Quantifying the Welfare Gains from History Dependent Income Taxation

Transcription

1 Quantifying the Welfare Gains from History Dependent Income Taxation Marek Kapička CERGE-EI, Prague February 10, 2017 Abstract I quantify the welfare gains from introducing history dependent income tax in an incomplete markets framework where individuals face uninsurable random walk idiosyncratic shocks. I assume that the income tax paid is a function of a geometrical weighted average of past incomes, and solve for the optimal weights. I find that the optimal weights on past incomes decline geometrically at a rate equal to the discount rate. The welfare gains from history dependence are large, about 1.77 percent of consumption. I decompose the total effect into an efficiency effect that increases labor supply, and an insurance effect that reduces volatility of consumption and find that, quantitatively, the insurance effect dominates the efficiency effect. The optimal tax increases consumption insurance by trading higher tax progressivity with repect to past incomes for a reduced tax progressivity with respect to the current income. J.E.L Codes: E6, H2 Keywords: optimal taxation, income taxation, incomplete markets 1 Introduction Recent research on dynamic optimal taxation, starting with Golosov et al. (2003), has shown one robust feature of optimal tax systems in economies with idiosyncratic shocks: 1

2 they should in general depend on the full history of individual s incomes. 1 Yet, our understanding of how specifically the income taxes should depend on past incomes is limited. How important is one s income last year relative to one s income ten years ago? What are the welfare gains from history dependence, and where do the welfare gains come from? What parameters of the environment are key for determining the gains from history dependence? How much is lost by restricting history dependence to a limited number of periods? Robust answers to questions like these have not yet been provided. This paper answers those questions in an analytically tractable framework similar to the one used by Heathcote et al. (2014a). Individuals face uninsurable random walk productivity shocks, and the government uses a nonlinear income tax of the form T t (y t ) = y t λȳ 1 τ t, where y t is a current income, and ȳ t is an aggregator of past incomes: ȳ t = t j=0 (y t j) θ j. The tax is thus history dependent, with the relative importance of past incomes embodied in the coefficients {θ j }. The framework is flexible enough to incorporate history independent income taxes, income taxes that depend on a finite number of lags, and income taxes that depend on the full history of incomes. It also retains tractability that characterizes this environment with history independent taxes. Why is history dependence useful? In principal, history dependence could either help reduce distortions of labor supply (incentive effect), or provide a more efficient consumption insurance (insurance effect). One can write the parameters of the tax system in a way that the incentive effect is determined only by the parameter τ: higher τ drives a wedge between wage and marginal rate of substitution, and reduces labor supply. The overall progressivity of the tax system is thus determined by τ, and I call τ the progressivity wedge. The extent of consumption insurance is, on the other hand, driven by both the progressivity wedge τ and the history dependence parameters θ. I show that the history dependence parameters θ are chosen in a very simple way: to minimize the cross-sectional dispersion of consumption subject to a constraint that the overall progressivity of the tax system does not change. This is achieved by, choosing a tax system that is regressive with respect to the current income, and mildly progressive with respect to the past income, with past incomes discounted at a rate proportional to the rate of time preference. By making the tax system less regressive with respect to the current income and more progressive with respect to past incomes, the government trades insurance of 1 Exceptions to this rule involve some economies where all uncertainty is resolved in the initial period and, to some extent, economies with IID shocks, where history dependence can be replaced by dependence on assets (Albanesi and Sleet (2006)). 2

3 current consumption with insurance of past consumption while maintaining labor supply incentives. The trade-off is resolved by increasing insurance of past consumption. I also show that the weights θ depend only on a small number of parameters. Notably, up to a first-order approximation, they are independent of the variance of the underlying shocks, or of the progressivity parameter τ. While the history dependence parameters θ are chosen independently of the progressivity wedge τ, the reverse is not true. The progressivity wedge is chosen to balance the distortions of labor supply and, again, a reduction in the cross-sectional consumption dispersion. Since history dependence already reduces consumption dispersion, the government responds by reducing the progressivity wedge τ relative to the case with history independent taxation. Thus, history dependence in the end reduces both the progressivity of the tax system, and the variance of consumption. I calibrate the model to the U.S. economy. I follow Heathcote et al. (2014a) who show that the U.S. tax system can be well approximated by a (history independent) tax system with a progressivity parameter τ = I show that the welfare gains from history dependence are large: they are about 1.76 percent in consumption equivalents. Quantitatively, almost all of the benefits of history dependent income taxation, about 92 percent of it, come from the fact that it reduces the cross-sectional variance of log consumption, which is reduced by about a third. The remaining 8 percent comes from efficiency gains, i.e. from the reduction of the progressivity wedge τ. In the optimum, τ is increased modestly from to 234. In contrast, if the tax system is history independent, then the current U.S. tax system should be much more progressive, since the parameter τ should be increased to I also compute welfare gains from a limited history dependence. There are large welfare gains even from relatively short history dependence: Taxes that depend only on the past 10 annual incomes generate about 50 percent of the potential welfare gains from an unlimited history dependence, while taxes that depend on the past 16 incomes generate about 75 percent of the potential welfare gains. Finally, I compare the welfare gains from a history dependent taxes to taxes that depend only on age. Age dependent taxation achieves a similar reduction of consumption dispersion, by increasing tax progressivity with age. This, however, introduces unnecessary variations in the progressivity wedge over time - something that history dependent taxation is able to evade. As a result, the welfare gains from age dependent taxation, while still large, are only about 48 percent of the overall gains. The paper connects two strands of the existing literature. On one hand, it uses in- 3

4 sights from the recent dynamic public finance literature (Golosov et al. (2003), Kocherlakota (2005), Farhi and Werning (2012) and many others) that shows that history dependence in income taxation is optimal. On the other hand, in order to achieve tractability, the paper does not use a standard mechanism design approach to gain insights about the optimal policies. Instead, it follows the tractable analytical framework pioneered by Constantinides and Duffie (1996) and further extended by Heathcote et al. (2014a) and Heathcote et al. (2014b), who extend it by including insurable transitory shocks as well as labor supply decision. As in Benabou (2002), who studies educational decisions in a related framework, I assume that the agents cannot borrow and save to self-insure to gain tractability. Each of the last three papers assumes that income tax function is a power function but, importantly, none of them allows for history dependence. 2 Setup The model is a perpetual youth overlapping generation model. There is a measure one of agents in the population. Each period, a fraction 1 δ of each generation dies and is replaced by a new generation of size 1 δ as well. The measure of age-j agents is thus (1 δ)δ j. The utility function takes the form ( w 0 = E 0 (1 βδ)(βδ) j ln c j φ ) 1 + η h1+η j, 0 β < 1. (1) j=0 The parameter η 0 is inverse of the Frisch elasticity of labor supply, while φ determines the relative weight of disutility from working. The parameter η is identical for all agents, while φ is a random variable, drawn once at age zero, with a cumulative distribution function F φ. At each age, agents received labor productivity z j Z (0, ). Their productivity, together with hours worked h j determines the output y j = z j h j. Log productivity z j follows a random walk, ln z j = ln z j 1 + ω j, and ω j is an iid random variable that has a cumulative distribution function F ω and density f ω. The distribution is normalized so that E(e ω ) = 1, but is not necessarily lognormal. Both the initial seed z 1 and the initial shock ω 0 are equal to zero, implying 4

5 that the initial productivity z 0 is equal to one for everyone. 2 Market Structure. I assume that there is no insurance against the permanent shocks z. That includes self-insurance: the agents are not allowed to save to hedge against the permanent shocks. This is a strong assumption, but it allows me to get closedform solution for the equilibrium allocations even for the tax systems that are history dependent. Benabou (2002) makes the same assumption for the same reasons. I will revisit this assumption later. 2.1 A Tax with History Dependence The government taxes individual incomes by using an income tax that is history dependent: the tax paid depends on individual s history of earnings. The tax function has the following functional form: an individual of age j with a history of incomes y 0, y 1,..., y j pays taxes ) 1 τ T j (y 0, y 1,..., y j ) = y j λ j (ȳj, (2) where ȳ t is a weighted geometric average of current and past incomes, ȳ j = j k=0 (y j k ) θ k. The history dependence parameters θ = (θ 0, θ 1,...) represent how the current tax paid depends on past income realizations. Specifically, the parameter θ k represents the weight on income of lag k. A history independent tax is a special case with θ 0 = 1 and θ k = 0 otherwise. The tax function T j depends on age directly through the level tax parameter λ. Age specific λ allows the government to choose a trend in consumption independently of the remaining parameters. The income tax also depends on age indirectly, because the length of individual histories depends on age, although both the history dependence parameters and the progressivity wedge are time and age invariant. 3 The parameters τ and θ jointly determine the progressivity of the tax system. The average income- 2 The model can be easily generalized to allow for initial heterogeneity in productivity, without changing the main results. 3 If one assumes that incomes before being born are all equal to one, then the tax function can be written as a time and age invariant function of an infinite history of incomes. 5

6 weighted marginal tax with respect to the current income is 1 (1 τ)θ 0, and so if θ 0 is sufficiently large, the tax system may be regressive with respect to the current income. The average income-weighted marginal tax with respect to the income of lag k is (1 τ)θ k. If θ k is positive then the tax system is regressive with respect to past incomes, while if θ k is negative, the tax system is progressive. Progressivity keeping constraint. There is one degree of freedom in the tax parameters τ and θ, and I normalize the coefficients θ by requiring that the following progressivity keeping constraint holds: k=0 (βδ) k θ k = 1. (3) As I shall now show, this normalization implies that labor supply decisions of the agents are independent of the coefficients θ. Substituting consumption c t = λ (ȳ t ) 1 τ into the utility function (1) and rearranging yields the lifetime utility of an agent j k=0 w 0 = (1 βδ)e 0 (βδ) [ln j λ j + (1 τ) j=0 [ = (1 βδ)e 0 (βδ) j ln λ j + (1 τ) ln y j j=0 θ k ln y j k ] φ ( ) 1+η yj /z 1 + η j φ ] ( ) 1+η yj /z 1 + η j, (4) where the second equality collects all the terms involving y t and uses (3). Each individual, when choosing hours worked, takes into account the effect of all future taxes paid from current income. If the normalization (3) holds, reducing marginal tax rates in the current period by reducing θ 0 by one unit must be exactly offset by an increase in the marginal tax rate in some period k by (βδ) k units. Since future marginal tax rates are effectively discounted at rate βδ, this trade-off does not change incentives to supply labor. The progressivity keeping constraint thus implies that the effective progressivity of the tax system with respect to the current income is determined exclusively by the parameter τ, henceforth called progressivity wedge, and independent of the parameters θ. The effective progressivity of the tax system must be distinguished from the nominal progressivity with respect to current income, ˆτ 0 = 1 (1 τ)θ 0. If there is no history dependence then ˆτ 0 = τ and both measures of progressivity coincide. dependence, they will be distinct. With history 6

7 2.2 Allocations It follows from the last form of the expected utility function (4) that, given the assumption of no savings, the agent s choice of earnings is now static, with the effective tax progressivity wedge 1 τ. Taking first-order conditions yields the optimal hours worked ln h j = 1 [ln(1 τ) ln φ]. (5) 1 + η Hours worked are independent of the productivity shocks, because the income and substitution effects cancel out. Heterogeneity in the preference parameter φ is the only source of heterogeneity in hours worked in this model. Due to the progressivity keeping constraint (3), hours worked are independent of the history dependence parameters θ, and only depend on the progressivity wedge 1 τ, and on the preference parameter φ. Individual consumption can be obtained by using the budget constraint c j = λ j (ȳj ) 1 τ, and by substituting in the equilibrium hours worked: ln c j = ln λ j + 1 τ 1 + η j k=0 θ k [ln(1 τ) ln φ] + (1 τ) j k=0 θ k ln z j k. (6) Consumption depends on all past productivities only because taxes paid depend on past incomes. The key in determining the nature of history dependence are, of course, the history dependence parameters θ. Note that consumption in general can move predictably with age, first because λ depends on age and, second, because the expected value of the weighted average of past incomes ȳ j is changing with age. Specifically, if j k=0 θ k is decreasing in j, then the expected consumption will be increasing with age. 3 Partial Equilibrium I will start by analyzing a government s problem in a partial equilibrium, where prices are exogenous, and there is only one generation, born at time zero. The government maximizes the following social welfare function: W (1 αδ) (αδ) j E 0 (u j ). (7) j=0 7

8 where u j is period utility, and α [β, 1) is the government s discount factor. The government is therefore discounting future at a rate that may be different from the agent s discount rate. This assumption can be taken as a primitive assumption about the social welfare function. In that case, the social welfare function is non utilitarian. But, as we shall see later, the assumption can be also justified in an overlapping generation setting, where the government has a utilitarian social welfare function for each generation, but cares more about future generations than about the current generation (in which case α > β). In any case, the additional flexibility of allowing for a different discount factor for the government will prove useful for clarifying what matters for history dependence. The tax system imposes aggregate cost equal to the present value of aggregate consumption minus the present value of aggregate earnings: P (1 qδ) j=0 (qδ) j E 0 (c j y j ), (8) where the interest rate q (0, 1) is exogenously given. The government chooses the tax parameters λ, τ and θ to maximize the social welfare function (7) subject to the resource constraint P = 0 and the progressivity keeping constraint (3), taking the policy functions (5) and (6) as given. Substituting the policy functions (5) and (6) into the social welfare function (7) and the resource constraint and optimizing with respect to the level parameters λ yields the following optimality condition: the parameters λ should be chosen in a way that the aggregate consumption is growing at a rate equal to α/q: E 0 (c j ) = E 0 (c 0 ) ( ) α j. (9) q Aggregate consumption is thus constant if α = q, and the government s discount factor is exactly offset by the rate of return. Note that individual discount factor β does not play any role in determining the trend in aggregate consumption because an individual cannot shift resources over time on his own. This is reminiscent of Calvo and Obstfeld (1988) findings that aggregate consumption in the Pareto optimum of an overlapping generations economy is determined by the social planner s rate of time preference, and individual s time preference does not play any role. It is shown in Appendix A that if the parameters λ are set optimally, the aggregate welfare can be written as a function of 8

9 τ and θ only, and is given by ( ) 1 αδ + αδ (ln α ln q) 1 qδ 1 αδ W(τ, θ) = ū(τ) + ln [ (αδ) j αδ ln B ω ((1 τ) j k=0 θ k j=0 + (1 τ) ) ( + (1 αδ) ln B φ j 1 τ ) ] k=0 1 + η θ k ( αδ 1 αδ Eω η E ln φ) (αδ) k θ j, (10) j=0 where B ω (a) = E(e aω ) and B φ (a) = E(e a ln φ ) are moments of the underlying distributions, and u(τ) = 1 1+η [ln(1 τ) 1 + τ] is the representative agent s period utility.4 The government s problem is now to maximize the objective function (10) by choosing the tax parameters τ and θ subject to the progressivity keeping constraint (3). 3.1 Lognormal Distribution The government s problem will be greatly simplified if both the productivity shocks and the preference shocks are lognormally distributed. To that end, assume that ω N ( σ2 ω ) 2, σ2 ω ( ln φ N (1 + η) σ2 φ 2, (1 + η)2 σφ 2 The mean and variance of the taste shock distribution are normalized so as to simplify algebra later. Then B ω (a) = e 1 2 a(1 a)σ2 ω and B φ (a) = e 1 2 (1+η)a(1+(1+η)a)σ2 φ and the government s objective function (10) can be simplified to W(τ, θ) = u(τ) 1 ( ) αδ 2 (1 τ)2 P(θ; α) 1 αδ σ2 ω + σφ 2, (11) 4 In principle, there is an additional constant term reflecting the aggregate production gains from the dispersion in φ. This term is zero due to the normalization of the taste shock distribution. ). 9

10 where the function P(θ; α) is a quadratic function of the history dependence parameters only, and is given by P(θ; α) = k=0 ( ) (αδ) k θk k 1 l=0 θ lθ k. The functional form of the social welfare function (11) shows a very important point: The choice of the history dependence parameters θ is independent of the choice of the progressivity wedge τ (the reverse is clearly not true). This substantially simplifies the problem, both technically, and substantially. It is easy to see that, under history independence, the value of P equals one. The welfare effects of a history dependent tax policy can then be measured by how much it reduces the value of P. The size of the welfare effects is furthermore proportional to the variance of the shocks, and to the square of the progressivity wedge τ. Before solving for the optimum of this model, it is however worth considering a simple example that sheds light on why, and under what conditions, is history dependence in income taxation welfare improving. 3.2 Why History Dependence? To understand why it is optimal to introduce income taxes with history dependence, consider a special case with only one lag, K = 1. Consumption at the initial age then depends only on the current income, while consumption in all other periods depends on both current and on previous income. Since permanent productivity shocks enter symmetrically with preference shocks in the objective function (11), assume furthermore that σ 2 ω = 0, so that people are heterogeneous only in their preferences. Since the intertemporal price does not affect the choice of the history dependence parameters, set q = α. Finally, set τ = 0 to reduce notation. 5 To separate the effects on the aggregate means from the effect on the aggregate variance, I will use ψ = 1 2 σ2 φ < 0 to represent the mean of ln ψ. The welfare function (7) can be rewritten as W(0, θ) = u(0) + (1 αδ) ln λ 0 + αδ ln λ 1 + ψ(θ 0 + αδθ 1 ), 5 None of these assumptions has any effect on the underlying intuition. 10

11 while the resource constraint is (1 αδ)λ 0 e ψθ σ2 φ θ2 0 + αδλ 0 e ψ(θ 0+θ 1 )+ 1 2 σ2 φ (θ 0+θ 1 ) 2 = 1. The progressivity keeping constraint (3) is θ 0 + βδθ 1 = 1, and it needs to hold under any tax reform. The optimal choice of the level parameters λ 0 and λ 1 ensures that aggregate consumption is constant over time, and is equal to one in every period. Consider the following perturbation to a history independent tax with θ 0 = 1 and θ 1 = 0. Suppose that θ 0 increases by dθ 0 > 0. An increase in θ 0 has two effects. First, it makes the tax system less progressive at all ages, and directly changes welfare by ψ dθ 0 < 0. Second, a change in θ 0 requires an adjustment in the tax parameters λ 0 and λ 1 to keep mean consumption unchanged, and the government budget constraint holds. This changes welfare by (ψ + σ 2 φ ) dθ 0 < 0. Taken together, the level effects cancel each other out and an increase in θ 0 on net decreases welfare by σ 2 φ dθ 0. The effect is negative, because an increase in θ 0 reduces consumption insurance at every age. An increase in θ 0 is, however, compensated by a decrease in θ 1. The progressivity keeping constraint implies that θ 1 must change by dθ 1 = (βδ) 1 dθ 0 < 0. The effects now have the opposite sign: a decrease in θ 1 changes utility directly by αδψ dθ 0 > 0 and indirectly through adjustment in λ 1 by αδ(ψ + σ 2 φ ) dθ 1, with a positive net effect αδσ 2 φ dθ 1 > 0. The positive net effect now comes from an increased consumption insurance for agents with age one and more; that s why its size is αδ times smaller than what a perturbation in θ 0 produces. The overall change in welfare is dw = σ 2 φ dθ 0 δασ 2 φ dθ 1 = ( ) α β 1 σφ 2 dθ 0. If α > β then the total effect is positive, and it is optimal to increase θ 0 and decrease θ 1. The key to understanding the result lies in understanding the role of history dependence. By changing history dependence, the government can change consumption insurance across ages without changing labor supply incentives (as long as the progressivity keeping constraint holds). The net effect depends on two things: at what rate can consumption insurance be traded across ages, and how the government values consumption insurance at different ages. An increase in θ 0 decreases consumption insurance at all ages, while a decrease in θ 1 increases consumption insurance in the future (at ages one and on). The trade-off is determined by the individual discount factor βδ. Since 11

12 βδ < 1, an increase in θ 0 allows for a proportionally larger reduction in θ 1, and the above perturbation decreases consumption insurance at age zero while increasing consumption insurance in the future. Now, how important this trade-off is for the government is determined by the government s discount factor αδ. If the government is more patient than the agent, an increase in future consumption insurance dominates, and the trade-off is worthwhile. If β = α, then a perturbation in θ 0 or θ 1 each has nonzero first-order effect on welfare, and the trade-off between consumption insurance across ages still exists. But the effects exactly cancel each other in the social welfare function, and a history independent tax policy is an optimal one. If, on the other hand, σ φ = 0 then there is no trade-off. any change in θ is exactly compensated by changes in λ 0 and λ 1, and any perturbation in θ 0 or θ 1 has zero welfare effect. It is again optimal to have history independent tax because it smooths consumption for the representative agent. An analogous intuition will extend to the more general case with infinite history dependence. What the example does not show, is what happens if the government discount factor is not equal to the intertemporal price of consumption. In that case, as was shown in (constime), aggregate consumption will no longer be constant over time. But beyond that, the intuition from this example will carry over. It will also carry over to the case when there is idiosyncratic uncertainty about productivity shocks, rather than preference shocks. 3.3 Optimal History Dependence The social welfare function shows that the optimal history dependence parameters are a solution to the following simple maximization problem: θ = arg min θ P(θ; α) s.t. (3). This minimization problem makes it clear that the optimal history dependence coefficients depend only on the discount factors β and α, and on the survival rate δ. Notably, they are independent of the progressivity wedge τ, and of the variance of the shocks σ 2 ω 12

13 and σ 2 φ. The first-order condition in θ k is k θ l + l=0 (αδ) l k θ l = l=k+1 ( ) β k ζ, α where ζ is an (appropriately normalized) Lagrange multiplier on (3). This is a first-order difference equation that, together with (3), can be solved for the coefficients θ and for the Lagrange multiplier ζ. The solution is given in the next Proposition: Proposition 1. The welfare maximizing coefficients θ are θ0 = 1 β2 δ ( α 1 βδ, θ k = 1 β ) ( ) β k 1 θ0 α α, k > 0, and the value of P(θ ) is P(θ ; α) = ( ) (1 δα) 1 β2 δ α (1 βδ) 2. Proposition 1 offers a very simple and practical characterization of the optimal progressivity coefficients. It immediately implies that the optimal coefficient on the current income is positive and greater than one, while the coefficients on past incomes are negative. Relatively to a history independent tax, the tax system thus becomes less progressive with respect to the current income and more progressive with respect to past incomes. The results, and the intuition, are both analogous to the example above. The coefficients on past incomes are geometrically increasing, and converge to zero. The rate at which they are decreasing is given by the ratio of both discount factors, β/α. They will typically be significantly smaller than the coefficient on the current income: for example, the coefficient on the previous period income is only 1 β/α the coefficient on the current income. Even for α = 1, it will be only around 2-4 percent of what the coefficient on the current income is. Turning to the progressivity wedge τ, an increase in τ weighs the costs in terms of labor supply distortions against the benefits of reduction in the dispersion of consumption (see 11). Since history dependence reduces consumption dispersion by reducing P below one, the benefits from higher τ are reduced, and the balance shifts in favor of lower labor supply distortion: 13

14 Proposition 2. The optimal progressivity wedge τ decreases when history dependence is allowed. Reinterpreting History Dependent Tax. The history dependent tax function (2) can, for the optimum coefficients θ, be reinterpreted in two useful ways. First, by rearranging the terms, one can write the geometric average of past incomes as T t = y t t k=0 ( yt k y t k 1 ) β k (1 τ)θ 0 α. That is one the optimal tax can be written as a function of a weighted average of past income growth rates, as opposed to a weighted average of the levels of income. result follows intuitively from the fact that the productivity shocks are essentially shocks to the growth rate of productivity, as opposed to shocks to the level of productivity. Alternatively, and perhaps more usefully from a practical perspective, one can define an average past income as x t by x t = t k=0 y(1 β/α)(β/α)k t k. Then the tax paid is The ( ) (1 τ)θ0 yt T t = y t λ. x t 1 The past income average is simply updated recursively according to x t = y 1 β/α t x β/α t 1, and every agent starts with x 0 = 1. This formulation is useful since one needs only one state variable, the average of past incomes x t 1, to characterize the current tax function. Labor supply elasticity. It seems intuitive that the gains from history dependence should increase with (Frisch) labor supply elasticity: history dependence decreases the progressivity wedge τ, and the gains from reduced distortions should be larger if labor supply elasticity is larger. This intuition is correct. Since the welfare is linear in P, one can express the complementarity between history dependence and labor supply elasticity by 2 W P η ( ) αδ τ = (1 τ) 1 αδ σ2 ω + σφ 2 η. History dependence will then be complementary to labor supply elasticity if and only if τ increases with η (i.e. τ decreases with labor supply elasticity). A direct differentiation 14

15 yields that this is indeed the case: τ/ η > 0. History dependence is thus always complementary to labor supply elasticity, and the gains are higher for higher values of labor supply elasticity. 3.4 Limited History Dependence What if the government is only allowed to use income history of length K? Denote by P K (θ; α) to be the function P where all the history dependence coefficients θ k are restricted to be zero for k > K. Next proposition shows the closed form solution to the truncated case: Proposition 3. Assume that θ k = 0 for k > K, for some K > 0. Then the welfare maximizing coefficients are and the value of P K (θ ) is θ0 = 1 β2 δ α 1 βδ θ k = ( 1 β α 1 θk = 1 ( 1 β 1 βδ α ( 1 β α ) 2 δα (1 βδ) 2 ( β 2 δ α 1 ) K ) ( ) β k 1 θ0 α, k = 1,..., K 1 ) ( ) β K 1 θ0 α. P K (θ ; α) = ( (1 δα) 1 β2 δ α (1 βδ) 2 ) 1 ( 1 β α ) 2 δα (1 βδ) 2 ( β 2 δ α 1 ) K. The coefficients in the full history dependence problem are just a limiting case with K going to infinity. On the other hand, there is a substantial difference between a tax that includes only current and past income (K = 1), and a tax under full history dependence. For example, when τ = 0.2, α = 1, β = 0.96 and δ = 0.97 then 1 (1 τ)θ0 = The average marginal income tax rate with respect to the current income is thus -75 percent. In contrast, the current income is taxed only at a rate of -23 percent with full history dependence. There is also a substantial difference between a tax with no history dependence (K = 0) and a tax that includes only current and past income. However, 15

16 including one period lag reduces variance of consumption by only about 5 percent, significantly less than under full history dependence. Although the value of P K (θ ; α) is decreasing in K, one needs more than one lag to capture most of the potential benefits from history dependence. 3.5 Age Independent Tax Function Up until now, the government was allowed to use a tax function that depended directly on age through the parameters λ j. What happens if the government does not have the opportunity to choose age specific λ? I derive a full solution in Appendix A. there, I show several results. First, the inability to transfer resources across age through variations in λ means that the parameters τ and θ will, at least in part, be chosen so as to substitute for age varying λ. The optimal history dependence parameters θ now also depend on the intertemporal price of consumption q. The exact form of the objective function can be found in the appendix. Here I only mention that an exercise analogous to the perturbation exercise in section 3.2 yields that a perturbation of θ 0 = 1 by dθ 0 and a corresponding perturbation of θ 1 = 0 by dθ 1 = (βδ) 1 dθ 0 will yield a welfare change dw = ( q β 1 + α q ) σφ 2 dθ 0. 2β Hence, it will be welfare improving if, first, q > β and, second, if α > q. The first requirement is that the government is more patient than the agent, but it is now the intertemporal price of consumption, rather than the government discount factor, that is being compared to the agent s discount factor α. The second condition requires that the government is patient relatively to it s the discount factor. We know from (9) that, in that case, the government wants to implement an increasing pattern of consumption over time. It cannot do it directly now. However, a progressive history dependent tax acts as an imperfect substitute for that, because the progressivity parameters accumulate and increase consumption as individual ages. SHOW THIS. It is easy to see that if α = q then the condition collapses to the one in section 3.2. In fact, I show more than that. Whenever α = q and the distribution is lognormal, the welfare function (11) is approximately correct, and so are the optimal welfare coefficients. More specifically, a first-order approximation to the time invariant parameter λ yields the welfare function identical to (11). As it turns out, for empirically relevant parameter 16

17 values, namely standard deviations of the idiosyncratic shocks, the approximation is very close to the unrestricted solution. In those cases, an age independent tax function is a very good approximation of the optimal one. 4 General Equilibrium I will now analyze a problem for a full overlapping generation economy, and show how it can be malled to the partial equilibrium framework. The social welfare function aggregates lifetime utilities of generations that are not yet born, and the remaining lifetime utilities of generations that are currently alive. The government weights the welfare of a generation born in period t by π t for some Pareto weight parameter π (0, 1). This includes generations not yet born, but also generations that are currently alive, i.e. born before time t = 0. Their Pareto weights are then whose weight is then greater than one. The social welfare function is then W (π βδ)(1 π) π(1 βδ) [ π t E 0 (w t ) + t=0 ] 1 π t E 0 (ŵ t ), (12) t= where E 0 (w t ) is the expected lifetime utility of someone who will be born at time t 0, E 0 (w t ) = (1 βδ) j=0 (βδ) j E 0 (u j ) t 0, (13) and E 0 (ŵ t ) is the remaining lifetime utility of someone who is currently alive and was born at time t < 0. Importantly, the remaining lifetime utility is discounted back to their birthdate: E 0 (ŵ t ) = (1 βδ) (βδ) j E 0 (u j ) t < 0. (14) j= t Discounting everyone s period utilities back to their birthdates and discounting every generation s lifetime utilities at rate π ensures that everyone is treated symmetrically, and implies that the resulting social welfare function is time consistent (Calvo and Obstfeld (1988)). The expected period utilities E 0 (u j ) depend on age, but not explicitly on time, or one s cohort. That is because the tax function is not explicitly dependent on time (al- 17

18 though it can depend on age or individual history). Then all individuals of age j have the same utility u j. Substituting the expressions (13) and (14) into the welfare function (12) yields that the welfare is a weighted average of period utilities, with future utilities discounted at a rate determined jointly by the discount factor, the survival rate, and the Pareto parameter: W = ( 1 βδ π ) j=0 ( ) βδ j E 0 (u π j ). (15) The general equilibrium case can thus be mapped into the partial equilibrium model with α = β/π. If the benchmark case when the government discounts at the same rate as the agents, π = β and the welfare is simply the average of period utilities weighted by the survival rate δ. 6 Similar calculations can be made regarding the resource costs. The annualized present value of the aggregate costs is the sum of the discounted expected cost of the generations not yet born, and the expected costs of the currently living generations for the reminder of their life: P (1 q)(1 δ) 1 qδ [ q t E 0 (p t ) + t=0 ] E 0 ( ˆp t ), t= 1 where E 0 (p t ) is the expected present value of costs of a generation born in period t discounted back to the present value at rate q, E 0 (p t ) = (1 qδ) j=0 (qδ) j E 0 (c j y j ) t 0, (16) and E 0 ( ˆp t ) for t < 0 is the expected present value of of costs for the currently living generation for the remainder of their lifetimes, E 0 ( ˆp t ) = (1 qδ)δ t j=0(qδ) j E 0 (c j t y j t ) t < 0. (17) If the period cost again do not depend on time, then the aggregate cost are independent of the intertemporal price of consumption, and are simply equal to the period costs, 6 One interpretation is that, conditional on being alive, all period utilities are weighted equally. 18

19 where the cost of older generations is discounted at rate δ because of a decreasing generation size: P = (1 δ) δ j E 0 (c j y j ). (18) j=0 The general equilibrium case can again be mapped into the partial equilibrium model with q = 1. Overall, the solution to the general equilibrium model is identical to the solution to the partial equilibrium model with α = β/π and q = A Benchmark Case I will now assume that the government weights the generations at the discount rate π = β. The welfare function (15) is now W(τ, θ) = u(τ) 1 2 (1 τ)2 P(θ; 1)(σ 2 z + σ 2 φ), (19) where σz 2 = 1 δ δ σ2 ω is a cross-sectional variance of productivity shocks. One implication of the fact that welfare can be represented by (19) is that the preference shocks ln φ and the productivity shocks z enter symmetrically; in fact, only the sum of both variances matters for aggregate welfare. This is intuitive, since both shocks affect the agent s ability to produce output similarly, and each innovation in the agent s productivity ω is permanent, just like the preference shock. Since the cross-sectional variance of log consumption is Var (ln c) = (1 τ) 2 P(θ; 1)(σ 2 φ + σ 2 z ), the welfare function in the benchmark case is simply a welfare from a representative agent s allocation ū(τ), minus one half of the cross-sectional variance of log consumption Var (ln c). The optimal history dependence parameters can be characterized as follows: Proposition 4. The coefficients θ maximize social welfare if and only they minimize the crosssectional variance of log consumption subject to the progressivity keeping constraint (3). The optimal history dependence parameters are given in Proposition 3, with α set equal to one. The coefficients on past incomes are thus geometrically increasing at rate β. The left panel of Figure 1 plots the value of the coefficient θ 0 for a range of values of β and for three values of δ. The value of θ 0 is significantly greater than one. In fact, for 19

20 δ = 0.96 δ = 0.97 δ = θ0 1.4 P(θ ) δ = 0.96 δ = 0.97 δ = β β Figure 1: The value of the history dependence parameter θ 0 (left panel) and the reduction in the crosssectional variance of consumption P(θ ; 1) (right panel) for various values of β and δ. plausible value of the progressivity wedge τ, the value of nominal progressivity with respect to the current income 1 (1 τ)θ 0 is negative. Since the marginal tax rate with respect to the past incomes is positive, the tax system subsidizes current income, and then taxes it for the rest of the agent s life. The value of P(θ ; 1) converges to one as β approaches unity (more precisely, when β converges to α = 1), and so for extremely low discount rates the reduction in the crosssectional variance of consumption will not be very high. For more plausible values the reduction can be significant: for β around 0.96 the variance of log consumption will be reduced by about a third, as the right panel of Figure 1 illustrates. Consumption Dynamics. The optimal income tax changes dramatically the dynamics of individual consumption. While consumption follows a random walk when taxes are history independent, it is a moving average process under the history dependent tax 20

21 system: ln c j = ln c j 1 τ j η βj ln φ + (1 τ)θ 0 β k ω j k, k=0 where ln c j is a deterministic component of consumption. The left panel of Figure 2 shows a typical response to a unit increase in the innovation to the permanent component ω for a given progressivity wedge. Under history independent taxation, the effect on consumption is permanent. Under full history dependence, consumption responds more in the short run, but then the effect of the shocks disappears. Clearly, in order to maintain the same progressivity wedge, there needs to be a trade-off, and consumption cannot always respond less under history dependence. But the government can convert the long-run consumption risk (under history independence) into a short-run risk. Figure 2 also shows the consumption impulse response function and the variance of log consumption under partial history dependence for K = 5. The result combines features of both extremes: Not all long-run risk is eliminated, although it is significantly reduced. The variance of log consumption initially increases faster than under history independence, and then grows without bounds, although at a smaller rate than under history independence. This has important implications for the nature of consumption dispersion. The right panel of Figure 2 shows the variance of log consumption by age. Under history independent taxation, the variance of individual consumption Var (ln c j ) grows over time linearly without bounds. 7 Under the optimal history dependent scheme, on the other hand, the variance of log consumption is bounded away from infinity, Var (ln c j ) = (1 τ) 2 θ0 2 1 β 2j 1 β 2 σ2 ω, (20) and converges over time to a finite limit θ 2 0 /(1 β2 )(1 τ) 2 σ 2 ω. Since θ 0 > 1, the optimal policy increases the immediate impact of the shock and, in the short run, the variance of log consumption is higher than under history independence. But then the shock s impact decays at a rate equal to the discount rate. Welfare gains. What are the welfare gains from optimal history dependent tax? Previous discussion shows that history dependence increases welfare in two ways: first, 7 What keeps the cross-sectional consumption from growing without bounds is that individuals die at rate δ. 21

22 ln c j Var ln c j age j age j Figure 2: Response of log consumption to a unit increase in ω (left panel) and variance of log consumption by age (right panel). Blue lines represent taxes dependent only on current income. Red lines represent taxes with full history dependence. Green lines represent taxes with limited history dependence (K = 5). it reduces consumption dispersion for any given progressivity wedge τ and, second, it reduces the progressivity wedge itself. The contribution of each of those components will be examined quantitatively later. The welfare gain from the reduction of consumption dispersion is, however, easy to characterize. Let Ŵ 0 (τ) be welfare under a history dependent tax with progressivity wedge τ, and Ŵ(τ) be welfare under a history dependent tax where the coefficients θ are chosen optimally, but the progressivity wedge is still τ. Using the fact that with log utility the welfare gains are approximately equal to the difference between both values of welfare, we get Ŵ(τ) Ŵ 0 (τ) = 1 2 (1 τ)2 [1 P(θ ; 1)] (σ 2 φ + σ 2 z ). If τ = τ0, the optimal progressivity wedge under the history independent tax system, then Ŵ Ŵ 0 represents a lower bound on the welfare gain from history dependence since there is an additional welfare gain from replacing τ 0 by τ. If τ equal to a value that approximate U.S. tax code, then Ŵ Ŵ 0 is the lower bound on the welfare gains 22

23 from reforming the current U.S. tax code. The latter calculation can in fact be done very easily, since (1 τ US ) 2 (σφ 2 + σ2 z ) is the measured variance of consumption under the U.S. tax code and is approximately Thus, the welfare gains are at least 0.07 [1 P(θ )]. If the variance of consumption is reduced by a third, then the welfare gains are at least 2.3 percent in consumption equivalents. As we shall see, this is a pretty accurate estimate of the welfare gains. 5 A Quantitative Example The model is parameterized to match U.S. experience. The parameterization is similar to Heathcote et al. (2014b), although differences are necessary given that the model of this paper does not include the education sector. The U.S. tax code is history independent and its progressivity parameter is estimated by Heathcote et al. (2014b) to be τ = I set η = 2, implying a Frisch elasticity of 0.5, β = to match an interest rate of 4 percent in the calibrated economy, and π = β. The survival rate is set to δ = 0.971, to match an expected working life of 35 years. Under the U.S. tax code, the measured cross-sectional variance of consumption is given by Var ( ) (ln c) = (1 τ) 2 σz 2 + σφ 2 + ξ c, (21) where ξ c is the measurement error in consumption. To set the cross-sectional variance of the permanent productivity shock σ 2 z and the variance of the taste shock σ 2 φ, I use equation (21) as a restriction on feasible parameter values. Based on the Panel Study of Income Dynamics and Consumer Expenditure Survey data from 2000 to 2006, Heathcote et al. (2014b) report empirical moments Var (ln c) = 0.18 and its measurement error ξ c = Equation (21) then provides a restriction on plausible values of σ 2 z and σ 2 φ.8 Heathcote et al. (2014b) find σ 2 ω = in their baseline calculations. Other estimates reported in the literature are larger: Heathcote et al. (2010) find σ 2 ω = by using a longer time period , although in a framework that does not include taste shocks. 9 I will set the benchmark values to σ 2 ω = 0.005, but I will consider other values 8 There are corresponding expressions for the variance of hours worked and earnings. Unlike equation (21) they potentially depend on the variance of insurable productivity shocks (which are not modeled in this paper). For this reason I do not use them to parameterize the model. 9 Older estimates are provided by Heathcote et al. (2005) who estimate σ 2 ω = and Storesletten 23

24 satisfying (21) as well. The remaining benchmark values are σ 2 z = and σ 2 φ = All the benchmark parameters are summarized in Table 1. Table 1: Benchmark Parameters β δ η σ 2 ω σ 2 ε σ 2 φ Figure 3 plots the aggregate welfare as a function of the progressivity wedge τ for both the tax system that depends only on the current income (blue line), and for a tax system with optimally chosen full history dependence (red line). The US tax system, being history independent, is represented by a point on the blue line. Since the utility is logarithmic, the welfare gains in consumption equivalents are well approximated by the differences in the welfare. The welfare maximizing progressivity wedge in a tax system based only on current income τ HI is 0.296, a significant increase in progressivity relative to the current U.S. value of Introduction of history dependence decreases the progressivity wedge τ FH substantially, down to 0.234, although still more than the current U.S. value. The welfare gains from replacing the best tax system based only on the current income by the optimal fully history dependent tax system are large: they are equal to 1.77 percent of consumption equivalents. The welfare gain from replacing the current U.S. tax system with the optimal fully history dependent tax system is naturally even larger, 2.48 percent of consumption. In contrast, the welfare gain from replacing the current U.S. tax system with the best tax system based on the current income only is only 0.72 percent of consumption. To understand the source of the welfare gains, recall than one can decompose the optimal tax reform into two parts: a choice of the history dependence parameters for any given progressivity wedge τ, and a choice of the welfare maximizing progressivity wedge τ. The first part is represented in Figure 3 by a vertical move from the blue line to the red line, and the second part is represented by a move along the red line. Starting with the best history independent tax system, the figure shows that most of the gains are realized from the introduction of history dependence: 1.63 percent of consumption, which represents is 92.1 percent of all gains. The remaining 7.9 percent of the gains, that et al. (2004) who estimate σω 2 = , both by using data up to

25 τ FH = % c.e % c.e % c.e. τ HI = W τ US = Current Income Only History Dependence τ Figure 3: Welfare gains from history dependence is 0.14 percent of consumption equivalents, is then realized by reducing the progressivity wedge from τ HI to τ FH. Introducing a limited history dependence reduces the welfare gains from the optimal tax reform. The left panel of Figure 4 plots the welfare gains as a function of the length of history dependence, ranging from 0 to K = 30. About 16 percent of the overall welfare gains can be obtained when the tax is allowed to depend only on the current and previous income. To reach 50 percent of the overall welfare gains, one needs to include 10 past incomes; including 16 past incomes allows the government to capture 75 percent of the gains. Given the size of the overall welfare gains, conditioning income taxes on even a few past incomes has large benefits, although longer lags are needed to capture the majority of benefits. The progressivity wedge τ gradually decreases as the length of history dependence increases, as the right panel of Figure 4 shows. The average marginal tax rate out of the current income, on the other hand, is not monotone. It equals the progressivity wedge of 29.5% when K = 0, then increases to 57.9% for K = 3 as the preference shock considerations dominate, then decreases before increasing again. 25