Tagging and income taxation: theory and an application

Transcription

1 Tagging and income taxation: theory and an application Helmuth Cremer Toulouse School of Economics (University of Toulouse and Institut universitaire de France) Firouz Gahvari Department of Economics University of Illinois at Urbana-Champaign Jean-Marie Lozachmeur Toulouse School of Economics (CNRS) April 2008 Revised, December 2008 We thank Sören Blomquist, Ray Rees, to referees of this journal, and the Editor, Alan Auerbach, for helpful comments and advice.

2 Abstract We derive a set of analytical results for optimal income taxation ith tags using quasilinear preferences and a Ralsian social elfare function. Secondly, assuming a constant elasticity of labor supply and log-normality of the skills distribution, e analytically identify the inners and losers of tagging. Third, e prove that if the skills distribution in one group first-order stochastically dominates the other, tagging calls for redistribution from the former to the latter group. Finally, e calibrate our model to the US orkers using gender as tag. Welfare implications are dramatic: Only male high-age earners lose; everyone else gains and some substantially. Key ords: Tagging, optimal income taxation, hazard rate, gender, race. JEL Classification: H21, H23.

3 A generally accepted tenet of optimal tax theory is that redistribution should be based solely on taxpayers earning potentials; other inherent characteristics, such as one s race, height, age and the like, are not considered relevant for redistributive purposes. In a classic paper, George A. Akerlof (1978) argued that even if one accepts that such characteristics are not, in and out of themselves, pertinent for redistribution, they still have a role to play in designing optimal tax schemes provided that they are correlated ith earning potentials. Akerlof considered a model in hich high- and lo-ability orkers could be grouped into to categories on the basis of an exogenously observable characteristic at no cost. One category consisted of lo-ability types only and the other of both lo- and highability types. 1 He shoed that, ithin his setup, conditioning taxes on income and a tag indicating the taxpayer s category increases social elfare (assumed to be utilitarian). The solution makes individuals in the group consisting only of lo-ability types better off relative to a tax scheme that depends on income alone. It also implies that to lo-ability persons end up ith different utility levels, ith the person in the group consisting only of lo-ability types enjoying a higher utility level. The basic insight is that tagging reduces the cost of income redistribution, although it does violate the principle of horizontal equity. After thirty years, and many studies, the question of ho tagging might change the properties of an optimal general income tax schedule remains, to a large extent, an open one. 2 Nor do e kno much beyond Akerlof s original findings on ho gains or loses as a result of tagging. This paper aims to fill in these gaps, albeit under certain strong assumptions. It considers a model ith a continuum of individuals ho can 1 The other characteristic of Akerlof s model is the discrete labor supply decision: one either has a hard job or an easy job, ith a fixed disutility associated ith the hard job. 2 The papers in this area include Nicholas Stern (1982), Andre W. Dilnot, John A. Kay, and C.N. Morris (1984), John Bennett (1987), Donald O. Parsons (1996), Ritva Immonen, Ravi Kanbur, Michael Keen, and Matti Tuomala (1998), Michael Kremer (2001), Viard (2001a, b), Jonathan Hamilton and Pierre Pestieau (2004), Robin Boaday and Pestieau (2006), Alberto Alesina, Andrea Ichino, and Loukas Karabarbounis. (2007), N. Gregory Manki and Matthe Weinzierl (2007), Sören Blomquist and Luca Micheletto (2008), and Weinzierl (2008). 1

4 be divided into to groups ith different ability distributions over the same support. Assuming that preferences are quasi-linear and the social elfare function is Ralsian, it derives a set of analytical results for optimal income taxation ith tags. The tagging is based on a publicly and costlessly observable exogenous characteristic. To additional assumptions, namely log-normality of the skills distribution and a constant elasticity of labor supply, allo us to analytically identify the inners and losers of tagging. Specifically, e sho that if the hazard rates in the to tagged groups do not cross, every individual in the group ith loer average skills ould benefit from tagging (in terms of consumption and utility levels and compared to the pooled equilibrium solution). They ill also consume more and enjoy higher utility levels than their counterparts in the group ith higher average skills. Members of the latter group may lose, as ell as gain, from tagging. If the hazard rates cross, gainers and losers sitch sides after the crossing. Our other results include the finding that if the skills distribution in one group firstorder stochastically dominates the other, tagging calls for redistribution from the former to the latter group. This result holds for a general utility function as long as the social elfare function is Ralsian. It also holds for a utilitarian social elfare function ith decreasing eights in skills provided that preferences are again quasi-linear. In terms of analytical results, the closest paper to ours is Boaday and Pestieau (2006). They study a model ith to ability types here the population can be separated into to groups, one of hich has a higher proportion of lo-ability individuals than the other (as opposed to Akerlof s formulation here one group consists only of the lo-ability persons). As ith our model, this formulation allos for a to-sided error here not only does a lo-ability person in the not-favored group get a less favorable treatment than his counterpart in the favored group, but also that a high-ability person in the favored group gets a more favorable treatment than his counterpart in the notfavored group. Boaday and Pestieau (2006) prove, assuming quasi-linear preferences, that tagging entails redistribution from the group ith a higher proportion of high- 2

5 ability persons to the group ith a higher proportion of lo-ability persons, 3 improves the social elfare ithin each group, and makes the high-ability individuals in the notfavored group the group ith a higher proportion of high-ability persons orse off. They interpret this last result as to imply that the tax system is more progressive in the tagged group ith the highest proportion of high-ability types. 4 To supplement our analytical results, e study the implications of using gender tags in designing optimal income taxes in the US. According to PSID data, the median earnings in 1993 ere approximately $30,000 for men as compared to $20,000 for omen, ith corresponding mean logarithmic deviations of 0.25 and There are of course many other differences beteen these groups beyond those due to average earnings. We ignore them all, assuming that all groups have lognormal distribution of skills that differ in mean incomes only. Additionally, e assume thatallorkershaveaconstantage elasticity of labor supply (hich e set equal to 0.5). The reason for these assumptions is consistency ith our theoretical model hich concentrated on the differences in average earnings alone. These somehat unrealistic features must be borne in mind hen interpreting our quantitative results. In particular, the numbers are not meant to sho hat ill actually happen in the US as a result of tagging. They are simply suggestive. We find that tagging on the basis of gender has dramatic implications for elfare and optimal tax rates. It increases the elfare of the least ell-off male and female orkers alike by $1,174 per year (equivalent to an increase of 6% in terms of utility). At higher age levels, omen gain at all deciles. The gains range from $1,677 per year for a female orker at the loest decile to $13,728 per year at the highest decile. Interestingly, lo-earning male orkers also gain ith those at the first decile gaining by as much as 3 The result is derived for a Ralsian social elfare function and, more generally, for any social elfare function that exhibits a constant absolute aversion to inequality. 4 Boaday and Pestieau (2004) also state that Intuition suggests that if the density of ability distributions across pairs of groups crossed only once, and if all ability-types ere in all groups, the analog of the above results should apply [to a case ith n ability types (as opposed to to)]. Social elfare ithin each group should be increased by inter-group transfers and group-specific tax structures, and tax schedules should be more progressive in groups that have more skill intensive ability distributions (p.16). 3

6 $935 per year. The gains decrease for higher deciles, turning negative for the fifth. After that, male orkers lose more and more as their earning ability increases, ith the loss reaching $8,397 per year mark at the tenth decile. Concerning tax rates, e derive, for all income brackets beteen $10,000 and $175,000, the values of the optimal marginal and average income tax rates for males and females as ell as for hen the population is pooled. The marginal tax rates are alays decreasing in income. The highest marginal tax rate faced by men is 80% percent versus only 26% for omen. Average tax rates increase sharply initially and then level off. The highest average tax rates are 26% for men versus to less than one per cent for omen. I The model The economy is inhabited by a continuum of individuals ho differ on the basis of one immutable and publicly observable characteristic, or tag, as ell as in skill levels. The average level of skills in one group is higher than in the other. We shall refer to the group ith the loer average skills as group l, and to the other group as group h. Labor is the only factor of production and the aggregate production level is linear. The age rates, hich also represent the skill levels, are distributed according to the distribution function F l () in group l, andf h () in group h. The support of the to distributions are the same and given by [, ]. The corresponding density functions, f j () =Fj 0(), j = l, h, are assumed to be strictly positive and differentiable for all [, ]. Individuals have identical preferences that depend on consumption, c, positively, and on labor supply, L, negatively. Preferences are represented by the quasi-linear utility function u = c ϕ (L). (1) The social elfare criterion is Ralsian (maxi-min); it is being implemented through a purely redistributive income tax system. The quasi-linearity of preferences and the maxi-min criterion make the problem analytically tractable. They also imply that the 4

7 government is in effect maximizing the transfers to the least ell-off individuals hile keeping the excess burden of taxation at a minimum. Nothing else matters. We ill further elaborate on these points later. To understand ho tagging may improve the redistributive poer of the tax system, and ho it affects the properties of the income tax structure, e derive the optimal income tax structure once ignoring the tag and a second time conditioned on it. In deriving the latter, e find it useful to follo a to-stage procedure, as in Immonen et al. (1998). In the first stage, one derives the optimal income tax structure for the to groups separately. This requires the government to achieve its objective function for each group separately, hile assigning each its on separate tax revenue constraint. In the second stage, one determines the size of the surplus, or deficit, for each group so that the overall government s budget constraint is satisfied for the to groups taken together hile simultaneously ensuring that the social elfare objective is attained for the entire population. The to-stage procedure developed belo is particularly simple, given our Ralsian objective function and quasi-linear preferences. The to assumptions together imply that the size of the government s external revenue requirement enters the society s optimal income tax function additively. Any increase or decrease in the revenue requirement is met by a lump-sum tax, or rebate, levied uniformly on all taxpayers; no other aspects of the tax function is affected. As a consequence, the determination of the revenue requirements in the second stage, ould leave the properties of the optimal tax schedule derivedinthefirst stage intact. A The generic optimal income tax problem As a background to the optimal tax problem ith tagging, e first sketch the generic tax problem ith quasi-linear preferences and a Ralsian objective function. Bernard Salanié (2003) and Boaday and Laurence Jacquet (2008) give details. Let c(),i() = L(), and u() denote consumption, income, and utility of an individual of type. 5

8 Denote the net taxes collected from the population by R. It follos from the quasi-linear specification (1) that R Z [I() c()] f()d = Z I() u() ϕ µ I(). (2) Integrating the local incentive compatibility constraint, du/d = I()/ 2 ϕ 0 (I()/), e have Z µ I(s) I(s) u() =u + s 2 h0 ds, s here u = u() is the utility of the poorest individual. One can then easily sho that R = u + Ψ(F ), (3) here Ψ(F ) Z I() ϕ µ I() Z f()d I() 2 ϕ0 µ I() [1 F ()]d. (4) Observe that expression (4) measures the maximum tax revenue the government can, in the second best and for a given I(), extract from a population ith the distribution function F ( ). Thefirst term is the total surplus; that is, sum of all individuals utilities if they pay no taxes to the government. The second term is the sum of informational rents noted above (aggregated over all individuals ith ages >). Assuming that the government s external revenue requirement is R, and given the skills distribution F ( ), the utility of the poorest individual is given by u R, F = R + Ψ(F ), (5) With a Ralsian social elfare function, the optimal income tax problem reduces to the choice of I() to maximize Ψ(F ), or tax revenues. Maximize Ψ(F ) ith respect to I() andsimplifythefirst-order condition to get 1 1 ϕ0 (I()/) 1 ϕ0 (I()/) = 1 F () f() 6 1+ I() ϕ 00 (I()/) ϕ 0. (6) (I()/)

9 Denote the solution to (6) by I (, F) and rite Ψ (F ) for the maximized value of Ψ(F ). One can then establish that u R, F = R + Ψ (F ), (7) u, R, F = u R, F Z I µ (s, F )(s) I + s 2 ϕ 0 (s, F ) ds, (8) s c, R, F = u, R, F µ I (, F) + ϕ. (9) Equations (8) (9) tell us that once the optimal value of I() hich is independent of the value of R is chosen, a one unit increase (or decrease) in R loers (or increases) the values of u R, F,u, R, F, and c, R, F by one unit. To determine the implication of the solution u ( ),u ( ),c ( ) for the shape of the implementing tax function, denoted by T (I (, F)), assume that T ( ) is differentiable. No, hen an individual of type maximizes u = c ϕ (I/) subject to the budget constraint c = I T (I), he ill have to solve the first-order condition, 1 T 0 (I) ϕ 0 (I/) / =0. Thus, the implementing tax function is determined according to T 0 (I (, F)) = 1 ϕ 0 (I (, F)/) /. Rearranging this equation and introducing, H(; F ) 1+ L (, F)ϕ 00 (L (, F)) ϕ 0 (L, (, F)) (10) Ω() 1 F () f(), (11) the implementing tax function is characterized by T 0 1 T 0 = H(, F) Ω(). (12) With ϕ 0 ( ) > 0 and ϕ 00 ( ) > 0, it follos from (10) that H(, F) > 1. Equation (12) then implies T 0 ( ) > 0 as long as F () < 1; it also implies that T 0 ( ) =0at = here F () =1. Taxpayers preferences enter (12) only through the H(, F) term. Observe also that L ( )ϕ 00 ( )/ϕ 0 ( ) is equal to the inverse of the age elasticity of labor supply for a -type individual. Thus equation (12) tells us the more elastic is the labor 7

10 supply of the -type, the loer must be the marginal income tax rate they face. On the other hand, Ω() depends solely on the properties of the distribution function, F (). Specifically, Ω() is equal to the inverse of the hazard rate (the Mills ratio) divided by. If the elasticity of labor supply is constant, the marginal income tax rate is determined entirely by the shape of the hazard rate and the size of the age rate. The hazard rate for lognormal distribution, has an initially increasing segment folloed by a decreasing part; see Dale W. Jorgenson John J. McCall, and Roy Radner (1967). The corresponding marginal income tax rate must then be initially decreasing in, but it may eventually become increasing. 5 Salanié (2003, p. 95) notes that, given quasilinearity and the Ralsian social elfare function, the second-order condition for the optimal income tax problem is satisfied ifandonlyifω() is nonincreasing in. II Optimal taxation ith and ithout tagging We start ith the characterization of optimal tax solution hen the information from the tag is discarded. Then, e proceed to examine ho conditioning the tax on the tag changes the properties of the optimal tax solution. A Optimal solution ithout tagging Denote the distribution function for the entire population by F hl () and its corresponding density by f hl (). They are related to the distribution and the density functions of the to groups that comprise it according to: F hl () = F l()+f h (), 2 (13) f hl () = f l()+f h (). 2 (14) 5 Pareto distribution has an increasing hazard rate leading to a constant marginal income tax rate (an increasing hazard rate and an increasing result in a constant Ω()). Thehyperbolicfunction, f() = λe λ, has a constant hazard rate resulting in a marginal income tax rate that is alays decreasing in (Ω() is decreasing because is increasing). 8

11 The optimal tax structure in this case is precisely the same as the one derived for the generic optimal tax problem, ith F hl () and f hl () replacing F ( ) and f( ). Denote the optimal solution ithout tagging, and ith an external revenue requirement R =0, by u hl = u (0,F hl ),u hl () =u (, 0,F hl ), and c hl () =c (, 0,F hl ). To be consistent, e denote the optimal values of I() and Ψ(F hl ), hich are independent of R, by Ihl () =I (, F hl ) and Ψ hl = Ψ (F hl ). The optimal solutions are thus characterized by u hl = Ψ hl, Z (15) u hl () =u hl + Ihl (s) µ I s 2 ϕ 0 hl (s) ds, (16) s µ I c hl () =u hl ()+ϕ hl (). (17) B Optimal solution under tagging Given our Ralsian social elfare function, e continue to be concerned ith the maximization of the elfare of the least ell-off individuals in the entire population. With tagging, e can rite this as maximization of min[u l,u h ]. As observed earlier, it is simpler to derive the tagged solution in a to-stage manner. The first stage corresponds to the generic optimal income tax problem of Section I herein I j (),c j (; R j ) are derived to maximize u j under the constraint that R j = R j,j= l, h, treating R j as fixed. This yields Ij () =I (; F j ) ith a maximal u j equal to u j Rj u R j,f j = Rj + Ψ (F j ). Inthesecondstage,echooseR l, R h to maximize min u l Rl,u h Rh subject to R l + R h =0. This requires u l Rl and u h Rh to be equalized. Using (7), the optimal values of R l, R h are found as the solution to R l + Ψ (F l ) = R h + Ψ (F h ), R l + R h = 0, 9

12 and denoted by R l and R h. We have, R l = Ψ (F l ) Ψ (F h ), (18) 2 R h = Ψ (F h ) Ψ (F l ). (19) 2 This procedure highlights the fact that the interaction beteen groups occurs through the transfers only. ³ ³ Denote u l R l = u h R h u. The value of u can then be determined from equation (7) using the values of R l and R h from equations (18) (19). Subsequently, one can determine the values of u j () and c j (), j= l, h, from equations (8) (9). We have u = Ψ (F e )+Ψ (F i ), (20) 2 Z u j() = u Ij + (s) µ I s 2 ϕ 0 j (s) ds, (21) s µ I c j() = u j () j()+ϕ. (22) The folloing proposition summarizes the results of this section. Proposition 1 Assume preferences are quasi-linear and the social elfare function is Ralsian. There are to groups of individuals of equal size, l and h, each ith a continuum of skills distributed over the same support [, ]. Let Ij (), j= l, h, hl, denote the solution to equation (6) hen F j () replaces F (). The optimal utility and consumption levels for every -type individual are given by equations (15) (17) hen the to groups are pooled together, and by equations (20) (22) under tagging. C Marginal income tax rates On the basis of equation (12), and denoting H(, F j ) by H j (), onecanritethe marginal income tax rates for the tagged and pooled solution as 6 T 0 j 1 T 0 j = H j () Ω j (), j = l, h, hl. (23) 6 Kremer (2001) derives a similar result. 10

13 Observe that under tagging, the marginal tax rate for any given individual type depends only on the characteristics of the group to hich he belongs (specifically on the elasticity of the labor supply and the hazard rate). 7 Second, the profile of the marginal tax rates ithin any given group does not depend on the characteristics of the second group ith hich it is combined. Specifically, it does not matter hether this second group is richer or poorer. Observe, hoever, that this property applies only to the marginal tax rates and not the average tax rates. Third, if one of the tagged groups has a degenerate skills distribution ith a single age level, as in the ordinal Akerlof s (1978) example, this group should be subjected to a lump sum tax and face a zero marginal tax rate (regardless of ho high or lo the age level is). In all cases, the marginal income tax rate is characterized by equation (23). We end this section by demonstrating a relationship beteen the hazard rates of the to groups and that of the entire population. This ill be found useful later on. Lemma 1 Assume there are to groups of individuals, l and h, hose populations are of the same size, and hose skill levels are distributed over the same support [, ] according to the distribution functions F l () and F h (). Then, the inverse of the hazard rate for an individual of type in the entire population is bracketed by the inverse of the individual s hazard rates in the to groups. Proof. It follos from equations (13) (14) and the definition of Ω() that Ω hl () = 1 F hl() = 1 [F l()+f h ()] /2 f hl () [f l ()+f h ()] /2 f l () = f l ()+f h () Ω f h () l()+ f l ()+f h () Ω h(). (24) 7 With our quasi-linear specification, external revenue requirements do not affect the values of the marginal tax rates. With more general preferences, this ill not be the case. Thus marginal tax rates may depend on the presence of other groups, but only through the revenue requirement. 11

14 III Welfare and redistribution There are to questions concerning redistribution. One is ho tagging impacts the elfare of lo- and high-ability individuals in the to groups; the second is the redistribution beteen groups. Ultimately, of course, it is the first question that underlies a government s tax policy. Tagging is done not ith a vie to transfer resources from one group to another; rather, it is undertaken in order to more efficiently transfer resources from high- to lo-ability persons. We examine this question here, leaving the question of transfers beteen groups to the next section. It is of course a trivial observation that tagging can never loer social elfare. One can alays offer the pooled solution to the to groups. In our notation, Ihl () and the corresponding, u hl,u hl (),c hl (), constitute a feasible allocation to the optimal tax problem ith tagging as ell. Consequently, if the tagged solution differs from the untagged solution, the former must yield a better allocation (i.e., u >u hl ). Observe also that the first-stage of our to-stage optimal tax problem yields a solution Ij () = Ihl (), j= l, h, if F l() =F h (). Put differently, hen the to groups have different distribution of skills, one can alays increase social elfare by offering the to groups different tax schedules. Given that the Ralsian social elfare is represented by the utility of the least elloff persons of the society as a hole, tagging in our setup redistributes to the least able individuals in both groups leaving them equally ell off. 8 Hoever, one cannot a priori determine hat happens to the elfare of the other individuals in either group. To shed light on this, e have to make other simplifying assumptions. The first assumption e make is that labor supply exhibits a constant age elasticity. Thus set ϕ(l) =L 1+1/, here is the labor supply elasticity. 9 Observe that the strict convexity of ϕ(l) implies >0. This assumption leads to a closed-form solution for 8 With other types of social elfare function, say utilitarian, the utility of less able individuals in the tagged and untagged groups ill be different. 9 Thereisnoneedtoassignacoefficient to L 1+1/ as one can alays adjust the unit of measurement for c. 12

15 optimal incomes hich, for j = l, h, hl, are given by 10 µ µ Ij () = Ω j (). (25) 1+ With Ω hl () being a eighted average of Ω l () and Ω h (), equation (25) tells us that Ihl () alays lies beteen I l () and I h (). Moreover, I h () T I l () Ω h() S Ω l (). That is, Ih () T I l () if and only if the hazard rate for the -type in group h is greater than/equal to/smaller than the hazard rate for the -type in group l. This makes perfect sense. With the hazard rate indicating the proportion of -type in the segment of population ith ages and above, a higher hazard rate implies that e ant to assign a higher income level to the -types. From the expression for Ij () in (25), the definition of Ω j(), and the relationship beteen F l (),F h (), and F hl (), it then also follos that Z I l () f l()d + Z I h () f h()d = 2 Z I hl () f hl()d. (26) That is, tagging increases aggregate output. The closed-form solution for Ij () allos us to also derive the expressions for u hl (), c hl (), u j (), and c j (): µ Z µ u hl () =u hl + s Ω hl (s) 1 ds, (27) 1+ µ c hl () = u hl ()+ 1+ µ Z u j() = u + 1+ c j() = µ u j() µ Ω hl () 1, (28) µ s Ω j (s) 1 ds, j = l, h, (29) µ 1+ 1 Ω j () 1, j = l, h. (30) It is clear from expressions (27) (30) that one should kno the relationship beteen Ω h () and Ω l () for every [, ], to be able to determine ho gains and ho 10 Setting ϕ(l) =L 1+ 1 simplifies the expression for Ψ(Fj) in (4). Rearranging the first-order condition for the maximization of Ψ(F j ) ith respect to I j () then yields (25). 13

16 loses from tagging. This, in turn, requires one to make assumptions on the distribution of skills. We assume that skills ithin each group have a lognormal distribution, and that group h has a higher mean age than group l. Otherise, the to lognormal distributions have identical other moments. Given the shape of lognormal distributions, hen is close to zero, the hazard rate is higher in group l. Thus, initially, Ω l () < Ω hl () < Ω h (). As increases, hoever, Ω() decreases for lognormal distributions (at least initially). To possibilities arise. First, Ω h () may remain alays above Ω l (). 11 In this case the Ω l () < Ω hl () < Ω h () property is alays satisfied. Second, it is possible that as increases, at some point, Ω h () crosses Ω l (). Interestingly, hoever, if this happens, Ω h () ill never curl back to cross Ω l () again. That is, after the crossing, Ω h () ill alays remain belo Ω l (). Wehavethefolloinglemma. Lemma 2 Assume the distribution of skills in the to groups is lognormal ith one group, h, having a higher mean age than the other group, l. Let Ω j () denote the inverse of the hazard rate divided by for distribution F j (), here j stands for groups l, h, and hl (the to groups pooled together). Then, Ω h () cannot cross Ω l () more than once. by Proof. When the skills distribution is lognormal, the expression for Ω() is given Ω() = σ 2π 2 " 1 2 Z # ln μ σ 2 e t2 dt π 0 (ln μ)2 e 2σ 2, here μ and σ 2 are the mean and variance of ln. Differentiating Ω() ith respect to yields dω d = 1 " 2π + (ln μ) 1 2 Z # ln μ σ 2 e t2 (ln μ)2 dt e 2σ 2σ π 2 0 = 1 Ωσ (ln μ) 1. (31) 2 11 At high values of, f l () <f h () but F l () >F h () so that the hazard rate in group h can remain alays belo the hazard rate in group l. 14

17 T 0 l <T0 hl <T0 h 0 Everypersoningroupl gains Figure 1: Ω h () and Ω l () do not intersect No denote the expression for Ω by Ω l hen μ = μ l and by Ω h hen μ = μ h, here, by assumption, μ h >μ l. It then follos from equation (31) that d d (Ω h Ω l )= 1 σ 2 [(ln μ h) Ω h (ln μ l ) Ω l ]. (32) Equation (32) tells us that henever Ω h = Ω l (= Ω), d d (Ω h Ω l )= Ω σ 2 [μ l μ h ] < 0. But this can not happen to consecutive times. Lemma2tellsusthatifapoint D exists at hich Ω h () =Ω l (), then Ω l () < Ω hl () < Ω h () for all [, D ) and Ω l () > Ω hl () > Ω h () for all ( D, ). Of course, e also kno that if Ω h () and Ω l () do not cross, it is alays the case that Ω l () < Ω hl () < Ω h (). No, as long as Ω l () < Ω hl () < Ω h (), equations (27) (29) imply that u l () u hl () > 0 and u l () u h () > 0. These results are quite comforting in that they tell us it is not just the poorest persons ho become better off ith tagging. Either all or the relatively poorer people in the tagged group l also become better off. Observe, hoever, that one does not kno ho u h () and u hl () compare because hile u >u hl, the second expression in u h () is smaller than the second expression in u hl (). This means that it is possible for a person ith a lo in group h to become orse off as a result of tagging. Similarly, e have from (28) (30), c l () c hl () > 0, and c l () c h () > 0; but that the comparison beteen c h () and c hl () is ambiguous. 15

18 0 T 0 l <T0 hl <T0 h One gains if in group l D T 0 l >T0 hl >T0 h One gains if in group h Figure 2: Ω h () and Ω l () intersect Finally, observe that ith a constant and identical elasticity of labor supply, the marginal income tax rate formula in (23) is simplified to Tj 0 µ 1 Tj 0 = 1+ 1 Ω j (), j = l, h, hl. (33) It then immediately follos from (24) that the marginal income tax rate in the pooled equilibrium solution is a eighted average of the marginal income tax rates in the to tagged groups. Specifically, the folloing relationship holds beteen Tl 0,T0 h, and T hl 0 : T 0 hl T 0 l f l () f h () 1 Thl 0 = f l ()+f h () 1 Tl 0 + f l ()+f h () 1 Th 0. (34) That is, the marginal income tax rate in the pooled equilibrium case is a eighted average of the marginal tax rates for the to tagged groups and thus bracketed by them. Equation (34) implies that Tl 0(I()) <T0 hl (I()) <T0 h (I()). If the hazard rates intersect at D, equations (27) (30) imply that for all ( D, ),u h () u hl () > 0,u h () u l () > 0,c h () c hl () > 0, and c h () c l () > 0; but e do not kno ho u l () compares ith u hl () and c l () ith c hl (). Thatis, tagging may have the unintended consequence of making even the richer individuals in the rich group h better off. Finally, e also have that Tl 0(I()) >T0 hl (I()) >T0 h (I()). Figures 1 2 illustrate the to possible cases hen Ω h () and Ω l () do not intersect andhentheydo,andproposition2summarizesourresults. Proposition 2 In addition to the assumptions of Proposition 1, assume that the age elasticity of labor supply is constant and identical for the groups h and l. Then 16 T 0 h

19 (i) Income of -type orkers in l and h groups under the tagged solution bracket the income that they ould earn if the entire population is pooled. (ii) Tagging increases aggregate output. Additionally, assume that the distribution of skills in the to groups is lognormal ith group h having a higher mean than group l. Then: (iii) Either all individuals in group l or those ith earning ability [, D ), assuming there exists a D at hich the hazard rates for group l and group h intersect, benefit from tagging (in terms of consumption and utility levels and compared to the pooled equilibrium solution). These individuals also earn more, consume more, and have higher utility levels than corresponding -type persons in group h. Similar individuals in group h may lose, as ell as gain. (iv) Every individual ( D, ) in group h, assuming there exists a D at hich the hazard rates for group l and group h intersect, benefit from tagging. These individuals also earn more, consume more and have higher utility levels than corresponding -type persons in group l. Similar individuals in group l may lose, as ell as gain. (v) The pattern of marginal income tax rates are Tl 0(I()) <T0 hl (I()) <T0 h (I()) either for all, or for all [, D ). In the latter case, Tl 0(I()) >T0 hl (I()) > Th 0 (I()) for all (D, ]. IV Redistribution beteen groups We no turn to the question of redistribution beteen groups. Interestingly, the result e ill derive belo holds for a setting more general than hat e have considered thus far. The basic idea of tagging is that if one group contains relatively more disadvantaged people hom the government ants to help, then by favoring that group one succeeds in redistribution toards the disadvantaged people in more effective ay. In the context of our model, the question is hether tagging entails redistribution from group h to group l. Akerlof s (1978) original example as based on the assumption 17

20 that one of the groups consisted of only lo-ability individuals. It as then by ay of redistributing toards this group that he as able to sho that every member of the group received more transfers and became better off (as compared to the pooled equilibrium solution). 12 Note that redistribution from the not-favored to the favored group is essentially a redistribution from the total population to the favored group. To shed light on this particular question, again one has to kno more about the skills distributions beyond the fact that they are different. Hoever, one need not resort to log-normality of skills assumption. The question can be ansered for any distribution of skills as long as one stochastically dominates the other. The direction of transfers ill be from the dominant group to the dominated one. Interestingly too, quasi-linearity of preferences is not needed for this result as long as the social elfare function is Ralsian. On the other hand, ith quasi-linearity, the result holds ith more general social elfare functions. Proposition 3 states and proves these results. Proposition 3 There are to groups of individuals of equal size, l and h, eachitha continuum of skills distributed according to the distribution functions F l () and F h () over the support [, ]. Group h first-order stochastically dominates group l so that F h () F l () for all [, ] and F h () <F l () for some. Then tagging implies redistribution from group h to group l under either of these conditions: (i) The social elfare function is Ralsian. (ii) Preferences are quasi-linear and the social elfare function is eighted utilitarian, R γ()v () f () d. Proof. Part (i): Under a Ralsian social elfare function, the objective of the government is to maximize tax revenues subject to incentive compatibility and resource constraints. No observe that incentive compatibility does not depend on the distribution of types so that the allocation (Il (),c l ()) is incentive compatible in h as ell 12 Hoever, one could not determine the impact that tagging had on the elfare of lo-ability persons in the not-favored group. This ambiguity does not arise ith a Ralsian social elfare function as the least able individuals ould get the same treatment in the favored and not-favored groups. 18

21 as in l. But, given that (Ih (),c h ()) is the optimal solution in h, e must have13 Z [I h () c h ()] df h() = Z [I l () c l ()] df h(). (35) No denote the taxes paid by a person of type by t() T (I()). Differentiating t() ith respect to yields dt() d = T 0 (I) di() d = 0, here the sign follos from the signs of T 0 (I) and di()/d in Mirrlees optimal tax problem. With t() being non-decreasing in, itfollosfromthedefinition of firstorder stochastic dominance that Z [I l () c l ()] df h() > Inequalities (35) (36) then imply that Z [I h () c h ()] df h() > Z Z [I l () c l ()] df l(). (36) [I l () c l ()] df l(). (37) This means that the government can extract more tax revenue from group h than from group l so that the direction of transfers is from group h to group l. Part (ii): The government maximizes R γ()v j () f j () d for each group subject to local incentive compatibility constraint and the revenue requirement R j in the first stage, and then equalizes the shado cost of public funds across the groups in the second stage. Denote the optimized value of V j () by Vj () and the Lagrange multipliers associated ith the revenue requirement by μ j. We have μ j = d dr j Z γ()v j () f j () d. Thus there ill be redistribution from group h to group l ifandonlyifμ l >μ h. No, ith quasi-linear preferences, dvj () /dr j = 1. Substituting in the above equation 13 Remember that ith our assumptions the optimal tax schedule maximizes tax revenues. 19

22 for μ j and integrating by parts, one can rite the expression for μ j as μ j = Z Subtracting μ h from μ l, μ l μ h = γ()f j () d = γ() Z Z γ 0 ()F j () d. γ 0 ()[F h () F l ()] d. The assumptions that F h () first-order stochastically dominates F l () and γ 0 () < 0 imply that μ l μ h > 0. V Simulations This section applies our theoretical model to the taxation of US orkers using PSID data for 1993 ith gender as tag. Assume that the distribution of skills in each of the group of orkers is truncated lognormal over the same support. The computed mean logarithmic deviation of income is 0.25 for men and 0.20 for omen. Configure the means of the skills distributions in such a ay that the median gross income ill be $30,000 for men and $20,000 for omen. Also assume, in line ith our theoretical construct, that the age elasticity of labor supply is constant and identical for the to tagged groups; setting its value at 0.5 hich is a figure used in this literature. Figures 3 presents the graph of the optimal marginal income tax rates for men versus omen. Observe that, the to tagged groups marginal tax rates bracket the pooled population s tax rate at every income level ith men facing the higher rates. At the income level of $10,000, the marginal income tax rate is 80% for men, 26% for omen, and 50% for the pooled population. The rates decrease all the ay to zero for the top income level of $175,000. Unlike the marginal income tax rates, average tax rates are increasing in income indicating the progressivity of the tax schedules; see Figure 4. The rates are close across the groups up to an income level of $20,000 after hich the tax rate for the 20

23 margi nal tax group female male income Figure 3: Optimal marginal income tax rates by gender. pooled population is bracketed by those of men (at a higher rate) and omen (at a loer rate). The higher average tax rates faced by male high-age earners sho that they are the losers in the tagging procedure. To get a clearer picture of these elfare effects, Table 1 reports ho men and omen are affected by tagging at different decile levels of skills. Observe first that tagging improves the elfare of the least ell-off male and female orkers alike by $1,174 per year. This is the Compensating Variation measure of the elfare change (hich, given our quasi-linear specification for the preferences, is also equal to the Equivalent Variation measure). In terms of utility, this amounts to an increase of 6%. At higher age levels, omen gain at all deciles. The gains range from $1,677 per year for a female orker at the loest decile to $13,728 per year at the highest decile. Lo-age male orkers also gain ith those at the first decile gaining by as much as $935 per year. The gains decrease for higher deciles, turning negative for the fifth. After that, male orkers lose more and more as their age increases, ith the loss increasing to $8,397 per year at the tenth decile. Given our Ralsian perspective, the gains and losses to orkers beyond least ell- 21

24 average tax income group female male -0.3 Figure 4: Optimal average income tax rates by gender. off individuals are not part of the calculus of social elfare. The improvement in social elfare is the $1,174 per year gain enjoyed by the least ell-off orkers, females and males alike. Hoever, that all lo-age orkers gain, regardless of their tag, attests to the efficiency-enhancing poer of tagging. VI Conclusion The optimal marginal income tax rate in Mirrlees depends on many factors. This makes the study of tagging somehat difficult. This paper has restricted the analysis to quasilinear preferences, and a maxi-min social elfare criterion, to reduce the determinants of the tax rate to the labor supply elasticity and the hazard rate of skills. Considering an economy that can be tagged into to different groups, each consisting of individuals ith a continuum of skills over the same support, and every person having the same constant age elasticity of labor supply, e have been able to derive a set of analytical results for optimal income taxation ith tags. Secondly, to additional assumptions, namely log-normality of the skills distribution and a constant elasticity of labor supply, 22

25 Table 1. Welfare implications of tagging on the basis of gender as compared to the pooled population solution (monetary figures in dollars per year) female male Utility Compensating Utility Compensating change % variation change % variation Least ell-off 6 1, , 174 Skills deciles 1 9 1, , , , , , , , , , , , , ,397 have alloed us to analytically identify the inners and losers of tagging. Specifically, e have shon that if the hazard rates in the to tagged groups do not cross, every individual in the group ith loer average skills ould benefit from tagging (in terms of consumption and utility levels and compared to the pooled equilibrium solution). They ill also consume more and have higher utility levels than their counterparts in the group ith higher average skills. Members of the latter group may lose, as ell as gain, from tagging. If the hazard rates cross, the direction of gainers and losers change after the crossing. Third, e have demonstrated that if the skills distribution in one group first-order stochastically dominates the other, tagging calls for redistribution from the former to the latter group. This result holds for a general utility function as long as the social elfare function is Ralsian. It also holds for a utilitarian social elfare function ith decreasing eights in skills provided that preferences are quasi-linear. 23

26 Finally, e have studied the implications of tagging on the basis of gender, assuming that the skills distribution in each group is truncated lognormal. The distributions are calibrated such that average earnings and mean logarithmic deviation of income in each group are those of the US in 1993 according to the PSID data. We have found that optimal marginal income tax rates are about 80% for men and 26% for omen at the $10,000 income level, steadily decreasing to zero at an income level of $175,000. We have also found that tagging has dramatic impacts on elfare. The elfare of the least ell-off male and female orkers increase by the equivalent of $1,174 per year (6% in utility terms). Given our Ralsian perspective, this also measures the gain in social elfare. Nevertheless, e have also calculated the gains and losses to orkers at all deciles of skills distribution. Most interestingly, all lo-age orkers gain regardless of their tag. The only people ho lose are male high-earners hose losses are more than compensated by the gains to females of equal ability. In interpreting these numbers, the shortcomings of our calibrated model should be borne in mind. The distributions are calibrated to reflect the differences in average earnings and mean logarithmic deviation of incomes across groups; hoever, e have not alloed for differential labor supply responses. Yet the finding that all lo-age orkers gain suggests that tagging enhance the efficiency of the tax system considerably. This merits further investigation. 24

27 References [1] Akerlof, George A The economics of tagging as applied to the Optimal income tax, elfare programs, and manpoer planning. American Economic Revie, vol 68, [2] Alesina, Alberto, Ichino, Andrea, and Loukas Karabarbounis Gender based taxation and the division of family chores. Unpublished manuscript, Harvard University. [3] Bennett, John The second-best lump-sum taxation of observable characteristics. Public Finance/Finances Publiques. vol [4] Blomquist, Sören and Luca Micheletto (2008). Age related optimal income taxation. Scandinavian Journal of Economics, vol. 110, [5] Boaday, Robin and Laurence Jacquet Optimal marginal and average income taxation under maxi-min. Journal of Economic Theory, forthcoming. [6] Boaday, Robin and Pierre Pestieau Tagging and redistributive taxation. Annales d Economie et de Statistique 83/84, [7] Dilnot, Andre W., Kay, John A., and C.N. Morris The Reform of Social Security. Oxford: Clarendon Press. [8] Hamilton, Jonathan and Pierre Pestieau Optimal income taxation and the ability distribution: implications for migration equilibria. International Tax and Public Finance vol 12, [9] Immonen, Ritva, Kanbur, Ravi, Keen, Michael, and Matti Tuomala Tagging and taxing: the optimal use of categorical and income information in designing tax/transfer schemes. Economica, vol 65, [10] Jorgenson, Dale W., McCall, John J., and Roy Radner Replacement of Industrial Equipment Mathematical Models. Amsterdam: North Holland Publishing Company. [11] Kremer, Michael Should taxes be independent of age? Unpublished manuscript, Harvard University. 25

28 [12] Manki, N. Gregory and Matthe Weinzierl The optimal taxation of height: A case study of utilitarian income redistribution. Unpublished manuscript, Harvard University. [13] Parsons, Donnald O Imperfect tagging in social insurance programs. Journal of Public Economics, vol 62, [14] Salanié, Bernard The Economics of taxation. Cambridge, Massachusetts: MIT Press. [15] Stern, Nicholas Optimum taxation ith errors in administration. Journal of Public Economics, vol 17, [16] Viard, Alan D Some results on the comparative statics of optimal categorical transfer payments. Public Finance Revie, vol 29, [17] Viard, Alan D Optimal categorical transfer payments: the elfare economics of limited lump-sum redistribution. Journal of Public Economic Theory, vol 3, [18] Weinzierl, Matthe The surprising poer of age-dependent taxes. Unpublished manuscript, Harvard University. 26

29 Appendix Derivation of the expression for I j () in equation (25): Setting h(l) =L1+ 1 simplifies the expression for Ψ(F j ) in (4), j = l, h, hl, to ( µ Ψ(F j )= I j () µ Ij () 1+ 1 ) Ω j () f j ()d. (A1) Rearranging the first-order condition for the maximization of Ψ(F j ) ith respect to I j () then yields µ µ Ij () =L j() = Ω j (), j = l, h, hl. (A2) 1+ Derivation of the expressions for u hl in (27) and u in (29): First, substitute the expression for Ij () from (A2) into (A1) and simplify to get, for j = l, h, hl, Z µ Ψ (F j )= (1 + ) Ω j () f j ()d. (A3) Second, given the expression for Ψ (F hl ) and using (15), e have Z µ u hl = Ψ (F hl )= (1 + ) Ω hl () f hl ()d. Finally, given the expression for Ψ (F j ),j= l, h, and using (20), e have u = 1 2 (1 + ) 1 ( Z + Z µ µ Ω h () f h ()d ) Ω l () f l ()d. Derivation of equations (27) (30): With h(l) =L 1+ 1, e have, for j = l, h, hl, Z Ij (s) µ I µ s 2 h 0 j (s) ds = 1+ 1 Z s 1 L s j(s) 1+ 1 ds. 1

30 Substitute for L j () from (A2) into above. This gives Z I j (s) s 2 h 0 µ I j (s) s µ Z µ ds = s Ω j (s) 1 ds. 1+ Given this expression, one can simplify the expressions for u hl () and u j (), j= l, h, derived in (16) and (21), to (27) and (29). Similarly, one simplifies the expressions for c hl () and c j (), j = l, h, derived in (17) and (22), to (28) and (30). Proofofinequality(26):Let = Z Z Z Ih () f h()d + Il () f l()d 2 [I h () I hl ()] f h()d + Z Z I hl () f hl()d [I l () I hl ()] f l()d. Substitute for Ii (), j = l, h, hl, from (A2) in above and simplify. We have µ 1+ = Z ( µ µ Ω h () ) Ω hl () f h ()d + Z ( µ µ Ω l () ) Ω hl () f l ()d. No subtract the expression for u hl from the expression for u : u u hl = 1 2 (1 + ) 1 ( Z Z µ (1 + ) 1 Z µ Ω h () f h ()d + ) Ω l () f l ()d µ Ω hl () f hl ()d (A4) 2

31 = 1 2 (1 + ) 1 ( Z Z 1 µ ( Z 1 2 (1 + ) Z µ = 1 2 (1 + ) 1 Z µ Ω h () f h ()d ) Ω hl () [f h ()+f l ()] d + µ Ω l () f l ()d ) Ω hl () [f h ()+f l ()] d ( µ Ω h () µ Ω hl () )f h ()d + 12 Z (1 + ) 1 ( 1+µ 1+ 1 µ Ω l () Ω hl () ) 1+ f l ()d. (A5) Substituting from (A5) into (A4), e get =2(1+)(u u hl ) = 0. 3