OPTIMAL TAXATION AND SOCIAL INSURANCE IN A LIFETIME PERSPECTIVE

January 26 OPTIMAL TAXATION AND SOCIAL INSURANCE IN A LIFETIME PERSPECTIVE A. Lans Bovenberg, Tilburg University, CEPR and CESifo ) Peter Birh Sørensen, University of Copenhagen, EPRU and CESifo ) Abstrat Advanes in information tehnology have improved the administrative feasibility of redistribution based on lifetime earnings reorded at the time of retirement. We study optimal lifetime inome taxation and soial insurane in an eonomy in whih redistributive taxation and soial insurane serve to insure (ex ante) against skill heterogeneity as well as disability risk. Optimal disability benefits rise with previous earnings so that publi transfers depend not only on urrent earnings but also on earnings in the past. Hene, lifetime taxation rather than annual taxation is optimal. The optimal tax-transfer system does not provide full disability insurane. By offering imperfet insurane and struturing disability benefits so as to enable workers to insure against disability by working harder, soial insurane is designed to offset the distortionary impat of the redistributive labor inome tax on labor supply. Keywords: Optimal lifetime inome taxation, optimal soial insurane JEL Code: H2, H55 ) Corresponding author: CentER and Netspar, Tilburg University, P.O. Box 953, 5 LE Tilburg, The Netherlands. E-Mail: A.L.Bovenberg@uvt.nl ) Department of Eonomis, University of Copenhagen, Studiestraede 6, 455 Copenhagen K, Denmark. E-mail: Peter.Birh.Sorensen@eon.ku.dk

OPTIMAL TAXATION AND SOCIAL INSURANCE IN A LIFETIME PERSPECTIVE A. Lans Bovenberg and Peter Birh Sørensen. Introdution Muh of the inequality in the distribution of annual inomes stems from people having different earnings apaities in various stages of the life yle. Hene, in the presene of well-funtioning apital markets enabling onsumers to smooth onsumption over the life yle, redistributive taxes and transfers should address inequalities in the distribution of lifetime inomes. Yet, in pratie, taxes and transfers are mostly onditioned on annual inome, with little or no regard to a person s longer-run earnings apaity. The explanation is mainly administrative beause governments rarely keep systemati reords of the earnings histories of their itizens. Moreover, sine a person s lifetime labor earnings are not fully known until the time he or she retires, the authorities annot base taxes and transfers on lifetime inome. However, it is possible to ondition publi retirement benefitsonaperson spreviousearnings. Theeffetive marginal and average tax rate on inome earned earlier in life thus beomes dependent on earnings in other periods of life. In fat, retirement benefits in many ountries do to some extent depend on previous earnings. Moreover, with modern information and ommuniation tehnologies, information on individual earnings histories beomes muh easier to gather and store. The question whether an optimal tax-transfer system should exploit information on lifetime earnings therefore beomes relevant. This paper addresses this issue. In partiular, we study whether soial insurane benefits aimed at ompensating for a loss of earnings apaity should depend on previous labor inome. Although for the sake of onreteness we label the shok to earnings apaity as disability, our analysis applies also to other types of idiosynrati shoks to human apital. In our model, people partiipate in the labor market for two periods, but some people beome disabled in the seond period. The government wants to redistribute inome for two reasons: first, to redue inequalities stemming from exogenous differenes 2

in produtivities at the beginning of the working life and, seond, to ompensate unluky individuals who beome disabled during their areer. In the late stage of life, able individuals reeive an ordinary retirement benefit, while the disabled ollet a speial disability benefit. Both types of benefits may be onditioned on previous earnings. We show that the optimal disability benefit should inrease more strongly with previous inome than the ordinary retirement benefit. In this way, the government an provide disability insurane to not only the low-skilled but also to the high-skilled, while at the same time improving the first-period labor-supply inentives of the high-skilled. By thus basing seond-period transfers on first-period earnings, the optimal tax-transfer system involves lifetime taxation rather than annual taxation. In the presene of distortionary labor taxes aimed at redistribution from the high-skilled to the low-skilled, optimal disability insurane is only imperfet. The reason is that imperfet disability insurane enourages young workers to inrease their first-period earnings by working harder. By raising their labor supply, workers an improve their insurane against disability beause the disability benefit inreases more strongly with previous inome than the ordinary retirement benefit olleted by able workers. Our analysis thus shows that full disability insurane is not optimal. Thus, even though the private market ould implement full disability insurane (sine moral hazard is absent in our model), this would not be optimal beause private insurers would fail to internalize the external effets of additional disability insurane on the base of the redistributive labor tax. The government thus faes an inentive to prevent private insurane ompanies from fully insuring disability. Indeed, a mix of a publi tax-transfer system offering less than full insurane and self insurane through preautionary saving is optimal. The optimal tax literature has onsidered linear as well as non-linear tax systems. Real-world tax systems are typially piee-wise linear. In fat, reent deades have witnessed a trend towards more linearity, as governments have flattened their tax shedules and redued the number of inome brakets to simplify the tax system. Against this bakground, we onsider a linear tax-transfer system with a onstant marginal tax rate. However, by tying soial insurane benefits to previous earnings, the poliy maker in our model an differentiate the effetive marginal tax rate on labor inome aording to lifetime earnings apaity. Our analysis shows that it is indeed optimal to exploit 3

opportunities for suh differentiation. The literature on lifetime inome taxation is quite sparse. Vikrey (939, 947) made early ontributions to the normative theory of lifetime inome taxation. He was onerned about the overtaxation of flutuating as opposed to stable inomes under a progressive annual inome tax with a marginal tax rate that rises with the level of inome. Vikrey therefore proposed an inome-averaging sheme in whih annual inome taxes are in fat olleted as a form of withholding for lifetime inome tax alulations that are ompleted only upon death. Diamond (23, h. 3 and 4) analyzes lifetime inome taxation in a two-period setting, but without allowing for early retirement due to disability. He finds that the optimal nonlinear lifetime inome tax tends to imply greater equality of onsumption levels among retirees than among workers, assuming that the elderly tend to be more risk averse than younger people. Intuitively, when the marginal utility of onsumption delines faster for the elderly, the soial planner is more eager to avoid inequality of onsumption opportunities among the elderly than among younger people. A paper more losely related to the present one is that of Diamond and Mirrlees (978), who analyze optimal soial insurane in a two-period model in whih agents an hoose their retirement age endogenously, but may also be fored to retire early due to an exogenous risk of disability. One of the results derived by Diamond and Mirrlees is that agents who suffer disability early in life should reeive a larger net transfer from the government than those able to work until later in life. The optimal soial insurane sheme subsidizes those who retire early, although only to the extent ompatible with maintaining inentives to work. This result is onsistent with the analysis in the present paper. In some respets, the model of Diamond and Mirrlees (op.it.) is more general than the one presented here, sine they allow for a fully non-linear tax sheme (inluding a apital inome tax). However, whereas Diamond and Mirrlees assume that all able workers feature the same produtivity, we allow for different skill levels. In our model, the government thus employs its redistributive poliy instruments to insure against not only skill heterogeneity but also disability risk. We thus integrate the onventional analysis of optimal redistributive taxation with the analysis of optimal soial insurane. Moreover, by employing Epstein-Zin preferenes (see Epstein and Zin (989)), we are 4

able to provide a detailed haraterization of the optimal tax and subsidy rates. Reent ontributions to the literature on soial insurane based on mandatory individual savings aounts also onsider redistribution poliy in a lifetime perspetive (see, e.g., Fölster (997, 999), Orszag and Snower (997), Feldstein and Altman (998), Fölster et al. (22), Stiglitz and Yun (22), Sørensen (23) and Bovenberg and Sørensen (24)). These papers analyze poliy shemes in whih workers must ontribute a fration of their earnings to an individual savings aount that is debited when the owner draws ertain soial insurane benefits. At the time of retirement, any surplus on the aount is onverted into an annuity and added to the ordinary publi retirement benefit. If the aount is negative, the owner is still guaranteed a minimum publi pension. Bovenberg and Sørensen (op.it.) show that the introdution of suh a system as a supplement to the onventional tax-transfer system improves the equity-effiieny trade-off by reduing the distortionary impat of those taxes and transfers that mainly serve to redistribute inome over the individual s own lifeyle. Mandatory individual savings aounts for soial insurane introdue an element of lifetime inome taxation by effetively onditioning retirement benefits on the individual s prior labor market performane. Intertemporally optimizing agents who are able to aumulate a surplus on their aount at the time of retirement fae redued marginal tax rates on labor effort. Individuals who end up with a surplus on their aounts and who will therefore fae stronger inentives to supply labor tend to be onentrated in the low-risk segments of the working population. This is in ontrast to the optimal tax-transfer system in the eonomy modelled here, where people who end up with a relatively low lifetime inome due to disability atually fae a lower marginal effetive tax rate on labor inome earned early in life. The ontradition is only superfiial, however. The system of mandatory savings aounts is designed for soial insurane benefits that involve a signifiant risk of moral hazard and relatively little redistribution from high to low lifetime inomes (as opposed to redistribution over the lifeyle). The present paper, however, fouses on optimal redistribution of lifetime inomes in a setting with exogenous idiosynrati shoks to human apital. In any ase, the individual aounts onsidered by Bovenberg and Sørensen (24) and the soial insurane sheme analyzed here are based on the same fundamental priniple: net benefits reeived at a later stage in life 5

vary positively with labor inome earned earlier in life so as to redue the distortions to labor supply aused by a redistributive tax-transfer system. 2. The model Individuals live for two periods. Everybody is able to work in the first period, but in the seond period individuals fae the risk of beoming disabled. Disabled individuals must finane their onsumption by saving undertaken in the first period and by a publi transfer that may be onditioned on their previous earnings. Able individuals work during (part of) the seond period. The leisure onsumed by able workers in period 2 may be interpreted as time voluntarily spent in retirement. Larger seond-period labor supply an thus be viewed as a higher retirement age. The government transfer olleted by able workers in the seond period orresponds to an ordinary retirement benefit. Also this benefit may be onditioned on previous earnings, and it may be differentiated from the disability benefit. We distinguish two skill groups (the low-skilled and the high-skilled) earning different real wage rates refleting exogenous differenes in labor produtivity. Also the real interest rate is exogenous. Indeed, our eonomy an be viewed as a small open eonomy with perfet apital mobility. 2.. Individual behavior This setion desribes the behavior of a low-skilled worker; the behavior of the highskilled is given by fully analogous relationships. A low-skilled worker s labor supply in the first period is, and his onsumption during that period is C. Ifheisabletowork in the seond period, he supplies labor 2 and onsumes an amount C2 a. If he beomes disabled in period 2, his onsumption is C2 d. His expeted lifetime utility U is given by the nested utility funtion U = U (C g ( )) + δf (E [U 2 ]), U >, U <, f >, (2.) E [U 2 ]=pu C d 2 +( p) u (C a 2 h ( 2 )), <p<, g >, g >, h >, h >, 6

where U ( ) denotes utility during the first period of life, δ adisountfator,e [U 2 ] expeted utility during the seond period, and p the probability of beoming disabled in the seond period. Utility during the first period depends on first-period onsumption, adjusted for the disutility of first-period work effort, g ( ). Similarly, for an able worker, the seond-period utility u (C2 a h ( 2 )) depends on his onsumption orreted for the disutility of his seond-period labor supply, h ( 2 ). A disabled worker obtains utility u C2 d. The speifiation in (2.) is suffiiently flexible to allow the degree of intertemporal substitutability in onsumption to deviate from the reiproal of the degree of relative risk aversion, as suggested by Epstein and Zin (989). For later purposes, we define Ud U δ p C d 2 = f pu C d 2 +( p) u (C a 2 h ( 2 )) u C d 2 >, (2.2) Ua U δ ( p) C a 2 U Ua p = f pu C d 2 +( p) u (C a 2 h ( 2 )) u (C a 2 h ( 2 )) >. d p Ua C2 a U a Ud C2 d = f u C2 d 2 f u C d + 2 p (2.3), (2.4) = f [u (C2 a h ( 2 ))] 2 + f u (C2 a h ( 2)), (2.5) p U d Uda = = f u C p C2 d p C 2 d u (C a 2 a 2 h ( 2 )). (2.6) In the speial ase in whih the reiproal of the intertemporal substitution elastiity oinides with the oeffiient of relative risk aversion, f =so that the (ex ante) marginal utility of disabled onsumption does not depend on able onsumption (i.e. Uda =). f is positive (negative) if the degree of risk aversion is greater (smaller) than the inverse of the intertemporal substitution elastiity so that the marginal utility of disabled onsumption rises (falls) with able onsumption. During the first period, the onsumer s budget onstraint amounts to C = w ( t) + G S, (2.7) where w represents the real wage rate of a low-skilled worker, t the onstant marginal tax rate on labor inome, G a lump-sum transfer, and S saving of the low-skilled worker. In the seond period, an able worker reeives a benefit onsisting of a lump-sum omponent B plus a omponent amounting to a fration s a of his earnings during the first period. 7

With r denoting the real interest rate, an able worker therefore faes the following seondperiod budget onstraint: C a 2 =(+r) S + w ( t) 2 + B + s a w. (2.8) A disabled worker reeives a benefit equal to the onstant b plus a fration s d of his previous labor inome, so his seond-period budget onstraint is: C d 2 =(+r) S + b + s d w. (2.9) The onsumer maximizes (2.) subjet to (2.7) through (2.9). Optimal seond-period labor supply implies that the marginal disutility of work equals the marginal after-tax real wage: h ( 2 )=w ( t). (2.) The first-order ondition for optimal saving is given by h i δ( + r) pu d +( p)u a U =, (2.) where U represents the marginal utility of first-period onsumption of the low-skilled worker. U d and U a are defined in (2.2) and (2.3), respetively. The first-order ondition for optimal first-period labor supply amounts to h i [w( t) g ( )] U + δw ps d U d +( p)s a U a =. (2.2) Part of the benefit of first-period labor supply arues in the seond period if disability and retirement benefits rise with earnings (i.e. s a,s d > ). Substituting(2.)into(2.2) to eliminate U, we an write (2.2) as where with w( ˆt )=g ( ), (2.3) µ ˆp ˆt s d +( ˆp )s a = t, (2.4) +r pu d ˆp = pu d +( p)u. (2.5) a The variable ˆp an be viewed as the risk-neutral probability of beoming disabled for the low-skilled worker, so that ˆt may be interpreted as a risk-adjusted (ertaintyequivalent) marginal effetive tax rate on first-period labor inome for the low-skilled 8

worker. The risk-neutral probabilities differ from real-world probabilities if agents are risk-averse and not perfetly insured (so that U d 6= U a ). If, for example, sd >s a and U d >U a, the individual an enhane the insurane against disability risk by raising firstperiod labor supply. Ex post, the effetive marginal tax rate on first-period inome for a ³ disabled worker t sd then differs from the orresponding effetive marginal tax rate +r for an able worker t +r sa.bydifferentiating s d from s a, the government thus makes the marginal tax rate on first-period inome depend on seond-period inome. In other words, marginal and average tax rates depend on lifetime earnings. A key issue addressed in this paper is whether suh lifetime inome taxation s d 6= s a is in fat optimal and if so, whih fators determine the optimal gap between s d and s a. For welfare analysis, we employ the onsumer s indiret lifetime utility funtion, whih exhibits the form V = V G, b, B, t, s d,s a, (2.6) with the derivatives (denoted by subsripts and found by applying the Envelope Theorem): V G = U, V b = δpu d, V B = δ ( p) U a, (2.7) V t = w U δw 2 ( p) U a, V s d = δpw U d, V s a = δ ( p) w U a. (2.8) 2.2. The government Setting aside issues of intergenerational redistribution, we assume that the present value of the taxes levied on eah generation equals the present value of transfers paid to that generation. This implies that the generational aount of eah ohort is zero. The highskilled are paid the wage rate W>w, and a high-skilled worker s labor supply is denoted by L. The exogenous fration of low-skilled individuals in eah ohort is α. Both skill types fae the same probability p of disability in the seond period of life. Normalizing the size of the ohort to unity, and using subsripts to indiate time periods, we an write the onstraint that a ohort s generational aount must be zero as generational aount of a low-skilled worker z µ } µ { p p b α tw + (tw 2 B s a w ) + s d w G + +r +r 9

generational aount of a high-skilled worker z µ } µ { p p b ( α) tw L + (tw L 2 B s a WL ) + s d WL G =. +r +r (2.9) Assuming that disability annot be verified, the government also faes the inentive ompatibility onstraint that an able worker should have no inentive to mimi a disabled worker. In other words, the seond-period utility of a mimiker should be no higher than the seond-period utility of a non-mimiker. For low-skilled workers, the resulting nonmimiking onstraint is given by u ( + r) S + w 2 ( t)+b + s a w h ( 2 ) u ( + r) S + b + s d w Z w 2 ( t) h ( 2 )+B b + s a s d w, (2.2) and for high-skilled workers the analogous onstraint amounts to Z h WL 2 ( t) h (L 2 )+B b + s a s d WL. (2.2) The government maximizes the utilitarian sum of expeted lifetime utilities. With V and V h indiating the utility of a low-skilled and that of a high-skilled worker, respetively, we write the utilitarian soial welfare funtion (SWF) as SWF = αv G, b, B, t, s d,s a +( α) V h G, b, B, t, s d,s a, (2.22) whih must be maximized with respet to the poliy instruments G, b, B, t, s d,s a,subjet to the onstraints (2.9), (2.2) and (2.2). 3. Optimal taxation and soial insurane 3.. The optimality of soial insurane through lifetime inome taxation The first-order onditions for the solution to the poliy problem stated in the previous setion are given in setion A.3 of the appendix. Before exploring the impliations of these optimality onditions, we demonstrate that a lifetime inome tax, rather than an annual inome tax, is optimal. In partiular, the government an generate a Pareto improvement Sub-setion 3.4 shows that the non-mimiking onstraint is typially met in the optimum.

by moving from a onventional tax-transfer system based on annual inomes only (i.e. s d = s a ) towards lifetime inome taxation with s d >s a. Indeed, with s d >s a, the expost effetive marginal tax rate on first-period labor inome depends on lifetime earnings apaity. Moreover, seond-period transfers are based not only on the earnings in that period, but also on the earnings in the first period. Hene, the government implements lifetime inome taxation. To prove these results, we start out from a situation with annual inome taxation (s = s d = s a ), where the government has optimized the other poliy instruments in a manner respeting the non-mimiking onstraints (2.2) and (2.2). With annual inome taxation, it is optimal to inrease b and to redue B in a balaned-budget manner suh that the non-mimiking onstraint for the low-skilled worker beomes binding. The reason is that enhaning disability insurane in this way does not affet labor-supply inentives if s a = s d (sine (A.2) and (A.3) in the appendix imply that labor supply does not respond to b and B with annual inome taxation). In the absene of a trade-off between inentives and insurane, full disability insurane for the low-skilled is optimal. With s d = s a, only the low-skilled an be fully insured against disability (i.e. Z implies Z h > (and hene U dh U ah > ), sine WL 2 ( t) h (L 2 ) >w 2 ( t) h ( 2 )). 2 Intuitively, ompared to the low-skilled, the high-skilled lose more earnings in ase of disability, but reeive the same ompensation b B if s d = s a. Starting from an equilibrium with annual taxation, we onsider a poliy experiment involving an inrease in s d and a derease in s a alibrated so as to keep the average subsidy rate es ps d +( p) s a onstant, that is, a poliy hange satisfying µ p des = = ds a = ds d, ds d >. (3.) p At the same time, the government adjusts the poliy instrument b to satisfy the binding non-mimiking onstraint (2.2). Realling that s d = s a initially, and using (3.) to eliminate ds a, this requires db w ds d ds a µ w = = db = ds d. (3.2) p Finally, G is adjusted to keep the utility of the low-skilled agents onstant, given the poliy hanges speified in (3.) and (3.2). Using the expressions for V G and V b given in 2 The Envelope Theorem implies that the surpluses WL 2 ( t) h (L 2 ) and w 2 ( t) h ( 2 ) are inreasing in the pre-tax wage rate. W>wthus implies that WL 2 ( t) h (L 2 ) >w 2 ( t) h ( 2 ).

(2.7), and noting from (2.) that full insurane implies that Ud = U a = U /δ ( + r), we find that the required hange in G is dg = p db. (3.3) +r Using (2.9), one an easily show that the poliy hanges desribed by (3.) through (3.3)havenodiretimpatonnetgovernmentrevenuesothattherevenueeffet of the poliy reform depends only on labor supply responses. With a binding non-mimiking onstraint (2.2) (and thus full disability insurane of the low-skilled (i.e. U d = U a )), (2.4) implies that the hanges in s d and s a satisfying (3.) will not affet the effetive tax rate bt and hene will not affet, aording to (2.3). Furthermore, sine t is unhanged, it follows from (2.) that also 2 and L 2 are onstant, while (A.2) and (A.4) in the appendix imply that L = L =when b G sd = s a. Aording to (A.6) and (A.7) in the appendix, the hanges in s d and s a will affet the first-period labor supply of high-skilled workers in the following manner: µ µ L bp h L s = d +r, L bp h L s = a +r, (3.4) where L < is the ompensated response of first-period high-skilled labor supply to a hange in the ordinary tax rate t. Using (3.), (2.), and (2.5), and realling that Udh U ah >, we an write the (unompensated) labor-supply response as L dl = s + L µ µ ds a ds d = pδ L U dh Uah ds d >. (3.5) d s a ds d Uh Thus, high-skilled labor supply expands. Intuitively, when disability insurane is linked more losely to first-period labor effort, high-skilled workers an enhane their disability insurane by working harder. The improved labor-supply inentives benefit the government budget as long as t = t s >. +r Atthesametime,thehangesinb, G, s a, and s d inrease the lifetime utility of high-skilled workers, sine it follows from (2.7), (2.8) and (3.) through (3.3) that 3 dv h = pδ(wl w )(U dh U ah) ds d >. (3.6) We onlude that moving from annual to lifetime taxation in this way enhanes both labor-market inentives and disability insurane for the high-skilled. Lifetime taxation 3 We use the fat that the derivatives of the indiret utility funtion of the high-skilled are given by expressions analogous to (2.7) and (2.8). 2

thus improves the trade-off between insurane and inentives. Even without redistributive motives (i.e. t =), lifetime inome taxation dominates annual taxation beause of the possibility to offer better disability insurane for the high-skilled without violating the non-mimiking onstraint for the low-skilled. These arguments are strengthened if redistributive taxation distorts labor supply. In that ase, lifetime taxation not only improves disability insurane, but also alleviates the labor-market distortions imposed by redistributive taxation. 3.2. The suboptimality of full insurane We now proeed to show that full disability insurane of both skill groups an never be optimal, even though separate linear tax shedules for the high-skilled and the low-skilled allow for full insurane. To prove this result, we show that starting from an equilibrium with full insurane of both skill groups, we an design a poliy reform that leaves the utility levels of both groups unaffeted, while at the same time raising publi revenue. We start by noting that if both skill groups are fully insured (so that the nonmimiking onstraints are both met with equality), we may add (2.2) and (2.2) to obtain (WL 2 w 2 )( t) [h(l 2 ) h( 2 )] = (s d s a )(WL w ). (3.7) Sine the left-hand side is positive (see footnote 2), and first-period skilled earnings exeed the orresponding unskilled earnings (i.e. WL >w ), this expression implies that s d >s a. Intuitively, ompared to the low-skilled, high-skilled households fae a larger inome loss if they beome disabled. Hene, if low-skilled agents are fully insured against disability risk, the disability benefit must rise more with earnings than the retirement benefit does, so as to ensure that also the high-skilled agents are not hurt should they beome disabled. We now make disability insurane less than perfet by reduing b and inreasing G. We redue disability insurane in suh a way that the lifetime utility of both households remains onstant. Using the expressions for VG and Vb given in (2.7), along with the analogous expressions for the high-skilled group, and noting from (2.) that full insurane (i.e., U d = U a)impliesthatu d = U /δ ( + r) for both skill groups, we find that suh a poliy reform must satisfy expression (3.3). From the government s perspetive, 3

the effetive marginal tax rate on first-periodlaborinomeis(see(2.9)) t t es +r, es psd +( p) s a, (3.8) where es denotes the expeted seond-period subsidy rate on first-period inome. With this definition of the first-period marginal tax rate, the overall impat of the poliy reform on the government budget (2.9) an be written as t w[αd +( α)dl ]. While (3.3) ensures that the diret effet on the budget is zero, (2.) implies that seond-period labor supply remains onstant beause the tax rate t is unaffeted. The government budget thus improves if the first-period labor supply of both skill types inreases (under the assumption t > ; sub-setion 3.3 below shows that t is indeed typially positive in the optimum). Given the relationship between dg and db implied by (3.3), labor supply does in fat inrease, beause setion A.2 of the appendix establishes that p +r G b > and p L +r G L b > for sd >s a. (3.9) The improvement of the publi budget resulting from the utility-preserving poliy reform (3.3) would enable the government to engineer a Pareto improvement (say, by raising G by more than implied by (3.3)). This shows that the starting point haraterized by full insurane of both skill groups annot be a soial optimum. The intuition for this result is the following: by reduing disability insurane through autinb, the government stimulates labor supply and thus expands the base of the labor tax beause agents an partly undo the worsening of disability insurane by working harder in the first period if s d >s a a ondition that must be met in the initial equilibrium with full insurane. Given an initial equilibrium with full disability insurane, the welfare loss from redued insurane is only seond order, whereas the expansion of the labor inome tax base generates a first-order welfare gain if t >. In other words, disability insurane should be less than perfet if the government also wants to insure against skill heterogeneity through a positive labor inome tax rate redistributing resoures from high-skilled to low-skilled agents. The government thus faes an inentive to prevent private insurane ompanies from fully insuring disability. This enourages individuals to self-insure through preautionary individual saving and to improve their benefits from publi disability insurane through additional work effort when young (if s d >s a ). Although we do not model moral hazard 4

in disability insurane, full insurane is thus not optimal. The reason is that private insurane against disability generates a negative fisal externality on the base of the distortionary tax offering insurane against skill heterogeneity. With endogenous labor supply and lak of publi information on individual work effort, this publi insurane of skill heterogeneity does generate moral hazard. 4 3.3. The optimal marginal tax rates Expressions for the optimal (effetive) marginal inome tax rates are derived in setion A.3 of the appendix. If the high-skilled are less than fully insured against disability, the optimal marginal tax rate on seond-period labor inome is given by 5 t t = ( β 2) α h 2 ( α), (3.) ε 2 β 2 w 2, ε 2 αβ WL 2 ε 2 +( α) ε 2h, α h 2 δ ( + r) U ah + t W 2 λ µ +r L p B. The variable α h 2 in (3.) measures the marginal soial valuation of seond-period inome for an able high-skilled worker (aounting for the impat on the publi budget through the indued inome effet on labor supply). ε 2 and ε h 2 denote the wage elastiities of seond-period labor supply for the low-skilled and the high-skilled, respetively, so that ε 2 is a weighted elastiity of seond-period labor supply. β 2 measures the degree of inequality in the distribution of seond-period pre-tax labor inome. The optimal value of t depends only on variables relating to the seond period. The reason is that first-period labor supply is determined by bt rather than t. By varying s d and s a, the government an manipulate bt independently from t (see (2.3) and (2.4)). The optimal effetive marginal tax rate on first-period labor inome (defined in(3.8)) is given by an analogous expression if both skill groups are less than fully insured against 4 For the external effets between insurers in the presene of moral hazard, see Pauly (974) and Greenwald and Stiglitz (986). 5 The next sub-setion shows that the onditions for both skill groups to be less than fully insured in the optimum are weak. If the non-mimiking onstraint for the high-skilled would nevertheless be ³ ³ binding, we must define α h 2 δ(+r)u ah +r λ + t W L p B + µh +r α p W (s d s a ) L B,where µ h is the shadow prie assoiated with the non-mimiking onstraint for the high-skilled. All other expressions are unaffeted. 5

disability: 6 t = ( β ) α h ( α) t ε, (3.) β w, ε αβ WL ε +( α) ε h, α h U h λ + t W L G. The inequality in the distribution of first-period labor inome enters through the variable β. During the first period, both inome and substitution effets affet labor supply. Nevertheless, the optimal marginal tax rate depends only on substitution effets, aptured by the weighted average (ε ) of the ompensated skill-speifi labor supply elastiities, ε and ε h.thevariableαh measures the marginal soial evaluation of first-period inome for a high-skilled worker taking the tax-base effet into aount. Inthenormalase,thegovernmentwishestoredistributeinomesothatα h i <, i =, 2. 7 (3.) and (3.) then imply that the optimal marginal tax rates are positive. Moreover, eteris paribus the elastiities and the marginal soial evaluations, these optimal tax rates inrease with the degree of inequality in the distribution of pre-tax inome. Furthermore, a larger fration of high-skilled workers in the labor fore α broadens the base for redistribution, making it worthwhile to impose a higher marginal tax rate. 8 Aording to (3.) and (3.), the government typially wants to impose different marginal effetive tax rates on inome in the two periods by hoosing a non-zero value of 6 As already mentioned, the next sub-setion shows that the onditions are weak for both skill groups to be less than fully insured in the optimum. 7 Expressions (A.6) and (A.8) in the appendix imply that the marginal soial evaluation averaged over the low- and high-skilled is unity: α α i +( α) αh i =(where α i is defined analogously as αh i ³ : α U λ + t w G and α 2 δ(+r)u a +r λ + t w p B ). 8 Although derived in an intertemporal ontext, the formulas (3.) and (3.) are losely related to the formula for the optimal linear inome tax obtained by Dixit and Sandmo (977) for the ase with many skill groups in a one-period setting. In the Dixit-Sandmo world, the optimal marginal tax rate on labor inome is given by t α i t = ov,w i L i E (w i L i ε i ) where ov α i,w i L i is the ovariane between the marginal soial evaluation of inome for skill group i (aounting for the impat on the publi budget via the indued inome effet on labor supply) and the pre-tax labor inome w i L i of that skill group, and E w i L i εi is the inome-weighted average ompensated labor supply elastiity aross skill groups. In fat, (3.) and (3.) an be written in this form by using expressions (A.6) and (A.8) in the Appendix, whih imply that the marginal soial evaluation averaged over the low- and high-skilled is unity (see the previous footnote). 6

the average subsidy rate es (sine t t s,whileβ +r, α h and ε generally differ from β 2, α h 2 and ε 2 ). Ceteris paribus β i and α h i ; i =, 2, if the labor supply of older workers is more wage elasti than that of younger workers (i.e ε 2 >ε ), effiieny onsiderations ause the optimal t to be below the optimal t. Ceteris paribus the elastiities and the marginal soial evaluations, α h i, distributional onsiderations reinfore this tendeny if first-period labor inome is more unequally distributed than seond-period labor inome (i.e. β <β 2 ). 3.4. The optimal level of soial insurane The previous sub-setion assumed that neither the low-skilled nor the high-skilled were fully insured. This sub-setion states the onditions under whih imperfet insurane of both skill groups is indeed optimal. Setion A.4 in the appendix employs the firstorder onditions for the solution to the optimal tax problem to derive expressions for the marginal utility differentials Ud U a and U dh U ah, assuming that no skill group faes a binding non-mimiking onstraint, i.e., that no group is fully insured. If the resulting expressions are positive, this validates the initial assumption of imperfet insurane. For the low-skilled group, the assumption that no group faes a binding non-mimiking onstraint gives rise to (see setion A.4 of the appendix) Ud U a = s d s a µ ½ µ µ µ λt h ¾ ( β Ψ ) wω λt ε + h α wω + W Ω, t Uh α µ µ (3.2) λt ε Ψ β + h β ε, t Uh U where Ω and Ω h are positive magnitudes that depend on the properties of the utility funtion (see eq. (A.37) in the appendix). Sub-setion 3. demonstrated that the optimal poliy involves s d >s a. The expression on the right-hand side of (3.2) is therefore positive if Ψ is positive. In view of the definition of Ψ, the onditions on ε h and ε for this to be the ase are very weak, sine β < and U >U h. Aordingly, the low-skilled are imperfetly insured against disability as long as t >. Redistributive taxation thus makes imperfet disability insurane optimal. For high-skilled workers, the assumption that no skill group faes a binding non- 7

mimiking onstraint implies that (see setion A.4 of the appendix) Udh U ah = s d s a µ ½ µ µ λt ( β Ψ ) W Ω h λt β ε t Inserting (3.) into (3.3) to eliminate t t,weobtain ½ ( β ) W Ω h α h U dh U ah = s d s a µ λt Ψ U µ β λε U ε µ α α wω + W Ω h ¾. (3.3) αwω +( α) W Ω h ¾. (3.4) The onditions for the right-hand side of (3.4) to be positive are weak, sine W>w, α h, and U /λ > (if L ). In partiular, the ondition is met if G Ω does not greatly exeed Ω h (implying that imperfet insurane of the low-skilled does not provide muh stronger inentives than imperfet insurane of the high-skilled) and inequality is high so that β is small. Intuitively, high inequality drives up the marginal tax rate, thus distorting labor supply. To offset this distortion, the government finds it optimal to offer only imperfet disability insurane to skilled agents in order to indue these agents to work harder in the first period so as to obtain better disability insurane in the seond period. Indeed, equations (3.2) and (3.3) show that full disability insurane is optimal if the government does not employ distortionary taxes to redistribute aross skills (i.e. if t =beause β =,α h =, or α =). Hene, disability insurane is imperfet to the extent that it helps to alleviate the labor-market distortions imposed by redistributive taxation. In the absene of these distortions, the government would struture its publi transfers so as to provide full disability insurane to both skills. 4. Conluding remarks This paper studied optimal lifetime inome taxation and soial insurane in an eonomy where publi poliy insures (from behind the veil of ignorane ) both skill heterogeneity and exogenous disability risk. Although the government has at its disposal suffiient poliy instruments to insure both skill groups fully against disability, and even though moral hazard in disability is absent, full disability insurane is not optimal. Instead, by offering imperfet insurane and struturing disability benefits so as to enable workers to improve their insurane against disability by working harder, the government an alleviate the distortionary impat of the redistributive labor inome tax. Speifially, optimal 8

disability insurane should allow disability benefits to vary positively with previous earnings. Hene, the effetive marginal tax rate depends on the taxpayer s lifetime earnings apaity, and redistribution is based on lifetime inomes. Lifetime taxation improves the trade-off between insurane and inentives. It provides better disability insurane for the high-skilled and enhanes their inentives to supply labor, thereby alleviating the labor-market distortions imposed by redistributive taxation. To allow a detailed haraterization of the optimal tax and subsidy rates, we have restrited the analysis to a linear tax-transfer system with ertain non-linear elements. We did not study the potential seond-best role of apital inome taxation in the overall tax-transfer system. Sine preautionary saving allows people to partly insure against shoks to their human apital, the government may hoose to distort saving. In future work we plan to extend the analysis to a fully non-linear tax system that also allows for apital inome taxation distorting saving behavior. 9

Tehnial Appendix This appendix derives the effets of the various poliy instruments on individual labor supply and the first-order onditions for the solution to the optimal tax problem. We then use these relationships to prove some results reported in the main text. A.. The effets of taxes and transfers on labor supply We onsider the labor supply of the low-skilled group; the labor supply of high-skilled workers is haraterized by ompletely analogous expressions. For onveniene, we drop the subsript in terms involving derivatives of the utility funtion. To find the elastiities of first-period labor supply and saving with respet to the poliy variables, we totally differentiate (2.) and (2.2) to arrive at a G ( + r)(a b + a B ) a G s w a b s d w a B s a w a 2G ( + r)(a 2b + a 2B ) g ( )U a 2G s w a 2b s d w a 2B s a w = S, ds d (A.) where S a G dg + a b db + a B db +(a G w a B w 2 )dt + a b w ds d + a B w ds a, L a 2G dg + a 2b db + a 2B db +(wu + a 2G w a 2B w 2 )dt +(a 2b w δwpu d)ds d +(a 2B w δw( p)u a)ds a, s ˆp s d +( ˆp )s a, a G U, a 2G = swu, +r a b δ( + r)p [pud +( p)uda], a 2b δwp ps d Ud +( p)s a Uda, a B δ( + r)( p)[pu da +( p)u a ], a 2B δw( p) ps d U da +( p)s a U a. Applying Cramer s Rule to the system (A.), we an find the various labor-supply effets from the system 2

ds d = g ( )U a 2G s w a 2b s d w a 2B s a w a G s w + a b s d w + a B s a w a 2G +(+r)(a 2b + a 2B ) a G ( + r)(a b + a B ) S (A.2) where the determinant of the Jaobian is positive beause of the seond-order ondition for individual optimization. From this solution, we find = w G w 2 B = + w s d s d b (A.3) = + w s a s a B (A.4) = [a G +(+r)(a b + a B )] wu (A.5) s = δwpu d [a G +(+r)(a b + a B )] = δpu d d U = ˆp +r, (A.6) where the last equality follows by substituting (2.) to eliminate U and using (2.5). Similarly, we find s = δw( p)u a[a G +(+r)(a b + a B )] a = δ( p)u a U = ˆp +r, (A.7) whilethevariousinomeeffets are given by so that G = ( + r)[a 2G(a b + a B ) a G (a 2b + a 2B )], (A.8) b = a 2Ga b a G a 2b +(+r)[a 2B a b a B a 2b ], (A.9) B = a 2Ga B a G a 2B ( + r)[a 2B a b a B a 2b ], (A.) µ G =(+r) b +. (A.) B 2

Note that with s = s a = s d (so that first-period labor supply does not at as insurane against disability), we have a 2i = sw a (+r) i, i= G, b, B and thus = = =. G b B Intuitively, saving rather than labor supply is adjusted to realloate onsumption intertemporally. This is also the intuition behind (A.): if the onsumer reeives additional lump-sum inome in both states in the seond period (i.e. db = db > ), shewill respond in the same way as if that inome omes in the first period (disounted properly with +r so that dg = db = db ). The onsumer will simply undo realloation of +r +r lump-sum inome dg = db = db over the life yle through saving behavior as long +r +r as the generational aount is not affeted. By substituting the definitions of a ij into the solutions for the inome effets on labor supply, we find: b = s a )p 2 ( p) δw(sd B = δw(sd s a )p( p) 2 Ua a U d G (pu d +( p)u a) ³ U da U a U d + Ud δ( + r) 2 ( p)[ua Ud (Uda )2 ] Ua a U d G (pu d +( p)u a) ³ U da U d U a + Ua δ( + r) 2 p [Ua Ud (Uda )2 ] (A.2) (A.3) µ G = (+r) d db + d db = δ( + r)w(sd s a ) a G p( p)uau d pu d +( p)u a ½ pu d Ud ( p)u a U a Uda[ p Ua ( p) ] Ud ¾ (A.4) From (2.2) through (2.6) one an show that U da = u C2 d > and Ua U d Ud pu Uda C2 d U U a d Ua Moreover, onavity of the utility funtion implies that U = u (C a 2 h ( 2)) ( p)u (C a 2 h ( 2)) >. a Ud (Uda )2 >. It then follows from (A.2) that a higher transfer to the disabled (b) redues labor supply if s d >s a. Intuitively, labor supply helps to insure disability if s d >s a. In that ase, more insurane through a higher b makes labor supply less attrative. Similarly, a higher transfer to the able (B) implies that disability is less well insured, and aording to (A.3) labor supply therefore inreases to better insure disability (if s d >s a so that labor supply helps to insure disability). Note that there are two terms in the expressions for d db and d db. The 22

term inluding a G depends on intertemporal substitution (and also on risk aversion), while the other term (inluding Ua Ud (Uda )2 ) depends only on risk aversion. With higher b, the onsumer wants to spread the welfare gain to the able state if risk aversion is positive (this is the term with Ua Ud (Uda )2 )andtothefirst period (via inreased first-period onsumption of leisure as well as material goods) if intertemporal substitution is finite. The latter effet is aptured by the term with a G, whih is positive only if risk aversion is orrespondingly positive; otherwise, the onsumer an better realloate resoures to the first period through dissaving rather than by lowering first-period labor supply. A higher first-period transfer G depresses first-period labor supply if higher inome boosts utility (espeially in the disabled state (U d >U a)) and onsumption in the two states are omplements (i.e. Uda > beause risk aversion exeeds the inverse of intertemporal substitution), and the intertemporal substitution elastiity is finite (i.e., a G > ). If Uda =, a higher transfer may atually raise first-period labor supply if additional seondperiod inome espeially leads to a rapid fall in utility in ability (i.e., ( Ua ) is large ompared to ( Ud )) so that it beomes attrative to realloate inome to the disabled state. Notethatthesignoftheinomeeffet on labor supply is different from normal. This is beause labor supply has an insurane funtion. A.2. The suboptimality of full insurane We may now derive the result stated in eq. (3.9) whih was used to demonstrate that full insurane of both skill groups annot be optimal. From (A.2) through (A.4), we have = δw(sd s a )p 2 ( p) 2 p +r ½ G b = p B ( p) b ag U au d X pu d +( p)u a X U da U d + U da U a + δ( + r) 2 U a U d (U da) 2 ¾, U d U d U a. U a Using the definitions in (2.2) through (2.6), we find that " X u (C2 a = h ( # 2)) ( p)u (C2 a h ( 2)) + u C d 2 pu >. C2 d (A.5) Sine onavity of the utility funtion implies Ua Ud (Uda )2 >, it then follows from 23

(A.5) that p +r G b as reported in (3.9). A.3. The optimal labor inome tax rates > for sd >s a. A similar result holds for the high-skilled group, The optimal tax problem is to maximize the soial welfare funtion (2.22), subjet to the onstraints (2.9), (2.2) and (2.2). Using (2.7) and (2.8) together with the results (A.3) through (A.7), we may write the first-order onditions for the solution to this problem as follows (where the subsript (h) refers to the low-skilled (high-skilled), the supersript indiates a ompensated labor supply response, and λ, µ,andµ h are the shadow pries assoiated with the government budget onstraint and the non-mimiking onstraints for the low-skilled and the high-skilled, respetively (note that seond-period labor supply is not affeted by inome effets)): 9 G: αu +( α) Uh + λt αw G +( α) W L G = λ + µ w s d s a G + µh W s d s a L G, (A.6) b : δp [αu d +( α) U dh]+λt αw = pλ +r + µ +w s d s a b + µ h b +( α) W L b +W s d s a L b, (A.7) = B: δ ( p)[αua +( α) Uah]+λt αw B +( α) W L B µ w s d s a µ h ( p) λ +r B W s d s a L B, (A.8) t: α [w U + δw 2 ( p) Ua ]+( α)[wl Uh + δwl 2 ( p) Uah] ½ µ ¾ p = λ αw +( α) WL + [αw 2 +( α) WL 2 ] +r µ +λα t w w G w 2 + tw B µ p +r 2 9 (A.6), (A.7) and (A.8) are not independent equations. To see this, add (A.7) and (A.8), multiply the result by ( + r), and use (A.) and (2.) to arrive at (A.6). The government thus has only two independent lump-sum instruments. 24

= µ µ L +λ ( α) t W WL L G WL L p 2 + tw B +r µ w 2 µ w s d s a µ w G w 2 B s d : µ h WL 2 µ h W s d s a µ L WL L G WL 2 δp [αw Ud +( α) WL Udh]+λαt w w +λ ( α) t W µ L bp h WL b +r L B µ bp b +r L µ pλ [αw +( α) WL ]+µ w + µ w s d s a w +r b s a : µ ( p) λ = +r +µ h WL + µ h W s d s a µ L bp h WL b +r L δ ( p)[αw Ua +( α) WL Uah]+λαt w w B +λ ( α) t W WL L B µ bp h L +r [αw +( α) WL ] µ w + µ w s d s a w L2, (A.9) µ bp +r, (A.2) µ bp +r µ bp B +r µ h WL + µ h W s d s a µ L bp h L WL B, (A.2) +r where t is definedin(3.8).inadditiontomeetingthesefirst-order onditions, the solution to the optimal tax problem must also satisfy the omplementary slakness onditions: µ, Z, µ Z =, (A.22) µ h, Z h, µ h Z h =. (A.23) To find the optimal marginal tax rate on seond-period labor inome (t), we start by adding the first-order onditions (A.2) and (A.2), multiplying by +r, and using (A.) and (2.) (for both households) to obtain αu w +( α) UhWL + λt αw G w +( α) W L G WL λt αw +( α) W L = λ [αw +( α) WL ] 25

+µ w s d s a µ w G + µ h W s d s a µ L WL G L. (A.24) Now insert (A.24) into (A.9) to find µ p αw 2 δ ( p) Ua λ + λt w +r B +( α) WL 2 δ ( p) U ah λ µ µ p t +r = λ t µ w 2 w s d s a µ h WL 2 B ε 2 µ p +r 2 w ( t), ε h 2 w ( t) 2 + λt W L B αw 2 ε 2 +( α) WL 2 ε h 2 W s d s a L B L 2 W ( t). W ( t) L 2, (A.25) Multiplying (A.8) by w 2,weobtain µ p αw 2 δ ( p) Ua λ + λt w +r B µ p = ( α) w 2 δ ( p) Uah λ + λt W L w 2 µ + µ h +r B +w 2 s d s a µ µ w B + µh W L B. (A.26) Substituting (A.26) into (A.25) to eliminate U a, dividing through by λw L 2( α)( p) +r and rearranging, we end up with t t = ( β 2) α h 2 ( α) + ε 2 µ µ h ( + r)( β 2 ) λ ( p) ε 2 where α h 2, β 2 and ε 2 are defined in eq. (3.) in the main text. W s d s a L B, (A.27) Next we derive the optimal effetive marginal tax rate on first-period labor inome (t ). Multiplying (A.6) by w,weobtain µ αw U λ + λt w µ = ( α) w Uh λ + λt W L G G +w s d s a µ µ w G + µh W L, (A.28) G while (A.24) implies µ αw U λ + λt w = λt αw G +( α) W L 26

µ ( α) WL Uh λ + λt W L G +µ w s d s a µ w G + µ h W s d s a µ L WL G L. (A.29) Equating the right-hand sides of (A.28) and (A.29), using the fats (from the definition of t )that = = w w ( t ), L = L = W L W ( t ), (A.3) and dividing by WL,weget µ ( β )( α) Uh λ + λt W L G µ s d s a + t ε =( β ) µ h W s d s a L G µ µ β ε + µ h ε t h λ [αβ t ε +( α) ε h], w ( t ), ε h w ( t ) L W ( t ). W ( t ) L Dividing through by λ [αβ ε +( α) ε h ] in (A.3) and rearranging, we find (A.3) µ s d s a µ + λε β ( t ) ε + µ h ε h t = ( β ) α h ( α) t ε + µ h W ( β ) s d s a λε L G, (A.32) where α h and ε are definedin(3.)inthemaintext. Whennoneofthetwononmimiking onstraints are binding (that is, when it is optimal to offer less than full insurane to both skill groups), we have µ = µ h =, and (A.27) and (A.32) then redue to eqs. (3.) and (3.) in the text, respetively. A.4. The optimal level of soial insurane Finally, we derive the expressions for the optimal level of soial insurane reported in sub-setion 3.4. To investigate the onditions under whih the optimal insurane level is less than perfet, we set µ = µ h =. Dividing (A.2) by p and (A.2) by p, and subtrating the latter equation from the former, we obtain µ +λαwt w p b p δαw (Ud Ual)+δ ( α) WL (Udh Uah) L + λ ( α) Wt WL B p b 27 µ p L B