DYNAMIC STRUCTURAL MODELS OF RETIREMENT AND DISABILITY

Transcription

1 DYNAMIC STRUCTURAL MODELS OF RETIREMENT AND DISABILITY John Rus, Universiy of Maryland and NBER Moshe Buchinsky, UCLA and NBER Hugo Beníez-Silva, SUNY--Sony Brook February 2003 SUMMARY We propose o use all available waves of he Healh and Reiremen Survey (HRS) and AHEAD Survey o esimae a comprehensive dynamic programming (DP) model of behavior a he end of he life cycle ha provides a deailed reamen of he Social Securiy Adminisraion's (SSA) Old Age and Survivors (OASI), Supplemenal Securiy Income (SSI) and Disabiliy Insurance (DI) programs. Major changes o hese programs are being conemplaed. Ye, we currenly lack a unified model of social insurance a he end of he life cycle ha can help us evaluae he behavioral and disribuional impacs of hese policies. Paricular aenion is paid o developing, esimaing, and esing a muli-sage dynamic programming (DP) model of he SSI and DI applicaion, appeal, and award process, for (possibly) heerogeneous agens. We are developing a racable empirical model ha capures an individual's decisions regarding (1) labor supply and reiremen (2) applicaion for OA, DI and SSI benefis, and (3) consumpion and savings. The resuling model will allow us o derive predicions of he behavioral and welfare implicaions of policy changes. While here is a large lieraure using reduced-form and saic srucural models ha has invesigaed some of hese issues, i suffers from wo major shorcomings. Firs, reduced-form models canno be used for welfare analysis or o predic behavior responses o policy changes. Second, saic srucural models do no accuraely reflec he level of complexiy and uncerainy facing individual decision makers, nor do hey capure he imporan dynamic elemens of he decision processes. The DP model we are developing will circumven hese shorcomings, providing a racable framework for analyzing individual behavior and well-being, and forecasing heir response o a wide range of policy changes. Our model could provide new insighs ino a number of puzzling aspecs abou disabiliy in he U.S. One puzzle is o deermine he facors responsible for he pronounced swings in DI incidence raes in recen years. Anoher puzzle is o deermine why he fracion of Americans receiving SSI and DI benefis coninues o increase despie overwhelming epidemiological evidence of seady improvemens in various objecive indicaors of healh saus. The SSA is currenly conemplaing significan changes o he disabiliy award process, in order o reduce delays, and reduce large unexplained sae-level differences in award raes. We will use deailed healh and funcional saus indicaors from he HRS o evaluae wheher or no here are alernaive screening rules ha can reduce he level of classificaion errors in he DI award process. Our esimaed DP model will produce deailed predicions of he behavioral and welfare effecs of changes in benefi levels, delays, he probabiliy of being awarded benefis, and he probabiliy ha a DI beneficiary will be audied. This framework will allow us o develop mehodologies for characerizing efficien policies, i.e., hose ha minimize he expeced discouned cos of providing a sream of social insurance benefis subjec o he consrain ha individuals' expeced discouned uiliies are a leas as high as under he saus quo.

2 1. INTRODUCTION AND SPECIFIC AIMS The life-cycle model has been a cornersone of economics for over 50 years, ye surprisingly only wihin he las few years have economiss begun o consider how o esimae and es i. Alhough economiss such as Bernheim (1993, 1994, 1995, 1996, and 1997) have claimed ha observed behavior is inconsisen wih he predicions of he life-cycle model since These sudies consisenly find ha baby boomers are saving a 33 o 38 percen of he rae required o cover heir expeced coss of reiremen. (Bernheim 1997, p. 43). We argue ha mos economiss do no really undersand he implicaions and richness of he full life cycle model because hey have been unable o solve realisic versions of i. Insead hey have relied on limied, ofen incorrec inuiions obained from overly simplified deerminisic/sochasic specificaions ha yield analyic soluions. Virually all empirical work has avoided solving he full life-cycle model by esimaing and esing essenially saic parial characerizaions of opimaliy such as he Euler necessary condiion ha implies ha (for inerior soluions) he marginal rae of subsiuion beween consumpion and labor supply equals he curren wage rae. Recenly, he adven of improved algorihms and compuer hardware have allowed economiss o solve more realisic specificaions of he life cycle model, resuling in a much deeper undersanding of is implicaions and he realizaion ha versions of he life-cycle model migh be capable of explaining observed behavior. For example, Engen, Gale, and Uccello (1999) solved and simulaed a calibraed version of he life-cycle model and compared he prediced levels of saving and wealh accumulaion o levels observed in he Healh and Reiremen Survey (HRS) and he Survey of Consumer Finances. They conclude ha, conrary o Bernheim, observed saving behavior appears o be adequae in he sense ha he prediced levels of saving and wealh accumulaion in heir calibraed life-cycle model appear o be roughly consisen wih observed levels. They find ha Because of he uncerainy of earnings, he model generaes a disribuion of opimal wealh-earnings raios among households ha are observaionally equivalen. This disribuion implies ha some households ha have very low wealh-earnings raios are noneheless saving opimally for reiremen. (p. 141). A limiaion of Engen e al. (1999) is ha labor earnings are reaed as an exogenous sochasic process. In realiy i is mosly an endogenous process resuling from an individual's labor supply decision. As we show below, once we model labor supply and consumpion joinly, he predicions of he life-cycle become much richer. Surprisingly low levels of wealh accumulaion can be opimal if an individual expecs o reire laer han normal, an expecaion ha surveys have found o be common among baby boomers. The close ineracion beween reiremen expecaions and prereiremen savings was noed as far back as Feldsein (1974), bu o our knowledge he magniude of he effec of reiremen age expecaions on he level of prereiremen savings has no been empirically demonsraed. I is one of he quesions we will aemp o answer in his research. Finally, i has also been known ha low levels of saving can be opimal in he presence of Social Securiy and oher forms of social insurance. For example Hubbard, Skinner and Zeldes (1995) showed ha i can be opimal for individuals wih he lowes earnings capaciies o hold almos no wealh when asse esing is a precondiion for he receip of Medicaid and welfare benefis. All hese consideraions sugges ha a considerably richer version of he life-cycle model --- wih join reamen of consumpion and labor supply and a realisic reamen of social insurance insiuions --- will be necessary before we can judge wheher or no individuals are behaving opimally and in accordance wih he predicions of he life-cycle model. To dae, mos empirical work on he life-cycle model has been based on relaively informal comparisons of observed behavior wih he simulaed behavior from calibraed versions of he life-cycle model. Calibraion syle mehods specify values for he parameers of he model, parly because i is oo hard o wrie down a formal likelihood funcion for he model, and parly because i is oo compuaionally

3 demanding o repeaedly resolve he life-cycle model in a formal search for bes fiing values of he model's unknown parameers. Recenly pioneering work by French (2001), building on previous mehods developed by Rus and Phelan (1997), has provided he firs economeric esimaes of life-cycle models where consumpion and labor supply are joinly endogenous and social insurance insiuions are carefully modeled (alhough economeric mehods for esimaing he life-cycle model can be raced as far back as Heckman 1974, Heckman's early work was based on deerminisic models of he life-cycle and absraced from social insurance insiuions). We are no aware of any readily available economeric mehods ha economiss can use o compue, simulae, esimae, and es he life-cycle model ha incorporaes a coninuous decision over consumpion and a discree or coninuous decision over labor supply. The purpose of his proposal is o develop new, compuaionally efficien mehods for doing his, and o apply hese echniques o he sudy of behavior in he las half of he life-cycle, paying paricularly close aenion o how consumpion and labor supply behavior are affeced by social insurance, pension, and privae insurance insiuions. Specifically, we propose o develop a unified empirical model of social insurance a he end of he lifecycle using daa from he HRS and he AHEAD Survey. Our model will include a deailed reamen of he following componens of he U.S. Social Securiy sysem: (1) Old Age and Survivors Insurance (OASI); (2) Disabiliy Insurance (DI) and Supplemenal Securiy Income (SSI) benefis; (3) Medicare and Medicaid, and (4) Unemploymen Insurance (UI). We will also pay close aenion o modeling axes including: (1) he Federal income ax and Earned Income Tax Credi (EITC); (2) he esae ax; and (3) sae and local income axes, sales axes, and propery axes. The resul will be a model ha will enable us o analyze a wide variey of ax and ransfer policies, paricularly hose associaed wih Social Securiy reform. Because of he aging of he U.S. sociey, he Social Securiy program is no in long run acuarial balance. Evenually, Congress will have o decide on wheher significan ax increases, benefi cus, or even more drasic changes, will be needed. A number of imporan changes will begin o ake effec in coming years as a resul of he 1983 Social Securiy amendmens, which was inended o creae incenives for delayed reiremen in order o resore he long run balance in he OASI program. In addiion o increasing conribuion raes, he 1983 amendmens increased he normal reiremen age (NRA) from 65 o 67, increased he delayed reiremen credi (DRC) from 1% o 8%, and decreased he reiremen es ax on pos-reiremen earnings, firs from 50% o 33%, and in he las year of he Clinon Adminisraion, all he way o 0% for people over 65. In recen years much more radical changes o he Social Securiy program have been proposed, including he inroducion of individual accouns, and major changes in he way he DI program is adminisered. These, and oher policy measures, can be examined in our framework. Our research will place paricularly high prioriy on developing a fully dynamic model of he SSI and DI programs. We do his in par because he DI program has never been previously modeled in a fully dynamic conex, and also because he DI program provides a unique opporuniy o subjec he esimaed life-cycle model o a srong experimenal es. The Projec Leader, John Rus, has been appoined as an advisor o he Social Securiy Adminisraion (SSA) o assis in he implemenaion of a large scale demonsraion projec mandaed under he 1999 Ticke o Work Ac and Work Incenives Improvemen Ac (TWWIIA). The demonsraion projec will essenially be a huge conrolled experimen in which cerain DI recipiens will be randomly assigned alernaive benefi formulae. A paricular alernaive was specified in he TWWIIA Ac: i is known inernally a he SSA as he 2 for 1 benefi offse. Under he curren rules, a DI recipien who reurns o work and earns more han a se-aside amoun known as he subsanial gainful aciviy level (SGA) (currenly equal o $700 per monh), loses all of her DI benefis if she coninues o earn more han he SGA level beyond a 9 monh rial work period. The 2 for 1 offse proposal would reduce he effecive 100% ax on earnings in excess of he SGA o 50%. The SSA also has he auhoriy o change he SGA level, i.e., o change he disregard level a which his 50% ax kicks in. The demonsraion projec

4 provides a unique opporuniy o pu he life-cycle model o a rigorous es: we propose o esimae a lifecycle model using he HRS and AHEAD daa and use he esimaed model o generae predicions of he behavioral response of he DI recipiens who are assigned he reamen in he TWWIIA demonsraion projec. If our life-cycle model is able o accuraely predic he acual response of he DI recipiens who are randomly assigned he 2 for 1 offse, i will have much more credibiliy for use in a variey of oher imporan policy forecasing asks confroning he SSA in coming years. Ever since he work of Lalonde (1986) and Lalonde and Maynard (1987) here has been some skepicism abou he reliabiliy of complicaed economeric models for use in policy forecasing. The skepicism is par of he reason why he Congress has mandaed he use of experimenal mehods for policy evaluaion. However, Heckman, Hoz and Dabos (1987) poined ou a number of severe limiaions of he experimenal approach o policy evaluaion, no he leas of which are he huge coss and delays involved in implemening large scale social experimens. An example of a serious limiaion ha he SSA is confroning in is mandae o carry ou he TWWIIA projec is relaed o a phenomenon known as he induced enry effec. This effec arises because he 2 for 1 offse proposal amouns o a liberalizaion of he DI benefi rules: more benefis will be paid o DI recipiens if he 2 for 1 offse plan is in place if hey should choose o reurn o work. Ex ane, a more liberal DI program should induce addiional enry by individuals who are considering wheher or no o apply o he program. A number of iniial inelligen guessimaes produced by he SSA Office of he Acuary have suggesed ha he 2 for 1 offse policy could significanly increase he cos of he DI program because he addiional benefis paid o new recipiens due o induced enry would exceed he reducion in benefis o DI recipiens who would leave he roles as a consequence of induced exi. Alhough he change in he probabiliy of applying for DI benefis is hough o be very small, a small increase in he probabiliy of enry spread ou over a large populaion of poenial enrans can resul in a subsanial increase in DI roles and coss in he long run. For similar reasons, saisicians advising he SSA have esimaed ha i would ake exremely large sample sizes --- on he order of hundreds of housands or even millions of subjecs --- o generae saisically reliable esimaes of he induced enry effec. Due o he huge cos of such a sudy, i is very unlikely ha he SSA will be able o use experimenal mehods o measure he full budgeary impac of he 2 for 1 offse policy. This is a case where a credible behavioral lifecycle model may be one of he only ways for he SSA o provide predicions of he oal impac of he policy change. Our life-cycle model will also be useful for analyzing a number of oher changes o he DI program ha have recenly been conemplaed. Many of he proposed changes resuled from he huge increases in processing delays and backlogs following he rapid growh in applicaions and appeals during he early 1990s. As par of is Disabiliy Process Redesign (DPR) plan, he SSA has considered implemening major changes in he muli sage applicaion and appeal process in order o reduce he long delays beween an iniial applicaion and an ulimae award, including possibly muliple levels of appeals if he iniial applicaion is rejeced by one of he 54 sae-run disabiliy deerminaion services (DDS). The SSA is also considering he use of sandardized funcional impairmen indices in order o reduce he large sae-o-sae variaions in award raes. All of hese proposed policy changes will have significan effecs on he srucure of he Social Securiy program ha canno be accuraely prediced and examined using reduced-form mehods ha are only able o esimae behavioral relaionships ha hold under he saus quo bu which may no coninue o be valid afer a significan policy change. Currenly, here is no unified behavioral model ha he SSA can rely on o forecas he behavioral and welfare implicaions of any of hese policy changes. The only comprehensive way o deal wih hese muliude of quesions is by modeling explicily he decision processes by boh he individuals and he SSA, under realisic assumpions governing heir behavior. Forunaely, in recen years increasingly realisic dynamic srucural models have been formulaed and esimaed. These models include Rus and Phelan (1997), which esimaed a deailed dynamic programming

5 model of he OASI and Medicare program. In conras o reduced-form papers in he lieraure, he innovaion of heir work is ha hey showed ha a number of previously puzzling aspecs of reiremen behavior are simply arifacs of paricular deails of he Social Securiy rules. In paricular, hey showed ha OASI and Medicare benefis have complex ineracing incenive effecs, and ha seriously misleading policy conclusions can be drawn from sudies ha aemp o sudy OASI and Medicare in isolaion from each oher. The Rus-Phelan model was able o provide coheren economic explanaions for a wide variey of phenomena observed in he daa, including he pronounced peaks in he disribuion of reiremen ages a 62 and 65. These resuls illusrae he poenial payoffs o developing an inegraed dynamic model of social insurance a he end of he life-cycle. We will relax some of he key limiaions of he Rus-Phelan model (paricularly heir assumpion ha consumpion equals income) and use his more general life-cycle model o analyze a number of imporan policy-relaed quesions and issues including: 1) Why does he fracion of Americans on he DI and SSI roles coninue o increase when epidemiological sudies find ha healh of older Americans has improved over ime? 2) Wha is he relaive imporance of changes in award raes, unemploymen raes, welfare reform, and social facors in he large swings in DI incidence raes in recen years? 3) Wha impac do delays in he DI award process have on incenives o apply or appeal? Will proposals o speed up his process increase he number of applicaions and awards? 4) How would reiremen incenives and individual welfare be affeced by an inroducion of individual accouns similar o Presiden Bush's proposed plan? 5) Will he 1983 Social Securiy Amendmens, paricularly he increase in he NRA and DRC, cause individuals o significanly delay he age a which hey apply for OA benefis? How would individuals' be affeced if he Medicare eligibiliy age (MEA) were also increased? 6) Will he increase in he normal reiremen age (NRA) increase he incenive o apply for SSI and DI benefis prior o he NRA? If so, o wha exen will any reducion in he coss of he OA program due o he increased NRA be offse by an increase in he cos of he SSI/DI program? Our model can also address a wide range of policy issues conneced wih axaion, such as predicing he impac of he recen changes in he esae ax on savings and beques decisions. However, we will devoe mos of our aenion o modeling he dynamics of disabiliy, moraliy, and healh, and he facors influencing decisions o apply for SSI and DI benefis, since hese are relaively volaile programs ha have grown a unsusainable raes in recen years. 2. BACKGROUND AND SIGNIFICANCE There is a large empirical lieraure sudying he facors affecing DI applicaions and awards, and a somewha smaller lieraure on he SSI program. This lieraure, (e.g. Rupp and Sapleon, 1996, or Sapleon e al. 1994) has idenified a number of imporan facors: (1) benefi levels; (2) program leniency as measured by award probabiliies and audi raes; (3) srengh of he demand for labor; (4) he availabiliy of alernaive sources of suppor; and (5) social aiudes and sigma associaed wih receiving DI benefis. However, he relaive imporance of hese facors is sill no well undersood, hampering he SSA's abiliy o do policy analysis and shor and long erm forecasing. Figures 1 and 2 illusrae some of he key hisorical and forecased rends in he DI program. Figure 1 summarizes he hisorical and projeced rends in he size and cos of he DI program, measured by he DI prevalence rae and by he raio of DI expendiures o GDP. The righ hand side of Figure 1 shows a rapid rise in he cos of he DI program since is incepion in 1956 unil he mid 1970s, inerruped only by a decrease in he cos of he program during a period of rerenchmen from 1977 o 1990, and a decrease during he economic boom of recen years. The Acuary forecass coninued growh in he program over he nex 75 years, opping ou a roughly 0.9% of GDP by The lef hand panel of Figure 1 plos hisorical and projeced prevalence of DI over he period 1988 o We see ha prevalence raes have increased seadily over he period 1988 o 1996, pausing briefly in 1997 and The Acuary forecass a paricularly

6 rapid increase in DI prevalence unil 2030, by which ime mos of he baby boom generaion will have reached normal reiremen age. Thereafer prevalence coninues o grow a a more moderae rae reaching 7% of he insured populaion by The adjused prevalence curve is based on he assumpion ha he age disribuion of he U.S. remains a is 1998 values. While he unadjused prevalence raes increase from 4% o 7% beween 1999 and 2075, he adjused prevalence rae increases o only 5%. Thus, populaion aging accouns for only abou one hird of he projeced increase in prevalence of DI in he nex 75 years. Figure 1: Hisorical and Forecased Growh in SSDI Roles and Coss Figure 2 illusraes some of he hisorical volailiy in applicaion and award raes. The lef hand panel plos he rend in he crude accepance rae---he raio of he number of new DI awards o he oal number of applicaions and appeals files in a given year. The righ hand panel plos he raio of DI applicaions and awards o he DI insured populaion. The award rae reached is lowes level in 1982 during he Reagan Adminisraion, during a clamp-down on he DI and SSI programs. There was a large increase in audis, also known as Coninuing Disabiliy Reviews (CDR), during his period. The combined effec was o srongly discourage individuals from applying for DI benefis. On he oher hand, DI applicaions and awards peak in 1974 due o several facors: (1) a recession in he early 1970s; (2) a rapid increase in benefi levels due o an error in he 1972 Social Securiy amendmens which resuled in an inadveren double indexing of Social Securiy benefis o inflaion; and (3) a lenien policy owards DI applicans. SSI was also inroduced in 1974, so he public may have acually perceived he SSA as encouraging applicans, reducing perceived hassle coss o applying for DI or he sigma associaed wih receiving benefis. DI applicaion raes began growing rapidly again in he early 1990s following a susained period of growh in award raes. The causes of his rapid burs of growh are no fully undersood, bu high unemploymen raes in he early 1990s, and a cuback in sae General Assisance (GA) programs are hough o be imporan conribuing facors. The passage of he Americans wih Disabiliies Ac in 1990 was designed o force employers o accommodae workers wih disabiliies and hus reduce he incidence and prevalence of individuals receiving DI benefis.

7 Figure 2: Hisory of SSDI Applicaion and Award Raes, Roles, and Coss Applicaion raes declined equally quickly afer This was also he peak year for enrollmens in he AFDC program, and a period of high social sigma owards welfare recipiens may have been one of he mos imporan facors moivaing he ough 1996 Welfare Reform Ac. While par of he decline in applicaion raes migh be ascribed o an increase in real or perceived sigma owards AFDC and SSI recipiens, he years afer 1993 have also consiued he longes peaceime economic boom in recorded hisory. Only wihin a srucural model can one make an aemp o disenangle he relaive conribuions of hese wo possible explanaions. The paradox ha DI prevalence raes have grown while he objecive healh saus of Americans has improved, suggess ha he concep of disabiliy used by he SSA is no based on an absolue objecively deerminable measure of physical saus, bu is raher more akin o a socially defined concep whose absolue sandards may change over ime wih changes in he poliical, social, and echnological climae. Clearly, he naure of physical/menal condiions ha are regarded as disabling is very differen in oday's informaion economy han hey were in an indusrial/agrarian economy early in he cenury. I is no surprising herefore ha he SSA documens significan changes in he disribuion of impairmens ha are lised as he primary reasons being awarded DI benefis. 1 We believe ha he way he SSA adminisers he DI award process can help o creae a social sandard ha has a powerful impac on he public's percepion of he hresholds for menal and physical impairmens ha are sufficienly severe o consiue disabiliy. Indeed, we have shown (see Beníez-Silva e al. 2001) ha self-repored disabiliy saus is an unbiased indicaor of he SSA's ulimae award decision. This finding suggess he possibiliy ha ighening or loosening of DI award raes may have a double effec. Is direc effec is on he individuals' incenives for applying for benefis since i affecs heir chances of success. The indirec effec is hrough he individuals' self-percepions of wheher or no hey believe hey are, in fac, disabled. Our curren version of he model incorporaes he direc effec and allows disabiliy o be a social sandard ha evolves slowly over ime, bu no he indirec effec, which requires we plan o analyze and if feasible incorporae in our model. The DI and SSI programs can be viewed as a game beween applicans and he governmen. The oucome of his game depends on he objecives of he governmen, and he preferences of he individuals. An addiional complicaion is ha he governmen is no a single decision maker, bu raher a hierarchical 1 See Gruber and Kubik (1997), and Auhor and Duggan (2001) for discussions.

8 bureaucracy. The DI sysem of he SSA is divided among 54 sae Disabiliy Deerminaion Services (DDS's) ha process he iniial applicaions and reconsideraions. There are also more han 1,000 Adminisraive Law Judges (ALJ's) and Appeals Board ha process appeals. The complee DI process has been modeled, in Hu e al. (2001) and Lahiri e al. (1995), using he SIPP panel daa, and in Beníez-Silva e al. (1999) using he HRS. The laer paper esimaed a deailed reduced-form muli-sage model of an individual's decision o apply and appeal for SSDI and SSI benefis, and he muli-sage award decision by he DDS, ALJ, and Appeals Board. The paper finds ha he ulimae award rae rises from abou 46%, a he firs sage decision made by he DDS, o abou 73% when he opion o appeal is considered. However, his increased award rae comes a he cos of subsanial delays ha dissuade opporunisic behavior. 2 However, hese reduced-form resuls canno be used o forecas he effecs of policy changes, such as changes in award raes, audi raes, or benefi levels. This mus be incorporaed in a srucural economeric approach ha models he individual's applicaion decision. There are a number of saic srucural models of he DI applicaion process, such as Halpern and Hausman (1978), Kreider (1999), and Kreider and Riphan (2000). The problem wih hese models is ha hey are incapable of capuring he dynamic aspecs of he applicaion and appeal process. This moivaes he model inroduced in he nex secion. This model will be capable of addressing a hos of dynamic policy issues regarding he DI program, including predicing he behavior impac of: (a) changing benefi levels; (b) changing award raes; (c) changing raes a which DI beneficiaries are audied in so-called coninuing disabiliy reviews (CDRs); and (d) changing he delays beween applicaion and award decisions. Economic analysis suggess ha changes in hese parameers will induce complicaed paerns of selfselecion ha will affec who chooses o apply for DI benefis, who chooses o appeal, and so forh. Our model can be used o address he exen o which he curren sysem is incenive compaible, i.e., he exen o which hose who have he mos disabling condiions are he ones who acually apply for and receive disabiliy benefis. 3. PRELIMINARY RESULTS, RESEARCH DESIGN, AND METHODS 3.1. Solving he Life-Cycle Model wih OASI, SSI, DI, and Medicare This secion oulines our plans o esimae a dynamic programming (DP) model of male and female labor supply and social insurance applicaion decisions a he end of he life-cycle. The following key aspecs of he U.S. Social Securiy program and privae insurance and pensions will be modeled: (1) Old Age and Survivors (OASI); (2) Disabiliy, including Supplemenal Securiy Income (SSDI, SSI); (3) Medicare/Medicaid; (4) privae healh insurance; (5) privae pensions and annuiies; (6) Unemploymen and Worker's Compensaion (UI, WC); and (7) join decisions of couples in a household. We firs will focus on iems (1) o (4), and in fuure work will incorporae iems (5) o (7). Our ulimae goal is o develop a DP model where decisions are made on a monhly basis. A monhly decision inerval may be necessary o accuraely model cerain deails of he DI applicaion and appeal process. Our iniial work will focus on building a DP model where decisions are aken a annual inervals. However, an exended model will evenually be developed ha will incorporae monh by monh decisions on all seven iems menioned above. 3 2 The Beníez-Silva e al. (1999) sudy also revealed he imporance of opporunisic economic facors and policy incenives affecing an individual's decision o apply for DI benefis. For example, only very few individuals who are over 62 apply for DI benefis, specifically because hey can ge early Social Securiy reiremen benefis a his age. Even hough DI pays a benefi equal o he full Primary Insurance Amoun (PIA) payable a normal reiremen age, in conras o he early reiremen benefis, he coss associaed wih long delays ouweigh he difference in benefis. 3 The DP model has been programmed in C, and using he Parallel Virual Machine (PVM) library we can disribue he compuaion over several neworked Unix worksaions locaed a Maryland, UCLA, and SUNY. We also plan o apply for supercompuer ime.

9 A his poin he bes way o describe he DP model and illusrae he feasibiliy of our approach is o formulae, solve, and simulae a specific prooype of he DP model ha we are planning o esimae and es using economeric mehods described in secion 5. Since we srongly advocae he full soluion approach o srucural esimaion, he majoriy of he proposal will focus on demonsraing ha i really is feasible o solve and esimae models of he ype we are describing. Alhough simplified in several respecs, our illusraive model already consiues one of he mos ambiious and deailed compuaional models of lifecycle behavior ha has ever been formulaed and solved. We pay special aenion o providing a fairly realisic reamen of he main feaures of he U.S. Social Securiy sysem, including he Disabiliy Insurance program. However, we emphasize ha our simplified prooype model is preliminary and is presened mainly o provide a concree illusraion of how a relaively simple and parsimoniously parameerized life-cycle model can yield richly deailed, inuiively plausible soluions. We hink hese iniial resuls are exremely promising boh compuaionally and empirically, since even his relaively simple iniial prooype model appears consisen wih mos of he broad sylized facs of behavior a he end of he life-cycle ha we observe in he HRS/AHEAD daa ses. These resuls make us opimisic ha i will indeed be possible o develop models ha will acually resul in improved undersanding of a variey of behaviors, and will provide a credible framework for conducing a wide array of policy analyses. However, before his can be done we will need o develop more sophisicaed economeric mehods o esimae and es our models. We emphasize ha we will be subjecing our models o very rigorous in-sample and ou-of-sample saisical ess and would no consider relying on he raher crude iniial calibraion resuls ha we presen below in any formal policy forecasing exercise. We solve he life-cycle model by backward inducion, saring from he erminal age of 100 and working backward unil age 21, when we assume individuals ener he labor force. Agens in our model make hree decisions a he sar of each period, denoed by l, c, ssd. Here, l denoes leisure, ha is, he amoun of waking ime devoed o non-work aciviies, normalized o 1. Thus we define, l = 1 o denoe no working a all, l =. 543 corresponds o full ime work, while l =. 817 corresponds o par ime work. These laer quaniies correspond o he amoun of waking ime spen in leisure, assuming ha a full ime job requires 2000 hours per year and a par-ime job requires 800 hours per year (his is how we ge he leisure values l =. 543 = (12* ) /(12*365) and l =. 817 = (12* ) /(12*365) corresponding o full and par ime-work respecively). The quaniy c denoes consumpion expendiures, which is reaed as a coninuous decision variable. The quaniy ssd denoes he individual's Social Securiy decision, and assumes hree possible values where ssd = 1 denoes he decision o apply for Old Age benefis, ssd = 2 denoes he decision o apply for DI benefis, and ssd = 0 denoes he decision no o apply for benefis. Some of hese choices may be infeasible under cerain circumsances. For example, individuals who are below he early reiremen age (denoed by ERA, currenly se a 62) are no allowed o receive OA benefis. Hence, heir choice se reduces o ssd {0,2}. Also if a person is already receiving OA benefis hey canno re-apply for addiional benefis, so hey face no furher choices unless heir age saisfies ERA < NRA, in which case hey sill have he opion o apply for DI benefis, even while receiving OA benefis. The sae of an individual a any poin in ime can be summarized by four variables: Curren age, ne (angible) wealh w, he individual's Social Securiy sae ss, and he individual's average wage, aw. The ss variable can assume up o en muually exclusive values: ss = 0 (no eniled o benefis), ss = 62 (eniled o OA benefis a he early reiremen age), and ss = 63, K70 represen he remaining 8 Social Securiy saes corresponding o firs becoming eniled for benefis a each of he ages 63, K, 70, respecively. The reason hese saes are required is ha under he SSA benefi formula, he individual's All compuer code ha is developed in his projec will be made freely available o oher researchers via our web sie by he end of he gran period.

10 monhly old age benefi is based on heir primary insurance amoun (PIA) (a piece-wise linear concave funcion of average indexed earnings) and a permanen acuarial adjusmen facor ha depends on he age a which he person was firs eniled o OA benefis. If he age of firs enilemen is before he normal reiremen age (NRA) here is a permanen acuarial reducion: if i occurs afer he NRA here is a permanen increase in benefis due o he delayed reiremen credi (DRC). Noe ha a person mus be a leas 62 in order o be eniled o OA benefis. Therefore i is impossible for ss > 0 if < 62 unless he person is awarded DI benefis. We le ss = NRA denoe he even ha a person is eniled o DI benefis. The reason for his noaion is ha under he SSA rules, if a person is younger han he NRA and is awarded DI benefis, hey receive he same cash benefi as hey would ge if hey had already reached he NRA and applied for OA benefis, wih he excepion ha Medicare benefis are only payable afer a wo year delay. Once a DI recipien reaches he NRA, heir DI benefis auomaically conver o OA benefis. Thus, we can differeniae beween someone who is on he DI program and someone who applied for OA benefis a he NRA by considering heir age: if hey are younger han he NRA bu ss = NRA, hen he person is on DI, oherwise hey are receiving OA benefis. We se he upper bound on he Social Securiy saes o age 70 due o he fac ha here are no furher increases in reiremen benefis under he curren delayed reiremen credi for delaying reiremen pas age 70. Also, due o he SSA provision for auomaic recompuaion of benefis and he fac ha here is no earnings es for individuals who are over 70 years old (and effecive 2000, for individuals over 65 years old), i can be shown ha i would never be opimal o delay applying for reiremen benefis beyond age 70. The average wage, aw, is a key variable in he DP model, serving wo roles: (1) i acs as a measure of permanen income ha serves as convenien sufficien saisic for predicing he evoluion of annual wage earnings; and (2) i is key o accuraely modeling he rules governing paymen of Social Securiy benefis. An individual's highes 35 years of earnings are averaged (or if here is less han 35 years of earnings when he person firs becomes eligible, hen he 5 lowes years of earnings are dropped and he remaining wages are averaged) and he resuling Average Indexed Earnings is wha we refer as average wage and denoe as aw. The poenial Social Securiy benefi rae for reiring a he normal reiremen age (NRA), he so-called Primary Insurance Amoun (PIA), is a piece-wise linear, concave funcion of aw, whose value is denoed by pia ( aw ). In order no o carry as sae variables he enire pas earnings hisory, we approximae he evoluion of average wages in a Markovian fashion, i.e., nex period average wage aw + 1 is prediced using only age,, curren average wage, aw, and curren period earnings, y. Specifically, we use he observed sequence of average wages as regressors o esimae he following ( misspecified ) log-normal regression model of an individual's annual earnings: 2 log( y) α1 α2log( aw) α3 α4 η. = (1) While his regression need no correspond o he rue process governing y, using he hisory of 2 earnings from he resriced HRS daa se we obained an R above 0.9. Also, a log-normal regression model for average wages akes he form: 2 log( aw 1) γ1 γ 2log( y) γ3log( aw) γ4 γ5 ε. + = (2) 2 The R for his ype of regression is also close o 0.9, wih an exremely small esimaed sandard error, resuling from he low variabiliy of he { aw } sequences. This finding is highly encouraging, since i is a key resul for an imporan compuaional simplificaion ha allows us o accuraely model he Social Securiy rules in our DP model wih minimal number of sae variables.

11 Our DP model also accouns for he oher key deails of he Social Securiy rules. For example, here is a penaly for reiring prior o he normal reiremen age. Tha is, an individual's PIA is permanenly reduced by an acuarial reducion facor of exp( g1k), where k is he number of years prior o he NRA (o a maximum of NRA-ERA) ha he individual firs sars receiving OA benefis. Our DP model uses he acuarial reducion rae g 1 = ha is currenly in effec in he U.S. If a person is acceped ino he DI program, he/she receives he full PIA regardless of his/her age. To increase he incenives o delay reiremen, he 1983 Social Securiy reforms gradually increased he NRA from 65 o 67 and increased he delayed reiremen credi. This is a permanen increase in he PIA by a facor of exp( g 2 k), where k denoes he number of years afer he NRA ha he individual delays receiving OA benefis. The rae g 2 is being gradually increased over ime. In he simulaions below we use he curren value of g 2 = The maximum value of k is MRA-NRA, where MRA denoes a maximum reiremen age (currenly 70), beyond which furher delays in reiremen yield no furher increases in PIA. As noed above i is no opimal o delay applying for OA benefis beyond he MRA, because due o moraliy furher delays generally reduce he presen value of OA benefis he person will collec over heir remaining lifeime. The final aspec of he Social Securiy rules concerns axaion of benefis. Individuals whose combined income (including Social Securiy benefi) exceeds a given hreshold mus pay Federal Income axes on a porion of heir Social Securiy benefis. We incorporae hese rules in our model as well as he 15.75% Social Securiy payroll ax, in addiion o he Federal income ax, on wage earnings. In addiion o hese axes, we accoun for he Social Securiy earnings es. 4 Our model also incorporaes a deailed model of axaion of oher income, including he progressive Federal income ax schedule (including he negaive ax known as he EITC -- Earned Income Tax Credi), and sae and local income, sales and propery axes. Finally, our model provides a simplified accoun of he DI award/appeal process. We assume ha, even if a person is no working, here is only a probabiliy p 1(, h, w ) [0,1] ha a person of age, healh h and wealh w will be awarded benefis. Even if benefis are awarded here is a six monh waiing period before hey can be paid. Adding on he ypical delays in he applicaion and appeal process, we assume ha if a person applies for DI a he beginning of year ha hey will only be noified wheher hey will sar receiving benefis a year + 1. We do no model he rial work period a his sage. We assume ha he SSA also randomly audis DI recipiens who are no working, and wih probabiliy p 2 (, h, w ) a DI recipien can be removed from he rolls. We have esimaed hese probabiliies using he HRS microdaa. To complee he specificaion of he DP model, we need o make some assumpions abou individuals' healh, moraliy, and preferences for leisure and consumpion. We assume ha he maximum possible age for any individual is 100. Currenly we use age specific deah raes from he U.S. Decennial Life Tables (1997). 5 We inroduce healh as an exogenous sae variable (denoed by h = 0, 1 or 2 ) ha akes on hree values: good, poor, and disabled. The ransiion probabiliies for healh are esimaed from he HRS. Our esimaion resuls were consrained so ha moraliy, weighed by he various proporions of individuals in each of he 3 healh saes a each age, equals he aggregae U.S. moraliy raes from he Decennial life ables. This ensures ha he esimaed survival curve from our simulaions always provides a very close esimae of he populaion average survival curve implied by he Decennial life ables. We assume ha he individual's uiliy is given by 4 If a person reires beween he ERA and NRA, each dollar of earnings above a cerain hreshold (currenly $10,800) resuls in a 50 cen reducion in Social Securiy benefis. Beween he NRA and MRA he implici earnings es ax rae falls o 33% for earnings above a higher hreshold (currenly $17,000). For individuals who are above he MRA, here is no earnings es. The earnings es provision has been recenly eliminaed for individuals 65 and over. However, i was sill in place during he ime he HRS daa was colleced and herefore we include i in our model. 5 We are also well under way in pursuing esimaion of moraliy probabiliies using he HRS/AHEAD daa. These probabiliies depend on healh, wealh, marial saus, educaion, ec.

12 if ssd = 2, oherwise u ( c, l, ssd, h, age) = log( c) + φ( age, h, aw) log( l) 2h K (3) u ( c, l, ssd, h, age) = log( c) + φ( age, h, aw)log( l) 2h (4) where φ ( age, h, aw) is a weigh ha can be inerpreed as he relaive disuiliy of work. We assume ha φ an increasing funcion of age and healh saus (i.e., individuals in worse healh have a higher disuiliy of work) and is a decreasing funcion of aw, reflecing he fac ha individuals wih higher permanen income ypically have more ineresing and physically less demanding jobs, and hus a lower disuiliy of work han a blue collar worker who ypically performs more physically demanding. 6 The final parameer k represens he hassle and sigma coss involved in applying for DI benefis. The k parameer can acually be a funcion of unobserved heerogeneiy and oher observed covariaes such as age, average wage, and so forh. We assume here are no ime or financial coss o applying for OA benefis. In subsequen work we will include hese coss as addiional parameers o be esimaed, as well as allowing for sigma coss o being on he DI roles (as opposed o simply applying). Figure 3 plos he φ funcion ha we used in he simulaions below. The lef panel shows how φ varies as a funcion of age and healh he righ panel show how i varies wih aw. Figure 3: Relaive Weigh on Leisure as a Funcion of Age, Healh and Average Wage Regarding wage earnings, we model he sochasic evoluion of full-ime wages, for full-ime workers, via he regression model in equaion (1). These wages are based on 2,000 hours of work per year. Par-ime workers are assumed o work 800 hours per year, and a a wage rae ha is a fracion, /(1 + h / 4), of he full-ime wage rae. We le healh saus ener he realized wage rae (and also he full ime wage rae) o reflec he likelihood ha individuals who are in poor healh or who are disabled will expec o earn a lower wage rae han individuals who are in good healh. In fuure work we will esimae a more realisic specificaion of he wage rae relaionship using he HRS and AHEAD microdaa. 6 In he subsequen economeric analysis we will allow he disuiliy o conain parameers reflecing unobserved heerogeneiy for leisure, and le he daa ell he disribuion of he disuiliy of work condiional on he average wage and oher observable variables.

13 Le V ( w, aw, ss, h) denoe he individual's value funcion, he expeced presen discouned value of uiliy from age onward for an individual wih curren wealh w, average wage aw, in Social Securiy sae ss and healh sae h. We solved he DP problem via numerical compuaion of he Bellman recursion for V given by V ( w, aw, ss, h) = max V ( w, aw, ss, c, l, ssd, h), where (5) 0 c w l {.54,.81,1} ssd A ( ss) [ ] 1 V( w, aw, ss, c, l, ssd, h) = u( c, l, ssd, h) + β 1 d( h) EV+ ( w, aw, ss, c, l, ssd, h), (6) + d ( h) EB( w, aw, ss, c, l, ssd, h). (7) where A (ss) denoes he se of feasible Social Securiy choices for a person of age in Social Securiy sae ss and d (h) denoes he age and healh-specific deah rae, B (w) is he beques funcion, and EB denoes is condiional expecaion. The beques funcion is simple, i only depends on he absolue size of he wealh lef a he end of he period in which he individual dies, which follows wha has been called he egoisic model of beques. As we noed before, we used he HRS and AHEAD daa o esimae age and healh-specific deah raes, bu since here is lile daa on individuals over 80 years old we make parameric smoohness assumpions on he d (h) funcion (basically a logi funcional form ha is polynomial in and has dummy variables for he various healh saes h ) and subjec he esimaes o he furher resricion ha for each he expeced hazard over h should equal he uncondiional age-specific deah raes given in he 1997 ediion of he U.S. Decennial life ables. The funcion EV + 1 denoes he condiional expecaion of nex period's value funcion, given he individual's curren sae ( w, aw, ss, h) and decision ( c, l, ssd). Specifically, we have = y + 1 h = 0 ss = 0 EV ( w, aw, ss, c, l, ssd, h) V ( wp ( w, aw, y, ss, ssd), awp ( aw, y ), ss ) f ( y awk ) ( h hg ) ( ss awwssssddy,,, ), (8) where awp ( aw, y) is he Markovian updaing rule ha approximaes Social Securiy's exac formula for updaing an individual's average wage, and wp summarizes he law of moion for nex period's wealh, ha is, wp( w, aw, y, ss, ssd) = R w + ssb( aw, y, ss, ssd) + y τ ( y, w) c, (9) where R is he reurn on saving, and τ ( y, w) is he ax funcion, which includes income axes such as Federal income axes and Social Securiy axes and poenially oher ypes of sae/local income and propery/wealh axes. The awp funcion, derived from (2), is given by 2 2 { γ1 γ2 γ3 γ 4 γ5 σ } awp ( aw, y) = exp + log( y) + log( aw) / 2 (10) where σ is he esimaed sandard error in he regression (2). Noe here is a poenial Jensen's inequaliy problem here due o he fac ha we have subsiued he condiional expecaion of w + 1 ino he 2 nex period value funcion V + 1 over w + 1 and aw + 1 joinly. However, as noed above, he R for he

14 regression of aw + 1 on aw and w + 1 is almos 1 wih an exremely small esimaed sandard error. In his case here is virually no error resuling from subsiuing wha is an essenially deerminisic mapping deermining aw + 1 from w + 1 and aw. 7 Finally, f ( y aw) is a log-normal disribuion of curren earnings, given curren age and average wealh aw, ha is implied by (1) under he addiional assumpion of ' normaliy of errors η. The discree condiional probabiliy disribuions g ( ss aw, w, ss, ssd) and k ( ' h h) reflec he ransiion probabiliies in he Social Securiy and healh saes, respecively. A each ime period he explici opimizaions in equaion (5) were performed over a grid of 375 poins in he ( w, aw) sae space (25 grid poins for w and 15 grid poins for aw ), where w ranges from $1 o $1,000,000, and aw ranges from $3,000 o $72,600. Two-dimensional inerpolaion was used o compue approximae values for V ( w, aw, ss, h) a ( w, aw) poins ha are no on he predefined grid. A oal of more han 15 million evaluaions of he expeced value funcion EV + 1 were required o compue he opimal decision rule for consumpion, labor supply, and he Social Securiy pension decision for all he ( w, aw ) grid poins, and all 10 Social Securiy saes and he 3 healh saes and he 80 ages beween 21 and 100. The DP problem solves in approximaely 5 minues on a 2.4Ghz Penium IV compuer. We are currenly implemening changes in he soluion algorihm and compuer code ha should bring his ime down o less han 1 minue for a single soluion of he life-cycle problem. Alhough we do no have space o go ino deails abou our plans in his area, suffice i o say ha we will be using polynomial approximaion mehods and Mone Carlo inegraion echniques ha will conver he backward inducion problem ino a sequence of linear regression problems, where he coefficiens in hese regressions represen he projecion of he value funcion V (evaluaed a a random grid of poins in he sae space) on a suiably chosen se of basis funcions. Figures 4 o 9 illusrae he rich ypes of behavior ha he DP model predics. Each of he curves is an average of 300 IID simulaions, wih each simulaion corresponding o a separae person followed from age 21 unil heir deah. The averages were compued a each age, for he subpopulaion of survivors who lived unil a leas ha age. The lef hand panel of Figure 4 shows he employmen saus from he HRS/AHEAD daa as a funcion of age. There is a clear decline in labor force paricipaion saring a abou he age of 54. There is also significan increase in par-ime work afer he age of 60. The simulaion resuls shown on he righ hand panel of Figure 1 exhibis a very similar paern, excep ha he DP model predics far less par ime work a older ages han is observed in he daa. Par of his is an arifac of our classificaion of par ime work in he HRS: we classified individuals who worked beween 100 and 1400 hours in a given year as par ime workers. Many individuals in his caegory may be individuals who reired from a full ime job par way ino a calendar year. I would make more sense o classify such individuals as full ime workers: once we adjus for his we expec he observed raes of par-ime work will be far less han hose shown on he lef panel of Figure 1. A he same ime we will be adjusing our specificaion of he DP model o increase he raes of par ime work among older individual o beer mach he (correced) raes observed in he HRS and AHEAD. Figure 5 depics acual and simulaed healh saus. Again, he simulaed healh saus on he righ panel of he figure is exremely close o he acual paern on he lef panel of he figure. This is an encouraging resul, since healh is a vial variable in our model. Figure 6 provides he resuls for Social Securiy saus. On he lef panel of he figure we presen he acual HRS daa from he firs wo waves of he survey, while on he righ panel we depic he simulaion resuls. The model capures he main feaures of he daa wih reasonable accuracy. The percenage of 7 Noe we are unable o repor he parameer esimaes in his proposal since hey are based on confidenial resriced daa on Social Securiy earnings records. We are currenly seeking approval from he NIH human subjecs commiee o repor hese esimaes in published aricles since here is virually no way any confidenial individual-level informaion can be inferred from our regression esimaes.

15 populaion on disabiliy is around 10%. Furhermore, in boh panels almos all individuals sar receiving OA benefis saring a age 62. Figure 4: Acual vs. Simulaed Labor Force Paricipaion Figure 5: Acual vs. Simulaed Healh Saus Figure 7 shows acual and simulaed rajecories for wages, and wealh. On he righ panel of he figure we also provide he simulaed rajecories of Social Securiy benefis and consumpion over he life-cycle. Firs, we see ha wages increased over he firs par of he individuals' life-cycle and sar dropping in heir lae 50's in boh panels of he figure. During he firs 30 years, individuals consumed only abou 70% of heir wage earnings, resuling in a rapid buildup of ne worh ha peaks a age 60 in our simulaions, and slighly laer in he acual daa. The maximum level of wealh accumulaion is abou he same in he daa and he simulaions, bu he life-cycle model predics a more peaked rajecory for wealh: building up much faser han we observe in he HRS before age 60 and decumulaing a a faser rae han we observe afer age 60 for older individuals in he HRS and AHEAD. Also he acual disribuion of wealh is more skewed han we

16 observe in he simulaions, so ha he mean levels of wealh a age 60 is more han wice as high as he median wealh a age 60. The life-cycle model does no resul in such a skewed disribuion of wealh. We hink par of he reason for his discrepancy is ha we have ignored income from oher sources in he life cycle model, such as spousal income and inheriances. We hink ha a more realisic version of he model, which would ake ino accoun hese oher sources of income will produce more skewed disribuions of wealh as observed in he daa. We also believe ha accoun for oher risks such as he risk of job loss (involunary unemploymen) and uninsured medical coss, he life-cycle model will predic subsanially higher precauionary savings raes han we observe in he curren model where he main risks are loss of job due o healh problems, moraliy, and uncerainy abou fuure wage raes. We also hink ha more careful reamen of asse illiquidiy and ransacions coss associaed wih many durable goods such as housing will also resul in larger, more skewed wealh disribuions. Also, modeling liquidiy consrains and ransacions coss associaed wih housing will probably lead o much slower raes of decumulaion of wealh afer reiremen. 8 Figure 6: Acual vs. Simulaed Social Securiy Saus Figure 7: Wealh, Earnings, Social Securiy Benefis, and Consumpion 8 Fernández-Villaverde and Krueger (2001), and Berola, Guiso, and Pisaferri (2001) presen models ha include durable goods in life cycle models wih uncerainy.

17 Overall we hink he simulaion resuls provide a very good firs approximaion o he daa. Similar o he Engen e al. (1999) simulaion findings, we see lile evidence ha individuals in he HRS cohors are undersaving for reiremen, a conclusion suppored by direc empirical analyses of wealh and pension accumulaions in he HRS by Gusman and Seinmeier (1999). The life-cycle model does show how wealh accumulaion plays an imporan role in life-cycle consumpion smoohing. In paricular, he rapid decumulaion of wealh afer reiremen allows individuals o mainain a relaively smooh paern of consumpion over heir life cycle (wih a drop around he age of reiremen counerbalanced in uiliy erms by a subsanial increase in leisure), even when major changes in labor supply, and available resources, occur. Figure 8 compares simulaed and acual disribuions of age of firs enilemen o OA benefis. Again, boh he acual and he prediced daa have he same main feaure: mos individuals sar receiving benefis a age 62. In he acual daa here also a second, much smaller, peak a age 65, which he life-cycle model overpredics. Noice he clear rend oward early reiremen compared wih he evidence from he 1970s, analyzed for example in Rus and Phelan (1997). This change has also been menioned by Gusman and Seinmeier (2002), however, hey do no use enilemen raes bu a measure of reiremen from he labor force, which do no necessarily coincide. Figure 8: Acual vs. Simulaed Disribuion of Age of Firs Enilemen o OA Benefis Noe ha here is no heerogeneiy in he life-cycle model oher han ha which is produced by randomly evolving incomes, average wages, and healh. We feel ha our life cycle model maches he observed disribuion of reiremen ages quie well given he limied amoun of heerogeneiy currenly in he model. In fuure work we expec ha inclusion of addiional sources of observed and unobserved heerogeneiy will enable us o mach he observed disribuion of reiremen ages more precisely. Figure 9 compares he disribuion of ages of firs enilemen for DI benefis. In his case our simulaions are qualiaively similar o he acual disribuion, excep ha he life-cycle model underpredics he mean age of firs receip of DI benefis by 3 years. We believe ha wih addiional experimenaion on he form of he ime and hassle/sigma coss associaed wih applying for DI benefis we will be able o provide a much more accurae approximaion o he disribuion of ages of eligibiliy for DI benefis.

18 Figure 9: Disribuion of Age of Firs Enilemen o DI Benefis We conclude wih Figure 10, which shows wo more ineresing implicaions of he life-cycle model. The lef panel shows he disribuions of bequess. I is highly skewed, wih a small number of relaively large bequess. However, none of he bequess in our simulaion of 300 relaively lower income individuals exceeded he $600,000 exempion ha would have subjeced hem o Federal esae axes. Alhough here is no direc daa on he size of bequess in he HRS, here has been some work using he firs wo waves of he AHEAD by Hurd and Smih (2001), which indicaes ha our simulaions are no unreasonable, and ha he disribuion of bequess is indeed skewed, bu seem o undersae he level of bequess. We plan o updae par of heir work in order o be able o appropriaely characerize his aspec of he model. If necessary, we also plan o seek ou oher sources of daa (including probae records) o check he predicions of our model. Figure 10: Implied Disribuions of Bequess and IRRs on Social Securiy Conribuions The righ panel of Figure 10 plos he disribuion of inernal raes of reurn (IRR) on Social Securiy conribuions implied by our model. For each of he 300 people in our simulaion, we compued he IRR on Social Securiy conribuions---i.e., he ineres rae which equaes he discouned value of Social Securiy

19 axes (including he employer share) o he presen value of Social Securiy benefis (including disabiliy benefis). We can see ha consisen wih oher sudies using acual Social Securiy benefis, he average inernal rae of reurn on Social Securiy is quie low---less han 2%. This low inernal rae of reurn is he main cos of coninuing o operae a fundamenally pay-as-you-go Social Securiy sysem in a rapidly aging sociey. I is quie easy o re-solve he life-cycle model wihou any Social Securiy a all. Individuals lose he valuable risk-sharing feaures of he curren Social Securiy program bu are freed from he resricion of having o make forced conribuions o a program ha offers very poor raes of reurn. I is no difficul o compue compensaing variaions as a funcion of an individual's curren sae: his is he amoun a person would pay (or need o be paid) as a bribe o be released from he curren Social Securiy sysem. Due o space consrains we do no show hese calculaions here, bu noe ha our preliminary calculaions show jus he paern we would expec: young people (i.e., hose younger han age 50) would pay a subsanial fracion of heir wealh o be released from Social Securiy. However, old people, having already paid mos of heir Social Securiy conribuions and who are looking forward o receiving heir Social Securiy benefis would pay an even larger share of heir wealh in order o keep he saus quo. We can average hese person-specific bribes over he populaion disribuion of consumers and deermine wheher he young in aggregae are willing o pay enough o he old o compensae hem for abandoning he sysem and moving o a fully privaized sysem. We are no suggesing here ha a fully privaized social securiy is he bes possible alernaive o he saus quo. Our poin here is simply o illusrae how he life-cycle model can be used o conduc welfare analyses and analyze he disribuional consequences and relaive efficiency of alernaive proposed policies (assuming lump sum ransfers are possible) in a way ha is no heavily dependen on subjecive inerpersonal uiliy comparisons. We go ino furher deail abou his issue in secion 6. We believe hese resuls show he richness and he insighs ha can be obained by aking he effor o formulae and solve a reasonably realisic version of he life-cycle model. We also believe hese resuls show ha he model is already empirically promising. If we define an empirically realisic model as one ha is consisen wih he main sylized facs of observed behavior, hen our model seems a he very leas o be he ballpark. We inend o subjec our model o much more rigorous ess ha can be viewed as he saisical equivalens of he Turing es : i.e., will an average researcher be able o disinguish arificial daa generaed from sochasic simulaions of our life-cycle model from real daa such as we have in he HRS/AHEAD? We have a lo of work o do before we can hink of posing such ess, since we have noed a number of aspecs where his simple model appears o be inconsisen wih he daa. Much of he focus of he firs year of our research will be on exending he model in order o relax some of he resricive predicions of his prooype model. We bear in mind a resul of Rus (1994) ha he dynamic programming model is non-paramerically unidenified. Tha is, if we are free o choose any specificaion for individual's preferences and beliefs we will always be able o consruc a sufficienly complicaed life-cycle model ha can raionalize any observed paern of behavior. Moreover here will generally be infiniely many disinc combinaions of preferences and beliefs ha will succeed in raionalizing observed behavior. We are no deerred by his heoreical argumen abou possible lack of idenificaion for wo reasons. Firs, alhough i is in principle possible o raionalize any observed se of daa using a sufficienly complicaed life-cycle model, in pracice compuaional consrains do no afford us he luxury of being able o formulae, solve, and esimae arbirarily complicaed models. Alhough our research represens a big sep forward, our models will sill remain highly simplified and unrealisic in many respecs, herefore, subjec o sandard goodness of fi ess ha compare he predicions of he model wih in-sample and ou of sample daa. Those sandard ess ofen indicae ha we have solved misspecified models, his is no oo surprising, since he models under consideraion are jus simplified represenaions of a very complex realiy. We do no view he exended life-cycle model as an exac lieral descripion of an individual's rue behavior, bu only as

20 a convenien approximaion. The relevan quesion is wheher he life-cycle model provides a parsimonious represenaion of behavior ha resuls in beer predicions han oher compeing saisical models: we believe ha he exended life-cycle framework represens he bes curren framework for modeling he wide range of behaviors we observe in he daa. Because we are aware of he idenificaion problem, in he process of underaking our planned exensions of our model we will be concerned abou he possibiliy of overfiing, i.e., of selecing paricular funcional forms ha enable us o fi he daa well in-sample. The concern is ha such specificaion searching will no resul in a model ha forecass well ou-of-sample. As we noed in he inroducion, we plan o subjec our model o perhaps one of he mos demanding possible ypes of ou-ofsample predicive ess, namely, using he DP model o predic he response of he subjecs receiving he 2 for 1 reamen in he TWWIIA demonsraion projec. If our models fi well boh in-sample and in ou-offsample predicive ess, his is abou he mos ha we can hope for. However, he fac ha our models may fi and predic well does no prove ha people are acually behaving as if hey had solved he life-cycle problem, bu i does give us more confidence ha he models can be given more weigh in policy forecasing exercises. Ulimaely he model ha predics he bes and is easies o use will be he winner. By coming forward and submiing his proposal we are being ha he life-cycle model will emerge as he winner --- a leas for he foreseeable fuure Economeric Implemenaion We briefly describe how our DP models will be esimaed and esed economerically. We pariion an individual's sae a ime, s, ino observed, x, unobserved, ε, componens, ha is s = ( x, ε ). The individual observes he full sae s, bu he economerician only observes x. One mus accoun for unobserved saes mainly because a model wihou unobserved sae variables would be saisically degenerae, i.e., i would predic ha cerain observed combinaions of saes and decisions would have zero likelihood of occurring. Assume, for he momen, ha all choices are discree, i.e., he individual chooses a decision d from a finie se of alernaives C ( x ). Then, here is a well-developed racable economeric approach (summarized in Rus 1994) for incorporaing unobserved saes and esimaing he unknown parameers of he DP model. Le he number of componens of ε be he same as he number of choices in he choice se C ( x ). Then ε (d) can be inerpreed as a componen of uiliy ha depends on he decision d and oher unobserved saes of he agen. For reasons of compuaional racabiliy, we impose he assumpion ha x and ε are condiionally independen px (, ε x, ε, d) = px ( x, d) q( ε ). (11) We also assume ha ε is an IID Type III exreme value process. Under hese assumpions Rus (1994) showed ha he DP recursions ake he following form: V( x, ε ) = max u( x, d) + ε( d) + β V 1(, ) (, ) ( ) x ε p dx x d q dε + d C( x) (12) ε x We can rewrie equaion (12) as follows: where V ( x, ε) = max V ( x, d) + ( d), (13) d C x [ ] ε ( )