Household Welfare, Precautionary Saving, and Social Insurance under Multiple Sources of Risk

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Household Welfare, Precautionary Saving, and Social Insurance under Multiple Sources of Risk Ivan Vidangos 2009-14 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Household Welfare, Precautionary Saving, and Social Insurance under Multiple Sources of Risk (cid:3) Ivan Vidangos y Federal Reserve Board November 6, 2008 Abstract This paper assesses the quantitative importance of a number of sources of income risk for household welfare and precautionary saving. To that end I construct a lifecycle consumption model in which household income is subject to shocks associated with disability, health, unemployment, job changes, wages, work hours, and a residual component of household income. I use PSID data to estimate the key processes that drive and a(cid:11)ect household income, and then use the consumption model to: (i) quantify the welfare value to consumers of providing full, actuarially fair insurance against each source of risk and (ii) measure the contribution of each type of shock to the accumulation of precautionary savings. I (cid:12)nd that the value of fully insuring disability, health, and unemployment shocks is extremely small (well below 1/10 of 1% oflifetimeconsumptioninthebaselinemodel). Thegainsfrominsuringshockstothewageand totheresidualcomponentofhouseholdincomearesigni(cid:12)cantlylarger(above1%and2%oflifetimeconsumption,respectively). Thesetwoshocksaccountformorethan60%ofprecautionary wealth. (cid:3)I am indebted to Joe Altonji for his continuous support and encouragement. I would also like to thank Tony Smith, Fabian Lange, George Hall, Giuseppe Moscarini, Luigi Pistaferri, Bjoern Bruegemann, Melissa Tartari, and MaryDalyforhelpfuldiscussions. Iamresponsibleforallremainingshortcomingsofthepaper. Theviewsexpressed herearemyownanddonotnecessarilyrepresentthoseoftheBoardofGovernorsorthesta(cid:11)oftheFederalReserve System. yE-mail: ivan.vidangos@frb.gov. 1

1 Introduction A great deal of attention has been devoted to modeling dynamics and measuring risk in individual and household income. On one front, there is a substantial empirical literature in labor economics which has studied the dynamic properties of labor income.1 On a di(cid:11)erent front, a large number of studies in macroeconomics and public (cid:12)nance, which require the speci(cid:12)cation of an income process and measures of income uncertainty, have used a variety of strategies to quantify income risk.2 In spite of these large bodies of work, surprisingly little is known about the speci(cid:12)c sources underlying income risk and their importance for households’ economic well-being. Almost all existing studies model income risk by means of one or two statistical innovations in a univariate time-series income process. Such innovations are di(cid:14)cult to interpret since they capture variation in income that may be due to a number of rather di(cid:11)erent factors including unemployment, illness, job changes, unexpected changes in wages and hours of work, etc. It is impossible to tell from such formulations which of the many possible sources drive the unexplained variation in income, or what the relative importance of the di(cid:11)erent sources is. Yet, identifying the sources of variation and risk is essential for addressing questions about insurance of risk, since most insurance programs{ whether public or private{typically address speci(cid:12)c risks, as in the case of unemployment insurance and disability insurance. Identifying the various sources of risk and quantifying their e(cid:11)ects on household welfare necessitates extending the usual treatment and modeling of income. First, we need to explicitly account for the fact that households face a large number of risks which are likely to di(cid:11)er in key aspectssuchastheirpredictability, insurability, relativeimportanceoverthelifecycle, etc. Second, one needs to recognize that households can a(cid:11)ect their income by adjusting their behavior along a numberof margins, including theirlabor supply, saving, job-search e(cid:11)ort, andtiming ofretirement. Unfortunately, introducing both multiple risks and multiple choice variables into a decision model as the one considered here makes the computational cost of solving the model too burdensome. 1ThisliteratureincludesLillardandWillis(1978),LillardandWeiss(1979),Hause(1980),MaCurdy(1982),Hall and Mishkin (1982), Abowd and Card (1987, 1989), Topel (1991), Baker (1997), Geweke and Keane (2000), Haider (2001), Meghir and Pistaferri (2004). 2Models that require a measure of income uncertainty or risk have been used to study a wide range of issues, includingthefollowing. Inmacroeconomics: consumption and precautionary saving (Carroll,1992,1997;Gourinchas and Parker, 2002; Cagetti, 2003); the distribution of wealth and consumption (Huggett, 1996; Krusell and Smith, 1998;Castan~eda,D(cid:19)(cid:16)az-Gim(cid:19)enezandR(cid:19)(cid:16)os-Rull,2003;Storesletten,Telmer,andYaron,2004a);asset pricing (Heaton andLucas,1996;KrusellandSmith,1997;Storesletten,Telmer,andYaron,2007). Inpublic(cid:12)nance: theadequacyof private saving (Engen,Gale,andUccello,1999;Scholz,Seshadri,andKhitatrakun,2006);tax-sheltered accounts and saving (EngenandGale,1993;Engen,Gale,andScholz,1994);wealth accumulation (Hubbard,Skinner,andZeldes, 1994, 1995; Dynan, Skinner, and Zeldes, 2004). 2

This paper thus follows most previous studies in treating income as exogenous, and advances the existing literature by considering a signi(cid:12)cantly richer speci(cid:12)cation of income risk. Morespeci(cid:12)cally, Iproposealifecycleconsumptionmodelinwhichhouseholdsoptimallychoose consumptionandsaving,andwherehouseholdincomeissubjecttoshocksassociatedwithdisability, health, unemployment, job changes, wages, hours, and a residual component of household income.3 Thespeci(cid:12)cationallowsforcomplexdynamicrelationshipswhicharedi(cid:14)culttocaptureinasimple reduced-formapproach. Anunemploymentshock,forinstance,a(cid:11)ectshouseholdincomeviaseveral channels: it has a direct negative e(cid:11)ect on current earnings operating through work hours; it also has a positive e(cid:11)ect on household income which re(cid:13)ects both the labor supply response of other household members as well as unemployment insurance bene(cid:12)ts; it a(cid:11)ects the new wage of the worker upon (cid:12)nding a new job after the spell of unemployment; and (cid:12)nally, it a(cid:11)ects social security bene(cid:12)ts during retirement through the e(cid:11)ect on lifetime average earnings. I use PSID data to estimate the processes which govern the joint evolution of variables that driveanda(cid:11)ectincome. Estimationiscomplicatedbythepresenceofbothdiscreteandcontinuous variables, state dependence in several equations, and the need to control for unobserved heterogeneityinallequations. Iaddresstheseissuesbyusingavarietyofestimationtechniquesincluding generalized indirect inference.4 The parameterized model is then used to quantify the welfare value to consumers (in terms of the equivalent variation in lifetime consumption) of providing full, actuarially fair insurance against each of the sources of uninsured income risk, and to measure the contribution of each source of uncertainty to the accumulation of precautionary savings.5 Veryfewpreviousstudieshaveanalyzedthewelfaree(cid:11)ectsofmultiplesourcesofriskinauni(cid:12)ed framework. One exception is Low, Meghir, and Pistaferri (2006), who consider a consumption model with endogenous participation and job-mobility decisions, and two sources of uncertainty: unemployment and wage risk. They (cid:12)nd that wage risk has important welfare e(cid:11)ects but that unemployment risk does not. Onerelatedlineofresearchhasexaminedavarietyofmechanismstoinsurehouseholdconsump- 3I also consider out-of-pocket medical expenditure shocks. However, a proper treatment of these shocks requires introducing means-tested transfers, and this complicates the solution of the consumption model signi(cid:12)cantly. The current version of the model thus makes the simplifying assumption that medical expenditures can wipe out current income but cannot wipe out accumulated wealth, and excludes means-tested transfers. This simpli(cid:12)es the solution of the model but it also restricts the potential importance of medical expenditures signi(cid:12)cantly. Hence, the results relatedtomedicalexpendituresareviewedhereasverypreliminaryandareexcludedfrommuchoftheanalysisand discussion. A full treatment, including means-tested transfers, is left for future research. 4TheimplementationofgeneralizedindirectinferenceinthispaperbuildsontheimplementationusedinAltonji, Smith, and Vidangos (2008). 5The insurance considered here is additional to already-existing insurance provided by government transfers, self-insurance through saving, and insurance within the family. This will be further discussed below. 3

tion, includingunemploymentinsurance(Gruber, 1997; BrowningandCrossley, 2001), foodstamps (Blundell and Pistaferri, 2003), and Medicaid (Gruber and Yelowitz, 1999).6 Some of the empirical studies in this literature have looked at the e(cid:11)ects of speci(cid:12)c income shocks and of speci(cid:12)c forms of insurance (such as unemployment and unemployment insurance) on the levels of consumption. One important di(cid:11)erence between this paper and those studies is that this paper captures, and focuses on, the uncertainty in consumption introduced by speci(cid:12)c sources of income risk, rather than on the levels of consumption. My main (cid:12)ndings are as follows. The welfare gains from fully insuring disability, health, and unemployment shocks are extremely small (well below 1/10 of 1% of lifetime consumption in the baseline model). The gains from insuring wage shocks and the additional component of household income are signi(cid:12)cantly larger (above 1% and 2% of lifetime consumption, respectively). These two shocks account for more than 60% of precautionary wealth. The paper is organized as follows. The next section presents the lifecycle consumption model and discusses its implementation. Section 3 describes the data. Section 4 presents estimation results and an evaluation of the (cid:12)t of the model. Section 5 discusses the solution of the parameterized lifecycle model. Section 6 presents the welfare and precautionary saving analysis, and section 7 concludes. 6Other mechanisms that have been studied include spousal earnings (Cullen and Gruber, 2000; Stephens, 2002), thedelayofdurablegoodspurchases(BrowningandCrossley,2001),mortgagere(cid:12)nancing(HurstandSta(cid:11)ord,2004), progressive income taxation (Kniesner and Ziliak, 2002), and unsecured debt (Sullivan, 2008). 4

2 Model 2.1 Overview This section introduces a lifecycle model of consumption in which households face multiple income shocks. The decision unit is the household and a period corresponds to one year. Households are part of the labor force for T years, retire at an exogenously speci(cid:12)ed date, and live in retirement W foruptoT years.7 Yearsinthemodelareindexedbythevariablet, wheret = 1indicatesthe(cid:12)rst R period of a worker’s career. During working years, t thus represents potential experience, although sometimes t will also be referred to as age, especially when refering to the retirement years. After retirement, households face mortality risk. Thechoiceproblemfacingthehouseholdisstandard: everyperiod,householdschooseconsumption and saving with the objective of maximizing expected discounted utility over their remaining periods of life: E [ T (cid:12)s t(cid:25) u(c )], where E [ ] is the expectations operator conditional on infort s=t (cid:0) s s t (cid:1) mation available inPperiod t, (cid:12) 1 is the discount factor, (cid:25) t are conditional survival probabilities8, (cid:20) u is the per-period utility function satisfying u > 0, u 0, and lim u(c ) = , c is household 0 00 c 0 0 t t (cid:20) ! 1 consumption, and T = T +T is the maximum age attainable. The household is assumed to W R have no bequest motives. The dynamic budget constraint is z = (1+r)(z c )+y , where z is cash on hand, r is t+1 t t t+1 t (cid:0) therealinterestrate,andy istotalhouseholdnonasset income. Eachperiod,householdsreceive t+1 an exogenous stream of nonasset income y . During the working years, y should be thought of t t as encompassing all labor and transfer income of the household.9 During the retirement years, y t consists of social security bene(cid:12)ts.10 Nonasset income depends on a number of stochastic variables and shocks: y = f(s ;" ), where " is a vector of shocks and s is a vector of state t+1 t+1 1;t+1 1;t+1 t+1 variables which describe household characteristics such as health, employment, wage rate, etc. State variables s evolve stochastically over time according to the law of motion s = g(s ;" ), t t+1 t 2;t+1 where " is a vector of shocks. I next describe separately, and in more detail, the problem faced 2;t+1 by the household in the working years and in the retirement years. 7In implementing the model, T and T will be set to 43 and 25, respectively. W R 8Survivalprobabilitiesmaybeallowedtodependnotonlyonage,butalsoonanyotherstatevariableconsidered in the model, such as health status. 9The appropriate measure of income to use here is after-tax income. The current version of the model does not distinguishbetweenpre-taxandafter-taxincome. Futureversionswillmakethisdistinction,anduseafter-taxincome, by introducing estimated tax functions that approximate e(cid:11)ective income taxes. Such tax functions are estimated, for instance, by Gouveia and Strauss (1994, 2000) for the years 1966-1989. 10One could also allow y to include a de(cid:12)ned bene(cid:12)t pension. t 5

2.2 Working Years During the household’s working years, the state variables that a(cid:11)ect income and are included in vectors are: (i)anindicatorofdisabilityD ; (ii)anindicatorofotherhealthlimitationsH ; (iii)an t t t indicatorofemploymentstatus E ; (iv)a persistent component ofthe wage pw; and(v)a persistent t t y component of household income p , which captures all residual variation in household income not t explainedbyearningsoftheheadorstatevariablesD , H , andE . Thisresidualcomponentturns t t t out to play a large role in explaining the variation of household income in the PSID.11 Households in the model take into account their current health status, employment status, and non-transitory aspectsofwagesandhouseholdincomewhenformingexpectationsaboutuncertainfuturenonasset income. The household’s decision problem during the working years is: Maximize T E [ (cid:12)s t(cid:25) u(c )] t (cid:0) s s s=t X subject to D = g (t;D ;H ;"D ;(cid:28)) (1) t+1 D t t t+1 H = g (t;D ;D ;H ;"H ;(cid:28)) (2) t+1 H t+1 t t t+1 E = g (t;D ;H ;E ;"EE;"UE;"DE;(cid:28)) (3) t+1 E t+1 t+1 t t+1 t+1 t+1 pw = g (E ;E ;pw;"w ;"J ;(cid:28)) (4) t+1 w t+1 t t t+1 t+1 y y y p = g (p ;" ;(cid:28)) (5) t+1 y t t+1 z = (1+r)(z c )+y (t;D ;H ;E ;pw ;p y ;"J ;"h ;(cid:28)) (6) t+1 t (cid:0) t t+1 t+1 t+1 t+1 t+1 t+1 t+1 t+1 c 0; (z c ) b t t t (cid:21) (cid:0) (cid:21) (cid:0) Above, equation (1) describes the evolution of the indicator of disability D . The value of t+1 D depends on age t, on whether the household (head) was disabled in the previous period D , t+1 t on any other previous-period health limitations H , and on an iid shock "D . Index (cid:28) indicates a t t+1 11After accounting for variation in household income due to earnings of the head, household size and composition (includingvariablesthatmighta(cid:11)ectthelaborsupplyofaspouse,suchasnumberofchildrenunder6),apolynomial inage,education,race,yearindicators,disability,health,employment,andpermanentheterogeneitytheresidualthat remainsinhouseholdincomeislargeandpersistent. Thiscomponentisrepresentedbypy andismodeledhereasan t AR(1) process. This component consists primarily of unexplained variation in spousal labor earnings and transfers from public programs. 6

speci(cid:12)c household type.12 Equation (2) describes the evolution of the health limitations indicator H , which depends on age, lagged disability, lagged health, and an iid shock "H . In addition, t+1 t+1 H dependsoncurrentdisability, sinceH isde(cid:12)nedtobe1wheneverD equals1. Equation t+1 t+1 t+1 (3) describes the evolution of employment status E , which depends on potential experience t, t+1 current disability status, current health limitations, previous-period employment status, and a set of iid shocks.13 Equation (4) determines the evolution of the persistent wage component pw , which depends t+1 on current and previous-period employment status (i.e., on the employment transition between the two periods), on its own lagged value, on an iid shock "w , and on whether there was a job change t+1 between periods t and t 1.14 Equation (5) shows the dependence of the persistent component p y (cid:0) t+1 on its own lagged value and a stochastic shock. Equation (6) describes the evolution of cash on hand, making explicit that the value of household nonasset income y depends on the realizations t+1 of variables t, D , H , E , pw , p y , "J , and "h .15 Finally, households cannot borrow t+1 t+1 t+1 t+1 t+1 t+1 t+1 more than amount b in any given period. 2.3 Retirement Years In all periods following retirement labor income is zero. Retired households receive nonasset income from social security only. The level of social security bene(cid:12)ts, SS , is determined in the t+1 last year of work, according to the formula SS = PIA(ALE(D ;H ;E ;pw;p y )); (7) t+1 t t t t t where PIA stands for principal insurance amount and ALE stands for average lifetime earnings. In the last working year, state variables D , H , E , pw, and p y are used to predict average lifetime t t t t t earnings. PredictedALE arethenused, alongwiththerulesoftheSocialSecurityAdministration, to determine the PIA. Households are assumed to receive a level of bene(cid:12)ts equal to their PIA. Details of the calculation of both ALE and PIA are given in Appendix 2. Once the level of social security bene(cid:12)ts has been determined, it is assumed to stay constant for as long as the household is 12Typescanbede(cid:12)nedaccordingtopermanentobservedcharacteristics(suchaseducationandrace)orunobserved characteristics (such as unobserved permanent heterogeneity in ability or preferences). The baseline model will be evaluated for one speci(cid:12)c household type, which will be de(cid:12)ned below. 13These three shocks will determine the transition into employment from three possible states in the previous period: employment, unemployment, and disability. 14Shock "J is a job-change shock. Section 4 discusses how job changes are determined. t+1 15Some of these variables, in turn, depend on the shocks "D , "H , "EE, "UE, "DE, "w , and "y . t+1 t+1 t+1 t+1 t+1 t+1 t+1 7

alive.16 Households also receive asset income, which is determined endogenously within the model. Retiredhouseholdsinthemodelmayadditionallyfaceout-of-pocketmedicalexpendituresM , t+1 which reduce net income disposable for consumption. The household’s decision problem during the retirement years is thus: Maximize T E [ (cid:12)s t(cid:25) u(c )] t (cid:0) s s s=t X subject to H = g (t;D ;D ;H ;"H ;(cid:28)) (2) t+1 H t+1 t t t+1 pM = g (pM;"M ;(cid:28)) (8) t+1 M t t+1 SS = g (SS ) (9) t+1 S t z = (1+r)(z c )+SS M (t;H ;pM )+I (10) t+1 t t t+1 t+1 t+1 t+1 t+1 (cid:0) (cid:0) Above, equation (8) describes the evolution of pM , the persistent component of out-of-pocket t+1 medical expenditures. Equation (9) describes the evolution of social security bene(cid:12)ts. Finally, equation (10) describes the evolution of cash on hand during retirement, re(cid:13)ecting the fact that exogenous nonasset income now comes from social security bene(cid:12)ts SS and that there may t+1 be out-of-pocket medical expenses M which would reduce income available for consumption. t+1 Additionally, households may receive insurance transfers I . t+1 Transfers for retired households are intended to provide a minimum level of consumption after accounting for medical expenditures. These transfers capture the combined e(cid:11)ects of programs such as Food Stamps, Supplemental Security Income, and Medicaid. A common speci(cid:12)cation for such transfers is I = max 0;c+M [SS +(1+r)(z c )] (see Hubbard, Skinner, and t+1 t+1 t+1 t t f (cid:22) (cid:0) (cid:0) g Zeldes (1994, 1995) and Scholz, Seshadri, and Khitatrakun (2006)). The introduction of medical expenditures and transfers I , however, complicates the solution of the consumption decisions t+1 because they introduce nonconvexities which lead to the existence of multiple local maxima in the solution of the Bellman equation. This problem can be addressed by using a global search in the optimization involved in the solution of the Bellman equation as in Hubbard, Skinner, and Zeldes (1994, 1995), but it increases the computation time required to solve the problem signi(cid:12)cantly. 16In the current version of the model, y during retirement is calibrated using only social security bene(cid:12)ts of the t household head. A later section discusses how this might a(cid:11)ect the analysis and results presented here. 8

This will be left for future research. This version makes the simplifying assumption that medical expenditures can wipe out current income but cannot a(cid:11)ect accumulated wealth. This assumption preserves the concavity of the righ-hand-side of the Bellman equation and guarantees that the unique local maximum is also the global maximum. It also preserves the monotonicity (in wealth) of the consumption policy functions. On the other hand, this simplifying assumption signi(cid:12)cantly restricts the potential importance of medical expenditures. The treatment of medical expenditures hereisthuspreliminaryandtheresultsshouldbeinterpretedwithcaution. Severalstudiessuggest thatmedicalexpendituresareimportantforsavingbehaviorandpotentiallyforwelfare. Examples include Palumbo (1999) and De Nardi, French, and Jones (2006). 2.4 Model Implementation This section discusses two points about implementing the model presented above. The (cid:12)rst point regards index (cid:28). As mentioned earlier, (cid:28) indexes the household type. Types can be de(cid:12)ned based on observed characteristics (education, race) and unobserved characteristics (unobserved permanent heterogeneity). Di(cid:11)erent household types will face di(cid:11)erent processes (di(cid:11)erent parameter values) governing the evolution of the various state variables and income. The parameterized lifecycle model will be evaluated here for one speci(cid:12)c household type: households whose head is white, who have the mean level of education in the PSID sample, and who are at the mean of the distribution of the unobserved permanent heterogeneity components. One important assumption maintained throughout the analysis is that unobserved heterogeneity is known at the beginning of a worker’s career. Under this assumption, these permanent components do not constitute risk (i.e., uncertainty). The second point regards the de(cid:12)nition of the state variables in the model and their correspondence to variables in the PSID. Most of the variables included in the state vector, such as those refering to health or employment, will refer to characteristics of the household head in the PSID. The reason is that these variables are likely to be the most important determinants and predictors of income. Household income, on the other hand, will refer to a household aggregate in the PSID which includes labor income and transfer income from all members of the household. I use this variable to construct predicted income for a household of the average size and composition in the PSID sample. The household income process is estimated using this predicted income measure (which has been purged from variation due to di(cid:11)erences in household size and composition). 9

2.5 Model Parameterization All parameters that appear in the parameterized form of transition equations (1) - (5) and (8) are estimated using PSID data. Estimation is discussed in section 4. On the other hand, model parameters such as the coe(cid:14)cient or relative risk aversion, the real interest rate, and the discount factor are chosen based on values found elsewhere in the literature. The sensitivity of results to alternative assumptions about these parameters is examined in a later section. Preferences are assumed to be of the constant relative risk aversion (CRRA) form, u(c ) = t c1 t(cid:0) (cid:11) (cid:0) 1 , where (cid:11) is the coe(cid:14)cient of relative risk aversion. The baseline model assumes a value 1 (cid:11) (cid:0) of (cid:11) = 3:0.17 The interest rate is assumed to be (cid:12)xed at the value r = 0:0344 (Gourinchas and Parker, 2002). The discount factor (cid:12) is set such that the discount rate equals the interest rate, as in Low, Meghir, and Pistaferri (2006) and many other studies.18 Conditional survival probabilities are obtained from the Life Tables published by the Center for Disease and Control Prevention of the U.S. Department of Health and Human Services. Finally, households in the baseline model are assumed to be credit-constrained (b = 0). 3 Data Estimation is conducted using data from the PSID. I start here by giving a brief description of the key variables. Appendix 1 provides a detailed explanation of all variables used. The disability indicator D equals 1 if an individual is disabled and 0 otherwise. It is constructed from the t respondent’s self-reported employment status at the survey date, where \disabled" is one of the possibleanswersinthequestionnaire. Acurrentlydisabledindividualisbyde(cid:12)nitionnot currently employed. This variable is thus likely to capture severe forms of disability. Indicator H , on the t other hand, is constructed based on the survey question: \Do you have any physical or nervous condition that limits the type of work or the amount of work you can do?" H equals 1 when a t respondent answers \yes", and 0 otherwise. This variable is thus likely to capture both serious and less serious health limitations, including temporary illness and other conditions that a(cid:11)ect work. Additionally, H is set to 1 whenever D equals 1. Employment E , like disability, is based on t t t self-reported employment status at the survey date. It equals 1 for employed and temporarily 17Therangeofvaluesusuallyconsideredempiricallyplausibleis0:5 5:0. GourinchasandParker(2002)estimate (cid:0) (cid:11) to be around 0:5 1:4 in a lifecycle consumption model. Cagetti (2003) obtains considerably higher estimates, (cid:0) around 4:0. Chetty (2006), using a model with consumption and leisure in the utility function, estimates (cid:11) to be around 1:0 and argues that values of (cid:11) above 2:0 are inconsistent with the evidence on labor supply behavior. 18Noticethatmodelswith(cid:12)nite horizondonotfacetherestrictionsontherelativevaluesofthediscountfactorand the real interest rate that in(cid:12)nite horizon models have. Smaller values of the discount factor (greater impatience), however, will lead to less saving. 10

laido(cid:11) workers, and zero for disabled and unemployed individuals. Finally, household income y t includes all labor and transfer income of the household head and, if present, of the spouse and any other family members. As was discussed in the previous section, I construct and use predicted income forahouseholdoftheaveragesizeandcompositioninthePSIDsample, inordertoaccount for heterogeneity in household size and composition which is not present in the consumpion model. Aswillbediscussedbelow,estimationisconductedinfourparts,whereeachofthefollowingfour subsets of equations is estimated separately: (i) disability; (ii) health; (iii) employment, wage, and householdincome;(iv)medicalexpenditures. Somesamplerestrictionsimposedvaryslightlyacross the di(cid:11)erent estimation samples. Estimation of all equations other than medical expenditures uses data from the 1975-1997 PSID waves. Medical expenditures, on the other hand, use the 1999, 2001, and 2003 waves. The reason is that earlier waves did not contain information on medical expenses (note also that interviews have been conducted only every two years since 1997). In all cases, the data include members of both the SRC and SEO samples, as well as nonsample members who married PSID sample members. I consider only households with a male head who is living in the household at the time of the interview. Both single and married individuals are included. I exclude a small fraction of person-year observations in which the head reports being a full-time student or \keeping house" at the time of the interview. These observations are discarded because in the lifecycle model household heads cannot be out of the labor force except when disabled. I also exclude individuals who are missing data on education. Only observations where the head is at least 18 years old are used. Some additional sample restrictions, including restrictions based on potential experience (or age) di(cid:11)er according to the process being estimated. For estimation of the health limitations process, I use observations where potential experience ranges from 1 to 65. The corresponding age range is 18 to 87. This sample has 87;979 observations. Table 1A (bottom panel) displays the number of observations and the mean of the health limitations indicator for di(cid:11)erent levels of potential experience. The column labeled All t refers to all levels of potential experience, while the next columns report values for the ranges of potential experience indicated in the top row. The overall mean of the H indicator is 0:136 and increases from 0:052 for low experience (1 t 10) (cid:20) (cid:20) to 0:497 for high experience (61 < t < 65). Estimation of disability and of the joint process of employment, wage, and household income excludes retired individuals and observations where age exceeds 64 (the resulting range of potential experience is 1 48, with very few values above 45. Table 1A (top panel) displays the number (cid:0) 11

of observations and the mean of the disability indicator for di(cid:11)erent levels of potential experience. The overall mean of the D indicator is 0:022 and increases from 0:022 for 1 t 10 to 0:110 for (cid:20) (cid:20) t 41. (cid:21) In addition, for estimation of the employment, wage, and household income equations (which also uses data on work hours and labor earnings, as will be discussed below), I censor reported work hours at 4000, add 200 to reported hours before taking logs to reduce the impact of very low values on the variation in the logarithm, and I restrict observed wage rates and household income (in levels) to increase by no more than 500% and decrease to no less than 20% of their previousyear values.19 I also censor the wage to be no less than $3.50/hour and household income to be no less than $1,000/year in year-2000 dollars.20 Table 1B displays the number of observations, mean, standard deviation, minimum, and maximum values of all key variables used in the estimation of the joint model of income. The second-to-last row displays the raw household income data, while the last row displays the predicted value for a household of the average size and composition, which is the variable actually used in estimation. Finally, estimation of the out-of-pocket medical expenditures process uses observations where potential experience is above 43, in correspondence with the model’s assumption that medical costs are zero during working years. The PSID variable used here consists of all out-of-pocket payments made by the household over the course of the two years prior to the interview year (see Appendix 1). Table 1C provides summary statistics in thousands of year-2000 dollars. The sample consists of 2,831 observations and has a mean of 3:25, a standard deviation of 10:76, a 99th percentile of 38:63, and a maximum value of 317:47. 4 Parameter Estimation Thissectionpresentstheparametricmodelsspeci(cid:12)edforthevarioustransitionequationsintroduced in section 2 and discusses their estimation. As was mentioned above, the estimation strategy involves estimating the various equations of the model in four parts. Speci(cid:12)cally, the equations for medical expenditures, health limitations, and disability are estimated separately from the rest of the equations (employment, wage rates, and household income). The reasons for estimating these equationsseparatelyarethefollowing: (i) Out-of-Pocket Medical Expenditures: Asdiscussedabove, 19Increases above 500% and declines to less than 20% of the previous value are very uncommon and are likely to represent reporting errors in most cases. 20Given the existence of minimum-wage legislation, reported hourly wages below $3.50/hour are likely to be misreports. Similarly, for a measure of household income as broad as the one used here, very low values of household income are likely to be misreports. 12

estimationofmedicalexpendituresusesPSIDwaves1999, 2001, and2003, whileallotherequations use data from waves 1975-1997. (ii) Health: In the lifecycle model, health appears as a state variable both before and after retirement. Estimation of the health process is therefore based on a samplewhichincludesalllevelsofpotentialexperience(andage). Mostotherstatevariablesinthe consumption model (including disability) relate to the labor market and their estimation therefore excludes high levels of potential experience (and age). (iii) Disability: I initially attempted to estimate the disability process jointly with all other labor market and income processes. However, thefactthatthedisabilityindicatorinthePSIDsampleiszeromorethan97%ofthetimeintroduces numerical instabilities in the implementation of indirect inference.21 Estimating the disability equation individually allows the use of standard maximum likelihood methods and sidesteps the numerical di(cid:14)culties. The following subsections introduce the parametric forms used and discuss estimation as well as estimation results. 4.1 Disability The transition equation for the disability indicator D is assumed to have the following form t D = I[(cid:13)D +(cid:13)D(t+1)+(cid:13)D(t+1)2+(cid:13)D(t+1)3+(cid:13)DD +(cid:13)DH +(cid:16)D +"D > 0]; (11) t+1 0 1 2 3 4 t 5 t t+1 where (cid:16)D is an individual-speci(cid:12)c, permanent component. In estimation, it is assumed that (cid:16)D N(0;(cid:27)2 )and"D N(0;1)soequation(11)isadynamicprobitwithpermanentunobserved (cid:24) (cid:16)D t+1 (cid:24) heterogeneity. The parameters of equation (11) are estimated by maximum likelihood. The unobserved component (cid:16)D is integrated out of the (conditional) likelihood function by Gauss- Hermite quadrature. The initial condition for D is assumed to be zero, since in the PSID sample t D = 0 for all individuals for t = 1. t Table 2A presents estimation results for equation (11). The probability that D = 1 is t+1 monotonically increasing in experience (the slope of the polynomial is positive for t between 1 and 40). The coe(cid:14)cient of (cid:13)^D = 1:611 on D indicates a fairly high degree of state dependence.22 4 t Unobserved heterogeneity, with an estimated standard deviation of (cid:27)^ = 0:934, also plays a (cid:16)D signi(cid:12)cant role. 21See discussion of estimation mechanics in Appendix 4. The introduction of disability leads to (cid:13)at regions near the global optimum of the objective function. 22To get a sense of the magnitude of this coe(cid:14)cient, note that the transitory shock "D in the equation has a t+1 standard deviation of 1. 13

4.2 Health The transition equation for the health indicator is assumed to take the form H = I[(cid:13)H +(cid:13)H(t+1)+(cid:13)H(t+1)2+(cid:13)H(t+1)3+(cid:13)HH +(cid:16)H +"H > 0]; (12) t+1 0 1 2 3 4 t t+1 where (cid:16)H is an individual-speci(cid:12)c permanent component.23 In addition, H is set to 1 whenever t+1 D equals1.24 Theparametersofequation(12)arealsoestimatedbymaximumlikelihood, where t+1 (cid:16)H is assumed to be N(0;(cid:27)2 ) and is integrated out of the conditional likelihood by numerical (cid:24) (cid:16)H quadrature. The equation is estimated on the sample of observations where D = 1 (since H t+1 t+1 6 is de(cid:12)ned to be 1 whenever D equals 1). The initial realization of H is assumed to be t+1 t+1 independent of (cid:16)H 25, but the distribution of H in the model is set such that its mean equals the 1 mean of H in the PSID. 1 Table2Bpresentsestimationresults(whichareconditionalonanindividualnot beingdisabled). The probability that H = 1 increases monotonically with age. Lagged health limitations t+1 have a fairly large and strongly signi(cid:12)cant e(cid:11)ect on the probability of current health limitations ((cid:13)^H = 1:119). Unobserved heterogeneity also plays an important role ((cid:27)^ = 0:961). 4 (cid:16)H 4.3 Employment, Wage Rates, and Household Income Employment, wage rates, and household income are determined jointly by a set of recursive equations. The system also includes an equation for job changes and an equation for work hours. Although job changes and work hours are not state variables in the lifecycle model, they are key variables in the determination of income. Therefore, the lifecycle model includes two shocks which capture job changes and innovations in work hours, respectively. The evolution of job changes and hours, and their e(cid:11)ects on income, are thus estimated as part of the recursive system that drives household income. The following subsection presents the joint model of employment, job changes, wage rates, work hours, and household income. Additional details are provided in Appendix 3. 23SincetheequationsforD andH areestimatedseparately,Idonotestimatethecorrelationbetween(cid:16)D and(cid:16)H. t t This is of no consequence here, however, since I consider households at the mean of the distribution of the various permanent heterogeneity components, which is normalized to zero. The important issue is to control for variation duetopermanentcomponents,bothbecausethisvariationshouldnotrepresentriskandtoavoidbiasincoe(cid:14)cients on lagged dependent variables. 24TheequationcanthusbeexpressedasH =D +(1 D ) I[(cid:13)H+(cid:13)H(t+1)+(cid:13)H(t+1)2+(cid:13)H(t+1)3+ (cid:13)HH +(cid:16)H+"H >0]. t+1 t+1 (cid:0) t+1 (cid:1) 0 1 2 3 4 t t+1 25Methods that deal with the initial-conditions problem such as that proposed in Wooldridge (2005) are not applicable here because of the highly unbalanced nature of the sample. I experimented with correcting for initial conditionsusingthemethodsuggestedinHeckman(1981),butthereduced-formapproximationoftheinitialvalueof thelatentvariableinvariablyshowedverylittleexplanatorypower,andhencewasnothelpful. Alldynamicequations other than equation (12), however, deal with the initial-conditions problem (see Appendix 3). 14

4.3.1 Functional Forms Employment - Employment Transition Conditional on being employed, next-period employment is determined by E = I[(cid:13)EE +(cid:13)EEt+(cid:13)EEt2+(cid:13)EEH +(cid:13)EEED +(cid:16)EE +"EE > 0]: (13) t+1 0 1 2 3 t+1 4 t t+1 Employment-employment transitions are thus determined by a latent variable which depends on a quadratic polynomial in potential experience t, current health limitations H , employment t+1 duration ED , and the error term (cid:16)EE+"EE. The de(cid:12)nition of ED and its treatment is discussed t t+1 t below. The error component (cid:16)EE is an individual-speci(cid:12)c permanent component and "EE is an iid t+1 idiosyncratic shock. It is assumed that (cid:16)EE N(0;(cid:27)2 ) and "EE N(0;1). Component (cid:16)EE (cid:24) (cid:16)EE t+1 (cid:24) is allowed to be correlated with unobserved permanent components in the other equations. The factor structure of the permanent components in the di(cid:11)erent equations is also speci(cid:12)ed below. Unemployment - Employment Transition Conditional on being unemployed, next-period employment is given by E = I[(cid:13)UE +(cid:13)UEt+(cid:13)UEt2+(cid:13)UEH +(cid:13)UEUD +(cid:16)UE +"UE > 0]: (14) t+1 0 1 2 3 t+1 4 t t+1 Transitions from unemployment into employment are thus determined by a latent variable which depends on a quadratic polynomial in potential experience t, current health limitations H , t+1 unemployment duration UD (discussed below), an individual-speci(cid:12)c permanent component (cid:16)UE, t and the iid idiosyncratic shock "UE N(0;1). The term (cid:16)UE is assumed to be N(0;(cid:27)2 ) and t+1 (cid:24) (cid:24) (cid:16)UE may be correlated with the permanent components in the other equations. Disability - Employment Transition Transitions from disability back into employment are rather infrequent in the PSID sample. These transitions are modeled as E = I[(cid:13)DE +"DE > 0] where "DE N(0;1). (15) t+1 0 t+1 t+1 (cid:24) The fact that these transitions are infrequent in the data makes it di(cid:14)cult to estimate their dependence on experience or on permanent unobserved components. 15

Job Changes Conditional on being employed in both t and t+1, the occurrence of a job change between the two periods is determined by J = I[(cid:13)J +(cid:13)Jt+(cid:13)Jt2+(cid:13)JJD +(cid:16)J +"J > 0]; (16) t+1 0 1 2 3 t t+1 whereJD isjobduration(discussedinthenextparagraph),(cid:16)J N(0;(cid:27)2 )isanindividual-speci(cid:12)c t (cid:24) (cid:16)J permanent component, and "J N(0;1) is iid. t+1 (cid:24) Employment-, Unemployment-, and Job Duration Estimation of equations (13), (14), and (16) controls for duration dependence in employment and job mobility by including the variables ED , UD , and JD , respectively. Here, ED is t t t t de(cid:12)ned as the number of consecutive periods that an employed individual has been employed up to period t, UD is the number of consecutive periods that an unemployed individual has been t unemployed up to period t, and JD is the number of periods that an employed individual has t been at their current job. It would be straightforward to introduce ED , UD , and JD as state t t t variables in the lifecycle consumption model from section 2. This is not done here because of the computational burden of the additional state variables. Instead, the approach is to control for duration dependence in estimation, but to set the duration variables to their sample mean (by year of potential experience) when parameterizing the employment and job-change equations. I also experimented with estimating equations (13), (14), and (16) without controlling for duration dependence. In this case the potential-experience pro(cid:12)les of the transition probabilities do not match the data well, but the results for welfare and precautionary saving are not a(cid:11)ected. Wage Equation26 Log wages are assumed to follow the process lnwage = (cid:12)wXw +w ; t+1 t+1 t+1 26The speci(cid:12)cation of the wage process proposed here is similar to one of the speci(cid:12)cations studied in Altonji, Smith, and Vidangos (2008). A more detailed description of the process is provided in that paper. That paper also studies alternative speci(cid:12)cations of the wage process which include job-speci(cid:12)c components. The reason that speci(cid:12)cations with job-speci(cid:12)c wage components are not used in this paper is that the job-speci(cid:12)c wage component would introduce an additional state variable in the consumption model, adding to the computational requirements of its solution. 16

where wage is a latent wage which is equal to the actual wage for employed individuals, but is t+1 also de(cid:12)ned for individuals who are not employed.27 Vector Xw is a vector of exogenous variables t+1 including a polynomial in experience, and w is speci(cid:12)ed as t+1 w = (cid:13)wH +pw : t+1 1 t+1 t+1 In the above equation pw is a persistent component of the wage and is given by t+1 pw = (cid:26) (1+(cid:30) (cid:9) )pw +(cid:13)wJ +(cid:13)w(1 E )+(cid:16)w +(1+(cid:30) (cid:9) ) "w ; (17) t+1 w 1 t+1 t 2 t+1 3 t+1 2 t+1 t+1 (cid:1) (cid:0) (cid:1) (cid:1) where (cid:9) J +E (1 E ) and "w N(0;(cid:27)2). t+1 t+1 t+1 t t+1 w (cid:17) (cid:1) (cid:0) (cid:24) The persistent wage component pw depends on the previous-period (latent) wage via the autoret+1 gressive coe(cid:14)cient (cid:26) . The degree of dependence on pw is allowed to change according to whether w t indicator (cid:9) equals 0 or 1. Indicator (cid:9) equals 1 if either (i) there is a job change between t+1 t+1 periods t and t + 1 or (ii) a worker who is unemployed or disabled at t is reemployed at t + 1. The variance of the shock "w also depends on the value of (cid:9) . This dependence captures the t+1 t+1 increased level of wage uncertainty associated with new jobs, which is present whether the worker was previously employed or not. The term (cid:16)w N(0;(cid:27)2 ) is a person-speci(cid:12)c permanent compo- (cid:24) (cid:16)w nent which is allowed to be correlated with the permanent components present in the employment, job-change, hours, and household income equations.28 A job change and a job loss are also allowed to a(cid:11)ect the mean of the persistent component pw via the coe(cid:14)cients (cid:13)w and (cid:13)w. t+1 2 3 Hours Equation Log hours are assumed to follow the process lnhours = (cid:12)hXh +h ; t+1 t+1 t+1 where Xh is de(cid:12)ned similarly to Xw and h is given by t+1 t+1 t+1 h = (cid:13)h+(cid:13)hE +(cid:13)hw +(cid:13)hD +(cid:13)hH +(cid:16)h+"h : (18) t+1 0 1 t+1 2 t+1 3 t+1 4 t+1 t+1 27For a discussion of the concept of a latent wage used here, see Altonji, Smith, and Vidangos (2008). 28Notice that the permanent component is inside the autoregressive part of the persistent wage component. Consequently, its e(cid:11)ect on the wage may change with t. This speci(cid:12)cation may be thought of as an alternative to the "heterogeneous-pro(cid:12)le" types of models for wages and earnings often used in the literature. For a discussion of the "heterogeneous pro(cid:12)les" literature in earnings dynamics see Baker (1997) or Guvenen (2006). 17

That is, annual hours of work are allowed to depend on employment status at the survey date, the wage rate, disability, and health. The term (cid:16)h is person-speci(cid:12)c and time-invariant, and may be correlated with the unobserved permanent components in the previous equations. The error "h N(0;(cid:27)2) is iid. t+1 (cid:24) h Household Income Equation Log household income is assumed to follow the process y y lnincome = (cid:12) X +y ; t+1 1 t+1 t+1 where X y is de(cid:12)ned similarly to Xw and Xh , and y is given by t+1 t+1 t+1 t+1 y = (cid:13) y +(cid:13) y w +(cid:13) y h +(cid:13) y D +(cid:13) y H +(cid:13) y U +(cid:16)y +p y ; (19) t+1 0 1 t+1 2 t+1 3 t+1 4 t+1 5 t+1 t+1 y y y p = (cid:26) p +" : t+1 y t t+1 Above, U is an indicator of unemployment de(cid:12)ned by U = 1 E D . The household t+1 t+1 t+1 t+1 (cid:0) (cid:0) income equation states (cid:12)rst that household income depends on the wage and work hours of the head.29 The reason for this dependence is simply that labor earnings of the head are typically the main component of household income. In addition, (19) allows D , H , and U to t+1 t+1 t+1 a(cid:11)ect household income via components of household income other than the head’s earnings, such as public and private transfers received by the family or labor income of other family members. y y y Parameters(cid:13) ,(cid:13) ,and(cid:13) arethuslikelytocaptureinsurancetoshockstothehead’sabilitytowork, 3 4 5 health, or employment status. The household-speci(cid:12)c permanent component (cid:16)y is allowed to be correlated with the permanent components in the employment transitions, job changes, wage rate, and work hours equations. The factor structure of the various unobserved permanent components y is described below. The component p captures the residual unexplained variation in household t+1 income. This residual exhibits important persistence in the data and is thus modeled as an AR(1) y process. The shock " N(0;1) is iid. t+1 (cid:24) 29Recall that when taking the model to the data, w , h , D , H , and U refer to the head of the t+1 t+1 t+1 t+1 t+1 household. 18

Permanent Unobserved Heterogeneity The permanent unobserved components in the above equations are assumed to follow the factor structure (cid:16)EE = (cid:14)EE(cid:22)+(cid:14)EE(cid:17) (cid:22) (cid:17) (cid:16)UE = (cid:14)UE(cid:22)+(cid:14)UE(cid:17) (cid:22) (cid:17) (cid:16)JC = (cid:14)JC(cid:22)+(cid:14)JC(cid:17) (20) (cid:22) (cid:17) (cid:16)w = (cid:14)w(cid:22) (cid:22) (cid:16)h = (cid:14)h(cid:22)+(cid:14)h(cid:17) (cid:22) (cid:17) (cid:16)y = (cid:14) y (cid:21); (cid:21) where (cid:22) N(0;1), (cid:17) N(0;1), and (cid:29) N(0;1) (introduced below) are mutually independent, (cid:24) (cid:24) (cid:24) individual-speci(cid:12)cpermanentcomponents, andall(cid:14) coe(cid:14)cientsarefactorloadingstobeestimated. Factor (cid:22) is de(cid:12)ned as the unobserved heterogeneity component in wages, but it is also allowed to in(cid:13)uence employment, job changes, and hours. Factor (cid:17) also a(cid:11)ects employment, job changes, and hours, but is assumed to have no in(cid:13)uence on wages. The latter component is intended to capture factors related to labor supply and to job and employment mobility preferences. Factor (cid:21) in the income equation is de(cid:12)ned as (cid:21) = (cid:20)(cid:22)+p1 (cid:20)2(cid:29) and is thus N(0;1) and correlated with (cid:22), with (cid:0) (cid:24) correlation coe(cid:14)cient (cid:20) (which is also estimated). 4.3.2 Estimates of the Income Model The employment, job mobility, wage rate, work hours, and household income equations presented above are estimated jointly by generalized indirect inference.30 Appendix 4 describes implementation of the estimation method used here.31 In order to estimate the e(cid:11)ects of disability and health in equations (13) - (19) I simulate D and H using the estimates of equations (11) and (12) t t presented above. Estimation in this section is thus conditional on the estimated parameters of the disability and health processes. 30The coe(cid:14)cients on the variables in vectors Xw, Xh, and Xy, however, are not estimated by indirect inference. t t t VariationduetovariablesintheX vectorsisremovedfromthedatausinga(cid:12)rst-stageregressionpriortoestimation t by indirect inference. Vectors Xw and Xh contain a polynomial in experience, education, race, and year indicators. t t Vector Xy contains a number of additional variables (see Appendix 3). The experience polynomial and the sample t average of most of the variables in vector Xy are added back to the household income equation when calculating t levels of household income to be used in the consumption model. 31For a general discussion of the method see Keane and Smith (2003). 19

Estimation results for equations (13) - (19) are presented in Table 2C. I will not discuss the estimated parameters of all equations here. Instead, I will focus on the main features of the estimates of the wage and household income equations only. The next section will use simulations toprovideaninformalevaluationofthe(cid:12)tofallestimatedequationsandwillthusillustratesomeof the implications of the estimated parameters in the employment, job change, and hours equations. Panel (d) in Table 2C presents estimates of the wage equation. The most important features of the estimates are the following. Persistence in the wage rate is high but well below unity (the autoregressive coe(cid:14)cient (cid:26)^ is 0:939). The standard deviation (cid:27)^ of the wage shock "w for w w t+1 job stayers is fairly large (0:097). When wage shocks involve a new job (whether the job change involvesgoingthroughaperiodofnonemploymentornot)thestandarddeviationofthewageshock more than triples to 0:298 ((cid:30)^ = 2:054), and the dependence on the lagged wage falls to 0:794 1 ((cid:30)^ = 0:154). New jobs are thus associated with substantial wage risk. Finally, nonemployment 2 (cid:0) is negatively related to the persistent wage component through the coe(cid:14)cient (cid:13)w = 0:140. 3 (cid:0) Panel (f) presents the estimated parameters of the household income equation. The most b important features of the estimates are the following. Wages and hours have a strong positive y y association with household income (coe(cid:14)cients (cid:13)^ and (cid:13)^ are 0:592 and 0:454, respectively). Con- 1 2 y ditional on the wage and hours, disability is positively related to household income (coe(cid:14)cient (cid:13)^ 3 is 0:186). This positive relationship is likely to re(cid:13)ect transfers from disability insurance but could also re(cid:13)ect a positive labor supply response of a spouse or other family members to disability of the head. In either case, the positive coe(cid:14)cient suggests the presence of substantial insurance against disability shocks. Contrary to disability, health limitations do not have a positive association with household y income conditional on the wage and hours (coe(cid:14)cient (cid:13)^ is 0:007). There is thus no evidence 4 (cid:0) of insurance against health limitations captured by H. Unemployment does have have a positive y relationship, but the coe(cid:14)cient is small ((cid:13)^ = 0:027). 5 The results further indicate that permanent unobserved heterogeneity in household income is important(^(cid:14) y = 0:248)andnegativelycorrelatedwithunobservedheterogeneityintheemployment, (cid:21) job mobility, wage rate, and hours equations ((cid:20)^ = 0:148). One possible explanation is that (cid:0) households who do permanently better in the labor market may also receive permanently less transfers from public programs. Another possible explanation is that permanently higher earnings of the head may permanently reduce the labor supply of a spouse if present. Finally, the serially y correlated error in household income p (i.e., the residual household income component) has an t 20

autoregressive coe(cid:14)cient of (cid:26)^ = 0:449 and a large standard deviation of (cid:27)^ = 0:168. y y 4.4 Medical Expenditures Out-of-pocket medical expenditures in old age are assumed to be given by32 M = exp((cid:13)M +(cid:13)M(t+1)+(cid:13)M(t+1)2+(cid:13)M(t+1)3+(cid:13)MH +pM ) (21) t+1 0 1 2 3 4 t+1 t+1 pM = (cid:26) pM +"M ; t+1 M t t+1 (cid:1) where Var["M ] = (cid:27)2 . The log of medical expenditures lnM thus depends on a deterministic t+1 M t+1 polynomialinage,healthstatus,andapersistentcomponentpM whichfollowsanAR(1)process.33 t+1 The (cid:13) coe(cid:14)cients in equation (21) are estimated by (cid:12)tting a least-squares regression of lnM . t+1 Any unobserved permanent component a(cid:11)ecting medical expenditures is assumed to be captured by the permanent component in health.34 Parameters (cid:26) and (cid:27)2 are estimated by (cid:12)tting sample M M autocovariances of the least-squares regression residuals to theoretical autocovariances implied by the (cid:12)rst-order autoregressive assumption on the error term. Autocovariances are (cid:12)tted using an equally-weighted minimum distance estimator.35 Table 2D presents the estimated parameters. The age pro(cid:12)le is not very pronounced: the polynomialintisinitiallyincreasing,thendecreasing,andthenincreasingagain. Healthlimitations have a signi(cid:12)cant positive relationship with medical expenditures. If one sets the persistent shock pM to zero, for instance, expected medical expenditures at t = 50 are $413 for someone in good t+1 health and $640 for someone in poor health. Uncertainty in medical expenditures is large ((cid:27)^ = M 0:936) and the shocks are fairly persistent (the autoregressive coe(cid:14)cient is (cid:26)^ = 0:745).36 M 32Similar speci(cid:12)cations are used in Hubbard, Skinner, and Zeldes (1994, 1995) and Scholz, Seshadri, and Khitatrakun (2006), among others. 33Notice that the disability indicator D does not enter equation (21). The reason is that D is a variable related t t to employment status and is not a state variable in old age in the lifecycle model. 34Equation (21) does not allow for permanent unobserved heterogeneity other than that entering through health. Thereasonisthatfromthepointofviewofthelifecycle,medicalexpendituresinoldageareassumedtobeunknown early in life and are thus treated as risk. 35ForevidenceinfavorofusingequalweightsratherthananoptimalweightingschemeseeAltonjiandSegal(1996). 36Scholz, Seshadri, and Khitatrakun (2006) use a model similar to equation (21) and a measure of medical expenditures similar to the one used here but constructed from the Health and Retirement Study. They estimate the persistence parameter to be around 0.84 and 0.86, and a standard deviation of the shock which ranges from 0:512formarriedhouseholdswithcollegeeducationto2:081forsinglehouseholdswithnocollege. Reportedmedical expenditures of zero are set in their analysis to $1.00 (i.e., 0 in logs). I (cid:12)nd that setting expenditures of zero to $1 considerably in(cid:13)ates the persistence of shocks. The reason is that very low values of expenditures (such as zero expenditures) tend to be more persistent than large values. This turns out to a(cid:11)ect the estimates of persistence signi(cid:12)cantly. Setting zero expenditures to $1 in a log speci(cid:12)cation also implies that percentage changes at very low levels of expenditures have the same e(cid:11)ect as similar percentage changes at high levels of expenditures. It would generally be more appropriate to set zero expenditures in a log model to some higher minimum level, say $300. 21

4.5 Evaluation of Fit This section provides an informal evaluation of the (cid:12)t of the processes estimated in the previous section by simulating data from the estimated processes, and then comparing sample statistics of the simulated data against sample statistics of the PSID data. I simulate data from the estimated equations for a large number of individuals and then randomly select a subsample of the simulated data in such a way that its demographic pattern matches that of the PSID sample. Table 3A presents the sample mean of the disability and health limitations indicators for different levels of potential experience. The column under the "Overall" heading displays statistics for all levels of potential experience. The next columns report results for the level of t indicated in the top row. For each reported level of experience, the results combine data for periods t 1, (cid:0) t, and t+1. For instance, t = 5 uses data for t = 4;5; and 6. As the (cid:12)gures show, the simulated data match the PSID data fairly closely. The overall mean of D is 0:02 in both the PSID and the simulated data. The probability that D equals 1 increases steadily with experience. For health limitations, the overall mean is 0:13 in the PSID and 0:14 in the simulations. The probability that H = 1 also increases steadily with experience. For high levels of experience (t around 60), these probabilities are 0:50 in the PSID and 0:54 in the simulated data. Table3Bpresentssimilarstatisticsforallvariablesusedintheestimationofthejointhousehold income process.37 I will not attempt to discuss all statistics here. The main points to notice in Table 3B are the following. (i) The overall (cid:12)t of the simulated data is good. (ii) The main aspect that is missed by the simulations is a fairly strong and steady increase in the standard deviation of log hours with potential experience. This increase translates into a similar rise in the standard deviation of labor earnings of the head and, to a lesser extent, of household income. This feature of the PSID data appears to re(cid:13)ect the fact that exceptionally low levels of reported hours become more common with large values of potential experience. This, in turn, is likely to be the result of workers retiring gradually and signi(cid:12)cantly reducing their hours of work prior to full retirement. Finally, Table 3C presents statistics for the estimated medical expenditures process. Overall, the (cid:12)t seems fair given the somewhat erratic pattern observed in the data. The overall mean is 1:576 in the PSID and 1:342 in the simulated data. The overall standard deviation is 3:758 in the However,inthecaseofmedicalexpenditures,thisdistortstheestimatessigni(cid:12)cantlybecauseofthelargenumberof zeros in the distribution. The approach taken here is to use only positive values of expenditures in the estimation. This will overestimate the probabilities of facing medical expenditures. However, this approach still implies smaller expenditures (and smaller persistence) than setting $0 observations to any amount that is below about $300. 37The table also displays statistics for employment duration, unemployment duration, job duration, and labor earnings. Even though these variables do not appear in the consumption model, they are used in estimation of the joint model of employment, job changes, wage, hours, and income. See Appendix 3 and Appendix 4. 22

PSID and 3:519 in the simulations. 5 Solution to the Lifecycle Model Once parameterized, the lifecycle model is solved by numerical dynamic programming. The model gives rise to three di(cid:11)erent forms of Bellman equations, corresponding to (i) the working years, (ii) the transition between work and retirement, and (iii) the retirement years, respectively. The Bellman equations are as follows: Working years V(t;D ;H ;E ;pw;p y ;z ) = max u(c ) t t t t t t t f +(cid:12)E[V(t+1;D ;H ;E ;pw ;p y ;z ) t+1 t+1 t+1 t+1 t+1 t+1 t;D ;H ;E ;pw;p y ;z ] ; (22) t t t t t t j g where the expectation, given transition equations (1) - (6), is taken over shocks "D , "H , "EE, t+1 t+1 t+1 "UE, "DE, "J , "w , "h , " y . t+1 t+1 t+1 t+1 t+1 t+1 Last working year V(t;D ;H ;E ;pw;p y ;z ) = max u(c ) t t t t t t t f +(cid:12)(cid:25) E [V(t+1;H ;pM ;SS ;z ) t t t+1 t+1 t+1 t+1 t;D ;H ;E ;pw;p y ;z ] ; (23) t t t t t t j g where the expectation, given transition equations (2), (7), and (10) is taken over shocks "H and t+1 "M . t+1 Retirement years V(t;H ;pM;SS ;z ) = max u(c ) t t t t t f +(cid:12)(cid:25) E [V(t+1;H ;pM ;SS ;z ) t t t+1 t+1 t+1 t+1 t;H ;pM;SS ;z ] ; (24) t t t t j g where the expectation, given transition equations (2) and (8) - (10) is taken over "H and "M . t+1 t+1 23

In solving the model, cash on hand z and the persistent wage component pw are treated as t t continuous state variables. Consumption c is also continuous. State variables p y and pM are t t t approximated by Markov chains, and SS is also discretized. Since the dynamic programming t+1 problem has a (cid:12)nite horizon, the Bellman equations are solved by value function iteration. The largest computational costs of solving the Bellman equations in this model arise from computing the expectations of the next-period value function. Expectations are computed as follows: For state variables D , H , and E , I use estimated equations (11)-(15), and pseudo-random draws of t t t "D , "H , "EE, "UE, "DE, to simulate the joint behavior of D , H , and E . From the simulation, t+1 t+1 t+1 t+1 t+1 t t t I compute matrices of transition probabilities for a vector (D , H , E ) and use these transition t t t y matrices to compute the expectations. For state p , I use the transition probabilities associated t with the Markov chain approximation. Finally, I use Gauss-Hermite quadrature to compute expectations with respect to "J , "h , and "w . t+1 t+1 t+1 Treating cash on hand and the persistent wage component as continuous requires the use of an interpolation scheme to evaluate next-period’s value function at arbitrary values of the continuous statevariables. Iusebicubicinterpolation. Thispreservesdi(cid:11)erentiabilityoftheright-handsideof the Bellman equation and allows solving each optimization problem using Newton-Raphson, which is convenient because of its fast (quadratic) convergence. All programs are written in Fortran 90 and parallelized using MPI (Message Passing Interface). 5.1 Optimal Consumption Behavior This section discusses some of the properties of the solution to the lifecycle consumption model presented above. The data were treated, and the model was parameterized, so that income and consumption are in thousands of year-2000 dollars. The model corresponds to a household of mean size and composition, with mean years of education, whose head is white, and who is at the mean of the distribution of permanent unobserved heterogeneity components. All aggregate risk is abstracted from in the model.38 Figure 1 presents mean experience pro(cid:12)les for household nonasset income and consumption, both simulated from the baseline lifecycle model. The nonasset income pro(cid:12)le has a humped shape, with a signi(cid:12)cant drop at the time of retirement. The drop at retirement is particularly pronounced because the current parameterization of nonasset retirement income in the model uses only social security bene(cid:12)ts of the household head.39 The mean pro(cid:12)le of (optimal) consumption 38Year e(cid:11)ects are removed from the wage, hours, earnings, and household income data prior to estimation. 39It is straightforward to include social security bene(cid:12)ts of a spouse and dependents. One could also consider a secondtypeofhouseholdwhichadditionallyreceivesde(cid:12)nedbene(cid:12)tpensions. Theinclusionofadditionalcomponents 24

is also hump-shaped, but smoother than income. Mean consumption in the (cid:12)gure never exceeds mean income. This is because of the presence of borrowing constraints along with the assumption in the simulations that households begin their career with zero initial assets.40 Figures2-5displayoptimalconsumptionrulesandtheirdependenceonthevalueofthevarious state variables in the model. Figure 2 exhibits optimal consumption as a function of cash on hand for an employed household in good health in period 1 ((cid:12)rst year of career) who is at the mean of the distribution of the persistent wage and household income components pw and p y . At low t t levels of cash on hand, households are credit-constrained and consume their entire wealth. Above a certain threshold, households begin to save. This behavior is typical of consumption models with precautionary motives.41 Figure 3 illustrates the dependence of optimal consumption on the level of the residual compoy nent of household income p in period 1 ((cid:12)rst year of career) for an employed individual in good t health who is at the mean of the wage distribution. The (cid:12)gure displays consumption policy funcy tions for values of p ranging from 2 standard deviations above to 2 standard deviations below the t mean. The (cid:12)gure indicates that, conditional on having $20,000 in cash on hand in its (cid:12)rst year of y career, a household with component p two standard deviations above the mean will spend about t y $3,000 more on consumption than a household with p two standard deviations below the mean. t Figure 4 shows the dependence of consumption on the persistent wage component pw. The t (cid:12)gure refers to a household who is employed, in good health, and at the mean of the household y income component p in period 1. A higher current wage leads to a higher level of consumption t for a given level of resources. The (cid:12)gure corresponds to values of pw ranging from 2.8 standard t deviations above to 2.8 standard deviations below the mean. Conditional on having $30,000 in total resources in its (cid:12)rst year of career, a household with component pw 2.8 standard deviations t above the mean will spend about $12,000 more on consumption than a household with pw 2.8 t standard deviations below the mean. Figure 5 displays the optimal consumption rule over the entire range of possible realizations that pw may take in the simulations.42 t ofretirementincomewillmakethedroplesspronouncedandwilltendtoreducesavingforretirementpurposes. Itwill nota(cid:11)ectuncertainty,however,whichisthemainobjectofinteresthere. Theimportantaspectsofretirementnonasset income here are that (i) income drops at the time of retirement, and (ii) there is no uncertainty after retirement in social security receipts. One possible extension that would introduce an additional source of uncertainty during retirementwhichseemsrelevantwouldbetoaccountforuncertaintyinratesofreturnandhenceinendogenousasset income. Oneimportantaspecttoconsiderinsuchanextensionisthatahighdegreeofheterogeneityinparticipation inthestockmarketimpliesthatuncertaintyinratesofreturnislikelytobehighlyheterogeneousacrosshouseholds. 40All assumptions used in the simulations are discussed in more detail in the next section. 41See, for instance, Deaton (1991), Carroll (2001), Gourinchas and Parker (2002), or Cagetti (2003). Optimal behavior is qualitatively similar whether borrowing constraints are exogenously imposed or arise as "natural" constraints, using the terminology introduced by Aiyagari (1994). 42Recall that both the persistent wage pw and cash on hand z are treated as continuous state variables in the t t 25

The (cid:12)gures give a sense of the e(cid:11)ects on optimal consumption of changes in a particular state variable, holding the rest constant. Changes in most state variables, such as a change in employment status, however, lead to same-period changes in other state variables. The next section uses simulations to analyze various implications of the lifecycle model, accounting for all interactions among the di(cid:11)erent variables. 6 Simulation Analysis Thissectionusessimulationstoinvestigatetheimplicationsoftheconsumptionandincomemodels for the importance of the various sources of income risk for household welfare and precautionary saving. Theanalysisisbasedonnumericallysolvingandthensimulatingtheconsumptionmodelfor a large number of households under a variety of scenarios. The scenarios di(cid:11)er in the number and typeofshocksfacingthehouseholds,whobehaveoptimallyundereachscenario. Inallsimulations, households are assumed to begin their career with zero initial assets. The determination of initial conditions for the key simulated processes is discussed in more detail in Appendix 3. 6.1 Welfare Gains of Insuring Speci(cid:12)c Sources of Risk Thissubsectionevaluatesthewelfaregainstothehouseholdfromfullyinsuringagainsteachspeci(cid:12)c source of income risk. We will consider two di(cid:11)erent insurance schemes. In the (cid:12)rst, which will be called the unadjusted case, full insurance means the following: For any given shock, an insured household is compensated (by a lumpsum transfer) upon the realization of the shock in such a way that the realization of the shock has no e(cid:11)ect on the realization of income. For instance, insuring unemploymentriskmeansthatintheeventofunemploymentthehouseholdreceivesatransferthat exactly o(cid:11)sets the income lost due to the unemployment shock. Thus, income after the insurance transfer is exactly the same as it would have been had the worker remained employed. As another example, fully insuring wage risk means that the insured household’s income after the transfer will be the same, regardless of the actual realization of the wage shock, as it would have been had the realization of the wage shock been zero. Insuring risk in this way has two e(cid:11)ects. First, insurance reduces the variance (uncertainty) in income associated with a particular source of risk. Second, insurance may also a(cid:11)ect the mean of income.43 This bring us to the second insurance scheme, which will be called the adjusted case. numerical solution of the model. 43Thereasonwhyinsurancea(cid:11)ectsmeanincomevariessomewhatacrossdi(cid:11)erentshocks. Forinstance,theoccurrence of disability, health, and unemployment shocks always a(cid:11)ects income negatively. Hence, fully insuring against these risks raises expected income. Wage, hours, and household income shocks, on the other hand, a(cid:11)ect expected 26

In this case, mean income in the insured scenario will be adjusted so that it equals mean income in the uninsured scenario for each year in the lifecycle. That is, the household is required to pay an actuarially fair premium for insurance, and as a result expected household income is the same under both the insured and uninsured scenarios. In this sense the insurance considered here is actuarially fair. Insurance is also full or complete in that it eliminates all uncertainty in income created by the presence of a particular source of risk. Most of the discussion that follows will focus on the adjusted case, although for completeness, I will also present results for the unadjusted case (where mean income in the insured scenario is allowed to be di(cid:11)erent from mean income in the uninsured scenario). Two additional points about the insurance experiment and welfare analysis are worth stressing. First, the insurance considered here is in addition to already existing insurance mechanisms which are captured by the household nonasset income process estimated on PSID data.44 Second, in all cases, households in the model adjust their behavior optimally to the provision of the additional insurance. Thewelfaregainsofinsurancearecalculatedintermsofthe"equivalentcompensatingvariation" in consumption. That is, the welfare calculations ask the following question: What percentage of currentlifetimeconsumptionwouldhouseholdsbewillingtopayinordertobefullyinsuredagainst a particular source of risk? The metric used for welfare comparisons is expected lifetime utility at time zero. This is the expected lifetime utility right before an individual begins their career and before any uncertainty (other than the household’s type) is resolved. This metric is given by: T T W = E [ (cid:12)t 1(cid:25) u(c ((cid:10) ))] = [ (cid:12)t 1(cid:25) u(c ((cid:10) ))]d(cid:8); (25) 0 (cid:0) t (cid:3) t (cid:0) t (cid:3) t t=1 Z t=1 X X where (cid:10) is the state vector, c ((cid:10) ) is the consumption policy function, which prescribes the t (cid:3) t optimal level of consumption at any given point in the state space (i.e., at any possible contingency that the householdmay encounter), and(cid:8) denotes the jointdistributionof the random state vector (cid:10) = ((cid:10) ;:::;(cid:10) ).45 1 T wage,hoursandhouseholdincomebecausethesevariableshavelog-normaldistributions. Hence,althoughtheshocks are symmetric around zero, a change in the variance of the processes also a(cid:11)ects their mean. 44Recallthatthehouseholdnonassetincomedatausedtoestimatetheincomeprocessincludeslaborincomeofall household members and transfers from outside the household, whether from public or private sources. 45Let W denote welfare under the full-uncertainty scenario and W denote welfare under the insured scenario. U I Asexplainedabove, theinsuredscenariocompensatesthee(cid:11)ectsofaparticularsourceofrisk suchthattherisk has no e(cid:11)ect on net income. The "equivalent compensating variation" is then de(cid:12)ned as the value of parameter (cid:24) which 1 s T o h lv is es is th th e e eq m u e a a t s i u o r n e W of ( w (1 el + fa (cid:24) re )c g (cid:3) a ((cid:10) in t s )) of = in W su I r . a S n o c l e vi ( n a g lte fo rn r a (cid:24) ti y v i e e l l y d , s w (cid:24) e = lfare W W U c I o + + s K K ts o 1 (cid:0) f (cid:11) ris (cid:0) k) 1; us w e h d er b e el K ow = . 1 (cid:0) 1 (cid:11) T t=1 (cid:12)t (cid:0) 1(cid:25) t . (cid:0) (cid:1) P 27

6.1.1 Results Table 4 presents results for the welfare gains of insuring each source of risk for the baseline consumption model. Panel (a) presents adjusted results (the case of actuarially fair insurance), while panel (b) shows unadjusted results. The entries in columns (1), (2), and (3) of panel (a) indicate that the welfare gains of fully insuring disability, health, and unemployment risk are extremely small. According to the table, households are willing to pay no more than 0:04 of 1% of lifetime consumption in exchange for full insurance against these risks. The welfare value of insurance is small even if one does not adjust for the e(cid:11)ect of insurance on mean income. As panel (b) shows, the value of insuring disability and health risks in the unadjusted case is still no larger than 0:04 of 1% of lifetime consumption. The value of insuring unemployment in the unadjusted case is 0:62 of 1% of lifetime consumption. Column (5) displays the results for wage shocks. These shocks are innovations in the hourly wage which are not related to changes in employment status or in employer. As the table shows, households in the baseline model would be willing to pay up to 1:20% of their lifetime consumption in order to be insured against such shocks. Columns (6), (7), and (8) present the gains of insuring shocks associated with job changes, hours of work, and the residual component of household income. The equivalent compensating variation in these cases is 0:72%, 0:45%, and 2:19%, respectively.46 I don’t discuss the results for medical expenditures here, as these are preliminary (see discussion in section 2). 6.1.2 Discussion Overall, the results in Table 4 indicate that the value of insuring most sources of risk in the model is small. Particularly striking is how minuscule consumers’ willingness to pay for insurance against disability, health, and unemployment risks turns out to be. It is also remarkable how much more valuableinsuranceagainstwageshocksis. Theseresults,however,areconsistentwithLow,Meghir, and Pistaferri (2006), who study the welfare e(cid:11)ects of unemployment and wage shocks, and (cid:12)nd that wage risk is important, but that unemployment risk is not. The shock to the residual component of household income turns out to play a very important role. This is perhaps not surprising, considering that this shock captures all variation in household income which is not explained by earnings, disability, health, or unemployment of the head, and 46Noticethatjob-mobilityshocksalsooperatemostlythroughane(cid:11)ectonthewagerate. Thedi(cid:11)erencewithwage shocks is that the latter do not involve a change of employer. 28

that household income includes spousal labor income and all transfer income.47 An important part ofthisresidualcomponentisthuslikelytocapturefactorsthatarenotreallyrisk, butratherre(cid:13)ect choice, such as spousal labor supply. The approach used here does not allow to determine what part of a given shock actually represents risk. It is interesting, nevertheless, that the shock to residual household income turns out to play the largest role of all shocks in the welfare analysis, which suggests that this component constitutes an important part of the shock estimated in simple univariate models of household income, where income is driven by a single shock to the income process. Regarding the very small welfare value of insuring disability, health, and unemployment risk, it should be stressed again that all insurance considered here is insurance over and above existing insurance provided by government transfers (already included in the household income process), self-insurance provided by saving, and insurance within the family (also included in the household incomeprocess). TheresultsfromTable4thusseemtosuggestthatexistinginsurancemechanisms doagoodjobofprotectinghouseholdsagainstthesourcesofriskconsideredhere. Theyalsosuggest that even a small deadweight loss created by additional insurance would be su(cid:14)cient to wipe out much of the gains of such additional insurance. One important caveat that should be emphasized is that the value of insurance is measured here from an ex-ante point of view, that is, before any uncertainty other than a household’s type is realized. Measuring the value of insurance conditional on, for instance, being disabled or unemployed would yield higher bene(cid:12)ts of additional insurance. Still,thevalueofinsuringsomeoftheserisksmayseemsurprisinglylowinlightoftheestimation results, which would appear to imply that their e(cid:11)ects are important. In the case of disability, for instance, the estimates presented in Table 2C suggest that becoming disabled has a very strong negative e(cid:11)ect on earnings (D enters the equation of log work hours with coe(cid:14)cient (cid:13)h = 0:896 3 (cid:0) and the persistent wage equation with coe(cid:14)cient (cid:13)w = 0:280).48 3 (cid:0) b However,evenifdisabilityhasalargee(cid:11)ectonincomewhen itoccurs,itisstillalow-probability b event and hence, from an ex-ante point of view, it does not account for much of the variation in lifetime income and does not contribute greatly to income uncertainty. The same is true of unemployment. Health limitations, as measured here, are on the other hand more frequent, but their estimated e(cid:11)ect on income is rather small. It should also be noted that the framework used 47Recall that variation due to potential experience, education, race, and other demographic variables was also removed from the data in a (cid:12)rst-stage regression. 48Recall, however, that there is also an important insurance component captured by the coe(cid:14)cient (cid:13)y =0:186 in 3 the household income equation. b 29

here only considers the e(cid:11)ects of disability and health on income and abstracts from direct e(cid:11)ects on utility. Such e(cid:11)ects are likely to be important for welfare. Oneadditionalreasonthathelpsexplainthesmallvalueofinsuranceobtainedandwhichshould be mentioned here is that for the broad measure of household income taken from the PSID, income very rarely falls below $1;000 (in year-2000 dollars). Consequently, income shocks in the lifecycle model are discretized in such a way that household income in the model can never fall below $1;000. This is important for welfare because under constant relative risk aversion (CRRA) utility the really painful events occur when consumption drops to extremely low levels. The presence of an income (cid:13)oor e(cid:11)ectively provides a consumption (cid:13)oor, ruling out situations where consumption is near zero. The view taken here is that the (cid:13)oor of $1;000 per year used in the calculations is very conservative, given that people have access to a wide array of transfers. Thus, a more precise way to interpret the results in this paper may be that, if one is willing to assume that constant relative risk aversion utility is a reasonable representation of preferences and that existing safety nets provide a minimum level of income and consumption as low as $1;000 a year, then the value of insuring risk over and above already existing insurance is small. It may also be worth mentioning that ignoring some margins of choice in the model, such as labor supply, has an ambiguous impact on the value of insurance. In the simulated model, treating labor supply as (cid:12)xed forces consumption to take on the full e(cid:11)ects of the shocks, which thus turn out to be more painful than they would otherwise be. However, keeping labor supplied (cid:12)xed in estimation of the income process also a(cid:11)ects the estimates of risk. A given observed variation in income may re(cid:13)ect demand shocks that are partially absorbed and o(cid:11)set by an adjustment in labor supply, and this adjustment has a welfare cost. Thus, allowing for (cid:13)exible labor supply in the estimation and in the simulation could make the results change either way. Finally, Table 7A presents results obtained under alternative assumptions regarding the household’s degree of risk aversion and patience. Panel (a) (second row) displays results for households with a coe(cid:14)cient of relative risk aversion of 5:0.49 As should be expected, the higher degree of risk aversion increases the value of insurance for all sources of risk. However, the increase in the coe(cid:14)cient of relative risk aversion to a value as high as 5:0 does not change any of the main results orconclusionsdrawnfromtheanalysisabove. Inparticular, thevalueofinsuringdisability, health, and unemployment risk remains minuscule, and the relative importance of the various sources of 49Asnotedabove,thevalue5:0isatthehighendofvaluesgenerallyconsideredempiricallyplausible. Seediscussion innote17. Idonotconsidervaluesofthecoe(cid:14)cientofriskaversionsmallerthan3:0becausetheywouldleadtoan even smaller value of insurance than in the baseline case. 30

risk does not change. Panel (b) displays results for di(cid:11)erent values of the discount factor. The numbers in the table suggest that the importance of the residual component of household income increases as impatience increases, while the wage component appears to become less important. One possible reason for this is that the wage a(cid:11)ects social security bene(cid:12)ts, and thus may a(cid:11)ect future periods more than the residual component of household income does. Most importantly, the results regarding disability, health, and unemployment do not change signi(cid:12)cantly. Overall, we conclude that the most important features of the results are robust to alternative, empirically plausible assumptions about household preferences. 6.2 Precautionary Saving This section investigates the contribution of the various sources of uncertainty to the accumulation of precautionary savings. Figure 6 displays the mean level of assets held by simulated households, by year of potential experience. The upper curve represents mean asset holdings under the fulluncertaintyscenario, whilethelowercurvedisplaysmeanassetholdingsundernouncertainty. The meanpro(cid:12)leofnetincomeisidenticalunderbothscenarios. Thedi(cid:11)erencebetweenthetwocurves is mean precautionary wealth, that is, the mean level of assets held only because of the presence of uncertainty. Figure 7 displays this di(cid:11)erence.50 Figure 8 displays mean precautionary wealth as a fraction of total wealth. The (cid:12)gure shows that precautionary savings make up the entirety of savings for the (cid:12)rst 11-12 years of a worker’s career. Workers start saving for lifecycle reasons (retirement) only after this point. Even 20 years into a worker’s career, precautionary wealth continues to make up 50% or more of total savings. The exact point at which retirement saving begins to matter will depend on how substantial the drop in income at retirement is. The current version of the lifecycle model analyzed here does not account for de(cid:12)ned bene(cid:12)t pensions or social security bene(cid:12)ts received by family members other than the household head. As a result, income drops signi(cid:12)cantly at retirement and workers start saving for retirement relatively early in their career. Table 5 decomposes precautionary wealth into components attributable to various sources of risk. Speci(cid:12)cally, the table calculates the di(cid:11)erence between mean precautionary wealth under full uncertainty and mean precautionary wealth under a scenario in which one particular source of risk is fully insured, and then expresses this di(cid:11)erence as a percentage of mean precautionary wealth. Contributions are normalized to sum to 100% of precautionary wealth51. The results in the table 50Like most consumption models, the model used here does not distinguish between liquid and illiquid assets. In particular, the model does not separately allow for real estate wealth. Intergenerational transfers are also ignored. 51Without the normalization, their sum slightly exceeds 100% because of interactions among the di(cid:11)erent sources 31

show that the largest contribution by far is made by shocks to the residual component of household income (43% of precautionary wealth). Wage shocks are also very important, contributing almost 20% of precautionary wealth. Together, these two sources thus account for about 63%. The individual contribution of all other shocks is below 20%. Among these, job-mobility shocks make the largest contribution (9:34%) and disability shocks the smallest (2:34%). Table7Bpresentsresultsfromasimilardecompositionunderalternativeassumptionsaboutthe coe(cid:14)cient of relative risk aversion and discount factor. As above, Panel (a) considers households with a coe(cid:14)cient of risk aversion of 5.0, while Panel (b) considers households with an increasing degree of impatience. As the numbers in the table show, the relative importance of the various sources of risk for the accumulation of precautionary saving does not appear to be sensitive to alternative assumptions about risk aversion and patience. More generally, the results for precautionary saving seem consistent with the welfare results from the previous section, and the general discussion presented in that section also applies here. 7 Conclusions and Research Agenda This paper uses a lifecycle consumption model to quantify the e(cid:11)ects of a number of sources of income risk on household welfare and precautionary saving. The model includes income shocks associated with disability, health, unemployment, job changes, wages, work hours, and a residual component of household income. I estimate the processes driving the evolution of these variables using PSID data, accounting for permanent unobserved heterogeneity {which is assumed to be known at the beginning of a worker’s career{ and a rich set of dynamic interactions among the variables. I then use the consumption model to quantify the welfare value of providing full, actuarially fair insurance against each source of risk and measure the contribution of each shock to the accumulation of precautionary savings. The main (cid:12)ndings are that: (i) the value of insuring disability, health, and unemployment shocks is extremely small (well below 1/10 of 1% of lifetime consumption in the baseline model); (ii) the gains from insuring shocks to the wage and to the residual component of household income are signi(cid:12)cantly larger (above 1% and 2% of lifetime consumption, respectively); and (iii) the latter two shocks account for more than 60% of precautionary wealth. The insurance evaluated in this paper is insurance over and above existing insurance provided by government transfers, self-insurance through saving, and insurance within the family, all of of uncertainty (for instance, the dependence of disability on lagged health limitations). 32

which are already captured in the baseline model. The results thus seem to suggest that existing insurance mechanisms do a good job of protecting households against the sources of risk considered here. They also suggest that even a small deadweight loss created by additional insurance would be su(cid:14)cient to wipe out much of the gains of such additional insurance. It should be noted, however, that the value of insurance is measured from an ex-ante point of view, thatis, beforeanyuncertaintyotherthanahousehold’stypeisrealized. Measuringthevalue ofinsuranceconditionalon, forinstance, beingdisabledorunemployedshouldyieldhigherbene(cid:12)ts. Even more importantly, the model analyzed in this paper corresponds to households who are at the mean of the distribution of the permanent unobserved components. Although the framework developed here permits analyzing households over the entire distribution of these components, this is left for future research because of the computational costs of these calculations. It is important to bear in mind, however, that a clear understanding of the role of heterogeneity is required in order to draw de(cid:12)nitive conclusions. An interesting question that arises from the results presented here regards the value and relative role of di(cid:11)erent existing insurance mechanisms. For instance, to what extent are transfers from disability insurance and unemployment insurance responsible for the small welfare e(cid:11)ects of disability and unemployment risk? This particular question may be addressed using this paper’s framework in the following way: (i) remove payments received from disability insurance (alternatively, unemployment insurance) from the income data constructed from the PSID; (ii) estimate the household income equation presented in section 4 using the modi(cid:12)ed household income data; (iii) solve the lifecycle consumption model using the new estimated income process and perform a welfare analysis. An analysis of the role of these and other sources of insurance is left for future research. One potentially important source of risk that is only partially addressed in this paper is the risk associated with catastrophic medical-expenditure shocks in old age. As discussed earlier, the treatment in this paper does not allow medical-expenditure shocks to wipe out a household’s accumulated wealth. Some studies, including Palumbo (1999) and De Nardi, French, and Jones (2006), suggest that the risk of catastrophic medical expenditures may be important for saving behavior and welfare. A more (cid:13)exible treatment of medical expenditures and their e(cid:11)ects on accumulated wealth will be pursued in future work. 33

8 Appendix 1: De(cid:12)nition of PSID Variables Potential Experience: Potential labor market experience is de(cid:12)ned as t = age max(education; (cid:0) 10) 5 where education is years of education. Labor market experience obtained with less than 10 (cid:0) yearsofeducation(whichisunusualinthedataandtypicallycorrespondstoveryyoungindividuals) is not counted as work experience. Employment Status: TheemploymentindicatorE isconstructedfromthereportedemployment status of the head of household at the survey date. The "employment status" PSID variable has eight possible categories: (1) working now; (2) only temporarily laid o(cid:11), sick leave or maternity leave;(3)lookingforwork,unemployed;(4)retired;(5)permanentlydisabled;temporarilydisabled; (6) keeping house; (7) student; (8) other; "workfare"; in prison or jail. Indicator E is set to 1 for categories (1) and (2); it is set to 0 otherwise. Disability: ThedisabilityindicatorD isalsobasedonreportedemploymentstatusatthesurvey date. Indicator D is set to 1 whenever employment status is (5) and it is set to 0 otherwise. Health Limitations: ThehealthlimitationsindicatorH isconstructedfromthesurveyquestion: \Do you have any physical or nervous condition that limits the type of work or the amount of work you can do?" Indictor H is set to 1 when a respondent answers "yes" to the above question, and it is set to 0 otherwise. Wage: For hourly workers, the wage variable used is the reported hourly wage at the survey date. For salaried workers, the variable is constructed from reported weekly, monthly, or yearly salary, divided by an appropriate standard number of hours. The measure used here further accounts for the fact that the PSID variable is capped at $9.98 per hour prior to 1978. This is done by replacing capped values for the years 1975-1977 with predicted values constructed by Altonji and Williams (2005). Predicted values are based on a regression of the log of the reported wage on a constant and the log of annual earnings divided by annual hours using the sample of individuals in 1978 for whom the reported wage exceeds $9.98. Hours: The hours variable is the reported total annual hours of work (in all jobs). Household income: Household income is de(cid:12)ned as the sum of (i) total labor income of the head; (ii) total labor income of the wife; (iii) total transfer income of the head and wife; (iv) taxable income of others; and (v) total transfers of others. In most waves, the PSID does not separately report labor and asset income of other family unit members. Out-of-Pocket Medical Expenditures: Total out-of-pocket medical expenditures are the sum of out-of-pocket payments for (i) nursing home and hospital bills; (ii) doctor, outpatient surgery, 34

dental bills; and (iii) prescriptions, in-home medical care, special facilities, and other services. The payments refer to the two-year period prior to the survey year. Detailed medical expenditures data are provided by the PSID starting with the 1999 wave. 35

9 Appendix 2: Determination of Social Security Bene(cid:12)ts This appendix describes the determination of the level of social security bene(cid:12)ts. Social security bene(cid:12)ts in the model are determined in the last year of work according to the formula: S = PIA(ALE(D ;H ;E ;pw;p y )) t+1 t t t t t wherePIAstandsforprincipalinsuranceamount andALE standsforaveragelifetimeearnings. Households are assumed to receive a level of bene(cid:12)ts equal to their PIA. ALE and PIA are determined as follows. Average Lifetime Earnings: In the last working year, the state variables are used to predict average lifetime earnings according to a forecasting equation. The coe(cid:14)cients of the forecasting equation are determined by simulations of the income (earnings) model using the following procedure: 1. Use the model of disability, health, employment, job changes, wages and hours (which imply earnings) to simulate a large number of careers. 2. Use the simulated earnings data to compute average lifetime earnings for each simulated career as calculated by the Social Security Administration. In particular, yearly earnings are censored from above to the maximum yearly earnings subject to the social security tax. In 1996, for instance, the maximum taxable yearly amount was $62,700. ALE are then computed as the average of such censored earnings over the 35 years of highest earnings. 3. Regress ALE against all (simulated) variables that would be known to the agent in the last year of career in the lifecycle model. These include all state variables in the last working year (disability, health, employment, persistent wage, persistent household income) as well as all permanent heterogeneity components (which de(cid:12)ne the household type). This regression uses a (cid:13)exible speci(cid:12)cation which includes a number of higher-order terms and interactions among the di(cid:11)erent variables. Several alternatives speci(cid:12)cations were tried and the regression with the best (cid:12)t was selected as the forecasting equation. Estimation results for this forecasting regression are presented in Table 6. Notice in particular the high R2 in the regression (0:88). Principal Insurance Amount Once average lifetime earnings (ALE) have been determined, the principal insurance amount (PIA) is determined by the rules of the Social Security Administration. I use "bend points" for the year 1996 (the last year of the sample used in estimation). The 1996 monthly "bend points", in current dollars, are $437 and $2;635. The corresponding yearly bend points in thousands of 36

year-2000 dollars are b = 5:606 and b = 33:801. The PIA, in thousands of year-2000 dollars, is 1 2 then calculated as PIA = 0:90 min ALE;b +0:32 min max ALE b ;0 ;b b +0:15 max ALE b ;0 : 1 1 2 1 2 (cid:1) f g (cid:1) f f (cid:0) g (cid:0) g (cid:1) f (cid:0) g 37

10 Appendix 3: Further Details of Model Speci(cid:12)cation and Estimation - Employment, Job Changes, Wage, Hours, and Household Income This appendix provides some additional details about the joint model of employment, job changes, wage rate, work hours, and household income, its estimation, and its relation to the income process in the lifecycle model. First Stage Regression and Household Income in Levels Recall that log household income is given by lnincome = (cid:12)yX y +y : t+1 t+1 t+1 y y y y y y Let X = [X ;X ;X ] be a partition of X where (i) X contains a quadratic t+1 1;t+1 2;t+1 3;t+1 t+1 1;t+1 y polynomial in age (of the head); (ii) X contains years of education of the head and wife as 2;t+1 well as variables describing household size and composition (number of major adults, number of additional adults, number of children under 6, and number of children between 6 and 18); and y (iii) X contains year indicators and race indicators for black and other nonwhite. Vector 3;t+1 (cid:12)y = [(cid:12) y ;(cid:12) y ;(cid:12) y ] is estimated in a (cid:12)rst-stage least-squares regression. Variation in income due to 1 2 3 y y [X ;X ] is then removed from the data prior to estimation by GII (where the coe(cid:14)cients of 2;t+1 3;t+1 component y in equation (19) are estimated). The level of household nonasset income in the t+1 lifecycle model Y is determined as Y^ = exp(lnincome ) = exp((cid:12)^y X y +(cid:12)^y X y +y^ ), t+1 t+1 t+1 1 1;t+1 2 2;t+1 t+1 y y where X is the average of vector X in the PSID sample and where y^ is the predicted 2;t+1 2;t+1 d t+1 value of component y . t+1 Initial Conditions Initial conditions in employment, job changes, the wage, hours, and household income are explicitly modeled and estimated. For a discussion of related models and further details about estimation, see Altonji, Smith, and Vidangos (2008). Employment: Initial employment status is assumed to be determined by E = I[^b +(cid:14)EE(cid:22)+ 1 0 (cid:22) (cid:14)EE(cid:17)+"EE > 0] where ^b ^b (cid:27)^ , (cid:27)^ (^(cid:14) EE )2+(^(cid:14) EE )2+1, (cid:14)EE(cid:22) and (cid:14)EE are de(cid:12)ned as (cid:17) 1 0 (cid:17) (cid:3)0(cid:1) E1 E1 (cid:17) (cid:22) (cid:17) (cid:22) (cid:17) before, "EE N(0;1),and^b isthecoe(cid:14)cient q estimatefromaProbitofE onaconstantestimated 1 (cid:24) (cid:3)0 t on PSID data for t 3. I use the (cid:12)rst three years here rather than just the (cid:12)rst year because (cid:20) there are relatively few observations when t = 1. Wages: The initial condition of the persistent wage component is modeled as pw = (cid:18) (cid:16)w+u ; 1 w w1 38

where (cid:18) is a free parameter estimated with GII and u N(0;(cid:27)2 ); where (cid:27)2 is set such that w w1 (cid:24) w1 w1 Var((cid:18) (cid:16)w +u ) equals the variance of the residual of regression for low levels of t . w w1 Household Income: The initial condition of the residual component of household income is determinedasp y = (cid:18) (cid:16)y+u ;where(cid:18) isafreeparameterestimatedwithGIIandu N(0;(cid:27)2 ); 1 y y1 y y1 (cid:24) y1 where (cid:27)2 is set such that Var((cid:18) (cid:16)y+u ) equals the variance of the residual of regression for low y1 y y1 levels of t. Measurement Error Estimation accounts for measurement error in wages, hours, earnings, and household income. The treatment of measurement error here follows the treatment in Altonji, Smith, and Vidangos (2008). For household income, measurement error is set to account for 25% of the variance of the (cid:12)rstdi(cid:11)erenceinobservedhouseholdincome,afteraccountingforthee(cid:11)ectofmeasurementerrorin earnings. Since earnings are a component of household income, one additionally needs to account for the e(cid:11)ect of measurement error in earnings on observed household income. Altonji, Smith, and Vidangos (2008) shows that there is a residual component in earnings which is not accounted for by wages and hours. This component is assumed here to consist entirely of measurement error. The system estimated by generalized indirect inference includes an auxiliary equation of earnings in order to identify this component. Estimates of the household income process account for the e(cid:11)ect of this measurement error component. 39

11 Appendix 4: Estimation Mechanics - GII Theequationsthatdeterminetheevolutionofemployment,jobchanges,wage,hours,andhousehold income are estimated by generalized indirect inference (GII). The implementation of GII used here is akin to that used in Altonji, Smith, and Vidangos (2008), which provides a detailed discussion. This appendix gives some further details of the implementation used in this paper. The auxiliary model consists of a system of seemingly unrelated regressions (SUR) with 8 equations and 27 covariates that are common to all 8 equations. The model is implemented under the assumption that the errors follow a multivariate normal distribution with unrestricted covariance matrix. The auxiliary model e(cid:11)ectively used52 may be written as y = Z (cid:5)+u ; u N(0;(cid:6)); u iid over i and t; (26) it it it it it (cid:24) where y = [E E ;E U ;J E E ; (27) it it i;t 1 it i;t 1 it it i;t 1 (cid:0) (cid:0) (cid:0) w ;h ;e ;ln(1+w 2);y ]; it it it it it 0 52In practice, the auxiliary model actually used in estimation included a few additional covariates. Due to an error in one of the programs, however, the sorting of the data in the additional covariates was di(cid:11)erent from the sorting of the data in the rest of the variables, rendering the additional covariates uncorrelated with the rest of the variables in the SUR system. Consequently, the auxiliary model e(cid:11)ectively used is the model described in this appendix. Excluding the additional covariates altogether from estimation yields estimates that are very close to the ones reported in this paper. The additional covariates that were initially intended to be included are E , J , ED , i i i UD ,JD ,w w ,h h ,e e ,andy y ,whereabaroveravariableindicatesaperson-speci(cid:12)c i i i;t 1 i i;t 1 i i;t 1 i i;t 1 i (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) time average. 40

and Z = [cons;(t 1);(t 1)2;ED ;UD ;JD ; (28) it i;t 1 i;t 1 i;t 1 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) E E ;E E ; i;t 1 i;t 2 i;t 2 i;t 3 (cid:0) (cid:0) (cid:0) (cid:0) E U ;E U ; i;t 1 i;t 2 i;t 2 i;t 3 (cid:0) (cid:0) (cid:0) (cid:0) J E E ;J E E ; i;t 1 i;t 1 i;t 2 i;t 2 i;t 2 i;t 3 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) w ;w ;h ;h ;e ;e ;y ;y ; i;t 1 i;t 2 i;t 1 i;t 2 i;t 1 i;t 2 i;t 1 i;t 2 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) w (t 1);w (t 1)2;w J i;t 1 i;t 1 i;t 1 it (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) D ;D ;H ;H ]: it i;t 1 it i;t 1 0 (cid:0) (cid:0) Above, U is an indicator of unemployment de(cid:12)ned as U = 1 E D . All other variables it it it it (cid:0) (cid:0) are de(cid:12)ned as before. 41

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Table 1A Descriptive Statistics - Disability and Health Samples All t 1 ≤ t ≤ 10 11 ≤ t ≤ 20 21 ≤ t ≤ 30 31 ≤ t ≤ 40 41 ≤ t ≤ 50 51 ≤ t ≤ 60 61 ≤ t ≤ 65 Disability Obs. 79,545 16,966 29,696 17,872 10,451 4,560 Mean 0.022 0.002 0.009 0.025 0.048 0.110 Health Obs. 87,979 16,940 29,686 17,963 10,990 8,206 3,725 469 Mean 0.136 0.052 0.081 0.125 0.187 0.313 0.417 0.497 The table displays the number of observations and mean of the disability and health limitations indicator, by cells of potential experience. All cells contain ten years except for the last one, which contains only 5. The statistics displayed correspond to the samples used in estimation of the disability and health models, respectively. The disability sample excludes retired individuals, and all individuals above 64 years of age. After this restriction, the maximum level of potential experience in the sample is 49. All variables are constructed from the PSID.

Table 1B Descriptive Statistics - Employment, Job Changes, Wage, Hours, and Household Income Variable Obs. Mean StDev Min Max E 41,840 0.94 0.24 0 1 t J 41,840 0.09 0.28 0 1 t wage 39,337 16.80 9.07 3.50 145.20 t hours 41,840 2101 646 0 4000 t earnings 41,840 38.84 25.43 0 785.83 t income (raw) 41,840 56.77 33.11 0 809.75 t income (predicted) 41,840 48.77 24.20 0.65 618.53 t The table presents descriptive statistics for variables in the PSID sample used in the estimation of the joint model of employment, job changes, wage, hours, and household income. "Predicted" household income is the level of household income predicted for a household of the average size and composition in the PSID sample. This is the household income variable used in estimation (see Appendix 3).

Table 1C Descriptive Statistics - Medical Expenditures Obs. Mean St. Dev. Min Max 2,831 3.25 10.76 0.00 317.47 Percentiles 1% 5% 10% 25% 50% 75% 90% 95% 99% 0.00 0.00 0.00 0.25 1.10 2.92 6.47 10.21 38.63 The table displays descriptive statistics for reported out-of-pocket medical expenditures in the 1999, 2001, and 2003 PSID waves. Expenditures refer to total outlays over the two calendar years preceding the survey year and are measured in thousands of year- 2000 dollars. The sample was restricted to individuals with more than 43 years of potential experience.

Table 2A Point Estimates and Standard Errors - Disability Indicator (D ) t+1 Variable cons t+1 (t+1)2/100 (t+1)3/1000 D H ζD t t Parameter γD γD γD γD γD γD σ 0 1 2 3 4 5 ζD Estimate -4.6743 0.0808 -0.1794 0.0251 1.6107 1.1744 0.9343 SE (0.2377) (0.0283) (0.1129) (0.0137) (0.0694) (0.0459) (0.0545) Obs. 87,626 The table displays parameter estimates and standard errors of the disability indicator equation. All parameters were estimated by maximum likelihood. The person-specific permanent unobserved component was integrated out of the conditional likelihood function by numerical quadrature. The sample excludes retired individuals, and individuals above 64 years of age. Table 2B Point Estimates and Standard Errors - Health Limitations Indicator (H ) t+1 Variable cons t+1 (t+1)2/100 (t+1)3/1000 H ζH t Parameter γH γH γH γH γH σ 0 1 2 3 4 ζH Estimate -2.5656 0.0103 0.0793 -0.0061 1.1191 0.9613 SE (0.0597) (0.0070) (0.0248) (0.0026) (0.0203) (0.0198) Obs. 85,295 The table displays parameter estimates and standard errors for the health limitations indicator equation. All parameters were estimated by maximum likelihood. The person-specific permanent unobserved component was integrated out of the conditional likelihood function by numerical quadrature. The sample was restricted to observations where the disability indicator is zero.

Table 2C Point Estimates and Standard Errors - Employment, Job Mobility, Wage, Hours, and Household Income Panel (a) - Employment-to-Employment Transitions (E ) t+1 Variable cons t t2/100 H t+1 ED t μ η Parameter γEE γEE γEE γEE γEE δEE δEE 0 1 2 3 4 μ η Estimate 2.3858 0.0327 -0.0482 -0.0622 -0.0337 0.8720 -0.6124 SE (0.0682) (0.0089) (0.0206) (0.0045) (0.0595) (0.0568) (0.0514) Panel (b) - Unemployment-to-Employment Transitions (E ) t+1 Variable cons t t2/100 H t+1 UD t μ η Parameter γUE γUE γUE γUE γUE δUE δUE 0 1 2 3 4 μ η Estimate 0.9398 0.0090 -0.0480 -0.0456 -0.0556 0.7598 0.4925 SE (0.0737) (0.0011) (0.0350) (0.0263) (0.0448) (0.1138) (0.1014) Panel (c) - Job Changes (J ) t+1 Variable cons t t2/100 JD t μ η Parameter γJ γJ γJ γJ δJ δJ 0 1 2 3 μ η Estimate -0.6054 -0.0111 -0.0374 -0.1034 -0.3882 0.2828 SE (0.0553) (0.0051) (0.0166) (0.0120) (0.0433) (0.0310) The table displays point estimates for the Employment, Job Mobility, Wage, Hours, and Household Income equations. All equations were estimated jointly by generalized indirect inference. Parametric bootstrap standard errors are in parentheses. When parameterizing the employment and job-change equations in the lifecycle model, the duration variables ED, UD, and JD t t t are evaluated at their sample mean (by year of potential experience).

Table 2C (continued) Point Estimates and Standard Errors - Employment, Job Mobility, Wage, Hours, and Household Income Panel (d) - Wage (w ) t+1 Variable H t+1 pw t Ψ t+1 J t+1 1-E t+1 μ εw Ψ t+1 Parameter γw 1 ρ w Φ 1 γw 2 γw 3 δw μ σ w Φ 2 Estimate -0.0015 0.9389 -0.1538 0.0197 -0.1400 0.0160 0.0975 2.0535 SE (0.0421) (0.0029) (0.0093) (0.0045) (0.0089) (0.0019) (0.0020) (0.2766) Panel (e) - Work Hours (h ) t+1 Variable cons E t+1 w t+1 D t+1 H t+1 μ η εh Parameter γh γh γh γh γh δh δh σ 0 1 2 3 4 μ η h Estimate -0.5258 0.5921 -0.1948 -0.8957 -0.1085 0.2436 0.1284 0.2269 SE (0.0089) (0.0080) (0.0160) (0.0192) (0.0065) (0.0093) (0.0114) (0.0014) Panel (f) - Household Income (y ) t+1 Variable cons w t+1 h t+1 D t+1 H t+1 U t+1 λ py t εy Parameter γy 0 γy 1 γy 2 γy 3 γy 4 γy 5 δy λ ρ y σ y κ Estimate 0.1447 0.5919 0.4535 0.1862 -0.0068 0.0269 0.2478 0.4486 0.1677 -0.1478 SE (0.0046) (0.0096) (0.0113) (0.0327) (0.0024) (0.0085) (0.0877) (0.0965) (0.0088) (0.0134) The table displays point estimates for the Employment, Job Mobility, Wage, Hours, and Household Income equations. All equations were estimated jointly by generalized indirect inference. Parametric bootstrap standard errors are in parentheses.

Table 2D Point Estimates and Standard Errors - Medical Expenditures (lnM ) t+1 Variable cons t+1 (t+1)2 (t+1)3 H Family Size year 2001 year 2003 t+1 Parameter γM γM γM γM γM γM γM γM ρ (1) σ (1) 0 1 2 3 4 5 6 7 M M Estimate -19.9013 1.1011 -0.0209 0.0001 0.4394 -0.1266 0.1391 0.2520 0.7455 0.9356 SE (7.5830) (0.3903) (0.0066) (0.0000) (0.0612) (0.0367) (0.0712) (0.0715) Obs. 2,426 Adj R-squared 0.09 The table displays point estimates and standard errors for the medical expenditures equation. The parameters were estimated by leastsquares, unless indicated otherwise. Parameters indicated by (1)were estimated by equally-weighted minimum distance matching the 0th, 2nd, and 4th order autocovariances of the residuals of the least-squares equation to those impied by an AR(1) process (the data on medical expenditures are available in two-year intervals only). The sample was restricted to individuals who were retired or had more than 43 years of potential experience. The least-squares regression includes year indicators and controls for family size.

Table 3A Evaluation of Fit: Disability and Health Descriptive Statistics - PSID Sample and Simulated Data Sample Statistic Model Overall t=5 t=10 t=20 t=30 t=40 t=50 t=60 PSID 0.02 0.001 0.004 0.016 0.030 0.075 Mean Disability Simulated 0.02 0.002 0.004 0.014 0.034 0.090 PSID 0.13 0.05 0.06 0.11 0.14 0.24 0.37 0.50 Mean Health Limitations Simulated 0.14 0.05 0.06 0.10 0.17 0.27 0.40 0.54 The table presents descriptive statistics of the PSID sample, and of data simulated from the estimated model. The descriptive statistics of simulated data are based on a simulated sample which is 10 times as large as the PSID sample, but has the same demographic structure (by potential experience) as the PSID sample. All statistics are computed using 3-year windows around the indicated value of t. For instance, t=10 corresponds to sample moments computed over all observations where t=9,10,11. The only exception is t=60, which uses only two years: t=59,60.

Table 3B Evaluation of Fit: Working-Years Variables Descriptive Statistics - PSID Sample and Simulated Data Sample Statistic Model Overall t=5 t=10 t=20 t=30 t=40 PSID 0.93 0.94 0.94 0.94 0.94 0.88 Mean Employment Simulated 0.93 0.93 0.94 0.93 0.92 0.86 PSID 0.97 0.98 0.97 0.97 0.98 0.96 Mean Emp. To Emp. Transition Simulated 0.96 0.96 0.96 0.96 0.96 0.93 PSID 0.55 0.56 0.59 0.55 0.46 0.42 Mean Unemp. To Emp. Transition Simulated 0.53 0.57 0.58 0.57 0.47 0.37 PSID 0.08 0.21 0.15 0.07 0.05 0.04 Mean Job Change if Employed Simulated 0.10 0.18 0.15 0.09 0.05 0.02 PSID 11.87 4.21 6.61 12.49 17.34 21.22 Mean Employment Duration Simulated 14.49 5.05 8.78 15.57 20.85 22.55 PSID 1.88 1.65 1.82 2.11 1.75 1.70 Mean Unemployment Duration Simulated 1.35 1.68 1.73 1.42 1.07 0.68 PSID 9.60 3.00 4.78 9.88 14.98 19.26 Mean Job Duration Simulated 9.35 2.44 4.53 9.73 15.18 18.14 PSID 0.39 0.35 0.37 0.40 0.40 0.41 St. Dev. Log Wage Simulated 0.41 0.39 0.40 0.41 0.41 0.42 PSID 0.49 0.35 0.42 0.49 0.55 0.75 St. Dev. Log Hours Simulated 0.45 0.43 0.43 0.44 0.48 0.57 PSID 0.78 0.55 0.68 0.79 0.85 1.09 St. Dev. Log Earnings Simulated 0.70 0.63 0.66 0.70 0.75 0.82 St. Dev. Log Household Nonasset PSID 0.49 0.45 0.47 0.50 0.51 0.56 Income Simulated 0.48 0.45 0.47 0.49 0.49 0.50 The table presents descriptive statistics of the PSID sample, and of data simulated from the estimated model. The descriptive statistics of simulated data are based on a simulated sample which is 10 times as large as the PSID sample, but has the same demographic structure (by potential experience) as the PSID sample. All statistics are computed using 3-year windows around the indicated value of t. For instance, t=10 corresponds to sample moments computed over all observations where t=9,10,11. The only exception is t=40, which uses only two years: t=39,40.

Table 3C Evaluation of Fit: Out-of-Pocket Medical Expenditures Descriptive Statistics - PSID Sample and Simulated Data Sample Statistic Data Overall 44 ≤ t ≤ 48 49 ≤ t ≤ 53 54 ≤ t ≤ 58 59 ≤ t ≤ 63 64 ≤ t ≤ 68 PSID 1.576 1.446 1.290 1.651 1.904 1.886 Mean Simulated 1.342 1.397 1.352 1.266 1.260 1.436 PSID 3.758 2.260 1.781 4.266 5.847 4.494 Standard Deviation Simulated 3.519 3.540 3.916 3.117 3.363 3.606 The table presents descriptive statistics of the PSID sample, and of data simulated from the estimated model. The numbers refer to yearly figures, measured in thousands of year 2000 dollars. The descriptive statistics of simulated data are based on a simulated sample which is 10 times as large as the PSID sample, but has the same demographic structure (by potential experience) as the PSID sample. Statistics are displayed for the entire sample and for 5-year cells of potential experience. For instance, 44 ≤ t ≤ 48 includes all years between 44 and 48. For the PSID sample, only observations with strictly positive levels of medical expenditures are included in the table (only this subset of the data was used in estimation).

Table 4 Welfare Gains of Full Insurance: Equivalent Variation in Lifetime Consumption - Baseline Case (1) (2) (3) (4) (5) (6) (7) (8) Panel (a) - Effect ADJUSTED for Variation in Mean Income Profile (Actuarially Fair Insurance) Insured Source of Risk Disability Health Unemp. MedCosts Wage Job Change Hours Res. Income Equivalent Variation 0.02% 0.00% 0.04% 0.01% 1.20% 0.72% 0.45% 2.19% (% of Lifetime Consumption) Panel (b) - Effect NOT Adjusted for Variation in Mean Income Profile Insured Source of Risk Disability Health Unemp. MedCosts Wage Job Change Hours Res. Income Equivalent Variation 0.04% 0.01% 0.62% 0.24% 0.62% -0.35% -0.13% 0.28% (% of Lifetime Consumption) The table presents the welfare gains from fully insuring individual sources of risk. For each source, the calculation compares expected lifetime utility under the full-uncertainty scenario versus a scenario in which one particular source of risk is fully insured so that its realization has no effect on net income. Welfare gains are expressed in terms of the "equivalent compensating variation", that is, the percentage variation of lifetime consumption (in the full-uncertainty world) that would make the household indifferent between living in that compensated full-uncertainty world versus living in the alternative insured scenario. Panel (b) adjusts mean income under the insured scenario so that it equals mean income under the uncompensated, full-uncertainty scenario.

Table 5 Contribution of Individual Sources of Risk to Precautionary Saving - Baseline Case Column (1) (2) (3) (4) (5) (6) (7) (8) Source of Risk Disability Health Unemp. MedCosts Wage Job Change Hours Res. Income Contribution 2.34% 8.56% 3.77% 5.57% 19.13% 9.34% 8.15% 43.14% The table displays the contribution of individual sources of risk to the accumulation of precautionary wealth. Precautionary wealth is defined here as the difference between mean precautionary wealth under full uncertainty and mean precautionary wealth under no uncertainty. The calculation then computes the difference between mean precautionary wealth under full uncertainty and mean precautionary wealth under a scenario in which one particular source of risk is fully insured so that its realization has no effect on net income. This difference is then expressed as a percentage of mean precautionary wealth. The contributions are normalized to sum to 100%.

Table 6 Average Lifetime Earnings Regression Dependent Variable: (Simulated) Average Lifetime Earnings Variable cons ζD (ζD)2 (ζD)3 ζH (ζH)2 μ μ2 μ3 η η2 pw T (pw T )2 (pw T )3 D T H T E T μ*pw T Estimate 3.358 -0.003 -0.016 -0.006 -0.015 -0.007 0.357 -0.038 0.005 0.149 -0.007 0.240 0.008 -0.018 -0.108 0.000 -0.040 -0.011 SE (0.004) (0.001) (0.001) (0.000) (0.001) (0.001) (0.002) (0.001) (0.000) (0.001) (0.001) (0.004) (0.005) (0.004) (0.005) (0.003) (0.004) (0.004) Obs. 48,030 Root MSE 0.184 Adj R-squared 0.88 212 The table presents estimation results for the average lifetime earnings regression. The regression uses data simulated from a joint model of labor earnings to determine the predictive power of variables known in the last year of a worker's career for his/her average lifetime earnings, as calculated by the Social Security Administration (taking, for instance, only the 35 years with highest earnings to calculate the average). The dependent variable is simulated average lifetime earnings. The independent variables are all permanent variables variables in the joint model of earnings, all time-varying variables in the last year of career only (year T), and some interactions. After experimentation with different specifications, only the interactions with a significant predictive power were included in the equation.

Table 7A Welfare Gains of Full Insurance: Sensitivity to Degree of Risk Aversion and Discount Factor Equivalent Variation in Lifetime Consumption - Effect Adjusted for Variation in Mean Income Profile (Actuarially Fair Insurance) Source of Risk Disability Health Unemp. MedCosts Wage Job Change Hours Res.Income Panel (a) Coefficient of Relative Risk Aversion α = 3.0 0.02% 0.00% 0.04% 0.01% 1.20% 0.72% 0.45% 2.19% α = 5.0 (1) 0.03% 0.01% 0.07% 0.02% 1.95% 1.17% 0.72% 4.79% Panel (b) Discount Factor β = 0.967 (2) 0.02% 0.00% 0.04% 0.01% 1.20% 0.72% 0.45% 2.19% β = 0.94 (1) 0.01% 0.00% 0.05% 0.02% 0.97% 0.75% 0.44% 3.15% β = 0.90 (1) 0.01% 0.00% 0.05% 0.02% 0.73% 0.77% 0.42% 4.19% The table presents the welfare gains from fully insuring individual sources of risk (see Table 4.4A, Panel a) for different values of the coefficient of risk aversion α and the discount factor β. (1) All other parameters are held at their baseline value. (2) β = 1/(1+r).

Table 7B Contribution of Individual Sources of Risk to Precautionary Saving: Sensitivity to Degree of Risk Aversion and Discount Factor Source of Risk Disability Health Unemp. MedCosts Wage Job Change Hours Res.Income Panel (a) Coefficient of Relative Risk Aversion α = 3.0 2.34% 8.56% 3.77% 5.57% 19.13% 9.34% 8.15% 43.14% α = 5.0 (1) 2.26% 8.14% 3.21% 5.11% 19.71% 9.37% 8.16% 44.04% Panel (b) Discount Factor β = 0.967 (2) 2.34% 8.56% 3.77% 5.57% 19.13% 9.34% 8.15% 43.14% β = 0.94 (1) 2.14% 10.44% 2.93% 8.77% 16.82% 6.91% 8.03% 43.96% β = 0.90 (1) 2.03% 10.79% 2.80% 8.81% 15.62% 7.61% 7.86% 44.48% The table presents the contribution of individual sources of risk to the accumulation of precautionary wealth (see Table 4.5A) for different values of the coefficient of risk aversion α and the discount factor β. (1) All other parameters are held at their baseline value. (2) β = 1/(1+r).

Figure 1 Mean Nonasset Income and Consumption by Potential Experience 45 40 35 30 25 20 15 10 5 0 1 6 11 16 21 26 31 36 41 46 51 56 61 66 Potential Experience (years) $ 0002-raeY fo sdnasuohT Income Consumption The figure displays simulated mean nonasset income and simulated (optimal) mean consumption. The mean profiles are based on simulating the lifecycle model for 50,000 households. Period 43 is the exogenous retirement date. Income was parameterized such that income and consumption are measured in thousands of year-2000 dollars. Figure 2 Optimal Consumption at t=1 by Cash on Hand 30 25 20 15 10 5 0 0 10 20 30 40 50 60 70 80 90 100 Cash on Hand in Thousands of Year-2000 $ fo sdnasuohT ni noitpmusnoC $ 0002-raeY The figure displays optimal consumption in the first period of career as a function of cash on hand for an employed, healthy individual with mean persistent wage and household income components. The figure corresponds to the baseline model, in which individuals face credit constraints.

Figure 3 Optimal Consumption at t=1 by Cash on Hand and Persistent Household Income Component 25 20 15 10 5 0 0 10 20 30 40 50 60 Cash on Hand in Thousands of Year-2000 $ fo sdnasuohT ni noitpmusnoC $ 0002-raeY The figure displays optimal consumption in the first period of career as a function of cash on hand for different levels of the persistent household income component, for an employed, healthy individual with mean (persistent) wage. The figure corresponds to the baseline model, in which individuals face credit constraints. Figure 4 Optimal Consumption at t=1 by Cash on Hand and Persistent Wage 40 35 30 25 20 15 10 5 0 0 10 20 30 40 50 60 70 80 90 100 Cash on Hand in Thousands of Year-2000 $ fo sdnasuohT ni noitpmusnoC $ 0002-raeY The figure displays optimal consumption in the first period of career as a function of cash on hand for different levels of the persistent wage component, for an employed, healthy individual with mean persistent household income component. The figure corresponds to the baseline model, in which individuals face credit constraints.

Figure 5 Optimal Consumption at t=1 by Cash on Hand and Persistent Wage (for wider range of values of persistent wage) The figure displays optimal consumption in the first period of career as a function of the two continuous state variables in the numerical solution to the household consumption problem: cash on hand and the persistent wage. The figure refers to an employed individual in good health and with mean persistent household income component. It corresponds to the baseline model, in which individuals face credit constraints. Figure 6 Mean Asset Holdings by Potential Experience 350 300 250 200 150 100 50 0 1 6 11 16 21 26 31 36 41 46 51 56 61 66 Potential Experience (years) $ 0002-raeY fo sdnasuohT Full Uncertainty No Uncertainty The figure displays the mean level of assets held by households in simulations of the lifecycle model. The curve above represents mean asset holdings under the full-uncertainty scenario, while the curve below displays mean asset holdings under no uncertainty. The mean profile of net income is identical under both scenarios.

Figure 7 Mean Precautionary Wealth by Potential Experience 35 30 25 20 15 10 5 0 1 6 11 16 21 26 31 36 41 46 51 56 61 66 Potential Experience (years) $ 0002-raeY fo sdnasuohT The figure displays mean precautionary wealth by year in the lifecycle. The figures are based on a simulation of a large cross-section of households. Mean precautionary wealth is the difference in mean asset holdings between the full-uncertainty scenario and the no-uncertainty scenario in which net income follows its mean profile deterministically. Figure 8 Ratio of Precautionary Wealth to Total Wealth by Potential Experience 1.20 1.00 0.80 0.60 0.40 0.20 0.00 1 6 11 16 21 26 31 36 41 46 51 56 61 66 Potential Experience (years) oitaR The figure displays the ratio of mean precautionary wealth to mean total wealth by year in the lifecycle. The figure is based on simulation of a large cross-section of households. Mean total wealth is the mean of asset holdings under the full-uncertainty world. Mean precautionary wealth is the difference in mean asset holdings between the full-uncertainty scenario and the no-uncertainty scenario in which net income follows its mean profile deterministically.