is τ k µ = (1 ε (γ lim RP t t βµ / (F k βγ and τ l = 0. Proof. It can easily be shown that introducing lump-sum transfers into workers budget constraint, (1, and government s budget constraint, (10, the time periodzero implementability constraint of workers becomes βt {m (c t, l t u c,t T t } = u c,0 b 0, where T t denotes the lump-sum transfer. Optimizing with respect to T t, requires that the Lagrange multiplier on the time period-zero implementability constraint of workers, (11, to be λ w = 0. Substituting this in (26, results in τ l = 0. In this case, the Ramsey government can use lump-sum taxation to fully finance government expenditures, but it cannot use them to correct for the ineffi ciency induced by the binding collateral constraint. The reason why lump-sum taxes cannot affect the bindness of the collateral constraint is because they do not affect agents marginal decisions. Indeed, the tax on capital income remains intact regardless of whether lump-sum taxes are available or not; on the other hand, the tax on labor income becomes zero. Then, one can conclude that the tax on labor income is used to fully finance the government expenditures, whereas the tax on capital income is there to alleviate the ineffi ciencies induced by the binding collateral constraint. 11 Even though, the assumption of lump-sum taxes may appear unnatural, its purpose is to help in analyzing the direction in which the second goal of the government (alleviating the ineffi ciencies drive the tax rates. 18

20 An alternative way to see the differential role of labour and capital income taxes is to remove the effect of financial frictions. This could be done, for example, by taking the discount factor of capitalists arbitrarily close to the one of workers, i.e., γ β, or by allowing capitalists to borrow against the total value of their capital investment, i.e., ɛ 1. In both cases, the tax on capital income goes to zero, while the tax on labour income is given by (25. This result reverts back to the standard Chamley-Judd result, i.e. when the economy is frictionless the tax on capital income is zero and the tax on labor income is positive in the steady state to finance the expenditures of the government. Yet, the fact that the Ramsey planner generally sets positive capital income taxes in the long-run may seem counterintuitive given that financial frictions restrict capital investment in the first place. The planner would want to support capital investment if it had the ability to relax the collateral constraint. 12 In the absence of additional policy instruments targeted at removing financial frictions, the planner cannot accumulate enough capital such that allocations are decentralized by a subsidy. Hence, the collateral constraint binds in the planner s solution. Although the planner cannot remove the financial frictions, there is an alternative way to tackle them. Capital investment carries a collateral premium, which pushes the marginal product of capital below the socially optimal level. 13 A positive capital tax reduces the demand for capital (hence also borrowing, diminishes its collateral premium, and increases its marginal product. The way capitalists will change the capital investment in response to a capital tax crucially depends on their motive to smooth consumption inter-temporally (i.e. the EIS parameter. The Ramsey planner accounts for this dependence reflected in term Ψ t in (20, which is used to derive the value of µ RP t. As σ c 0 and capitalists consumption smoothing motives diminish, the planner does not need to impose a positive tax on capital income (i.e. t k 0 in order to induce capitalists to decrease their capital investment. In such case, a positive tax on capital income would only hurt output and wages. On the other hand, as capitalists consumption smoothing motive increases (i.e. EIS 0, the planner needs to impose positive (and decreasing in EIS tax on capital income in order to discourage capital investment and hence increase the marginal product of capital. This feature of 12 This can be seen from equation (23 when lim t µ RP t / 0, i.e. when the collateral constraint does not bind in the planner s economy. In this case, the Ramsey planner would opt for a subsidy on capital to allow capitalists to accumulate a suffi cient amount of collateral to effectively relax the binding constraint. 13 This is ineffi cient for the planner, because the shadow value of agents consumption in the Ramsey problem,, grows at β, while capitalits evaluate intetemporal consumption at rate γ. 19

21 the tax on capital income in the steady state is summarized by the following proposition. Proposition 11. The tax on capital income in the steady state is decreasing in the EIS parameter. Proof. See Appendix. 5 Conclusions Chamley (1986 and Judd (1985 established the result on the optimality of zero-tax on capital income in the long-run, which rests on the assumption that private agents can freely shift consumption inter-temporally in absence of any market frictions. This paper recasts the Ramsey policy problem in a Judd (1985- type economy with two classes of agents, households and capitalists, in which the latter face a collateral constraint. The planner has access to capital and labor income distortionary taxes. The tax on capital income depends on the risk attitudes of capitalists and it is positive at the steady state as long as capitalists are risk-averse. The labor income tax is used to fully finance government expenditure; where as the tax on capital income is used by the planner to affect the ineffi ciencies induced by the binding collateral constraint. REFERENCES [1] Alvarez, Fernando and Urban Jermann, "Effi ciency, equilibrium, and asset pricing with risk of default." Econometrica 68(4, pp [2] Aguiar, Mark and Manuel Amador (2016. "Fiscal policy in debt constrained economies. Journal of Economic Theory 161, pp [3] Aiyagari, Rao S. (1995. "Optimal Capital Income Taxation with Incomplete Markets, Borrowing Constraints, and Constant Discounting." Journal of Political Economy 103(6, pp [4] Aiyagari, Rao S., Albert Marcet, Thomas J. Sargent and Juha Seppala. (2002. "Optimal Taxation without State-Contingent Debt." Journal of Political Economy, 110(6, pp

22 [5] Atkeson, Andrew, Varadarajan V. Chari and Patrick J. Kehoe (1999. "Taxing capital income: a bad idea. Q. Rev. - Fed. Reserve Bank Minneap. 23, pp [6] Biljanovska, Nina (forthcoming. "Optimal Policy in Collateral Constrained Economies." Macroeconomic Dynamics. [7] Chamley, Christophe (1986. "Optimal Taxation of Capital Income in General Equilibrium with Infinite Lives." Econometrica 54(3, pp [8] Chamley, Christophe (2001. "Capital income taxation, wealth distribution and borrowing constraints." Journal of Public Economics 79 (1, pp [9] Chari, Varadarajan V., Lawrence J. Christiano and Patrick J. Kehoe (1994. "Optimal Fiscal Policy in a Business Cycle Model." Journal of Political Economy 102(4, pp [10] Chari, Varadarajan V. and Patrick J. Kehoe (1999. "Optimal fiscal and monetary policy." In Handbook of Macroeconomics, Vol.1C. Eds. J.B.Taylor and M.Woodford. Elsevier Science, North-Holland, Amsterdam. [11] Chari, V.V., Juan Pablo Nicolini, Pedro Teles (2016. "Optimal Capital Taxation Revisited." Mimeo, Catolica-Lisbon SBE and Banco de Portugal. [12] Conesa, Juan Carlos, Sagiri Kitao, and Dirk Krueger (2009. "Taxing Capital? Not a Bad Idea after All!" American Economic Review 99(1, pp [13] Fostel, Ana, and John Geanakoplos (2008. "Leverage Cycles and the Anxious Economy." American Economic Review 98(4, pp [14] Geanakoplos, John and William R. Zame (2014. "Collateral Equilibrium, I: A Basic Framework." Economic Theory 56(3, pp [15] Hart, Oliver and John Moore (1994. "A theory of debt based on the inalienability of human capital. Quarterly Journal of Economics 109(4, pp [16] Iacoviello, Matteo "House Prices, Borrowing Constraints, and Monetary Policy in the Business Cycle." American Economic Review 95(3, pp [17] Itskhoki, Oleg and Benjamin, Moll (2015. "Optimal development policies with financial frictions." NBER Working Paper

23 [18] Jermann, Urban and Vincenzo Quadrini (2012. "Macroeconomic Effects of Financial Shocks." American Economic Review 102(1, pp [19] Judd, Kenneth L. (1985. "Redistributive taxation in a simple perfect foresight model." Journal of Public Economics 28(1, pp [20] Kiyotaki, Nobuhiro and John Moore (1997. "Credit Cycles." Journal of Political Economy pp.105( [21] Lansing, Kevin J. (1999. "Optimal redistributive capital taxation in a neoclassical growth model." Journal of Public Economics 73(3, pp [22] Lucas, Robert Jr. and Nancy L. Stokey (1983. "Optimal fiscal and monetary policy in an economy without capital." Journal of Monetary Economics 12(1, pp [23] Park, Yena (2014. "Optimal Taxation in a Limited Commitment Economy." Review of Economic Studies 81 (2, pp [24] Reis, Catarina (2012. "Social discounting and incentive compatible fiscal policy." Journal of Economic Theory 147(6, pp [25] Straub, Ludwig and Ivan Werning (2014. "Positive long run capital taxation: Chamley-Judd revisited. NBER Working Paper Appendix 5.1 Proof of Lemma 5 To prove necessity, we show that the competitive equilibrium conditions outlined in section 2 imply the conditions outlined in Lemma 5, (i (v. The key step in this part of the proof is the derivation of the set of implementability constraints characterizing the competitive equilibrium with a collateral constraint, (11 (13. To derive (11, multiply the budget constraint of workers, (1, with their marginal utility of consumption. Plugging in the optimality conditions of workers, imposing market clearing conditions, summing over t, and imposing the trasnsversality condition yields the following inter-temporal condition β t [u c,t c t + u l,t l t ] = β t [βu c,t+1 (1 + r (c t, c t+1 u c,t ] b t+1 + u c,0 b 0. 22

24 The first term on the right hand side is zero because it equals workers Euler condition; then we arrive to eq. (11. Subsequently, to derive (12, multiply the budget constraint of capitalists with their marginal utility of consumption. Plugging in the optimality conditions of capitalists, imposing market clearing conditions, and summing over all t periods, yields the following inter-temporal condition γ t u c c,tc c t = γ { [ ( ] } t γu c c,t+1 Fk,t+1 1 τ k t+1 + (1 δ u c c,t kt+1 u γ {γu c } t c c,t c,t+1 b t r (c t, c t+1 [ ( ] Fk,0 1 τ k 0 k0 + (1 δ k 0 b 0. +u c c,0 After substituting in capitalists consumption-capital investment and consumptionborrowing Euler conditions, i.e. (9 and (8, respectively, the equation reduces to γ t u c c,tc c t = γ t u c b t+1 c,tµ t ( εk t r (c t, c t+1 [ ( ] Fk,0 1 τ k 0 k0 + (1 δ k 0 b 0. +u c c,0 The first term on the right hand side of the equation above equals the complementarity slackness condition of capitalists, which is zero and it drops out; then we get eq. (12. Finally, to derive (13, consider any period s 0. Then, multiply the budget constraint of the capitalists by their marginal utility of consumption and plug in their optimality conditions. Iterate one period forward and sum over all t s periods to get γ t+1 s u c c,t+1c c t+1 = t=s [ ( ] γ t+1 s u c c,t+1 Fk,t+1 1 τ k t+1 + (1 δ kt+1 t=s γ t+1 s u c c,t+1k t+2 + t=s γ t+1 s u c c,t+1b t+1. t=s γ t+1 s u c b t+2 c,t r (c t+1, c t+2 t=s b s+1 Add u c c,sk s+1 and u c c,s 1+r(c s,c s+1 on both sides of the equation above. Following the same steps as in the derivation of the period-zero implementability condition of the capitalists, yields eq. (13. The remaining of the necessity proof is obvious since the resource constraint and the collateral constraint exactly correspond to 23