Double limit analysis of optimal personal income taxation

Double limit analysis of optimal personal income taxation Abstract A double limit analysis is developed to determine the optimal personal income tax based on the principle that moving a marginal dollar of tax revenue from one income interval to another should not raise social welfare. The first order condition can be differentiated with respect to an interval boundary to yield a second order differential equation for the optimal income tax that can be solved to yield specific solutions. Application of an optimal income tax to a broader income range in general reduces tax revenues and requires greater subsidies at low-income levels. Optimal solutions are provided assuming Cobb-Douglas or Constant Elasticity of Substitution (CES) utility and lognormal or Pareto productivity distributions. 1. Introduction Mirrlees (1971) established a framework for the analysis of optimal income tax schedules that has provided a basis for much of the development since then. The problem for a social planner is to choose a tax schedule based on income that maximizes utilitarian social welfare subject to a constraint on tax revenues generated and consistent with endogenous utility-maximizing choice of labour supply. Formulating this as an optimal control problem, with labour supply as the control variable, Mirrlees derives conditions that determine the optimal income tax in the general case (1971, eqs 15 and 27, pp. 180–1), but adds specific assumptions in Sections 6 and 7, including additive utility, to allow numerical solution. Diamond (1998) extends this analysis by developing a case in which a U-shaped pattern of marginal tax rates arises from a utility function for which income effects on labour supply are zero. Scheuer and Werning (2016) relate the Mirrlees solution to the linear commodity tax model of Diamond and Mirrlees (1971). As an alternative to the Mirrlees optimal control solution, Saez (2001) derives a condition on the marginal tax rate generated by constant elasticities of labour supply applied to tax changes on intervals rather than by specific utility functions. Saez does not solve the original Mirrlees problem, which is to determine the optimal nonlinear income tax given an exogenous productivity distribution, worker utility functions, and a social welfare function. Instead, Saez applies the resulting condition (eq. 14 in Proposition 1) to calibrate the skill distribution that would yield the observed earnings distribution given the assumed utility function and the actual tax schedule (see the discussion in Tuomala, 2016, p. 118). Tuomala (2016) and Piketty and Saez (2013) survey the literature on optimal income taxes and demonstrate applications of the results (see also Tuomala, 1990, Chapter 6; Salanié, 2003, Chapter 4; Kaplow, 2008, pp. 65–79; Mirrlees et al., 2011, Chapter 3; and Boadway, 2012). This paper pursues a different strategy that provides a method of numerically deriving optimal income tax solutions without the necessity of imposing the restrictive assumptions used by Mirrlees, Diamond, Saez, and others. A general first order condition, consistent with the Mirrlees framework but leading to a different analysis, is that shifting a dollar of tax collection from one interval to another (thereby keeping tax revenue the same) should not raise social welfare. This occurs if a marginal dollar of tax revenue generates the same ratio of change in social welfare to change in tax revenue in every interval of income. A common calculation in the literature is to derive the changes in social welfare and tax revenue at higher income levels generated by an increase in the marginal tax rate over a small interval. This paper develops a double limit analysis to derive the rates of change of social welfare and tax revenue with respect to a change in the marginal tax rate at a point instead of along an interval. With the double limit methodology, it is possible to describe how tax revenue could be raised by a marginal dollar within an arbitrary interval without affecting labour responses at income levels outside the interval. The results also lead to a second order differential equation that, together with initial conditions, completely describes the optimal personal income tax. This strategy yields new insights on optimal income taxation. The rate of change of the marginal tax rate (the second derivative of the tax level with respect to income) plays a central role in influencing labour supply responses. The double limit analysis differs from previous approaches by solving for the second derivative of the tax rate instead of the first derivative. The solution shows the distinct contributions of the labour responses to income and substitution effects in determining the optimal tax. The paper provides a unified analysis of reasons for the marginal tax rate to approach zero at upper and lower income limits in terms of the requirement that the substitution effects approach zero. The paper reveals the problem that the ratio of change in social welfare to change in tax revenue must decline as the optimal income tax is extended to higher income levels, requiring increasing subsidization of workers at lower income levels. The double limit methodology proceeds as follows. On an interval from y1 to the maximum income ymax, the marginal tax rate is increased by a factor k1 on the subinterval y1 to y1+ε, with no change in the marginal tax rate from y1+ε to ymax. The changes in social welfare and tax revenues are calculated for the two subintervals and the ratio of change in social welfare to change in tax revenue is formed and determined in the double limit as k1 approaches 1 and as ε approaches 0. In this way, the ratio of change in social welfare to change in tax revenue can be determined for the interval y1 to ymax. From y1 to an arbitrary upper limit y2 less than ymax, an increase in the marginal tax rate in the subinterval y1 to y1+ε is combined with a reduction in the marginal tax rate from y2 to y2+ε. Then in the limit, the consequences of the marginal tax changes are isolated to y1 to y2, and the ratio of changes is determined for the interval y1 to y2. This double limit analysis makes it possible more generally to consider movements of the tax burden from one income level to another. Once the ratio of change in social welfare to change in tax revenue is determined for intervals from an arbitrary income level to the maximum income, the first order condition requires that these ratios equal a common magnitude, λ. Second order conditions are also established. First order conditions can be specified for income intervals from the minimum income to an arbitrary income level, and from the minimum income level to the maximum income level. Corresponding to these intervals are ratios for the trade-off between social welfare and tax revenue that may be different. Proposition 2 in Section 2.3.4 presents the relationship among these three ratios and the conditions under which they are equal. Analysis of the ratios as boundary income approaches the maximum or minimum income provides results in the proposition that are relevant to the marginal tax rate at the highest and lowest incomes. In Section 3, differentiation of the first order condition for incomes from y1 to ymax with respect to the lower boundary yields a second order differential equation expressing the rate of change of the marginal tax rate with respect to income in terms of the marginal tax rate, the level of income and taxes and values of utility and social welfare functions at that income level. The differential equation determines a family of solutions that vary with initial conditions, for example the marginal tax rate and tax level at some income level. The derivation of the differential equation does not rely on restricted functional forms, assumptions of constant elasticities of labour supply, or absence of income effects on labour supply. Section 4 derives numerical solutions for the optimal income tax determined by the differential equation in Section 3. In general, Proposition 2 implies that the marginal tax rate at the maximum income will be zero, and this result is used in the determination of initial conditions for the solution. One possible exception to the conclusion of a zero-marginal tax rate at the highest income is an unlimited upper income, which is not considered in Proposition 2. In a proof made possible by the double limit analysis, Proposition 3 reveals a significant problem in the application of an optimal income tax to an unlimited interval. As the upper income increases, the trade-off declines in examples considered in the section. Since the trade-off must be the same at all income levels, the decline in the trade-off from a greater upper income limit reduces tax revenues and requires greater subsidies to low-income individuals. In the numerical solutions for finite wage intervals, the marginal tax rate follows a common form when a Cobb-Douglas utility function is combined with either lognormal or Pareto wage distributions: the marginal tax rate starts at zero, rises rapidly, and then declines to zero at the maximum income. Similar results arise using a Constant Elasticity of Substitution (CES) utility function. Section 5 describes the relation between double limit analysis and previous approaches in more detail, how the double limit analysis could be applied to optimal tax determination in more general economic contexts, and implications for the relationship between taxation and inequality. 2. Derivation of ratio of social welfare change to tax revenue change 2.1 Assumptions Let t[y] be the amount of taxes paid by an individual with income y, and let t′[y] and t′′[y] be the first and second derivatives of the tax level with respect to income. Following the standard approach in the literature, the condition for an optimal tax will be found by considering a tax change in which the marginal tax rate increases by a factor k1 in the interval y1 to y1+ε and then remains the same in the interval y1+ε to the maximum income, ymax.1 In a departure from the standard literature, the consequences of this tax change for social welfare and tax revenues (calculated as derivatives with respect to k1) will then be considered in the limit as k1 approaches one and as ε approaches zero. With k1 approaching one, the tax change is a deviation from the optimal income tax as used in Euler’s equation to solve a problem in the calculus of variations. For an income level in the interval y1 to y1+ε, the marginal tax rate will be k1t′[y] and the new level of taxes at income y in the interval will be:   t[y1]+k1∫y1yt′[z]dz=t[y1]+k1(t[y]−t[y1]) (1) In the interval y1+ε to ymax, the marginal tax rate at a level of income y will not change and the level of taxes will be:   t[y]+k1∫y1y1+εt′[z]dz=t[y]+k1(t[y1+ε]−t[y1]) (2) where t[y] is the tax level before the tax change. The marginal tax rates and tax levels for the change in taxes are summarized in Table 1. A tax increase over an arbitrary interval y1 to y2 can be obtained by matching the tax change from y1 to ymax with a second tax change from y2 to ymax in which the marginal tax rate decreases by a factor k2 from y2 to y2+ε. The factor k2 is chosen so that:   (k1−1)(t[y1+ε]−t[y1])+(k2−1)(t[y2+ε]−t[y2])=0 (3) With these matched tax changes, the marginal tax rate and the tax levels are unchanged in the interval y2+ε to ymax. In the limit analysis that follows, with ε approaching 0, the tax changes and labour supply responses would then be confined to the interval y1 to y2. A worker’s labour supply will depend on the tax system. As in the standard optimal tax literature, it is assumed that worker productivity and wages are exogenously determined, and the labour supply is heterogeneous with respect to wages. Assume wages cover an interval from wmin to wmax and assume the aggregate measure of workers is 1. Distributions with unlimited upper wage and income levels are analysed later in Section 4. Let f[w] be the probability density of workers. Suppose the utility for a worker with wage w is a continuous function of non-negative leisure, 1−h, and after-tax income, y−t[y], which is entirely spent on consumption: u[1−h,y−t[y]], the same for all individuals. Assume that the utility function has positive marginal utilities of consumption and leisure and is strictly concave. Income y is given by the product of wage rate and labour supply, wh. The optimal labour supply can be found by maximizing utility with respect to the labour supply after linearizing the budget constraint determined by the tax system. From the first order condition on h, obtained by differentiating u with respect to h, it is often (but not always) possible to determine labour supply as an explicit analytical function of (1−t′[y])w, the increase in income after taxes by providing an extra unit of labour, and y−t[y], income after taxes. This procedure is described in more detail for the specific cases considered in Section 4. An important condition on labour supply is that workers with higher wages should choose labour supply levels that yield higher levels of income.2 Discontinuities in labour supply can also arise if marginal tax rates change too rapidly (see Tuomala, 1990, pp. 89–92 and Diamond, 1998, p. 86, for discussions of bunching and gaps that arise from a nonconvex budget set for workers). The double limit analysis applied here does not generate discontinuities in the tax rates that would need to be incorporated into the calculations. The consequences of the tax deviation are expressed as the ratio of the derivative of social welfare with respect to k1 divided by the derivative of tax revenue with respect to k1. The rates of change of social welfare and tax revenue are calculated as k1 increases from one (that is, the derivatives are evaluated at k1=1) so that no actual departure from the optimal income tax is required. As a result, no kinks or discontinuities are introduced by the calculation of the ratio of derivatives in double limit analysis. With no kinks or discontinuities, the rates of change of labour supply with respect to k1 for workers outside the interval w1 to wmax would be zero and no bunching would occur at w1. Rapid but continuous changes in the marginal tax rate also need to be considered. The derivative of income with respect to the wage rate, dy/dw, will be obtained in the following section and will depend on the rate of change of the marginal tax rate, t′′[y]. A jump in labour supply and income could occur if dy/dw becomes infinite, and this can be routinely checked in numerical solutions. Assume that the individual’s optimal labour supply can be represented as h[(1−t′[y])w,y−t[y]] so that expressions in the analysis can be understood in terms of income and substitution effects. The labour supply h is assumed to be a continuous function of its arguments and have continuous first derivatives h1 and h2. The social welfare function is given by:   SWF=∫wminwmaxG[u[1−h,y−t[y]]f[z]dz (4) where G is a continuous concave function of u and income y is understood to be a function of the wage rate. 2.2 Double limit analysis of tax rate changes This section considers the consequences for tax revenues and social welfare of an increase in the marginal tax rate by a factor k1 on the interval y1 to y1+ε. The consequences are calculated separately for the intervals from y1 to y1+ε and from y1+ε to ymax. The derivatives are then calculated in the limit as k1 approaches 1 and as ε approaches 0. The complete derivation is provided in Appendix 1. Let T^ be aggregate tax revenues net of any subsidies or transfers that occur as part of the tax. An important feature of the double limit analysis is that as the size of the first interval ε approaches zero, dSWF/dk1 and dT^/dk1 both approach zero, so that the ratio will be indeterminate. Since the underlying functions are continuous, l’Hospital’s rule can be applied so that:   limε→0dSWF/dk1dT^/dk1=limε→0ddε(dSWF/dk1)ddε(dT^/dk1) (5) Corresponding to the income intervals (y1,y1+ε) and (y1+ε,ymax) are the wage intervals (w1,w1+εw1) and (w1+εw1,wmax). Both dSWF/dk1 and dT^/dk1 (calculated with ε>0) involve integrals over wage intervals in which εw1 enters the limits of integration. In taking the derivative with respect to ε in applying l’Hospital’s rule in (5), it is then necessary to include the term for dεw1/dε, which is the inverse of dy/dw. The general expression for dy / dw can be found as follows:   dydw=d(wh)dw=h+wdhdw  =h+w(∂h∂((1−t′[y])w)d((1−t′[y])w)dw+∂h∂(y−t[y])d(y−t[y])dw)  =h+wh1(1−t′[y])−w2h1t′′[y]dydw+wh2(1−t′[y])dydw (6) Solving for dy / dw yields   dydw=h+wh1(1−t′[y])1−wh2(1−t′[y])+w2h1t′′[y] (7) Define λ{w1,wmax} as the ratio of the change in social welfare to the change in tax revenue generated by an increase in the marginal tax rate at income y1 (corresponding to wage rate w1). This trade-off between social welfare and tax revenue, derived using the double limit analysis, is provided in the following proposition (with arguments of the social welfare function G, utility u, and labour supply h and its derivatives dropped to simplify notation). Proposition 1 Given the assumptions in Section 2.1:   λ{w1,wmax}=limε→0(dSWF/dk1dT^/dk1)=−∫w1wmaxG′u2f[z]dz(−w12h1t′[y1]h+w1h1(1−t′[y1]))f[w1]+∫w1wmax1+z2h1t′′[y[z]]−zh21+z2h1t′′[y[z]]−zh2(1−t′[y[z]])f[z]dz (8) A first order condition for maximizing social welfare for a given level of tax revenues is that this ratio should have the same value for arbitrary values of w1:   λ{w1,wmax}=λ{w2,wmax}, w1,w2∈(wmin,wmax) (9) where λ{w1,wmax}=λ{w2,wmax}<0. The derivation of (8) is provided in the Appendix 1, and (9) holds because otherwise it would be possible to move tax collections between intervals to increase social welfare while holding tax revenue constant. The condition (9) can be generalized to an arbitrary income interval y1 to y2, corresponding to a wage interval w1 to w2. Consider an increase in the marginal tax rate at y1 combined with a reduction in the marginal tax rate at y2 that cancels out the tax increases on that interval from the first change, so that the marginal tax rate and tax level stay the same on the interval y2 to ymax. Combining these two changes yields the general condition:   λ{w1,w2} =−∫w1w2G′u2f[z]dzw22h1t′[y2]h+w2h1(1−t′[y2])f[w2]+−w12h1t′[y1]h+w1h1(1−t′[y1])f[w1]+∫w1w21+z2h1t′′[y[z]]−zh21+z2h1t′′[y[z]]−zh2(1−t′[y[z]])f[z]dz (10) It is also possible to construct a condition on the ratio of changes in social welfare to changes in tax revenues for an interval from ymin to y2, corresponding to wmin to w2. In analogy to the procedures for the ratio on the interval w1 to wmax, let:   λ{wmin,w2}=−∫wminw2G′u2f[z]dzw22h1t′[y2]h+w2h1(1−t′[y2])f[w2]+∫wminw21+z2h1t′′[y[z]]−zh21+z2h1t′′[y[z]]−zh2(1−t′[y[z]])f[z]dz (11) An important question concerns whether λ{wmin,w1}=λ{w1,wmax} for all w1 within the interval wmin to wmax. This equality requires an additional condition. Let:   λ{wmin,wmax}=−∫wminwmaxG′u2f[z]dz∫wminwmax1+z2h1t′′[y[z]]−zh21+z2h1t′′[y[z]]−zh2(1−t′[y[z]])f[z]dz (12) This ratio describes the consequences of changing the level of tax revenues collected by the same amount for all individuals without any change in a marginal tax rate. It corresponds to a lump-sum tax on all individuals. The first order conditions in (9) and (11) are necessary for the maximization of social welfare for a given level of tax revenues. The condition in (12) is not necessary for this maximization problem but is instead relevant to the determination of the optimal level of tax revenue. The trade-off λ{wmin,wmax} is the social marginal cost of a marginal dollar of tax revenue raised by lump-sum taxation (Dahlby, 1998, 2008, pp. 22–4). This cost enters the Atkinson-Stern first order condition for the optimal provision of public goods that determines the optimal level of public expenditure and required tax revenue (Atkinson and Stern, 1974). The relationships among the three ratios λ{w1,wmax}, λ{wmin,w1} and λ{wmin,wmax} are described in the following proposition. Proposition 2 Let λ{w1,wmax}, λ{wmin,w1} and λ{wmin,wmax} be the trade-offs between social welfare and tax revenue given in (8), (10), and (12) and let w1 be a wage rate in the interval (wmin,wmax). Then:   1λ{wmin,wmax}=1λ{wmin,w1}∫wminw1G′u2f[z]dz∫wminwmaxG′u2f[z]dz+1λ{w1,wmax}∫w1wmaxG′u2f[z]dz∫wminwmaxG′u2f[z]dz (13) so that the ratio 1/λ{wmin,wmax} is a weighted average of the ratios 1/λ{wmin,w1} and 1/λ{w1,wmax}, with the weights given by the proportions of changes in social welfare above and below w1 ; (i) If any two of the three ratios are equal, they equal the third ratio; (ii) Since the weights vary with w1, it is impossible that both λ{w1,wmax} and λ{wmin,w1} are constant for all values of w1 unless they are equal; (iii) If λ{wmin,w1} and λ{w1,wmax} are unequal, then 1/λ{wmin,wmax} lies between 1/λ{wmin,w1} and 1/λ{w1,wmax}; (iv) Comparing (8) and (12), the limit:   limw1→wminλ{w1,wmax}=λ{wmin,wmax} (14) holds if and only if:   limw1→wmin(w12h1t′[y1]h+w1h1(1−t′[y1])f[w1])=0 (15) for interior labour supply solutions (not boundary constrained); and (v) Similarly, comparing (11) and (12), the limit:   limw1→wmaxλ{wmin,w1}=λ{wmin,wmax} (16) holds if and only if:   limw1→wmax(w12h1t′[y1]h+w1h1(1−t′[y1])f[w1])=0 (17) for interior labour supply solutions (not boundary constrained). The proof in Appendix 1 proceeds by multiplying λ{wmin,wmax} by the denominator in the ratio in (12) and breaking down the resulting integral of changes in social welfare into integrals on the two subintervals (wmin,w1) and (w1,wmax). Rearranging the resulting equality yields (13). The remaining results follow from this relationship. Define a uniformly optimal tax as a tax that satisfies λ{w1,wmax}=λ{wmin,w1}=λ{wmin,wmax} for any w1 in the open interval wmin to wmax. The statement of Proposition 2, made possible by the double limit analysis, provides a rigorous explanation for the conditions under which the marginal tax rate should be zero at the highest and lowest incomes, as well as the exceptions. There is an extensive literature on optimal tax rates at the highest and lowest incomes based on possible Pareto improvements from reducing marginal tax rates at the highest or lowest incomes to zero.3 In contrast, results in (iv) and (v) of Proposition 2 provide limit results that describe conditions close to the highest and lowest incomes: a uniformly optimal tax requires that the substitution effects in (15) and (17) (arising from an individual’s responses to a change in the marginal tax rate) approach zero as income approaches either the highest or lowest income. If the densities of the wage distribution are positive at the highest and lowest wage rates, then the marginal tax rates at the corresponding incomes will be zero. This result describes the marginal tax rates for more than just the highest and lowest incomes and provides an exception if the density of wage rates approaches zero at the highest or lowest wage rates. By providing both a basis for the zero-marginal tax rate result as well as exceptions, this paper is unlikely to change firmly held convictions. However, the results of Proposition 2 may be useful in resolving or at least narrowing conflicts. A practical consequence of Proposition 2 is that nonlinear income taxes determined by consistency with a first order condition over a subinterval of workers may not be optimal. Assuming the trade-off between social welfare and tax revenue is constant on a subinterval w1 to w2, the trade-offs λ{w,wmax} for w in the subinterval will be unequal if the condition λ{w1,wmax}=λ{wmin,w1}=λ{wmin,wmax} does not hold. The tax would then be suboptimal even within the interval w1 to w2. Proposition 2 provides a means of checking whether a proposed tax, however derived, is optimal, and this check has been applied to the solutions derived in this paper. 2.3 Second order conditions By the first order conditions, moving a marginal dollar of tax collection from one interval to another cannot raise social welfare. The second order condition requires that as one increases the level of tax collections moved between the intervals beyond the marginal dollar, social welfare does not start to increase. The second order condition for maximization of social welfare can be established as follows. For an arbitrary interval y1 to ymax, calculate the derivative of social welfare with respect to k1 and the derivative of tax revenues with respect to k1 for the subintervals y1 to y1+ε and y1+ε to ymax as in Section 2.2, but do not evaluate these derivatives at k1=1. Then with k1≠1, l’Hospital’s rule can be applied to calculate the derivatives of social welfare and tax revenue with respect to k1 as ε approaches zero, since the ratio of the derivative of social welfare to derivative of tax revenue will be indeterminate. Let λ[k1]=(λ{w1,wmax})k1≠1 be the resulting ratio when k1≠1. The second order condition will then be established from the derivative of λ[k1] with respect to k1. Following this procedure (the detailed derivation is provided in Appendix 1):   λ[k1]=−∫w1wmaxG′u2f[z]dz−h1w12k12t′[y1]1−w1h2(1−t[y1])+w12h1t′[y1]f[w1]+∫w1wmax1+z2h1t′′[y[z]]−zh21+z2h1t′′[y[z]]−zh2(1−t′[y[z]])f[z]dz (18) This is identical to the ratio in (8) except for the appearance of k1 in the denominator. To consider what happens to λ[k1] as tax revenues on the interval increase or decrease beyond a marginal dollar, let k1=1+η(δt′), where δt′ is a change in the marginal tax rate at y1 and η is a nonnegative variable. Suppose δt′>0 so that tax revenues increase on the interval as η increases. Then:   (dλ[k1]dη)η=0=(dλ[k1]dk1)k1=1dk1dη=(dλ[k1]dk1)k1=1δt′<0 (19) where dλ[k1]/dk1<0 from (18). Then as the taxes collected from an interval increase (by increasing k1 beyond 1 or equivalently η beyond 0), λ[k1] (which is negative) declines, so that the social welfare cost of a dollar of taxation increases as taxes are raised in the interval. Suppose instead that δt′<0, so that the marginal tax rate at y1 declines and tax revenues on the interval go down as η increases. Then:   (dλ[k1]dη)η=0=(dλ[k1]dk1)k1=1dk1dη=(dλ[k1]dk1)k1=1δt′>0 (20) Then as η is increased beyond zero, tax revenue declines on the interval and the trade-off λ[k1] increases. Since λ[k1] is negative, an increase in λ[k1] means that it declines in absolute value, and the social welfare cost of a dollar of tax revenue declines. Now consider moving more than a marginal dollar of tax collections from one interval to another. Social welfare would rise in the interval where tax collections are reduced, but the increase would be less than the decline in social welfare in the interval where tax collections increased (because of the changes in the ratios λ[k1] in the two intervals). Therefore, moving tax collections among intervals by more than marginal amounts cannot raise social welfare while keeping tax revenues the same, and the second order condition for an optimal income tax system is satisfied. 3. Second order differential equation An important advantage of the double limit analysis is that the results can be used to derive a second order differential equation determining the optimal income tax in the general case. This section constructs the differential equation for the tax system by differentiating the ratio in (8) with respect to the lower boundary of the wage interval. Rearranging the ratio yields:   −∫w1wmaxG′u2f[z]dz=λ{w1,wmax}(−h1w12t′[y1]1−w1h2(1−t[y1])+w12h1t′[y1]f[w1]+∫w1wmax1+z2h1t′′[y[z]]−zh21+z2h1t′′[y[z]]−zh2(1−t′[y[z]])f[z]dz) (21) Differentiation of both sides with respect to w1, with λ{w1,wmax} constant, yields:   G′u2f[w1]=λ{w1,wmax}(ddw1(−h1w12t′[y1]1−w1h2(1−t[y1])+w12h1t′[y1]f[w1])−1+w12h1t′′[y1]−w1h21+w12h1t′′[y1]−w1h2(1−t′[y1])f[w1]) (22) Solving yields:   ddw1(w12h1t′[y1]h+w1h1(1−t′[y1])f[w1])f[w1]=−G′u2λ{w1,wmax}−1+w12h1t′′[y1]−w1h21+w12h1t′′[y1]−w1h2(1−t′[y1]) (23) Both sides involve the derivative t′′[y1] so that the condition will be a second order differential equation involving only values of functions at y1. Expanding the left-hand side using the chain rule yields:   f′[w1]f[w1]w12h1t′[y1]h+w1h1(1−t′[y1])+ddw1(w12h1t′[y1]h+w1h1(1−t′[y1])) (24) The distribution of wages therefore enters into the differential equation only in the single term f′[w1]/f[w1]. Before developing the differential equation in specific cases, consider the interpretation of the three terms. On the right-hand side of (23), G′u2 is the contribution of a dollar of after-tax income to social welfare for the individual earning w1. Dividing by the trade-off λ{w1,wmax} yields the reduction in tax revenue that would be equivalent to a gain in social welfare of G′u2. The second term is the amount of tax revenue generated for an individual with the wage at the interval boundary, w1, from the original tax shift and the additional taxes from the labour response to the income effect. The difference in amounts on the right-hand side must be made up by the change in the substitution effect on the left side as the wage and income levels for the lower boundary of the interval increase (i.e. as w1 increases). The differential equation obtained in (23) will not by itself completely determine the optimal tax system. A general solution to a differential equation will be a family of solutions that depend on one or more constants of integration. The constants of integration can be determined by initial conditions that specify the values of the tax level and the marginal tax rate at a given level of income. If f[wmax]>0, the marginal tax rate at the highest income will be zero for a uniformly optimal income tax, providing one of the initial conditions. If the marginal tax rate at the lowest income is also zero, the tax level at the highest income can be varied until the second condition, at the lowest income, is satisfied. 4. Examples The examples in this section demonstrate how the uniformly optimal income tax can be determined using the methods above. Sections 4.1 and 4.2 assume a Cobb-Douglas utility function combined with a lognormal and Pareto wage distribution, respectively. Section 4.3 assumes a CES utility function combined with a lognormal wage distribution.4 4.1 Cobb-Douglas utility, lognormal wage distribution Assume utility takes the Cobb-Douglas form:   u[1−h,y−t[y]]=(1−h)α(y−t[y])1−α (25) where labour supply h varies between 0 and 1. The functional form for the labour supply that maximizes utility can be determined using standard methods (provided in Appendix 1) and is given by:   h[(1−t′[y])w,y−t[y]]=1−α(y−t[y])(1−α)(1−t′[y])w (26) Two alternative forms for the weighting function G in the social welfare function are the exponential (−e−γu), associated with a constant Coefficient of Absolute Risk Aversion when applied to a utility function, and the polynomial form (uγ/γ), associated with a constant Coefficient of Relative Risk Aversion when applied to a utility function. The latter form has the advantage that an optimal tax system would yield positive levels of utility for all individuals since the marginal contribution to social welfare of an extra dollar to an individual with zero utility would be infinite (essentially the Inada condition). On this basis, assume:   G[u]=uγ/γ, 0<γ<1 (27) so that G′[u]=uγ−1. Finally, suppose wage rates are distributed lognormally on the interval wmin to wmax. With this distribution, f′[w]/f[w]=−Log[w]/w. Proposition 2 in Section 2.2 and the implications for marginal tax rates at the highest and lowest incomes only apply to finite wage and income intervals. A possible exception to the conclusion that the marginal tax rate at the highest income is zero then arises if the wage distribution is unlimited. However, as a result of the double limit analysis, a significant problem becomes apparent when extending an optimal income tax to an increasing range of incomes. This difficulty is stated in the following proposition. Proposition 3 Assume utility is given by (25), the weighting function G in social welfare function is given by (27), and wages are distributed lognormally. If limy→∞t′[y]=τ<1,  limwmax→∞(limw1→wmaxλ{w1,wmax})=0 The proof in Appendix 1 proceeds by applying the assumed functional forms to determine the ratio λ{w1,wmax} in the limit. The significance of the result is that applying an optimal income tax to a wage interval with a very high maximum wage will require imposing a low trade-off between social welfare and tax revenue on all individuals (that is, a dollar of additional tax revenue could only impose a low loss of utility and social welfare at all income levels). As a consequence, tax revenues could be low or negative, and low-income individuals could require substantial subsidization. Potential difficulties with high wage boundaries described above do not rule out uniformly optimal tax solutions for finite wage intervals that are not too large. The condition that t′[ymax]=0 provides one initial condition at the upper boundary, and a given value of the tax level t[ymax] would provide the second initial condition needed to solve the second order differential equation in (23). Then t[ymax] can be varied until t′[ymin]=0 at the lower wage boundary. For the solution worked out here, λ{w1,wmax}=−0.36.5 Figure 1 shows the optimal tax solution. The tax level starts out negative at the lowest income level, corresponding to a transfer to low-income workers. Although it is not apparent from Fig. 1, the marginal tax rate starts out at zero at the lowest income level, rises rapidly, and then declines over most of the income range to zero at the highest income level, as shown in Fig. 2. This figure also shows that the average tax rate declines over a large income interval after being crossed by the marginal tax rate. Figure 3 shows the marginal and average tax rates at income levels such that a given percentile of workers earn less than that income. The figure shows that marginal and average tax rates rise with income for almost all workers. Fig. 1 View largeDownload slide Tax level Fig. 1 View largeDownload slide Tax level Fig. 2 View largeDownload slide Marginal and average tax rates Fig. 2 View largeDownload slide Marginal and average tax rates Fig. 3 View largeDownload slide Marginal and average tax rates by worker percentile Fig. 3 View largeDownload slide Marginal and average tax rates by worker percentile Define the crossover income level as the income at which the average tax rate starts to decline after the marginal tax rate intersects it from above. In Fig. 3, the crossover income occurs at the income level such that 96.7% of workers earn less. Although the marginal tax rate declines and lies below the average tax rate over a long income range in Fig. 2, it accounts for only a small percentage of workers in Fig. 3 and does not characterize the tax policy. Instead of the zero-marginal tax rate characterizing policy for high income individuals, the progressivity of the tax policy is more meaningfully described by the crossover income level and corresponding percentile. The relevant question is then where the crossover income should occur and how it compares to the crossover income in a given tax code. The labour supply consequences of the optimal tax are shown in Fig. 4. Labour supply is positive over the entire income interval, declining initially and then increasing rapidly at lower income levels and continuing to rise over the income interval in this specific case. Despite the presence of income effects, individuals with higher incomes provide higher levels of labour supply because of declines in the marginal tax rate. Fig. 4 View largeDownload slide Labour supply Fig. 4 View largeDownload slide Labour supply The ability to derive specific solutions of the second order differential equation for the optimal income tax provides a laboratory to test hypotheses concerning optimal income taxation. For example, an important question regarding taxation is who should pay higher taxes when greater tax revenue is needed. Choosing a greater trade-off between social welfare and tax revenue, λ{w1,wmax}, raises the social welfare lost per dollar of tax revenue and increases the aggregate tax revenue collected. The consequences of increasing the aggregate level of tax revenues can be examined further using methods developed in this paper. 4.2 Cobb-Douglas utility, Pareto wage distribution In this example, the lognormal wage distribution is replaced by a Pareto distribution. Let:   f[w]=Ifθwmin(wminw)θ+1 (28) where wmin is the lower boundary of the wage distribution and:   If=(∫wminwmaxθwmin(wminx)θ+1dx)−1 (29) (sothat∫wminwmaxf[x]dx=1). Using the Pareto probability density of wage rates, f′[w]/f[w] equals −(1+θ)/w. Corresponding to the proof for Proposition 3, Appendix 1 shows that in the Pareto case limwmax→∞limw1→wmaxλ{w1,wmax}=0 if limy→∞t′[y]<1. On a finite wage range, the uniformly optimal tax solution as a function of income for a Pareto wage distribution does not differ in any significant way from the results using a lognormal distribution, as shown in Fig. 5.6 While the marginal tax rate follows the same general pattern as for a lognormal wage distribution, the marginal and average tax rates as functions of worker percentile take a different form, as shown in Fig. 6. In this figure, the crossover income occurs at a level such that 99.97% of workers earn less. At lower incomes, the average tax rate is positive instead of negative as in the lognormal case. The major cause of the different shapes is the distributional term f′[w]/f[w], which takes the form −Log[w]/w in the lognormal case and −(1+θ)/w in the Pareto case. In the lognormal case, f′[w]/f[w] starts out positive for low values of w and then becomes negative, but at high values of w declines in absolute value at a slower rate than for the Pareto case. In the Pareto case, the more rapid decline in −(1+θ)/w reduces the significance of the substitution effect in (24) and therefore also reduces the gain from lowering marginal tax rates at high income levels. Besides establishing that the double limit methodology can be applied to the Pareto distribution, the example shows that the Pareto case generates increasing marginal tax rates over nearly the entire distribution of workers, with a decline only at the top percentile. The example underscores that the location of the crossover income is a more relevant description of the optimal tax than the decline in the marginal tax rate at the highest incomes. The relevance of a distribution of productivities entirely described by the Pareto distribution is limited by its applicability to only the upper tail. A distribution more consistent with observed income distributions would be either the Champernowne distribution (see Tuomala, 2016, p. 110) or a mixture of lognormal and Pareto distributions, with the weight for the latter increasing with productivity. The double limit analysis could be applied to either of these distributions. Fig. 5 View largeDownload slide Marginal and average tax rates Fig. 5 View largeDownload slide Marginal and average tax rates Fig. 6 View largeDownload slide Marginal and average tax rates by worker percentile Fig. 6 View largeDownload slide Marginal and average tax rates by worker percentile 4.3 Constant elasticity of substitution utility function, lognormal wage distribution This section provides optimal tax solutions assuming a CES utility function (Tuomala, 2016, p. 120, also obtains simulations using a CES utility function). Assume that utility takes the form:   uCES[1−h,y−t[y]]=(δ(1−h)−ρ+(1−δ)(y−t[y])−ρ)−1/ρ (30) with elasticity of substitution σu=1/(1+ρ). Then the labour supply is:   hCES[(1−t′[y])w,y−t[y]]=1−(y−T)((1−δδ(1−t′[y])w)−1/(1+ρ)) (31) (The derivation is provided in Appendix 1.) As ρ approaches zero, the CES function approaches the Cobb-Douglas function with exponent δ in place of α, allowing a comparison with the previous examples. For values of ρ equal to -.0909 and .1111, corresponding to σu equal to 1.1 and .9, Figs 7 and 8 show the marginal tax rates and average tax rates from optimal income tax solutions that yield aggregate tax revenue equal to the tax revenue in the Cobb-Douglas example in Section 4.1.7 A notable feature of the CES optimal income tax solutions is that both the marginal and average tax rates are lower when the elasticity of substitution is higher. With a higher elasticity of substitution, workers can more easily substitute income for leisure and therefore provide higher labour supply for given values of parameters and the marginal and average tax rates. As a result, labour supply is higher, incomes are higher, and lower tax rates are needed to generate the same level of tax revenue. Lower tax rates then generate even higher labour supply levels, as shown in the labour supplies in Fig. 9. In this figure, the maximum income level for ρ=−.0909 arises because of the greater labour supply multiplied times the maximum wage. Fig. 7 View largeDownload slide Marginal tax rates, CES utility Fig. 7 View largeDownload slide Marginal tax rates, CES utility Fig. 8 View largeDownload slide Average tax rates, CES utility Fig. 8 View largeDownload slide Average tax rates, CES utility Fig. 9 View largeDownload slide Labour supply Fig. 9 View largeDownload slide Labour supply 5. Conclusion 5.1 The contribution of double limit analysis The double limit analysis developed in this paper makes several contributions to the study of optimal income taxation. These include the expression for the trade-off between social welfare and tax revenue in Proposition 1, the relations among alternative trade-offs in Proposition 2, the derivation of the second order differential equation in Section 3, and the demonstration in Proposition 3 that the trade-off between social welfare and tax revenue declines to zero as the upper wage boundary increases indefinitely in some of the examples considered in Section 4. The usefulness of the double limit analysis is not restricted to the classical problem of determining the optimal income tax that maximizes social welfare subject to a constraint on tax revenue. It can be applied in any context in which the optimization principle requires equalization at all income levels of trade-offs between a social objective and a budget constraint that can be derived using the methods developed here. These applications include the use of a non-utilitarian social welfare function, Pareto efficient taxation and policy improvements (Werning, 2007; Hendren, 2014), heterogeneous labour responses (Jacquet et al., 2013; Jacquet and Lehmann, 2015), and endogenous wage rates (Kroft et al., 2016; Sachs et al., 2016). In each of these contexts, if the trade-off can be expressed in terms of integrals with a lower boundary, the trade-off can be differentiated to obtain a differential equation as in Section 3. A problem considered by Mirrlees (1971, p. 207), but not examined here, is whether some low-income individuals should be left out of the labour market and supported directly by rather than through the tax system. In the examples considered by Mirrlees and in this paper (in which the minimum productivity is positive), the optimal income tax draws all workers into the labour market. Instead, workers who would receive the lowest incomes could be tagged (or themselves choose to be bunched at zero labour supply) to receive income support outside the tax system, with the optimal income tax being applied to a wage (or productivity) interval that begins at a higher level (see the discussion in Tuomala, 2016, pp. 90–1, and Chapter 8). The decision of which workers to support outside of the tax system could be incorporated into a model with a social welfare function related to Rawls’ theory of justice that would maximize the well-being of the least-favoured individuals (Rawls, 1971, and discussions in Tuomala, 2016, pp. 72–7 and Salanié, 2003, pp. 84–7). With workers differing in preferences for leisure in addition to productivity, individuals could choose not to participate in the labour force at higher levels of productivity, yielding labour responses at the extensive margin in addition to hours of work (Tuomala, Chapter 6; Piketty and Saez, 2013, pp. 441–3; Salanié, 2003, pp. 40–1; Jacquet et al., 2013). The presence of heterogeneous characteristics affecting labour supply is treated as the multi-dimensional screening problem in contract theory (Armstrong and Rochet, 1999; Salanié, 2005, pp. 78–82; and Tuomala, 2016, Chapter 10, pp. 249–58). In the context of nonlinear pricing by a multiproduct firm, the principle in the principle–agent problem usually excludes some consumers (Armstrong, 1996) and induces some bunching in choosing products (Rochet and Choné, 1998). Problems also arise in comparing contributions to social welfare across individuals (Tuomala, 2016, p. 32). In the context of optimal income taxes with heterogeneous preferences for leisure, it is likely that the solution will also involve exclusion of some workers (see Jacquet and Lehman, 2015, for optimal income taxation with heterogeneous labour responses). 5.2 Relation to previous approaches Both the Mirrlees approach and the double limit analysis provide general conditions for optimal income taxation. The consistency of the two approaches is demonstrated in Appendix 1 by showing that as a result of imposing the semi-linear utility function with no income effects used in Diamond (1998), the trade-off in (8) simplifies to the Mirrlees-Diamond condition on the marginal tax rate. The two approaches instead differ in the additional assumptions necessary to derive numerical solutions. Mirrlees introduces the assumption that utility is additively separable and this assumption has been used routinely in numerical derivations since then. In their original forms, the Cobb-Douglas and CES functions are not additively separable in their arguments. Taking the logarithm of the Cobb-Douglas function and raising the CES function to the power 1/−ρ yield additively separable functions. The derivation of the trade-off in (8) and the condition for the differential equation in (23) do not depend on the assumption of a separable utility function. However, the double limit analysis is simplified in cases where the first order condition for the differential equation can be solved for t′′[y]. Absence of income effects has also been imposed (Diamond, 1998; Piketty and Saez, 2013, p. 435). With income effects, high incomes can induce individuals to reduce their labour supplies. Then an optimal income tax with high average income taxes would lead higher income individuals to maintain their labour supply levels. In the absence of income effects, labour supply would only depend on the after-tax wage rate, (1−t′[y])w. As w increases, individuals would then increase their labour supplies without limit unless the marginal tax rate t′[y] approaches one. Without income effects, solutions for the marginal tax rate that differ from one generally require that labour supply increases indefinitely (Saez, 2001, p. 223). Although it is not used to solve the original Mirrlees problem, the elasticity approach developed by Saez (2001) has the advantage that it relates the skill distribution and income tax to empirical estimates of labour supply responses. However, assuming a constant elasticity of labour supply with respect to the wage determines t′′[y] in lieu of the condition in (23). By replacing the solution for t′′[y] from the first order condition, the assumption of a constant elasticity imposes a restriction on the solution that is unnecessary with the double limit analysis. The double limit analysis differs from previous approaches in providing methods of testing for the optimality of a tax solution. As noted in Section 2, a solution that satisfies the first order condition on part of the income interval will not necessarily be optimal. Proposition 2 provides a test for a uniformly optimal tax by calculating the three ratios. By calculating labour supply, dy / dw and t′′[y] at each income, it is also possible to check whether discontinuities or singularities occur in the solution. The double limit analysis can be used to compare tax revenue and social welfare generated by a tax system to check whether an alternative solution is better. Calculation of the trade-off in (8) can be used to determine how tax collections should be shifted among income intervals. To derive numerical optimal income tax solutions, the double limit analysis imposes some restrictions that are absent in the original Mirrlees analysis. These include the assumption that utility and the social welfare weighting function are continuous functions of their arguments and that labour supply can be derived as an analytic function of (1−t′[y])w and y−t[y]. 5.3 Inequality In the strategy pursued in this paper, the first order condition for an optimal income tax promotes equality by imposing the same trade-off between social welfare and tax revenue on individuals at all income levels, a form of vertical equity at the margin. Both rich and poor are linked by a common trade-off between contributions to social welfare and tax revenues and by a common differential equation. Extending an optimal income tax to higher levels of income may require greater subsidies to individuals at low incomes. The role of subsidies to low-income individuals is not simply to raise their incomes in response to their high marginal contributions to social welfare, but to allow an efficient rapid increase in the marginal tax rate. The optimal income tax promotes equality by using a social welfare function (with lower weightings for higher income individuals) as its objective and promotes efficiency through the requirement for optimization. A basic question is therefore whether an optimal income tax can substantially reduce inequality or whether by seeking optimality it promotes efficiency at the cost of disregarding income differences. Redistribution in response to the weights of the social welfare function is limited by the costs generated by substitution effects. Questions regarding the interaction between optimal income taxation and inequality remain to be addressed in future work.  Table 1 Marginal tax rate and tax levels Interval  Marginal tax rate  Tax level  (y1,y1+ε)  k1t′[y]  t[y1]+k1(t[y]−t[y1])  (y1+ε,ymax⁡)  t′[y]  t[y]+k1(t[y1+ε]−t[y1])  Interval  Marginal tax rate  Tax level  (y1,y1+ε)  k1t′[y]  t[y1]+k1(t[y]−t[y1])  (y1+ε,ymax⁡)  t′[y]  t[y]+k1(t[y1+ε]−t[y1])  Supplementary Material Appendices 1 , 2, and 3 are available online at the OUP website. Footnotes 1 See for example Tuomala (2016, Fig. 4.3, p. 77), and Piketty and Saez (2013, Fig. 3, p. 435). Tuomala cites an early use of the tax perturbation method by Christiansen (1981) in a different context. Golosov et al. (2014) generalize tax perturbations to a dynamic setting. 2 See Assumption B in Mirrlees (1971, p. 182), and the discussion by Salanié (2003, pp. 87–91) and Tuomala (2016, pp. 65–6). The simple condition ensures that individuals with higher wages will earn higher incomes. Salanié and Tuomala relate this condition to the Spence-Mirrlees condition in contract theory, agent monotonicity, and incentive compatibility. The utility function can be re-expressed as a function of consumption and income divided by the wage rate (equal to the labour supply). The condition requires that the indifference curve between consumption and income become flatter as the wage rate increases. Then an indifference curve tangent to a budget constraint will rotate clockwise as the wage increases, so that the new point of tangency will be at a higher income level. See also Seade (1982) and Tuomala (1990, p. 87). 3 Early papers include Mirrlees (1971), Sadka (1976, p. 266), Seade (1977, p. 231, footnote; and 1982). See also conflicting conclusions in Diamond and Saez, (2011, pp. 188–9) and Mankiw et al. (2009, pp. 151–5), and the discussion by Tuomala (2016, pp. 77–83). 4 The Mathematica notebooks that derive the numerical solutions and generate the figures are provided in Appendix 2 for Cobb-Douglas utility and Appendix 3 for CES utility. 5 The other parameters are α=.3, γ=.8, wmin=1,andwmax=100. The parameters of the lognormal distribution are μ=1 and σ=1. Tax revenue is .964 and production is 3.307. The average elasticity of labour supply with respect to (1−t′[y])w is .65, and the average elasticity of labour supply with respect to y−t[y] is -.65. Detailed procedures for finding the solution by varying t[ymax] are provided in Appendix 2. 6 The solution assumes θ=2.0 and λ=−.51, with wages ranging from 1 to 100. The utility function is the same as for the lognormal example. Tax revenue is .300 and production is 1.331. The average elasticity of labour supply with respect to (1−t′[y])w is .46 and the average elasticity of labour supply with respect to y−t[y] is -.46. 7 The optimal tax solutions are derived assuming δ=.3, with λ equal to -.37296 for σu=1.1 and λ equal to -.34668 for σu=.9 in order to yield equal tax revenue of .9640. Other parametric assumptions (including the lognormal distribution of wage rates) are the same as in Section 4.1. Acknowledgements The author acknowledges helpful comments by the editor and referees as well as Henning Bunzel, Michael Jerison, John B. Jones, and Kwan Koo Yun. Any remaining errors are the responsibility of the author. References Armstrong M. ( 1996) Multiproduct nonlinear pricing, Econometrica , 64, 51– 75. Google Scholar CrossRef Search ADS   Armstrong M., Rochet J.-C. ( 1999) Multi-dimensional screening: a user’s guide, European Economic Review , 43, 959– 79. Google Scholar CrossRef Search ADS   Atkinson A., Stern N. 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Google Scholar CrossRef Search ADS   Kaplow L. ( 2008) The Theory of Taxation and Public Economics , Princeton University Press, Princeton, NJ. Kroft K., Kucko K., Lehmann E., Schmieder J. ( 2016) Optimal Income Taxation with Unemployment and Wage Responses: A Sufficient Statistics Approach, Discussion Paper 9718, IZA, Bonn. Mankiw N.G., Weinzierl M.C., Yagan D. ( 2009) Optimal Taxation in Theory and Practice, Journal of Economic Perspectives , 23, 147– 74. Google Scholar CrossRef Search ADS   Mirrlees J.A. ( 1971) An Exploration in the Theory of Optimal Income Taxation, Review of Economic Studies , 38, 175– 208. Google Scholar CrossRef Search ADS   Mirrlees J.A., Adam S., Besley T., Blundell R., Bond S., Chote R., Gammie M., Johnson P., Myles G., Poterba J. (eds) ( 2011) Tax by Design: The Mirrlees Review , Institute for Fiscal Studies, Oxford University Press, Oxford. Piketty T., Saez E. ( 2013) Optimal Labor Income Taxation, in Auerbach A.J., Chetty R., Feldstein M., Saez E.. 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( 2016) Mirrlees meets Diamond-Mirrlees, Working Paper 22076, National Bureau of Economic Research, Cambridge, MA. Seade J.K. ( 1977) On the Shape of Optimal Tax Schedules, Journal of Public Economics , 7, 203– 36. Google Scholar CrossRef Search ADS   Seade J.K. ( 1982) On the Sign of the Optimum Marginal Income Tax, Review of Economic Studies , 49, 637– 43. Google Scholar CrossRef Search ADS   Tuomala M. ( 1990) Optimal Income Taxation and Redistribution , Clarendon Press, Oxford. Tuomala M. ( 2016) Optimal Redistributive Taxation , Oxford University Press, Oxford. Google Scholar CrossRef Search ADS   Werning I. ( 2007) Pareto Efficient Income Taxation, MIT Working Paper, Cambridge, MA. © Oxford University Press 2017 All rights reserved http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Oxford Economic Papers Oxford University Press

Double limit analysis of optimal personal income taxation

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Abstract

Abstract A double limit analysis is developed to determine the optimal personal income tax based on the principle that moving a marginal dollar of tax revenue from one income interval to another should not raise social welfare. The first order condition can be differentiated with respect to an interval boundary to yield a second order differential equation for the optimal income tax that can be solved to yield specific solutions. Application of an optimal income tax to a broader income range in general reduces tax revenues and requires greater subsidies at low-income levels. Optimal solutions are provided assuming Cobb-Douglas or Constant Elasticity of Substitution (CES) utility and lognormal or Pareto productivity distributions. 1. Introduction Mirrlees (1971) established a framework for the analysis of optimal income tax schedules that has provided a basis for much of the development since then. The problem for a social planner is to choose a tax schedule based on income that maximizes utilitarian social welfare subject to a constraint on tax revenues generated and consistent with endogenous utility-maximizing choice of labour supply. Formulating this as an optimal control problem, with labour supply as the control variable, Mirrlees derives conditions that determine the optimal income tax in the general case (1971, eqs 15 and 27, pp. 180–1), but adds specific assumptions in Sections 6 and 7, including additive utility, to allow numerical solution. Diamond (1998) extends this analysis by developing a case in which a U-shaped pattern of marginal tax rates arises from a utility function for which income effects on labour supply are zero. Scheuer and Werning (2016) relate the Mirrlees solution to the linear commodity tax model of Diamond and Mirrlees (1971). As an alternative to the Mirrlees optimal control solution, Saez (2001) derives a condition on the marginal tax rate generated by constant elasticities of labour supply applied to tax changes on intervals rather than by specific utility functions. Saez does not solve the original Mirrlees problem, which is to determine the optimal nonlinear income tax given an exogenous productivity distribution, worker utility functions, and a social welfare function. Instead, Saez applies the resulting condition (eq. 14 in Proposition 1) to calibrate the skill distribution that would yield the observed earnings distribution given the assumed utility function and the actual tax schedule (see the discussion in Tuomala, 2016, p. 118). Tuomala (2016) and Piketty and Saez (2013) survey the literature on optimal income taxes and demonstrate applications of the results (see also Tuomala, 1990, Chapter 6; Salanié, 2003, Chapter 4; Kaplow, 2008, pp. 65–79; Mirrlees et al., 2011, Chapter 3; and Boadway, 2012). This paper pursues a different strategy that provides a method of numerically deriving optimal income tax solutions without the necessity of imposing the restrictive assumptions used by Mirrlees, Diamond, Saez, and others. A general first order condition, consistent with the Mirrlees framework but leading to a different analysis, is that shifting a dollar of tax collection from one interval to another (thereby keeping tax revenue the same) should not raise social welfare. This occurs if a marginal dollar of tax revenue generates the same ratio of change in social welfare to change in tax revenue in every interval of income. A common calculation in the literature is to derive the changes in social welfare and tax revenue at higher income levels generated by an increase in the marginal tax rate over a small interval. This paper develops a double limit analysis to derive the rates of change of social welfare and tax revenue with respect to a change in the marginal tax rate at a point instead of along an interval. With the double limit methodology, it is possible to describe how tax revenue could be raised by a marginal dollar within an arbitrary interval without affecting labour responses at income levels outside the interval. The results also lead to a second order differential equation that, together with initial conditions, completely describes the optimal personal income tax. This strategy yields new insights on optimal income taxation. The rate of change of the marginal tax rate (the second derivative of the tax level with respect to income) plays a central role in influencing labour supply responses. The double limit analysis differs from previous approaches by solving for the second derivative of the tax rate instead of the first derivative. The solution shows the distinct contributions of the labour responses to income and substitution effects in determining the optimal tax. The paper provides a unified analysis of reasons for the marginal tax rate to approach zero at upper and lower income limits in terms of the requirement that the substitution effects approach zero. The paper reveals the problem that the ratio of change in social welfare to change in tax revenue must decline as the optimal income tax is extended to higher income levels, requiring increasing subsidization of workers at lower income levels. The double limit methodology proceeds as follows. On an interval from y1 to the maximum income ymax, the marginal tax rate is increased by a factor k1 on the subinterval y1 to y1+ε, with no change in the marginal tax rate from y1+ε to ymax. The changes in social welfare and tax revenues are calculated for the two subintervals and the ratio of change in social welfare to change in tax revenue is formed and determined in the double limit as k1 approaches 1 and as ε approaches 0. In this way, the ratio of change in social welfare to change in tax revenue can be determined for the interval y1 to ymax. From y1 to an arbitrary upper limit y2 less than ymax, an increase in the marginal tax rate in the subinterval y1 to y1+ε is combined with a reduction in the marginal tax rate from y2 to y2+ε. Then in the limit, the consequences of the marginal tax changes are isolated to y1 to y2, and the ratio of changes is determined for the interval y1 to y2. This double limit analysis makes it possible more generally to consider movements of the tax burden from one income level to another. Once the ratio of change in social welfare to change in tax revenue is determined for intervals from an arbitrary income level to the maximum income, the first order condition requires that these ratios equal a common magnitude, λ. Second order conditions are also established. First order conditions can be specified for income intervals from the minimum income to an arbitrary income level, and from the minimum income level to the maximum income level. Corresponding to these intervals are ratios for the trade-off between social welfare and tax revenue that may be different. Proposition 2 in Section 2.3.4 presents the relationship among these three ratios and the conditions under which they are equal. Analysis of the ratios as boundary income approaches the maximum or minimum income provides results in the proposition that are relevant to the marginal tax rate at the highest and lowest incomes. In Section 3, differentiation of the first order condition for incomes from y1 to ymax with respect to the lower boundary yields a second order differential equation expressing the rate of change of the marginal tax rate with respect to income in terms of the marginal tax rate, the level of income and taxes and values of utility and social welfare functions at that income level. The differential equation determines a family of solutions that vary with initial conditions, for example the marginal tax rate and tax level at some income level. The derivation of the differential equation does not rely on restricted functional forms, assumptions of constant elasticities of labour supply, or absence of income effects on labour supply. Section 4 derives numerical solutions for the optimal income tax determined by the differential equation in Section 3. In general, Proposition 2 implies that the marginal tax rate at the maximum income will be zero, and this result is used in the determination of initial conditions for the solution. One possible exception to the conclusion of a zero-marginal tax rate at the highest income is an unlimited upper income, which is not considered in Proposition 2. In a proof made possible by the double limit analysis, Proposition 3 reveals a significant problem in the application of an optimal income tax to an unlimited interval. As the upper income increases, the trade-off declines in examples considered in the section. Since the trade-off must be the same at all income levels, the decline in the trade-off from a greater upper income limit reduces tax revenues and requires greater subsidies to low-income individuals. In the numerical solutions for finite wage intervals, the marginal tax rate follows a common form when a Cobb-Douglas utility function is combined with either lognormal or Pareto wage distributions: the marginal tax rate starts at zero, rises rapidly, and then declines to zero at the maximum income. Similar results arise using a Constant Elasticity of Substitution (CES) utility function. Section 5 describes the relation between double limit analysis and previous approaches in more detail, how the double limit analysis could be applied to optimal tax determination in more general economic contexts, and implications for the relationship between taxation and inequality. 2. Derivation of ratio of social welfare change to tax revenue change 2.1 Assumptions Let t[y] be the amount of taxes paid by an individual with income y, and let t′[y] and t′′[y] be the first and second derivatives of the tax level with respect to income. Following the standard approach in the literature, the condition for an optimal tax will be found by considering a tax change in which the marginal tax rate increases by a factor k1 in the interval y1 to y1+ε and then remains the same in the interval y1+ε to the maximum income, ymax.1 In a departure from the standard literature, the consequences of this tax change for social welfare and tax revenues (calculated as derivatives with respect to k1) will then be considered in the limit as k1 approaches one and as ε approaches zero. With k1 approaching one, the tax change is a deviation from the optimal income tax as used in Euler’s equation to solve a problem in the calculus of variations. For an income level in the interval y1 to y1+ε, the marginal tax rate will be k1t′[y] and the new level of taxes at income y in the interval will be:   t[y1]+k1∫y1yt′[z]dz=t[y1]+k1(t[y]−t[y1]) (1) In the interval y1+ε to ymax, the marginal tax rate at a level of income y will not change and the level of taxes will be:   t[y]+k1∫y1y1+εt′[z]dz=t[y]+k1(t[y1+ε]−t[y1]) (2) where t[y] is the tax level before the tax change. The marginal tax rates and tax levels for the change in taxes are summarized in Table 1. A tax increase over an arbitrary interval y1 to y2 can be obtained by matching the tax change from y1 to ymax with a second tax change from y2 to ymax in which the marginal tax rate decreases by a factor k2 from y2 to y2+ε. The factor k2 is chosen so that:   (k1−1)(t[y1+ε]−t[y1])+(k2−1)(t[y2+ε]−t[y2])=0 (3) With these matched tax changes, the marginal tax rate and the tax levels are unchanged in the interval y2+ε to ymax. In the limit analysis that follows, with ε approaching 0, the tax changes and labour supply responses would then be confined to the interval y1 to y2. A worker’s labour supply will depend on the tax system. As in the standard optimal tax literature, it is assumed that worker productivity and wages are exogenously determined, and the labour supply is heterogeneous with respect to wages. Assume wages cover an interval from wmin to wmax and assume the aggregate measure of workers is 1. Distributions with unlimited upper wage and income levels are analysed later in Section 4. Let f[w] be the probability density of workers. Suppose the utility for a worker with wage w is a continuous function of non-negative leisure, 1−h, and after-tax income, y−t[y], which is entirely spent on consumption: u[1−h,y−t[y]], the same for all individuals. Assume that the utility function has positive marginal utilities of consumption and leisure and is strictly concave. Income y is given by the product of wage rate and labour supply, wh. The optimal labour supply can be found by maximizing utility with respect to the labour supply after linearizing the budget constraint determined by the tax system. From the first order condition on h, obtained by differentiating u with respect to h, it is often (but not always) possible to determine labour supply as an explicit analytical function of (1−t′[y])w, the increase in income after taxes by providing an extra unit of labour, and y−t[y], income after taxes. This procedure is described in more detail for the specific cases considered in Section 4. An important condition on labour supply is that workers with higher wages should choose labour supply levels that yield higher levels of income.2 Discontinuities in labour supply can also arise if marginal tax rates change too rapidly (see Tuomala, 1990, pp. 89–92 and Diamond, 1998, p. 86, for discussions of bunching and gaps that arise from a nonconvex budget set for workers). The double limit analysis applied here does not generate discontinuities in the tax rates that would need to be incorporated into the calculations. The consequences of the tax deviation are expressed as the ratio of the derivative of social welfare with respect to k1 divided by the derivative of tax revenue with respect to k1. The rates of change of social welfare and tax revenue are calculated as k1 increases from one (that is, the derivatives are evaluated at k1=1) so that no actual departure from the optimal income tax is required. As a result, no kinks or discontinuities are introduced by the calculation of the ratio of derivatives in double limit analysis. With no kinks or discontinuities, the rates of change of labour supply with respect to k1 for workers outside the interval w1 to wmax would be zero and no bunching would occur at w1. Rapid but continuous changes in the marginal tax rate also need to be considered. The derivative of income with respect to the wage rate, dy/dw, will be obtained in the following section and will depend on the rate of change of the marginal tax rate, t′′[y]. A jump in labour supply and income could occur if dy/dw becomes infinite, and this can be routinely checked in numerical solutions. Assume that the individual’s optimal labour supply can be represented as h[(1−t′[y])w,y−t[y]] so that expressions in the analysis can be understood in terms of income and substitution effects. The labour supply h is assumed to be a continuous function of its arguments and have continuous first derivatives h1 and h2. The social welfare function is given by:   SWF=∫wminwmaxG[u[1−h,y−t[y]]f[z]dz (4) where G is a continuous concave function of u and income y is understood to be a function of the wage rate. 2.2 Double limit analysis of tax rate changes This section considers the consequences for tax revenues and social welfare of an increase in the marginal tax rate by a factor k1 on the interval y1 to y1+ε. The consequences are calculated separately for the intervals from y1 to y1+ε and from y1+ε to ymax. The derivatives are then calculated in the limit as k1 approaches 1 and as ε approaches 0. The complete derivation is provided in Appendix 1. Let T^ be aggregate tax revenues net of any subsidies or transfers that occur as part of the tax. An important feature of the double limit analysis is that as the size of the first interval ε approaches zero, dSWF/dk1 and dT^/dk1 both approach zero, so that the ratio will be indeterminate. Since the underlying functions are continuous, l’Hospital’s rule can be applied so that:   limε→0dSWF/dk1dT^/dk1=limε→0ddε(dSWF/dk1)ddε(dT^/dk1) (5) Corresponding to the income intervals (y1,y1+ε) and (y1+ε,ymax) are the wage intervals (w1,w1+εw1) and (w1+εw1,wmax). Both dSWF/dk1 and dT^/dk1 (calculated with ε>0) involve integrals over wage intervals in which εw1 enters the limits of integration. In taking the derivative with respect to ε in applying l’Hospital’s rule in (5), it is then necessary to include the term for dεw1/dε, which is the inverse of dy/dw. The general expression for dy / dw can be found as follows:   dydw=d(wh)dw=h+wdhdw  =h+w(∂h∂((1−t′[y])w)d((1−t′[y])w)dw+∂h∂(y−t[y])d(y−t[y])dw)  =h+wh1(1−t′[y])−w2h1t′′[y]dydw+wh2(1−t′[y])dydw (6) Solving for dy / dw yields   dydw=h+wh1(1−t′[y])1−wh2(1−t′[y])+w2h1t′′[y] (7) Define λ{w1,wmax} as the ratio of the change in social welfare to the change in tax revenue generated by an increase in the marginal tax rate at income y1 (corresponding to wage rate w1). This trade-off between social welfare and tax revenue, derived using the double limit analysis, is provided in the following proposition (with arguments of the social welfare function G, utility u, and labour supply h and its derivatives dropped to simplify notation). Proposition 1 Given the assumptions in Section 2.1:   λ{w1,wmax}=limε→0(dSWF/dk1dT^/dk1)=−∫w1wmaxG′u2f[z]dz(−w12h1t′[y1]h+w1h1(1−t′[y1]))f[w1]+∫w1wmax1+z2h1t′′[y[z]]−zh21+z2h1t′′[y[z]]−zh2(1−t′[y[z]])f[z]dz (8) A first order condition for maximizing social welfare for a given level of tax revenues is that this ratio should have the same value for arbitrary values of w1:   λ{w1,wmax}=λ{w2,wmax}, w1,w2∈(wmin,wmax) (9) where λ{w1,wmax}=λ{w2,wmax}<0. The derivation of (8) is provided in the Appendix 1, and (9) holds because otherwise it would be possible to move tax collections between intervals to increase social welfare while holding tax revenue constant. The condition (9) can be generalized to an arbitrary income interval y1 to y2, corresponding to a wage interval w1 to w2. Consider an increase in the marginal tax rate at y1 combined with a reduction in the marginal tax rate at y2 that cancels out the tax increases on that interval from the first change, so that the marginal tax rate and tax level stay the same on the interval y2 to ymax. Combining these two changes yields the general condition:   λ{w1,w2} =−∫w1w2G′u2f[z]dzw22h1t′[y2]h+w2h1(1−t′[y2])f[w2]+−w12h1t′[y1]h+w1h1(1−t′[y1])f[w1]+∫w1w21+z2h1t′′[y[z]]−zh21+z2h1t′′[y[z]]−zh2(1−t′[y[z]])f[z]dz (10) It is also possible to construct a condition on the ratio of changes in social welfare to changes in tax revenues for an interval from ymin to y2, corresponding to wmin to w2. In analogy to the procedures for the ratio on the interval w1 to wmax, let:   λ{wmin,w2}=−∫wminw2G′u2f[z]dzw22h1t′[y2]h+w2h1(1−t′[y2])f[w2]+∫wminw21+z2h1t′′[y[z]]−zh21+z2h1t′′[y[z]]−zh2(1−t′[y[z]])f[z]dz (11) An important question concerns whether λ{wmin,w1}=λ{w1,wmax} for all w1 within the interval wmin to wmax. This equality requires an additional condition. Let:   λ{wmin,wmax}=−∫wminwmaxG′u2f[z]dz∫wminwmax1+z2h1t′′[y[z]]−zh21+z2h1t′′[y[z]]−zh2(1−t′[y[z]])f[z]dz (12) This ratio describes the consequences of changing the level of tax revenues collected by the same amount for all individuals without any change in a marginal tax rate. It corresponds to a lump-sum tax on all individuals. The first order conditions in (9) and (11) are necessary for the maximization of social welfare for a given level of tax revenues. The condition in (12) is not necessary for this maximization problem but is instead relevant to the determination of the optimal level of tax revenue. The trade-off λ{wmin,wmax} is the social marginal cost of a marginal dollar of tax revenue raised by lump-sum taxation (Dahlby, 1998, 2008, pp. 22–4). This cost enters the Atkinson-Stern first order condition for the optimal provision of public goods that determines the optimal level of public expenditure and required tax revenue (Atkinson and Stern, 1974). The relationships among the three ratios λ{w1,wmax}, λ{wmin,w1} and λ{wmin,wmax} are described in the following proposition. Proposition 2 Let λ{w1,wmax}, λ{wmin,w1} and λ{wmin,wmax} be the trade-offs between social welfare and tax revenue given in (8), (10), and (12) and let w1 be a wage rate in the interval (wmin,wmax). Then:   1λ{wmin,wmax}=1λ{wmin,w1}∫wminw1G′u2f[z]dz∫wminwmaxG′u2f[z]dz+1λ{w1,wmax}∫w1wmaxG′u2f[z]dz∫wminwmaxG′u2f[z]dz (13) so that the ratio 1/λ{wmin,wmax} is a weighted average of the ratios 1/λ{wmin,w1} and 1/λ{w1,wmax}, with the weights given by the proportions of changes in social welfare above and below w1 ; (i) If any two of the three ratios are equal, they equal the third ratio; (ii) Since the weights vary with w1, it is impossible that both λ{w1,wmax} and λ{wmin,w1} are constant for all values of w1 unless they are equal; (iii) If λ{wmin,w1} and λ{w1,wmax} are unequal, then 1/λ{wmin,wmax} lies between 1/λ{wmin,w1} and 1/λ{w1,wmax}; (iv) Comparing (8) and (12), the limit:   limw1→wminλ{w1,wmax}=λ{wmin,wmax} (14) holds if and only if:   limw1→wmin(w12h1t′[y1]h+w1h1(1−t′[y1])f[w1])=0 (15) for interior labour supply solutions (not boundary constrained); and (v) Similarly, comparing (11) and (12), the limit:   limw1→wmaxλ{wmin,w1}=λ{wmin,wmax} (16) holds if and only if:   limw1→wmax(w12h1t′[y1]h+w1h1(1−t′[y1])f[w1])=0 (17) for interior labour supply solutions (not boundary constrained). The proof in Appendix 1 proceeds by multiplying λ{wmin,wmax} by the denominator in the ratio in (12) and breaking down the resulting integral of changes in social welfare into integrals on the two subintervals (wmin,w1) and (w1,wmax). Rearranging the resulting equality yields (13). The remaining results follow from this relationship. Define a uniformly optimal tax as a tax that satisfies λ{w1,wmax}=λ{wmin,w1}=λ{wmin,wmax} for any w1 in the open interval wmin to wmax. The statement of Proposition 2, made possible by the double limit analysis, provides a rigorous explanation for the conditions under which the marginal tax rate should be zero at the highest and lowest incomes, as well as the exceptions. There is an extensive literature on optimal tax rates at the highest and lowest incomes based on possible Pareto improvements from reducing marginal tax rates at the highest or lowest incomes to zero.3 In contrast, results in (iv) and (v) of Proposition 2 provide limit results that describe conditions close to the highest and lowest incomes: a uniformly optimal tax requires that the substitution effects in (15) and (17) (arising from an individual’s responses to a change in the marginal tax rate) approach zero as income approaches either the highest or lowest income. If the densities of the wage distribution are positive at the highest and lowest wage rates, then the marginal tax rates at the corresponding incomes will be zero. This result describes the marginal tax rates for more than just the highest and lowest incomes and provides an exception if the density of wage rates approaches zero at the highest or lowest wage rates. By providing both a basis for the zero-marginal tax rate result as well as exceptions, this paper is unlikely to change firmly held convictions. However, the results of Proposition 2 may be useful in resolving or at least narrowing conflicts. A practical consequence of Proposition 2 is that nonlinear income taxes determined by consistency with a first order condition over a subinterval of workers may not be optimal. Assuming the trade-off between social welfare and tax revenue is constant on a subinterval w1 to w2, the trade-offs λ{w,wmax} for w in the subinterval will be unequal if the condition λ{w1,wmax}=λ{wmin,w1}=λ{wmin,wmax} does not hold. The tax would then be suboptimal even within the interval w1 to w2. Proposition 2 provides a means of checking whether a proposed tax, however derived, is optimal, and this check has been applied to the solutions derived in this paper. 2.3 Second order conditions By the first order conditions, moving a marginal dollar of tax collection from one interval to another cannot raise social welfare. The second order condition requires that as one increases the level of tax collections moved between the intervals beyond the marginal dollar, social welfare does not start to increase. The second order condition for maximization of social welfare can be established as follows. For an arbitrary interval y1 to ymax, calculate the derivative of social welfare with respect to k1 and the derivative of tax revenues with respect to k1 for the subintervals y1 to y1+ε and y1+ε to ymax as in Section 2.2, but do not evaluate these derivatives at k1=1. Then with k1≠1, l’Hospital’s rule can be applied to calculate the derivatives of social welfare and tax revenue with respect to k1 as ε approaches zero, since the ratio of the derivative of social welfare to derivative of tax revenue will be indeterminate. Let λ[k1]=(λ{w1,wmax})k1≠1 be the resulting ratio when k1≠1. The second order condition will then be established from the derivative of λ[k1] with respect to k1. Following this procedure (the detailed derivation is provided in Appendix 1):   λ[k1]=−∫w1wmaxG′u2f[z]dz−h1w12k12t′[y1]1−w1h2(1−t[y1])+w12h1t′[y1]f[w1]+∫w1wmax1+z2h1t′′[y[z]]−zh21+z2h1t′′[y[z]]−zh2(1−t′[y[z]])f[z]dz (18) This is identical to the ratio in (8) except for the appearance of k1 in the denominator. To consider what happens to λ[k1] as tax revenues on the interval increase or decrease beyond a marginal dollar, let k1=1+η(δt′), where δt′ is a change in the marginal tax rate at y1 and η is a nonnegative variable. Suppose δt′>0 so that tax revenues increase on the interval as η increases. Then:   (dλ[k1]dη)η=0=(dλ[k1]dk1)k1=1dk1dη=(dλ[k1]dk1)k1=1δt′<0 (19) where dλ[k1]/dk1<0 from (18). Then as the taxes collected from an interval increase (by increasing k1 beyond 1 or equivalently η beyond 0), λ[k1] (which is negative) declines, so that the social welfare cost of a dollar of taxation increases as taxes are raised in the interval. Suppose instead that δt′<0, so that the marginal tax rate at y1 declines and tax revenues on the interval go down as η increases. Then:   (dλ[k1]dη)η=0=(dλ[k1]dk1)k1=1dk1dη=(dλ[k1]dk1)k1=1δt′>0 (20) Then as η is increased beyond zero, tax revenue declines on the interval and the trade-off λ[k1] increases. Since λ[k1] is negative, an increase in λ[k1] means that it declines in absolute value, and the social welfare cost of a dollar of tax revenue declines. Now consider moving more than a marginal dollar of tax collections from one interval to another. Social welfare would rise in the interval where tax collections are reduced, but the increase would be less than the decline in social welfare in the interval where tax collections increased (because of the changes in the ratios λ[k1] in the two intervals). Therefore, moving tax collections among intervals by more than marginal amounts cannot raise social welfare while keeping tax revenues the same, and the second order condition for an optimal income tax system is satisfied. 3. Second order differential equation An important advantage of the double limit analysis is that the results can be used to derive a second order differential equation determining the optimal income tax in the general case. This section constructs the differential equation for the tax system by differentiating the ratio in (8) with respect to the lower boundary of the wage interval. Rearranging the ratio yields:   −∫w1wmaxG′u2f[z]dz=λ{w1,wmax}(−h1w12t′[y1]1−w1h2(1−t[y1])+w12h1t′[y1]f[w1]+∫w1wmax1+z2h1t′′[y[z]]−zh21+z2h1t′′[y[z]]−zh2(1−t′[y[z]])f[z]dz) (21) Differentiation of both sides with respect to w1, with λ{w1,wmax} constant, yields:   G′u2f[w1]=λ{w1,wmax}(ddw1(−h1w12t′[y1]1−w1h2(1−t[y1])+w12h1t′[y1]f[w1])−1+w12h1t′′[y1]−w1h21+w12h1t′′[y1]−w1h2(1−t′[y1])f[w1]) (22) Solving yields:   ddw1(w12h1t′[y1]h+w1h1(1−t′[y1])f[w1])f[w1]=−G′u2λ{w1,wmax}−1+w12h1t′′[y1]−w1h21+w12h1t′′[y1]−w1h2(1−t′[y1]) (23) Both sides involve the derivative t′′[y1] so that the condition will be a second order differential equation involving only values of functions at y1. Expanding the left-hand side using the chain rule yields:   f′[w1]f[w1]w12h1t′[y1]h+w1h1(1−t′[y1])+ddw1(w12h1t′[y1]h+w1h1(1−t′[y1])) (24) The distribution of wages therefore enters into the differential equation only in the single term f′[w1]/f[w1]. Before developing the differential equation in specific cases, consider the interpretation of the three terms. On the right-hand side of (23), G′u2 is the contribution of a dollar of after-tax income to social welfare for the individual earning w1. Dividing by the trade-off λ{w1,wmax} yields the reduction in tax revenue that would be equivalent to a gain in social welfare of G′u2. The second term is the amount of tax revenue generated for an individual with the wage at the interval boundary, w1, from the original tax shift and the additional taxes from the labour response to the income effect. The difference in amounts on the right-hand side must be made up by the change in the substitution effect on the left side as the wage and income levels for the lower boundary of the interval increase (i.e. as w1 increases). The differential equation obtained in (23) will not by itself completely determine the optimal tax system. A general solution to a differential equation will be a family of solutions that depend on one or more constants of integration. The constants of integration can be determined by initial conditions that specify the values of the tax level and the marginal tax rate at a given level of income. If f[wmax]>0, the marginal tax rate at the highest income will be zero for a uniformly optimal income tax, providing one of the initial conditions. If the marginal tax rate at the lowest income is also zero, the tax level at the highest income can be varied until the second condition, at the lowest income, is satisfied. 4. Examples The examples in this section demonstrate how the uniformly optimal income tax can be determined using the methods above. Sections 4.1 and 4.2 assume a Cobb-Douglas utility function combined with a lognormal and Pareto wage distribution, respectively. Section 4.3 assumes a CES utility function combined with a lognormal wage distribution.4 4.1 Cobb-Douglas utility, lognormal wage distribution Assume utility takes the Cobb-Douglas form:   u[1−h,y−t[y]]=(1−h)α(y−t[y])1−α (25) where labour supply h varies between 0 and 1. The functional form for the labour supply that maximizes utility can be determined using standard methods (provided in Appendix 1) and is given by:   h[(1−t′[y])w,y−t[y]]=1−α(y−t[y])(1−α)(1−t′[y])w (26) Two alternative forms for the weighting function G in the social welfare function are the exponential (−e−γu), associated with a constant Coefficient of Absolute Risk Aversion when applied to a utility function, and the polynomial form (uγ/γ), associated with a constant Coefficient of Relative Risk Aversion when applied to a utility function. The latter form has the advantage that an optimal tax system would yield positive levels of utility for all individuals since the marginal contribution to social welfare of an extra dollar to an individual with zero utility would be infinite (essentially the Inada condition). On this basis, assume:   G[u]=uγ/γ, 0<γ<1 (27) so that G′[u]=uγ−1. Finally, suppose wage rates are distributed lognormally on the interval wmin to wmax. With this distribution, f′[w]/f[w]=−Log[w]/w. Proposition 2 in Section 2.2 and the implications for marginal tax rates at the highest and lowest incomes only apply to finite wage and income intervals. A possible exception to the conclusion that the marginal tax rate at the highest income is zero then arises if the wage distribution is unlimited. However, as a result of the double limit analysis, a significant problem becomes apparent when extending an optimal income tax to an increasing range of incomes. This difficulty is stated in the following proposition. Proposition 3 Assume utility is given by (25), the weighting function G in social welfare function is given by (27), and wages are distributed lognormally. If limy→∞t′[y]=τ<1,  limwmax→∞(limw1→wmaxλ{w1,wmax})=0 The proof in Appendix 1 proceeds by applying the assumed functional forms to determine the ratio λ{w1,wmax} in the limit. The significance of the result is that applying an optimal income tax to a wage interval with a very high maximum wage will require imposing a low trade-off between social welfare and tax revenue on all individuals (that is, a dollar of additional tax revenue could only impose a low loss of utility and social welfare at all income levels). As a consequence, tax revenues could be low or negative, and low-income individuals could require substantial subsidization. Potential difficulties with high wage boundaries described above do not rule out uniformly optimal tax solutions for finite wage intervals that are not too large. The condition that t′[ymax]=0 provides one initial condition at the upper boundary, and a given value of the tax level t[ymax] would provide the second initial condition needed to solve the second order differential equation in (23). Then t[ymax] can be varied until t′[ymin]=0 at the lower wage boundary. For the solution worked out here, λ{w1,wmax}=−0.36.5 Figure 1 shows the optimal tax solution. The tax level starts out negative at the lowest income level, corresponding to a transfer to low-income workers. Although it is not apparent from Fig. 1, the marginal tax rate starts out at zero at the lowest income level, rises rapidly, and then declines over most of the income range to zero at the highest income level, as shown in Fig. 2. This figure also shows that the average tax rate declines over a large income interval after being crossed by the marginal tax rate. Figure 3 shows the marginal and average tax rates at income levels such that a given percentile of workers earn less than that income. The figure shows that marginal and average tax rates rise with income for almost all workers. Fig. 1 View largeDownload slide Tax level Fig. 1 View largeDownload slide Tax level Fig. 2 View largeDownload slide Marginal and average tax rates Fig. 2 View largeDownload slide Marginal and average tax rates Fig. 3 View largeDownload slide Marginal and average tax rates by worker percentile Fig. 3 View largeDownload slide Marginal and average tax rates by worker percentile Define the crossover income level as the income at which the average tax rate starts to decline after the marginal tax rate intersects it from above. In Fig. 3, the crossover income occurs at the income level such that 96.7% of workers earn less. Although the marginal tax rate declines and lies below the average tax rate over a long income range in Fig. 2, it accounts for only a small percentage of workers in Fig. 3 and does not characterize the tax policy. Instead of the zero-marginal tax rate characterizing policy for high income individuals, the progressivity of the tax policy is more meaningfully described by the crossover income level and corresponding percentile. The relevant question is then where the crossover income should occur and how it compares to the crossover income in a given tax code. The labour supply consequences of the optimal tax are shown in Fig. 4. Labour supply is positive over the entire income interval, declining initially and then increasing rapidly at lower income levels and continuing to rise over the income interval in this specific case. Despite the presence of income effects, individuals with higher incomes provide higher levels of labour supply because of declines in the marginal tax rate. Fig. 4 View largeDownload slide Labour supply Fig. 4 View largeDownload slide Labour supply The ability to derive specific solutions of the second order differential equation for the optimal income tax provides a laboratory to test hypotheses concerning optimal income taxation. For example, an important question regarding taxation is who should pay higher taxes when greater tax revenue is needed. Choosing a greater trade-off between social welfare and tax revenue, λ{w1,wmax}, raises the social welfare lost per dollar of tax revenue and increases the aggregate tax revenue collected. The consequences of increasing the aggregate level of tax revenues can be examined further using methods developed in this paper. 4.2 Cobb-Douglas utility, Pareto wage distribution In this example, the lognormal wage distribution is replaced by a Pareto distribution. Let:   f[w]=Ifθwmin(wminw)θ+1 (28) where wmin is the lower boundary of the wage distribution and:   If=(∫wminwmaxθwmin(wminx)θ+1dx)−1 (29) (sothat∫wminwmaxf[x]dx=1). Using the Pareto probability density of wage rates, f′[w]/f[w] equals −(1+θ)/w. Corresponding to the proof for Proposition 3, Appendix 1 shows that in the Pareto case limwmax→∞limw1→wmaxλ{w1,wmax}=0 if limy→∞t′[y]<1. On a finite wage range, the uniformly optimal tax solution as a function of income for a Pareto wage distribution does not differ in any significant way from the results using a lognormal distribution, as shown in Fig. 5.6 While the marginal tax rate follows the same general pattern as for a lognormal wage distribution, the marginal and average tax rates as functions of worker percentile take a different form, as shown in Fig. 6. In this figure, the crossover income occurs at a level such that 99.97% of workers earn less. At lower incomes, the average tax rate is positive instead of negative as in the lognormal case. The major cause of the different shapes is the distributional term f′[w]/f[w], which takes the form −Log[w]/w in the lognormal case and −(1+θ)/w in the Pareto case. In the lognormal case, f′[w]/f[w] starts out positive for low values of w and then becomes negative, but at high values of w declines in absolute value at a slower rate than for the Pareto case. In the Pareto case, the more rapid decline in −(1+θ)/w reduces the significance of the substitution effect in (24) and therefore also reduces the gain from lowering marginal tax rates at high income levels. Besides establishing that the double limit methodology can be applied to the Pareto distribution, the example shows that the Pareto case generates increasing marginal tax rates over nearly the entire distribution of workers, with a decline only at the top percentile. The example underscores that the location of the crossover income is a more relevant description of the optimal tax than the decline in the marginal tax rate at the highest incomes. The relevance of a distribution of productivities entirely described by the Pareto distribution is limited by its applicability to only the upper tail. A distribution more consistent with observed income distributions would be either the Champernowne distribution (see Tuomala, 2016, p. 110) or a mixture of lognormal and Pareto distributions, with the weight for the latter increasing with productivity. The double limit analysis could be applied to either of these distributions. Fig. 5 View largeDownload slide Marginal and average tax rates Fig. 5 View largeDownload slide Marginal and average tax rates Fig. 6 View largeDownload slide Marginal and average tax rates by worker percentile Fig. 6 View largeDownload slide Marginal and average tax rates by worker percentile 4.3 Constant elasticity of substitution utility function, lognormal wage distribution This section provides optimal tax solutions assuming a CES utility function (Tuomala, 2016, p. 120, also obtains simulations using a CES utility function). Assume that utility takes the form:   uCES[1−h,y−t[y]]=(δ(1−h)−ρ+(1−δ)(y−t[y])−ρ)−1/ρ (30) with elasticity of substitution σu=1/(1+ρ). Then the labour supply is:   hCES[(1−t′[y])w,y−t[y]]=1−(y−T)((1−δδ(1−t′[y])w)−1/(1+ρ)) (31) (The derivation is provided in Appendix 1.) As ρ approaches zero, the CES function approaches the Cobb-Douglas function with exponent δ in place of α, allowing a comparison with the previous examples. For values of ρ equal to -.0909 and .1111, corresponding to σu equal to 1.1 and .9, Figs 7 and 8 show the marginal tax rates and average tax rates from optimal income tax solutions that yield aggregate tax revenue equal to the tax revenue in the Cobb-Douglas example in Section 4.1.7 A notable feature of the CES optimal income tax solutions is that both the marginal and average tax rates are lower when the elasticity of substitution is higher. With a higher elasticity of substitution, workers can more easily substitute income for leisure and therefore provide higher labour supply for given values of parameters and the marginal and average tax rates. As a result, labour supply is higher, incomes are higher, and lower tax rates are needed to generate the same level of tax revenue. Lower tax rates then generate even higher labour supply levels, as shown in the labour supplies in Fig. 9. In this figure, the maximum income level for ρ=−.0909 arises because of the greater labour supply multiplied times the maximum wage. Fig. 7 View largeDownload slide Marginal tax rates, CES utility Fig. 7 View largeDownload slide Marginal tax rates, CES utility Fig. 8 View largeDownload slide Average tax rates, CES utility Fig. 8 View largeDownload slide Average tax rates, CES utility Fig. 9 View largeDownload slide Labour supply Fig. 9 View largeDownload slide Labour supply 5. Conclusion 5.1 The contribution of double limit analysis The double limit analysis developed in this paper makes several contributions to the study of optimal income taxation. These include the expression for the trade-off between social welfare and tax revenue in Proposition 1, the relations among alternative trade-offs in Proposition 2, the derivation of the second order differential equation in Section 3, and the demonstration in Proposition 3 that the trade-off between social welfare and tax revenue declines to zero as the upper wage boundary increases indefinitely in some of the examples considered in Section 4. The usefulness of the double limit analysis is not restricted to the classical problem of determining the optimal income tax that maximizes social welfare subject to a constraint on tax revenue. It can be applied in any context in which the optimization principle requires equalization at all income levels of trade-offs between a social objective and a budget constraint that can be derived using the methods developed here. These applications include the use of a non-utilitarian social welfare function, Pareto efficient taxation and policy improvements (Werning, 2007; Hendren, 2014), heterogeneous labour responses (Jacquet et al., 2013; Jacquet and Lehmann, 2015), and endogenous wage rates (Kroft et al., 2016; Sachs et al., 2016). In each of these contexts, if the trade-off can be expressed in terms of integrals with a lower boundary, the trade-off can be differentiated to obtain a differential equation as in Section 3. A problem considered by Mirrlees (1971, p. 207), but not examined here, is whether some low-income individuals should be left out of the labour market and supported directly by rather than through the tax system. In the examples considered by Mirrlees and in this paper (in which the minimum productivity is positive), the optimal income tax draws all workers into the labour market. Instead, workers who would receive the lowest incomes could be tagged (or themselves choose to be bunched at zero labour supply) to receive income support outside the tax system, with the optimal income tax being applied to a wage (or productivity) interval that begins at a higher level (see the discussion in Tuomala, 2016, pp. 90–1, and Chapter 8). The decision of which workers to support outside of the tax system could be incorporated into a model with a social welfare function related to Rawls’ theory of justice that would maximize the well-being of the least-favoured individuals (Rawls, 1971, and discussions in Tuomala, 2016, pp. 72–7 and Salanié, 2003, pp. 84–7). With workers differing in preferences for leisure in addition to productivity, individuals could choose not to participate in the labour force at higher levels of productivity, yielding labour responses at the extensive margin in addition to hours of work (Tuomala, Chapter 6; Piketty and Saez, 2013, pp. 441–3; Salanié, 2003, pp. 40–1; Jacquet et al., 2013). The presence of heterogeneous characteristics affecting labour supply is treated as the multi-dimensional screening problem in contract theory (Armstrong and Rochet, 1999; Salanié, 2005, pp. 78–82; and Tuomala, 2016, Chapter 10, pp. 249–58). In the context of nonlinear pricing by a multiproduct firm, the principle in the principle–agent problem usually excludes some consumers (Armstrong, 1996) and induces some bunching in choosing products (Rochet and Choné, 1998). Problems also arise in comparing contributions to social welfare across individuals (Tuomala, 2016, p. 32). In the context of optimal income taxes with heterogeneous preferences for leisure, it is likely that the solution will also involve exclusion of some workers (see Jacquet and Lehman, 2015, for optimal income taxation with heterogeneous labour responses). 5.2 Relation to previous approaches Both the Mirrlees approach and the double limit analysis provide general conditions for optimal income taxation. The consistency of the two approaches is demonstrated in Appendix 1 by showing that as a result of imposing the semi-linear utility function with no income effects used in Diamond (1998), the trade-off in (8) simplifies to the Mirrlees-Diamond condition on the marginal tax rate. The two approaches instead differ in the additional assumptions necessary to derive numerical solutions. Mirrlees introduces the assumption that utility is additively separable and this assumption has been used routinely in numerical derivations since then. In their original forms, the Cobb-Douglas and CES functions are not additively separable in their arguments. Taking the logarithm of the Cobb-Douglas function and raising the CES function to the power 1/−ρ yield additively separable functions. The derivation of the trade-off in (8) and the condition for the differential equation in (23) do not depend on the assumption of a separable utility function. However, the double limit analysis is simplified in cases where the first order condition for the differential equation can be solved for t′′[y]. Absence of income effects has also been imposed (Diamond, 1998; Piketty and Saez, 2013, p. 435). With income effects, high incomes can induce individuals to reduce their labour supplies. Then an optimal income tax with high average income taxes would lead higher income individuals to maintain their labour supply levels. In the absence of income effects, labour supply would only depend on the after-tax wage rate, (1−t′[y])w. As w increases, individuals would then increase their labour supplies without limit unless the marginal tax rate t′[y] approaches one. Without income effects, solutions for the marginal tax rate that differ from one generally require that labour supply increases indefinitely (Saez, 2001, p. 223). Although it is not used to solve the original Mirrlees problem, the elasticity approach developed by Saez (2001) has the advantage that it relates the skill distribution and income tax to empirical estimates of labour supply responses. However, assuming a constant elasticity of labour supply with respect to the wage determines t′′[y] in lieu of the condition in (23). By replacing the solution for t′′[y] from the first order condition, the assumption of a constant elasticity imposes a restriction on the solution that is unnecessary with the double limit analysis. The double limit analysis differs from previous approaches in providing methods of testing for the optimality of a tax solution. As noted in Section 2, a solution that satisfies the first order condition on part of the income interval will not necessarily be optimal. Proposition 2 provides a test for a uniformly optimal tax by calculating the three ratios. By calculating labour supply, dy / dw and t′′[y] at each income, it is also possible to check whether discontinuities or singularities occur in the solution. The double limit analysis can be used to compare tax revenue and social welfare generated by a tax system to check whether an alternative solution is better. Calculation of the trade-off in (8) can be used to determine how tax collections should be shifted among income intervals. To derive numerical optimal income tax solutions, the double limit analysis imposes some restrictions that are absent in the original Mirrlees analysis. These include the assumption that utility and the social welfare weighting function are continuous functions of their arguments and that labour supply can be derived as an analytic function of (1−t′[y])w and y−t[y]. 5.3 Inequality In the strategy pursued in this paper, the first order condition for an optimal income tax promotes equality by imposing the same trade-off between social welfare and tax revenue on individuals at all income levels, a form of vertical equity at the margin. Both rich and poor are linked by a common trade-off between contributions to social welfare and tax revenues and by a common differential equation. Extending an optimal income tax to higher levels of income may require greater subsidies to individuals at low incomes. The role of subsidies to low-income individuals is not simply to raise their incomes in response to their high marginal contributions to social welfare, but to allow an efficient rapid increase in the marginal tax rate. The optimal income tax promotes equality by using a social welfare function (with lower weightings for higher income individuals) as its objective and promotes efficiency through the requirement for optimization. A basic question is therefore whether an optimal income tax can substantially reduce inequality or whether by seeking optimality it promotes efficiency at the cost of disregarding income differences. Redistribution in response to the weights of the social welfare function is limited by the costs generated by substitution effects. Questions regarding the interaction between optimal income taxation and inequality remain to be addressed in future work.  Table 1 Marginal tax rate and tax levels Interval  Marginal tax rate  Tax level  (y1,y1+ε)  k1t′[y]  t[y1]+k1(t[y]−t[y1])  (y1+ε,ymax⁡)  t′[y]  t[y]+k1(t[y1+ε]−t[y1])  Interval  Marginal tax rate  Tax level  (y1,y1+ε)  k1t′[y]  t[y1]+k1(t[y]−t[y1])  (y1+ε,ymax⁡)  t′[y]  t[y]+k1(t[y1+ε]−t[y1])  Supplementary Material Appendices 1 , 2, and 3 are available online at the OUP website. Footnotes 1 See for example Tuomala (2016, Fig. 4.3, p. 77), and Piketty and Saez (2013, Fig. 3, p. 435). Tuomala cites an early use of the tax perturbation method by Christiansen (1981) in a different context. Golosov et al. (2014) generalize tax perturbations to a dynamic setting. 2 See Assumption B in Mirrlees (1971, p. 182), and the discussion by Salanié (2003, pp. 87–91) and Tuomala (2016, pp. 65–6). The simple condition ensures that individuals with higher wages will earn higher incomes. Salanié and Tuomala relate this condition to the Spence-Mirrlees condition in contract theory, agent monotonicity, and incentive compatibility. The utility function can be re-expressed as a function of consumption and income divided by the wage rate (equal to the labour supply). The condition requires that the indifference curve between consumption and income become flatter as the wage rate increases. Then an indifference curve tangent to a budget constraint will rotate clockwise as the wage increases, so that the new point of tangency will be at a higher income level. See also Seade (1982) and Tuomala (1990, p. 87). 3 Early papers include Mirrlees (1971), Sadka (1976, p. 266), Seade (1977, p. 231, footnote; and 1982). See also conflicting conclusions in Diamond and Saez, (2011, pp. 188–9) and Mankiw et al. (2009, pp. 151–5), and the discussion by Tuomala (2016, pp. 77–83). 4 The Mathematica notebooks that derive the numerical solutions and generate the figures are provided in Appendix 2 for Cobb-Douglas utility and Appendix 3 for CES utility. 5 The other parameters are α=.3, γ=.8, wmin=1,andwmax=100. The parameters of the lognormal distribution are μ=1 and σ=1. Tax revenue is .964 and production is 3.307. The average elasticity of labour supply with respect to (1−t′[y])w is .65, and the average elasticity of labour supply with respect to y−t[y] is -.65. Detailed procedures for finding the solution by varying t[ymax] are provided in Appendix 2. 6 The solution assumes θ=2.0 and λ=−.51, with wages ranging from 1 to 100. The utility function is the same as for the lognormal example. Tax revenue is .300 and production is 1.331. The average elasticity of labour supply with respect to (1−t′[y])w is .46 and the average elasticity of labour supply with respect to y−t[y] is -.46. 7 The optimal tax solutions are derived assuming δ=.3, with λ equal to -.37296 for σu=1.1 and λ equal to -.34668 for σu=.9 in order to yield equal tax revenue of .9640. Other parametric assumptions (including the lognormal distribution of wage rates) are the same as in Section 4.1. Acknowledgements The author acknowledges helpful comments by the editor and referees as well as Henning Bunzel, Michael Jerison, John B. Jones, and Kwan Koo Yun. Any remaining errors are the responsibility of the author. References Armstrong M. ( 1996) Multiproduct nonlinear pricing, Econometrica , 64, 51– 75. Google Scholar CrossRef Search ADS   Armstrong M., Rochet J.-C. ( 1999) Multi-dimensional screening: a user’s guide, European Economic Review , 43, 959– 79. Google Scholar CrossRef Search ADS   Atkinson A., Stern N. 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