Designing Optimal Disability Insurance: A Case for Asset Testing

Transcription

1 Designing Opimal Disabiliy Insurance: A Case for Asse Tesing Mikhail Golosov Massachuses Insiue of Technology and Naional Bureau of Economic Research Aleh Tsyvinski Harvard Universiy and Naional Bureau of Economic Research We analyze an implemenaion of an opimal disabiliy insurance sysem as a compeiive equilibrium wih axes. An opimum is implemened by an asse-esed disabiliy sysem in which a disabiliy ransfer is paid only if an agen has asses below a specified maximum. The logic behind his resul is ha an agen who plans o falsely claim disabiliy (a) finds doing so unaracive if he does no adjus his savings and (b) canno collec disabiliy insurance if he does adjus his savings in he desired direcion (upward). For a calibraed economy, we find ha welfare gains from asse esing are significan. I. Inroducion The Social Securiy Disabiliy Insurance (SSDI) program is one of he larges social insurance programs in he Unied Saes. In 2001, he We hank Nancy Sokey, he edior, and four anonymous referees. We also hank audiences a he Minneapolis Federal Reserve Bank, Chicago Graduae School of Business, Berkeley, Princeon, Harvard, Massachuses Insiue of Technology, Norhwesern, Universiy of Pennsylvania, Carnegie Mellon, Wharon Business School, Columbia Business School, McGill, Rocheser, Universiy of California a Los Angeles, he 2003 Sanford Insiue of Theoreical Economics conference, he 2003 NBER Summer Insiue, and he 2001 Sociey for Economic Dynamics conference. We are graeful o Daron Acemoglu, George-Marios Angeleos, Andy Akeson, Marco Basseo, Moshe Buchinsky, Hal Cole, Amy Finkelsein, Hugo Hopenhayn, Larry Jones, Parick Kehoe, Rober E. Lucas Jr., Casey Mulligan, Lee Ohanian, Chris Phelan, Ivan Werning, and especially V. V. Chari and Narayana Kocherlakoa for heir commens. [ Journal of Poliical Economy, 2006, vol. 114, no. 2] 2006 by The Universiy of Chicago. All righs reserved /2006/ $

2 258 journal of poliical economy program provided income o more han 6 million individuals, who accouned for 14 percen of Social Securiy beneficiaries. The program cos $61 billion and consiued 15 percen of Social Securiy benefis. The size of he program far surpasses spending on unemploymen insurance or food samps (SSA 2000). As in he classic work of Diamond and Mirrlees (1978, 1986), we assume ha i is impossible o know wheher an individual is ruly disabled and ha disabiliy is a permanen sae. We hen solve a dynamic mechanism design problem and provide heoreical and numerical characerizaions of he social opimum. The firs goal of he paper is o find a ax sysem ha implemens he opimal allocaion. By implemenaion we mean finding a ax sysem such ha a soluion o a compeiive equilibrium problem wih axes coincides wih he opimal soluion. We firs show ha a sysem conjecured by Diamond and Mirrlees (1978), consising of a linear ax equal o he ineremporal wedge in he opimal allocaion, does no implemen he opimum. Then we propose a ax sysem implemening he opimum: an asse-esed disabiliy program. An asse es is a form of a means es in which a person receives a disabiliy ransfer only if his asses are below a specified hreshold. The logic behind he resul is ha an agen who plans o falsely claim disabiliy (a) finds doing so unaracive if he does no adjus his savings and (b) canno collec disabiliy insurance if he does adjus his savings in he desired direcion (upward). We hen numerically characerize feaures of he opimal allocaions and welfare gains of asse esing. To evaluae advanages of asse esing, we provide esimaes of he welfare gain obained by shifing from he opimal program wihou asse esing o he opimal program wih asse esing. The opimal program wihou asse esing is equivalen o he soluion of he opimal program wih hidden savings. The welfare gain from asse esing is hus he difference in welfare beween he opimal program wih and wihou hidden savings. In a calibraed model economy, we find a significan welfare gain from using asse esing equal o 0.5 percen of consumpion. Several papers are closely relaed o our work. Golosov, Kocherlakoa, and Tsyvinski (2003) provide a characerizaion of he opimal allocaion in an economy wih dynamic, sochasic, privae skills. Unlike his paper, heir work characerizes he opimal ineremporal wedge bu does no derive how o implemen he opimum wih a ax sysem. Albanesi and Slee (2006) and Kocherlakoa (2005) consider a ax-based implemenaion of a dynamic Mirrlees problem. Albanesi and Slee derive an implemenaion wih labor and wealh axes in an environmen wih independenly and idenically disribued skill shocks. In heir environmen, wealh summarizes agens pas hisories of shocks and allows he

3 opimal disabiliy insurance 259 definiion of a recursive ax sysem ha depends only on curren wealh and effecive labor. Their implemenaion does no work in our seup in which disabiliy is a persisen, in fac permanen, skill shock. Kocherlakoa allows for a general process for skill shocks and derives an implemenaion wih linear axes on wealh and arbirarily nonlinear axes on he hisory of effecive labor. The opimum in our model can be implemened using axes similar o he axes in ha paper. Tha implemenaion would enail a regressive wealh ax schedule in which an agen who becomes disabled has o pay a high ax whereas an able agen receives a subsidy for his savings. 1 Anoher difference from he papers by Albanesi and Slee (2006) and Kocherlakoa (2005) is ha we also focus on he quaniaive evaluaion of he welfare gains from asse esing in a calibraed muliperiod model by comparing opimal sysems wih and wihou asse esing. This paper also conribues o he sudy of opimal dynamic social insurance programs (Wang and Williamson 1996; Hopenhayn and Nicolini 1997). These papers focused on finding opimal allocaions raher han on ax sysems implemening hem. Sudying implemenaion is imporan, since he exising lieraure on opimal social insurance ofen sops a characerizing an opimal allocaion wihou sudying axes ha implemen he opimum. The difficuly ha we highligh in his paper of consrucing ransfer sysems is also presen in oher dynamic mechanism design models, such as models of opimal unemploymen insurance. The echniques ha we develop in his paper can be used in hose seings. The key o our analysis is an assumpion ha disabiliy is unobservable and permanen. In pracice, deermining disabiliy saus proves o be very difficul. Muliple medical and vocaional facors are aken ino accoun when deermining wheher an individual is eligible for disabiliy benefis. However, even he deerminaion of medical facors is ofen subjecive. In 2001, abou 50 percen of awards wen o applicans wih difficul-o-verify disabiliies, such as menal disorders (mainly menal sress; reardaion is excluded) and diseases of he musculoskeleal sysem (ypically back pain). Disabiliy is a fairly permanen sae. For example, less han 1 percen of hose who sar receiving disabiliy benefis reurn o work. 2 1 If we use his implemenaion for he economy ha we compue in our paper, he wealh axes on he disabled range from 55 percen early in life o 10 percen lae in life; he subsidy o savings of he able ranges from 0.1 percen early in life o 0.5 percen lae in life. 2 A low number of disabled reurning o work does no necessarily mean ha disabiliy is a permanen sae. I could indicae, e.g., generosiy of benefis. However, a very low number of hose reurning o work gives us confidence in modeling disabiliy as a permanen sae. For a deailed discussion of difficulies in deermining disabiliy and he daa on he number of people leaving disabiliy, see Bound and Burkhauser (1999).

4 260 journal of poliical economy The res of he paper is srucured as follows: In Secion II, we describe he seup of he model. In Secion III, we provide a heoreical characerizaion of he opimum. In Secion IV, we discuss implemenaion of he opimum. In Secion V, we provide numerical resuls. In Secion VI, we discuss he robusness of hese heoreical resuls and he role of our assumpions. II. Seup An agen lives for T periods and has preferences represened by a uiliy funcion T 1 E b [u(c ) v(l )], p1 where E denoes an expecaion operaor, 0! b! 1, and cand ldenoe, respecively, he period consumpion and labor of he agen. We assume ha u 1 0, u! 0, and v! 0. An agen can become disabled in period, and his skill, v, is equal o zero. We assume ha disabiliy is an absorbing sae and ha once disabled, an agen says disabled for he res of his life. Skills of able agens evolve deerminisically over ime. We use he following noaion for probabiliies. Le p p Pr [able a p 1], 1 p p Pr [able a Fable a 1] for p 2,, T, P p p 7 7p p Pr [able a Fable a s 1], s, s P p Pr [able a ] for p 1,, T, P p 1. 0 Because disabiliy is an absorbing sae, we need o keep rack only of he agen s age and he age a which he became disabled. We denoe consumpion of an able agen a age by c, his labor by l, and con- sumpion of an agen who became disabled a s by x s. An agen who was able a 1 learns a he beginning of period wheher or no he has become disabled. This informaion is privae: i is never observed by anybody else. Labor l is also privae informaion. Only effecive labor supply y p vl is observable o ousiders. If y 1 0, an ousider can infer ha he agen is able. If y p 0, an ousider canno ell if he agen is disabled or able bu exering no effor. A disabled worker does no exer effor since i reduces his uiliy, and y p 0 even if he exers himself. Le v (0) p 0 be he uiliy from exering no effor.

5 opimal disabiliy insurance 261 We consider a seing in which he ne ineres rae R and he wage w are consan over ime and we assume ha b p 1/ (1 R). An allo- caion of consumpion and labor (c, l, x) isfeasible if and only if T T T T 1 1 s 1 s 1 s p1 sp1 ps p1 b Pc P (1 p ) b x b Pwvl. (1) This condiion saes ha he expeced presen value of consumpion allocaions is smaller han he expeced presen value of oupu. Allocaions mus respec incenive compaibiliy condiions, since disabiliy saus is privae informaion. In paricular, since disabiliy is an absorbing sae and an agen who claims disabiliy would no laer claim o be able, 3 here are T incenive consrains. These consrains require ha in each period he expeced uiliy of working be higher han he uiliy of claiming disabiliy: T s s s s 1, ps 1 T T i s P s 1, 1(1 p) b u(x i) ps 1 ip T s s ps [u(c ) v(l )] b P [u(c ) v(l )] b u(x ) Gs, 1 s T, (2) where Pi,k p 1 if i! k. A social planner maximizes he expeced uiliy of he represenaive agen and solves he following programming problem: T T T 1 1 s s 1 s c,l,x 0 p1 sp1 ps max b P[u(c ) v(l )] P (1 p ) b u(x ) (P) subjec o he feasibiliy (1) and he incenive compaibiliy (2) consrains. III. Characerizing Pareo Opima In his secion, we provide a heoreical characerizaion of an opimal allocaion. A useful benchmark is a case in which disabiliy saus is perfecly observable. In his case, a social planner can achieve full insurance. Tha s is, for all, s ( s ), c* p x * p c ; ha is, consumpion is consan over 3 An agen previously claiming disabiliy and laer working reveals ha he has lied; hence, he planner can preven such deviaion.

6 262 journal of poliical economy ime, and consumpion of he able and disabled is equalized. The consumpion-labor margin is also no disored: 1 v (l*) p u (c*)w. v We now proceed o characerize he opimal soluion when disabiliy is unobservable. We define an allocaion (c, l, x) obeinerior if l 1 0 for all. This assumpion is saisfied when skill v is sufficienly high. In he res of he paper, we assume ha he opimum is inerior. I is easy o show ha, in an opimal allocaion, he incenive consrains in each period and he feasibiliy consrain hold wih equaliy. Subracing he firs-order condiions for x from hose for c, we also derive ha c 1 x for all. The proposiion ha follows provides a characerizaion of he opimal allocaion. We show ha he consumpion-labor margin is no disored for able agens. This resul is reminiscen of he resul ha in a saic environmen, labor decisions of he highes-skilled agen are undisored (Mirrlees 1976). The ineremporal margin, however, is characerized by he inverse Euler equaion as in Golosov e al. (2003). Afer an agen becomes disabled, all uncerainy is resolved, and here is no need o disor his ineremporal decision. Since we assumed ha b p 1/ (1 R), he consumpion of he disabled is herefore consan. Proposiion 1. Suppose ha ( c*, l*, x* ) solves (P). 1. For each period, he consumpion-labor margin of an able agen is no disored: 1 v (l*) p u (c*)w. 2. For each period! T, he inverse Euler equaion holds: 1 p 1 1 p 1 p. 1 u (c*) u (c* ) u (x *) v Consumpion of a disabled agen is consan: s s x * p x * for 1 s! T. The proof of he proposiion summarizing he characerizaion of he opimum follows from examinaion of he firs-order condiions of he planner s problem. Suppose ha he fuure disabiliy saus of an able agen is no perfecly predicable. Then we apply Jensen s inequaliy o he inverse Euler

7 opimal disabiliy insurance 263 equaion o prove ha any opimal soluion involves a wedge beween he ineremporal marginal rae of subsiuion and he ineres rae. Corollary 1. Suppose ha ( c*, l*, x* ) solves (P). Then, if he probabiliy of becoming disabled is inerior (0! p! 1), 1 u (c*)! p 1u (c* 1) (1 p 1)u (x 1*). (3) IV. Implemenaion of he Opimum In his secion we propose a ax sysem ha implemens he opimal allocaion and includes only axes and ransfers similar o hose already in he U.S. ax code. Since he only resricions on he social planner s problem are incenive compaibiliy and feasibiliy, we implicily allow for a very large se of possible axes. Because of he generaliy of axes, he social planner s allocaion can be implemened in muliple ways, he mos obvious of which is a direc mechanism. However, he direc mechanism may include axes ha have never been used in pracice. We firs illusrae a difficuly in consrucing a ax sysem wih an example of a linear savings ax as in Diamond and Mirrlees (1978). This ype of implemenaion is common in he Ramsey lieraure of opimal axaion (see a review in Chari and Kehoe [1999]). We show ha such a ax does no implemen he opimum, since i canno preven agens from overaccumulaing asses and falsely claiming disabiliy. We hen propose a ax/ransfer sysem ha implemens he opimum: an asse-esed disabiliy sysem. The firs feaure of his sysem is ha disabiliy ransfers depend on he lengh of predisabiliy work hisory. The second feaure of he sysem is ha i conrols negaive effecs of savings on incenives: disabiliy ransfers should be asseesed, ha is, paid only o agens who have asses below a prespecified minimum. 4 Asse-esed programs, such as Medicaid, Temporary Assisance o Needy Families, and many ohers, are used widely in he U.S. social insurance sysem. 5 Firs, we formally define a compeiive equilibrium wih general axes. Definiion 1. Given a ax sysem { }, allocaions of consumpion, labor supply, and savings ( c, l, x, k ) consiue a compeiive equilib- 4 Empirical evidence suppors our argumen ha persons who falsely claim disabiliy have higher savings han disabled persons. A comprehensive sudy of disabiliy applicans and recipiens by Beniez-Silva, Buchinsky, and Rus (2004) finds ha nondisabled awardees of disabiliy insurance have significanly higher asses ($87,017) han disabled recipiens ($73,911) (see able 4 in heir paper). 5 We do no imply ha curren asse-esed programs are opimal.

8 264 journal of poliical economy rium if hey solve he following problem: subjec o T T T 1 1 s s 1 s (c,l,x) 0,k p1 sp1 ps max b P[u(c ) v(l )] P (1 p ) b u(x ) c k wvl (1 R)k 1 ({vl i i, k i 1} ip1) G, s s s s 1 T s s T x k (1 R)k 1 (({vl i i} ip1,{vl i ip 0} ips), ({k i 1} ip1,{k i} ips)) for s, s where ks 1 p ks 1, and feasibiliy (1) is saisfied. We say ha a ax sysem { } implemens he opimal allocaion ( c*, l*, x*) if he opimal allocaion is equal o he compeiive equilibrium allocaion ( c, l, x ) defined above. A. Linear Savings Tax Does No Implemen he Opimum In his subsecion, we presen a wo-period example ha demonsraes ha a linear savings ax canno implemen he opimum. We consider a seup in which agens live for wo periods and are able in he firs period of heir lives; his is a special case of he more general model wih T p 2 and p1 p 1. When an agen is able, he has skill v p 1. In he second period of his life, an agen is able wih probabiliy p and disabled wih probabiliy 1 p. Denoe firs- and second-period consumpion of an able agen by c1and c2, respecively, second-period con- sumpion of a disabled agen by x, and allocaions of labor of able agens in periods 1 and 2 by l1 and l2, respecively. We define he opimal allocaion (c*, l*, x*) p {(c*, 1 c*, 2 x*), (l*, 1 l*)} 2. One can conjecure (as in Diamond and Mirrlees [1978]) ha a linear savings ax ha is equal o he ineremporal wedge in equaion (3), combined wih correcly chosen lump-sum axes, implemens he opimal allocaion. We show ha his conjecure is false since here exiss a profiable deviaion for an agen. Consider a ax sysem ha consiss of a savings ax, lump-sum axes T1 in period 1, Ta if an agen provides a posiive amoun of effecive labor in period 2, and T d if an agen does no work in period 2. We now show ha his sysem of axes does no implemen he opimal allocaion. Proposiion 2. The opimal allocaion canno be implemened wih any ax sysem ha uses only a linear ax on savings.

9 opimal disabiliy insurance 265 Proof. Suppose he conrary. Then he savings ax mus saisfy u (c*) 1 { b[pu (c*) (1 p)u (x*)] } 2 p 1 1 R. (4) The lump-sum axes T1, Ta, and Tdmus saisfy c* k* p wl* T, (5) Z and c* p [1 R(1 )]k* wl* T, (6) 2 2 a x* p [1 R(1 )]k* T, (7) d for some level of capial k*. An agen planning o claim disabiliy in he second period, regardless of his rue ype, solves he following problem. Problem 1. subjec o and max u(c ) v(l ) bu(c ) (c,l,k) c k p wl T c p [1 R(1 )]k T. 2 d Firs noe ha ( c* 1, x*, l* 1), he allocaion of a disabled agen under he opimum, is feasible for his problem. I is no a soluion, however. To see his, noice ha he firs-order necessary condiion fails: u (c*) 1 p [1 (1 )R]b[pu (c*) 2 (1 p)u (x*)]! [1 (1 )R]bu (x*). Hence, he maximized uiliy in problem 1 exceeds he (ex pos) realized uiliy, under he opimum, of an agen who is disabled in period 2. Then noice ha because he incenive consrain binds in an opimal allocaion, he agen s (ex ane) expeced uiliy under ha allocaion is he same as his (ex pos) realized uiliy condiional on being disabled: u(c*) v(l*) bu(x*) p u(c*) v(l*) b{p[u(c*) v(l*)] (1 p)u(x*)}.

10 266 journal of poliical economy Fig. 1. A linear savings ax and asse esing Hence he maximized value in problem 1 exceeds he ex ane expeced uiliy under he opimal allocaion. An analogous proof would hold for he case wih an arbirary number of periods. QED Inuiively, a linear savings ax is no sufficien o implemen he opimal allocaion because i is designed o preclude single deviaions. Given ha an agen ells he ruh, a linear savings ax guaranees ha he chooses he correc amoun of savings. Given ha an agen chooses a correc amoun of savings, an agen chooses o ell he ruh. However, we have shown above ha a join deviaion in which an agen decides o boh lie and change he amoun of savings may be profiable. 6 We illusrae his inuiion graphically. An agen who plans o claim disabiliy in period 2 has uiliy u(c 1) bu(x). In figure 1, we plo an indifference curve for such an agen. By he incenive compaibiliy 6 A similar resul is also derived by Albanesi and Slee (2006) in heir environmen.

11 opimal disabiliy insurance 267 consrain, he uiliy of claiming disabiliy in he social planner s problemu(c*) 1 bu(x*), is equal o he uiliy of elling he ruh and here- fore is he uiliy of he opimal soluion. Poin A represens his choice of ( c* 1, x* ). In a compeiive equilibrium wih a linear ax, an agen s budge line, represened by he dashed line, has a slope of [1 R(1 )]. Noe ha he slope of he indifference curve a poin A is u (c*)/bu 1 (x*). Therefore, he budge line inersecs he indifference curve, and a poin beer han poin A can be found by he agen. One can also see how asse esing would work: if a budge is consruced such ha he indifference curve ouches he budge se only a poin A, hen poin A would be chosen. The solid line in he figure is an example of such a budge se. B. Asse-Tesed Disabiliy Sysem Implemens he Opimal Soluion We formally define an asse-esed disabiliy insurance program. Definiion 2. An asse-esed disabiliy insurance sysem ( k, S, T a ) consiss of (1) a sequence of asse ess k(i), i p 1,, T; (2) a sequence of lump-sum axes of he form S d(, i) p T d(i) wvl, 1 i T, where S d(, i) is he ransfer received in period by a consumer who became newly disabled in period i wih asses no exceeding k(i) ; and (3) a lumpsum ax T a ha is paid each period by a consumer who is sill working or who had asses exceeding k(i) when he declared disabiliy. The heorem ha follows saes he main heoreical resul of he paper: how o consruc an asse-esed disabiliy sysem ha implemens he opimum. Theorem 1. For any consrained opimal allocaion ( c*, l*, x* ), here exiss an asse-esed disabiliy insurance program ( k, S, T a ) for which ( c*, l*, x* ) is a compeiive equilibrium. Proof. See he Appendix. The logic behind he resul is as follows. Consider an able agen a age who in period 1 plans o work if able or o claim disabiliy if he becomes disabled. In period 1, he receives income from savings and, in addiion, income from working (if he remains able) or from disabiliy ransfers (if he becomes disabled). If insead he agen were o claim disabiliy in period 1 even if able, he would receive disabiliy ransfers insead of income from working. If hose ransfers are less han he income from working, an agen considering a false claim of disabiliy for period 1 has an incenive in period o accumulae higher asses han if he planned o behave honesly. An asse es deers false claims by penalizing he sraegy of oversaving and no working. In figure 1, we illusrae he inuiion behind asse esing on he woperiod example considered above. For c ( 1 1 c* 1 k 1 k), asse esing shifs he budge line down. The budge line is now represened by a solid

12 268 journal of poliical economy TABLE 1 Share of Disabled Populaion (%) Age Group Model CPS (Soddard e al. 1998) NA SIPP (McNeil 1997) line and has a disconinuiy a poin A. An agen who plans o claim disabiliy in period 2, herefore, chooses poin A. Poin A gives he same uiliy as he uiliy of elling he ruh under he opimal allocaion. Therefore, an agen chooses he opimal allocaion under he asseesed disabiliy sysem. V. Quaniaive Resuls We firs describe how we deermine parameers of he model. We proceed o characerize a soluion o he social planner s problem. We hen evaluae he welfare benefis ha a sysem wih asse esing has over he opimal sysem wihou asse esing. A. Parameerizaion We choose he probabiliy of becoming disabled using he daa from McNeil (1997), who repors he number of self-repored disabled people by age groups. 7 We hen calculae a condiional probabiliy of becoming disabled and inerpolae he daa o one-year inervals by fiing an exponenial funcion. Table 1 repors he share of disabled people in our model by various age groups. We assume ha 4 percen of he populaion is disabled a age 25, before enering he labor force. We compare he numbers we calculaed o hose repored in he Survey of Income and Program Paricipaion (SIPP) and he Curren Populaion Survey (CPS). The SIPP esimaes he number of people wih severe disabiliies, and he CPS repors he number of people wih work dis- 7 Disabiliy applicans may have a srong incenive o misrepor heir disabiliy saus o he SSA, bu here is significanly weaker reason for respondens o misrepresen heir informaion in anonymous surveys since any informaion hey repored canno have any impac on he saus of heir disabiliy benefis. One indicaion of respondens ruhfulness is provided by he fac ha nearly 20 percen of disabiliy recipiens repored ha hey do no have a healh problem ha prevens hem from working, and 5 percen of hese recipiens repored labor earnings in excess of he $500 per monh limi imposed by he SSA. Eiher of hese self-repors consiues prima facie evidence for erminaion of benefis (see Beniez-Silva e al. 2000).

13 opimal disabiliy insurance 269 abiliies. The CPS does no have informaion abou work disabiliies of people who are over age 65. A period is one year, and each agen begins life a age 25 and lives o 75. The uiliy funcion is ln (c) a ln (1 l), where a p 1.5 is he relaive disuiliy of labor. The ineres R p 0.043, so he discoun facor is b p 1/ (1 R) p The aggregae producion funcion is Cobb- a 1 a Douglas, F(K, Y ) p KY, wih a p 0.33 as capial s share. Wih hese values for R and a, he wage is w p The lifeime skill profile is obained by fiing a quadraic funcion o he daa in Rios-Rull (1996). The skill level peaks a age 50, a which poin an agen is 45 percen more producive han a age 25. Afer age 50, skills decline; a age 75, hey are roughly equal o heir level a age 25. B. Opimal Sysem and Implemenaion In his subsecion we numerically characerize an opimal disabiliy insurance sysem and is implemenaion for he parameerized economy described above. We acknowledge ha here are various reasons for reiremen ha are ouside he scope of his model. As he paper focuses on disabiliy insurance, we force agens o reire a age 64 by seing p40 p 1. We choose reiremen benefis for ages opimally since hey affec he dynamic incenives o claim disabiliy a ages prior o heir reiremen. We repor opimal consumpion profiles in figure 2a. The upper solid line represens consumpion ( c ) for agens who were able all heir lives. This consumpion is increasing wih he duraion of he agen s work hisory, since he social planner rewards he agen for working in period by allocaing him a higher coninuaion uiliy, which implies higher consumpion a fuure daes. The lower solid line represens consumpion x of a newly disabled agen. Noe ha we do no plo consumpion xs ( s 1 ) afer an agen becomes disabled since i is consan and equal o x. The significan fall in consumpion afer an agen becomes disabled is necessary o ensure ha able agens do no deviae and claim disabiliy. There are wo effecs ha deermine consumpion of an agen who becomes disabled. Firs, efficiency requires ha more skilled agens work more, and herefore, he consumpion drop should be larger for such agens. We can see ha agens who become disabled a ages receive lower consumpion han hose who become disabled a he age of 25. The second effec comes from he ineremporal provision of he incenives. The planner rewards an agen for working by increasing he coninuaion uiliy when an agen becomes disabled. This effec calls for higher consumpion for agens who become disabled laer in life and dominaes he firs effec once an agen reaches age 32. The second effec increases

14 Fig. 2. Opimal disabiliy programs wih asse esing (solid lines) and wihou asse esing (dashed lines): a, consumpion; b, labor; c, disabiliy ransfers; d, asse limis.

15 opimal disabiliy insurance 271 wih work hisory, so ha he consumpion of disabled agens laer in he life cycle rises more seeply han ha for able agens. The solid line in figure 2b represens he opimal labor allocaions l ha are influenced by effecs similar o he consumpion profiles. On he one hand, i is opimal o require more producive agens o work more, so ha labor supply inheris he hump-shaped form of he skills profile. Agens who are years old (i.e., he highes-skilled agens) spend abou percen of heir ime working. Younger and older people are no as producive and work less. On he oher hand, ineremporal provision of incenives calls for an increase of he coninuaion uiliy of an able agen, which can be parially achieved by reducing he amoun of labor. 8 An imporan feaure of he model is he ineremporal disorion, which we define as { } u (c*) D p 1 1 R. b[p u (c* ) (1 p )u (x *)] The ineremporal disorion depends on hree facors: he probabiliy of becoming disabled, skill profile, and lengh of work hisory. The probabiliy of becoming disabled increases for older agens, hus making heir fuure consumpion more unpredicable, which increases he disorion. For higher-skilled agens he incenive problems are more severe, and hey face a higher ineremporal wedge. The hird facor, he lengh of work hisory, decreases he wedge. Agens wih a longer work hisory provide less labor and have a smaller variance of consumpion. We find he ineremporal disorion o be quaniaively significan. The wedge grows from slighly below 3 percen a age 24 o 7 percen a age 50 and decreases almos o zero by age 63. From he proof of heorem 1, we calculae and plo ransfers o he disabled wih he solid line in figure 2c and asse limis in figure 2d. Noe ha we plo disabiliy ransfers only for newly disabled agens. Transfers are consan afer an agen becomes disabled; for example, an agen who sops working a age 40 receives approximaely 0.35 uni of consumpion for he res of his life. Asse limis evenually increase because agens become wealhier as hey accumulae more capial. Tha is also he reason why disabiliy ransfers evenually decrease, since agens receive a larger proporion of heir income from savings. One inerpreaion of his sysem is ha individuals who became disabled 8 We also compue he opimal sysem in which he skill level of he able is he same for all ages. In ha case, here is only he second effec, and he consumpion of able and disabled agens monoonically increases wih he lengh of heir work hisory, since here is no reason o require middle-aged agens o work more. Labor supply in ha model monoonically decreases wih work hisory (see Golosov and Tsyvinski 2005b). Z

16 272 journal of poliical economy early in life receive large ransfers, whereas hose who become disabled laer in life are supposed o supplemen heir lower disabiliy ransfers wih savings accumulaed while able. C. Welfare Benefis of Asse Tesing In his subsecion we numerically compare he welfare of he bes program wihou asse esing wih ha of he opimal insurance sysem. The opimal disabiliy sysem wihou asse ess is a soluion o he social planner s problem wih hidden savings, an example of which is Diamond and Mirrlees (1995). Absence of asse esing implies ha he planner does no have an abiliy o disor an ineremporal margin. The model wih hidden savings is also similar o ha of Werning (2001) and Abraham and Pavoni (2003). However, he dynamic firs-order approach in hese papers of imposing he Euler equaion on he social planner s problem is invalid in our seup. 9 Our compuaional mehod for he model of hidden savings is similar o ha in Golosov and Tsyvinski (2005a). For each lifeime allocaion of consumpion and labor ha a planner offers o an agen, we compue T opimal join deviaions in which an agen claims disabiliy and chooses he opimal level of hidden savings, and an addiional deviaion in which an agen ells he ruh bu chooses a level of savings differen from ha prescribed by he planner. This mehod allows us o find a globally opimal soluion o he social planner s problem wih hidden savings. In figure 2, we plo he soluion o he model wihou asse esing using dashed lines. In a comparison wih he soluion wih asse esing, here are four main differences, all of which conribue o he welfare loss. Firs, he consumpion profile of an able agen sars a a lower level and increases more rapidly. This rapid increase reduces welfare compared o he opimal sysem, since agens prefer smooher consumpion profiles. Second, he consumpion penaly for disabled agens who are years old is larger. A large penaly is needed o ensure ha an agen does no falsely claim disabiliy before becoming mos producive. In he absence of asse esing, he planner has o penalize agens who declare disabiliy early by giving hem lower consumpion han hey can achieve when asse esing is available. Third, he consumpion profile of he disabled is less smooh han when asse esing is available. In paricular, consumpion allocaions of he disabled rise seeply afer age 36. The fourh difference involves labor profiles and oal oupu. Labor profiles for boh cases are virually idenical unil abou age 40. Afer age 40, he absence of asse esing implies ha i is more difficul o provide incenives o work, and less labor is provided. 9 See also Kocherlakoa (2004) for a discussion of he firs-order approach.

17 opimal disabiliy insurance 273 Hence here is less oupu in oal, which appears mainly as lower redisribuion o he disabled. As figure 2c shows, asse esing allows a significan increase in he level of disabiliy ransfers a mos ages. Comparing he wo allocaions, we find ha asse esing yields a welfare gain of 0.5 percen. Specifically, under he opimal sysem wihou asse esing, a proporional increase in consumpion by 0.5 percen for each hisory produces he same lifeime uiliy as he lifeime uiliy in an opimal sysem wih asse esing. This number provides a lower bound on he welfare gains of swiching o an opimal sysem, since i represens gains solely of asse esing. D. Robusness of Quaniaive Resuls We also considered a model of Social Securiy as an opimal disabiliy insurance program. One of he explanaions for he exisence of he Social Securiy sysem is is role as opimal reiremen insurance. Diamond and Mirrlees (1978, ) view a seup similar o ours as a general way of modeling he Social Securiy sysem, including he old-age porion (see also Mulligan and Sala-i-Marin 1999). The Social Securiy sysem can be viewed as mandaory governmen insurance agains becoming disabled (no being able o work) a old ages. While Social Securiy benefis are condiioned on reiremen, in his modificaion of he model we condiion benefis on a more fundamenal risk, disabiliy. In his model, agens live for 75 years, he probabiliy of becoming disabled is compued o his age, and here is no mandaory reiremen. In fac, all agens who are able a ages work in he example ha we compue. However, he model feaures endogenous reiremen, since older agens end o work significanly less han younger agens wih similar skills. Agens who are 75 years old and are sill producive spend less han 10 percen of heir ime working. We find ha he welfare gains of asse esing increase modesly o 0.65 percen of consumpion, since agens aged provide a low amoun of labor, even wihou mandaory reiremen. We already discussed a modificaion of he model in which he skill profile for he able is consan over he lifeime. The welfare benefis of asse esing in his model are equal o abou 0.35 percen. We also calculaed a model in which he probabiliy of disabiliy is half of ha used in his paper. The welfare gain of asse esing is similar o he one derived in he benchmark model and is equal o 0.3 percen, since he size of he informaional fricion decreases wih he smaller probabiliy of disabiliy. We also calculaed a sylized curren social insurance sysem o compare wih he opimal sysem described above. Since disabiliy insurance is an inegral par of he social insurance sysem, we modeled he curren

18 274 journal of poliical economy social insurance sysem as a join disabiliy and reiremen sysem. An agen can sop working eiher because he is ruly disabled or because social insurance ransfers creae a disincenive o work. If an agen does no work, he receives a social securiy ransfer. An agen can save a a rae R and is axed a a rae. When an agen sops working, he receives a disabiliy ransfer T d ha is independen of age. In he supplemen o his paper (Golosov and Tsyvinski 2005b) we provide a deailed descripion of he sylized curren sysem. The welfare gain of a swich o he opimal insurance sysem from he sylized curren Social Securiy sysem is equivalen o an increase of consumpion by 2.8 percen for each hisory. The larger welfare gain mainly comes from he increase in benefis o agens who became disabled relaively early in heir lives. VI. Final Remarks, Robusness, and he Role of Assumpions In his paper we consider he problem of implemenaion of opimal disabiliy insurance when disabiliy saus is unobservable and show wha insrumens can implemen he opimum. Asse esing allows conrol of join deviaions in which an agen, in anicipaion of falsely claiming disabiliy, increases his savings compared o hose implied by he opimal allocaion. We hen provide numerical resuls ha sugges ha asse esing may be quaniaively imporan. We made wo significan assumpions ha are imporan for characerizing implemenaion of he opimum. Firs, disabiliy is an absorbing sae. This assumpion reduces he number of hisories ha we need o consider. We have o keep rack only of an agen s age and he age a which he claimed disabiliy. An ineresing exension would be o sudy an economy in which disabiliy is no permanen bu here is a small probabiliy of recovery. In ha case, opimal disabiliy benefis also have o encourage individuals who recover from disabiliy o leave disabiliy rolls. If skills follow a more general process such as nonpermanen disabiliy, a axaion mechanism of Albanesi and Slee (2006) modified o he case of persisen shocks or he mehod of Kocherlakoa (2005) may be needed o implemen he opimum. The second assumpion we made is ha a disabled agen has zero skill. This assumpion allows us no o consider deviaions in which a disabled agen preends o be able or more complicaed deviaions in which an agen undersaves and works oo much. We conjecure ha if he skill of a disabled agen is sufficienly close o zero, he implemenaion ha we derived sill remains valid. We also reaed governmen as he only provider of disabiliy insurance wihou considering insurance ha is provided by privae markes. This assumpion seems o be close o realiy. Excep for SSDI, few oher opions provide proecion agains disabiliy risk. For example, only 25 percen of privae-secor employees receive long-erm disabiliy coverage

19 opimal disabiliy insurance 275 (SSA 2001). In Golosov and Tsyvinski (2005a) we showed ha in an environmen in which consumpion is observable, publicly provided insurance is as efficien as insurance provided by privae markes. In paricular, if all insurance is provided by privae inermediaries, hen insurance conracs would feaure exacly he same asse-esed disabiliy benefis as he ones described in his paper. The heoreical resuls of he model are robus o wo exensions. Firs, consider a case of observable heerogeneiy. Suppose ha here are i ypes of agens who observably differ in he probabiliy of becoming disabled, discoun facors, or skill profiles when able. I is easy o show using he same proof as in he paper ha an asse es condiional on ype implemens he opimal allocaion. Examples of insurance ha is condiioned on an agen s ype such as gender abound in he pracice of privae insurance. We can also consider an environmen in which here are muliple unobservable levels of skills ha follow a general sochasic process bu here exiss an absorbing disabiliy sae in which disabled agens canno work. Assume ha allocaions of consumpion and effecive labor for all hisories excep for disabiliy saes are provided by a direc mechanism. Then i is easy o show ha an asse-esed disabiliy insurance implemens he opimum. Moreover, he asse es has o be condiioned on he asses of he marginal agen. This model can be inerpreed as a join sysem of opimal axaion and disabiliy insurance. Insurance for all skill shocks wih he excepion of disabiliy is accomplished hrough a direc mechanism ha sands in for he income and wealh ax sysem. Disabiliy insurance is achieved hrough an asseesed disabiliy sysem. The resuls in our model as well as in oher models of opimal dynamic axaion are no robus o inclusion of unobservable heerogeneiy such as, for example, differenial unobserved discoun raes. The main echnical difficuly is ha, even in he saic model, he problem becomes one of mulidimensional screening. In he case of one unobservable characerisic, i is easy o show ha incenive compaibiliy consrains are binding from he high o he low ypes. The major difficuly wih mulidimensional screening is deermining a paern of binding incenive consrains. 10 While he described exensions are ineresing, he magniude of he welfare gains from asse esing gives us confidence ha he forces we have capured in his paper are significan from boh heoreical and policy perspecives. 10 For a review of mulidimensional screening, see Armsrong and Roche (1999).

20 276 journal of poliical economy Appendix Proof of Theorem 1 The heorem is proved by consrucion. Choose he ax on he able, T, o saisfy 11 a Le he ransfers o he disabled, T T T a p1 p1 p1 c*b p wvl*b T b. T d(j), saisfy j 1 T j 1 T 1 j a d p1 pj p1 pj c*b x *b p (wvl* T )b T (j) b. Finally, asse limis k are defined recursively: k c* k p wvl* 1 Ta b wih k 1 p 0. Using his policy, we can rewrie he feasibiliy consrain: T T T 1 1 a s 1 s d p1 sp1 ps b PT P (1 p) b T (s) 0. (A1) Firs, we prove ha he expeced presen value of ransfers for an able agen is lower han he presen value of ransfers o a disabled agen. Tha is, {T a, T ()}, defined above, saisfy, for all p 1,, T, d { [ ] } T T ipt s i i a 1,i 1 i d 1,i a d ip 1 spi ip0 T P (1 p) T (i)b P T b T ()b. (A2) Noe ha (A2) is equivalen o [ ] ( ) T T T i s i i 1,i 1 i s 1,i i i i i ip 1 spi ip0 c* wvl* P (1 p) x *b P (c* wvl*)b x * b. (A3) Suppose ha equaion (A3) did no hold for some. Then he social planner ipt could give he consumpion of he disabled {x i *} ip0 o agens who are sill able in period and se heir labor o zero. Since he period incenive consrain holds wih equaliy, he uiliy of he agen does no change. The new allocaion is sill incenive compaible, bu he feasibiliy consrain is relaxed. The social planner can furher improve on such an allocaion; herefore, ( c*, l*, x* ) canno be an opimum. Nex we show ha (A2) implies Ta T d() for all. For p T, his fac is immediae from (A2). For p T 1, (A2) says ha Ta b[p T a (1 p T)T d(t)] T d(t 1)(1 b). 11 Noe ha agens who do no receive disabiliy ransfers face a ax regardless of T a heir age. Wihou loss of generaliy, we could have assumed ha hese axes are indexed by age. In ha case he levels of asse ess would no be uniquely pinned down.

21 opimal disabiliy insurance 277 Since Ta T d(t), he above equaion implies Ta T d(t 1). Coninue by inducion o esablish he claim for all. Consider he asse-esed sysem consruced as described above. Pick any allocaion (c, l, x) and saving decisions ( k ) ha maximize an agen s uiliy. We will show ha he uiliy from such allocaions canno be higher hen he uiliy from ( c*, l*, x* ). Sep 1: There exiss a uiliy-maximizing allocaion (c, y, k) such ha an agen never claims disabiliy if he is able. Suppose ha an agen is sricly beer off by claiming disabiliy if he is able in some period j. The agen can claim disabiliy in period j only if his asses in ha period are k. Suppose ha j kj kjp kj. By consrucion, he maximum uiliy he agen can obain if his asses are k j and his axes are T(j) d wvl for j T j j all subsequen periods is u(x j*) b u(x T*), which is he uiliy ha he planner allocaes o he agen who becomes disabled in period j. Bu he agen wih asses k in period j can choose he fuure pah ({c*, l*, x*} T j pj) since i is in his budge consrain. By he incenive compaibiliy of he opimal allocaions, his fuure pah gives weakly higher uiliy han claiming disabiliy in period j. Alernaively, suppose ha k j! kj. The agen s uiliy maximizaion implies ha j j j x. The allocaion is uiliy maximizing in his case if j! x j* u (c j 1) p u (c j). If his Euler equaion did no hold, an agen could ransfer a small amoun of resources e ineremporally. Such a ransfer sill allows him o claim disabiliy j j in period j and gives sricly higher uiliy. Since x j! x j*, his ogeher wih corollary 1 implies ha u (c ) 1 u (c* ). The agen s budge consrain j 1 j 1 1 c k p wv l k b j 1 j j 1 j 1 j 1 and inraemporal opimaliy condiion 1 v (l ) p u (c )w j 1 v j 1 imply ha k. We can coninue backward o show ha j 1! kj 1 k! k for all! j. However, his implies ha k 1! k1p 0, which is impossible. We showed ha here exiss a uiliy-maximizing allocaion in which an agen never claims disabiliy when he is able. Sep 2: The consruced asse-esed sysem implemens he opimum. We show ha if he condiions of sep 1 are saisfied, he uiliy-maximizing allocaion mus be feasible and incenive compaible. Therefore, i canno give a higher uiliy han ( c*, l*, x* ). The allocaion is incenive compaible since i comes from he agen s maximizaion problem. From sep 1, he able agen always receives a ransfer T a. We showed ha Ta T d() for all, so ha his is he lowes possible ransfer (he highes ax since Ta 0) ha he agen can receive. Noe ha if an agen saves more han k i in some period i 1 and becomes disabled in period i, he receives ransfer Ta unil his savings fall below he asse limi, afer which he is eniled o T(i) d. The presen value of such ransfers is lower han he presen value of he ransfers o he agen who could claim disabiliy in period i, which is equal o T d(i) T i ( 1 b ). j 1

22 278 journal of poliical economy Therefore, he ex ane expeced value of ransfers canno be higher han {[ ] } T 1 1 P i 1(1 p)t i d(i)b PTab 0, p1 ip1 and from (A1), he allocaion ha has such ransfers mus be feasible. References Abraham, Arpad, and Nicola Pavoni Efficien Allocaions wih Moral Hazard and Hidden Borrowing and Lending. Working paper, Univ. Coll. London. Albanesi, Sefania, and Chrisopher Slee Dynamic Opimal Taxaion wih Privae Informaion. Rev. Econ. Sudies 73 (January): Armsrong, Mark, and Jean-Charles Roche Muli-dimensional Screening: A User s Guide. European Econ. Rev. 43 (April): Beniez-Silva, Hugo, Moshe Buchinsky, Hiu Man Chan, Sofia Cheidvasser, and John Rus How Large Is he Bias in Self-Repored Disabiliy? Working Paper no (February), NBER, Cambridge, MA. Beniez-Silva, Hugo, Moshe Buchinsky, and John Rus How Large Are he Classificaion Errors in he Social Securiy Disabiliy Award Process? Working Paper no (January), NBER, Cambridge, MA. Bound, John, and Richard V. Burkhauser Economic Analysis of Transfer Programs Targeed on People wih Disabiliies. In Handbook of Labor Economics, vol. 3C, edied by Orley Ashenfeler and David Card. Amserdam: Norh- Holland. Chari, V. V., and Parick J. Kehoe Opimal Fiscal and Moneary Policy. In Handbook of Macroeconomics, vol. 1C, edied by John B. Taylor and Michael Woodford. Amserdam: Norh-Holland. Diamond, Peer A., and James A. Mirrlees A Model of Social Insurance wih Variable Reiremen. J. Public Econ. 10 (December): Payroll-Tax Financed Social Insurance wih Variable Reiremen. Scandinavian J. Econ. 88 (1): Social Insurance wih Variable Reiremen and Privae Saving. Manuscrip, Massachuses Ins. Tech. Golosov, Mikhail, Narayana R. Kocherlakoa, and Aleh Tsyvinski Opimal Indirec and Capial Taxaion. Rev. Econ. Sudies 70 (July): Golosov, Mikhail, and Aleh Tsyvinski. 2005a. Opimal Taxaion wih Endogenous Insurance Markes. Working Paper no (March), NBER, Cambridge, MA b. Supplemen o Designing Opimal Disabiliy Insurance. Manuscrip, Harvard Univ. Hopenhayn, Hugo A., and Juan Pablo Nicolini Opimal Unemploymen Insurance. J.P.E. 105 (April): Kocherlakoa, Narayana R Figuring Ou he Impac of Hidden Savings on Opimal Unemploymen Insurance. Rev. Econ. Dynamics 7 (July): Zero Expeced Wealh Taxes: A Mirrlees Approach o Dynamic Opimal Taxaion. Economerica 73 (Sepember): McNeil, John M Americans wih Disabiliies: Curren Populaion Repors, Household Economic Sudies P70, no. 61. Washingon, DC: U.S. Dep. Commerce, Bur. Census.

23 opimal disabiliy insurance 279 Mirrlees, James A Opimal Tax Theory: A Synhesis. J. Public Econ. 6 (November): Mulligan, Casey B., and Xavier Sala-i-Marin Social Securiy in Theory and Pracice (II): Efficiency Theories, Narraive Theories, and Implicaions for Reform. Working Paper no (May), NBER, Cambridge, MA. Rios-Rull, José-Vícor Life-Cycle Economies and Aggregae Flucuaions. Rev. Econ. Sudies 63 (July): SSA (U.S. Social Securiy Adminisraion) Social Securiy Bullein: Annual Saisical Supplemen. Washingon, DC: Soc. Securiy Admin Social Securiy Bullein: Annual Saisical Supplemen. Washingon, DC: Soc. Securiy Admin. Soddard, Susan, L. Jans, J. Ripple, and Lewis E. Kraus Charbook on Work and Disabiliy in he Unied Saes, 1998: An InfoUse Repor. Washingon, DC: Ins. Disabiliy and Rehabiliaion Res. Wang, Cheng, and Sephen D. Williamson Unemploymen Insurance wih Moral Hazard in a Dynamic Economy. Carnegie-Rocheser Conf. Ser. Public Policy 44 (June): Werning, Ivan Opimal Unemploymen Insurance wih Hidden Savings. Manuscrip, Univ. Chicago.