Intertemporal Tax Wedges and Marginal Deadweight Loss (Preliminary Notes)

Transcription

1 Intertemporal Tax Wedges and Marginal Deadweight Loss (Preliminary Notes) Jes Winther Hansen Nicolaj Verdelin December 7, 2006 Abstract This paper analyzes the efficiency loss of income taxation in a dynamic setting. The marginal deadweight loss is expressed solely as a function of empirically observable quantities and elasticities and can be separated into effects from (a) the level of marginal tax rates and (b) the intertemporal profile of marginal tax rates. The bias from ignoring the dynamic nature of the income tax problem depends on the degree of intertemporal substitution in taxable earnings, which recent empirical evidence suggests is substantial. The efficiency of capital income taxation is also discussed.

2 1 Introduction Traditionally, studies of income taxation have restricted attention to static models. This ignores the fact that, with annual income taxation, people are confronted with the tax schedule each year. Annual income taxation gives rise to intertemporal tax wedges, that is, variation in individual marginal tax rates over time. In such a setting, the timing of earnings becomes important, and taxpayers have an incentive to smooththeirincomeovertimetoreducetheiroveralltax liability. The ability of individuals to substitute taxable income between periods depends on a number of factors such as the shape of the disutility of labor, the nature of wage compensation, and the functioning of the capital market. Empirically, income shifting over time is captured by the difference between the intertemporal elasticity of substitution in taxable earnings, the Frisch elasticity, and the conventional compensated elasticity. The size of this difference remains an open research question, but recent empirical evidence suggests that the Frisch elasticity is indeed sizable with estimates in the range (Kimball and Shapiro, 2003, Looney and Singhal, 2006). This, together with the consensus on the size of the compensated elasticity, indicates that intertemporal substitution in taxable earnings is non-negligible. This paper derives a simple, empirically applicable, expression for the marginal deadweight loss of income taxation in a dynamic setting. We show that the marginal deadweight loss can be separated into two effects: (a) a static effect depending on the level of marginal tax rates and (b) a dynamic effect from the intertemporal profile of marginal tax rates. These effects can be expressed solely by empirically observable quantities and elasticities. The static effect is driven by the static elasticity, i.e., the compensated response to a permanent change in a constant marginal tax rate, whereas the dynamic effect is driven by the degree of intertemporal substitution in taxable earnings. Ignoring the dynamic effect leads to an underestimate of the marginal deadweight loss if intertemporal tax wedges are increased and an overestimate if the intertemporal tax wedges are reduced. In some cases, a marginal tax increase can improve efficiency if it reduces the intertemporal wedges. The insights derived from the analysis can help to illuminate the costs of phase-in and phase-out of benefits (earned income tax credit, cash benefits, and time-limited transfers such as the U.S. child tax credit), as well as the progression of marginal tax rates. We plan to analyze specific tax reforms (actual or simulated) in order to gain knowledge on the quantitative importance of the dynamic effects. An important question concerns the relationship between the taxation of earned income and the taxation of capital income. Recently, a number of papers have analyzed income taxation in a dynamic Mirrleesian economy, deriving results on the desirability of capital income taxation (see Golosov et al., 2006, for a review). However, this literature focuses on optimal non-linear income taxation, and the complexity of the analysis makes interpretation difficult. This paper analyzes the efficiency of capital income taxation in a simple two-period model of comprehensive income taxation. We evaluate the marginal deadweight loss of including a fraction of capital income in the tax base, again solely as a function of observable quantities and elasticities. The efficiency of capital taxation depends on the relative responsiveness of consumption and savings, as well as the intertemporal profile of marginal tax 1

3 rates. 2 Earned Income Taxation 2.1 Economic Environment Each taxpayer has a finite planning horizon of N periods. Utility is time-separable. The instantaneous utility function is u n (c n,z n ),wherec is consumption and z is earned income. The taxpayer gets utility from consumption and disutility from earned income. Subscript n refers to the time index. The tax schedule is piece-wise linear with earned income as the tax base. The tax function, T (z), constitutes a net payment to the public sector, embodying both taxes and transfers, and is constant over time. No restrictions on savings between periods. No uncertainty. For simplicity, time discounting is assumed away and the interest rate is assumed to be zero. The model can readily be generalized to incorporate discounting. The taxpayer s expenditure minimization problem is " NX N # X min [c n z n + T (z n )] λ u n (c n,z n ) ū. {c n,z n} 2.2 Marginal Deadweight Loss We first consider the marginal deadweight loss from a change in the marginal tax rate in a single tax bracket j, t j. The marginal deadweight loss can be interpreted as the efficiency loss from fully expected, permanent changes in the tax schedule. In this sense, the model is similar to a traditional static model, except that the single static period is divided into N sub-periods. Another interpretation is that it is the efficiency loss from tax reform, from the moment the reform is announced. But, the model cannot describe the response to unexpected changes in the tax schedule. The change in t j implies that for each period where t j is the marginal tax rate, there: 1. is an intratemporal distortion of earned income within the period. 2. are changes in earned income in all other periods because of intertemporal substitution. Notation: - τ n is the marginal tax rate of the taxpayer in period n. 2

4 - Ω is the set of periods where t j is the marginal tax rate of the taxpayer (the taxpayer is taxed in bracket j ). The marginal deadweight loss is derived by differentiating the expenditure function less the tax revenue NX T (zn ) dc n dt j = t j d (1 t j ) +(1 τ dz n n) d (1 t j ) NX T (zn ) dz n t j τ n d (1 t j ) ( NX T (z n ) = t j X ) c n (1 τ m ) (1 τ z n n) (1 τ m ) m Ω " # NX T (z n ) X z n t j τ n. (1 τ m ) m Ω From the envelope theorem, behavioral responses have no first order effects on the minimized expenditure. As a result, the deadweight loss can be expressed as dt j = = NX X z n (1 τ m ) NX X τ n z n τ n m Ω m Ω ε nm 1 t j, (1) where ε nm 1 τ z m z n n (1 τ m ) is the compensated elasticity of earned income in period n with respect to the marginal tax rate in period m. Wehavealsousedthatτ m = t j for all m Ω by definition. It follows from (1) that the relevant behavioral responses are summarized by the compensated elasticities of taxable income. We need information on the total behavioral response in each period. The total response is comprised of both intratemporal (own-price) and intertemporal (cross-price) responses. Typically, we cannot observe the total response directly. Hence, we need to separate the intra- and intertemporal effects if we want to evaluate the marginal deadweight loss using empirically observable elasticities. This is the goal of the following sections. 2.3 Intertemporal Substitution The intertemporal responses can be expressed in terms of the intertemporal substitution elasticity of earnings the Frisch elasticity. The Frisch elasticity measures the response in earned income to expected changes in the net-of-tax marginal rate 1 τ. This response is usually different from the response to unexpected changes that also cause an intertemporal reallocation of earnings. 3

5 Mathematically, the Frisch elasticity is the response in earned income to changes in the net-of-tax rate for a fixed marginal utility of income. The relationship between earned income in any two periods, m 6= n, is ln z n =lnz m + γ [ln (1 τ n ) ln (1 τ m )] + n, where γ is the Frisch elasticity (assumed constant), and n is an error term assumed independent of the tax schedule. The cross-price elasticity can be found by differentiating with respect to 1 τ m 1 z n z n (1 τ m ) = = 1 z m z m (1 τ m ) γ 1 1 τ m ε nm = ε mm γ. (2) It can be shown theoretically that γ ε nn n (e.g., MaCurdy, 1981). Hence, the intertemporal cross-price response is negative: an increase in the marginal tax rate in period m increases earnings in period n, because earnings are substituted toward the relatively cheaper period n. The assumption that γ is constant implies that the cross-price responses are independent of the time interval between period m and n. This is not necessarily very realistic: cross-price responses are likely lower for long time horizons, e.g., because of uncertainty about future income, preferences, etc. It is possible to extend the analysis to incorporate this variation. 2.4 The Static Elasticity In order to separate the marginal deadweight loss of income taxation into static and dynamic effects, it is useful to introduce the static elasticity, η. The static elasticity is defined as the compensated response in earned income to a permanent change in the net-of-tax rate for a taxpayer, who faces the same marginal tax rate, τ, inallperiods η n 1 τ = z n NX ε nm m=1 dz n d (1 τ) = ε nn + X m6=n (ε mm γ), where we have used (2). It follows that η n is constant over time. This, together with the Law of Demand, implies that η 0. 4

6 2.5 Static and Dynamic Effects Inserting the cross-price responses, (2), and the definition of the static elasticity into the expression for the marginal deadweight loss, (1), allows us to separate the static and dynamic effects dt j = X " t j 1 t j z n γ + X # (ε mm γ) n Ω m Ω + X τ l 1 t j z X l (ε mm γ) l/ Ω m Ω = NX X l/ Ω + X l/ Ω t j 1 t j z nη t j 1 t j z l " NX γ + z n z l (ε ll γ) τ l t j 1 t j z X l (ε mm γ). (3) m Ω Static effect (first term): the marginal deadweight loss from a change in t j when the taxpayer is taxedinbracketj in every period. This distortion depends on the level of t j and disregards tax bracket mobility. Dynamic effect (second and third terms): the distortion from the intertemporal tax wedges. The second term adjusts for the fact that there is no intratemporal distortion from t j when the taxpayer is taxed outside bracket j. In addition, raising t j has an efficiency cost if τ l <t j, such that the intertemporal wedge is increased. Analogously, there is an efficiency gain if τ l >t j, such that the intertemporal wedge is decreased. This is captured by the third term. Alternatively, one can think of the expression in terms of (a) the distortion arising from the level of the marginal tax rate, t j,(first and second terms) and (b) the distortion due to the intertemporal variation in individual marginal tax rates (third term). Intuitively, an increase in the tax on earnings, t j, creates an efficiency loss, because the taxpayer substitutes away from earnings toward leisure. This is the static effect. In addition, earnings are substituted over time toward other tax brackets that are now relatively cheaper. As a consequence, there is a revenue loss from substitution toward periods with lower marginal tax rates and a revenue gain from substitution toward periods with higher marginal tax rates. This is the dynamic effect. Results: # 1. Thestaticeffect overestimates the marginal deadweight loss when the intertemporal tax wedges decrease. 2. In some cases, a marginal tax increase can improve efficiency if intertemporal tax wedges decrease. 5

7 3. The static effect underestimates the marginal deadweight loss when the intertemporal tax wedges increase. The degree of tax bracket mobility has ambiguous effects on the marginal deadweight loss. For instance, if the taxpayer is taxed in bracket j in nearly every period, the number of periods with dynamic effects is small, but the intertemporal distortion in each period is relatively large. A small degree of tax bracket mobility does not necessarily imply that the dynamic effects are negligible. There are no distortions from intertemporal tax wedges if utility is quasi-linear. It can be shown that η = ε = γ in this case, implying that cross-price elasticities are zero. 2.6 Marginal Deadweight Loss: Change in All Tax Brackets We now consider the marginal deadweight loss from a change in the marginal tax rates in all brackets. This implies that the taxpayer is affected in all periods. The marginal deadweight loss is dτ = NX + τ n 1 τ n z n η NX τ n z n N X m=1 µ 1 (ε mm γ) 1. 1 τ m 1 τ n The first term is similar to the static effect from (3). It captures the marginal deadweight loss in the absence of intertemporal substitution in taxable earnings. The second term reflects the marginal deadweight loss from intertemporal substitution in taxable earnings. It naturally depends on the intertemporal wedges in marginal tax rates. Interestingly, the second term is generally positive, implying that ignoring intertemporal substitution in taxable earnings leads to an underestimate of the marginal deadweight loss. 2.7 Numerical Examples Three simple numerical examples in Tables 1 and 2 illustrate the results. Each example is restricted to two periods, two taxpayers, a and b, and two tax brackets, L and H, witht L <t H. In each example, we compare the marginal deadweight loss under two different assumptions about tax bracket mobility: 1. DWL1 :taxpayera is taxed in the low bracket, L, inbothperiods;taxpayerb istaxedinthe high bracket, H, in both periods (no tax bracket mobility). 2. DWL2 : taxpayer a istaxedinbracketl in the first period and in bracket H in the second period; vice versa for taxpayer b. The within-period income distribution is assumed identical in DWL1 and DWL2. 6

8 It is assumed that η, ε, andγ are identical across taxpayers. Further, ε is assumed constant over time which allows us to identify ε from η. Marginal change in t L. In DWL1, only taxpayer a is affected by the change, and the marginal deadweight loss only consists of the static effect. Both taxpayers are affected in DWL2. The efficiency gain from the reduced intertemporal tax wedge dominates the static effect for nearly every set of parameters, when γ>η. Marginal change in t H. Only taxpayer b is affected by the change in DWL1, and the marginal deadweight loss only consists of the static effect. Again, both taxpayers are affected in DWL2. The static effect underestimates the efficiency loss because the intertemporal tax wedge is increased. Marginal changes in t L and t H. Both taxpayers are affected in DWL1 and DWL2. InDWL1, there is only a static effect because there is no tax bracket mobility. The static effect underestimates the marginal deadweight loss. 7

9 Income distribution DWL1 DWL2 t H =0.5 z1 a : 5 5 t L =0.2 z1 b : z2 a : 5 11 z2 b : 11 5 Change in t L η γ DWL1 : 0.25 DWL2 : Change in t H η γ DWL1 : 2.1 DWL2 : Table 1. Numerical Example: Change in a Single Tax Bracket 8

10 Income distribution DWL1 DWL2 t H =0.5 z1 a : 5 5 t L =0.2 z1 b : z2 a : 5 11 z2 b : 11 5 Changes in t L and t H η γ DWL1 : 2.35 DWL2 : Table 2. Numerical Example: Change in All Tax Brackets 9

11 2.8 Evaluating Tax Reforms We plan to apply the expression for the marginal deadweight loss, (3), to specific tax reforms in order to illuminate the quantitative importance of intertemporal tax wedges and tax bracket mobility. Simulated tax reforms: conduct simulated changes in benefit phase-in/phase-out rules, tax deduction rules, and the progression of marginal tax rates in the income tax schedule, in order to highlight reforms that affect the intertemporal tax wedges. Actual tax reforms: comparison of the static and dynamic effects for actual tax reforms. Requirements: panel data on taxable income/earned income and individual marginal tax rates. For simulated reforms, we need information on how individual marginal tax rates are affected. Time horizon: as mentioned above, cross-price responses are likely higher in the short run than in the long run, e.g., because of uncertainty about future earnings. Since there is considerable uncertainty about the magnitude of the long run cross-price responses, the model is better suited to shorter time horizons (e.g., 4 6 years) than very long time horizons (life spans). Calibration: elasticity estimates obtained from the response to tax reforms are typically a combination of intratemporal and intertemporal responses. Careful calibration is needed in order to separate these responses. 3 Capital Income Taxation Taxation of capital income has received considerable interest from public finance economists. Much of the discussion has centered around some well-known zero-tax results: - The Chamley-Judd result: Chamley (1986) and Judd (1985) show that the optimal tax on capital income in steady state is zero, when individuals are identical and labor taxes are available. The intuition is that even a small amount of capital taxation implies that the distortion between consumption at different dates becomes excessively large, when the time horizon is long. - The Atkinson-Stiglitz result: Atkinson and Stiglitz (1976) show that distortionary taxes on consumption goods cannot be optimal in the presence of an optimal non-linear income tax, when preferences are separable between earnings and consumption goods, and all individuals have the same subutility function over consumption goods. Laroque (2005) shows that this, in general, does not require that the income tax is optimal. This result has been applied to savings, leading to the conclusion that capital income should not be taxed if preferences are homogeneous and the income tax is sufficiently flexible. One concern with applying the Atkinson-Stiglitz result to capital income taxation is that the problem is not explicitly dynamic. It is, more or less implicitly, assumed that it is possible to tax 10

12 and redistribute lifetime income using a non-linear tax. This is not very realistic. By far, the most prevalent way of income taxation is based on taxing annual income, for obvious reasons of implementation. Annual income taxation can be supplemented by some redistribution of life-time income, e.g., through means-tested benefits for retirees. The recent literature on New Dynamic Public Finance has analyzed Mirrleesian, optimal non-linear income taxation in explicitly dynamic settings with uncertainty, e.g., Werning (2002), Golosov et al. (2003), and Kocherlakota (2005). Golosov et al. (2003) show that the optimal intertemporal marginal rates of substitution for consumption and earnings typically are not identical when there is individual uncertainty. Unfortunately, it is fairly difficult to draw conclusions about how to implement these conditions in an actual tax system. Golosov and Tsyvinski (2006) consider a simpler two-period model of optimal disability insurance. They provide an argument for asset testing of benefits, translating into an implicit tax on savings, to prevent so-called double deviations where agents claim disability insurance in the second period and use savings to keep their consumption above the benefit level. This section analyzes the efficiency of capital income taxation in a simple two-period model of comprehensive income taxation. We evaluate the marginal deadweight loss of including a fraction of capital income in the tax base and derive an expression consisting of empirically observable quantities and elasticities. In general, the presence of a tax on earned income speaks in favor of a non-zero tax on capital income. We characterize the behavioral responses that drive the result. 3.1 Economic Environment Two periods. Generalized tax base: z n e n +αrb n 1,wherez is taxable income, e is earned income, b is financial assets, r is the real interest rate, and α is a policy parameter. Earned income taxation corresponds to α =0, capital income is taxed if α>0, and capital income is subsidized if α<0. Assets accumulate according to b n = s n +(1+r) b n 1, where savings are s n e n c n T (z n ). Consumption in period 2, c 2, is chosen as the numeraire. Utility is discounted by the time discount factor δ>1. Government revenue is discounted by the real interest rate. The taxpayer s expenditure minimization problem is min {c 1,e 1,c 2,e 2} c 1 e 1 + T (z 1 )+c 2 e 2 + T (z 2 ) rb 1 λ [δu 1 (c 1,e 1 )+u 2 (c 2,e 2 ) ū]. 11

13 3.2 Marginal Deadweight Loss We consider the marginal deadweight loss from a change in α, i.e., from a marginal increase in the share of capital income that is included in the tax base = dc 1 [1 + (1 ατ 2) r] de 1 (1 τ 1)[1+(1 ατ 2 ) r]+ dc 2 (1 τ 2) de 2 + z 1 α τ 1 [1 + (1 ατ 2 ) r]+ z 2 α τ 2 ½ de 1 (1 + r) τ 1 + τ de αrτ 2 ½ z1 α τ 1 [1 + (1 ατ 2 ) r]+ z 2 α τ 2 (1 τ 1 ) de 1 ¾. dc ¾ 1 From the envelope theorem, there is no first order impact on the minimized expenditure from behavioral changes. Using the definition of s 1 yields µ de 1 = (1 + r) τ 1 τ de2 2 + αr ds 1. Define the net-of-tax return ρ 1+(1 ατ 2 ) r. A change in α only affects behavior through its effect on the net return. Hence, the marginal deadweight loss can be written as µ de 1 = rτ 2 (1 + r) τ 1 dρ + τ de2 2 dρ + αr ds 1. (4) dρ 3.3 Behavioral Responses The marginal deadweight loss of capital taxation can be made empirically applicable by rewriting it in terms of observable quantities and elasticities. The compensated elasticity of savings with respect to the net return is σ 1 ρ s 1 ds 1 dρ. The intertemporal elasticity of substitution for consumption is defined as ³ κ ρ c d 1 c 2 c 1 c 2 dρ. Inserting the definition of c 1 and differentiating, allows us to find the response in c 2 dc 2 dρ = c 2 (1 τ 1 ) de 1 c 1 dρ s 1σ 1 + c 1 κ. ρ From the utility constraint (1 τ 2 ) de 2 dρ = dc 2 dρ ρds 1 dρ. Substituting for de 2 /dρ and ds 1 /dρ in (4) gives ½ = rτ 2 (1 + r) τ 1 + c 2 de1 (1 τ 1 ) c 1 dρ τ 2 1 τ 2 µ s 1 σ 1 + c ¾ 2 s 1 σ 1 + c 1 κ s 1 σ 1 + αrτ 2. c 1 ρ ρ 12

14 It is convenient to express the changes in e 1 by rewriting the expression in terms of the covariation in savings and earnings ½ = rτ 2 (1 + r) τ 1 + τ 2 c 2 de1 ds 1 (1 τ 1 ) 1 τ 2 c 1 ds 1 dρ τ 2 ρ + c µ 2 1+ c ¾ 1κ s1 σ 1 1 τ 2 c 1 s 1 σ 1 ρ = +α (rτ 2 ) 2 s 1 σ 1 ρ ½ (1 + r) τ 1 1 τ 1 ρτ 2 1 τ 2 ds 1 d (1 τ 1 ) e 1 + τ 2 1 τ 2 c 2 ds 1 d (1 τ 1 ) e 1 1 rτ 2 s 1 σ 1 ρ c 1 1 ds 1 d (1 τ 1 ) e 1 µ 1+ c ¾ 1κ s 1 σ 1 + α (rτ 2 ) 2 s 1 σ 1 ρ. (5) Hence, evaluating the marginal deadweight loss requires information on the elasticity of savings, σ 1, the intertemporal elasticity of substitution, κ, and the marginal covariation in savings and net earnings when the net return changes, ds 1 /d (1 τ 1 ) e Is Capital Taxation Efficient? Our expression for the marginal deadweight loss can be used to shed light on whether it is efficient to tax or subsidize capital income. We can identify the qualitative effects of capital income taxation under reasonable assumptions: - Consumption is a normal good in both periods. - Earned income is an inferior good in both periods, corresponding to leisure being a normal good. These assumptions allow us to determine the sign of the compensated responses: an increase in ρ lowers consumption and leisure in period 1, which in turn stimulates savings. Consumption and leisure increase in period 2. This implies that de 1 /dρ 0, ds 1 /dρ 0, andde 2 /dρ 0. From (4), the marginal deadweight loss when α =0, i.e., when only earned income is taxed, is de 1 = rτ 2 (1 + r) τ 1 dρ + τ de 2 2. dρ Because the marginal deadweight loss is evaluated at α =0,theefficiency of capital income taxation is fully characterized by the effects on revenue from the tax on earned income. The overall efficiency effect of capital income taxation depends on: 1. Behavioral responses. Period 1 earnings decrease, while period 2 earnings increase in order to counteract the drop in savings. The net effect on tax revenue depends on the relative magnitude of these responses. 2. Intertemporal tax wedges. The intertemporal profile of marginal tax rates affects the net effect on revenue because of the intertemporal reallocation of taxable income. If τ 1 > τ 2, the decrease in period 1 earnings matters relatively more for tax revenue, which tends to increase 13

15 the marginal deadweight loss. Conversely, the marginal deadweight loss tends to be lower if τ 1 <τ 2, because the revenue increase from period 2 earnings bears more weight. The relative response in e 1 and e 2 depends on the relative response of savings and consumption to changes in the net return. A smaller response in consumption, relative to the response in savings, increases the positive response in period 2 earnings. Intuitively, if consumption in period 2 does not change at all, then earnings have to make up for the entire drop in savings. Hence, using our notation above, a numerically lower κ relative to σ 1 decreases the marginal deadweight loss of capital income taxation. 3.5 Numerical Examples Two numerical examples in Table 3 illustrate. We compute the marginal deadweight loss of capital income taxation, (5), for α =0and different values of κ and σ 1. A numerically higher κ for a fixed σ 1 implies a higher marginal deadweight loss. A numerically higher σ 1 amplifies the response in earnings for a fixed value of ds 1 /d (1 τ 1 ) e 1. This increases the importance of the intertemporal tax wedge: a larger behavioral response makes capital taxation more efficient if τ 1 <τ 2 and less efficient if τ 1 >τ 2. Parameters τ 1 =0.2 z 1 : 5 ds 1 d(1 τ 1)e 1 : 1.5 τ 2 =0.5 c 1 : 3 c 2 c 1 : 1.2 s 1 : 1 r : 3 σ 1 κ Parameters τ 1 =0.5 z 1 : 5 ds 1 d(1 τ 1 )e 1 : 1.5 τ 2 =0.2 c 1 : 3 c 2 c 1 : 1.2 s 1 : 0.5 r : 3 σ 1 κ Table 3. Numerical Examples: Capital Income Taxation 14

16 References [1] Atkinson, A.B. and J.E. Stiglitz (1976). The Design of Tax Structure: Direct versus Indirect Taxation. Journal of Public Economics. 6, pp [2] Chamley, C. (1986). Optimal Taxation of Capital Income in General Equilibrium with Infinite Lives. Econometrica. 54, pp [3] Golosov, M., N. Kocherlakota, and A. Tsyvinski (2003). Optimal Indirect and Capital Taxation. Review of Economic Studies. 70, pp [4] Golosov, M. and A. Tsyvinski (2006). Designing Optimal Disability Insurance: A Case for Asset Testing. Journal of Political Economy. 114, pp [5] Golosov, M., A. Tsyvinski, and I. Werning (2006). New Dynamic Public Finance: A User s Guide. Mimeo. [6] Judd, K.L. (1985). Redistributive Taxation in a Simple Perfect Foresight Model. Journal of Public Economics. 28, pp [7] Kimball, M.S. and M.D. Shapiro (2003). Labor Supply: Are the Income and Substitution Effects Both Large or Both Small? Mimeo, University of Michigan. [8] Kocherlakota, N. (2005). Zero Expected Wealth Taxes: A Mirrlees Approach to Dynamic Optimal Taxation. Econometrica. 73, pp [9] Laroque, G.R. (2005). Indirect Taxation Is Superfluous under Separability and Taste Homogeneity: A Simple Proof. Economics Letters, 87, pp [10] Looney, A. and M. Singhal (2006). The Effect of Anticipated Tax Changes on Intertemporal Labor Supply and the Realization of Taxable Income. Mimeo. [11] MaCurdy, T.E. (1981). An Empirical Model of Labor Supply in a Life-Cycle Setting. Journal of Political Economy. 89, pp [12] Werning, I. (2002). Optimal Dynamic Taxation. Ph.D. Dissertation, University of Chicago. 15