Social Security, Pensions and the Savings and Retirement Behavior of Households 1

Transcription

1 Social Security, Pensions and the Savings and Retirement Behavior of Households 1 Wilbert van der Klaauw University of North Carolina at Chapel Hill 2 and Kenneth I. Wolpin University of Pennsylvania 3 November 2002 Preliminary and Incomplete Abstract In this paper we formulate and estimate an explicit sequential decision model of savings and work behavior of elderly individuals and couples. More specifically, we develop a stochastic dynamic model of retirement behavior in which forward-looking individuals are making employment and savings decisions in a setting that accounts for uncertainty about future labor market opportunities, health insurance coverage, health status, survival, and the future generosity of the social security system. The choice environment is characterized by an incomplete health insurance market, the presence and characteristics of employer provided health insurance and private pension plans, the Social Security and Medicare insurance system, and a capital market which applies borrowing limits. The model is estimated using longitudinal data from the Health and Retirement Study. The estimates are used to evaluate the importance of potential earnings, health, social security benefits and employer provided health insurance coverage and pensions in explaining observed retirement behavior, and to simulate the impact of potential social security reforms on life cycle employment and savings behavior and on welfare. 1 We are grateful for financial support from the National Institute on Aging. 2 Mailing address: Department of Economics, Gardner Hall, CB#3305, UNC-Chapel Hill, Chapel Hill, NC vanderkl@ .unc.edu. 3 Mailing address: Department of Economics, 3718 Locust Walk, University of Pennsylvania, Philadelphia, PA wolpink@econ.sas.upenn.edu.

2 1 Introduction The literature on retirement behavior has grown enormously during the last fifteen years. Much of that growth has been due to recent methodological advances in the structural estimation of dynamic discrete choice models of behavior under uncertainty. Unlike earlier static lifetime models (e.g., Fields and Mitchell, 1984), dynamic models account better for the sequential nature of the retirement process in which individuals adjust their behavior as events unfold. Structural estimation of the fundamental parameters of preferences and constraints as opposed to reduced form analyzes permits the simulation of policy experiments that act directly on constraints and which may be outside of current or prior policy regimes. Among the first attempts to formulate and estimate a forward looking model of retirement behavior was by Gustman and Steinmeier (1986). The model was not really dynamic because it ignored uncertainty. In the life cycle formulation (no uncertainty), the age of retirement can be optimally decided at the beginning of the life cycle. The model also assumed that individuals could borrow and lend at a fixed interest rate and did not incorporate health constraints. However, Gustman and Steinmeier did model in detail the post-retirement budget constraint, carefully accounting for income flows derived from private pensions and social security. The model was quite successful in fitting the data and in particular was able to predict the two spikes in the retirement hazard at ages 62 and 65. However, as pointed out by Rust and Phelan (1997), with perfect capital markets, where individuals can smooth consumption by borrowing against pension and social security income, it is unclear why individuals would delay retirement and thus why we would observe spikes. Berkovec and Stern (1991) estimated a dynamic model of retirement in which a person chooses among three alternatives each period: to work full-time, part-time or not at all. Unlike previous models of retirement behavior, their model does not assume that retirement is an absorbing state, allowing for temporary retirement. The model incorporates uncertainty in future wage earnings as well as time-invariant unobserved heterogeneity in wages and preferences. However, Berkovec and Stern do not account for Social Security or private pensions, they assume that an individual s health status does not change over time, and they do not model the asset accumulation (and decumulation) process. Phelan and Rust (1991) introduced social security benefits into a dynamic retirement model similar to that of Berkovec and Stern. They assumed that workers had semi-rational expectations in that they correctly anticipated the increase in benefits in the 1970s, but did not expect any benefit changes thereafter. Unlike Berkovec and Stern s model, Phelan and 1

3 Rust specified a stochastic process for the evolution over time of the individual s health status. The model was found to perform well in predicting observed retirement patterns at ages 60 to 63 as well as the peak in early retirements at age 62. However, it was unable to fully capture the peak at the normal retirement age of 65. Lumsdaine, Stock and Wise (1992) analyzed the departure rates of older workers at a Fortune 500 company. Workers at this company were covered by a defined benefit pension plan that provided substantial incentives to remain with the firm until age 55 and then substantial incentives to leave the firm before age 65. In 1982 the firm had offered a temporary early retirement plan to its workers over age 55. Lumsdaine et al. used pre-1982 data to estimate four models and compared each model s policy forecast to the actual observed change in the pattern of retirement following the introduction of the new incentive plan. While all four models fit the pre-1982 equally well, only the structurally estimated models were able to predict the large increase in departure rates induced by the window plan. None of the models were able to explain the peak in departure rates at age 65, however. Similar problems in explaining the peak in the retirement hazard at that age in data on other firms were reported in a subsequent paper by Lumsdaine, Stock and Wise (1993). More recently Rust and Phelan (1997) provided an empirical analysis of how the Social Security and Medicare insurance system affect retirement behavior in a world of incomplete markets for loans, annuities, and health insurance. The model accounts for individual subjective uncertainty about mortality, marital and health status, employment, income and health expenditures. Using a RHS sample of men who did not have a private pension plan, the model was found to fit the data well and was able to explain the peaks in the retirement hazard at ages 62 and 65. The peak at age 62 is a result of borrowing constraints that prevent lower wage individuals with relatively little tangible wealth from retiring prior to age 62. The peak at 65 is partly due to incomplete health insurance markets that make it too risky for individuals who don t have access to employer provided retiree health insurance to retire prior to being eligible for Medicare at age 65, and partly due to the negligible delayed retirement credit that made social security substantially actuarially unfair after age 65 for these individual. In a pair of recent papers Blau and Gilleskie (2000, 2001) extended the analysis by Rust and Phelan by focusing on the incentives provided by employment conditioned health insurance in explaining the timing of retirement by individuals as well as by couples. Preliminary estimates of approximation models as well as a structural dynamic model suggest that while health insurance is an important determinant of retirement decisions, it plays a relatively 2

4 small role in explaining the observed correlation in labor force decisions of spouses. These empirical studies and others, including those not based on a structural dynamic modeling approach, have therefore identified a number of factors that are related to retirement behavior: health status, private and public pension availability and generosity, Medicare and employer provided health insurance coverage, market work opportunities, and wealth. However, mainly because of data limitations as well as the associated complexity in solving and estimating such models, none of the previous structural dynamic models of retirement decisions have been able to accommodate all of these features in any single estimable behavioral model. In particular, structural retirement models that have been estimated typically do not allow for savings decisions, but instead assume incomplete capital markets in which individuals can neither borrow or lend. 1 Although obviously literally false, it is not clear whether this is a reasonable approximation and thus to what extent the estimation results are sensitive to it. Other empirical studies have found the effect of wages and assets other than social security and pension annuities on the timing of retirement to be weak (Blau (1994), Diamond and Hausman (1984), Sickles and Taubman (1986)). However, all these studies were based on static or reduced form analyzes that do not capture the complex interactions that exist between savings, health status, social security benefits, health insurance coverage and work decisions. Moreover, many of these studies take accumulated savings or assets to be exogenous in their analysis. Several studies, such as those by Feldstein (1974) and Bernheim and Levin (1989) have found social security to depress savings, which suggests that the exogeneity assumption may be incorrect. In our analysis, in contrast, we allow for savings and limited net borrowing. known about the interactions between savings and retirement decisions. Little is However, in an environment with an incomplete health insurance market, capital market imperfections and uncertain future lifetimes, health and wages, savings can be expected to be an important device in smoothing consumption and in choosing an optimal retirement age. With uncertain lifetimes and a borrowing constraint, our model will include a consumption smoothing and precautionary savings motive and individuals and couples in our model may also save to leave a bequest. Modeling savings decisions, or equivalently consumer expenditure decisions, implies that 1 Similarly, in the extensive literature on consumption and savings behavior, work decisions have typically been ignored or treated as exogenous, while a few studies (such as Browning and Meghir (1991) have instrumented employment status in the estimation of conditional consumption equations. 3

5 one of the state variables (assets at the beginning of the period) and a choice variable (consumption) are both continuous and therefore may take on an infinite number of possible values. In this paper we extend recent methods introduced by Keane and Wolpin (1994) for solving dynamic optimization models that include discrete and continuous choice variables and state space elements, to deal with this well known complication in solving and estimating these types of models. In addition to savings and limited borrowing, our model incorporates many of the determinants of retirement behavior listed above. Although we treat health and mortality as not subject to direct choice (they are affected by other prior decisions), individuals do have uncertain lifetimes, and health status is allowed to shift the worker s mortality hazard, and to affect the ability to work and enjoy leisure. Similar to Rust and Phelan (1997) and Blau and Gilleskie (2001), we incorporate information about individual health insurance coverage. We also analyze the role of market work opportunities, Medicare and private pension plans. We carefully incorporate social security rules into the budget constraint. However, as discussed in more detail below, in contrast to existing studies, we will not assume that individuals treat social security policy as time-invariant. The assumptions of myopic beliefs about the social security system are clearly unrealistic given the long history of changes in the social security rules and benefit levels which have been enacted over the period, with major changes in 1972, 1977 and As Moffitt (1987) has argued, the magnitude of behavioral responses to policy changes will depend strongly on the extent to which these policy changes were anticipated. While Moffitt concluded that the changes in the early seventies were fully anticipated, Burtless (1986) concluded the opposite. As will be shown later, when discussing the different types of subjective expectations we plan to use, a considerable number of individuals in the HRS appear to consider a future policy change very likely. If those responses represent actual expectations, one should assume they are taken into account in the decision-making. provide a method for doing so. In our analysis we will consider single individuals as well as couples. We propose a structural dynamic model where labor supply and savings decisions are made jointly by both spouses, and which incorporates the various determinants discussed above for each individual. 2 2 Two recent studies have estimated structural joint retirement models. Gustman and Steinmeier (2000) estimated a model that was not really dynamic because it ignored uncertainty, so that optimal retirement decisions can be determined many years prior to retirement. In their model each spouse is assumed to maximize own utility given the other spouse s labor supply choice, leading to a non-cooperative equilibrium outcome. They found strong evidence for joint consumption of leisure. More recently Blau and Gilleskie (2001) estimated a dynamic stochastic family utility model of joint labor supply decisions of couples and like We 4

6 We explicitly incorporate the social security benefit rules which apply to couples, as well as allow for health insurance coverage through the spouse. We also allow retirement preferences of the husband and wife to be correlated because people who share the same tastes are more likely to marry. A final contribution of this paper, which we already eluded to a little earlier, concerns the use of subjective expectations data in estimating the model. During the past fifteen years or so there has been an increased interest in the analysis of self-reported expectations or intentions about future life events and choices, such as mortality, labor force behavior, income, schooling and occupation (for a survey, see Van der Klaauw (2000)). This research has shown that these frequently available data contain very valuable information on individual decision making and the expectations formation process. The Health and Retirement Study contains a set of probabilistic questions, on, among others, retirement and longevity expectations. For example, all those employed in 1992 were asked for their subjective probability that they would be working full-time after reaching ages 62 and Honig (1994) analyzed these expectations data to examine the extent to which women take into account their own economic opportunity set (their wages, employer provided coverage for health and disability insurance and changes in pension and social security wealth) in forming retirement plans. The estimates indicated a strong dependence of these probabilities on the expected net rewards to working. The same expectations data, as well as similar probabilistic data on subjective life expectancies, were analyzed by Hurd and McGarry (1995). Their examination of the responses indicated that these subjective probabilities were both internally consistent (in particular, they imply conditional probabilities in the unit interval) and that they correlated closely with observed retirement probabilities and life expectancies in the population. Subjective probabilities of survival beyond age 75 and 85 were found to covary with other variables (such as social status or smoking behavior) in the same way actual outcomes vary with these variables. They concluded that these subjective probabilities have great promise for making a substantial contribution to our understanding of intertemporal decision making under uncertainty. We believe that the potential value of expectations data as a means of understanding behavior has been overlooked. Reported expectations about future choices have precise inter- Gustman and Steinmeier attribute most of the positive correlation in labor force participation choices of older married couples to a strong preference for shared leisure. 3 Dominitz and Manski (1997) argue that wording subjective expectations to ask directly about probabilities greatly enhances the utility of such data. 5

7 pretations within the context of dynamic behavioral models. Just as current choices are taken to portray optimal behavior given current information, expectations about future choices portray optimal future behavior conditional on current information. The same model can explain both objective behavior and subjective expectations. As a result, subjective data provide useful information about the decision process in the same way as does objective data on behavior. A recent study by Van der Klaauw (2000) provides an early illustration. That study demonstrated empirically the potential for increased estimation precision from combining expectations data and data on current choices. Especially because the HRS is at present only a very short panel, integrating expectations data consistently and rigorously into a formal behavioral model can potentially provide crucial additional inferential information and greatly enhance its current value. The remainder of the paper is organized as follows. In the next section we discuss the model. In section 3 we discuss the solution method used to solve the model and section 4 describes the HRS data. The econometric specification and estimation method are discussed in section 5. Finally, preliminary estimation results will be discussed in Section 6. 2 The Model The model represents the decision problem of an individual of given gender or a married couple. The optimization problem, consistent with the data available for estimation, begins at a point in the middle of the household s life cycle. Initial conditions are those that prevail at that life cycle point; variation among agents in initial conditions are not explicitly considered until the model s solution and estimation method are discussed. 2.1 Choice Set An unmarried individual of gender j (j = m, f) at each discrete age a chooses consumption, Ca j C, where C is minimum consumption for a single individual, and hours worked in the labor market, h j a. Hours worked is allowed to take on only two values, part-time hours (h j = 1) and full-time hours (h j = 2). In addition, the employment decision is constrained by whether or not the individual worked in the previous period; an individual who was working at age a 1 may choose to work at age a in the old firm, h jo, or in a new firm, h jn. An individual who did not work in the previous period can only work in a new firm. Employment choices are further restricted in that all individuals are assumed to stop working (permanently) at 6

8 age A. As will be described later, besides a different wage, new jobs are also characterized by the health insurance they offer. Thus, the choice set at age a for an individual who worked in the previous period consists of all feasible combinations of C j a, h jo, h jn ; for an individual who did not work in the previous period the choice set consists of the feasible combinations of C j a, h jn. 4 The choice set at age a is denoted by D j a and a specific choice within that set by d j a. A married couple chooses the consumption and hours of work of each; the choice set is given by D mf a = D m a D f a and any choice element within the set by d mf a. The minimum consumption level for the couple is denoted by C mf. Households, singles or couples, are assumed to be able to borrow and lend and thus may smooth consumption over the life cycle, although net borrowing is restricted (see below). Net assets carried over from a (to a + 1), W a+1, is determined residually from the consumption and labor force status decision at a. Consumption, and thus net assets, is treated as continuous. 2.2 Preferences Each individual of gender j is assumed to have a well-defined preference function over own consumption and labor force status, namely U j a = U j (C j a, h j a; j, X j, Z j a, ɛ cj a, ɛ hj a, µ uj ) where X represents the individual s race, Z is an indicator of health, the ɛ s are age-varying shocks to the marginal utility of consumption and hours worked, and µ uj is a time-invariant unobserved individual specific heterogeneity component. The marriage decision is not explicitly modeled. 5 The decision model is assumed to pertain only to ages at and beyond some age a 0, from which time it is assumed that an unmarried individual will forever remain single. A married individual may, however, become single after that age due to the death of the spouse, but may not remarry. 2.3 Optimization Problem for Single Individuals An individual who is single at some age a, a a 0, having been single up to age a or having been previously married but currently widowed, maximizes the expected present discounted value of remaining lifetime utility. The time of death is uncertain, although there is a known finite maximum length of life, a = A. Individuals are assumed also to obtain utility from bequests. The utility obtained from making a bequest if the individual were to die at age a is 4 There are five possible hours combinations in the first case, given that working in a new firm and in an old firm are mutually exclusive alternatives, and three in the second. 5 Exact functional forms are presented later. 7

9 B j a 1(W a, µ qj ), where µ qj captures unobserved heterogeneity in utility obtained from leaving bequests. Note that, although the bequest actually occurs at a, the utility associated with the bequest is derived at a 1, while the individual is still alive. In the last potential decision period, at age A 1, the individual s total utility is therefore U j A 1 + B j A 1. 6 Given the state space at A 1, Ω j A 1 (see below), the individual chooses the level of consumption, and thus net assets carried forward, and labor force status that maximizes this terminal period total utility, i.e., d j A 1 = d j A 1(Ω j A 1). Thus, the maximized value of each of the two components of total utility at A 1 can be written as a function of the state space. We denote the maximized value of the first component of utility by G j A 1(Ω j A 1) and that of the second (bequest) component as Q j A 1(Ω j A 1). At any age a, the maximized expected present value of remaining total lifetime utility given the state space at a, denoted by V j a (Ω j a), is the sum of the maximized expected present values of the remaining utility associated with the two components, G j a(ω j a) and Q j a(ω j a). Each of these components satisfy a Bellman equation, as does their sum. Specifically, V j a (Ω j a) = G j a(ω j a) + Q j a(ω j a) [ = max U j a (Ω j a) + δπ sj Da j a E(G j a+1(ω j a+1) Ω j a, d j a(ω j a)) (1 πa sj )Ba(W j a+1 (Ω j a, d j a(ω j a))) + δπa sj E(Q j a+1(ω j a+1) Ω j a, d j a(ω j a)) ] [ = max U j (1) a (Ω j a) + (1 π sj Da j a )Ba(W j a+1 (Ω j a, d j a(ω j a))) + δπa sj E(Va+1(Ω j j a+1) Ω j a, d j a(ω j a)) ], where E is the expectations operator conditional on the individual s information set at age a, π sj a is the one-period survival rate (from a to a + 1) for a person of gender j and δ is the discount factor. 2.4 Optimization Problem for Married Couples In considering the objective function of married couples, to avoid notational complexity, assume that the husband and wife are of the same age. Then, if each is alive at age A 1, the couple chooses consumption levels and hours of work of each to maximize a weighted average of the individual expected values of the remaining lifetime utilities. Specifically, letting θ be the weight placed on the husband s utility, (2) V mf A 1(Ω mf [ A 1) = max θ(ω mf A 1) ( ) UA 1 m + BA 1 m + (1 θ(ω mf A 1)) ( U f A 1 + BA 1)] f, D mf A 1 6 At ages prior to A 1, the bequest is multiplied by the probability of not surviving to the next period. Note that the bequest is not discounted because, as noted, utility is received during the last period of life. 8

10 where Ω mf A 1 denotes the state space for a couple, i.e, the Cartesian product of the individual state spaces. Note that the weight in any period, as written, is time-varying as a function of the current period state space. With separate utility functions for husband and wife, this framework incorporates interactions between spouse s incentives and potential conflicts in their objectives in determining their savings and timing of retirement. Since wives are typically younger than their husbands and life expectancy for women exceeds that for men, wives would generally prefer to save more for retirement than do their husbands. Therefore we would expect households in which wives have greater relative bargaining power to accumulate greater net worth as they approach retirement. 7 In (2), θ can be interpreted as a measure of the husband s relative bargaining power, and will depend on the age difference between the husband and wife and may depend on other individual specific attributes, such as pension accruals and work histories, as well as unobserved individual characteristics µ θm and µ θf. In decision periods prior to A 1, the couple takes into account the possibility that either or both may not survive into future periods. The expected present discounted value of the couple s remaining lifetime utility is given by Va mf (Ω mf a ) = max (3) Da mf [ θu m a (Ω mf +(1 π sm a a +(1 πa sm )π sf +(1 π sf ) + (1 θ)u f a (Ω mf a )(1 πa sf )[θba m (W a+1 (Ω mf a )πa sm a δ[θe(q m a+1(ω f a+1 Ω mf a ) + πa sm πa sf δe(va+1(ω mf mf a, d mf a δ[(1 θ)e(q f a+1(ω m a+1 Ω mf a+1 Ω mf a, d mf a ) )) + (1 θ)ba f (W a+1 (Ω mf a, d mf a ))], d mf a )) + (1 θ)e(v f a, d mf a )) + θe(v m a+1(ω f a+1 Ω mf a a+1(ω m a+1 Ω mf a, d mf a ))], d mf a )) ] where the argument, Ω mf a, has been suppressed in θ and in d mf a for convenience. 8 The value function in (3) is the sum of (i) the share-weighted average of the current individual utilities; (ii) the probability that they both survive times the couple s expected remaining lifetime utility one-period ahead; (iii) the probability that neither the husband nor the wife survives beyond the period times the share-weighted average of their individual utilities from a bequest; (iv) the probability that the wife survives beyond the period but the husband does not multiplied by the sum of the husband s share times his expected utility of the bequest that the wife will make upon her death (which depends on her future savings decisions) and the wife s share times her expected remaining lifetime utility as a single individual; and (v) the probability that the husband survives beyond the period but the wife does not multiplied by the sum of the wife s share times her expected utility of the bequest that the husband will make upon 7 See Lundberg and Ward-Batts (2000) for an empirical test of this hypothesis. 8 In (3), the mortality hazards of the husband and wife are assumed to be independent (conditional on the state variables in Ω mf a ). Later, we allow them to be correlated through assortative mating on unobservables. 9

11 his death (which depends on his future savings decisions) and the husband s share times his expected remaining lifetime utility as a single individual. If the husband and wife are of different ages, the Bellman equations are combinations of (1) and (3). Specifically, if the age difference is k periods, then from the younger spouse s age A k to age A 1, the value function will be that of a single person as in (1). At the younger spouse s age A k 1, when the older spouse is age A 1, the value function for the couple will be given by (3) with the survival probability of the older spouse set to zero. In periods prior to the previous one, the value function is given exactly by (3). 2.5 Budget Constraint Define y j ae to be the amount of labor market earnings at age a of an individual of gender j and y j an the amount of non-earned income. Labor market earnings is the product of the hourly wage, ω j a, and hours worked. ỹ j ae denotes labor market earnings net of social security payroll taxes. The payroll tax rate is τ s (=7.65% in 1992) and is applied to earnings up to a maximum of (=$55,500 in 1992). Therefore ỹ j ae = ω j ah j a(1 τ s )[I(ω j ah j a y max c )] + [ωah j j a τ s yc max ]I(ωah j j a > yc max ) Non-labor income at age a is the sum of interest income (payments) on net assets carried over from the previous period, rw a, where r is the fixed (borrowing and lending) rate of interest, retirement income from social security, S j a, and retirement income from private pensions, P j a. In addition to a homogenous consumption good, the budget constraint also incorporates expenditures that arise from poor health. Specifically, an individual in poor health at age a, Z j a = 0, pays a cost of c z if the individual is not covered by health insurance; an individual in poor health who is covered by health insurance at age a (hi j a 1) is assumed to have no out-of-pocket expenses as is a person in good health, Z j a = 1 9. The budget constraint for a single individual of gender j is (4) C j a + W j a+1 = W j a + (1 τ)[ỹ j ae + rw j a + P j a ] + S j a c z [1 Z j a][1 I(hi j a 1)], where I( ) is an indicator function equal to unity it the expression inside the parentheses is true and zero otherwise, and we have assumed a proportional income tax rate of τ. Similarly, 9 We therefore assume that the level of health expenditures is not subject to choice and we will also assume below that health expenditures do not directly affect health status transitions. Health care costs are simply physical maintenance costs which are higher in worse health states. 10

12 the budget constraint for a married couple is (5) Ca+W j mf a+1 = Wa mf +(1 τ)[ j j ỹae+rw j a mf + j P j a ]+ j S j a j c z [1 Z j a][1 I(hi j a 1)] The individual or couple also faces a borrowing constraint, namely that W a+1 W a+1, where the lower bound on net assets is a function of age and may be negative. Given stochastic wage and health shocks, it is possible that an individual who is in debt would be unable to maintain minimum consumption and not violate the borrowing constraint. If this occurs we assume that the individual receives a transfer of just the amount necessary to (i) pay the interest on the loans, (ii) pay enough of the principle to meet the next period borrowing constraint, and (iii) meet the minimum consumption level. Therefore, an individual must consume their assets before being eligible for a positive transfer. 2.6 Wage Offers Wage offers are the product of a skill rental price, ρ, and an individual s stock of human capital, K. An individual accumulates human capital through general work experience and work experience specific to a job (tenure). The rental price of human capital has a firmspecific component, η, that is constant over tenure within the firm. The rental price also differs between part- and full-time employment. The wage offered to an individual is thus (6) ω j a = ρ(h j a, η)k j a(e j, H j a, T j a, a, j, X j, ɛ ω a, µ wj ), where E is years of schooling (assumed fixed at the initial age), H a is cumulative hours worked up to age a, T a is tenure as measured by the cumulative hours worked for the current employer up to age a, X represents the individuals race and ɛ ω a is a random shock to an individual s human capital at age a. In addition, µ wj captures time-invariant individual specific wage heterogeneity. As already noted, an individual who was working at a 1 receives a wage offer from the same firm as well as an offer from a new firm. The wage offer from the new firm differs from that of the old firm in that tenure is zero at the new firm (T = 0), there is a new firm-specific component to the rental price for the individual s human capital stock (η) and a different human capital shock (ɛ ω a ). 2.7 Social Security Income Individuals generally become eligible to apply for social security at age 62. To be eligible for benefits at that age on the basis of one s own employment history requires that the individual 11

13 has accumulated 40 quarters of covered earnings. In 1992, an individual accumulated one quarter for each $570 of annual earnings (up to a maximum of 4 quarters). Benefits (the primary insurance amount or PIA), given eligibility, depend on an individual s average indexed monthly earnings (AIME) calculated on an annual basis. 10 In calculating this average, there is a maximum amount for covered monthly earnings within a year, $4,625 in 1992 ($55,500 annual earnings is the maximum taxable earnings, ye max ), and the lowest five years of indexed earnings are dropped. Given that we observe an individual s AIME only in the middle of the life cycle, in order to avoid having to keep track of the entire history of earnings, in updating the AIME from that first observation it is assumed that the lowest five years of earnings that have already occurred will remain the lowest. The number of computation years that is used to calculate the AIME is the number of years since turning age 21. Thus, letting e a be the AIME at age a, the AIME is updated as follows: (7) e a+1 = [ e a (a 21 5) min(y ae, y max e ) ] /(a ), where all earnings figures are divided by 12 to reflect the monthly basis of the AIME. Social security benefits are a piece-wise linear function of AIME. Specifically, gross benefits are determined by S g 62 = γ 0 e 62 if e 62 < b 1, = γ 1 + γ 2 (e 62 b 1 ) if b 1 e 62 < b 2, = γ 3 + γ 4 (e 62 b 2 ) if b 2 e 62 < b 3, (8) = γ 5 otherwise. Gross benefits are zero if q 62 < Earnings above a minimum amount, ȳ (= $7440 annually, $620 monthly in 1992), are taxed at a 50% rate, so potential net benefits at age 62 are S62 n = S g 62.5 max[(y 62,e ȳ), 0] if S62 n > 0, (9) = 0 otherwise. For each year that the age of first receipt of benefits is postponed there is an actuarial adjustment where gross benefits increase by 6.67% up to age After age 65, gross benefits 10 The index used to calculate the AIME is based on the national average of total wages. 11 In 1992, the values of the parameters determining benefits were as follows: γ 0 =.9, γ 1 = 348.3, γ 2 =.32, γ 3 = 971.0, γ 4 =.15, γ 5 = , and b 1 = 387, b 2 = 2333, b 3 = The latter set of parameters are referred to as bend points. 12 If benefits are collected prior to age 65, the reduction in benefits continues past age 65, although it is recalculated at age 65 to account for months in which net benefits were zero. We ignore this recalculation in the model. 12

14 are increased by a variable amount ranging from 5% to 8% depending on the calendar year in which the individual reaches age In addition, the earnings tax is reduced to 33.3% at age 65 and to zero at age 70, and the minimum monthly earnings not subject to tax between age 65 and 69 increases to $ We do not model the decision to apply for social security. Instead, it is assumed that all individuals accept social security in the first year in which their benefit net of the earnings tax exceeds 25% of the benefit they would receive if they had zero earnings. Thus, an individual who would receive a social security benefit at age 62 of at least that amount is treated as if the individual had applied for social security at age 62. If the net benefit is less than this amount at age 62, then the benefits obtainable at age 63 are augmented to reflect the actuarial fair adjustment. A similar procedure is followed for all subsequent ages. Married individuals may elect to collect benefits based upon either their own AIME or that of their spouse who is retired. At age 62, the gross benefits available using the spouse s AIME is 37.5% of the spouse s gross benefits. If the age of first receipt is postponed to 63, gross benefits are 41.7% of the spouse s gross benefits, if postponed to 64, 45.8%, and if postponed to 65 or later, 50%. Widowed individuals can collect on their spouse s earnings record at age 60. At age 60, gross benefits are 71.5% of their spouse s gross benefits and at age 61, 77.2%, and at ages 62 through 65, 82.9%, 88.6%, 94.3%, and 100%. At any time after reaching age 62, a widow(er) may elect to switch to the gross benefits based on their own earnings record. Given the forward-looking nature of the model, it is necessary to make an assumption about what individuals forecast about future social security rules. We model this uncertainty as a discrete probability distribution over proportionate shifts in the social security rule (8) (in the γ parameters). 15 Thus, benefits, for any given AIME, might be forecasted to change proportionately by 0, 25, 50, or 75% with given probabilities. We allow only one change to be forecasted by the individual up to age 62 which will be in place from then on. For a couple, the one change would have to occur by age 62 for the older spouse. 16 In terms of the vector 13 An individual reaching age 62 in 1993 or 1994 receives an adjustment to gross benefits of 5%, with the adjustment increasing by 0.5% for each additional two years up to 8% for individuals reaching age 62 on or after In the model, we assume a 6.5% rate for all birth cohorts. 14 The earning tax has recently been eliminated for ages 66 to We assume that the most recent change in the earning test reducing the exemption age to 66 was unanticipated. 16 In the case of a widow(er), if the one change that is permitted in the model has not occurred prior to the spouse s death and the widow(er) is under the age of 62, then the widow(er) s expectation would be in force until reaching age

15 γ, we model the stochastic process as (10) γ a = U γ a 1 with probability π b γ a = γ a 1 with probability 1 π b where U represents a draw of a multinomial distribution taking the values 0, 0.25, 0.5, 0.75, each with probability π b (U, µ bj ), where µ bj captures heterogeneity in beliefs. 2.8 Pension Income We consider only defined benefit (DB) pension plans. 17 Eligibility and pension benefit amounts in such plans are determined by a combination of tenure, earnings history and age. pension plan characteristics include: ages of eligibility for early and normal retirement benefits, vesting provisions such as years of service, and the exact benefit formula as a function of the earnings history and years of tenure. DB plans differ across firms in this formula that maps how changes in these worker characteristics affect the pattern in which pension benefits are accrued. For example, a relatively simple pension might have the following structure. An individual who reaches age 55 with between 5 years and 29 years of tenure is able to collect benefits subject to a formula based on early retirement provisions. If the individual were to reach age 65 with the same span of tenure, benefits would be collected subject to a formula based on normal retirement provisions. Within the same plan, however, an individual who has 30 or more years of tenure can retire under normal retirement provisions at the age of 46. There are generally large spikes in benefit accruals when the worker satisfies the age and service requirements for vesting and for early and normal retirement benefits. After the initial age of eligibility has been attained, benefit accruals often begin to decline and can eventually become negative if the individual remains with the firm at older ages. In these cases, the worker loses pension wealth and suffers a reduction in total compensation by working too long. Therefore, different early and normal retirement ages and conditions, and benefit accrual formulas provide incentives to remain with and then retire from a firm at specific ages. We assume that new jobs acquired after age a 0 do not come with a pension Defined contribution plans would require introducing a second asset given the deferred tax nature of those plans. As discussed by Clark and Quinn (1999), defined contribution plans differ strongly from defined benefit plans in that these plans are generally much more mobile and are approximately age neutral in their effect on the timing of retirement. 18 It is not unreasonable to assume that new jobs obtained in late middle ages do not provide significant pension benefits. If new jobs potentially come with a pension, then one would have to explicitly model the, presumably joint, distribution of pension characteristics and wages associated with new job offers. Key 14

16 Incorporating DB plans into the model raises serious computational problems for two reasons. First, the solution of the optimization problem is specific to each DB plan represented in the sample. Thus, the decision problem must be solved separately for each plan. Second, the degree to which the earnings history matters in determining benefits affects the size of the state space of the optimization problem. In some cases, benefits are determined only by the last year of earnings, in others by earnings over a longer period. With respect to the second of these problems, for those individuals who have vested at the time we first observe them and thus for whom their current pension accrual is observed, we need only to specify the way pension benefits accrue with additional tenure. In those cases, we assume that the annuity value of the pension, i.e. the annuity that would flow from the pension if the individual retired from the firm at some given age and tenure, can be well approximated by a Markov process. Given the previous year s annuity value, the current annuity value (at age a) depends on current age and tenure, the previous year s earnings, whether the individual is eligible for normal retirement at age a (n a = 1 if eligible and zero otherwise) and whether this is the first year of eligibility for normal retirement (l a 1 n a = 1 where l a 1 = 1 if eligible for early retirement in the previous year). Specifically, an individual who retired at age a, was vested and had reached either the early or normal retirement age, would receive an annuity given by (11) P a = P (a, T a, y a 1,e, n a, l a 1 n a, P a 1 ), where the accrual function (11) differs across pensions. 19 An individual who was vested, but retired prior to reaching the age at which the pension could be collected, would begin to collect either upon reaching the early retirement age or the normal retirement age depending on the pension rules. We assumed that married individuals elected an annuity schedule in which the surviving spouse would receive 50% of the annuity amount received by the couple when both are alive. For married individuals, the pension accrual function also includes the age and gender of the spouse. For individuals who are currently working in a job with a DB pension, but are not yet vested, we need also to determine the pension amount that would be accrued at the time they vest. We assume that this initial pension annuity amount is a quadratic function of their earnings in the period prior to vesting. At that point, pension accrual follows (11). 19 The pension accrual function (11) for each pension is estimated from simulated data obtained from the actual pension rules governing accrual. The simulated data were obtained by using the Pension Calculator Program provided by the HRS. 15

17 Once an individual leaves the firm, disposable income is augmented by the annuity value of the pension. If the individual leaves before becoming age eligible, disposable income increases at the first eligibility age. 2.9 Mortality Risk Mortality is exogenous in the model, although the risk of mortality at any given age depends on an individual s state of health, sex, age and a heterogeneity component µ sj. Specifically, the probability that the individual survives to age a + 1 given survival to age a, the survival hazard, is (12) for j = m, f Health π sj a = π s (Z a, j, a, µ sj ) An individual s state of health is assumed to affect utility directly as in (1) (poor health may increase the disutility of working full-time and part-time) and mortality risk as in (12). As discussed earlier, it also affects health expenditures. As noted, health is assumed to be either good, Z a = 1, or poor, Z a = 0. The probability of being in good health at a + 1 is assumed to depend on age, health at age a, on whether the individual is covered by health insurance, the individual s race (X) and a heterogeneity component µ zj, namely (13) π zj a+1 = π z (a + 1, Z a, hi a, j, X j, µ zj ). The range of health insurance options is defined next Health Insurance An individual can be covered by health insurance either through a job if one works, through a prior job that provides health insurance upon retirement from the firm, through a spouse s job or prior job, or automatically through Medicare upon reaching age 65. We do not allow for the purchase of private health insurance nor do we distinguish between employer-provided insurance versus Medicare in it s effect on health. Specifically, health insurance at age a (<65) equals zero if the individual either does not work or works and is not covered, equals one if the individual is covered either by the employer or by Medicare, equals two if the employer-provided coverage continues after retirement and three if the spouse is also covered after retirement. 16

18 While at a particular firm, coverage, or lack of it, is assumed to continue without alteration until age 65 at which time Medicare coverage is substituted. Thus, hi a = hi a 1 if T a > 0 and a < 65, and hi a = 2 if a 65. We restrict the type of coverage that may be obtained upon taking a new job to the following three possibilities: no coverage, covered only if work full-time in the first year on the job, covered regardless of hours worked in the first year on the job. Thus, new jobs do not provide coverage upon retirement and given the hours choice h jn, we have hi a {0, 1}. The probability that a new employer receives health insurance depends on whether the individual works full-time or part-time, and also depends on the firm-specific component in the wage offer function, as in (14) π hi a Ta=0 = π hi a (h n a, η) Empirical Specifications In the empirical implementation of the model we specify the utility function as: Ua j ) = Cα(µuj j α(µ uj ) e[1+β 1h a+β 2 Z a+ɛ cj a ] + [β 3 + ɛ hj a ]h j a + β 4 Z a + β 5 Z a h j a + β 6 Xh j a where α < 1. The bequest function B a (W a, µ q ) is specified as a quadratic function: B a (W a, µ qj ) = ψ 1 (µ qj )W a + ψ 2 Wa 2 The wage offer function (6) will be specified as a log-wage function quadratic in work experience and tenure, linear in all its other arguments, and linearly additive in the wage errors ɛ w a and µ w. The stochastic utility and wage components ɛ c a, ɛ h a, ɛ w a and η a are assume to be joint normally distributed with mean zero and covariance matrix Ψ. Finally, the transition functions π s, π z and π hi in equations (12), (13) and (14) are specified as Logit functions. 3 Solution Method The model is numerically solved by backwards recursion. However, because the state space consists of elements that are continuous variables, e.g., the current level of household assets, the current level(s) of the AIME, the pension annuity value(s) P a, it is not possible to obtain exact solutions. Instead, we adopt an approximation method due to Keane and Wolpin (1994, 1997). The details of the solution procedure are provided for single individuals. 17

19 The procedure for married couples differs only in the number of alternatives available to the household and in the size of the state space. At age A 1, a single individual decides on consumption and employment status to maximize terminal total utility, V A 1 ( Ω A 1, ɛ A 1 ), where the state space, Ω A 1, is divided into a deterministic component containing the elements that are not random at the beginning of A 1, Ω A 1 (which includes the vector of unobserved heterogeneity components µ = [µ u, µ q, µ θ, µ w, µ b, µ s, µ z ]), and a shock component containing the vector of random (preference and human capital) shocks drawn at A 1, ɛ A 1, inclusive of the new job-specific wage component, η. For any given value of the deterministic and shock components of the state space, optimal consumption is obtained by solving the Euler equation, allowing for the corner solution at the lower bound for net assets, for each of the three possible choices of employment status. The employment status and associated optimal consumption that maximizes total utility is chosen for that value of the state space. That is, given any work choice h j A 1, we can find the optimal consumption or savings decision at age A 1 by numerically solving (15) du j A 1(C j A 1, h j A 1; Ω j A 1) + B j A 1(W A (C j A 1, h j A 1, Ω j A 1) dc j A 1 = 0 giving C j A 1(h j A 1, Ω j A 1). Because of the borrowing constraint and because of the minimum consumption constraint which limits savings, the solution to (15) does not necessarily yield the optimum. The value function at both corner solutions need to be calculated and compared to check if the (unique) solution to (15) violates constraints. Similarly, the optimal consumption levels and implied savings choice can be obtained for couples by solving a system of two first order conditions. Once optimal assets are determined for each employment alternative, the choice of employment alternative is based on a direct comparison of the hours specific value functions. At any deterministic state point, the expected value of maximum terminal utility is then obtained by Monte Carlo integration, that is, by taking draws from the (joint) shock vector distribution and averaging to obtain EV A 1 ( Ω A 1 ). 20 This expectation is calculated at a subset of the deterministic state points and the function is then approximated for all other state points by a polynomial regression. We denote this function as Emax(A 1). 20 An alternative solution method would be to discretize the continuous state space elements into a finite set, while simultaneously discretizing the continuous choice variable in the same way. The Emax values can then be obtained at all state variable values by evaluating the Emax function at a subset of the state space elements and using interpolation functions to fill in the values of the other state space elements. In the solution method we use here, we do not need to interpolate all of the values of the Emax function. Knowledge of the polynomial approximations is all that is required. 18

20 This procedure is repeated at age A 2. Using (1), substituting the Emax(A 1) function for the future component of the value function at A 2, the Euler equation is solved for optimal consumption at all possible employment statuses for a given value of the state space at A 2, yielding the optimal decision. Monte Carlo integration over the shock vector at A 2 provides EV A 2 ( Ω A 2 ) for a given deterministic state point. A polynomial regression over a subset of the state points provides an approximation to the function, denoted by Emax(A 2). Repeating the procedure back to the initial age provides the Emax polynomial approximation at each age. The set of Emax(a) functions fully describe the solution to the optimization problem. In the model described by (1) - (14), the state space for an unmarried individual of age a and gender j who does not have a current DB pension job includes, in addition to a and j, net assets (W a ), health status (Z a ) and health insurance status (hi a ), education (E), the person s race (X 4 ), work experience (H a ), job tenure (T a ), unobserved heterogeneity µ, and the job-specific wage component of the previous period s job (ηi(h a 1 0)), the AIME (e a ), the number of covered quarters of social security (q a ), whether the individual ever collected social security, and if so the age at which benefit receipt began, whether a change in social security benefit rules has occurred and its magnitude, whether the individual previously retired from a pension job and is age-eligible to collect and if so, the amount currently collected (P a ) and if not, the age at which the individual will become eligible to collect and the amount to be collected (P a ). For an individual who is currently working on a job with a DB pension plan, the state space also includes the accrued pension annuity (P a ) as well as the age-tenure requirements for early and normal retirement. For a widow(er), the state space also includes the deceased spouse s AIME, the amount of the accrual from the deceased spouse s DB pension (if any) and the age at which the widow(er) is eligible to collect. For married couples, all of the individual-specific components plus household net assets comprise the state space. 4 Data The data come from the first three waves of the Health and Retirement Study. The target population for the HRS in the 1992 first wave included all non-institutionalized adults living in a household within the contiguous U.S. born in the years Spouses of age-eligible individuals residing within the household are also HRS respondents. The HRS includes a representative core sample and oversamples of blacks, Hispanics and Florida residents. A total 19