OPTIMAL CATEGORICAL TRANSFER PAYMENTS: THE WELFARE ECONOMICS OF LIMITED LUMP-SUM REDISTRIBUTION

Transcription

1 OPTIMAL CATEGORICAL TRANSFER PAYMENTS: THE WELFARE ECONOMICS OF LIMITED LUMP-SUM REDISTRIBUTION A central component of most countries' tax-transfer systems is the provision of transfer payments to categories of individuals defined by exogenous (or nearly exogenous) characteristics, such as date of birth or disability. As a result, individuals net taxes or transfers usually depend partly upon their labor earnings and commodity purchases and partly upon these exogenous characteristics. Despite their prevalence, however, the role of these categorical transfer payments in an optimal redistributive system has received only limited attention in the literature, since most analyses assume that the social planner uses only uniform transfers, along with taxes on labor earnings and commodity purchases. In this paper, I provide a general framework for the analysis of categorical transfers, by linking them to the theory of first-best lump-sum redistribution. Because the use of categorical transfer payments allows individuals net taxes or transfers to partly depend upon exogenous characteristics, they constitute a limited form of lump-sum redistribution. In a first-best economy, where the planner directly observes individuals exogenous earnings abilities, the planner eliminates inequality in the social marginal value of income without excess burden, by using lumpsum taxes and transfers based on ability. Categorical transfers similarly allow the planner to eliminate (without excess burden) one component of inequality in the social marginal value of income, the component that is correlated with exogenous observable characteristics. Due to this similarity, it is useful to analyze categorical transfers by reference to the well-established results on first-best redistribution. I consider an economy in which the planner wishes to redistribute from individuals with high exogenous wage rates to individuals with low wage-rates, but cannot observe wage rates. In the standard optimal-income-tax problem, the planner observes and taxes only labor earnings and commodity purchases. In the constrained optimum, the planner uses these distortionary taxes to reduce inequality in the net social marginal value of income, but does not eliminate inequality because of the taxes' efficiency costs. I introduce categorical transfers by assuming that the planner can also observe exogenous characteristics that are correlated (perhaps only weakly) with wage rates. In the new optimum, the planner,

2 in addition to taxing earnings and purchases, provides categorical transfers that vary across groups with different exogenous observable characteristics, paying larger transfers to groups with generally lower wage rates. Optimal categorical transfers equalize the mean net social marginal value of income across the different groups, thereby eliminating the between-group component of inequality. Distortionary taxes are still used to reduce, but not eliminate, inequality within each of the exogenously defined groups. Optimal categorical transfers therefore address between-group, but not within-group, inequality in the same qualitative manner as first-best taxes and transfers. I describe additional similarities between categorical transfers and first-best redistribution, presenting both analytical results and numerical results from a variation of Stern's (1976) classic linear-income-tax calculations. I find that optimal categorical transfers and first-best policies result in quantitatively similar amounts of between-group redistribution. Also, while first-best redistribution reduces the optimal income tax rate to zero because it fully accomplish the planner's distributional goals, optimal categorical transfers reduce the optimal income tax rate (but not to zero) because they partially achieve the planner's distributional goals. While first-best taxes and transfers induce individuals with higher wages to provide more labor and to have lower utility than lowwage individuals, optimal categorical transfers induce similar patterns in the between-group variation of labor supply and utility with wage rates. Just as an increase in the planner's inequality aversion counterintuitively reduces the magnitude of first-best redistribution, it also reduces the use of optimal categorical transfers, even as it increases the optimal income tax rate. The paper is organized as follows. In Section I, I review the effect of categorical transfers on the linear-income-tax problem. In section II, I establish the fundamental economic similarity of optimal categorical transfers and the first-best tax-transfer policy. In section III, I further analyze the similarity of these policies. I discuss extensions in section IV and briefly conclude in Section V. 2

3 I. CATEGORICAL TRANSFERS IN THE LINEAR-INCOME-TAX MODEL In this section, I examine the effect of categorical transfers on the optimal-income-tax problem. For simplicity, I analyze the effects of adding categorical transfers to a linear tax on labor earnings. As noted by Atkinson and Stiglitz (1980, p. 372), Ebrahimi and Heady (1988, p. 84), Myles (1995, pp , ), and other authors, a linear tax on labor earnings is equivalent to a proportional tax on consumption, so that the linear-income-tax model is a special case of the linear-commodity-tax model, with uniform tax rates on commodities. At the cost of some complexity, it would be possible to analyze the introduction of categorical transfers into an economy with nonlinear taxes or differentiated commodity taxes, as in Deaton and Stern (1986) and Ebrahimi and Heady (1988). As discussed below, the same fundamental result concerning the equalization of the mean social marginal value of income arises in all of these contexts. I.1 Economic Specification I consider a specification similar to that of Diamond (1975), Atkinson and Stiglitz (1980, pp , ), Auerbach (1985, pp ), Hellwig (1986), Tuomala (1990, pp ), Myles (1995, pp ), and other authors. A continuum of individuals exist, each of whom has a utility function increasing in Error! Switch argument not specified.her consumption Error! Switch argument not specified.c and decreasing in her labor supply L, both of which must be non-negative. In the no-tax equilibrium, each individual's exogenous incomeerror! Switch argument not specified. y equals zero and each individual's wage rate w equals a non-negative parameter n that varies across individuals, with crosssectional density f(n). The planner observes this density function, but cannot observe individual values of n. Following Hellwig (1986), I impose the following assumptions, where subscripts on C and L denote derivatives of the ordinary consumption demand and labor supply functions: 3

4 (A1) All individuals have the same quasi-concave twice-differentiable utility function.4 (A2) Leisure is a normal good, L y < 0 if L > 0. (A3) Consumption is an increasing function of the wage rate, C w > 0, if L > 0.5 (A4) The planner maximizes the sum of a strictly concave representation of the common utility function, where this representation is defined only up to an increasing linear transformation. Letting V(w,y) denote the indirect utility function corresponding to this representation and letting subscripts denote derivatives, V y is the increase in social welfare from an individual's additional consumption and concavity implies V yy < 0. I.2 Linear Income Tax with Uniform Transfer I first review the standard linear-income-tax model without categorical transfers. The planner imposes a linear tax on labor earnings with marginal rate t and pays a uniform lump-sum transfer G (which can be interpreted as a fully refundable income-tax exemption equal to G/t). The value of G may be either positive or negative (the latter would imply a poll tax), but a positive value is generally optimal for the economies studied in the literature. The planner maximizes (1) V ( n, t, G) f ( n) dn [ R + G - t nl( n, t, G) f ( n) dn] µ, where R is an exogenous per-capita revenue requirement (net of transfer payments), µ is the shadow price of the planner's budget constraint, and V(n,t,G) is V(w,y) evaluated for w equal to n(1-t) and y equal to G. The first-order condition governing the uniform transfer is V + µ ntl f ( n) dn = µ. (2) [ y y ] Following Diamond (1975, p. 338), Atkinson and Stiglitz (1980, p. 387), Auerbach (1985, pp. 88, 107), Mirrlees (1986, p. 1226), Stiglitz (1987, p. 1016), Ebrahimi and Heady (1988, p. 86), Atkinson (1995, p. 32), and Myles (1995, p. 111), define each individual's "net social marginal value of income" α to be V y + µtnl y. Note that α measures the net social welfare impact of increasing an individual's income, including the shadow value of the revenue change due to her labor supply response. The first-order condition (2) can be rewritten as, 4

5 (3) α = µ, where α α( n) f ( n) dn 6is the population mean of α. The first-order condition for the marginal tax rate on labor earnings t is VynLf ( n) dn = µ n L + tnlw f ( n) dn. (4) ( ) Letting subscripts on $ L denote derivatives of the compensated (Hicksian) labor supply function and using the Slutsky decomposition $ L w = L w - L y L, the definition of α, and equation (3) yields nlf n dn nlf n dn (5) t = α ( ) α ( ) α nl$ f ( n) dn w. Note that the numerator of (5) is the cross-sectional covariance of the net social marginal value of income α and pre-tax labor earnings nl. Since the Hicksian labor supply derivative is positive,7 the optimal income tax rate has the opposite sign from this covariance. Diamond (1975) provided the economic interpretation of this first-order condition. If α covaries negatively (positively) with earnings, so that the planner wishes to redistribute from high-earnings individuals to low-earnings individuals (from low-earnings individuals to high-earning), the marginal income tax rate is positive (negative) to achieve the desired redistribution. Hellwig (1986) demonstrated that assumptions (A1) through (A4) actually permit the more definite statements that the covariance in the numerator of (5) is negative and the marginal tax rate is positive. A prominent finding in the optimal-income-tax literature is the limited effectiveness of income taxation as a redistributive instrument. Mirrlees (1971, p. 208) expressed surprise that the optimal income tax rates he calculated were relatively low. Although other authors found higher tax rates by assuming less elastic labor supply and greater inequality aversion, they also confirmed that the optimal income tax achieves much less redistribution than first-best taxes and transfers, while imposing significant efficiency costs. Atkinson (1995, pp. 9-14) provided a general discussion of this "redistribution pessimism." 5

6 However, this linear-income-tax framework overstates the severity of the equity-efficiency tradeoff confronting actual tax-transfer systems, because these systems also include transfers paid to recipients in categories defined by exogenous (or nearly exogenous) observable characteristics. Each individual's (or household's) net tax or transfer is a function of both labor earnings and these characteristics. Benefits and taxes often depend upon date of birth and disability, and sometimes gender or race. For example, of the $916 billion of fiscal 1997 federal entitlement spending in the United States, $601 billion goes to Social Security, Medicare, and Supplemental Security Income, which are limited to disabled and elderly recipients, U.S. Congressional Budget Office (1997, p.36). Ebrahimi and Heady (1988, p. 94) and Atkinson (1995, pp ) noted the importance of categorical transfers in the United Kingdom's taxtransfer system. Atkinson and Stiglitz (1980, pp. 362, ), Stern (1982, p. 182), Roberts (1984, p. 178), Mirrlees (1986, pp. 1199, 1215), Deaton and Stern (1986, p. 263), and Revesz (1989, pp. 462, 464, 468) made similar observations. I.3 Linear Income Tax with Categorical Transfers To analyze categorical transfers, I relax the assumption that the lump-sum transfer G is uniform, by allowing the planner to distinguish J nonoverlapping groups of individuals on the basis of exogenous observable characteristics. Each group j has a distinct distribution of earnings opportunities with density f j (n). By definition, Σ j f j (n)p(j) equals f(n), where p(j) is the fraction of the population in each group. I assume that the planner observes the wage-rate density function for each group. I now assume that the characteristics distinguishing the groups are fully exogenous, but I discuss in section IV the possibility that they may respond to incentives. The planner chooses a distinct transfer G(j) to the members of each group j. For simplicity, I assume that the marginal income tax rate t remains uniform. Each individual's net tax payment equals tnl-g(j) and therefore depends on both labor earnings and exogenous characteristics. The planner maximizes 6

7 (6) p ( j ) V n t G j f n dn - [ R p j G(j) - t p j nl n t G j f n dn j (,, ( )) j ( ) + ( ) ( ) (,, ( )) j j j ( ) ] µ, where V(n,t,G(j)) equals V(w,y), evaluated for w equal to n(1-t) and y equal to G(j). The first-order condition for the marginal tax rate on labor earnings t is the same first-order condition (4) that arose with uniform transfers, as noted by Deaton and Stern (1986, p. 265, eq. 11). The first-order conditions governing the categorical transfers G(j) are Vy + µ ntly f j ( n) dn = µ, j, or, from the definition of α, (7) [ ] (8) α ( j) = µ, j, where α( j) α( G( j), n) f ( n) dn 8is the mean value of α for group j. Since (8) holds for each group, the j population mean α 9 also equals µ, equation (3) remains valid, which implies that the marginal-tax-rate first-order condition can still be rewritten in the form of equation (5). The form of the first-order condition (8) is also unchanged with non-linear income taxes or differentiated excise taxes, as considered by some earlier papers. Various forms of the first-order conditions (8) have been derived by Mirrlees (1986, p. 1215, eq. (3.32)), Deaton and Stern (1986, p. 265, eq. 10), Bennett (1987, p. 230, eq. 6), Ebrahimi and Heady (1988, p. 85, eq. 5), Diamond and Sheshinski (1995, p. 6), and Parsons (1996, p. 192, eq. 13), and informally discussed by Atkinson (1995, pp.59-60). Also, Bernard (1977, p. 373) and Dasgupta and Stiglitz (1981, p. 90, eq. 1.19) derived a special version of (8) as a necessary condition for an optimal tax rate of less than 100-percent on exogenous pure profits. The first-order conditions (8) state that all groups' mean net social marginal values of income are equated to the shadow value of government revenue and therefore to each other. If two groups' mean net social marginal values of income differ, increasing the transfer to the group with higher mean net social marginal value of income, financed by a budget-balancing reduction in the other group's transfer, would 7

8 increase social welfare. Optimal categorical transfers therefore eliminate between-group inequality in the net social marginal value of income. 8

9 II. CATEGORICAL TRANSFERS AND FIRST-BEST REDISTRIBUTION Because categorical transfers linked to exogenous characteristics offer a limited opportunity to redistribute without efficiency cost, it is useful to compare them to the unlimited lump-sum redistribution that would occur in the first-best, where the planner directly observes individual wage rates. To make this comparison, I briefly review the theory of first-best redistribution. II.1 First-Best Redistribution The properties of the first-best tax-transfer system have been analyzed by a number of authors, including Mirrlees (1971, p. 201), Allingham (1975), Sadka (1976), Helpman and Sadka (1978a), Atkinson and Stiglitz (1980, p. 421), Blomquist (1981), Mirrlees (1986, pp ), and Stiglitz (1987, p ). In the first-best problem, the planner observes the value of n for each individual and specifies a transfer schedule G(n), with negative values of G denoting a tax. The planner maximizes (9) V ( n) f ( n) dn - µ [ R + ng( n) f ( n) dn] where V(n) equals the indirect utility function V(w,y), evaluated for w equal to n and y equal to G(n). The first-order conditions governing G(n) are (10) V y (n) = µ, n, so that the social marginal value of income at all wage rates are equated to the shadow value of the budget constraint and hence to each other. (Note that, since t equals zero, α simply equals V y ). Equation (10) states the fundamental result of the theory of first-best redistribution, noted in all of the papers cited above. A comparison of (10) to the first-order condition (8) for optimal categorical transfers establishes the fundamental economic similarity between first-best redistribution and categorical transfers. 9

10 II.2 Elimination of Between-Group Inequality One way to see this similarity is to decompose the population-wide variance of the net social marginal value of income into its within-group and between-group components, 2 j j 2, j j j (11) p( j) [ α( j, n) α] f ( n) dn = p( j) [ α( j, n) α( j)] f ( n) dn + p( j)[ α( j) α] The first term on the right-hand side is the within-group variance, while the second term is the betweengroup variance, arising from differences of the groups' means. First-best taxes and transfers equalize α across all individuals, so that both terms on the right-handside of (11) are zero. Since optimal categorical transfers equalize the mean value of α across groups, the second term on the right-hand side of (11) is zero, but the first term remains positive. With optimal categorical transfers, the between-group variance in the social marginal value of income is removed, but the within-group variance remains, since it can be addressed only by the distortionary income tax. II.3 Numerical Calculations I now investigate whether the qualitative similarity of the two policies' treatment of between-group inequality also results in a quantitative similarity, by presenting numerical results for an economy similar to that in Stern's (1976) classic linear-income-tax calculations. The planner maximizes the sum of the following representation of individuals' utilities, 1 ( ε 1) / ε ( ε )/ ε (12) = [ + 1 ] 1 ε 1 V ( n, j) p( j) C( n, j) k( L( n, j)) f j ( n) dn, ν j where ε > 0 is the elasticity of substitution between consumption and leisure in each individual's constantelasticity-of-substitution (CES) utility function, k > 0 is a parameter affecting the propensity to consume leisure, and ν10 < 1 is a parameter reflecting the curvature of the social welfare function. As discussed below, algebraically lower values of ν 11imply greater inequality aversion. Equations (A1) and (A2) in the appendix are the associated labor supply and consumption demand functions. υε 10

11 I assume that the population consists of two exogenously defined groups ("advantaged" and "disadvantaged"), with population shares z and 1-z, respectively. Each of the two groups has lognormal wage-rate distributions with the same standard deviation, but different means. Letting d denoting the difference in the group means of the log wage rate and σ denote the population-wide standard deviation of the log wage rate, the standard deviation of the log wage rate within each group equals σ 2 z( 1 z) d. Aside from the leisure parameter k, the economy is characterized by six parameters; the populationwide standard deviation of the log wage rate σ, the difference in the groups' mean log wage rates d, the size of the advantaged group z, the elasticity of substitution ε, the net revenue requirement R, and the inequalityaversion parameter ν. Throughout the paper, I report results for a benchmark parameterization, and for 19 alternative parameterizations in which a single parameter differs from its benchmark value. The alternatives include four values of inequality aversion, four values of the between-group mean-wage-rate difference, two values of the group size, three values of the overall standard deviation, four values of the elasticity of substitution, and two values of the net revenue requirement. In practice, characteristics that are reasonably close to being exogenous, such as date of birth, clearly observable disability, race, and gender, are only weakly correlated with wage rates, with some low-wagerate individuals present in all groups, including those with high mean wage rates. Therefore, I choose benchmark parameters that yield a low correlation between group characteristics and wage rates. I set σ at.39, following Mirrlees (1971) and Stern (1976), and I set z at.88 and d at.25, which roughly match the proportion of whites in the United States population and the difference in the median annual earnings of white and nonwhite full-time full-year workers. The correlation between the log wage rate and membership in the advantaged-group, z( 1 z) d / σ, is only.208. I consider alternative values of.2,.6, and 1 for σ, alternative values of.0625,.125,.5, and 1 for d, and alternative values of.12 and.5 for z. Without loss of generality, I normalize the mean wage rate to equal unity. 11

12 For simplicity, I set the benchmark net revenue requirement R equal to zero, so that taxes are purely redistributive, with all income tax revenues used to finance transfers rather than public goods. I consider alternative values for R of.1 and.2 (approximately one-sixth and one-third of output, respectively). I set the benchmark inequality-aversion parameter ν equal to -1, following Stern (1976, p. 141), and consider alternative values of -5, -2, 0, and.5. I discuss the effects of changing inequality aversion in detail in section III.3 below. I set the benchmark elasticity of substitution ε equal to.5, following Stern (1976) and Blomquist (1981), and consider alternative values of.1,.2..8, and 2. I set the leisure parameter so that k ε always equals.5, which implies, from equations (A1) and (A2) in the appendix, that an individual with the meanwage rate unity chooses labor supply and consumption of 2/3 in the no-tax equilibrium. From equations (A5) and (A6) in the appendix, the compensated labor-supply elasticity for this individual in the no-tax equilibrium is ε/3 and the corresponding uncompensated labor-supply elasticity is (ε-1)/3. In Table 1, I present optimal tax rates and transfers for this model economy. For the familiar incometax optimum in which the transfer is uniform ("the uniform optimum"), Table 1 reports the marginal tax rate on labor earnings and the uniform transfer. For the optimum in which transfers differ across the two groups (the "categorical optimum"), Table 1 reports the marginal tax rate on labor earnings, the transfer to members of the advantaged group G(A), and the transfer to members of the disadvantaged group G(D). Although wage rates and group membership are only weakly correlated under the benchmark parameterization and most alternatives, it is optimal to pay a significantly larger transfers to members of the disadvantaged group. In Viard (1997), I examine comparative statics and analyze the social welfare gain from the introduction of categorical transfers, finding that the variance of groups' transfers and the welfare gain are roughly proportional to between-group wage-rate variance. 12

13 I now compare categorical transfers to first-best redistribution in the CES economy, using the firstbest tax-transfer schedule given by equation (A8) in the appendix. In Table 2, I report the net transfers (as a fraction of output) paid to members of the disadvantaged group in the uniform optimum, the categorical optimum, and the first-best. Categorical transfers and first-best redistribution have similar quantitative, as well as qualitative, implications for between-group redistribution. Under each parameterization, the fraction of output transferred to the disadvantaged group in the categorical optimum is very similar to, but slightly smaller than, the fraction transferred in the first-best. Of course, the volume of within-group redistribution (not reported) is much smaller in the categorical optimum than in the first-best. Under each parameterization, between-group net transfers in the uniform optimum are much smaller than transfers in the categorical optimum and the first-best. These analytical and numerical results can be summarized in the following proposition, Proposition 1: Optimal categorical transfers and first-best taxes and transfers both eliminate between-group variation in the net social marginal value of income and both policies transfer similar fractions of output between groups. II.4 Covariance with Net Social Marginal Value of Income The first-order conditions (8) and (10) imply another similarity between optimal categorical transfers and first-best redistribution. First-best taxes and transfers have zero cross-sectional covariance with the social marginal value of income, since the latter quantity is equalized across individuals. Similarly, categorical transfers have zero cross-sectional covariance with the net social marginal value of income, since the transfers vary across groups with the same mean marginal value. In each case, this zero covariance implies that there are no distributionally beneficial budget-feasible adjustments around the optimum. Note that, since adjustments of lump-sum transfers (whether first-best or categorical) have no efficiency effects, any distributionally beneficial adjustments would be welfare-improving, which cannot be true for budget-feasible perturbations around the optimum. 13

14 In contrast, equation (5) implies that the cross-sectional covariance between income tax payments17 tnl and the net social marginal value of income α is negative, since the marginal tax rate has the opposite sign of the cross-sectional covariance between labor earnings and α. This negative covariance implies that an increase in the marginal income tax rate financed by a balanced-budget reduction in all groups' transfers would be distributional beneficial. However, such an increase (like all budget-feasible perturbations around the optimum) would not increase social welfare, because the distributional benefits would be offset by efficiency costs. This is an example of Sosnow's result (1975, pp ) that, at a constrained optimum, policies with marginal efficiency costs (benefits) must have marginal distributional benefits (costs), while policies with no marginal efficiency effects must be distributionally neutral on the margin. These results can be summarized in the following proposition, Proposition 2: Optimal categorical transfers and first-best taxes and transfers each have zero covariance with the net social marginal value of income. II.4 Relationship of Transfers to Wage Rates To further analyze first-best redistribution, I use Roy's Identity, V w = V y L and the equality (from application of the chain rule to Roy's Identity) V wy = V yy L + V y L y. Under assumptions (A2) and (A4), therefore, V wy is non-positive and is strictly negative if labor supply is positive. Since first-best transfers equalize V y across all values of n, they must satisfy G n = - V yw /V yy., which yields the rule stated by Helpman and Sadka (1978a, p. 248), (13) G n = - L - (L y V y /V yy ). Under assumptions (A2) and (A4), this expression is negative if labor supply is positive and is zero if labor supply is zero. In the first-best, individuals with higher wage rates receive smaller transfers or pay higher taxes (except that equal transfers are paid to any individuals who do not work). This property can be verified for the CES case, using equation (A8) in the appendix. This occurs because, under the concavity 14

15 assumption (A4), V y is a decreasing function of n in the no-tax equilibrium, so that the planner wishes to redistribute from high-wage-rate individuals to low-wage-rate individuals. Intuition similarly suggests that the planner will provide larger categorical transfers to groups with generally lower wage rates. Although the significance of each moment of the groups' wage-rate distributions depends on the functional forms of individual preferences and social welfare, a partial ordering of optimal categorical transfers can be obtained by imposing one more restriction on preferences, (A5) The marginal propensity to spend on leisure -L y w is non-decreasing in the net wage rate and in exogenous income. THEOREM: Under assumptions A1 through A5, if the wage-rate distribution for any group j dominates that for any group k,18 F j (n)<f k (n) for all n, where F is the cumulative distribution function, then, at the optimum, members of group j receive a smaller transfer than members of group k. PROOF: See Appendix. These results can be summarized in the following proposition, Proposition 3: In the first-best, individuals with higher wage rates receive smaller transfers and optimal categorical transfers are smaller for groups with dominating wage-rate distributions. III. Additional Similarities of Categorical Transfers and First-Best Redistribution In section II, I established the fundamental economic resemblance and the quantitative similarity of optimal categorical transfers and first-best redistribution with respect to between-group inequality. In this section, I discuss additional similarities between optimal categorical transfers and first-best redistribution, including their effects on the use of distortionary taxes and between-group differences in labor supply and utility, and comparative statics with respect to the planner's inequality aversion. III.1 Use of Distortionary Taxes 15

16 I first compare the effects of categorical transfers and first-best redistribution on the magnitude of the income tax rate. Since the Second Fundamental Theorem of Welfare Economics implies that the income tax rate is zero in the first-best, I did not include the tax rate as a choice variable in section II.1 above. However, introducing the income tax rate as a choice variable continues to yield a first-order condition of the form (5). In fact, this first-order condition arises whenever the government sets µ equal to α, including the uniform-transfer, categorical-transfer, and first-best cases. Differences in the value of the optimal marginal tax rate are attributable to changes in the endogenous quantities entering equation (5). In the first-best, equation (5) confirms that the optimal income tax rate is zero, because the crosssectional covariance in the numerator of (5) is zero. The social marginal value of income is equalized across individuals and necessarily has zero covariance with labor earnings. Since lump-sum taxes and transfers fully achieve the planner s distributional objectives, the income tax has no distributional benefit to offset its efficiency cost. To analyze the effect of categorical transfers, I decompose the covariance between labor earnings and the net social marginal value of income into its within-group and between-group components, (14) j p( j) [ α( j, n) α][ nl( n, j) E ] f ( n) dn j, = p( j) [ α( j, n) nl( n, j) α ( j) E ( j)] f ( n) dn + p( j)[ α( j) α][ E ( j) E ]] j j j where E denotes the mean value of nl in the population and E (j) denotes the mean value of nl for group j. In the first-best, both terms on the right-hand side of (14) equal zero. With optimal categorical transfers, the second term equals zero, but the first term remains positive. Categorical transfers eliminate the between-group covariance of α and nl by eliminating the between-group variance of α. Heuristically, since the use of categorical transfers eliminates the between-group component of the covariance in the numerator of (5), it should reduce the absolute value of that covariance and thereby reduce the magnitude of the marginal tax rate. The planner should have less need for the income tax, 16

17 because categorical transfers remove the between-group component of the inequality that the income tax is designed to alleviate. Unfortunately, this heuristic argument cannot be analytically verified, since the use of categorical transfers also changes the within-group covariance and the various endogenous quantities entering the denominator of equation (5) and these changes could, in principle, cause a "perverse" increase in the optimal marginal rate. 1 (However, Akerlof (1978, p. 14) analytically demonstrated that the marginal payoff to work rises when "tagged" categorical transfers are introduced into his model, under the simplifying assumption that earnings ability can take on only two values). The effect of categorical transfers on the marginal income tax rate must be determined numerically. Table 1 reveals that the marginal tax rate declines when categorical transfers are introduced, under each parameterization, a result that also holds in other parameterizations not reported here. In Table 3, I compare the marginal tax rate, the excess burden (measured as the sum of individuals' equivalent variations minus net revenues), and mean consumption (for the overall population) for the uniform optimum and the categorical optimum. Mean consumption rises when categorical transfers are introduced, as the lower marginal tax rate induces individuals to substitute consumption for leisure (the income effect of changing the marginal tax rate is roughly offset by the income effect of changing the transfers, consistent with the analysis of Sandmo (1983)). In Viard (1997), I analyze the comparative statics relating the magnitude of the declines in the marginal rate and excess burden to the parameters of the problem. These results can be summarized in the following proposition, Proposition 3: First-best taxes and transfers reduce the optimal income tax rate to zero, while optimal categorical transfers reduce the optimal income tax rate, but not to zero. III.2 Relationship of Labor Supply and Utility to Wage Rates 1 The same difficulties preclude analytical proof of the conjecture that a reduction in wage-rate inequality lowers the marginal rate, a conjecture also supported by intuition and numerical results. See Helpman and Sadka (1978, pp ). 17

18 Allingham (1975, p. 374), Sadka (1976, pp. 932, 934), and other authors have noted a clear-cut result concerning the relationship between wage rates and labor supply in the first-best. With first-best taxes and transfers set in accordance with (13), the variation of labor supply with wage rates is given by (15) dl/dn = L n + L y G n = L n - L y L - V y (L y ) 2 /V yy = L $ n - V y (L y ) 2 /V yy, where the last equality follows from the Slutsky decomposition. From the non-negativity of the Hicksian derivative and assumptions (A2) and (A4), this expression is positive if labor supply is positive. In the first-best, labor supply is an increasing function of the wage rate, except that individuals in a bottom interval of the wage-rate distribution may all not work. This property can be verified for the CES case, using equation (A9) in the appendix. Previous authors have provided an economic explanation of this result. The first-best policy equalizes marginal utility of the consumption good, since all individuals face the same price for consumption, but does not equalize the marginal utility of leisure, since the opportunity cost of leisure, the wage rate, varies across individuals. Instead, it is efficient for high-wage-rate individuals to work more, because they are more productive. Therefore, transfers decline (or taxes rise) sufficiently rapidly as wage rates rise so that higher-wage individuals choose greater labor supply. I now investigate whether optimal categorical transfers similarly induce higher mean labor supply for groups with higher mean wage rates. In Table 4, I report mean labor supply for members of the advantaged and disadvantaged groups and the log difference of these means, for the uniform optimum, the categorical optimum, and the first-best. The two groups have roughly equal mean labor supply in the uniform optimum, under each parameterization, but the mean labor supply of the advantaged group is greater than that of the disadvantaged group in the categorical optimum, as well as in the first-best. Also, under each parameterization, the log difference of the two groups' mean labor supplies in the categorical optimum is similar to, but somewhat larger than, the corresponding difference in the first-best. 18

19 These results can be summarized in the following proposition, Proposition 4: Individuals with higher wage rates have higher labor supply in the first-best, while groups with higher mean wages tend to have higher mean labor supply with optimal categorical transfers. Although the relationship between labor supply and wage rates in the first best is clear-cut, first-best policies may result in individuals with higher wage rates having higher, lower, or the same consumption as individuals with lower wage rates. However, the relationship of utility (which depends upon both consumption and labor supply) is clear-cut. When lump-sum taxes and transfers are set in accordance with (13), the variation in utility across individuals with different wage rates is given by 2 (16) dv / dn = V + V G = L V / V w y n y y yy. Under assumptions (A2) and (A4), this expression is negative if labor supply is positive and zero if labor supply is zero. In the first-best, utility declines with wage rates (except that all individuals not working have the same utility), as noted by Atkinson and Stiglitz (1980, p. 421), Blomquist (1981, pp ), Mirrlees (1986, p. 1212), Stiglitz (1987, p. 995), and others. Taxes and transfers that equalize marginal utility result in higher utility levels for individuals with low wage rates. I define a money-metric measure of each individual's utility (that is invariant to the planner's social welfare function) as the amount of consumption that would yield the individual's actual utility if the individual had no leisure (L=1). In Table 5, I report the median values of this money-metric utility measure for members of each group in the uniform optimum, the categorical optimum, and the first-best, using equation (A10) in the appendix. Under each parameterization, the advantaged group has higher median utility in the uniform optimum, which is a necessary implication of the group's higher median wage rate. However, under each parameterization except that in row 2, the advantaged group has lower median utility in the categorical optimum, as well as in the first-best. Furthermore, the log difference in the median utility levels is similar, but slightly smaller, than the difference in the first-best. 19

20 These results can be summarized in the following proposition, Proposition 5: In the first-best, individuals with higher wage rates have lower utility. With optimal categorical transfers, the distribution of utility levels for groups with higher wage rates tend to be less favorable than that for groups with lower wage rates. 20

21 III.3 Comparative Statics with Respect to Inequality-Aversion A final similarity between optimal categorical transfers and first-best redistribution concerns their comparative statics with respect to the planner's inequality aversion. As noted in assumption (A4), the representation of the individual utility functions whose sum the planner maximizes is defined only up to an increasing linear transformation. As noted by Hellwig (1976) and Helpman and Sadka (1978b), a strictly concave transformation of the utility function whose sum is maximized increases the planner's inequality aversion, by increasing the weight given to individuals with lower utility levels in the social welfare function. Helpman and Sadka also proved one of the few analytical results in the comparative statics of optimal linear income taxation, by demonstrating that such a strictly concave transformation increases the optimal marginal tax rate and the value of the optimal uniform transfer. The marginal income tax rate reaches its maximum value in the limiting maximin (Rawlsian) case, which arises as the absolute value of V yy increases without bound and the planner maximizes the welfare of the individual with lowest utility. However, the effects of greater inequality aversion on first-best lump-sum redistribution is quite different. Equation (13) indicates that the absolute value of G n, the rate at which transfers decrease (or taxes rise) as wage rates rise, is actually lower when the absolute value of V yy /V y. is higher, i.e., when the planner is more averse to inequality. This occurs because high-wage-rate individuals have lower utility in the first-best, so that a planner who places greater weight on the welfare of low-utility individuals actually treats high-wage-rate individuals more favorably. In the limiting maximin case, as the absolute value of V yy increases without bound, equation (13) reduces to G n = -L, as noted by Helpman and Sadka (1978a, p. 249). Equation (16) implies that utility is constant across wage rates in the maximin case, as noted by Sadka (1976, p. 932), Atkinson and Stiglitz (1980, p. 420), and Tuomala (1990, p. 55). Stiglitz (1987, p. 996) and Tuomala (1990, p. 55) similarly noted that high-wage-rate individuals are better off in the maximin case than under social welfare functions that are less inequality-averse. Atkinson and Stiglitz (1980, p. 422) noted that the difference between 21

22 income taxation and first-best redistribution was smaller with a maximin objective, due to both the greater progressivity of the income tax and the reduced progressivity of first-best redistribution. At the other extreme, as V yy approaches zero, equation (13) reveals that the progressivity of first-best taxes and transfers increases without bound. In the social welfare function (9) for the CES economy, a reduction in the algebraic value of ν is a strictly concave transformation that increases inequality aversion. The maximin case obtains in the limit as ν diverges to negative infinity, while the opposite extreme in which V yy equals zero obtains as ν approaches unity. Higher values of ν result in more progressive first-best redistribution. From equation (A8) in the appendix, the wage-rate elasticity of full income n+g(n) in the first-best depends (for any given elasticity of substitution) upon the parameter -ν/(1-ν), which approaches a maximum of unity as ν approaches negative infinity and diverges to negative infinity as ν approaches unity. From equation (A10), the wage-rate elasticity of the money-metric utility measure similarly depends upon -1/(1-ν), which approaches a maximum of zero as ν diverges to negative infinity and diverges to negative infinity as ν approaches unity. In Table 6, I present the first-best transfers and the associated money-metric measures of utility at five selected points in the wage-rate distribution (recall that the mean wage rate is unity), under eight values of ν, keeping the other parameters at benchmark values. These calculations, like the similar calculations in Blomquist (1981, p. 405), reveal the increased progressivity of the first-best transfers as ν approaches unity. Also, as ν approaches unity, Table 2 reveals that the volume of inter-group transfers increases, Table 4 reveals that the relative labor supply of the advantaged group rises, and Table 5 reveals that the relative utility of the advantaged group declines. Since optimal categorical transfers result in members of the advantaged group having generally lower utility levels, an increase in the planner's inequality aversion should result in more favorable treatment of the advantaged group, i.e., less extensive use of categorical transfers. This intuition is confirmed by Table 22

23 7. As υ approaches unity, the differentiation of transfers across groups increases, with the advantaged group's transfer approaching zero. Conversely, as υ becomes more negative, the two groups' transfers converge. In contrast, the marginal income tax rate rises as ν becomes more negative. The fact that transfers to different groups converge in the maximin limit was noted by Bennett (1987, p. 230). Due to the lognormal wage-rate distribution within each group, there are some individuals with arbitrarily low wage rates in the advantaged group as well as the disadvantaged group. Since these individuals do not work in equilibrium and consume solely from their transfer payments, any deviation from a uniform-transfer policy would reduce the utility of the lowest-wage members of the group receiving the smaller transfer. In contrast, as ν approaches unity, the planner places little weight on the well-being of low-wage members of the advantaged group and provides very small transfers to that group. These results can be summarized in the following proposition, Proposition 6: First-best taxes and transfers and optimal categorical transfers both tend to be less redistributive when the planner is more inequality-averse, although income tax rates are higher. IV. Extensions This analysis can be extended along several dimensions. The marginal tax rate, as well as the transfer, could vary across groups, as it does in many actual transfer programs. With differential marginal rates, the first-order condition for each group's marginal tax rate has the same form as (5), but the covariance and elasticity are evaluated separately for each group. As noted in the income-support literature, it may be desirable to set lower marginal tax rates (or lower benefit-phaseout rates) for groups with higher labor supply elasticities. Other possible extensions would examine categorical transfers in the presence of nonlinear income taxation and excise taxes on different commodities. Ebrahimi and Heady (1988) considered optimal 23

24 commodity taxation, when transfers vary based on family size and age of children. Similarly, Deaton and Stern (1986) considered a model in which preferences are related to exogenous characteristics and demonstrated that optimal categorical transfers strengthen the case for uniform commodity taxation. I have assumed that the characteristics of the different groups are exogenous and completely unresponsive to incentives, but a natural extension is to consider the more realistic assumption that they are somewhat price-sensitive. Most observable characteristics may be slightly price-sensitive in practice, especially when fertility choices are considered in a dynamic model. Favorable treatment of a particular racial group or a particular cohort could increase births in that group or year and favorable treatment of one gender could encourage selective abortion. Also, favorable treatment of the disabled may encourage more aggressive disability claims. Akerlof (1978, p. 15) and Roberts (1984, pp ) analyzed these issues. Blomquist (1984) examined the effects of a tax linked to hourly wage rates, when those rates can be influenced by educational choices. Ebrahimi and Heady (1988, p. 95) noted the possibility that family size, which they treated as exogenous in the body of their analysis, might actually respond to tax incentives and Diamond and Sheshinski (1995, p. 8) similarly discussed the economic implications of the government's imperfect observation of disability. Fortunately, the above analysis is approximately valid for characteristics with small price elasticities. If the observable characteristic is price-sensitive, the optimal covariance of categorical transfers with the social marginal value of income is no longer zero, but is given by a counterpart of equation (5). If the compensated price-elasticities of the characteristics are small, the covariances are also small and the above analysis in which the covariance is zero remains approximately valid. In a multi-period extension, it would be possible to distinguish exogenous characteristics from predetermined characteristics that individuals chose at an earlier date. For example, characteristics such as past earnings or previous educational attainment are predetermined, but can be altered in response to anticipated future taxes. Roberts (1984, pp ) and Revesz (1989, p. 466) demonstrated that there 24

25 is little or no long-run welfare gain from taxing predetermined characteristics and the planner should precommit to not tax these characteristics. V. Conclusion Categorical transfer payments linked to exogenous (or nearly exogenous) characteristics are an important part of most countries' actual tax-transfer systems. In this paper, I analyze such transfers as a limited form of lump-sum redistribution and confirm that categorical transfers should vary with exogenous observable characteristics in a manner that equalizes the mean social marginal value of income across groups with different characteristics. This property results in a number of similarities between categorical transfers and first-best taxes and transfers, which equalize the social marginal value of income across individuals. I find that the introduction of categorical transfers reduces the optimal value of the marginal income tax rate. Also, with optimal categorical transfers, members of high-mean-wage-rate groups tend to have greater labor supply and lower utility than members of low-mean-wage-rate groups. I also found that an increase in the planner's inequality aversion induces less use of categorical transfers, even as it reduces the optimal use of income taxation. These results have strong implications for the evaluation and design of categorical transfer programs. 25

26 References Akerlof, George A., "The Economics of 'Tagging' as Applied to the Optimal Income Tax, Welfare Programs and Manpower Planning," American Economic Review, 68(1), March 1978, pp Allingham, M.G., "Towards an Ability Tax," Journal of Public Economics, 4(4), November 1975, pp Atkinson, Anthony B., Public Economics in Action: The Basic Income/Flat Tax Proposal (Oxford: Clarendon Press, 1995) and Joseph E. Stiglitz, Lectures on Public Economics (New York: McGraw-Hill, 1980) Auerbach, Alan J., "The Theory of Excess Burden and Optimal Taxation," in Handbook of Public Economics, Vol. 1, ed. Alan J. Auerbach and Martin S. Feldstein (Amsterdam: North-Holland, 1985), pp Bennett, John, "The Second-Best Lump-Sum Taxation of Observable Characteristics," Public Finance/Finances Publiques, 42(2), 1987, pp Bernard, Alain, "Optimal Taxation and Public Production with Budget Constraints," in The Economics of Public Services, ed. Martin S. Feldstein and Robert P. Inman (New York: Stockton Press, 1977), pp Blomquist, N. Soren, "A Comparison of Tax Bases for a Personal Tax," Scandinavian Journal of Economics, 83(3), 1983, pp Dasgupta, Partha and Joseph Stiglitz, "On Optimal Taxation and Public Production," Review of Economic Studies, 39(1), January 1972, pp Deaton, Angus and Nicholas Stern, "Optimally Uniform Commodity Taxes, Taste Differences and Lump-Sum Grants," Economics Letters, 20(3), 1986, pp Diamond, Peter A., "A Many-Person Ramsey Rule," Journal of Public Economics, 4(4), Nov. 1975, pp and Eytan Sheshinski, "Economic Aspects of Optimal Disability Benefits," Journal of Public Economics, 57(1), May 1995, pp Ebrahimi, Ahmad and Christopher Heady, "Tax Design and Household Composition," Economic Journal, 98(390), 1988 supplement, pp Hellwig, Martin F., "The Optimal Linear Income Tax Revisited," Journal of Public Economics, 31(2), November 1986, pp Helpman, Elhanan and Efraim Sadka, "Optimal Taxation of Full Income," International Economic Review, 19(1), February 1978, pp (a), "The Optimal Income Tax: Some Comparative Statics Results," Journal of Public Economics, 9(3), June 1978, pp (b) 26