Optimal Capital Income Taxes in A Representative-agent Model with Progressive Tax Schedules

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1 Optimal Capital Income Taxes in A Representative-agent Model with Progressive Tax Schedules Been-Lon Chen Academia Sinica Chih-Fang Lai * Academia Sinica August 204 Abstract In an infinitely-lived representative-agent model with linear income tax schedules, the influential studies by Chamley (986) and Judd (985) have shown that the optimal capital tax is zero in the long run. Our paper shows that in the same model, if the tax schedule is sufficiently progressive, the long-run optimal capital income tax is positive with tax rates increasing in income tax progressivity. Our optimal capital income tax rate is negatively related to the elasticity of the marginal capital tax rate with respect to capital complement which complements that of Pietty and Saez (203). The welfare gain of tax reforms to optimal income taxes is larger than the case with linear income taxes. Our findings lend support to progressive income taxes adopted in developed countries since the late 9 th century. Keywords: one-sector model, optimal capital income taxation, progressive tax. JEL classification: E62; H2, H22 Earlier versions have benefitted from discussions with Yili Chien, Jang-Ting Guo, Kevin Lansing, Leonor Modesto, Florian Scheuer, Steve Turnovsy, Chong Kee Yip, participants at the American Economic Association Meeting (Philadelphia), the Asian Meeting of the Econometrics Society (Taipei) and the Public Economic Theory Meeting (Seattle) and seminar participants at the Chinese University in Hong Kong. Corresponding address: Been-Lon Chen, the Institute of Economics, Academia Sinica, 28 Academia Road Section 2, Taipei 529, TAIWAN; Phone: (886-2) ext. 309; Fax: (886-2) ; bchen@econ.sinica.edu.tw. *Institute of Economics, Academia Sinica, 28 Academia Road Section 2, Taipei 529, TAIWAN; cflai@econ.sinica.edu.tw.

2 Introduction Capital income is taxed in the US and the tax rate is progressive. Corporate profits in the US are first taxed at a flat rate of 40% at the corporate level. Then, when these profits are distributed as dividends, the income is taxed again at progressive rates from 0% to 35% at the household level. In fact, most developed countries have adopted comprehensive income tax systems with graduated marginal tax rates that date bac as early as the second half of the 9 th century. With progressive income taxes in most developed countries, an important question in tax policy analysis is whether it is optimal to tax capital income. In an infinitely lived, representative-agent model with linear capital and labor income tax rates, Chamley (986) and Judd (985) found that, in the long run, the optimal capital tax should be zero. 2 The purpose of our paper is to show that, by applying same model, if there is an income tax system that is sufficiently progressive, a positive capital tax will be optimal in the long run. 3 The finding is not only in sharp contrast to the zero capital tax Chamley-Judd result, but also lends support to a system of graduated marginal income tax rates adopted in most of the developed countries since the second half of the 9 th century. Our result is understood as follows. In the long run, the Ramsey planner chooses capital such that the time-preference rate equals the planner s post-tax marginal product of capital (hence, MPK), which includes post-tax returns to capital, plus gains in utility from increases in capital income tax revenues, minus losses in utility from decreases in post-tax returns to capital due to progressive tax rates. Moreover, in the long run the household chooses capital such that the time-preference rate equals the household s post-tax MPK, which includes only post-tax returns to capital. For the Ramsey planner s choice to be consistent with the household s choice, the planner s post-tax MPK needs to equal the household s post-tax MPK. In the case of linear capital income tax rates, there is no progressive tax and thus no loss in utility from decreases in post-tax returns to capital due to progressive tax rates. If the capital tax rate is positive, the gain in utility from increases in capital income tax revenues is positive and the planner s post-tax MPK is larger than the household s post-tax MPK, implying an under-accumulation of capital from the social perspective. It is thus optimal to decrease the capital tax to zero. Yet, in the case of progressive capital tax rates, there are losses in utility from decreases in post-tax returns to capital due According to Saez (203), the German states such as Prussia and Saxony introduced the modern income tax during the second half of the 9th century, Japan in 887, the UK in 909, the US in 93 and France in Several studies revisited the issue by relaxing ey assumptions and proved the zero capital tax result to be robust. See Lucas (990), Jones, Manuelli and Rossi (993, 997), Chari, Christiano and Kehoe (994), Chari and Kehoe (999) and Ateson, Chari and Kehoe (999). 3 The large gap between a zero optimal capital tax in theory and a positive capital tax in practice is viewed as one of the most important failures of modern public economics by Pietty and Saez (202, p.). Our study offers one of the methods to resolve the failure.

3 to progressive tax rates. In this case, under a zero capital tax rate, if the capital income tax schedule is sufficiently progressive, the gain in utility from increases in capital income tax revenues is smaller than the loss in utility from decreases in post-tax returns to capital due to progressive tax rates. Then, the planner s post-tax MPK is smaller than the household s MPK. As a result, the modified golden rule condition indicates that the level of capital chosen by the planner is smaller than the one chosen by the household and there is an over-accumulation of capital from the social perspective. An increase in capital income tax rates enlarges the gain in utility from increases in income tax revenues and decreases the loss in utility from decreases in post-tax returns to capital due to progressive tax rates. Therefore, it is optimal to tax capital income. Our paper is valuable as it justifies a positive capital tax even if there are no inherent distortions in an economy. The existing infinitely lived, representative-agent models relied on inherent distortions to obtain a positive flat capital tax. 4 To the best of our nowledge, the only exceptions are Lansing (999), Chen and Lu (203) and Lu and Chen (203). First, Lansing (999) studied the Judd (985) model and found a long-run positive capital tax, but his result requires a logarithmic capitalists utility with the restriction of a zero government debt issue. Next, in a two-sector model with physical and human capital, Chen and Lu (203) showed that the long-run capital tax is positive, but their result involves a specific learning technology wherein agents learning time and human capital are inseparable as was the case in Lucas (988) and Bond, Wang and Yip (996), as opposed to the learning technology in Lucas (990). Finally, in an infinitely lived, representative-agent model, Lu and Chen (203) also found a long-run positive capital tax, but their result is obtained under a fixed share of government expenditure in output which implicitly allows for changes in government expenditure as capital changes. Moreover, these three existing models all study linear capital and labor income taxes. Our model contributes to these existing studies in that it obtains a positive capital income tax rate without relying on any assumptions of a zero debt issue, a particular form of preferences and technologies, and variable government expenditure. 5 Our paper also adds value to existing heterogeneous-agent models that have explored optimal progressive capital income taxes as put forth by Saez (203), Conesa, Kitao and Krueger (2009) and Farhi, Sleet, Werning and Yeltein (202). 6 First, Saez (203) analyzed optimal progressive capital 4 For infinitely lived representative-agent models that found positive optimal capital taxes based on inherent distortions, see Guo and Lansing (999) and Chen (2007) who incorporated maret imperfections and productive public capital, and Aiyagari (995) and Chamley (200) who considered credit constraints. 5 Another strand of the literature has used overlapping generations (OLG) models to study optimal capital income taxes. In the OLG model without bequests, when taxes are linear, capital taxes are generally positive in the long-run, simply because capital accumulation is due to life-cycle savings for retirement. See Atinson and Sandmo (980), Garriga (200) and Erosa and Gervais (2002). 6 See also the heterogeneous-agent model of Benabou (2002) who constructed a model with human capital, 2

4 income taxation in an infinitely lived, heterogeneous-agent model and found that progressive capital income taxation is more effective than linear taxation in redistributing wealth. Our model adds value to the Saez model as the model retains the long-run vanishing capital tax result as in Chamley (986) and Judd (985). 7 Next, in an OLG model with linear capital and progressive labor income taxes wherein agents are heterogeneous due to different initial endowments and abilities, Conesa, Kitao and Krueger (2009) uncovered that the optimal capital tax is positive when there are borrowing constraints. Our model augments value to Conesa, Kitao and Krueger (2009) in that we find optimal progressive capital income taxes without assuming heterogeneous agents and borrowing constraints. Finally, in another OLG model with non-linear taxation of labor and capital income and political economy constraints, Farhi, Sleet, Werning and Yeltein (202) discovered that it is optimal to levy a progressive capital income tax when policies are not committed, but the capital income tax is zero when policies are fully committed. Our model adds value to Farhi et al. (202) in that, even if policies are fully committed, a positive capital income tax is optimal when the income tax schedule is progressive. Moreover, all of these three existing papers introduce heterogeneous agents and there is a tension between equity and the efficiency of capital accumulation. Even though the efficiency of capital accumulation is the only tension in our model, a positive progressive capital tax is optimal. Recently, in a heterogeneous-agent model with a discrete set of generations and with linear income tax structure, Pietty and Saez (203) studied optimal inheritance tax that captures the equity-efficiency trade-off. They derived long-run optimal inheritance tax formulas in terms of sufficient statistics including tax elasticity and distributional parameters. Their optimal inheritance tax formulas net of the zero capital tax Chamley-Judd result as a limiting case that arises when the long-run elasticity of aggregate bequest flows with respect to the bequest-tax rate is infinite. The smaller the elasticity is, the larger the inheritance tax rate. Our optimal positive capital tax is negatively related to the elasticity of the marginal capital-tax rate with respect to capital. When the capital income tax schedule is more progressive, the elasticity of the marginal capital-tax rate is smaller and our capital tax rate is larger. From the perspective, we view our result as complementary to the Pietty and Saez (203) result. Our model add two additional values to Pietty and Saez (203) in that our optimal capital tax rate is increasing in capital income which is consistent with the capital income tax in practice. Moreover, we obtain positive capital income taxes in a homogeneous-agent model and thus, the equity-efficiency instead of physical capital, and studied non-linear taxation of income as well as the heterogeneous-agent model of Farhi and Werning (202) who explored the related issue of non-linear estate taxation. 7 There are infinitely lived agent models that incorporated progressive income taxes but did not analyze optimal capital taxes. See Carroll and Young (2009) who examined the non-degenerated long-run distribution of capital holdings in a heterogeneous-agent model. See also the representative-agent models of Guo and Lansing (998) who explored the effect of tax progressivity on dynamic stability and Li and Sarte (2004) who analyzed the effect of tax progressivity on long-run economic growth. 3

5 trade-off is not required. Finally, we also study a quantitative version of our model calibrated to the system of progressive income taxes in the US. We find that, with a sufficiently progressive income tax schedule, the optimal capital tax rate is positive and increasing in the degree of income tax progressivity. In particular, the welfare gain of a tax reform from the current code toward the optimal income taxes is larger than that in the case with linear income tax schedules. Moreover, the more progressive the tax schedule, the larger the welfare gain of a tax reform to the optimal tax. We organize this paper as follows. In Section 2, we set up a model with progressive factor income tax schedules and analyze households optimizations. In Section 3, we study the optimal factor income tax incidence in the Ramsey second-best problem. Finally, concluding remars are offered in Section The model Our basic model is otherwise the same as the Chamley (986) model with the exception of progressive factor income tax schedules. There are infinitely lived and identical households and identical firms. Households supply labor and capital to firms, earn labor and capital income and decide consumption and savings. There is a government which taxes capital and labor income in order to pay for wasteful expenditure that is given. 2. The household s problem In each period, given a fixed time endowment normalized to unity, the representative household allocates the time endowment between wor and leisure. The household maximizes the following discounted utilities over sequences of consumption and leisure hours. t Max b uc ( t, lt), () { ct, t+, bt+, lt} t= 0 where c t is consumption, l t is hours wored and β (0,) is the discount factor. The felicity function u(c t, l t ) is assumed to be twice continuously differentiable and increasing and concave in c t and l t. In each period, the representative household faces the following flow budget constraint. ( ) ( )( ) ct+ t+ + bt+ = tlt( wl t t) wl t t+ tt(( γt δ) t) γt δ + t+ Rb t t+ πt, 0 and b 0 given, (2) in which t is physical capital at the beginning of period t, b t refers to one-period, real government bonds carried into period t and π t is the profit remitted from firms in period t. The wage rate is w t and the gross return to bonds is R t. The rental rate of capital is γ t and the depreciation of capital is δ. Capital depreciation expenses are tax-deductible in (2) in order to be consistent with the US tax code. The results are not changed if capital depreciation expenses are not tax-deductible. For simplicity, the 4

6 rental rate net of depreciation will be denoted as r t (γ t δ). In (2), labor and capital income are taxed at the rates according to the tax schedule of τ lt (w t l t ) and τ t (r t t ), respectively. Differing from the linear income tax schedule studied by Chamley (986) and Judd (985), the income tax schedules in (2) are progressive and depend on the income level. We assume that these tax schedules are continuously differentiable with respect to income. In particular, we assume an income tax system with strictly progressive tax rates. To be specific, we assume t lt > 0 and t t > 0, 8 where t = w and lt tlt ( wl t t ) tlt ( wl t t ) lt t ( wl tt) t = r denote the derivatives of the tax rate with t tt ( r t t ) tt ( r t t ) t t ( r t t ) respect to labor and capital, respectively. With households taing prices as given, these assumptions imply t lt ( wl t t ) ( wl ) 0 tt > and t t ( r t t ) ( r ) 0. t t > The tax rates are bounded by 00%, as otherwise the household would not have any incentive to wor or save. As such, we assume that the marginal tax rate is decreasing: t lt < 0 and t < 0. In the case of a flat income tax schedule, t = 0 and t = 0 and the model is reduced to the Chamley (986) model. t The representative household s dynamic programming problem is to choose a sequence of { c,, b, l } t t+ t+ t t= in order to maximize its lifetime utilities () subject to the constraint (2). When maing choices, the household taes prices w t, r t and R t as determined by the maret. It also taes tax schedules τ lt ( ) and τ t ( ) as given by the government. Yet, as the tax schedules are progressive, the household nows that its choices of hours wored and savings affect not only tax bases but also marginal tax rates. The first-order conditions give: ( ) u ( c, l ) = βu ( c, l ) + t t r t t t+ t+ t+ t+ t+ t+ ( t t ) t+ t+ t+ t+ t+ lt t (3a) R = + r, (3b) ( t t ) u ( c, l ) = u ( c, l ) l w, (3c) 2 t t t t lt lt t t t t along with the transversality conditions lim βl tt + = 0 and lim bl b + = 0, which ensure that t there is no Ponzi scheme. Eq. (3a) is the standard consumption-euler equation that is traded off against consumption in periods t and t+. Eq. (3b) is the non-arbitrage condition between bonds and 8 It is well understood that, without restrictions on non-linear taxes that the government can implement, the government can pic labor taxes and capital taxes such that they act exactly lie lump sum tax, which implement the best allocation. Specifically, if the tax rates are not strictly progressive, one can easily choose positive taxes t lt > 0 and t t > 0 along with regressive tax rates t lt < 0 and t t < 0 in order to meet the conditions in the household s optimization (cf. (3a)-(3c) below) tlt + t l lt t = 0 and tt + t tt =0, so that the government problem yields the first best outcome. Our restrictions to progressive tax rates t lt > 0 and t t > 0 rule out such a situation to arise. t t t 5

7 capital, and (3c) refers to the tradeoffs between leisure and consumption in period t. 2.2 The firm s problem The representative firm rents capital and hires labor and produces a single final good y t given by: y = f(, l ). (4) t t t The function f( ) is assumed to be twice continuously differentiable and is strictly increasing and concave in capital and labor. Taing factor prices as given, the firm chooses capital and labor in order to maximize profits. The optimal conditions are standard and are as follows: rt = f( t, lt) δ, (5a) w (, ). t = f2 t lt (5b) 2.3 The government The government finances an exogenous stream of expenditure by taxing factor income and issuing debt. Denote g t as the exogenous government expenditure which increases neither the firms productivity nor the households utilities and is thus a waste. The government s flow budget constraint is t wl + t r + b+ = g + Rb (6) lt t t t t t t t t t. 3. The Ramsey planner s problem The Ramsey planner s problem is to determine the optimal sequence of the factor income tax rates. In analyzing the Ramsey planner s problem, we will rule out the lump-sum taxation that would be first-best efficient. Lie Chamley (986), we also rule out consumption taxation in order to focus on the factor income tax incidence. We allow for the government to issue debt and thus the government does not have to run a balanced budget in each period. Furthermore, we assume that the income tax rates in the initial period are not the government s choice variables, but are rather given by their historical values, since taxing initial private assets is equivalent to a lump sum that would be first best. In order to avoid time inconsistency, we assume that the government can tae a full commitment of policies announced at t=0. We start the planner s problem by defining a competitive equilibrium which is a feasible allocation { ct, t, lt, gt} + t= 0, a price system { wt, rt, Rt} t= 0 and a government policy { gt, tt, tlt, bt + } t= 0 such that (i) given the price system and the government policy, the allocation solves both the firm s and the household s problem and satisfies the resource constraint; and (ii) given the allocation and the price 6

8 system, the government policy satisfies the sequence of government budget constraints. Different government policies would yield different competitive equilibria. Given 0 and b 0, the Ramsey planner s problem is to choose a competitive equilibrium that maximizes the welfare of the representative household. The existing studies use the primal approach to the Ramsey planner s problem which eliminates all prices and taxes so that the government is thought of as directly choosing a feasible allocation. To use the primal approach, we need to express taxes and prices in terms of the allocation and substitute these expressions into the household s present-value budget constraint in order to obtain the so-called implementability condition. However, we cannot eliminate all terms involving tax rates in these expressions, the reason being that, as capital income or labor income increases, in addition to original tax rates, there are marginal tax rates. Thus, we cannot use the primal approach. Following Chamley (986), we formulate the Ramsey problem as if the government chooses the after-tax rental rate of capital and the after-tax wage rate. Then, capital and labor chosen by the Ramsey planner would affect the after-tax rental rate of capital and the after-tax wage rate. The Ramsey planner s goal is to maximize the households welfare, subject to the resource constraint in the economy, the government budget constraint and the best responses of households. These three constraints are as follows. First, the resource constraint is: 9 ( δ ) c + + g = f(, l ). (7a) t t+ t t t t Next, the government s flow budget constraint is: 0 ( ) ( ) gt + Rb t t bt + = f( t, lt ) δt tlt wl t t tt r t t. (7b) Finally, the best responses of households are: u2( ct, lt) w ( t t l ) lt lt t t ( ) = βu( ct +, lt + ) + tt+ tt+ t+ rt +. Thus, the Lagrange equation of the Ramsey planner s optimization problem is: t L= b { uc ( t, lt) + θt f ( t, lt) ct t+ + ( δ) t gt t= 0 ( ) ( ) +Ψt f( t, lt ) δt tlt wl t t tt r t t gt + bt + Rb t t u + µ t bu( ct +, lt + ) + ( tt+ t t+ t+ ) r 2( ct, l ) t t+, ( tlt t ltlt ) wt (7c) (8) where θ t, Ψ t and µ t are the Lagrange multipliers associated with constraints (7a), (7b) and (7c), 9 The constraint is obtained by substituting (6) into the household s flow budget constraint (2). 0 The constraint is derived by substituting (5a) and (5b) into the government s flow budget constraint (6). The condition comes from combining the household s optimization conditions (3a) and (3c). 7

9 respectively. The first-order condition with respect to t+ is: 2 { ( f ) ( f r ) u ( ) r+ } θ = β θ δ + +Ψ ( δ ) µ 2 t + t, (9) t t t t t t t t t t t t where f t + and u t + denote f(, l ) ( t t ) t+ t+ t+ t+ t+ t+ t+ and t+ t+ r r denotes the post-tax return to capital. u ( c, l ), respectively, t and t 2 t t ( r t t ) 2 t Following Ljungqvist and Sargent (2004, P487), condition (9) may be interpreted as follows. The social shadow price of capital θ t >0 is the social marginal value of one unit of capital investment in period t. 3 An increase of one unit of capital investment in period t increases the quantity of goods available at time t+ by the amount (f t+ -δ+), which has a social marginal value θ t+. Moreover, there are increases in capital income tax revenues which enable the government to reduce other taxes or its debt by the same amount. With given tax rates, income tax revenues are increased by ( f δ ) r, and the reduction in the excess burden is equal to ( f δ r ) t+ t+ t+ t+ t+ Ψ ( ). However, due to progressive tax rates, one unit of capital investment in period t decreases the post-tax return to capital in period t+ equal to ( t t ) ( 2t t ) t + t+ t+ t+ t+ 2 t + t + t + r t + +, with the loss in terms of utility equal to u + r. As the shadow price of the marginal rate of substitution between consumption in the next period and leisure in this period is positive µ t>0, this loss in utility in terms of social values is µ ( 2t t ) u + r. In the optimum, the social shadow price of the initial t t + t+ t+ t+ t+ investment good in period t is equal to the discounted sum of these three benefits in period t+. 3. The steady-state optimal capital income tax rate In the steady state, it is straightforward to show that the shadow price of the government budget constraint equals the marginal utility of consumption, Ψ=u (cf. the Appendix). With the use of (5a), the first-order condition concerning the Ramsey planner s optimal choice of capital in (9) is rewritten as u ( ) ( ) θ u µ β = r + + θ r r θ 2 τ + τ r. (0) Eq. ( 0) equates the time-preference rate, ( β ), to the post-tax marginal product of capital which includes post-tax returns to capital, r, plus gains in utility from increases in income tax 2 The other first-order conditions of the Ramsey planner s problem are relegated to the Appendix. In deriving the first-order condition with respect to t+, as in the method used in Ljungqvist and Sargent (2004, Ch. 5, equation (5.4.), p. 486), the factor prices are treated as given by the Ramsey planner. 3 See the Appendix for the proof of the signs θ>0 and µ >0. 8

10 θ + revenues adjusted by the social shadow price of capital, ( r r) u, θ minus losses in utility from decreases in post-tax returns to capital arising from progressive tax rates adjusted by the social shadow µ u price of capital, ( 2 ) θ τ τ + r. In the steady state, the household s optimal choice of capital in (3a) yields: = β r, () which requires the time-preference rate to be equal to the post-tax marginal product of capital from the household s perspective which includes only post-tax returns to capital. In order for the Ramsey planner s choice of capital in (0) to be consistent with the household s choice in (), it is clear that gains in utility from increases in capital income tax revenues adjusted by the social shadow price of capital minus losses in utility from decreases in post-tax returns to capital arising from progressive tax rates adjusted by the social shadow price of capital must be zero. Thus, the following condition should be met. ( ) ( ) ( θ + u ) r r µ u 2τ + τ r = 0. (2) Case : The linear income tax rate In this case, the income tax schedule is flat as was the case in Chamley (986) and Judd (985). Thus, τ = τ = 0 for all t and there is no loss in utility due to decreases in post-tax returns to capital arising from progressive tax rates. Then, gains in utility from increases in income tax revenues should be zero and condition (2) is reduced to: ( ) ( θ + u ) r r = 0, (3) which yields the result of a zero capital income tax, r = r. To understand the reason, let the post-tax marginal product of capital from the social perspective on the right-hand side of (0) be denoted by MPK s and its counterpart from the household perspective on the right-hand side of () be denoted by MPK h. Suppose that the capital income tax rate is positive and thus, r r = τ 0r > 0. Then, MPK s θ + u is r + τ r while MPK h is r. A positive capital tax rate θ implies that the MPK h is smaller than the MPK s. See Figure. [Insert Figure here.] With a given time-preference rate, the modified golden rule condition and the relative position of MPK h and MPK h in Figure, there is an under-accumulation of capital from the social perspective. As a result, the efficiency is improved if the capital income tax rate is reduced from τ 0 >0, as this decreases the post-tax marginal product of capital from the Ramsey planner s perspective and increases its counterpart from the household s perspective. A zero capital income tax rate is optimal as MPK s then, 0 9

11 meets MPK h. Case 2: The progressive income tax rate When the capital income tax is progressive, τ > 0. As we will see, the optimal capital income tax rate is no longer zero but is positive. τ τ Let ( ) = > 0 denote the elasticity of the marginal capital tax rate with respect to ξ τ τ τ capital. As the household taes the rental rate as given by the maret, this also stands for the elasticity of the marginal capital tax rate with respect to capital income. For simplicity, we refer to marginal capital tax rate elasticity. Consider ξ τ as the Condition E (Marginal Capital Tax Rate Elasticity) ξ τ θ + u < 2. µ u Condition E requires a marginal capital tax rate elasticity that is not too large so that the capital tax rate schedule is sufficiently progressive. To illustrate this, we consider the following capital income tax r t t φ schedule: t ( r ) = η ( ), where r stands for the steady-state level of post-depreciation capital t t r income. This tax schedule is taen from Li and Sarte (2004) which was based on the form proposed by Guo and Lansing (998). In this tax schedule, η 0 controls for the limiting value of the average tax rate and φ 0 determines the degree of income tax progressivity. For φ =0, the tax schedule is flat and thus τ =η ; for φ >0, the tax schedule is progressive. The parametric tax schedule gives ξ = φ. τ θ + u Then, Condition E is met if φ > + ( ), which requires that the capital income tax schedule be µ u sufficiently progressive. Now, we can state our main results as follows. Proposition. In a system of progressive income taxes, if the capital income tax schedule is sufficiently progressive, the optimal tax rate on capital income is positive in the long run. To understand the reason, let the initial capital tax rate be τ 0. With progressive capital taxes, the MPK s r given by (0) is r + θ [( θ + u)( τ0 + τ ) µ u(2 τ + τ )] which is different from the MPK h in () and (MPK s -MPK h r )= θ [( θ + u)( τ0 + τ ) µ u(2 τ + τ )]. It is easy to see that if the initial capital tax rate is τ 0 =0, then the difference is reduced to ( θ u )( τ ) µ u ( 2τ τ ) + + which, under Condition E, is negative. Thus, a zero capital tax rate implies that the MPK s is smaller than the MPK h. 0

12 See Figure 2. [Insert Figure 2 here.] With a fixed time-preference rate, the modified golden rule condition and the relative position of the MPK s and the MPK h in Figure 2 indicates that the level of capital from the social perspective is smaller than the one from the household s perspective. That is, if the capital tax rate is initially zero, a progressive capital tax rate maes the Ramsey planner choose a level of capital that is smaller than the level chosen by the household. There is an over-accumulation of capital from the social perspective. Thus, it is optimal to tax capital income in order to reduce the level of capital chosen by the household. Intuitively, with a zero capital income tax rate initially, if the capital tax schedule is sufficiently progressive, a zero capital tax rate gives the gain in utility from increases in capital income tax revenues as being smaller than the loss in utility from lower post-tax returns to capital. An increase in the capital tax rate from zero enlarges the gain in utility from increases in capital tax revenues and decreases the loss in utility from lower post-tax returns to capital. The optimal capital tax rate is set at the level when the gain would completely offset the loss. We should mention that if government expenditure is not a waste but a lump sum transferred to the representative household, the optimal capital tax is still positive. The reason is that the effect of capital taxes wors through the post-tax marginal product of capital but the effect of the lump-sum transfer is neutral. As noted in the Introduction, in a heterogeneous agent model, Pietty and Saez (203) derived optimal inheritance tax formulas in terms of a sufficient statistics including tax elasticity and distributional parameters. Their optimal inheritance tax is positive when the elasticity of aggregate bequest flows with respect to the bequest-tax rate is less than infinite. The smaller the elasticity is, the µ u larger the capital tax rate. Our optimal capital tax rate formula is τ ( ) = τ ( )[ (2 ξ ) ] θ + u τ which depends negatively on the marginal capital-tax elasticity parameter ξ τ. If the capital income tax schedule is more progressive, the elasticity of the marginal capital tax rate with respect to capital income is smaller and then the optimal capital tax rate is larger. From this perspective, we view our result as complementary to the Pietty and Saez (203) result. Moreover, our model adds values to Pietty and Saez (203) in two more perspectives. First, our optimal capital tax rate is increasing in capital income and thus is consistent with graduated marginal capital income tax rates in practice. In addition, we obtain positive capital income taxes in a homogeneous-agent model without requiring the equity-efficiency trade-off. We have noted in the Introduction that, lie our paper, Lansing (999), Chen and Lu (203) and Lu and Chen (203) have obtained a positive capital tax. Our result adds value to these studies in that, even

13 with a debt issue, under a fixed amount of government expenditure, and with a general functional form of preferences and technologies, we obtain a positive capital income tax rate. In particular, we find that the use of progressive tax rates is optimal which rationalizes a system of graduated marginal income tax rates adopted in most of the developed countries since the second half of the 9 th century. Moreover, our paper adds value to Saez (203), Conesa et al. (2009) and Farhi et al. (202) who analyzed optimal progressive capital income taxes in models with heterogeneous agents. In Saez (203), the optimal capital income tax is zero in the long run. In Conesa et al. (2009), the optimal capital tax is zero when there are no borrowing constraints. In Farhi et al. (202), the capital income tax is zero with a non-linear tax on labor income when policies are fully committed. Our paper adds value to these studies in that, in a system of progressive income taxes, a positive capital income tax rate is optimal in the long run even when there is no borrowing constraint and the policy is fully committed. In particular, all these existing papers study a dynamic Mirrleesian model with heterogeneous agents and there is a tension between equity and the efficiency of capital accumulation. By contrast, our model has only homogeneous agents and thus the efficiency of capital accumulation is the only tension. 3.2 Quantitative Analysis To offer quantitative analysis, we calibrate our model to match the US annual data. First, we follow Conesa et al. (2009) to adopt the Cobb-Douglas production function, α α = (,) = and the y f l A l ν ν σ CES utility function uc (, l) = [ c( l) ]. 4 We also go along with these authors and tae σ A=, σ=4, l=/3, α=0.36, /y=2.7, I/y=0.255, g/y=0.7. The values of /y=2.7 and I/y=0.255 give a value of the capital depreciation rate of δ=9.44%. Next, we tae the form of tax rate schedules that was used by Li and Sarte (2004) and mentioned xit φi in subsection 3. above: ti( xit ) = ηi( x ), i=, l, where x it is factor i s income in period t and x i is its i steady-state level. Thus, x t is r t t and x lt is w t l t. While these authors set φ =φ l =φ=0.75, we will start with φ=0.5 in the baseline parameterization. The tax schedule is thus progressive for both capital and labor income taxes. 5 Our analytical results above indicate that the optimal capital tax rate depends on the degree of income tax progressivity. We will carry out the sensitivity analysis to see how the optimal capital tax rate depends on the degree of income tax progressivity. With the tax series from McDaniel 4 The utility function is consistent with steady-state growth in a deterministic version of the real business cycle model (c.f., King and Rebelo, 999). 5 Conesa, Kitao and Krueger (2009) used a non-linear labor income tax and a linear capital income tax to ensure computational feasibility. In our paper, we adopt a more general strategy and employ non-linear tax schedules for both capital and labor income. 2

14 (2007), 6 the average tax rates on capital income and labor income in the US during were around 0.3 and 0.2, respectively. Thus, we choose initial average income tax rates equal to t =30% and t l =20%. This pins down the parameter values η =0.3 and η l =0.2. In the baseline parameterization, b 0 is calibrated so as to balance the government budget (6) in the initial steady state and we obtain b 0 = By using the foregoing parameter values, we utilize (5a) to compute the initial steady-state rental rate of capital equal to γ 0 = We then use the initial steady-state values of r 0 and l 0 to compute the initial steady-state capital equal to 0 =.5736, the initial steady-state output equal to y 0, and the initial steady-state government expenditure equal to g 0 =(g/y) y 0 = We employ (7a) to compute c 0 /y 0 = which, with the value of y 0, gives c 0 = Finally, we calibrate the discount factor β=0.979 from (3a) and the preference parameter ν= from (3c). Thus, the allocation in the initial steady state is (c 0, l 0, 0 )=(0.335, ,.5736). We are now ready to quantify the incidence of the Ramsey optimal factor income tax. In the exercise, the government expenditure is fixed at its initial level of g 0 = Our quantitative results provide a tax schedule with average rates of optimal factor income taxes (t, t l )=(27.74%, 6.57%) associated with the new steady state (c *, l *, * )=(0.4027, ,.8609). See Table. Thus, the optimal capital tax rate is positive in the long run. The optimal income tax schedule suggests the following tax reform: a small decrease in the average capital tax rate by 2.26 percentage points from the current t =30% level with a large decrease in the average labor tax rate by 3.43 percentage points from the current t l =20% level. The results reveal that moving away from the current income tax code in the US toward the optimal income tax would increase consumption, labor supply and capital accumulation. The reform would have a welfare gain of 6.3% in terms of changes in consumption equivalence. As compared to those results in the existing literature, the welfare gain is large. For example, a similar welfare gain of a factor income tax reform in terms of changes in consumption equivalence is 5.5% in Lucas (990) in which case human capital accumulation is exogenous, 3.4% in Conesa, Kitao and Krueger (2009) in which labor supplies are elastic, and.7% in Conesa and Krueger (2006). [Insert Table here.] To understand how the optimal capital income tax rate depends on the degree of income tax progressivity, we change the degree of income tax progressivity. 7 First, we increase the degree. When the degree of income tax progressivity is increased from 0.5 to 0.75, the average rate of the optimal 6 McDaniel (2007) calculated a series of average tax rates on consumption, investment, labor and capital using national account statistics in 5 OECD countries. The data has been used by Rogerson (2008) and others. 7 When the degree of the tax progressivity is changed below, we maintain all parameter values in the baseline except for the values of b 0, β and ν which are recalibrated in order to be consistent with (3a), (3c) and (6). 3

15 capital income tax is increased from 27.74% to 32.24% while the average rate of the optimal labor income tax is decreased from 6.57% to.47%. Our results indicate that the largest degree of income tax progressivity is 0.85 when labor income taxes remain positive in which case the average rate of optimal capital taxes is increased to 32.83%. Next, we decrease the degree of income tax progressivity. When the degree is decreased from 0.5 to 0.25, the average rate of optimal capital income taxes is decreased from 27.74% to 7.49% with the average rate of optimal labor income taxes increasing from 6.57% to 20.65%. The smallest degree of income tax progressivity is 0.23 when capital income taxes remain positive. Thus, our results indicate that if the income tax schedule is sufficiently progressive, it is optimal to tax capital income with the average tax rate increasing in the degree of income tax progressivity. Moreover, with a larger degree of income tax progressivity, a tax reform from the current income tax code to the optimal tax gives a larger welfare gain. When the income tax schedule is progressive, is the welfare gain of a tax reform toward the optimal income tax larger than that of an income tax reform when the income tax schedule is linear? When the income tax schedule is linear, the degree of income tax progressivity is decreased to zero and this is the case studied by Chamley (986). In this case, the optimal capital tax is zero and a tax reform from the current income tax code to the optimal tax gives a welfare gain of.84% in terms of changes in consumption equivalence. See the bottom row in Table. Such a welfare gain is smaller than those in cases with positive optimal capital taxes when the degree of income tax progressivity is larger than or equal to Is the required threshold degree of income tax progressivity 0.23 too high? Recently, using the data from the Internal Revenue Service, the U.S. Census Bureau, and the Bureau of Economic Analysis, Mathews (204) constructed the degree of federal income tax progressivity in the US over the period He constructed annual tax concentration curves and income concentration curves with respect to income, which are lie the well-nown Lorenz curve. Based on the measure proposed by Suits (977), the degree of income tax progressivity is calculated as the ratio of the area between the income concentration curve and the tax concentration to the area below the income concentration curve. While the constructed degrees of income tax progressivity vary over the years, the median degree is 0.46 in the period under study. Earlier, Li and Sarte (2004) used Individual Income Tax Returns publications of the Internal Revenue Service and pinned down the degree of the income tax progressivity. They found the degree of the income tax progressivity at These two values indicate that the degree of income tax progressivity in the US is above the threshold value This thus indicates that it is optimal to tax capital income. 4

16 5. Conclusion In an infinitely lived, representative-agent model with only income taxes, by restricting the income tax schedule to be linear, Chamley (986) and Judd (985) found that the optimal capital tax is zero in the long run. In the same model, we find that, if the income tax schedule is progressive, the optimal capital tax is positive in the long run. The result provides a rationale for the use of a system of graduated marginal income tax rates in most developed countries adopted since the second half of the 9 th century. Our result emerges because there is a tension from the social perspective between gains in utility from increases in capital income tax revenues and losses in utility from decreases in post-tax returns to capital due to progressive tax rates. We show that with a sufficiently progressive income tax schedule, if the initial capital tax rate is zero, the gain in utility from increases in capital income tax revenues would be smaller than the loss in utility from decreases in post-tax returns to capital due to progressive tax rates. As a result, the level of capital chosen by the Ramsey planner is smaller than the level chosen by the household. There is thus an over-accumulation of capital from the social perspective. An increase in the capital tax rate would increase the gain in utility from increases in capital income tax revenues and decrease the loss in utility from decreases in post-tax returns to capital due to progressive tax rates. Therefore, it is optimal to tax capital income. The optimal capital tax rate is increasing in the degree of income tax progressivity. By calibrating our model to the US economy, we find that the welfare gain of a tax reform to the optimal income tax is larger than that in the case with linear income tax schedules. Moreover, the more progressive the tax schedule, the larger the welfare gain of a tax reform to the optimal tax. Mathematical Appendix. Derivation of the Ramsey planner s problem The first-order conditions for c t, t+, b t+, and l t for the Ramsey planner s problem in (8) are u2t ut + µ t ut ( t t t tt ) r µ t t t 0, ( t t l ) w θ + = lt lt t t { + ( f + ) + ( f ) r + + u ( ) r+ } β θt t δ + +Ψt t δ t µ t t 2tt + tt t t θt = 0, Ψ Ψ = t β t + Rt + 0, 5

17 ( ) u2t + θt f2t +Ψt f2t tlt tltlt wt u2t ( 2t lt + t ltlt ) 2 ( ) u 22t + µ t µ t u2t ( tt tt ) rt 0. ( tlt tltlt ) w + = t tlt t ltlt wt In the steady state, these conditions and the constraints give u 2 θ = u + µ u + ( τ τ )( f δ), ( τl τ ll) f2 ( β r) = ( +Ψ)( r r) u( + ) r = ( +Ψ )( + ) u( + ) ( f ) θ θ µ 2τ τ θ τ τ µ 2 τ τ δ, (A2) Ψ=, (A3) u ( 2τ + τ l) µ u2 l l u2 = θ f2 +Ψ ( τl + τ ll) f2+ u22 µ u2 + ( τ τ )( f δ), ( tl t ll) f2 tl t ll (A4) c= f(,) l δ g, (A5) ( ) ( ) ( )( f ) ( τ τ ). (A) τl fl 2 + τ τ τ b f δ = g, (A6) = β + τ τ δ, (A7) u = u l f (A8) 2 l l 2 2. Derivation of the sign of µ>0 and θ>0 First, substituting (A), (A7) and (A8) into (A4) yields ( )( ) ( 2τ τ ) u + u + l f + u f u f + u u+ψ τl + τ ll f2 = µ + ( tl t ll) f2 b 22 l l The left-hand side of (A9) is positive and the expression in the bracets on the right-hand side of (A9) is positive, which gives µ>0. Next, for progressive income taxes, (A) gives u 2 u u2u θ = u + µ u + ( τ )( f δ ) = u + µ, (A0) ( τl ) f2 β u2 where (A7) and (A8) are used in the second equality. As claimed by Chamley (986), θ>0 and thus the shadow price of capital is positive in the economy with linear income taxes. For non-linear income taxes, (A) gives u 2 u u2u θ = u + µ u + ( τ τ )( f δ ) = u + µ, (A) ( tl t ll) f2 b u2 where (A7) and (A8) are also used in the second equality. Notice that the right-hand side of the second equality in (A) is the same as the right-hand side of the second equality in (A0). Thus, θ>0 and the shadow price of capital is still positive even when the income tax rates are progressive.. (A9) 6

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