Abstract. 1. Introduction

Transcription

1 OPTIMAL CATEGORICAL TRANSFER PAYMENTS: THE WELFARE ECONOMICS OF LIMITED LUMP-SUM REDISTRIBUTION ALAN D. VIARD Federal Reserve Bank of Dallas Abstract Despite their importance in tax-transfer systems, categorical transfer payments, based on ~nearly! exogenous characteristics such as disability or date of birth, have been deemphasized in optimal-tax analysis. I use the well-developed theory of first-best redistribution to clarify the welfare economics of categorical transfers, which are a form of limited lump-sum redistribution. The comparison to first-best redistribution explains how categorical transfers affect groups labor supplies and utility levels, why the use of categorical transfers is inversely related to the planner s inequality aversion, and why their use reduces the optimal income tax rate. 1. Introduction An important component of most countries tax-transfer systems is the provision of transfer payments to categories of individuals defined by ~nearly! exogenous characteristics, such as date of birth or disability. Atkinson ~1995, pp !, Kanbur and Keen ~1989!, Mirrlees ~1986, pp. 1199, 1215!, Atkinson and Stiglitz ~1980, pp. 362, !, and many other authors note the prevalence of these categorical transfer payments. However, the optimal-tax literature has downplayed the role of categorical transfer payments in an optimal redistributive system. In this paper, I provide further insights into optimal categorical transfers by linking them to the well-developed theory of first-best lump-sum redistribution. Alan D. Viard, Research Department, Federal Reserve Bank of Dallas, P.O. Box , Dallas, TX ~alan.viard@dal.frb.org!. I am grateful for helpful comments by Thomas A. Barthold, Stephen G. Cecchetti, Gene E. Mumy, V. Brian Viard, seminar participants at Vanderbilt University and Indiana University0 Purdue University at Indianapolis, and an associate editor and referee of this journal. The views expressed in this article are solely my own and do not reflect the views of the Federal Reserve System or the Federal Reserve Bank of Dallas. Received 11 August 1998; Accepted 14 December Blackwell Publishers, Inc. Journal of Public Economic Theory, 3~4!, 2001, pp

2 484 Journal of Public Economic Theory In the standard optimal-income-tax model, the planner wishes to redistribute from individuals with high exogenous earnings ability to those with low ability but observes labor earnings rather than ability. In the constrained optimum, the planner reduces inequality in the social marginal value of income by using an income tax to finance a uniform transfer payment. Because the income tax is distortionary, the planner does not fully eliminate inequality. However, a planner who also observes exogenous characteristics that are correlated with ability can achieve limited lump-sum redistribution through categorical transfer payments that vary across groups with different characteristics. Optimal categorical transfers equalize the mean social marginal value of income across these groups and eliminate the betweengroup component of inequality. The planner must continue to use income taxation to reduce inequality within each group. I confirm previous findings that optimal categorical transfers are higher for disadvantaged ~lower average ability! groups, that such transfers ~counterintuitively! vary less across groups when the planner is more averse to inequality, and that the use of such transfers reduces the optimal marginal income tax rate. I explain these results and others on utility and labor supply differences between groups by comparison to the theory of first-best redistribution. In the first best, the planner observes individual ability and eliminates inequality in the social marginal value of income through lumpsum taxes and transfers based on ability. Categorical transfers address betweengroup, but not within-group, inequality in a similar manner. With optimal categorical transfers ~and identical preferences!, advantaged groups tend to have generally higher labor supply and lower utility, paralleling the well-known result that higher-ability individuals supply more labor and have lower utility in the first best. Another wellestablished result is that, because utility decreases with ability in the first best, increased inequality aversion ~greater weight on low-utility individuals! reduces the amount of first-best redistribution from high-ability individuals to low-ability individuals. Similarly, increased inequality aversion reduces the extent to which the planner uses categorical transfers to redistribute from advantaged groups. Also, the reduction in the income tax rate from the use of categorical transfers resembles the reduction ~to zero! that occurs with first-best transfers. In Section 2, I review the effects of categorical transfers on the linearincome-tax problem. I establish the fundamental economic similarity of categorical and first-best transfers in Section 3. I examine labor supply and utility in Section 4, the implications of inequality aversion in Section 5, and the effect on the income tax rate in Section 6. I discuss extensions and conclude in Section Categorical Transfers in the Linear-Income-Tax Model I examine the effect of combining categorical transfers with a linear tax on income ~labor earnings!.

3 Optimal Categorical Transfer Payments Analytics of Optimal Categorical Transfers A continuum of individuals have identical utility functions increasing in consumption C and decreasing in labor supply L, both of which must be nonnegative. With no taxes, each individual s exogenous income y equals zero and each individual s wage rate w equals a nonnegative earningsability parameter n, which has cross-sectional density f ~n!. Following Hellwig ~1986!, I impose the following restriction on individual preferences and the social welfare function ~subscripts on C and L denote derivatives of ordinary consumption demand and labor supply functions!. The common utility function is quasi-concave and twice differentiable. Leisure is a normal good, L y 0, and consumption is an increasing function of the wage rate, C w L wl w 0, ifl 0. ASSUMPTION 1: The planner maximizes the sum of a strictly concave representation (defined only up to an increasing linear transformation, as discussed by Sen 1974) of the common utility function. V ~w, y! is the indirect utility function corresponding to this representation, and V y is the increase in social welfare from an individual s additional consumption. Strict concavity implies V yy 0. ASSUMPTION 2: To introduce categorical transfers, I allow the planner to distinguish J nonoverlapping groups of individuals on the basis of exogenous characteristics. Each group j has an ability distribution with density f j ~n!, where (j f j ~n!p ~ j! equals f ~n! and p ~ j! is the fraction of the population in each group. The planner observes each group s size and density function but does not observe individual values of n. The planner chooses a transfer G ~ j! for the members of each group j. For simplicity, the marginal tax rate t on labor earnings is uniform. Each individual s net tax payment, tnl G ~ j!, depends on both labor earnings and group membership. The standard linear-income-tax model is the special case in which J equals one ~the transfer payment is the same for all individuals!. The planner maximizes ( j p ~ j! V ~n~1 t!,g ~ j!!f j ~n!dn m R ( j p ~ j!g ~ j! t ( j p ~ j! nl~n, t,g ~ j!!f j ~n!dn, ~1! where R is an exogenous per capita net revenue requirement ~to finance public goods that enter utility separably! and m is the shadow price of the planner s budget constraint. At an interior solution, the categorical transfers G ~ j! satisfy *@V y mntl y # f j ~n!dn m, j. Following previous authors, define each individual s social marginal value of income a as V y mtnl y, the social welfare gain from increasing the individual s income, including the shadow value

4 T T Z 486 Journal of Public Economic Theory of the revenue change from the labor supply response. The first-order condition can then be written a~ T j! m, j, ~2! where a~ j! [*a~g~j!,n!f j ~n!dn is the mean value of a for group j. Letting a[*a~n!f~n!dn T denote the population mean of a, equations ~2! imply a m. The first-order conditions ~2! state that all groups mean social marginal values of income are equated to the shadow value of revenue and hence to each other. Optimal categorical transfers therefore eliminate between-group inequality in the social marginal value of income. Various forms of ~2! appear in Viard ~2001!, Immonen, Kanbur, Keen, and Tuomala ~1998!, Parsons ~1996!, Diamond and Sheshinski ~1995!, Blackorby and Donaldson ~1994!, Keen ~1992!, Kanbur and Keen ~1989!, Ravallion and Chao ~1989!, Beath, Lewis, and Ulph ~1988!, Bennett ~1987!, and Mirrlees ~1986, p. 1215!. Equalization of mean social marginal value of income remains optimal with nonlinear taxes, heterogeneous preferences, and commodity taxes. At an interior solution, the first-order condition for the marginal income tax rate t is *V y nlf ~n!dn m*n~l tnl w!f ~n!dn. Let subscripts on LZ denote derivatives of the compensated labor supply function and use the Slutsky decomposition LZ w L w L y L, the definition of a, and at m to obtain t anlf ~n!dn a nlf T ~n!dn a n T 2 LZ w f ~n!dn. ~3! As Diamond ~1975! notes, the numerator of ~3! is the cross-sectional covariance of the social marginal value of income a and pretax labor earnings nl. Since L w is positive, the optimal income tax rate has the opposite sign from this covariance. If a covaries negatively ~positively! with earnings, so that the planner wishes to redistribute from high ~low! earners to low ~high! earners, the marginal tax rate is positive ~negative! to achieve the desired redistribution. Hellwig ~1986! proves that, under Assumptions 1 and 2, the covariance is negative and the optimal tax rate is positive. 2.2 Illustrative Calculations of Optimal Categorical Transfers To obtain further insight, I now numerically compute optimal categorical transfers in a variant of Stern s ~1976! classic linear-income-tax calculations. The cross-sectional distribution of ability n is lognormal. Following Stern and subsequent authors, I set s, the standard deviation of log n, equal to.39. I normalize the mean value of n to unity ~without loss of

5 Optimal Categorical Transfer Payments 487 generality, since preferences will be homothetic!. I set net revenue R equal to.08, which is 13% of equilibrium output. This value is somewhat lower than the 17.5% share of government purchases in U.S. gross domestic product ~GDP! to reflect the fact that some of these purchases ~such as health and education! may be substitutes for private consumption. Tuomola ~1990, pp ! notes that such purchases should be treated as part of transfer payments rather than as part of R. As in Stern ~1976!, the planner maximizes the following function of individual utilities: V ~n, j! 1 1 A ( j A!e p ~ ~n, j! ~e 1!0e k~1 L~n,j!! ~e 1!0e # ~1 e 1 f j ~n!dn. ~4! In equation ~4!, e.0 is the elasticity of substitution between consumption and leisure in this constant-elasticity-of-substitution ~CES! utility function, while k 0 is a parameter affecting the propensity to consume leisure. Stern ~1976, pp. 136, 152! concludes that empirical evidence supports a value of about.4 for e, while Blomquist ~1981, p. 399! cites evidence for a value of.5. Tuomala ~1990, p. 50! concludes that empirical studies support a range of.2 to.6 but that a higher value may be warranted if dimensions of labor supply other than hours worked are considered. In general accordance with these authors, I set e equal to.5. I set the leisure share parameter k equal to.25. In the no-tax equilibrium, L and C both equal.667 for the mean-ability individual, and the local compensated and uncompensated labor-supply elasticities are.167 and.167, respectively. The parameter A. 0 reflects the curvature of the social welfare function and the extent of the planner s aversion to inequality. An increase in A is a strictly concave transformation that increases the planner s inequality aversion and gives greater weight to low-utility individuals. The maxi-min case obtains in the limit as A approaches infinity. Stern ~1976, pp ! favors a value of 2 for this parameter. I consider three values, 6, 2, and 1. The population consists of two groups ~ advantaged and disadvantaged!, characterized by two parameters, the population share of the disadvantaged group z and the difference of the group means of log ability d. I consider the 15 cases obtained by combining three values of z ~.2,.5,.8! with five values of d ~.1,.2,.3,.4,.5!. Details are available upon request. The correlation between log n and an indicator variable for the advantaged group ranges from.077 ~d is.1, z is.1 or.9! to.635 ~d and z are.5!. Each of the three panels of Table 1 presents results for a different value of A. In the top row of each panel, I report the optimal marginal tax rate and transfer payment and the mean consumption level under the optimal linear income tax ~ uniform optimum!. In the body of each

6 Table 1: Optimal Taxes and Transfers d.1 d.2 d.3 d.4 d.5 t G~Dis! G ~Ad! t G~Dis! G ~Ad! t G~Dis! G ~Ad! t G~Dis! G ~Ad! t G~Dis! G ~Ad! ~a! A 6 Uniform income tax optimum: Tax rate.599, transfer.273 ~mean consumption.510! z z z ~b! A 2 Uniform income tax optimum: Tax rate.467, transfer.210 ~mean consumption.540! z z z ~c! A 1 Uniform income tax optimum: Tax rate.385, transfer.164 ~mean consumption.555! z z * z * * * Corner solution in which advantaged group receives no transfers. 488 Journal of Public Economic Theory

7 Optimal Categorical Transfer Payments 489 panel, I report, for each of the 15 cases, the optimal marginal tax rate and the optimal categorical transfers G ~Dis! and G ~Ad! for the disadvantaged and advantaged groups ~ categorical optimum!. Three designated cases in panel ~c! are corner solutions with zero transfers to the advantaged group ~since a strictly negative transfer would induce negative consumption by low-ability members of the group and discontinuously reduce social welfare to negative infinity, these solutions are not characterized by the first-order conditions derived above!. Table 1 confirms that it is optimal to give a larger transfer to the disadvantaged group and that the difference in transfers rises with the difference of ability distributions. Both of these conclusions are robust to parameter variation. Extensive calculations for alternative values of s, e, R, A, z, and d are available from the author upon request. I now explain the economic similarity of the limited lump-sum redistribution permitted by categorical transfers to the first-best lump-sum redistribution that occurs when the planner observes individual abilities. 3. Fundamental Economic Similarity of Categorical and First-Best Transfers Tuomala ~1990!, Stiglitz ~1987, pp !, Mirrlees ~1986, pp !, Blomquist ~1981!, Atkinson and Stiglitz ~1980, p. 421!, and Helpman and Sadka ~1978a! analyze the first-best problem. The income tax rate is zero, and the planner chooses a transfer schedule G ~n!, with negative values denoting taxes, to maximize V ~n,g ~n!!f ~n!dn m R ng ~n!f ~n!dn. ~5! The first-order conditions require that V y ~n! equal m at each n. Since t equals zero, a is V y, implying a~n! m, n. ~6! First-best transfers equate social marginal values of income across ability levels. To see the similarity to categorical transfers, decompose the cross-sectional variance of a into within-group and between-group components. While first-best transfers equalize a across all individuals and set both components equal to zero, optimal categorical transfers equalize the group means of a and set the between-group component equal to zero. First-best and optimal categorical transfers each have zero crosssectional covariance with a, implying that these transfers ~which have no efficiency costs! are employed until their distributional benefits are exhausted. In contrast, as Diamond ~1975! notes, ~3! implies that income tax payments tnl have negative cross-sectional covariance with a. An incometax rate increase has distributional benefits, but they are offset by its

8 490 Journal of Public Economic Theory efficiency costs. This relationship between distributional and efficiency effects accords with the general analysis of Sosnow ~1974!. Since first-best transfers equalize V y across all values of n, they satisfy G n V wy 0V yy. Since V wy V yy L V y L y ~from differentiation of Roy s identity V w V y L!, G n L ~L y V y 0V yy!, ~7! as Helpman and Sadka ~1978a! note. Under Assumptions 1 and 2, V wy, 0 and G n, 0ifL.0. In the first best, higher-ability individuals receive smaller transfers ~or pay higher taxes!, except that all nonworkers receive equal transfers. Because V y is a decreasing function of n in the no-tax equilibrium, the planner redistributes from high-ability to low-ability individuals. Table 1 suggests that the planner similarly pays larger categorical transfers to groups with generally lower ability. I now confirm this relationship analytically. To do this, it is necessary to impose an additional assumption on preferences. The marginal propensity to spend on leisure L y w is nondecreasing in w and y. ASSUMPTION 3: The CES utility function satisfies Assumption 3 when e is less than one, since this propensity is then independent of y and increasing in w, and also when e equals one ~Cobb Douglas utility!, since the propensity is then constant. I also introduce the following definitions. Let n q ~ j! denote the ability level at the q quantile of group j s distribution, F j ~n q ~ j!! q. For example, n.5 ~ j! is median ability in group j. Similarly, for any variable x, let x q ~ j! denote the value of x at the q quantile of group j s ability distribution. DEFINITION 1: Group j has higher ability than group k if n q ~ j! n q ~k!, q, with the inequality strict for some q. Groups j and k have noncomparable ability if q, n q ~ j! n q ~k! and q, n q ~ j! n q ~k!. DEFINITION 2: Theorem 1 demonstrates that groups transfer payments are inversely related to ability. THEOREM 1: Under Assumptions 1, 2, and 3, if group j has higher ability than group k, then group j s transfer payment is smaller than group k s at an interior categorical optimum, G ~ j! G ~k!. Proof: As described above, V y is decreasing in n and y. Under Assumption 3, mtnl y is nonincreasing in n and y. Therefore, a is decreasing in both y and n. Suppose, contrary to the theorem, that the optimal transfer to group j is less than or equal to group k. Then, the social marginal value of income a is strictly lower for group j than for group

9 Optimal Categorical Transfer Payments 491 k at every quantile, contradicting the equality of the means of a required by ~2!. This analysis establishes the fundamental similarity of first-best and categorical transfers. I now use results from first-best theory to clarify the effects of categorical transfers on between-group differences in utility and labor supply, comparative statics with respect to the planner s inequality aversion, and effects on the income tax rate. 4. Effects on Utility and Labor Supply I first show that categorical and first-best transfers have similar implications for between-group differences in utility and labor supply. 4.1 Analytical Results Tuomala ~1990!, Stiglitz ~1987, p. 995!, Mirrlees ~1986, p. 1212!, Stern ~1982!, Blomquist ~1981!, and Atkinson and Stiglitz ~1980, p. 421! note the negative relationship between ability and utility in the first best. Income differences that equalize marginal utility result in lower utility levels for higher-ability individuals. Under the tax-transfer schedule ~7!, the variation of utility with ability is dv0dn V w V y G n L y V y 2 0V yy. ~8! Under Assumptions 1 and 2, this derivative is negative when labor supply is positive. Utility declines with ability, except that all nonworkers have the same utility. In contrast, in the no-tax and uniform optima, higher-ability individuals have higher utility because they have higher after-tax wage rates and the same exogenous income. With lower utility and higher wage rates, high-ability individuals must supply more labor in the first best ~since the compensated labor supply function is decreasing in utility and increasing in the wage rate!. The variation of labor supply with ability is dl0dn L w L y G n L w L y L V y ~L y! 2 0V yy LZ w V y ~L y! 2 0V yy, ~9! which is positive when labor supply is positive. In the first best, labor supply increases with ability, except that it may be zero throughout a bottom interval of the ability distribution. The economic intuition is that it is efficient for more productive individuals to work more. In the first best, the equalization of the social marginal value of income across individuals results in lower utility and greater labor supply for high-ability individuals. With categorical transfers, Theorem 2 states that the equalization of mean marginal value of income across groups induces similar effects in at least some quantiles of the ability distribution.

10 492 Journal of Public Economic Theory THEOREM 2: Under Assumptions 1, 2, and 3, if group j has higher ability than group k, then at an interior categorical optimum, there exist some quantiles at which utility is lower and labor supply is greater for group j than for group k, q, s.t.v q ~ j! V q ~k!, L q ~j! L q ~k!. Proof: Suppose, contrary to the theorem, that utility for group j is equal or higher at every quantile. Then, as shown above, V y is lower for group j at every quantile. Since mtnl y is equal or lower for group j at every quantile under Assumption 3, a is lower for group j at some quantiles and higher at none, contradicting the necessary condition that the mean of a is equal for all groups. So, utility must be lower for group j at some quantiles. At these quantiles, labor supply must be greater for group j, since the compensated labor supply function is decreasing in utility and increasing in the wage rate. Theorem 3 provides a more specific statement about relative utility levels when the optimal income tax results in some low-ability individuals performing zero labor. THEOREM 3: Under Assumptions 1, 2, and 3, if group j has higher ability than group k, and if, at an interior categorical optimum, the bottom X~ j! 0 quantiles of group j and the bottom X~k! 0 quantiles of group k provide zero labor, utility for members of group j is lower than that of group k at the overlapping quantiles, V q ~ j! V q ~k!, q min~x~ j!, X~k!!. Proof: Nonworkers have consumption equal to their group s transfer payment. Since group j s transfer payment is smaller ~Theorem 1!, its nonworkers have lower consumption and, since leisure is constant at unity, lower utility. Theorem 4 uses an additional assumption to provide a broader result for groups relative labor supply. THEOREM 4: If the uncompensated labor supply is nondecreasing in the wage rate, and if group j has higher ability than group k, then, at an interior categorical optimum, labor supply for members of group j is greater than that of group k at every quantile, L q ~ j! L q ~k!, q, except possibly that labor supply may be zero for both groups at some quantiles. Proof: At each quantile, group j has a higher wage rate and ~by Theorem 1!, lower exogenous income. Since labor supply is nondecreasing in the wage rate and is decreasing in exogenous income ~except when labor supply is zero!, labor supply is greater for group j. I now return to the numerical calculations for additional insight into how categorical transfers affect groups relative utility and labor supply levels.

11 Optimal Categorical Transfer Payments Numerical Results In Table 2, for each value of A, I consider 5 of the 15 cases from Table 1, those in which the groups are of equal size, z.5. I report the log differences of the two groups median values of a money-metric utility measure and their median labor supplies, for the uniform and categorical optima and the first best. The categorical optimum always results in a median-utility ratio that is less favorable to the advantaged group than the ratio in the uniform optimum and generally similar to ~but never quite as unfavorable as! the ratio in the first best. In panel ~a!, where the advantaged group has only slightly lower median utility in the first best, it has slightly higher median utility in the categorical optimum, but much less so than in the uniform optimum. In the other two panels, the advantaged group has lower median utility in every case. Similarly, while the difference between the two groups median labor supplies is small and of varying sign in the uniform optimum, the advantaged group always has higher median labor supply in the categorical optimum and first best. The log difference of the medians is strikingly similar across the latter two equilibria. These results confirm the quantitative, as well as qualitative, similarity of categorical and first-best transfers. The conclusion that categorical transfers result in advantaged groups having higher labor supply and lower utility is robust to parameter variation. Extensive calculations for alternative values of s, e, R, A, z, and d are available from the author upon request. 5. Comparative Statics with Respect to Inequality Aversion Changes in the planner s inequality aversion ~the relative weight placed on low-utility individuals! have sharply different implications for the volume of first-best lump-sum redistribution and the volume of redistribution accomplished through the income tax. I now show that the implications of inequality aversion for categorical transfers are similar to those for first-best redistribution. 5.1 Analytical Results Equations ~7! and ~8! state that the absolute values of G n and dv0dn are lower when the planner is more inequality averse ~the absolute value of V yy 0V y is higher!. As inequality aversion vanishes, the extent of first-best redistribution and the relative utility of the less able increase without bound. This inverse relationship between inequality aversion and the extent of first-best redistribution contrasts with the direct relationship between inequality aversion and income taxation. Helpman and Sadka ~1978b!

12 Table 2: Relative Utility and Labor Supply ~Log of Advantaged Group s Median Money-Metric Utility Minus Log of Disadvantaged Group s Median Money-Metric Utility! ~Log of Advantaged Group s Median Labor Supply Minus Log of Disadvantaged Group s Median Labor Supply! d.1 d.2 d.3 d.4 d.5 U F C U F C U F C U F C U F C ~a! A 6 Utility Labor supply ~b! A 2 Utility Labor supply ~c! A 1 Utility Labor supply U uniform optimum; F first best; C categorical optimum. Elasticity of substitution e equals.5, and leisure share parameter k equals.25. Revenue requirement is.08. Log of ability n is normally distributed with standard deviation.39. Disadvantaged group s population share z is.5. Advantaged group s mean log n exceeds disadvantaged group s mean log n by d. 494 Journal of Public Economic Theory

13 Optimal Categorical Transfer Payments 495 prove that increased inequality aversion increases the optimal marginal tax rate and uniform transfer. The different implications of inequality aversion for first-best and income-tax redistribution are due to differences in relative utility levels. Since lower-ability individuals have lower utility in the uniform optimum, they are treated more generously when the planner places greater weight on low-utility individuals. But, since lower-ability individuals have higher utility in the first best, they are treated less generously when the planner places greater weight on low-utility individuals. Since optimal categorical transfers generally induce lower utility levels for advantaged groups, as shown above, greater inequality aversion similarly results in more favorable treatment of advantaged groups and less redistribution. Theorem 5 demonstrates that differences in groups categorical transfers disappear as inequality aversion increases to the maximin limit. THEOREM 5: Under Assumptions 1 and 2, if some members of each group supply zero labor at the categorical optimum, the differential in groups transfer payments approaches zero as the social welfare function approaches maxi-min. Proof: Suppose, contrary to the theorem, that transfers differ across two such groups in the maxi-min limit. Nonworkers in the group with the lower transfer have the minimum utility level. The minimum utility level is increased by a budget-neutral change that increases this group s transfer and reduces the other group s transfer Numerical Results While the theorem applies to only the maxi-min limit, Table 1 provides broader evidence that transfer differences are inversely related to inequality aversion. As the inequality aversion parameter A declines from 6 to 2 to 1, the difference between G ~Dis! and G ~Ad! consistently rises for each given combination of d and z ~except in some of the corner solutions!. Table 2 similarly reveals that the disadvantaged group s relative utility level in the categorical optimum consistently rises as A declines. The conclusion that a decline in A widens the transfer differences and improves the disadvantaged group s relative utility level is robust to parameter variation. Extensive calculations for other ranges of A and alternative values of s, e, R, z, and d are available from the author upon request. Numerical results in Stern ~1982! and Blomquist ~1981! also reveal that inequality aversion and the use of categorical transfers are inversely related. 1 Bennett ~1987! notes that minimum incomes are equalized in the maxi-min case, but he does not draw out the implication that transfer payments are equalized in the presence of nonworkers or that inter-group redistribution is minimized.

14 496 Journal of Public Economic Theory 6. Effects on the Marginal Income Tax Rate I now show that categorical and first-best transfers also have similar effects on the marginal tax rate. 6.1 Analytical Results If desired, it is possible to treat the tax rate as a choice variable in the first-best problem and obtain a first-order condition of the form ~3!. Since a is equalized across individuals in the first best, it has zero covariance with labor earnings. The numerator of ~3!, and therefore the first-best income tax rate, is zero, as implied by the second fundamental theorem of welfare economics. First-best transfers achieve all desired redistribution, and the income tax has no distributional benefits to offset its efficiency costs. With first-best transfers, both the between-group and within-group covariances of a with nl are zero. With uniform transfers, both covariances are negative. With categorical transfers, the within-group covariance is negative, but the between-group covariance is zero. This decomposition suggests that categorical transfers reduce the absolute value of the overall covariance and lower the marginal tax rate. Income taxation is less necessary because categorical transfers remove the between-group component of the inequality that the income tax must otherwise alleviate. This argument is heuristic because it ignores income effects and changes in the within-group covariance of V y and a, which arise because V yy is not constant. However, it can be given analytical support under assumptions that remove these complications. ASSUMPTION 4: V yy is a negative constant ~V is quadratic in y), V yyw V yyy 0, Since V yyw V yyy 2V yy L y V y L yy, Assumption 4 implies that L y is zero, which requires a change in Assumption 1. ASSUMPTION 1 ' : Assumption 1 holds, except that L y 0. Theorem 6 then states that the use of categorical transfers reduces the optimal income tax rate. THEOREM 6: Under Assumptions 1 ', 2, and 4, if all groups can be ranked by ability (no pair has noncomparable ability), the optimal marginal income tax rate is lower in the categorical optimum than in the uniform optimum. Proof: See Appendix. The assumptions used in Theorem 6 are quite stringent. The assumption L y 0 is unrealistic ~but for another application see Diamond 1998! and requires that Theorems 3 and 4 be modified to refer to groups j and k having equal utility. Moreover, the quadratic form of V is compatible with a positive value of V y only over particular ranges of y. Akerlof ~1978,

15 T T T Optimal Categorical Transfer Payments 497 p. 14! also proves that categorical transfers reduce the optimal tax rate, but he restricts L and n to each take on only two values. Fortunately, numerical results offer broader evidence that categorical transfers reduce the income tax rate. 6.2 Numerical Results In Table 1, the tax rate is lower in the categorical optimum in every interior-solution case. 2 The decline is very small when group membership is weakly correlated with ability, but it is somewhat larger when the correlation is higher ~d is larger or z is closer to.5!. The conclusion that categorical transfers reduce the income tax rate is robust to parameter variation. Extensive calculations for alternative values of s, e, R, A, z, and d, which reveal a tax-rate decline in every interior-solution case, are available from the author upon request. Calculations in Viard ~2001! and Stern ~1982! show similar tax-rate declines. 7. Conclusion and Extensions Categorical transfer payments linked to ~nearly! exogenous characteristics are an important part of current tax-transfer systems. I analyze such transfers as a limited form of lump-sum redistribution. Optimal categorical transfers equalize the social marginal value of income across groups, while first-best transfers equalize this value across individuals. The two instruments treat between-group, but not within-group, inequality similarly. The comparison of these instruments clarifies existing and new results about optimal categorical transfers, including effects on the income tax rate and utility and labor supply and the implications of the planner s inequality aversion. Several extensions are possible. The policy instruments can be expanded to allow group-specific nonlinear income tax schedules, as in Immonen et al. ~1998!, or excise taxes, as in Ebrahimi and Heady ~1988! and Deaton and Stern ~1986!. Preferences or needs could vary and be correlated with the characteristics, as in Immonen et al. ~1998!, Blackorby and Donaldson ~1994!, Ebrahimi and Heady ~1988!, and Deaton and Stern ~1986!. The similarity of first-best and categorical transfers is likely to still hold. A natural extension is to allow characteristics to be slightly price sensitive ~such as disability classification and household composition!, as in Ebrahimi and Heady ~1988!, Kanbur ~1987!, Blomquist ~1984!, Roberts 2 However, the marginal tax rate rises in one of the three corner-solution cases. At a corner solution, m a~d!. a. a~a!, T where the latter two quantities are computed using the right-side derivative of V with respect to y at zero. Since m does not equal a, the tax-rate first-order condition cannot be written in the form ~3!, categorical transfers do not eliminate between-group inequality, and raising the tax rate is the only way to increase the disadvantaged group s transfer.

16 498 Journal of Public Economic Theory ~1984!, and Akerlof ~1978!. If characteristics are price elastic, the optimal covariance of categorical transfers with the social marginal value of income is not zero and depends upon the elasticity. If the elasticity is small, the covariance is small, and the above analysis is approximately valid ~Blomquist 1984!. In dynamic models, it is necessary to distinguish exogenous and predetermined characteristics. Revesz ~1989! and Roberts ~1984! demonstrate that there is little long-run welfare gain from taxing the latter ~such as past earnings or education! and the planner should precommit to not tax them. Recognition of the parallels to first-best redistribution can undoubtedly yield further results on the role of categorical transfer payments in an optimal redistributive system. Appendix Proof of Theorem 6: To conserve notation, let #[*x~n!f~n!dn and M # [ *x~n!f j ~n!dn, so that the M operator returns the crosssectional mean of a variable in the overall population and the M j operator returns the corresponding mean for group j. Consider a set of problems in which the tax rate t and the population-wide mean transfer G are chosen, subject to the standard budget constraint G R and to a set of constraints of the form G ~ j! G D~ j!, where D~ j! are exogenously specified parameters with (j p ~ j!d~j! 0. The uniform optimum is the special case in which the D parameters are all zero, and the categorical optimum is the special case in which they equal the differences of each group s optimal transfers from the population-wide mean. Since L y is zero, the tax-rate first-order condition ~3! simplifies to FOC [ y tn 2 L w # y nl# 0. Consider a small perturbation to the D parameters that satisfies (j p ~ j!dd~ j! 0. ~]FOC0]D~ j!!dd~ j!# ~]FOC0]t!dt ~]FOC0]G!dG 0. From the budget constraint, dg dt tn 2 L w #%. Therefore, dt ( ~]FOC0]D~ j!!dd~ j! j ~]FOC0]t! tn 2 L w #. The second-order condition for an interior maximum states that the denominator of this expression is negative. Therefore, sign~dt! sign (j ~]FOC0]D~ j!!dd~ j!. With V yy constant and L y equal to zero, the first two terms of FOC are unaffected by any perturbation for which (j p ~ j!d~j! 0. Differentiating the third term of FOC yields ( ~]FOC0]D~ j!!dd~ j! ( p~ j!m j!. j j

17 Optimal Categorical Transfer Payments 499 In words, the marginal tax rate is reduced by any perturbation ~or series of perturbations! that increases transfer differentials for groups with low earnings and reduces them for groups with high mean earnings. Under Assumption 1 ', groups mean earnings are monotonically increasing with their ability. Transfer differentials are zero at the uniform optimum and are monotonically decreasing with group s ability ~under Theorem 1, which remains valid with L y 0!. Therefore, the move from the uniform optimum to the categorical optimum reduces the marginal tax rate. References AKERLOF, G. A. ~1978! The economics of tagging as applied to the optimal income tax, welfare programs, and manpower planning, American Economic Review, 68, ATKINSON, A. B. ~1995! Public Economics in Action: The Basic Income0Flat Tax Proposal. Oxford: Clarendon Press. ATKINSON, A. B., and J. E. STIGLITZ ~1980! Lectures on Public Economics. New York: McGraw-Hill. BEATH, J. A., G. W. LEWIS, and D. T. ULPH ~1988! Policy targeting in a new welfare framework with poverty; in Surveys in Public Sector Economics, P. G. Hare, ed., Oxford: Basil Blackwell. BENNETT, J. ~1987! The second-best lump-sum taxation of observable characteristics, Public Finance0Finances Publiques, 42, BLACKORBY, C., and D. DONALDSON ~1994! Information and intergroup transfers, American Economic Review, 84, BLOMQUIST, N. S. ~1981! A comparison of tax bases for a personal tax, Scandinavian Journal of Economics, 83, BLOMQUIST, N. S. ~1984! The wage rate tax: An alternative to the income tax? Scandinavian Journal of Economics, 86, DEATON, A., and N. STERN ~1986! Optimally uniform commodity taxes, taste differences, and lump-sum grants, Economics Letters, 20, DIAMOND, P. A. ~1975! A many-person Ramsey rule, Journal of Public Economics, 4, DIAMOND, P. A. ~1998! Optimal income taxation: An example with a U-shaped pattern of optimal marginal tax rates, American Economic Review, 88, DIAMOND, P. A., and E. SHESHINSKI ~1995! Economic aspects of optimal disability benefits, Journal of Public Economics, 57, EBRAHIMI, A., and C. HEADY ~1988! Tax design and household composition, Economic Journal, 98, supplement, HELLWIG, M. F. ~1986! The optimal linear income tax revisited, Journal of Public Economics, 31,

18 500 Journal of Public Economic Theory HELPMAN, E., and E. SADKA ~1978a! Optimal taxation of full income, International Economic Review, 19, HELPMAN, E., and E. SADKA ~1978b! The optimal income tax: Some comparative statics results, Journal of Public Economics, 9, IMMONEN, R., R. KANBUR, M. KEEN, and M. TUOMALA ~1998! Tagging and taxing: The use of categorical and income information in designing tax0 transfer schemes, Economica, 65, KANBUR, R. ~1987! Transfers, targeting, and poverty, Economic Policy, 4, , KANBUR, R., and M. KEEN ~1989! Poverty, incentives, and linear income taxation; in The Economics of Social Security, A. DILNOT and I. WALKER, eds., Oxford: Oxford University Press. KEEN, M. ~1992! Needs and targeting, Economic Journal, 102, MIRRLEES, J. A. ~1986! The theory of optimal taxation; in Handbook of Mathematical Economics, vol. 3, K. J. ARROW and M. D. INTRILIGATOR, eds., Amsterdam: North-Holland. PARSONS, D. O. ~1996! Imperfect tagging in social insurance programs, Journal of Public Economics, 62, RAVALLION, M., and K. CHAO ~1989! Targeted policies for poverty alleviation, Journal of Policy Modeling, 11, REVESZ, J. T. ~1989! The optimal taxation of labor income, Public Finance0Finances Publiques, 44, ROBERTS, K. ~1984! The theoretical limits to redistribution, Review of Economic Studies, 51, SEN, A. ~1974! Informational bases of alternative welfare approaches: Aggregation and income distribution, Journal of Public Economics, 3, SOSNOW, N. D. ~1974!, Optimal policies for income redistribution, Journal of Public Economics, 3, STERN, N. H. ~1976! On the specification of models of optimum income taxation, Journal of Public Economics, 6, STERN, N. H. ~1982! Optimum taxation with errors in administration, Journal of Public Economics, 17, STIGLITZ, J. E. ~1987! Pareto efficient and optimal taxation and the new new welfare economics; in Handbook of Public Economics, vol. 2, A. J. AUERBACH and M. FELDSTEIN, eds., Amsterdam: North-Holland. TUOMALA, M. ~1990! Optimal Income Tax and Redistribution. Oxford: Clarendon Press. VIARD, A. D. ~2001! Some results on the comparative statics of optimal categorical transfer payments, Public Finance Review, 29,