No. 2012/18 Analyzing the Effects of Insuring Health Risks On the Trade-off between Short Run Insurance Benefits vs. Long Run Incentive Costs

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1 CFS WORKING P APER No. 212/18 Analyzing te Effects of Insuring Healt Risks On te Trade-off between Sort Run Insurance Benefits vs. Long Run Incentive Costs Harold L. Cole, Soojin Kim, and Dirk Krueger Center for Financial Studies Goete-Universität Frankfurt House of Finance Grüneburgplatz Frankfurt Deutscland Telefon: +49 () Fax: +49 () ttp:// info@ifk-cfs.de

2 Center for Financial Studies Te Center for Financial Studies, located in Goete University s House of Finance in Frankfurt, is an independent non-profit researc center, funded by te non-profitmaking organisation Gesellscaft für Kapitalmarktforscung e.v. (GfK). Te CFS is financed by donations and by contributions of te GfK members, as well as by national and international researc grants. Te GfK members comprise major players in Germany s financial industry. Establised in 1967 and closely affiliated wit te University of Frankfurt, it provides a strong link between te financial community and academia. CFS is also a contributor to policy debates and policy analyses, building upon relevant findings in its researc areas. Te CFS Working Paper Series presents te result of scientific researc on selected topics in te field of money, banking and finance. Te autors were eiter participants in te Center s Researc Fellow Program or members of one of te Center s Researc Projects. If you would like to know more about te Center for Financial Studies, please let us know of your interest. Prof. Micalis Haliassos, P.D. Prof. Dr. Jan Pieter Kranen Prof. Dr. Uwe Walz Center for Financial Studies Goete-Universität House of Finance Grüneburgplatz Frankfurt am Main Deutscland Telefon: +49 () Fax: +49 () ttp:// info@ifk-cfs.de

3 CFS Working Paper No. 212/18 Analyzing te Effects of Insuring Healt Risks * On te Trade-off between Sort Run Insurance Benefits vs. Long Run Incentive Costs Harold L. Cole 1, Soojin Kim 2, and Dirk Krueger 3 November 29, 212 Abstract Tis paper constructs a dynamic model of ealt insurance to evaluate te sort- and long run effects of policies tat prevent firms from conditioning wages on ealt conditions of teir workers, and tat prevent ealt insurance companies from carging individuals wit adverse ealt conditions iger insurance premia. Our study is motivated by recent US legislation tat as tigtened regulations on wage discrimination against workers wit poorer ealt status (Americans wit Disability Act of 29, ADA, and ADA Amendments Act of 28, ADAAA) and tat will proibit ealt insurance companies from carging different premiums for workers of different ealt status starting in 214 (Patient Protection and Affordable Care Act, PPACA). In te model, a trade-off arises between te static gains from better insurance against poor ealt induced by tese policies and teir adverse dynamic incentive effects on ouseold efforts to lead a ealty life. Using ouseold panel data from te PSID we estimate and calibrate te model and ten use it to evaluate te static and dynamic consequences of no-wage discrimination and no-prior conditions laws for te evolution of te cross-sectional ealt and consumption distribution of a coort of ouseolds, as well as ex-ante lifetime utility of a typical member of tis coort. In our quantitative analysis we find tat altoug a combination of bot policies is effective in providing full consumption insurance period by period, it is suboptimal to introduce bot policies jointly since suc policy innovation induces a more rapid deterioration of te coort ealt distribution over time. Tis is due to te fact tat combination of bot laws severely undermines te incentives to lead ealtier lives. Te resulting negative effects on ealt outcomes in society more tan offset te static gains from better consumption insurance so tat expected discounted lifetime utility is lower under bot policies, relative to only implementing wage nondiscrimination legislation. JEL Classifications: E61, H31, I18 Keywords: Healt, Insurance, Incentive * We tank Hanming Fang, Iourii Manovskii and participants at EUI, EIEF, te Penn Macro Lunc and te Annual Meeting for Society for Economic Dynamics 212 for many elpful comments. Krueger gratefully acknowledges _nancial support from te NSF under grant SES University of Pennsylvania and NBER 2 University of Pennsylvania 3 University of Pennsylvania, CEPR, CFS, NBER, and Netspar

4 Americans wit Disabilities Act and its Amendment in 29 sougt to restrict te ability of employers to employ and compensate workers differentially based upon ealt related reasons. In order to analyze te impact of tese policies we construct a dynamic model of ealt insurance wit eterogeneous ouseolds. As in Grossman (1972), ealt for tese ouseolds is a state variable. A ouseold s ealt state elps to determine bot teir productivity at work and te likeliood tat tey will be subject to adverse ealt socks. Our model features te two-way interaction between ealt and income tat as been empasized in te literature. Our model of ealt socks includes temporary ealt socks tat impact on productivity and can be offset by medical expenditures (as in Dey and Flinn 25), and catastropic ealt socks wic require nondiscretionary ealt expenditures to avoid deat. Healt status in our model is persistent and evolves stocastically. Tis evolution is affected by te ouseold s efforts to maintain teir ealt wic results in a moral azard problem as ealt related insurance reduces ouseolds incentives to maintain teir ealt. We explicitly model te coice of medical expenditure and tereby endogenously determine te ealt insurance policy and ow it responds bot to te ouseold s state in terms of ealt status, age and education. Te focus of our analysis is ow te distributions of ealt status, earnings and ealt insurance costs will evolve under different policy coices and te impact of tese coices on welfare. We consider several different policy regimes. Te first is a complete insurance bencmark in wic te social planner can dictate bot te ealt insurance contract, te effort made to maintain ealt and te extent of redistributive transfers tat provide full insurance against all ealt related socks. Te second is pure competition in wic workers enter into one-period employment and insurance contracts. Competition leads tese contracts to partially insure te worker against witin period temporary ealt socks, but not against is initial ealt status and te transition of tis status. Te second is a version of te no-prior conditions restriction on ealt insurance in wic ealt insurance companies compete to offer one-period ealt insurance contracts in wic tey cannot differentially carge based upon te worker s ealt status. Te tird is a version of te no-discrimination restrictions on employment in wic firms cannot differentially ire or pay workers based upon teir ealt status. In te fourt version we consider te impact of bot te no-prior conditions and te no-discrimination restrictions jointly. We study bot te static and te dynamic impact of tese policies. One of te key aspects of te dynamic analysis is te impact tese policies ave on individuals incentives to maintain teir ealt and te feedback tis creates between te ealt distribution of te population and te costs of ealt insurance and productivity of te workforce. We evaluate te quantitative impacts of te different policies on consumption insurance, incentives and aggregate outcomes, and, ultimately, welfare. To do so, we first estimate and calibrate te model using PSID data to matc key aggregate statistics on labor earnings, medical expenditures and observed pysical exercise levels. We ten use te parameterized version of te model as a laboratory to evaluate different policy scenarios. Our results sow tat a combination of wage non-discrimination law and no prior conditions law provides full insurance against ealt risks and restores te first-best consumption insurance allocation in te sort run, but leads to a severe deterioration of incentives and tus te population ealt distribution in te long run. Quantitatively evaluating te welfare consequences of tis trade-off we find tat even toug bot policies improve upon te laissez-faire equilibrium, implementing tem jointly is suboptimal, relative to introducing a wage nondiscrimination in isolation. 1.1 Institutional Background Te U.S. as a long istory of policy initiatives in relation to ealt risk. Implicitly Welfare programs, wic date back to te 193s and were greatly expanded by te Great Society in te 196s, insure workers against a variety of socks, implicitly including ealt related socks insofar tey affect earnings. Since 1965 Medicare as sougt to provide ealt insurance to te elderly and te disabled. Medicaid as sougt to provide ealt insurance to te poor since te 199s. Te last two decades legislation in te U.S. was passed tat limits te ability of employers to condition wages on te ealt conditions of employees, and to discriminate against applicants wit prior ealt conditions wen filling vacant positions. 2

5 1.1.1 Wage Based Discrimination In 199 Congress enacted te Americans wit Disabilities Act (ADA) to ensure tat te disabled ave equal access to employment opportunities. 1 At tis point a disability was interpreted as an impairment tat prevents or severely restricts an individual from doing activities tat are of central importance to one s daily life. In 29 te ADA Amendments Act (ADAAA) went into effect. Tis act rejected te strict interpretation of te ADA, broadening te notion of a disability. Tis included proibiting te consideration of measures tat reduce or mitigate te impact of a disability in determining weter someone is disabled. It also allowed people wo are discriminated against on te basis of a perceived disability to pursue a claim on te basis of te ADA regardless of weter te perceived disability limits or is perceived to limit a major life activity. Te ADAAA excludes from te definition of a disability tose temporary or minor impairments. 2 Under te ADAAA people can be disabled even if teir disability is episodic or in remission. For example people wose cancer is remission or wose diabetes is controlled by medication, or wose seizures are prevented by medication, or wo can function at a ig level wit learning disabilities are all disabled under te act. Before te ADA job seekers could be asked about teir medical conditions and were often required to submit to a medical exam. Te act proibited certain inquiries and conducting a medical exam before making an employment offer. However, te job could still be conditioned upon successful completion of a medical exam. 3 Te ADA permits an employer to establis job-related qualifications on te basis of business necessity. However, business necessity is limited to essential functions of te job. So impairments tat would only occasionally interfere wit te employee s ability to perform tasks cannot be included on tis list. 4 Ajob function is essential if te job exists to perform tat function or if te limited number of employees available at te firm requires tat te task must be performed by tis worker. Furtermore, a core requirement of te ADA is te obligation of te employer to make a reasonable accommodation to qualified disabled people Insurance Cost and Exclusion Discrimination In 1996, Congress passed te Healt Insurance Portability and Accountability Act (HIPAA) wic placed limits on te extent to wic insurance companies could exclude people or deny coverage based upon preexisting conditions. Altoug insurance companies were allowed exclusions periods for coverage of preexisting conditions, tese exclusion periods were reduced by te extent of prior insurance. In particular, if an individual ad at least a full year of prior ealt insurance and se enrolled in a new plan wit a break of less tan 63 days, se could not be denied coverage. However, insurers were still allowed to carge iger premiums based upon initial conditions, limit coverage and set lifetime limits on benefits. 6 Tere is evidence tat many patients wit pre-existing conditions ended up eiter being denied coverage, 7 or aving teir access to benefits limited. 8 Te Patient Protection and Affordable Care Act of 21 furter extended protection against pre-existing conditions. Beginning in 21 cildren below te age of 19 could not be excluded from teir parents ealt insurance policy or denied treatment for pre-existing conditions. Beginning in 214 tis restriction will apply to adults as well. Moreover, insurance companies will no longer be able to use ealt status to determine eligibility, benefits or premia. In addition, insurers will be prevented from limiting lifetime or annual benefits 1 Te ADA sets te federal minimum standard of protection. States may ave a more stringent level. 2 Under te ADAAA major life activities now include: caring for oneself, performing manual tasks, seeing, earing, eating, sleeping, walking, standing, lifting, bending, speaking, breating, learning, reading, concentrating, tinking, communicating, working, as well as major bodily functions. 3 For example, te Equal Employment Opportunity Commission (EEOC) as ruled tat an employee may be asked ow many days were you absent from work?, but not ow many days were you sick?. 4 For example, an employer cannot require a driver s license for a clerking job because it would occasionally be useful to ave tat employee run errands. Also qualification cannot be suc tat a reasonable accommodation would allow te employee to perform te task. 5 Tese accommodations include: a) making existing facilities accessible and usable b) job restructuring c) part-time or modified work scedules d) reassigning a disabled employee to a vacant position e) acquiring or modifying equipment or devices f) providing qualified readers or interpreters. 6 See ttp:// 7 See Kass et al. (27). 8 See Sommers (26). 3

6 or from taking away coverage because of an application mistake Summary It is our interpretation of tese legislative canges tat, relative to 2 years ago, it is muc more difficult now for employers to condition wages on te ealt status of teir (potential) employees and preferentially ire workers wit better ealt. In addition, current and pending legislation will make it increasingly difficult to condition te acceptance into, and insurance premia of ealt insurance plans on prior ealt conditions. Te purpose of te remainder of tis paper is to analyze te aggregate and distributional consequences of tese two legislative innovations in te sort and in te long run, wit specific focus on teir interactions. 1.2 Related Literature Our paper incorporates ealt as a productive factor, and studies te effect of labor and ealt insurance market policies on its evolution. We allow for a two-way interaction between ealt socks and earnings troug worker productivity. We model medical expenditures wic mitigate te impact of tese ealt socks. Tere ave been a number of studies tat empirically estimate te effect of ealt on wages. Tese papers (see te summary in Currie and Madrian, 1999) generally find tat poor ealt decreases wages, bot directly and indirectly troug a decrease in ours worked. Te effect of a ealt sock on wages ranges from 1% to as ig as 15%. Many studies consistently find tat te effects on ours worked is greater tan tat on wages. Specifically relevant for us is Cawley (24). Similarly to wat we do for working age individuals, Pijoan-Mas and Rios-Rull (212), use HRS data on self-report ealt status to estimate a ealt transition function from age 5 onwards. Tey find tat tere is an important dependence in tis transition function on socioeconomic status (most importantly education), and tat tis dependence is quantitatively crucial for explaining longevity differentials by socioeconomic groups. As we do Hai (212) and Prados (212) model te interaction between ealt and earnings over te life cycle, but focus on te implications of teir models for wage-, earnings- and ealt insurance inequality. 1 A relatively small literature examines te incentive linkages between ealt insurance and ealt status. Battacarya et al. (29) use evidence from a Rand ealt insurance experiment, wic featured randomized assignment to ealt insurance contracts, to sow tat access to ealt insurance leads to increases in body mass and obesity. Tey argue tat tis comes from te fact tat insurance, especially troug its pooling effect, insulates people from te impact of teir excess weigt on teir medical expenditure costs. Consistent wit tis, tey find te impact of being ealt-insured is larger for public insurance programs tan in private ones in wic te ealt insurance premium is more likely to reflect te individuals body mass. Tis paper contributes to te broad literature tat examines te macroeconomic and distributional implications of ealt, ealt insurance and ealt care policy reform. Important related contributions include Grossman (1972), Erlic and Becker (1972), Erlic and Cuma (199), Frenc and Jones (24), Hall and Jones (27), Jeske, and Kitao (29), Jung and Tran (21), Attanasio, Kitao and Violante (211), Ales, Hosseini and Jones (212), Halliday, He and Zang (212), Hansen, Hsu and Lee (212), Kopecky and Koreskova (212), Laun (212) and Ozkan (212), Pascenko and Porapakkam (212). Brügemann and Manovskii (21), wile endogenizing ealt, study te macroeconomic effects of te employer-sponsored ealt insurance system tat is unique to te US labor market. Concretely, tey determine te effect of PPACA on ealt insurance coverage, but do not study te incentive effects of te regulation tat we formalize in our model. Several papers investigate te impact of regulation designed to limit te direct effect of ealt on bot ealt insurance costs and on wages. Sort and Lair (1994) examine ow ealt status interacts wit insurance coices. Madrian (1994) studies te lock-in effect of employer provided ealt care. Dey and Flinn (25) estimate a model of ealt insurance wit searc, matcing and bargaining and argue tat employer provided ealt care insurance leads to reasonably efficient outcomes. Related to our study of wage non-discrimination laws is te literature tat studies te effect of te ADA legislation of 199 on employment, wages and labor ours of te disabled (see DeLeire (21) and Acemoglu 9 See again ttp:// 1 Bot papers also study te impact of compulsary ealt insurance legislation. 4

7 and Angrist (21), for example). Most find tat it as decreased te employment of te disabled. DeLeire (21) quantifies te effect of ADA on wages of disabled workers and reports tat te negative effect of poor ealt on te earnings of te disabled fell by 11.3% due to ADA. Finally, a recent literature examines te impact of ealt on savings and portfolio coice in life cycle models tat sare elements wit our framework. Tese include Yogo (29), Edwards (28) and Hugonnier et al. (212). Te latter study jointly portfolio of ealt and oter asset coices. In teir model ealt increases productivity (labor income) and decreases occurrence of morbidity and mortality sock arrival rates (as tey do in our model). Te paper argues tat in order to explain te correlation between financial and ealt status, tese sould be modelled jointly. 2 Te Model Time t =, 1, 2,...T is discrete and finite and te economy is populated by a coort of a continuum of individuals of mass 1. Since we are modeling a given coort of individuals we will use time and te age of ouseolds intercangeably. We tink of T as te end of working life of te age coort under study. 2.1 Endowments and Preferences Houseolds are endowed wit one unit of time wic tey supply inelastically to te market. Tey are also endowedwitaninitiallevelofealt and we denote by H = { 1,..., N } te finite set of possible ealt levels. Houseolds value current consumption c and dislike te effort e tat elps maintain teir ealt. We will assume tat teir preferences are additively separable over time, and tat tey discount te future at time discount factor β. We will also assume tat preferences are separable between consumption and effort, and tat ouseolds value consumption according to te common period utility function u(c) andvalue effort according to te period disutility function q(e). We will denote te probability distribution over te ealt status at te beginning of period t by Φ t (), and denote by Φ () te initial distribution over tis caracteristic. Assumption 1 Te utility function u is twice differentiable, strictly increasing and strictly concave. q is twice differentiable, strictly increasing, strictly convex, wit q() = q () = and lim e q (e) =. 2.2 Tecnology Healt Tecnology Let ε denote te current ealt sock. 11 In every period ouseolds wit current ealt remain ealty (tat is, ε = ) wit probability g(). Wit probability 1 g() te ouseold draws a ealt sock ε (, ε] wic is distributed according to te probability density function f(ε). Assumption 2 f is continuous and g is twice differentiable wit g() [, 1], and g () >,g () < for all H. An individual s ealt status evolves stocastically over time, according to te Markov transition function Q(,; e), were e is te level of exercise by te individual. We impose te following assumption on te Markov transition function Q Assumption 3 If e >eten Q(,; e) first order stocastically dominates Q(,; e ). 11 In te quantitative analysis we will introduce a second, fully insured (by assumption) ealt sock to provide a more accurate map between our model and te ealt expenditure data. 5

8 2.2.2 Production Tecnology A individual wit ealt status and current ealt sock ε tat consumes ealt expenditures x produces F (, ε x) units of output. Assumption 4 F is continuously differentiable in bot arguments, increasing in, and satisfies F (, y) = F (, ) for all y, and F 2 (, y) < as well as F 2 (, ε) < 1. Finally F 22 (, y) < for all y> and F 12 (, y). Te left panel of figure 1 displays te production function F (,.), for two different levels of te current ealt sock. Holding ealt status constant, output is decreasing in te uncured portion of te ealt sock ε x, and te decline is more rapid for lower levels of ealt ( <). Te rigt panel of figure 1 displays te production function as function of ealt expenditures x, for a fixed level of te sock ε, and sows tat expenditures x exceeding te ealt sock ε leave output F (, ε x) unaffected (and tus are suboptimal). Furtermore, a reduction of te sock ε to a lower level, ε, sifts te point at wic ealt expenditures x become ineffective to te left. Figure 1: Production Function F (, ε x) Figure 2: Production Function F (, ε x) forfixedε Te assumptions on te production function F imply tat ealt expenditures can offset te impact of a ealt sock on productivity, but not raise an individual s productivity above wat it would be if tere ad been no sock. In addition, te last assumption on F tat F 12 implies tat te negative impact of a given net ealt sock y is lower te ealtier a person is. 12 Te assumption F 2 (, ε) < 1 insures tat, if it by te worst ealt sock te cost of treating tis ealt sock, at te margin, is smaller tan te positive impact on productivity (output) tis treatment as. 2.3 Time Line of Events In te current period te timing of events is as follows 1. Houseolds enter te period wit current ealt status. 2. Houseolds coose e. 3. Firms offer wage w() and ealt insurance contracts {x(ε, ),P()} 13 to ouseolds wit ealt status wic tese ouseolds accept. 12 Tis is also te approac taken by Hugonnier et al. (212) and Erlic and Cuma (199). 13 Since we restrict attention to static contracts, weter firm offers contracts before or after te effort is undertaken does not matter. 6

9 ε t drawn according to g( t )andf(ε t ) t x t (ε t, t )spent t +1 t firms offer wage w( t ) and HI contract {x(ε t, t ),P( t )} t+1 determined by Q( t+1, t ; e t ) ouseolds coose e t ouseolds produce F ( t,ε t x t (ε t )) and consume c t ( t ) Figure 3: Timing of te Model 4. Te ealt sock ε is drawn according to te distributions g, f. 5. Resources on ealt x = x(ε, ) arespent. 6. Production and consumption takes place. 7. Te new ealt status of a ouseold is drawn according to te ealt transition function Q. 2.4 Market Structure witout Government Tere are a large number of production firms tat in eac period compete for workers. Firms observe te ealt status of a worker and ten, prior to te realization of te ealt socks, compete for workers of type by offering a wage w() tat pools te risk of te ealt socks and bundle te wage wit an associated ealt insurance contract (specifying ealt expenditures x(ε, ) and an insurance premium P ()) tat breaks even. Perfect competition for workers of type requires tat te combined wage and ealt insurance contract maximize period utility of te ouseold, subject to te firm breaking even. 14 In te absence of government intervention a firm specializing on workers of ealt type terefore offers a wage w CE ()(werece stands for competitive equilibrium) and ealt insurance contract (x CE (ε, ),P CE ()) tat solves U CE () = max u (w() P ()) (1) w(),x(ε,),p () s.t. P () = g()x(,)+(1 g()) f(ε)x(ε, )dε (2) w() = g()f (, x(,)) + (1 g()) f(ε)f (, ε x(ε, ))dε (3) Note tat by bundling wages and ealt insurance te firm provides efficient insurance against ealt socks ε, and te only source of risk remaining in te competitive equilibrium is ealt status risk associated wit 14 Note tat instead of assuming tat firms completely specialize by iring only a specific ealt type of workers we could alternatively consider a market structure in wic all firms are representative in terms of iring workers of ealt types according to te population distribution and pay workers of different ealt differential wages according to te scedule w CE (). In oter words ealt variation in wages and variation in ired ealt types are perfect substitutes at te level of te individual firm in terms of supporting te competitive equilibrium allocation. 7

10 . Tis risk stems bot from te dependence of wages w() as well as ealt insurance premia P () on in te competitive equilibrium, and tese are exactly te sources of consumption risk tat government policies preventing wage discrimination and proibiting prior ealt conditions to affect insurance premia are designed to tackle. 2.5 Government Policies We now describe in turn ow we operationalize, witin te context of our model, a policy tat outlaws ealt insurance premia to be conditioned on prior ealt conditions, and a policy tat limits te extent to wic firms can pay workers of varying ealt differential wages No Prior Conditions Law Under tis law ealt insurance companies are assumed to be constrained in terms of teir pricing, teir insurance scedule offers and teir applicant acceptance criteria. Te purpose of tese constraints is to prevent te companies from differentially pricing insurance based upon ealt status. 15 To be completely successful, tese constraints must lead to a pooling equilibrium in wic all individuals are insured at te same price. Te best suc regulation in addition assures tat te equilibrium ealt insurance scedule x(ε, ), given te constraints, is efficient. We now describe te regulations sufficient to acieve tis goal. Te first constraint on ealt insurers is tat a company must specify te total number of contracts tat it wises to issue, it must carge a fixed price independent of ealt status, and accept applications in teir order of application up to te sales limit of te company. In tis way, te insurance company cannot examine applications first and ten decide weter or not to offer te applicant a ealt insurance contract. Te second constraint regulates te ealt expenditure scedule. If te no-prior conditions law is to ave any bite te government needs to prevent te emergence of a separating equilibrium in wic te ealt insurance companies (or te production firms in case tey offer ealt insurance contracts) use te ealt expenditure scedule x(ε, ) to effectively select te desired ealt types, given tat tey are barred from conditioning te ealt insurance premium P on directly. Terefore, to acieve any sort of pooling in te ealt insurance market requires te government to regulate te ealt expenditure scedule x(ε, ). To give te legislation te best cance of being successful we will assume tat te government regulates te ealt expenditure scedule x(ε, ) efficiently. For te same reason, since risk pooling is limited if some ouseold types coose not to buy insurance, we assume tat all individuals are forced to buy insurance. Given tis structure of regulation and a cross-sectional distribution of workers by ealt type, Φ, te ealt insurance premium P carged by competitive firms (or competitive insurance companies, wo offer ealt insurance in te model), given te set of regulations spelled out above, is determined by P = [ ] g()x(,)+(1 g()) f(ε)x(ε, )dε Φ() (4) were x(ε, ) is te expenditure scedule regulated by te government. Tis scedule is cosen to maximize u(w() P )Φ() wit wages w() determined by (3) No Wage Discrimination Law Te objective of te government is to prevent workers wit a lower ealt status, and ence lower productivity, being paid less. As wit te no prior conditions law, te purpose of tis legislation is to elp insure workers against teir ealt status risk. However, if a production firm is penalized for paying workers wit low ealt status low wages, but not for preferentially iring workers wit a favorable ealt status (ig 15 Consistent wit tis restricted purpose, we will assume tat te government cannot use ealt insurance to offset underlying differences in productivity coming from, say, education. Tis will prove important in te quantitative section. 8

11 ), ten a firm can effectively circumvent te wage nondiscrimination law. Terefore, to be effective suc a law must penalize bot wage discrimination and iring discrimination by ealt status. Limiting wage dispersion wit respect to gross wages w() via legislation necessitates regulation of te ealt insurance market as well, in order to prevent te insurance gains from decreasing wage dispersion being undone troug te adjustment of employer-provided ealt insurance. For example, te firm could also offer ealt insurance and overcarge low productivity workers and undercarge ig productivity workers for tis insurance, effectively undermining te illegal wage discrimination. Tis suggests tat te government will need to limit te extent to wic te cost of a worker s ealt insurance contracts deviates from its actuarially fair value. However, tis will not be sufficient to make tis policy effective. Since te productivity of a worker depends upon te extent of is ealt insurance, workers wose expected productivity is below teir wage will face pressure to increase teir productivity troug increased spending on ealt (and ence better ealt insurance coverage) wile tose wose productivity is above teir wage will ave an incentive to lower teir ealt insurance purcases. To prevent tese distortions in te ealt insurance market and tereby acieve better consumption insurance across types, policy makers will need to regulate te ealt insurance directly as well. Te moderate version of ealt insurance regulation would be to ensure tat eac policy was individually optimal and actuarially fair. Te most extreme version of regulation would be to combine no-wage discrimination legislation wit no-prior conditions legislation and tereby acieve te static first-best, full insurance outcome. In tis case ealt insurance would be socially efficient and actuarially fair on average (tat is, across te insured population). We will analyze bot cases. It will turn out tat limiting wage dispersion wit respect to net wages, w() P (), avoids te negative incentive effects on te ealt insurance market. Te policy of combining bot no-wage discrimination and no-prior conditions can terefore be implemented troug a policy of limiting net wage dispersion. Te impact of te nondiscrimination law will, unfortunately, be sensitive to te way in wic te law is implemented, and in particular, to te form of punisment used. If te limitation in wage variation is acieved troug a policy tat penalizes te firms for discriminating, ten tese costs are realized in equilibrium, reducing overall efficiency in te economy. If, owever, te limitation on wage variation is acieved eiter troug te treat of punisment (e.g. troug grim trigger strategies in repeated interactions between firms and te government) or troug te delegation of iring in a union iring all type arrangement, ten costs from te wage nondiscrimination law will not be realized in equilibrium. 16 Since we wis to give te no wage discrimination law te best sot of being successful, in te main text we focus on te version of te policy in wic no costs from te policy are realized in equilibrium, leaving te analysis of te alternative case to appendix B.2 and B.3. In eiter case we only tackle te extreme versions of tese policies in wic tere is no wage discrimination (rater tan limited wage discrimination) in equilibrium for reasons of analytic tractability. Under te policy, te firm takes as given tresolds on te size of te gap in wages or employment sares tat will trigger te punisment. Assume tat te wage penalty will be imposed if te maximum wage gap witin te firm exceeds te tresold ε w. Since type = will receive te lowest wage in equilibrium, to avoid te penalty a firm as to offer a wage scedule tat satisfies: max w() w() ε w. Letting n() denote te number of workers of type ired by te firm, assume tat te iring penalty will be imposed if te employment sare of type deviates from te population average by more tan δ, and ence n() n() Φ() Φ() δ. We will assume tat te punisment is sufficiently dire tat te firm will never coose to violate tese tresolds. We analyze te more general case in appendix B.1, but ere focus on te limiting case in wic te tresolds ε w and δ converge to zero. In tis case, te firm will simply take as given te economy-wide wage w at wic it can ire a representative worker. We assume tat te government regulates te insurance 16 Te delegation metod is similar to te structure we assumed in te insurance market since insurance companies were restricted to serving teir customers on a first-come-first-serve basis. Tis assumption to us seems more problematic in te labor market because of te idiosyncratic nature of te benefits to te worker-firm matc. 9

12 market determining te extent of coverage by ealt type, x(e, ), subject to te requirement tat te offered ealt insurance contracts exactly break even, eiter ealt type by ealt type (in te absence of a no prior conditions law) or in expectation across ealt types (in te presence of te no prior conditions law). Perfect competition drives down equilibrium profits of firms to zero wic determines te equilibrium wage rate as w = { g()f (, x(,)) + (1 g()) } f(ε)[f (, ε x(ε, ))] dε Φ() (5) Te insurance premium carged to te ouseold is P () =g()x(,)+(1 g()) f(ε)x(ε, )dε (6) in te absence of a no-prior conditions law and P = [ g()x(,)+(1 g()) ] f(ε)x(ε, )dε Φ() (7) in its presence. Houseold consumption is given by c() = w P () or c = w P depending on weter a no prior conditions law is in place or not. Given a cross-sectional ealt distribution Φ te efficiently regulated ealt insurance contract x(ε, ) is te solution to max u(w P ())Φ() x subject to (5) and (6) if te no-prior conditions restriction is not imposed on ealt insurance, and subject to (7) instead of (6) if te no-prior conditions restriction is present. We now turn to te analysis of te model, starting wit a static version in wic by construction te coice of effort is not distorted in equilibrium. We will sow tat in tis case te competitive equilibrium implements an efficient allocation of ealt expenditures, but fails to provide efficient consumption insurance against prior ealt conditions, tat is against cross-sectional variation in. We ten argue tat a combination of a strict wage non-discrimination law and a no prior conditions law in addition results in efficient consumption insurance in te competitive equilibrium, restoring full efficiency of allocations in te regulated market economy. 3 Analysis of te Static Model We now turn to te analysis of te static version of our model, and we will caracterize bot efficient and equilibrium allocations (in te absence and presence of te nondiscrimination policies). Te purpose of tis analysis is two-fold. First, it will result in te caracterization of te optimal and equilibrium ealt insurance contract, a key ingredient for our dynamic model. Second, te analysis will demonstrate tat in te sort run (tat is statically) te combination of bot policies is ideally suited to provide full consumption insurance in te regulated market equilibrium, and tus restores full efficiency of te market outcome. Te static benefits of tese policies are ten traded off against te adverse dynamic consequences on te ealt distribution, as our analysis of te dynamic model will uncover in te next section. 3.1 Social Planner Problem Given an initial cross-sectional distribution over ealt status in te population Φ() te social planner maximizes utilitarian social welfare. Te social planner problem is terefore given by 1

13 U SP (Φ) = max e(),x(ε,),c(ε,) subject to { g()c(,)+(1 g()) { g()f (, x(,)) + (1 g()) { q(e()) + g()u(c(,)) + (1 g()) f(ε)c(ε, )dε + g()x(,)+(1 g()) } f(ε)f (, ε x(ε, ))dε Φ() } f(ε)u(c(ε, ))dε Φ() } f(ε)x(ε, )dε Φ() We summarize te optimal solution to te static social planner problem in te following proposition, wose proof follows directly from te first order conditions and assumption 4 (see Appendix A). Proposition 5 Te solution to te social planner problem {c SP (ε, ),x SP (ε, ),e SP ()} H is given by e SP () = c SP (ε, ) = c SP x SP (ε, ) = max [,ε ε SP () ] were te cutoffs satisfy F 2 (, ε SP ()) = 1, (8) and te first best consumption level is given by c SP = [ g()f (, ) + (1 g()) f(ε) [ F (, ε x SP (ε, )) x SP (ε, ) ] ] dε Φ() (9) Te optimal cutoff { ε SP ()} is increasing in, strictly so if F 12 (, y) >. Te social planner finds it optimal to not ave te ouseold exercise (given tat tere are no dynamic benefits from doing so in te static model) and to provide full consumption insurance against adverse ealt socks ε, but also against bad prior ealt conditions as consumption is constant in. Te optimal level of ealt expenditure and its implications on production is grapically presented in Figure 4. As sown in te previous proposition, optimal medical expenditures take a simple cutoff rule: small ealt socks ε< ε SP () are not treated at all, but all larger socks are fully treated up to te tresold ε SP (). Tese optimal medical expenditures are displayed in Figure 4(b) for two different initial levels of ealt 1 < 2 : below te -specific tresold ε SP () ealt expenditures are zero, and ten rise one for one wit te ealt sock ε. Te determination of te tresold itself is displayed in Figure 4(a). It sows tat under te assumption tat te impact of ealt socks on productivity is less severe for ealty ouseolds (F 12 (, y) >, reflected as a more concave curve for 1 tan for 2 in Figure 4(a)), ten te social planner finds it optimal to insure ealtier ouseolds less, in te sense of undoing less of te negative ealt socks ε troug medical treatment x(ε, ). Tis is reflected in a lower tresold (more insurance) for 1 tan for 2, tat is ε SP ( 2 ) < ε SP ( 1 ). Te optimal ealt expenditure policy function leads to a net of-ealt-treatment production function F (, ε x SP (ε, )) as sown in Figure 4(c). 3.2 Competitive Equilibrium As in te social planner problem tere is no incentive for ouseolds to exercise in te static model, and tus e() =. As described in section 2.4 te equilibrium wage and ealt insurance contract solves U CE () = max u (w() P ()) (1) w(),x(ε,),p () s.t. P () = g()x(,)+(1 g()) f(ε)x(ε, )dε (11) w() = g()f (, x(,)) + (1 g()) 11 f(ε)f (, ε x(ε, ))dε (12)

14 1 < 2 F(, -x) x(,) x*(,)=max{, - ()} F(, -x*(,)) -F 2 ( 2, )=1 -F 2 ( 1, )=1 -x -x ( 1 ) ( 2 ) max ( 1 ) ( 2 ) max ( 1 ) ( 2 ) max (a) Production Function (b) Healt Expenditure (c) Production Figure 4: Optimal Healt Expenditure and Production Te following proposition caracterizes te solution to tis problem: Proposition 6 Te unique equilibrium ealt insurance contract and associated consumption are given by and te cutoff satisfies x CE (ε, ) = max [,ε ε CE () ] (13) c CE (ε, ) = c CE () =w CE () P CE () (14) P CE () = (1 g()) Proof. See Appendix We immediately obtain te following ε CE () w CE () = g()f (, ) + (1 g()) f(ε) [ ε ε CE () ] dε (15) f(ε)f (, ε x(ε, ))dε (16) F 2 (, ε CE ()) = 1 (17) Corollary 7 Te competitive equilibrium implements te socially efficient ealt expenditure allocation since ε CE () = ε SP () for all H. Corollary 8 Te cutoff ε CE () is increasing in, strictly so if F 12 (, y) >. Wile it follows trivially from our assumptions tat te worker s net pay, w() P (), is increasing in, it is not necessarily true tat is gross wage, w(), is increasing in as well since optimal ealt expenditures are decreasing in ealt status. We analyze te beavior of gross wages w() wit respect to ealt status furter in Appendix C, were we provide a sufficient condition for te gross wage scedule to be monotonically increasing in. In any case, te previous results sow tat in te static case te only source of inefficiency of te competitive equilibrium comes from te inefficient lack of consumption insurance against adverse prior ealt 12

15 conditions. Tis can be seen by noting tat c SP = { g()f (, ) + (1 g()) f(ε) [ F (, ε x SP (ε, )) x SP (ε, ) ] } dε Φ() = [ w CE () P CE () ] Φ() = c CE ()Φ() In contrast to wat will be te case in te dynamic model, effort trivially is not distorted in te equilibrium, relative to te allocation te social planner implements (since in bot cases e SP = e CE =). Furtermore te equilibrium allocation of ealt expenditures is efficient, due to te fact tat te firm bundles te determination of wages and te provision of ealt insurance, and tus internalizes te positive effects of ealt spending x(ε, ) on worker productivity. Given tese results it is plausible to expect, witin te context of te static model, tat policies preventing competitive equilibrium wages w CE () to depend on ealt status (a wage non-discrimination law) and insurance premia P CE () to depend on ealt status (a no prior conditions law) will restore full efficiency of te policy-regulated competitive equilibrium by providing full consumption insurance. We will sow next tat tis is indeed te case, providing a normative justification for te two policy interventions witin te static version of our model. 3.3 Competitive Equilibrium wit a No Prior Condition Law As discussed above, in order to effectively implement a no prior conditions law te government as to regulate te ealt insurance provision done by firms or insurance companies. Given a population ealt distribution Φ te regulatory autority solves te problem: U NP (Φ) = max x(ε,) u(w() P )Φ() (18) s.t. P = [ g()x(,)+(1 g()) w() = g()f (, x(,)) + (1 g()) Te next proposition caracterizes te resulting regulated equilibrium allocation ] f(ε)x(ε, )dε Φ() (19) f(ε)f (, ε x(ε, ))dε (2) Proposition 9 Te equilibrium ealt expenditures under a no-prior condition law satisfies, for eac H x NP (ε, ) = max[,ε ε NP ( )] wit cutoffs uniquely determined by F 2 (, ε NP ( )) = u (w NP () P NP )Φ(). u (w( ) P NP ) Te equilibrium wage, for eac, is given by w NP ( ) =g( )F (, ) + (1 g( )) and te ealt insurance premium is determined as P NP = [ g()x NP (,)+(1 g()) f(ε)[f (, ε x NP (ε, ))]dε ] f(ε)x NP (ε, )dε Φ(). Moreover, te optimal cutoffs are increasing in ealt status. 13

16 Proof. See Appendix. Note tat te ealt expenditure levels are no longer efficient as te government provides partial consumption insurance against initial ealt status wen coosing te cutoff levels ε NP (), in te absence of direct insurance against low wages induced by bad ealt. In fact, as sown in te next proposition, it is efficient to over-insure ouseolds wit bad ealt status and under-insure tose wit good ealt status, relative to te first-best. Proposition 1 Let be te ealt status wose marginal utility of consumption is equal to te population average, i.e. for, F 2 (, ε( )) = u (w() P )Φ() = 1 (21) u (w( ) P ) olds. 17 Ten, ε NP () < ε SP (), ε NP () = ε SP (), ε NP () > ε SP (), for < for = for >, Te cutoffs ε() are strictly monotonically increasing in ealt status. Proof. See Appendix. Tis feature of te optimal ealt expenditure wit a no prior conditions law also indicates tat mandatory participation in te ealt insurance contract is an important part of government regulation, since in te allocation described above ealty ouseolds cross-subsidize te unealty in terms of insurance premia and tey are given a less generous ealt expenditure plan (iger tresolds) tan te unealty. 3.4 Competitive Equilibrium wit a No Wage Discrimination Law Te equilibrium wit a no wage discrimination law is determined by te solution to te program: U ND (Φ) = max u(w P ())Φ() (22) x(ε,) [ s.t. ] P () = g()x(,)+(1 g()) f(ε)x(ε, )dε w = { } g()f (, x(,)) + (1 g()) f(ε)f (, ε x(ε, ))dε Φ() Proposition 11 Te equilibrium ealt expenditures under a no-wage discrimination law alone satisfies, for eac H x ND (ε, ) [ ] =max,ε ε ND ( ) wit cutoffs determined by F 2 (, ε ND ( )) = Te equilibrium wage is given by w ND = [ g()[f (, )]+(1 g()) u (w ND P ( )) u (w ND P ())Φ(). f(ε) [ F (, ε x ND (ε, )) ] ] dε Φ() and te ealt insurance premium is given by, for eac, [ P ND ( ) = g( )x ND (, )+(1 g( )) ] f(ε)x ND (ε, )dε. 17 For te purpose of te proposition it does not matter weter H or not. 14

17 Proof. Follows directly from te first order conditions of te program (22). Unlike in te no prior conditions case, we cannot establis monotonicity in te cutoffs ε ND ( ). Note tat under a no prior conditions law te regulatory autority partially insures consumption of te unealty by allocating iger medical expenditure to tem. Under a no wage discrimination law instead, tere are two opposing forces, preventing us from establising monotonicity in cutoffs ε ND () across ealt groups. On one and, a one unit increase in medical expenditure P () is more costly to te unealty since marginal utility of consumption is iger for tis group. On te oter and, production efficiency calls for iger medical expenditure for te unealty, given our assumption of F 12 (as was te case for te no prior conditions law). Tus te cutoffs ε ND () need not be monotone in. 3.5 Competitive Equilibrium wit Bot Policies Finally, combining bot a no-wage discrimination law and a no-prior conditions legislation restores efficiency of te regulated equilibrium since bot policies in conjunction provide full consumption insurance against bad ealt realizations. Tis is te content of te next. Corollary 12 Te unique competitive equilibrium allocation in te presence of bot a no wage discrimination and a no prior conditions law implements te socially efficient allocation in te static model. Proof. Te equilibrium is te solution to u(w P )Φ() max x(ε,) s.t. P = w = [ g()x(,)+(1 g()) { g()f (, x(,)) + (1 g()) ] f(ε)x(ε, )dε Φ() } f(ε)f (, ε x(ε, ))dε Φ(). Te result ten follows trivially from te fact tat tis maximization problem is equivalent to te social planner problem analyzed above. Te no prior conditions law equalizes ealt insurance premia P across ealt types, te no wage discrimination law implements a common wage w across ealt types, and te (assumed) efficient regulation of te ealt insurance market assures tat te ealt expenditure scedule is efficient as well. 3.6 Summary of te Analysis of te Static Model Te competitive equilibrium implements te efficient ealt expenditure allocation but does not insure ouseolds against initial ealt conditions. Bot a no-prior conditions law and a no-wage discrimination law provide partial, but not complete, consumption insurance against tis risk, witout distorting te effort level. Te ealt expenditure scedule is distorted wen eac policy is implemented in isolation, relative to te social optimum, as te government provides additional partial consumption insurance troug ealt expenditures. Only bot laws in conjunction implement a fully efficient ealt expenditure scedule and full consumption insurance against initial ealt conditions, and terefore restore te first best allocation in te static model. Enacting bot policies jointly is tus fully successful in wat tey are designed to acieve in a static world (partially due to te fact tat additional government regulation severely restricted te options of firms to circumvent te government policies). 4 Analysis of te Dynamic Model We now study a dynamic version of our economy. Bot in terms of casting te problem, as well as in terms of its computation we make use of te fact tat tere is no aggregate risk (due to te continuum of agents cum law of large numbers assumption). Terefore te sequence of cross-sectional ealt distributions {Φ t } T t= 15

18 is a deterministic sequence. Furtermore, conditional on a distribution Φ t today te ealt distribution tomorrow is completely determined by te effort coice e t () of ouseolds 18 (or te social planner), so tat we can write Φ t+1 = H(Φ t ; e t (.)) (23) were te time-invariant function H is in turn completely determined by te Markov transition function Q( ;, e). Te initial distribution Φ is an initial condition and exogenously given. Under eac policy, given a sequence of aggregate distributions {Φ t } T t= we can solve an appropriate dynamic maximization problem of an individual ouseold for te sequence of optimal effort decisions {e t () H } T t= wic in turn imply a new sequence of aggregate distributions via (23). Our computational algoritm for solving competitive equilibria ten amounts to iterating on te sequences {Φ t,e t }. Witin eac period te timing of events follows exactly tat of te static problem in te previous section. 4.1 Social Planner Problem Te dynamic problem of te social planner is to solve { T V (Φ )= max β t U SP (Φ t ) {e t()} t= q(e t ())Φ t () } were {Φ t+1 } is determined by equation (23) and U SP (Φ) = max x(ε,),c(ε,) = u(c SP (Φ)) { g()u(c(,)) + (1 g()) is te solution to te static social planner problem caracterized in section 3.1: x SP (ε, ) =max [,ε ε SP () ] } f(ε)u(c(ε, ))dε Φ() wit cutoffs defined by F 2 (, ε SP ()) = 1 (24) and consumption of eac ouseold given by c SP (Φ) = [ g()f (, ) + (1 g()) f(ε) [ F (, ε x SP (ε, )) x SP (ε, ) ] ] dε Φ(). ε We now want to caracterize te optimal effort coice by te social planner, te key dynamic decision in our model bot in te planner problem and te competitive equilibrium. In contrast to ouseolds in te competitive equilibrium, te social planner fully takes into account te effect of effort coices today on te aggregate ealt distribution and tus aggregate consumption tomorrow. A semi-recursive formulation of te problem is useful to caracterize te optimal effort coice, but also to explain te computational algoritm for te social planner problem. For a given cross-sectional distribution Φ t at te beginning of period t te social planner solves: { } V t (Φ t ) = u(c t )+ max e t() H, s.t. c t = c SP (Φ t ) Φ t+1 ( ) = q(e t ())Φ t ()+βv t+1 (Φ t+1 ) Q( ;, e t ())Φ() (25) 18 We assert ere tat te optimal effort in period t is only a function of te current individual ealt status. We will discuss below te assumptions required to make tis assertion correct. 16

19 In appendix D we discuss ow we solve tis problem numerically, iterating on sequences {c t,e t (), Φ t ()} T t= from te terminal condition V T (Φ T )=u(c T ). To caracterize te optimal effort coice, for an arbitrary time period t we obtain te first order condition: q (e t ())Φ t () = β V t+1 (Φ t+1 ) Φ t+1 ( ) = β V t+1 (Φ t+1 ) Φ t+1 ( ) Φ t+1( ) e t () Q( ;, e t ()) Φ t (), e t () Tis simplifies to q (e t ()) = β V t+1 (Φ t+1 ) Φ t+1 ( ) Q( ;, e t ()). (26) e t () Tus te marginal cost of extra effort q (e t ()) is equated to te marginal benefit, te latter being given by te te benefit tat effort as on te ealt distribution tomorrow, Q( ;,e t()) e t(), times te benefit of a better ealt distribution Vt+1(Φt+1) Φ t+1( ) from tomorrow on. By assumption 1, q () =, and assumption 3 guarantees tat te rigt and side of equation (26) is strictly positive. Terefore te social planner finds it optimal to make every ouseold exert positive effort to lead a ealty life: e t () > for all t and all H. From te envelope teorem te benefit of a better ealt distribution is given by: V t (Φ t ) Φ t () = u (c t ) Ψ() q(e t ()) + β V t+1 (Φ t+1 ) Φ t+1 ( Q( ;, e t ()). (27) ) Here Ψ() denotes te expected output, net of ealt expenditures, tat an individual of ealt status delivers to te social planner Competitive Equilibrium witout Policy In our model, since absent wage and ealt insurance policies ouseolds do not interact in any way, we can solve te dynamic programming problem of eac ouseold independently of te rest of society. Te only state variables of te ouseold are er current ealt and age t, and te dynamic program reads as: { } were v t () =U CE ()+max e t() U CE () = max u(w() P ()) x(ε,),w(),p () s.t. w() = g()f (, x(,)) + (1 g()) P () = g()x(,)+(1 g()) q(e t ()) + β Q( ;, e t ())v t+1 ( ) f(ε)x(ε, )dε f(ε)f (, ε x(ε, ))dε (28) 19 Note tat [ [ ] ] Ψ() = g()f (, ) + (1 g()) f(ε) F (, ε x SP (ε, )) x SP (ε, ) dε ε is exclusively determined by te optimal cut-off rule ε SP () for ealt expenditures, wic is independent of c t or Φ t. 17

20 is te solution to te static equilibrium problem in section 3.2, wic was given by: x CE (ε, ) = max [,ε ε CE () ] c CE () = w CE () P CE () P CE () = (1 g()) ε CE () w CE () = g()f (, ) + (1 g()) f(ε) [ ε ε CE () ] dε f(ε)f (, ε x(ε, ))dε wit cutoff: F 2 (, ε CE ()) = 1 Note again tat te provision of ealt insurance is socially efficient in te competitive equilibrium. In contrast to te social planner problem, and in contrast to wat will be te case in a competitive equilibrium wit a no-wage discrimination law or a no-prior conditions law, in te unregulated competitive equilibrium tere is no interaction between te maximization problems of individual ouseolds. Tus te dynamic ouseold maximization problem can be solved independent of te evolution of te cross-sectional ealt distribution. It is a simple dynamic programming problem wit terminal value function v T () =U CE () and can be solved by straigtforward backward iteration. Given te solution {e t ()} of te ouseold dynamic programming problem and given an initial distribution Φ te dynamics of te ealt distribution is ten determined by te aggregate law of motion (23). Te optimal coice e t () solves te first order condition q (e t ()) = β Q( ;, e t ()) v t+1 ( ) (29) e t () Notetatattimetwen te decision e t () is taken te function v t+1 (.) is known. Furtermore, given knowledge of v t+1 and te optimal e t te period t value function v t is determined by (28). As in te social planner problem, by assumptions 1 and 3 effort e t () is positive for all t and. 4.3 Competitive Equilibrium wit a No Prior Condition Law As discussed above, we assume tat te government in every period t takes as given te ealt distribution Φ t and enforces te no prior condition law and regulates ealt insurance contracts efficiently, as in te static analysis of section 3.3. We now make explicit tat te solution of te static government regulation problem (18)-(2) is a function of te cross-sectional ealt distribution, wit cutoffs for eac H determined by x NP (ε, ;Φ t ) = max[,ε ε NP ( ;Φ t )] (3) F 2 (, ε NP ( ;Φ t ))u (w NP ( ;Φ t ) P NP (Φ t )) = u (w NP (;Φ t ) P NP (Φ t ))Φ t () :=Eu (Φ t ) (31) and w NP (;Φ t ) = g()f (, ) + (1 g()) f(ε)[f (, ε x NP (ε, ;Φ t ))]dε (32) P NP (Φ t ) = [ ] g()x NP (,;Φ t )+(1 g()) f(ε)x NP (ε, ;Φ)dε Φ t () (33) 18

21 In order for te ouseold to solve er dynamic programming problem se only needs to know te sequence of wages and ealt insurance premia {w t (),P t }, but not necessarily te sequence of distributions tat led to it. Given suc a sequence te dynamic programming problem of te ouseold ten reads as { v t () =u(w t () P t )+max q(e t ()) + β } Q( ;, e t ())v t+1 ( ) (34) e t() wit terminal condition v T () =u(w T () P T ). As before te optimality condition reads as q (e t ()) = β Q( ;, e t ()) v t+1 ( ). (35) e t () and tus equates te marginal cost of providing effort, q (e) wit te marginal benefit of an improved ealt distribution tomorrow. Altoug equation (35) looks identical to equation (29) from te unregulated equilibrium, te determination of te value functions tat appear on te rigt and side of bot equations is not (compare te first terms on te rigt and sides of equations (28) and (34)). Te difference in tese equations igligts te extra consumption insurance induced by te no-prior conditions law, in tat wit tis policy te ealt insurance premium does not vary wit. Tis extra consumption insurance, ceteris paribus, reduces te variation of v t+1 in and tus limits te incentives to exert effort in order to acieve a (stocastically) iger ealt level tomorrow. In appendix E we describe a computational algoritm to solve te dynamic model wit a no-prior conditions law. 4.4 Competitive Equilibrium wit a No Wage Discrimination Law Te main difference to te previous section is tat now te static ealt insurance contract and premium are given by ealt spending x ND (ε, ;Φ t ) = max[,ε ε ND ( ;Φ t )] (36) wit cutoffs for eac H determined by F 2 (, ε ND ())Eu t = u (w ND (Φ t ) P ND (, Φ t )) (37) were Eu t := u (w ND (Φ t ) P ND (, Φ t ))Φ t (). (38) Te equilibrium wage is given by w ND (Φ t )= { g()f (, ) + (1 g()) } f(ε)[f (, ε x ND (ε, ;Φ t ))]dε Φ t (). (39) Te equilibrium ealt insurance premium depends on weter a no prior conditions law is in place or not: Witout suc policy te premia are given as P ND (;Φ t )=P ND () =(1 g()) f(ε)x ND (ε, )dε (4) wereas wit bot policies in place te premium is determined by 2 P Bot (Φ t )= [ ] (1 g()) f(ε)x Bot (ε, )dε Φ t () (41) For a given sequence of wages {w t,p t ()} te dynamic problem of te ouseold reads as before: { v t () =u(w t P t ()) + max q(e t ()) + β } Q( ;, e t ())v t+1 ( ) e t() (42) and te terminal condition v T () =u(w T P T ()), first order conditions and updating of te value function for tis version of te model are exactly te same, mutatis mutandis, as under te previous policy. In appendix E we discuss te algoritm to solve tis version of te model. 2 Wages still take te form as in (39), but wit x Bot (ε, ) replacing x ND (ε, ). Recall from te static analysis tat x Bot (ε, ) =x SP (ε, ), tat is, te medical expenditure scedule is socially efficient. 19

22 4.5 Competitive Equilibrium wit Bot Laws If bot policies are in place simultaneously, we can give a full analytical caracterization of te equilibrium witout resorting to any numerical solution procedure. We do so in te next Proposition 13 Suppose tere is a no wage discrimination and a no prior condition law in place simultaneously. Ten e t () =for all, and all t. Te provision of ealt insurance is socially efficient. From te initial distribution Φ te ealt distribution in society evolves according to (23) wit e t (). Te proof is by straigtforward backward induction and is given in Appendix A. In te presence of bot policies tere are no incentives, eiter troug wages or ealt insurance premia, to exert effort to lead a ealty life. Since effort is costly, ouseolds won t provide any suc effort in te regulated dynamic competitive equilibrium. Tus in te absence of any direct utility benefits of better ealt te combination of bot policies leads to a complete collapse in incentives, wit te associated adverse long run consequences for te distribution of ealt in society. Equipped wit tese teoretical results and te numerical algoritms to solve te various versions of our model we now map our model to cross-sectional ealt and exercise data from te PSID to quantify te effects of government regulations on te evolution of te cross-sectional ealt distribution, as well as aggregate production, consumption and ealt expenditures. 5 Bringing te Model to te Data 5.1 Augmenting te Model Te model described so far only included te necessary elements to igligt te key static insurancedynamic incentive trade-off we want to empasize. However, to insure tat te model can capture te significant eterogeneity in ealt, exercise and ealt expenditure data observed in micro data we now augment it in four aspects. We want to stress, owever, tat none of te qualitative results derived so far rely on te absence of tese elements, wic is wy we abstracted from tem in our teoretical analysis. First, in te data some ouseolds ave ealt expenditures in a given year from catastropic illnesses tat exceed teir labor earnings. In te model, te only benefit of spending resources on ealt is to offset te negative productivity consequences of te adverse ealt socks ε. Tus it is never optimal to incur ealt expenditures tat exceed te value of a worker s production in a given period. In order to capture tese large medical expenditures in data and arrive at realistic magnitudes of ealt insurance premia we introduce a second ealt sock. Tis exogenous sock z stands in for a catastropic ealt expenditure sock, and wen ouseolds receive te z-sock, tey ave to spend z; oterwise, tey die (or equivalently, incur a proibitively large utility cost). Houseolds in te augmented model are assumed to eiter not receive any ealt sock, face eiter a z-sock, or an ε-sock, but not bot. We denote by μ z () temean of te ealt expenditure sock z, conditional on initial ealt, and by κ() te probability of receiving a positive z-sock. Houseolds tat received a z-sock can still work, but at a reduced productivity ρ<1 relative to ealty workers. As described in more detail in appendix F.1, te z-sock merely scales up ealt insurance premia by μ z () and introduces additional ealt-related wage risk (since z-socks come wit a loss of 1 ρ of labor productivity). Second, in our model so far all variation in wages was due to eiter ealt ( and ε x) or age t. Wen bringing te model to te data we permit earnings in te model to also depend on te education educ of a ouseold, and consequently specify te production function as F (t, educ,, ε x). Given tis extension we ave to take a stance on ow ouseolds of different education levels interact in equilibrium under eac policy. Since our objective is to igligt te insurance aspect of bot policies wit respect to ealtrelated consumption risks we assume tat even in te presence of a wage discrimination law individuals wit iger education can be paid more, and tat ealt insurance companies can carge differential premia to individuals wit eterogeneous education levels even in te presence of a no-prior conditions law. 2

23 Tird, for te model to ave a cange of generating te observed eterogeneity in exercise levels of individuals tat are identical in terms of teir age, ealt and education levels we introduce preference socks to te disutility from effort. Instead of being given by q(e), as in te teoretical analysis so far, te cost of exerting effort is now assumed to be given as γq(e), were γ Γ is an individual-specific preference sock tat is drawn from te finite set Γ at te beginning of life and remains constant during te individual s life cycle. 21 Note tat since γ only affects te disutility of effort wic is separable from te utility of consumption, te analysis of te static model in section 3 remains completely uncanged (and so do te optimal ealt insurance contracts and ealt expenditure allocations). In te analysis of te dynamic model, since γ is a permanent sock, all expressions involving q(.) turnintoγq(.) but te analysis is oterwise unaltered. Under te maintained assumption tat wages and insurance premia are allowed to differ across different γ-groups even in te presence of te laws (an assumption tat parallels te one made in te previous paragrap) tere is no interaction between te different (γ, educ) types and equilibrium allocations under all policies can be solved for eac (γ, educ) pair separately. 22 Tese assumptions again igligt te role of (γ, educ)-eterogeneity modeled ere: it is not te focal point of our insurance vs. incentives analysis, but rater allows us to capture some of te eterogeneity in outcomes in te data and tus avoids attributing all of tis observed eterogeneity to ealt differences. Ignoring tese oter sources of eterogeneity would quantitatively overstate likely bot te insurance benefits as well as te incentive costs of te policies we analyze in tis paper. Consistent wit te introduction of preference and skill (education) eterogeneity te initial distribution over ouseold types is now denoted by Φ (, γ, educ) and will be determined from te data (but exploiting predictions of te structural model). Te last, and peraps most significant departure from te teoretical model is tat we now endow te ouseold wit a ealt-dependent continuation utility v T +1 () from retirement. Te teoretical model implicitly assumed tat tis continuation utility was identically equal to zero, independent of te ealt status at retirement. Te vector v T +1 () will be determined as part of our structural model estimation. Endowing individuals wit nontrivial continuation utility at retirement avoids te counterfactual prediction of te model tat effort is zero in te last period of working life, T. Tis assumption also introduces a direct utility benefit from better ealt (albeit one tat materializes at retirement) and tus avoids te complete collapse of incentives to provide effort under bot policies (tat is, proposition 13 no longer applies). In te rest of tis section, we use te so extended version of our model to estimate parameters to matc PSID data on ealt, expenditure and exercise in In te main body of te paper, we describe te procedure we follow in a condensed form, relegating te detailed data description and estimation procedures to te Appendix F. Once te model is parameterized and its reasonable fit of te data establised, in section 6 we ten use it to analyze te positive and normative sort- and long-run consequences of introducing non-discrimination legislation. 5.2 Parameter Estimation and Calibration Te determination of te model parameters proceeds in tree steps. First, we fix a small subset of parameters exogenously. Second, parts of te model parameters can be estimated from te PSID data directly. Tese include te parameters governing te ealt transition function Q(, e), te probabilities (g(),κ()) of receiving te ε and z ealt socks, as well as te productivity effect of te z-socks given by ρ. Tird, (and given te parameters obtained in step 1 and 2) te remaining parameters (mainly tose governing te production function F, te ε-sock distribution f(ε) and preferences) are ten determined troug a metod of moments estimation of te model wit PSID wage, ealt and effort data. We now describe tese tree steps in greater detail. 21 It does not matter weter firms/ealt insurance companies observe a worker s preference parameter γ since tey engage only in sort-term contracts and since is observable (γ only affects effort and firms as well as ealt insurance companies do not care ow te individual s ealt evolves due to te restriction of attention to sort-term contracts). 22 In order to obtain a meaningful welfare comparison wit socially optimal allocations we also solve te social planner problem separately for eac (γ,educ) combination, terefore ruling out ex ante social insurance against bad initial (γ,educ) draws. 21

24 5.2.1 A Priori Cosen Parameters First, we coose one model period to be six years, a compromise between assuring tat effort as a noticeable effect on ealt transitions (wic requires a sufficiently long time period) and reasonable sample sizes for estimation (wic speaks for sort time periods). We ten select two preference parameters a priori. Consistent wit values commonly used in te quantitative macroeconomics literature we coose a risk aversion parameter of σ = 2 and a time discount factor of β =.96 per annum Parameters Estimated Directly from te Data In a second step we estimate part of te model parameters directly from te data, witout aving to rely on te equilibrium of te model. Healt Transition Function Q(, e) Te PSID includes measures of ligt and eavy exercise levels 23 starting in 1999 wic we use to estimate ealt transition functions. We denote by e l and e te frequency of ligt and eavy exercise levels, and assume te following parametric functional form for te ealt transition function: (1 + π(, e l,e ) αi() )G(, ), if = + i, i {1, 2} (1 + π(, e l,e ))G(, ), if =, > 1or = +1,=1 Q( ;, e l,e )= 1 Q( ;, e l,e ) G(, ) G(, ), if = 1,>1or =, =1 < were π(, e l,e )=φ()(δe l +(1 δ)e ) λ(). Since ligt and eavy pysical exercise can ave different effects on ealt transition, we give weigt δ on ligt exercise, and (1 δ) on eavy exercise. We tink of δe l +(1 δ)e as te composite exercise level e used in te teoretical analysis of our model. Healt Sock Probabilities g() and κ() In our model, g() represents te probability of not receiving any sock, and κ() is te probability of facing a z-sock. Since we assume tat ouseolds do not receive bot an ε-sock and a z-sock in te same period, te probability of facing an ε-sock is given by 1 g() κ(). From PSID, we first construct te probabilities of aving a z-sock and an ε-sock. We define ouseolds tat ave received a z-sock as tose wo were diagnosed wit cancer, a eart attack, or a eart disease 24 and tose wo spent more on medical expenditures tan teir current income wen it wit a ealt sock. Houseolds wit all oter ealt socks or tose wo missed work due to an illness are categorized as aving received an ε-sock. Impact ρ of a z-sock on Productivity Using te criterion for determining ε and z-socks specified above, we use mean earnings of tose wit a z-sock relative to tose witout any ealt sock to directly estimate ρ Parameters Calibrated witin te Model In a final step we now use our model to find parameters governing te production function, te ε- andz-sock distribution, te distribution of preference parameters for exercise, and te terminal value function v T +1 ( ). 23 Number of times an individual carries out ligt pysical activity (walking, dancing, gardening, golfing, bowling, etc.) and eavy pysical activity (eavy ousework, aerobics, running, swimming, or bicycling). 24 Tese tree diseases lead to te most mean medical expenditures, relative to oter ealt conditions reported in te data. 22

25 Te structure of our model allows us to calibrate te parameters in two separate steps. Te first part of te estimation consists of finding parameters for te production function and distribution of ealt socks, and only involves te static part of te model from section 3. Tis is te case since realized wages and ealt expenditures in te model are determined in te static part and are independent of effort decisions and te associated ealt evolution in te dynamic part of te model. In a second step we ten employ te dynamic part of te model to estimate te preference distribution for exercise and te terminal value of ealt. 25 Production Function and Healt Status tecnology: We assume te following parametric form for te production (k (ε x))φ(a,educ) F (t, educ,, ε x) =A(t, educ) +, <φ( ),ξ( ) < 1,A( ) >. ξ(a,educ) Te production function captures two effects of ealt on production: te direct effect (first term) and te indirect effect wic induces te marginal benefit of ealt expenditures x to decline wit better ealt (tat is F 12 < ). Te term A(t, educ) allows for eterogeneity in age and education of te effect of ealt on production and tus wages. Here age can take seven values, t {1, 2,..., 7} and we classify individuals into two education groups, tose tat ave graduated from ig scool and tose tat ave not: educ {less tan Hig Scool, Hig Scool Grad}. We also allow for differences in marginal effects of medical expenditures on production across education and two broad age groups troug parameters φ(a, educ) andξ(a, educ), were a {Young,Old}. We define Young as tose individuals between te ages of 24 and 41 and te rest as Old. Tis age classification divides our sample rougly in alf. We represent te functions A(t, educ),φ(a, educ) andξ(a, educ) by a full set of age and education dummies. Since in te unregulated equilibrium te production of individuals (after ealt expenditures ave been made) ( equals teir labor earnings, we use data on labor earnings of ouseolds wit different ealt status w(2 ) w( 1 ), w( 3) w( 1 ), w( ) 4) as well as relative average earnings of te Young and te Old to pin down te w( 1 ) ealt levels { 1, 2, 3, 4 } in te model. 26 Moreover, since A(t, educ) captures te effects of age t and education educ on labor earnings we use conditional (on age and education) earnings to pin down te 14 (7 2) parameters A(t, educ). In order to determine te values of te dummies representing φ( ) andξ( ) we recognize tat in te model tey determine te expenditure cutoffs for te ε-sock, as a function of individual ealt status. Tus we use medical expenditure data to estimate tese parameters. More specifically te four parameters representing φ(a, educ) are determined to fit te percentage of labor earnings spent on medical expenditure (averaged over ) for eac (a, educ)-group and te four parameters representing ξ(a, educ) are cosen to matc te percentage of labor earnings spent on medical expenditures (averaged over (a, educ) groups) for eac level H of ouseold ealt. 27 Distribution of Healt Socks In order to estimate te parameters governing te distribution of ealt socks ε we exploit te teoretical result from section 3 tat medical expenditures on tese socks is linear in te sock: x (ε, ) =max{,ε ε()}. Tus te distribution of medical expenditures x coincides wit tat of te socks temselves, above te endogenous ealt-specific tresold ε(). Frenc and Jones (24) argue tat te cross-sectional distribution of ealt care costs 28 can best be fitted by a log-normal distribution (truncated at te upper tail). We terefore assume tat te ealt socks ε follow a truncated log-normal 25 Even toug we describe te parameters and calibration targets of te different model elements in separate subsections below for expositional clarity, te parameters for production function and ealt sock distributions are calibrated jointly, using te targets in tese sections. Similarly, te parameters for exercise preference distribution and marginal value of ealt at terminal date are calibrated jointly, using te observations in bot subsections. 26 Te categories {Excellent, Very Good, Good, Fair} used in te data itself ave no cardinal interpretation. 27 Since tere is more variation in te data for labor earnings tan by ealt spending by age we decided to use a finer age grouping wen estimating A(t, educ) using wage data tan wen estimating ξ(a, educ) and φ(a, educ) using ealt (expenditure) data. 28 Tey use HRS and AHEAD data. Healt care costs include ealt insurance premia, drug costs and costs for ospital, nursing ome care, doctor visits, dental visits and outpatient care. 23

26 distribution: f(ε; μ ε,σ ε,ε, ε) = ( 1 ɛσ ε φ ln ε με Φ ( ln ε με σ ε ) Φ σ ε ) ( ) ln ε με σ ε were φ and Φ are standard normal pdf and cdf. We ten coose te mean and standard deviation (μ ε,σ ε ) of te socks suc tat te endogenously determined mean and standard deviation of medical expenditures in te model matces te mean and standard deviation of ealt expenditures for tose wit ε-socks from te data. For te catastropic ealt sock z, apart from te probability of receiving it (wic was determined in section 5.2.2), only te mean expenditures μ z () matter. We use te percentage of labor income spent on catastropic medical expenditures, conditional on ealt status, to determine tese. Distribution of Exercise Preference Parameters Wit estimates of te production function and ealt sock distributions in and we now calibrate te preference for exercise distribution, using te dynamic part of te model. We assume tat te effort utility cost function takes te form [ ] 1 γq(e) =γ (1 + e). 1 e Te functional form for q guarantees tat q (e) >, tat q() = q () = and tat lim e 1 q (e) =. We assume tat for eac education group te preference sock γ can take two (education-specific) values, γ {γ 1 (educ),γ 2 (educ)}. We treat tese values (4 in total) as parameters. Te initial joint distribution Φ over types (, educ, γ) is ten determined by te eigt numbers Φ (γ 1 educ, ) tat give te fraction of low cost (γ 1 ) individuals for eac of te eigt (educ, )-combinations. Tus we ave to a total of 12 parameters determining preference eterogeneity in te model. We coose te initial distribution Φ (γ 1 educ, ) so tat model effort levels matc mean effort levels in period 1 (ages 24-29), conditional on ealt (4 targets) and conditional on education (2 targets), and mean effort levels in period 7 (ages 6-65), conditional on education (2 targets) in te data. To pin down te four values γ(educ), we use te aggregate mean and standard deviation of effort in period 1, and te measure of ouseolds wit fair and excellent ealt in te last period, t =7. Marginal Value of Healt at Terminal Date As discussed above, absent direct benefits from better ealt upon retirement ouseolds in te model ave no incentive to exert effort, wereas in te data we still see a significant amount of exercise for tose of ages 6 to 65. By introducing a terminal and ealt dependent continuation utility v T +1 () tis problem can be rectified. Given te structure of te model and te parametric form of te ealt transition function Q(, e) only te differences in te continuation values Δ i = v T +1 ( i ) v T +1 ( i 1 ), for i =2, 3, 4 matter for te coice of optimal effort in te last period T. We coose te Δ 2, Δ 3, Δ 4 suc tat te model reproduces te ealt-contingent average effort levels of te 6 to 65 year olds, for 2, 3, 4. Te data targets and associated model parameters are summarized in Tables 9 and 1. Te estimated parameter values are reported in Table 11, togeter wit teir performance in matcing te empirical calibration targets. 5.3 Model Fit Our model is fairly ricly parameterized (especially along te production function/labor earnings dimension). It is terefore not surprising tat it fits life cycle earnings profiles well. We ave also targeted effort levels for very young and very old ouseolds (te latter by ealt status), but ave not used data on -specific effort levels (apart from at te final pre-retirement age) in te estimation. How well te model captures te age-effort dynamics is terefore an important test of te model. Figures 5 (for mean effort) and 27-3 in appendix G.1 (for effort by ealt status) plot te evolution of effort (exercise) over te life cycle bot in te data and in te model. Te dotted lines sow te one-standard deviation confidence bands. From 24

27 Figure 5 we see tat our model fits te average exercise level over te life cycle very well, and Figures 27-3 sow te same to be true for effort conditional on Very Good and Excellent ealt. For ouseolds wit Fair and Good ealt te model fit is not quite as good as tat for te Very Good and te Excellent ealt groups, but still witin te one-standard deviation confidence bands (wic are arguably quite wide toug, on account of smaller samples once conditioning bot on age and ealt) Average Effort in Equilibrium Model Data: Mean Data: Mean ± St.Dev.8 Average Effort Time Figure 5: Average Effort in Model and Data 6 Results of te Policy Experiments: Insurance, Incentives and Welfare After aving establised tat te model provides a good approximation to te data for te late 199 s and early 2 s in te absence of non-discrimination policies, we now use it to answer te main counterfactual question of tis paper, namely, wat are te effects of introducing tese policies (one at a time and in conjunction) on aggregate ealt, consumption and effort, teir distribution, and ultimately, on social welfare. Te primary benefit of te non-discrimination policies is to provide consumption insurance against bad ealt, resulting in lower wages and iger insurance premia in te competitive equilibrium. However, tese policies weaken incentives to exert effort to lead a ealty life, and tus worsen te long run distribution of ealt, aggregate productivity and tus consumption. In te next two subsections, we present te key quantitative indicators measuring tis trade-off: first, te insurance benefits of policies, and second, te adverse incentive effects on aggregate production and ealt. Ten, in subsection 6.3, we display te welfare consequences of our policy reforms. In te main text we focus on weigted averages of te aggregate variables and welfare measures across workers of different (educ, γ)-types, and document te disaggregated results (wic are qualitatively, and to a great extent, quantitatively similar to te averaged numbers) in appendix G Insurance Benefits of Policies Turning first to te consumption insurance benefits of bot policies, we observe from figure 6 tat te combination of bot policies is indeed effective in providing perfect consumption insurance. As in te social planner problem, witin-group consumption dispersion, as measured by te coefficient of variation, is zero for all periods over te life cycle if bot a no-prior conditions law and a no-wage discrimination law are in place (te lines for te social planner solution and te equilibrium under bot policies lie on top of one 29 For Fair and Good ealt, our model predicts iger exercise level between te ages of 3 and 54 tan in te data. Tis is partly due to a composition effect: in te second period of life, many workers wit low disutility for exercise ave fair ealt and exercise a lot, leading to an increase in te average exercise level for te fair ealt group. One mecanical way of rectifying tis problem would be to let te values te taste parameter γ can take on vary wit age, reflecting differences in taste for exercise at different stages of life. 25

28 anoter and are identically equal to zero). 3 Tis is of course wat te teoretical analysis in sections 3 and 4 predicted. Also notice from figure 6 tat a wage non-discrimination law alone goes a long way towards providing effective consumption insurance, since te effect of differences in ealt levels on wage dispersion is significantly larger tan te corresponding dispersion in ealt insurance premia. Tus, altoug a noprior conditions law in isolation provides some consumption insurance and reduces witin-group consumption dispersion by about 3%, relative to te unregulated equilibrium, te remaining ealt-induced consumption risk remains significant. Coefficient of Variation of Consumption Social Planner Comp. Eq. No Prior Conditions No Wage Discrimination Bot Policies Coefficient of Variation Time Figure 6: Consumption Dispersion Anoter measure of te insurance benefits provided by te non-discrimination policies is te level of crosssubsidization or implicit transfers: workers do not necessarily pay teir own competitive (actuarially fair) price of te ealt insurance premium or/and tey are not fully compensated for teir productivity. Under no-prior conditions policy, as establised teoretically in Proposition 1, te ealty workers subsidize te premium of te unealty. Similarly, wages of te unealty workers are subsidized by te ealty, productive workers under te no-wage discrimination policy. Moreover, under bot policies, tere is cross-subsidization in bot ealt insurance premia and wages. Cross Subsidy: Excellent Healt Cross Subsidy: Fair Healt.4 Premium: No Prior Premium: Bot Pol. Wage: No Wage Wage: Bot Pol Transfer.2 Transfer Time.6 Premium: No Prior Premium: Bot Pol. Wage: No Wage Wage: Bot Pol Time Figure 7: Cross Subsidy: Excellent Healt Figure 8: Cross Subsidy: Fair Healt Figures 7 and 8 plot te degree of cross-subsidization over te life cycle, bot for ouseolds wit excellent and tose wit fair ealt, and Table 13 in appendix G.2 summarizes te transfers for all ealt groups. Te 3 Due to te presence of eterogeneity in education levels and preferences te economy as a wole displays non-trivial consumption dispersion even in te presence of bot policies (as it does in te solution of te restricted social planner problem). 26

29 plots for te ealt insurance premium measures te differences between te actuarially fair ealt insurance premium a particular ealt type ouseold would ave to pay and te actual premium paid in te presence of eiter a no-prior conditions policy or te presence of bot policies. Similarly, te wage plots display te difference between te productivity of te worker (and tus er wage in te unregulated equilibrium) and te wage received under a no-wage discrimination policy and in te presence of bot policies. Negative numbers imply tat te worker is paying a iger premium, or is paid lower wage tan in a competitive equilibrium witout government intervention. Tus suc a worker, in te presence of government policies, as to transfer resources to workers of different (lower) ealt types. Reversely, positive numbers imply tat a worker is being subsidized, i.e., se is paying a lower premium and is paid iger wage. We observe from Figure 7 tat te workers wit excellent ealt significantly cross-subsidize te oter workers, bot in terms of cross-subsidies in ealt insurance premia as well as in terms of wage transfers. To interpret te numbers quantitatively, note tat average consumption of te excellent group is 1.4 wen young and 1.75 wen aged Tus te wage transfers delivered by tis group amount to 12 14% of average consumption wen young and close to 3% in prime working age (note tat te sare of workers in excellent ealt in te population as srunk at tat age, relative to wen tis coort of workers was younger). From figure 7 we also observe tat te implicit transfers induced by a no-prior conditions law are still significant (tey amount to 3-7% of consumption for young workers of excellent ealt, and 4-1% wen middle-aged), but quantitatively smaller tan tose implied by wage-nondiscrimination legislation. Figure 8 displays te same plots for ouseolds of fair ealt. Tese ouseolds are te primary recipients of te transfers from workers wit excellent ealt, 31 and for tis group (wic is small early in te life cycle but grows over time) te transfers are massive. In terms of teir average competitive equilibrium consumption, te implicit ealt insurance premium subsidies amount to a massive 37-6% and te wage transfers amount to a staggering 65-75% of pre-policy average consumption of tis group. Altoug tese transfers srink (as a fraction of pre-policy consumption) over te life cycle as te sare of ouseolds wit fair ealt increases and tat wit excellent ealt declines, tey continue to account for a significant part of consumption for ouseolds of fair ealt. Tese numbers indicate tat te insurance benefits from bot policies, and specifically from te wage nondiscrimination law, will be substantial. An interesting property of te subsidies is tat te level of subsidization implied by a given policy is iger wen only one of te non-discrimination laws is enacted, relative to wen bot policies are present. Tis is especially true for te no-prior conditions law and is due to te fact tat te government insures te workers wit bad ealt troug an inefficient level of medical expenditure. Tus far, we ave discussed te insurance benefits of te non-discrimination policies. In te next subsection, we analyze te aggregate dynamic effects of te policies on production and te ealt distribution. 6.2 Adverse Incentive Effects on Aggregate Production and Healt Te associated incentive costs from eac policy are inversely proportional to teir consumption insurance benefits, as figure 9 sows. In tis figure we plot te average exerted effort over te life cycle, in te socially optimal and te equilibrium allocations under te various policy scenarios. In a nutsell, effort is igest in te solution to te social planner problem, positive under all policies, 32 but substantially lower in te presence of te non-discrimination laws. More precisely, two important observations emerge from figure 9. First, te policies tat provide te most significant consumption insurance benefits also lead to te most significant reductions in incentives to lead a ealty life. It is te very dispersion of consumption due to ealt differences, stemming from ealt-dependent wages and insurance premia tat induce workers to provide effort in te first place, and tus te policies tat reduce tat consumption dispersion te most come wit te sarpest reduction in 31 Table 13 in te appendix sows tat ouseolds wit very good ealt are also called upon to deliver transfers, albeit of muc smaller magnitude, and workers wit good ealt are on te receiving side of (small) transfers. As te coort ages te sare of ouseolds in tese different ealt groups sifts, and towards te end of te life cycle te now larger group of ouseolds wit fair ealt receives subsidies from all oter ouseolds, at least wit respect to ealt insurance premia. 32 Recall tat, relative to te teoretical analysis, we ave introduced a terminal value of ealt wic induces not only effort in te last period even under bot policies, but troug te continuation values in te dynamic programming problem, positive effort in all periods. How quantitatively important tis effect is for younger ouseolds depends significantly on te time discount factor β. 27

30 incentives. Wereas a no-prior conditions law alone leads to only a modest reduction of effort, wit a wage nondiscrimination law in place te amount of exercise ouseold find optimal to carry out srinks more significantly. Finally, if bot policies are implemented simultaneously te only benefit from exercise is a better distribution of post-retirement continuation utility, and tus effort plummets strongly, relative to te competitive equilibrium. Te second observation we make from figure 9 is tat te impact of te policies on effort is most significant at young and middle ages, wereas towards retirement effort levels under all polices converge. Tis is owed to te fact tat te direct utility benefits from better ealt materialize at retirement and are independent of te nondiscrimination laws (but eavily discounted by our impatient ouseolds), wereas te productivity and ealt insurance premium costs from worse ealt accrue troug te entire working life and are strongly affected by te different policies. 33 Average Effort Social Planner Comp. Eq No Prior Conditions No Wage Discrimination Bot Policies Average Effort Aggregate Consumption Average Healt Over Time Social Planner Comp. Eq No Prior Conditions No Wage Discrimination Bot Policies Time Time Figure 9: Effort Figure 1: Average Healt Given te dynamics of effort over te life cycle (and a policy invariant initial ealt distribution), te evolution of te ealt distribution is exclusively determined by te ealt transition function Q( ;, e). Figure 1 wic displays average ealt in te economy under te various policy scenarios is ten a direct consequence of te effort dynamics from Figure 9. It sows tat ealt deteriorates under all policies as a coort ages, but more rapidly if a no-prior conditions law and especially if a wage nondiscrimination law is in place. As wit effort, te conjunction of bot policies as te most severe impact on public ealt. Figure 12 demonstrates tat te decline of ealt levels over te life cycle also induce iger expenditures on ealt (insurance) later in life. Te level of tese expenditures (and tus teir relative magnitudes across different policies) are determined by two factors, a) te ealt distribution (wic evolves differently under alternative policy scenarios) and b) te equilibrium ealt expenditures, wic are fully determined by te tresolds ε() from te static analysis of te model and tat vary across policies. Te evolution of ealt is summarized by figure 1, and figure 11 displays te ealt dependent tresolds ε() for te youngest ouseolds. 34 Recall from section 3 tat te tresolds ε() under te unregulated competitive equilibrium and te equilibrium wit bot policies are socially efficient and tus te tree graps completely overlap. Also observe tat, relative to te efficient allocation (=unregulated equilibrium) under te no-prior conditions law workers wit low ealt are strongly over-insured (tey ave lower tresolds, ε NP ( i ) > ε SP ( i )for i =1, 2) and workers wit very good and excellent ealt are sligtly under-insured. Tis was te content of Proposition 1, and it is quantitatively responsible for te finding tat ealt expenditures are igest under tis policy. Te reverse is true under a no-wage discrimination law: low ealt types are under-insured and ig types are over-insured, relative to te social optimum, but quantitatively tese differences are minor. Finally, figures 13 and 14 display aggregate production and aggregate consumption over te life cycle. Since te productivity of eac worker depends on er ealt and on te non-treated fraction of er ealt 33 In fact, absent te terminal (and policy invariant) direct benefits from better ealt te differences in effort levels across policies remain fairly constant over te life cycle. 34 Te figures are qualitatively similar for older coorts. 28

31 Cutoffs Healt Insurance Tresolds: Social Planner No Prior Conditions No Wage Discrimination Fair Good V.Good Ex. Healt Status Healt Sepnding Aggregate Dynamics of Healt Spending (Premia) Social Planner Comp. Eq. No Prior Conditions No Wage Discrimination Bot Policies Time Figure 11: Cutoffs Figure 12: Healt Spending sock, aggregate output is lower, ceteris paribus, under policy configurations tat lead to a worse ealt distribution and tat leave a larger sare of ealt socks ε untreated. From figure 13 we observe tat te deterioration of ealt under a policy environment tat includes a wage nondiscrimination policy is especially severe, in line wit te findings from figure 1. Interestingly, te more generous ealt insurance (for tose of fair and good ealt) under a no-prior conditions law alone leads to output tat even exceeds tat in te unregulated equilibrium, despite te fact tat te ealt distribution under tat policy is (moderately) worse. But ealt expenditures of course command resources tat take away from private consumption, and as figure 14 sows, resulting aggregate consumption over te life cycle under tis policy is substantively identical to tat under te wage discrimination law (and te consumption allocation is more risky under te no-prior conditions legislation). Relative to te unregulated equilibrium bot policies tus entail a significant loss of average consumption in society (in one case, because less is produced, in te oter case because more resources are spent on productivity enancing ealt goods); te same is even more true if bot policies are introduced jointly. 1.7 Aggregate Dynamics of Production Aggregate Dynamics of Consumption Aggregate Production Social Planner 1.1 Comp. Eq. No Prior Conditions No Wage Discrimination Bot Policies Time Aggregate Consumption Social Planner.7 Comp. Eq No Prior Conditions No Wage Discrimination Bot Policies Time Figure 13: Production Figure 14: Consumption Overall, te effect on aggregate effort, ealt, production and tus consumption suggests a quantitatively important trade-off between consumption insurance and incentives. Witin te spectrum of all policies, te unregulated equilibrium provides strong incentives at te expense of risky consumption, wereas a policy mix tat includes bot policies provides full insurance at te expense of a deterioration of te ealt distribution. Te effects of te no-prior conditions law on bot consumption insurance and incentives are modest, relative 29

32 to te unregulated equilibrium. In contrast, implementing a no wage discrimination law or bot policies insures away most of te consumption risk, but significantly reduces (altoug does not eliminate completely) te incentives to exert effort to lead a ealty life, especially early in te life cycle. In te next subsection we will now document ow tese two quantitatively sizable but countervailing effects translate into welfare consequences from ypotetical policy reforms. 6.3 Welfare Implications In tis section we quantify te welfare impact of te policy innovations studied in tis paper. For a fixed initial distribution Φ () over ealt status, 35 denote by W (c, e) te expected lifetime utility of a coort member (were expectations are taken prior to te initial draw of ealt) from an arbitrary allocation of consumption and effort over te life cycle. 36 Our consumption-equivalent measure of te welfare consequences of a policy reform is given by W (c CE (1 + CEV i ),e CE )=W (c i,e i ) were i {SP,NP,NW,Bot} denotes te policy scenario under consideration. Tus CEV i is te percentage reduction of consumption in te competitive equilibrium consumption allocation required to make ouseolds indifferent (ex ante) between te competitive equilibrium allocation 37 and tat arising under policy regime i. In order to empasize te importance of te dynamic analysis in assessing te normative consequences of different policies we also report te welfare implications of te same policy reforms in te static version of te model in section 3. Similar to te dynamic consequences we compute te static consumption-equivalent loss (relative to te competitive equilibrium) as U(c CE (1 + SCEV i )) = U(c i ) were U(c) is te expected utility from te period consumption allocation 38, under te cross-sectional distribution Φ, and tus is determined by te static version of te model. 39 Terefore SCEV i provides a clean measure of te static gains from better consumption insurance induced by te policies against wic te dynamic adverse incentive effects ave to be traded off. Te static welfare consequences reported in te first column of Table 1 tat isolate te consumption insurance benefits of te policies under consideration are consistent wit te consumption dispersion displayed in Figure 6. Perfect consumption insurance, as implemented in te solution to te social planner problem and also acieved if bot policies are implemented jointly, are wort close to 6% of unregulated equilibrium consumption. Eac policy in isolation delivers a substantial sare of tese gains, wit te no wage discrimination law being more effective tan te no-prior conditions law. 35 Recall tat we carry out our analysis for eac (educ, γ)-type separately and report averages across tese types. Tus in wat follows Φ suppresses te (policy-independent) dependence of te initial distribution on (educ, γ). 36 Tat is, using te notation from section 4, for te socially optimal allocation W (c SP,e SP )=V (Φ ) and for equilibrium allocations, under policy i, W (c i,e i )= v i ()dφ. 37 Recall tat even te social planner problem is solved for eac specific (γ,educ) group separately and tus also does not permit ex-ante insurance against unfavorable (γ,educ)-draws. We consider tis restricted social planner problem because we view te results are better comparable to te competitive equilibrium allocations. 38 In te static version of te model effort is identically equal to zero in te social planner problem and in te equilibrium under all policy specifications, and terefore disutility from effort is irrelevant in te static version of te model. 39 Tus, using te notation from section 3 U(c CE )=U CE (Φ )fori {SP,NP,NW,Bot} and U(c CE )= U CE ()dφ. 3

33 Static CEV i Dynamic CEV i Social Planner Competitive Equilibrium.. No Prior Conditions Law No Wage Discrimination Law Bot Policies Table 1: Aggregate Welfare Comparisons Turning now to te main object of interest, te dynamic welfare consequences (column 2 of Table 1) paint a somewat different picture. Consistent wit te static analysis, bot policies improve on te laissez-faire equilibrium, and te welfare gains are substantial, ranging from 6% to 9.5% of lifetime consumption. Te sources of tese welfare gains are improved consumption insurance (as in te static model) and reduced effort (wic bears utility costs), wic outweig te reduction in average consumption tese policies entail (recall Figure 14). Furtermore, as in te static model a wage nondiscrimination law dominates a no-prior conditions law. In ligt of Figures 14 and 6 tis does not come as a surprise: bot policies imply virtually te same aggregate consumption dynamics, but te no-prior conditions law provides substantially less consumption insurance. But wat we really want to stress is tat tere are crucial differences to te static analysis. First and foremost, it is not optimal to introduce a no-prior conditions law once a wage non-discrimination law is already in place. Te latter policy already provides effective (albeit not complete) consumption insurance, and te furter reduction of incentives and associated mean consumption implied by te no prior conditions law makes a combination of bot policies suboptimal. Te associated welfare losses of pusing social insurance too far amount to about 1.3% of lifetime consumption. 4 Finally we see tat in contrast to te static case te best policy combination (a wage nondiscrimination law alone) does not come close to providing welfare as ig as te social optimum: te gap between tese two scenarios turns out to about 7% of lifetime consumption. Tis gap is due to inefficiently little consumption insurance, inefficiently low aggregate consumption and an inefficient ealt expenditure allocation (see again Figure 11), altoug te latter effect is quantitatively modest. Tis effect is owever quantitatively crucial in explaining wy te noprior conditions law in isolations fares worse tan te wage nondiscrimination policies (and a combination of bot policies, wic restores efficiency in ealt expenditures, recall proposition 12). Fair Good Very Good Excellent Social Planner Competitive Equilibrium.... No Prior Conditions Law No Wage Discrimination Law Bot Policies Table 2: Welfare Comparison in te Dynamic Economy Conditional on Healt Te welfare consequences reported in Table 1 were measured under te veil of ignorance, before workers learn teir initial ealt level. Tey mask very substantial eterogeneity in ow workers feel about tese policies once teir initial ealt status in period as been revealed. Given te transfers across ealt types displayed in Figures 7 and 8 and te persistence of ealt status tis is ardly surprising. Table 2 quantifies tis eterogeneity by reporting dynamic consumption-equivalent variation measures, computed exactly as before, but now computed after te initial ealt status as been materialized. Broadly speaking, te lower a worker s initial ealt status, te more se favors policies providing consumption insurance. For te middle two ealt groups te ranking of policies coincides wit tat in te second column of Table 1; ouseolds wit excellent ealt prefer only te no prior conditions law (and tus only very moderate implicit transfers) to te unregulated equilibrium, wereas young ouseolds wit fair ealt would support te simultaneous 4 It sould be stressed tat tese conclusions follow under te maintained assumption tat a wage nondiscrimination law is indeed fully successful in curbing ealt-related wage variation, and does so completely costlessly. 31

34 introduction of bot policies. Te differences in te preference for different policy scenarios across different -ouseolds are quantitatively very large: wereas fair-ealt types would be willing to pay 54% of laissez faire lifetime consumption to see bot policies introduced, ouseolds of excellent ealt would be prepared to give up 4.5% of lifetime consumption to prevent exactly tis policy innovation. 7 Conclusion In tis paper, we studied te effect of labor and ealt insurance market regulations on evolution of ealt and production, as well as welfare. We sowed tat bot a no-wage discrimination law (an intervention in te labor market), in combination wit a no-prior conditions law (an intervention in te ealt insurance market) provides effective consumption insurance against ealt socks, olding te aggregate ealt distribution in society constant. However, te dynamic incentive costs and teir impact on ealt and medical expenditures of bot policies, if implemented jointly, are large. Even toug bot policies improve upon te laissez-faire equilibrium, implementing tem jointly is suboptimal (relative to introducing a wage nondiscrimination in isolation). We terefore conclude tat a complete policy analysis of ealt insurance reforms on one side and labor market (non-discrimination policy) reforms cannot be conducted separately, since teir interaction migt prove less favorable despite welfare gains from eac policy separately. Tese conclusions rest in part on our assumption tat bot policies can be implemented optimally at no direct overead cost. To us, tis assumption seems potentially more problematic for te no-wage discrimination policy tan te no-prior conditions policy because matc-specificity between a worker and a firm appears to be more important tan between a worker and a ealt insurance company. One can likely implement te no-prior conditions policy troug te ealt insurance excanges proposed by Obama Care in wic a government agency links tose seeking ealt insurance to ealt insurance providers and tereby overcomes, at low cost, te incentives of te ealt insurance companies to cerry-pick teir clients. However, a similar institution (e.g. someting akin to a union all type institution), is likely to demand iger costs, given te specificity in most worker-firm matces. In addition, te average output produced by a worker-firm pair is muc larger tan te expenses involved in ealt insurance (bot in our model as well as in te data). 41 Finally, our analysis of ealt insurance and incentives over te working life as ignored several potentially important avenues troug wic ealt and consumption risk affect welfare. First, te benefits of ealt in our model are confined to iger labor productivity, and tus we model te investment motives into ealt explicitly. It as abstracted from an explicit modeling of te benefits better ealt as on survival risk, altoug te positive effect of ealt on te continuation utility after retirement partially captures tis effect in our model, albeit in a fairly reduced from. Similarly, better ealt migt ave a direct effect on flow utility during working life. 42 Finally, in our analysis labor income risk directly translates into consumption risk, in te absence of ouseold private saving. We conjecture tat te introduction of self-insurance via precautionary saving against tis income risk furter weakens te argument in favor of te policies studied in tis paper. Future work as to uncover weter suc an extension of te model also affects, quantitatively or even qualitatively, our conclusions about te relative desirability of tese policies. 41 To put tese potential costs in perspective, from our quantitative results it follows tat if as little as 3% of production was consumed in implementing te no-wage discrimination policy (and te no prior conditions policy is cost-free), ten it is te latter policy tat would constitute te ex ante preferred policy option. 42 As we argue in appendix H at least in one extension of te model introducing a direct flow utility benefit from better ealt leaves our analysis qualitatively uncanged. 32

35 References [1] Acemoglu, D. and J. Angrist (21), Consequences of Employment Protection? Te Case of te Americans wit Disabilities Act. Journal of Political Economy, Volume 19, No. 5, [2] Ales, L., R. Hosseini, and L. Jones (212), Is Tere Too Muc Inequality in Healt Spending Across Income Groups? Working Paper. [3] Attanasio, O., S. Kitao and G. Violante (211), Financing Medicare: A General Equilibrium Analysis, Demograpy and te Economy (capter 9), J. Soven (ed.), University of Cicago Press. [4] Battacarya, J., K. Bundorf, N. Pace and N. Sood (29), Does Healt Insurance Make You Fat?, NBER Working Paper [5] Brügemann, B. and I. Manovskii (21), Fragility: A Quantitative Analysis of te US Healt Insurance System, Working paper, University of Pennsylvania [6] Cawley, J. (24), Te Impact of Obesity on Wages, Journal of Human Resources, 39(2), [7] Currie, J. and B. Madrian (1999), Healt, Healt Insurance and te Labor Market. In Handbook of Labor Economics, Volume 3C, edited by David Card and Orley Asenfelter, Amsterdam: Nort Holland, [8] DeLeire, T. (21). Canges in Wage Discrimination against People wit Disabilities: Journal of Human Resources, Volume 35, No. 1, [9] M. Dey and C. Flinn (25), An Equilibrium Model of Healt Insurance Provision and Wage Determination, Econometrica 73, [1] Edwards, R. (28), Healt Risk and Portfolio Coice, Journal of Business and Economic Statistics, 26, [11] Erlic, I. and G. Becker (1972), Market Insurance, Self-Insurance, and Self-Protection, Journal of Political Economy, 8, [12] Erlic, I. and H. Cuma (199), A Model of te Demand for Longevity and te Value of Life Extension, Journal of Political Economy, 98, [13] Frenc, E. and J. Jones (24), On te Distribution and Dynamics of Healt Care Costs, Journal of Applied Economics, 19, [14] Grossman, M. (1972), On te Concept of Healt Capital and te Demand for Healt, Journal of Political Economy, 8(2), [15] Hai, R. (212), Causes and Consequences of Increasing Inequality in Healt Insurance and Wages, Working Paper. [16] Hall, R. and C. Jones (27), Te Value of Life and te Rise in Healt Spending, Quarterly Journal of Economics, 122(1), [17] Halliday, T., H. He and H. Zang (212), Healt Investment over te Life Cycle, Working Paper, University of Hawaii. [18] Hansen, G., M. Hsu and J. Lee (212), Healt Insurance Reform: Te Impact of a Medicare Buy-In, NBER Working Paper [19] Hugonnier, J., F. Pelgrin and P. St-Amour (212), Healt and (Oter) Asset Holdings, Working Paper. [2] Jeske, K. and S. Kitao (29), U.S. Tax Policy and Healt Insurance Demand: Can a Regressive Policy Improve Welfare?, Journal of Monetary Economics, 56,

36 [21] Jung, J. and C. Tran (21), Healt Care Financing over te Life Cycle, Universal Medical Voucers and Welfare, Working Paper, TowsonUniversity. [22] Kass, N., A. Medley, M. Natowicz, S. Candros Hull, R. Faden, L. Plantinga and L. Gostin (27), Access to Healt Insurance: Experiences and Attitudes of Tose wit Genetic versus Non-genetic Medical Conditions, American Journal of Medical Genetic, 143A, [23] Kopecky, K. and T. Koreskova (212), Te Joint Impact of Social Security and Medicaid on Incentives and Welfare, Working Paper. [24] Laun, T. (212), Optimal Social Insurance wit Endogenous Healt, Working Paper. [25] Madrian, B. (1994), Employment-Based Healt Insurance and Job Mobility: Is Tere Evidence of Job-Lock?, Quarterly Journal of Economics, 19, [26] Ozkan, S. (212), Income Differences and Healt Care Expenditures over te Life Cycle, Working Paper, Federal Reserve Board. [27] Pascenko, S. and P. Porapakkarm (212), Quantitative Analysis of Healt insurance Reform: Separating Regulation from Redistribution, Review of Economic Dynamics, fortcoming. [28] Pijoan-Mas, J. and V. Rios-Rull (212), Heterogeneity in Expected Longevities, Working Paper. [29] Prados, M. (212). Healt and Earnings Inequality, Working Paper. [3] Sommers, A. (27), Access to Healt Insurance, Barriers to Care, and Service Use among Adults wit Disabilities, Inquiry, 43, [31] Yogo, M. (29), Portfolio Coice in Retirement: Healt Risk and te Demand for Annuities, Housing, and Risky Assets, NBER Working Paper

37 A Proofs of Propositions Proposition 5 Proof. Since exercise does not carry any benefits in te static model, trivially e SP =. Attacing Lagrange multiplier μ to te resource constraint, te first order condition wit respect to consumption c(ε) is u (c(ε, )) = λ and tus c SP (ε, ) =c SP for all ε E and H. Tus, not surprisingly, te social planner provides full consumption insurance to ouseolds. Te optimal ealt expenditure allocation maximizes tis consumption { } c SP =max x(ε,) g()[f (, x(,)) x(,)]+(1 g()) f(ε)[f (, ε x(ε, )) x(ε, )] dε Φ() Denoting by μ(ε, ) te Lagrange multiplier on te constraint x(ε, ), te first order condition wit respect to x(ε, ) readsas F 2 (, ε x(ε, )) + μ(ε, ) =1 Fix H. By assumption 4 F 22 (, y) < and tus eiter x(ε, ) =orx(ε, ) > satisfying F 2 (, ε x(ε, )) = 1 for all ε. Tus off corners ε x(ε, ) = ε SP () were te tresold satisfies F 2 (, ε SP ()) = 1. (43) Consequently x SP (ε, ) =max [,ε ε SP () ]. Te fact tat ε SP () is increasing in, strictly so if F 12 (, y) >, follows directly from assumption 4 and (43). Proposition 6 Proof. Attacing Lagrange multiplier μ() to equation (11) and λ() to equations (12) te first order conditions read as u (w() P ()) = λ() = μ() (44) λ()f 2 (, x(,)) μ() (45) = if x(,) > λ()f 2 (, ε x(ε, )) μ() (46) = if x(ε, ) > Tus off corners we ave F 2 (, ˆε x(ˆε, )) = F 2 (, ε x(ε, )) = K (47) for some constant K. Tus off corners ε x(ε, ) is constant in ε and tus medical expenditures satisfy te cutoff rule x CE (ε, ) =max [,ε ε CE () ]. (48) Plugging (48) into (46) and evaluating it at ε = ε CE () yields λ()f 2 (, ε CE ()) = μ(). (49) Using tis result in te second part of (44) delivers te caracterization of te equilibrium cutoff levels F 2 (, ε CE ()) = 1 for all H 35

38 wic are unique, given te assumptions imposed on F. Wages, consumption and ealt insurance premia ten trivially follow from (11) and (12). Proposition 9 Proof. Let Lagrange multipliers to equations (19) and (2) be μ and λ(), respectively. Ten, te first order conditions are: u (w() P )Φ() = μ u (w() P )Φ() = λ() (1 g())f(ε)[ F 2 (, ε x(ε, ))]λ() μ(1 g())f(ε)φ() = if x(ε, ) > g()[ F 2 (, x(,))]λ() μg()φ() = if x(,) > Tus, off-corners we ave F 2 (, ε x(ε, )) = F 2 (, ˆε x(ˆε, )) = K for some constant K and te cutoff rule is determined by u (w() P )[ F 2 (, ε NP ())] = u (w() P )Φ(). (5) Moreover, let us take te derivative of (5) wit respect to. u (w() P ) w() { F 2 + u ε NP } () (w() P ) F 12 + F 22 = { u (w() P ) εnp () w() ε NP () F 2 + u ε NP } () (w() P ) F 12 + F 22 εnp () = { } u w() (w() P )F 2 ε NP () + u (w() P )F 22 = u (w() P )F 12 Note tat as ε increases w() decreases, since F (, ε x(ε, )) is decreasing for ε< ε, and constant for ε ε. Tus, we ave ε NP () >. Proposition 1 Proof. From (21), we immediately obtain u F 2 (, ε NP (w() P )Φ() ()) = u (w() P ) < 1 ε NP () < ε SP () =1 ε NP () = ε SP () > 1 ε NP () > ε SP () as F 2 (, ε SP ()) = 1. Let us take L < < H, and suppose i.e. F 2 ( L, ε NP ( L )) > 1 > F 2 ( H, ε NP ( H )), (51) ε NP ( H ) < ε SP ( H ) w NP ( H ) >w SP ( H ) ε NP ( L ) > ε SP ( L ) w NP ( L ) <w SP ( L ), were w SP () =g()f (, ) + (1 g()) f(ε)f (, ε x(ε, ))dε. Ten, we ave u NP (c( H ) P ) <u SP (c( H ) P ) <u SP (c( L ) P ) <u NP (c( L ) P ), 36

39 were te second inequality follows from (54). Tis result, in combination wit (51) implies u NP (c( L ) P )[ F 2 ( L, ε NP ( L ))] >u NP (c( H ) P )[ F 2 ( H, ε NP ( H ))], a contradiction to (21). Proposition 13 Proof. Is by backward induction. Trivially e T () =. In period T, since bot policies are in place, te wage and ealt insurance premium of every ouseold is independent of. Tus v T () =u(w T P T )=v T and terefore te terminal value function is independent of. Now suppose for a given time period t te value function v t+1 is independent of. Ten from te first order condition wit respect to e t () weave q (e t ()) = βv Q( ;, e) t+1 e But since for every e and every, Q( ;, e) is a probability measure over we ave tus e t (, γ) = for all, on account of our assumptions on q (.). But ten { } v t () =u(w t P t )+ +βv t+1 Q( ;, ) = u(w t P t )+βv t+1 = v t Q( ;,e) e = and since Q( ;, ) = 1 for all. Tus v t is independent of. Te evolution of te ealt distributions follows from (23), and given tese ealt distributions wages and ealt insurance premia are given by (39) and (41). B Furter Analysis of te No-Wage Discrimination Case B.1 Healt Insurance Distortions wit No-Wage Discrimination Te firm s break-even condition is { g()f (, ) + (1 g()) } f(ε)[f (, ε x NP (ε, ))]dε w() Φ() =, and ence on average te production level of a worker will equal is gross wage. Taking ε w > andδ> as given, workers for wom te wage limits, max, w() w( ) ε w, bind will be paid eiter more or less tan teir production level depending on weter te wage discrimination bound binds from above or below. Te firm will optimally coose to ire less tan te population sare of any ealt type wose wage is above teir production level, and ence some of tese workers will be unemployed. Since we ave assume tat tere is no cost to working and workers pay for teir own insurance, competition over ealt insurance will lead tese workers to increase teir ealt insurance, x(e, ), so tat teir productivity is witin ε w of teir wage w(). In te limit as ε w, tis implies tat w() =g()f (, ) + (1 g()) f(ε)[f (, ε x NP (ε, ))]dε, (52) olds and tey are fully employed, or w() P () =. On te flip side, tere will be excess demand for workers wose expected production is more tan w(), tey will terefore find it optimal to eiter lower teir insurance, and in te limit as ε eiter (52) olds tey or set x(e, ) = if tey end up at corner wit respect to ealt insurance. Assuming tat neiter corner binds, tis implies tat te no-wage discrimination policy will be undone by adjustments in te ealt insurance market. Tis motivated our assumption tat te government will coose to regulate te ealt insurance market to prevent tis outcome as part of te no-wage discrimination policy. 37

40 For ealt types for wic te bounds do not bind, market clearing implies tat wile actuarial fairness implies tat w() =g()f (, ) + (1 g()) P () =(1 g()) f(ε)[f (, ε x NP (ε, ))]dε f(ε)x NP (ε, ))dε. Hence, an efficient ealt insurance contract for tis type will maximize w() P () =w CE () P CE (). Since w CE () P CE () is increasing in, it follows tat te wage bound binds for te lowest and igest ealt types. B.2 No-Wage Discrimination wit Realized Penalties in Equilibrium Here we assume tat te firm must pay a cost for aving wage dispersion conditional on ealt type or for aving te ealt composition of its work force differ from te population average. Te wage variation penalty is assumed to take te form C [w() w()] 2 n(), since ealt type will ave te lowest wage in equilibrium, and were C is te penalty parameter and n() is measure of type workers te firm ires. Note tat wit tis penalty function te penalty will apply to all workers wit ealt >. 43 Te penalty from aving one s composition deviate from te population average is given by [ n() D Φ() ] 2. n() Φ() Since tese penalties are small for small deviations, it will turn out tat penalty costs will be realized in equilibrium. Since bot of tese penalties are real we need to subtract tem from production. We will assume tat tere too te government will regulate te insurance market to prevent workers low ealt status workers raising teir productivity by over-insuring temselves against ealt risks and ig ealt status workers lowering teir productivity by under-insuring temselves. We begin analyzing tis case by assuming tat te penalties for wage discrimination C and iring discrimination D are bot finite and ten we examine te equilibrium in te limit as tey become large. Te firm takes as given te ealt policy of te worker and te equilibrium wage w() and cooses te measure of eac ealt type to ire n() so as to maximize max n() C [ g()[f (, x(,)) x(,)]+(1 g()) [w() w )] 2 n() [ n() Φ() ] 2, n() Φ() ] f(ε)[f (, ε x(ε, )) x(ε, )] dε w() n() 43 If we ave assumed tat te form of te penalty was C [w() w ] 2 ψ()d, were w is te average wage, tis would mean tat low productivity workers are more costly and less productive, wic will discourage iring tem. Hence, wit tis form te low productivity workers will only be employed because of te compositional penalty, wic means tat te iring penalty must bind at te margin. Hence te less tan average productivity workers will be in positive net supply in equilibrium, wic will complicate te analysis because some of tese workers will be employed and some will not be. 38

41 were w is taken ere to mean te lowest wage. Trivially, te firm will want to ire more tan te population sare of any type for wom [ ] N() g()[f (, x(,)) x(,)]+(1 g()) f(ε)[f (, ε x(ε, )) x(ε, )] dε w() C [w() w )] 2 is positive and less tat te population sare if N() is negative. Since all firms sare tis condition, tey will all coose te same relative sares of eac type of worker. Since workers are willing to work so long as w() P () >, it follows tat w() cannot be more tan w if N() is not positive. To see tis note tat tere would be excess supply of type workers and ence te labor market would not clear. Moreover, a firm would rater ire a worker of type at w ε tan for w for ε small. Hence, if w() =w, ten N() = so long as w P () >. Hence, for te labor market to clear for eac ealt type, eiter N() =fortype or N() > but w() P () =. Tis implies te following proposition. Proposition 14 If C and D are positive but finite, and w() P () > for all, ten in equilibrium all ouseolds are ired, all firms are representative, and te wage w() is equal to a worker s productivity less te cost of paying im. Since te government can set x(ε, ) = wic implies tat P () =, we assume tat w() P () > for all ealt types. B.3 Realized Penalties wit Bot Policies Since all tat workers care about is teir net wage w(), wic is also equal to teir consumption, it follows tat workers are indifferent over contracts tat offer combinations of a gross wage w() and medical costs P () forwic w() =w() P () is constant. Hence, it is natural to assume tat te firm takes te equilibrium net wage function w() as given and cooses te measure of eac ealt type to ire, n(), and its ealt plan, x(ε, ), to solve te following problem max n(),x(ε,) C [ g()[f (, x(,)) x(,)]+(1 g()) [ w() w()] 2 n() [ n() D Φ() ] 2. n() Φ() ] f(ε)[f (, ε x(ε, )) x(ε, )] dε w() n() Proposition 15 If C and D are positive but finite, ten in equilibrium all ouseolds are ired, all firms are representative, te net wage w() is equal to a worker s productivity less te cost of paying im more tan w(), and w() = w CE () P (). Te firm optimally sets x(ε, ) =x CE (ε, ). As C, w() w(). Proof. Te optimality condition for x(, ε) ifε =is F (, x(,)) 1 and if ε>is F (, ε x(ε, )) 1 w. equality if x(ε, ) >. Tese are te same conditions as in te competitive equilibrium. Next, we sow tat w() as to be increasing in and ence w() is te lowest paid type. Te wage penalty is w.r.t. to te lowest paid worker type, wic we denote by w. Given tat optimum insurance is te same as in te competitive equilibrium, it follows tat te net earnings per worker is w CE () P CE () w(), and from before w CE () P CE () is increasing in. Hence, for te firm to break even [ w CE () P CE () w() ] n() C [ w() w ] 2 n() [ n() D Φ() ] 2 =, n() Φ() 39

42 and te optimality condition for n() is [ w CE () P CE () w() ] C [ w() w ] 2 [ n() D n() Φ() ][ 1 n() ] Φ() n() 1 n() =. Tis condition implies tat a firm will ire more tat te population sare of any type for wom Ñ() w CE () P CE () w() C [ w() w ] 2 >, and less tan te population sare if te reverse is true. However any ealt type tat are not fully employed in equilibrium would ave excess members wo would be appy to be ired any positive wage. Hence, eiter type is paid te lowest equilibrium wage or tey are fully employed. Hence, any type for wom w() >w are fully employed. Any type receiving te lowest wage must be fully employed since te firm would be willing to ire more of tese workers if we lowered te bottom wage by ε. Since all workers are fully employed, it follows tat all firms will coose to be representative to avoid te iring penalty, and tat w() = w CE () = w and w() is increasing. Finally, since te marginal penalty for a deviation in a type s net wage from te economy-wide lowest type s wage is given by C [ w() w()] 2, and since tis cost goes to infinity as C for any positive wage gap, it follows tat as C becomes large w() w(), and all of te workers are paid as if tey were te lowest ealt status type and all of teir productivity gap is absorbed by te cost of discriminating on wages. Q.E.D. Te fact tat te productivity advantage of iger ealt status individuals is completely absorbed by te discrimination costs means tat te society as a wole gets no gain from teir productivity advantage. So te ealt expenditures tat raise teir productivity above te lowest type are inefficient. In addition, expenditure on te lowest ealt type relaxes te wage discrimination penalty on oter types. So tis equilibrium outcome is not socially efficient. C Wages in te Competitive Equilibrium To understand te implications of proposition 6 for te beavior of equilibrium wages, note tat our results imply tat te equilibrium competitive wage is given by Hence CE () w CE () = g()f (, ) + (1 g()) f(ε)f (, ε x(ε, ))dε +(1 g()) f(ε)f (, ε CE ())dε. ε CE () dw CE () d [ = g F (, ) ] ε CE () f(ε)f (, ε x(ε, ))dε () ε CE () f(ε)f (, εce ())dε CE () +g()f 1 (, ) + (1 g()) f(ε)f 1 (, ε x(ε, ))dε +(1 g()) f(ε)f 1 (, ε CE ())dε +(1 g()) ε CE () ε CE () f(ε)f 2 (, ε CE ()) d εce () dε, d since net effect of te cange in te integrand bounds generated by d εce () d optimality condition for ε CE (), (17), implies tat F 12 (, ε CE ())d + F 22 (, ε CE ())d ε CE () =, is zero. Next note tat our 4

43 and ence d ε CE () d Tis result, along wit (17), implies tat dw CE () d = F 12(, ε CE ()) F 22 (, ε CE ()). [ = g F (, ) ] ε CE () f(ε)f (, ε x(ε, ))dε () ε CE () f(ε)f (, εce ())dε CE () +g()f 1 (, ) + (1 g()) f(ε)f 1 (, ε x(ε, ))dε +(1 g()) f(ε)f 1 (, ε CE ())dε (1 g()) ε CE () ε CE () f(ε)f 2 (, ε CE ()) F 12(, ε CE ()) F 22 (, ε CE ()) dε. (53) All of te terms in (53) are trivially positive except te last, wic is negative since F 22 <. However, so long as te spillover ratio F 12 /F 22 evaluated at (, ε CE ()) is not too negative ten, ten wages will vary positive wit ealt status. Note tat tis is trivially implied if te direct effect of te cange in ealt status offsets te spillover, or F 1 (, ε CE ()) F 2 (, ε CE ()) F 12(, ε CE ()) F 22 (, ε CE >. (54) ()) Note tat tis is a condition purely on te fundamentals of te economy since ε CE () isgivenbyan (implicit) equation tat depends only on exogenous model elements. We summarize our results in te following proposition: Proposition 16 Te competitive wage is increasing in if (53) is positive. D Computation of te Social Planner Problem Te idea to solve te problems in (25) is to iterate on sequences {c t,e t (), Φ t ()}, using te first order condition (26) for te optimal effort coice and te envelope condition (27). To initialize te iterations, note tat V T (Φ T ) = u(c T ) [ V T (Φ T ) = u (c T ) g()f (, ) + (1 g()) f(ε) [ F (, ε x SP (ε, )) x SP (ε, ) ] ] dε Φ T () ε u (c T ) Ψ() (55) For tese expressions we only need to know c T, te term Ψ() is just a number tat depends on and is known once we ave solved te static insurance problem. Tis suggests te following algoritm to solve te dynamic social planner problem: Algoritm Guess a sequence {c t } T t= 2. Determine VT (ΦT ) Φ T () from (55) 3. Iterate on t to determine {e t ()} T 1 t= (a) For given Vt+1(Φt+1) Φ t+1( ) (b) Use c t,e t (), Vt+1(Φt+1) Φ t+1( ) use (26) to determine e t (). and (27) to determine Vt(Φt) Φ t() 41

44 4. Use te initial distribution Φ and {e t ()} T t= 1 to determine {Φ t} T t= and tus {c new t 5. If {c new t } T t= = {c t } T t= we are done. If not, set {c t } T t= = {c new t } T t= andgoto1. Tis algoritm is straigtforward to implement numerically, since we only ave to iterate on te aggregate consumption sequence, not on te sequence of distributions. In particular, te only moderately costly operation comes in step 2a) but even tere we only ave to solve one nonlinear equation in one unknown (altoug we ave to do it T card(h) times per iteration). } T t=. E Computation of te Equilibrium wit a No-Prior-Conditions Law and/or a No-Wage Discrimination Law Te algoritm to solve tis version of te model sares its basic features wit tat for te social planner problem, but differs in terms of te sequence of variables on wic we iterate: Algoritm Guess a sequence 44 {Eu t,p t } T t=. 2. Given te guess use equations (3)-(33) to determine ealt cutoffs and wages { ε NP t (),w t ()}. 3. Given {w t (),P t }, solve te ouseold dynamic programming problem (34) for a sequence of optimal effort policies {e t ()} T t=. 4. From te initial ealt distribution Φ use te effort functions {e t ()} T t= to derive te sequence of ealt distributions {Φ t } T t= from equation (23). 5. Obtain a new sequence {Eu new t,pt new } T t= from (32) and (33). 6. If {Eu new t,p new t } T t= = {Eu t,p t } T t= we are done. If not, go to step 1. wit new guess {Eu new t,pt new } T t=. Te algoritm for no-wage discrimination is a sligt modification of tat for no-prior conditions. Te algoritm iterates over {Eu t,w t } T t=. In Step 1 given te guess use equations (36)-(4) to determine ealt cutoffs and premia { ε NP t (),P t ()}. In Step 4 obtain a new sequence {Eu new t,wt new } T t= from (39) and (38). Wit bot policies, equation (41) replaces (4) in all expressions. F Details for Data and Calibration F.1 Details of te Augmented Model Analysis: Inclusion of te z-sock We assume tat ouseolds must incur te cost z, wen te z-sock its. Tis assumption and te fact tat ouseolds are risk averse imply tat te z-sock will be fully insured in te competitive equilibrium under any policy (and of course by te social planner). Moreover, we assume tat ouseolds receiving a z-sock can still work, but tat teir productivity is only ρ times tat of a ealty worker. Terefore, in a competitive equilibrium, te wage of a worker wit ealt status is given by w() =g()f (, ) + ρκ()f (, ) + (1 g() κ()) F (, ε x(ε, ))f(ε)dε and te ealt insurance premium is determined as P () =(1 g() κ()) x(ε, )f(ε)dε + μ z () Given our assumptions tere is no interaction between te z-socks and te ealt insurance contract problem associated wit te ε-sock since it is proibitively costly by assumption not to bear te z-expenditures. Te 44 Instead of {Eu t} one could iterate on {w t()} wic is more transparent, but significantly increases te dimensionality of te problem. 42

45 role of te z-expenditures is to soak up te most extreme ealt expenditures observed in te data associated wit catastropic illnesses, but to oterwise leave our teory from te previous sections unaffected. Te static analysis goes troug completely uncanged in te presence of te z-socks. In te dynamic analysis te benefits of iger effort e and tus a better ealt distribution Φ t () now also include a lower probability κ() of receiving a positive z-sock and a lower mean expenditure μ z () fromtatsockwit better ealt. Tis extension of te model leads to straigtforward extensions of te expressions derived in te analysis of te dynamic model in section 4, and does not cange any of te teoretical properties derived in sections 3 and 4. F.2 Descriptive Statistics of te PSID Data Before we proceed to descriptive statistics of te PSID data, we summarize, in Table 3, te mapping between variables in our model and data. Table 3: Mapping between Data and Model Model Description Data PSID Variable Actual Data Used x, μ z Medical Expenditure Average of total expenditure reported in 1999, 21, w Earning Average of total labor income reported in 1999, 21, ,2,22 Healt Status Self-reported Healt in Since our model period is six years, we take average of reported medical expenditure and wages over six year periods tat we observe. Moreover, we use ealt status data from 1997 (rater tan 1999) to capture te effect of ealt on wages and medical expenditure. Table 4 documents descriptive statistics of key variables from te 1999 PSID data tat we use in our analysis. Table 4: Descriptive Statistics of Key Variables in PSID Mean Std. Dev. Min Max Age Labor Income 3,17 4,573 1,153,588 if Labor Income > 32,76 41, ,153,588 Excellent 38,755 55,46 94,84 Very Good 32,768 4,351 1,153,588 Good 25,516 25,98 384,783 Fair 12,65 13,926 81,3 Medical Expenditure 1,513 4, ,815 Excellent 1,234 2,374 28,983 Very Good 1,647 5, ,815 Good 1,486 4,283 93,298 Fair 1,792 4,95 65,665 Healt Status Pysical Activity: fraction(number) of days in a year Ligt.63 (23.99).39 (142.28) 1 (365) Heavy.29 (15.69).35 (126.85) 1 (365) In te PSID, eac individual (ead of ouseold) self-reports is ealt status in a 1 to 5 scale, were 1 is Excellent, 2, Very Good, 3, Good, 4, Fair, and 5 is Poor. Even wit large number of observations, only about 1% of total individuals report teir ealt status to be poor. Tus, for our analysis, we will use four 43

46 levels of ealt status (merge poor and fair togeter). 45 Since PSID reports ouseold medical expenditure, we control for family size using modified OECD equivalence scale. 46 As we model working-age population, eac ouseold starts is life as a 24 year old and makes economic decisions until e is 65 years old. Our model time period is 6 years and tus tey live for 7 time periods. We coose six year time period to capture te effect of exercises on ealt transition. Since exercises tend to ave positive longer-term effects tan do medical expenditure, by allowing for a medium-term time period, we are able to quantify te impact of exercises in a more reliable way. Data on Healt Transitions Table 5 presents te transition matrix of ealt status over six years. We see tat ealt status is quite persistent. Table 5: Healt Transition over 6 years Excellent Very Good Good Fair Total Excellent 1, , % % 12.8 % 3.52 % 1 % Very Good 482 1,844 1, , % % % 7.18 % 1 % Good , , % % 5.9 % 2.36 % 1 % Fair , % 7.47 % % % 1 % Total 1,991 3,569 3,52 1,96 11, % % 31.77% % 1 % Pysical Activity Data Here, we report some statistics on pysical activity. Variation of Pysical Activity and Its Impact on Healt Transition Density of ligt and eavy pysical activity levels by ealt are summarized in Figures 15 and 16. From variations in ealt evolution by pysical activity and initial ealt status, we find tat about 3% of variance in ealt status in te future is explained by ealt status today, wereas, ligt and pysical activity explains about 8% and 14%, respectively. Moreover, bot initial ealt status and ligt (eavy) exercise explains 46% (41%) of variance in future ealt outcome. 47 Pysical Activity Over Time Ligt pysical activity as steadily decreased over time, wereas eavy pysical activity decreased for a wile, but started increasing in 25 (Figures 17 and 18). F.3 Healt Socks, Distribution of Medical Expenditures, and Discussion of Categorization of Healt Socks Before going into discussing te medical expenditure distribution in data, we briefly discuss te appropriate counterparts of data moments for our model. In our model, ouseolds do not consume medical care wen tey do not get a ealt sock (altoug, tey can coose not to spend any in case of ealt sock, since x (, ε) =max{, ε()}). Terefore, in data, we are interested in te distribution of medical expenditure conditional on aving gotten any ealt socks (wic we ave some information in PSID). 45 Labor income and medical expenditure data for fair ealt in Table 4 include poor (5) in data. 46 Eac additional adult gets te weigt of.5, and eac cild, From te law of total variance, we know var(y )=E(var(Y X)) + var (E(Y X)), were te former is te unexplained and te latter, explained component of te variance. 44

47 Figure 15: Density of Ligt Pysical Activity Figure 16: Density of Heavy Pysical Activity CDF of Ligt Pysical Activity: 1999 vs CDF of Heavy Pysical Activity: 1999 vs. 23 vs F(x).5 F(x) Ligt Pysical Activity (fraction of days in a year) Heavy Pysical Activity (fraction of days in a year) Figure 17: Ligt Activity Figure 18: Heavy Activity Table 6 summarizes medical expenditure by sock. Note tat all numbers reported are yearly average taken over six years ( ). We see tat cancer, eart attack, and eart disease incur te most medical expenditure, and tus we categorize tem to be catastropic socks (z-socks). Altoug te diseases PSID specifically reports information on are tose tat are common, tey are not, by all means, exaustive of te kind of ealt diseases tat one can be diagnosed wit. And tis is inted wen we look at te medical expenditure statistics for tose wo report to ave missed work due to illness. Te maximum amount of medical expenditure tey spend exceeds tose of te oters, and tis migt be due to some severe diseases for wic tey ad to be treated. Terefore, in addition to cancer, eart attack, and eart disease, we categorize tose wo ave spent more tan teir labor income on medical expenditure as aving ad a catastropic (z) ealtsock. 48 Tose wo ad a ealt sock tat were not cancer, eart attack, or eart disease, and wo spent less tan teir income on medical expenditure is considered to ave ad an ε-sock. 49 Figures 19-22, plot logs of medical expenditure distribution for all population, for tose wit ANY ealt sock, tose wit z-sock, and tose wit ε-sock. By definition, mean medical expenditure of z-sock ouseolds are iger tan tose of ε-sock, and so are standard deviations. 48 Categorizing catastropic ealt socks using expenditures as percentage of income is not new. Tere as been discussion on insuring catastropic ealt socks, and tey mostly refer to ig amount of expenditure as percentage of income. 49 In PSID sample, median of percentage of labor income spent on medical expenditure is 2%, and te mean, 132%. Only about 5% of ouseolds wit ealt socks spend medical expenditure in excess of teir labor income. 45

48 Table 6: Average Medical Expenditure by Healt Sock Categories Obs Mean Std. Dev. Min Max All 4,226 1,513 4, ,815 No Sock 1,419 1,35 4,447 11,952 Any Sock 2,87 1,595 4,71 127,815 Catastropic Disease Sock 168 3,745 9,363 93,298 Cancer 51 5,21 15,134 93,298 Heart Attack 46 3,334 4,75 27,161 Heart Disease 94 3,382 5,535 38,5 Ligt Sock 2,767 1,585 4, ,815 Diabetes 183 2,88 7,196 93,298 Stroke 33 2,2 4,95 27,161 Artritis 322 1,684 3,166 38,5 Hypertension 566 1,825 6,143 93,298 Lung Disease 63 1,75 2,476 12,595 Astma 61 1,135 1,444 7,17 Ill 2,351 1,637 5,4 127,815 z-sock 297 4,74 12, ,815 ε-sock 2,51 1,227 2,23 32,99 Figure 19: Average Medical Expenditure Distribution Figure 2: Average Expenditure wit Healt Sock Figure 21: Average Expenditure w/ z-sock Figure 22: Average Expenditure w/ ε-sock F.4 Estimation Results Healt Transition Using te functional form described in te main body of te paper, we estimate te ealt transition function in te following way. 46