Life Cycle Responses to Medicare

Transcription

1 Life Cycle Responses to Medicare Pascal St-Amour January 2, 23 Abstract Medicare provides subsidized health insurance access to U.S. elders whose responses to the program have been the subject of most research. Yet, changes in relative prices, net worth, as well as exposure to morbidity and mortality risks induced by Medicare should also affect younger agents through a backward induction argument. We compute and simulate the optimal life cycle responses of consumption, leisure and health expenditures, under various insurance statuses, in a setting where health is a durable good that lowers both morbidity and mortality risks. We find that Medicare and private insurance are close substitutes, and that most effects of Medicare stem from coverage of otherwise uninsured elders, rather than financial net worth. Most of pre-entitlement effects of Medicare occur around middle age, when health issues become more important, and concern financial, more than health variables. We document a greater inter-temporal substitution of leisure which falls prior to Medicare entitlement and increases afterwards. Pre-entitlement wealth is also lower and motivated by less pressing precautionary savings. Faculty of Business and Economics, University of Lausanne, Swiss Finance Institute, CIRANO and CIRPÉE. This project has benefited from very useful comments from and discussions with Eric French, and Jonathan S. Skinner.

2 Introduction Medicare subsidizes and guarantees access to health insurance for all U.S. citizens that are either aged 65 and over, or younger and disabled. In 2, the program covered more than 46 million beneficiaries, 86% of whom were elders (Henry J. Kaiser Family Foundation, 22). Medicare is financed in part by insurance premia, and by taxes on labor income but, with modest individual participation to costs, and mild eligibility conditions, the program remains very costly and deficit-prone, with a budget approaching Sweden s and Saudi Arabia s GDP, and ranking fourth in U.S. Federal outlays in 2 (3% of total expenditures, U.S. Census Bureau, 2, Tab. 473), and with a projected deficit of.35% of taxable payroll (OASDI Board of Trustees, 22). Given its target population, the bulk of academic research on Medicare has, expectedly, focused on elders. Documented effects on retirees behavior include an increase in the utilization of medical resources, and reduction in out-of-pocket (OOP) expenditures. Other researchers have also identified an increase in consumption and leisure, and general improvement in health status, longevity, and welfare. What remains puzzling however is the relative paucity of research on the effects of Medicare on younger generations. Indeed, a standard static optimum, and a backward induction arguments both establish that the pre-retirement optimality of financial and health capital accumulation should be altered by Medicare via (i) the distortionary taxes on income, (ii) the predictable changes in the post-retirement optimal path of wealth and health, and (iii) the ensuing evolution of morbidity, OOP and mortality risks. We can identify at least two broad pathways through which these effects on the optimal allocation of younger generations could take place: the budget constraint and the coverage channels. Medicare covers two main programs: Original Medicare, since 965 (Part A: inpatient care, Part B: outpatient care and Part D: Drugs), and Medicare Advantage, since 997 (MA, or Part C). Differences between the two programs relate to coverage, expenditures caps, and direct subsidies to private insurers under MA. Switches between the two are possible once a year, although the majority of beneficiaries remain under Original Medicare (Henry J. Kaiser Family Foundation, 22). Original Medicare is financed by a 2.9% tax on payroll (shared equally by employees and employers). The monthly premium for Part B is 99.9$. Medicare covers 8%, after deductible (,56$ for Part A) of admissible medical expenditures (Medicare.gov, n.d.). All persons aged 65+, who have resided legally in the U.S. for a minimum of 5 years, and who have paid Medicare taxes continuously for years or more are eligible.

3 First, Medicare induces changes in disposable resources through the taxation of labour income at all ages, and the subsidization of insurance premia for elders. On the one hand, income taxes distort the opportunity cost of leisure and affect the optimal consumption/leisure mix through the usual substitution and income effects. If the former effect dominates, more leisure and lower after-tax revenues ensue, with less disposable resources for savings, and therefore wealth accumulation, as well as for current medical expenses, and therefore health capital accumulation. The latter will be further impacted if healthy leisure as a subset of total leisure also determines investment in the health capital. For example, stronger income effects leading to a reduction in leisure will lower health investment and lead to a future deterioration in health and exposure to health-related risks. On the other hand, Medicare also subsidizes the insurance premia after retirement. To the extent that participation is compulsory, this is tantamount to an expected lump-sum transfer which will reduce the need to accumulate wealth reserves for old age. The foreseeable post-entitlement increase in disposable resources also affects the need to self-insure against high medical expenses when old by investing in the health capital through medical expenditures and healthy leisure when young. Second, Medicare provides health insurance coverage to otherwise uninsured elders, and thereby leads to an expected reduction in exposure to future OOP risks from the younger generations perspective. This mitigates the precautionary motives for building up wealth balances when young, and maintaining them when old. Furthermore, the expected future drop in the effective price of health care also alters the need to maintain high buffer stocks of health capital as self-insurance against future sickness risks. It can thus affect the optimal timing of medical expenses by postponing health care when young in favor of future expenses when Medicare coverage begins (i.e. piling up). Such inter-temporal substitution in health investment will alter the optimal path of the health stock, and, consequently, the path of the exposure to morbidity and mortality risks. Any change in the latter determines the expected longevity and therefore the motivation for savings when young. 2

4 These elements suggest the following observations. First, a Medicare-induced revision of the pre-entitlement optimal allocation cannot be ruled out, and undoubtedly cannot be investigated by focusing on elders exclusively. Second and related, the standard approach of assuming that the health and wealth statuses are pre-determined and Medicareindependent when studying the elders adjustments is hazardous at best. In particular, it remains open to Lucas critique that government policy alters not only the responses to, but also the levels of the state variables in a dynamic setting. Taken together, these factors make a compelling case for including the impacts on younger generations when performing any positive and normative analysis of Medicare. The purpose of this paper is therefore to shift attention on the effects of Medicare over the entire life cycle instead. Towards that aim, we modify a demand for health model in the spirit of Grossman (972) s human capital framework in order to account for convex adjustment costs, as well as (partially) endogenous sickness and death shocks. The agent selects consumption, leisure and health expenditures to solve a stochastic life cycle optimization, whereby the latter two augment the depreciable health stock, thus reducing morbidity and mortality risks. Time-varying optimal rules and welfare are induced through the finite maximal longevity, as well as the age dependency of (i) the deterministic and stochastic depreciation rates affecting the health capital, (ii) the premia of the health expenditures insurance contracts, and (iii) the wages and Social Security entitlement. Consistent with evidence that health expenditures insurance contracts when at all available are exogenously determined by employment or by the government, the insurance status is taken as given and includes private, and no coverage variants for both younger and older agents, as well as Medicare for elders. This characterization allows us to pinpoint the marginal allocative effects of young (resp. old) agent s health insurance status, by conditioning on that of the old (resp. young) individuals. Moreover, we can identify the marginal effect of Medicare, as opposed to private insurance, again conditioning on young individuals status. In the full insurance scheme, this marginal 3

5 effects allows us to neutralize the coverage effect in order to isolate the pure budget constraint effect of Medicare. The model admits no closed-form solution. It is therefore calibrated, solved numerically by backward iteration for the value function, and simulated forward over a wide population differentiated with respect to initial health and wealth statuses. Both the iterative results and the simulated life patterns with respect to allocations, wealth and health statuses, as well as corresponding expected longevity and survival rates, accurately reproduce a wide range of observed cross-sectional, population, and life cycle moments. Importantly, the differences in this output across the insurance and age dimensions can be isolated in order to gain insights on two questions. First, from a broader perspective, what are the life-cycle implications of being insured against medical expenses? Second, from a more specific perspective, how does Medicare alter these effects? Our first objective is to identify the marginal allocative effects of the health insurance status when young, and when old. Overall, our results are consistent with (i) a strong precautionary motive for building up wealth balances against uninsured medical expenses risks, and (ii) a greater inter-temporal substitution made possible when those risks are covered. In particular, we find that being insured yields better health, especially for elders, and that more cross-generational effects are observed when uninsured at other periods of life. This health improvement is however insufficient to generate better survival prior to retirement. It is achieved through more important health investment when health issues become more pressing after middle age, whereas exposure to OOP risks is significantly reduced. Importantly, we find no evidence of inter-temporal substitution in health expenses. Rather, cross-generational adjustments obtain through the leisure and consumption decisions; working hours are increased at middle age when wages are at their highest, and reduced when wages fall after retirement. Moreover, lower exposure to health risks implies that ex-ante precautionary wealth balances are lower, whereas ex-post wealth is higher for the insured. 4

6 We next turn to the allocative differences specifically induced by Medicare. We first ask whether Medicare is different from private insurance, conditional on young agents insurance statuses. We find that the marginal effects with respect to elders uninsurance are quite similar, suggesting that Medicare and private schemes are close substitutes. We next ask whether Medicare entitlement is net worth. We find that the pure budget constraint effects of subsidized premia and labour income taxes net out each other, such that the allocative effects are very modest, especially with respect to health outcomes. Rather, they concern the financial variables and are indicative of more leisure (due to the taxation of labor income) when young, and more wealth accumulation (due to the lump-sum transfers associated with subsidized premia) when old. The apparent weakness of the budget constraint effects confirms that the bulk of Medicare effects on allocations are attributable to the coverage of otherwise uninsured elders. Because of the close substitutability between Medicare and private insurance schemes, these life cycle effects mirror those associated with elders insurance status. We thus find little evidence of Medicare-induced inter-temporal substitution in health expenditures. These elements argue against the piling-up hypothesis of postponing medical expenses until after coverage begins. Finally, we ask whether Medicare is optimal from life cycle perspective. Our results confirm that exclusion from health insurance market is very detrimental after middle age, but not so for the younger individuals, casting doubts on the optimality of extending Medicare to all. This occurs because self insurance for young adults, when the health stocks are high and the depreciation rates are low is comparatively cheaper than market insurance prior to middle age where health-related issues start becoming more pressing. Nonetheless, individual optimality after that age should not be interpreted as social optimality as our modeling strategy abstracts from inter-generational considerations such as bequest motives. Our agents are therefore purely egoistic with respect to the share of current Medicare costs being supported by future generations. 5

7 This paper relates to the literature on the health considerations in the life cycles of consumption and of wealth accumulation (Hugonnier et al., 22; Scholz and Seshadri, 22; Baicker and Skinner, 2; French and Jones, 2; French, 25; Palumbo, 999), of leisure (Baicker and Skinner, 2; Rust and Phelan, 997; Palumbo, 999) and of health expenditures (Hugonnier et al., 22; Scholz and Seshadri, 22; Fonseca et al., 29; Hall and Jones, 27; Khwaja, 2). It expands on that literature by (i) simultaneously including consumption, leisure and health expenditures, (ii) endogenizing self insurance with respect to morbidity and mortality shocks (usually treated separately, when at all), and (iii) focusing on the (exogenous) health expenditures insurance status. Issues not considered in this paper relate to endogenous retirement (Fonseca et al., 29; French and Jones, 2; Blau and Gilleskie, 28; French, 25; Rust and Phelan, 997), or the social optimality of health spending and insurance (Baicker and Skinner, 2; Bhattacharya and Lakdawalla, 26; Hall and Jones, 27; McClellan and Skinner, 29). The rest of the paper proceeds as follows. Section 2 outlines the theoretical framework. The numerical methods are discussed in Section 3. Finally, the iterative and simulation results are presented and discussed in Section 4. All tables and figures are regrouped in the Appendix. 2 Model Health dynamics We consider the decisions of a finitely-lived risk-averse individual, aged t =, 2,..., T m, who is confronted with both sickness and death uncertainty. More precisely, let a prime ( ) denote a next-period variable, and λ : R + R ++ denote an ageinvariant, decreasing and convex intensity function of health (H). Health risks ɛ k {, } are a morbidity (k = s) or mortality shock (k = m) following stochastic processes, whose probabilities of occurrence are given by: Pr(ɛ k = H) = exp[ λ k (H)], k = m, s. () 6

8 Hence, an unhealthy agent faces higher risks of both sickness and death, and is subject to diminishing returns in risk reduction as health improves. The age at death T m is bounded above by T, the maximal biological longevity, and is the first nonzero occurrence of the mortality shock: T m = min{t m, T }, where t m = min{t : ɛ mt = }. The health capital is depreciable, and is depleted further upon occurrence of the health shock ɛ s =. It can be adjusted through investment, where I g : R + R + I R + is an age-invariant, increasing, and concave gross investment function of health, real investment (I), and leisure (l): H = [ δ(t)]h + I g (H, I, l) φ(t)hɛ s, (2) and where both depreciation rates δ, φ : N I = [, ] are increasing functions of age in order to induce finite optimal longevity (Ehrlich and Chuma, 99). The law of motion (2) is a standard health-as-capital specification in the demand-for-health literature (Grossman, 972). It is augmented to include convex adjustment costs (Ehrlich, 2; Ehrlich and Chuma, 99), healthy leisure inputs (Sickles and Yazbeck, 998), morbidity shocks (Hugonnier et al., 22), as well as time-varying deterministic and stochastic depreciation. In particular, age-increasing depreciation is intended to capture more pressing health issues for older agents, including the demand for long-term care by elders (Palumbo, 999). The non-negativity constraint for investment is standard and prevents agents from selling their own health in markets. Budget constraint The agent evolves in an incomplete financial markets setup comprising a risk-free asset, with gross rate R f and a health expenditures insurance contract; death risk is not insurable through markets but (partially) diversified through gross investments exclusively. The health expenditures insurance contract is defined by its 7

9 deductible D, co-payment rate ψ and premium Π(t) at age t. Deductibles and co-payment rates are assumed to be age-independent, and equal across insurance plans, whereas Medicare, when operational, renders the risk premia age-dependent. Participation in the insurance contract is assumed to be exogenous. The assumption of plan-independent deductibles and co-payments is made for tractability, yet is not unrealistic given that Medicare deductibles and typical co-payment are close to those of many private plans values. Exogenous participation can be rationalized by noting that health insurance is mainly decided upon and provided by employers and/or by government intervention, when the agent is not excluded altogether from health insurance markets because of moral hazard and adverse selection reasons (e.g. Blau and Gilleskie, 28; Currie and Madrian, 999). Let the indicator functions M, X (t) and R correspond to the Medicare, health insurance participation at age t, and old agent (t 65) status. Letting P I denote the price of medical expenditures, the agent s out-of-pocket payment OOP (I, t) and insurance premia functions Π(t) are obtained by: OOP (I, t) = P I I max {, X (t)( ψ) (P I I D)}, (3) Π(t) = Π M R (Π Π M ). (4) As illustrated in Figure, the contract (3) is standard and has the insured agent cover all medical expenditures P I I up to deductible D and pay a share of expenses ψ afterwards; the uninsured agent covers all medical expenses. The health insurance premium (4) captures the subsidization of market premium Π > Π M provided by Medicare for seniors. Denoting consumption C, wealth W, and labour income Y (l, t), a decreasing function of leisure, the budget constraint is given as: W = [W + Y (l, t) C OOP (I, t) X (t)π(t)] R f, (5) Y (l, t) = R y + ( M τ)w(t)( l), (6) 8

10 where y is fixed income, and w(t) are age-dependent wages. The income process (6) captures the effects of pension income (e.g. Social Security), the observed hump-shaped wage profiles, as well as the tax effects of Medicare. To summarize, being insured ( X = ) reduces disposable resources by the amount of the premia, but also frees up resources by lowering OOP for high medical spenders. Medicare ( M = ) reduces after-tax income for every worker, but also subsidizes the premia for insured elders only. 2 Preferences Let U : R + I R + denote an age-independent monotone increasing, and concave period utility function over consumption, and leisure, and V : R + R + N R + be a value function over wealth, health, and age. Taking the initial wealth, health, as well as insurance participation as given, the agent s problem is to select consumption, leisure, and health expenditures to maximize separable VNM utility: V (W, H, ) = max {C t,i t,l t} Tm t= { Tm } E exp[ ρt]u(c t, l t ) t= (7) where the optimization is subject to the health process (2), and the budget constraint (5), and where ρ > is a subjective discount rate. As discussed in Hugonnier et al. (22), non-negative preferences in (8) are necessary to ensure that the agent always exhibits a preference for life over death in endogenous mortality settings. Applying the Law of Iterated Expectations, and using the distributional assumption (), the agent s objective (7) can conveniently be rewritten as a recursive problem with deterministic horizon and endogenous discounting: V (W, H, t) = max {C,I,l} U(C, l) + {t<t } exp [ ρ m (H)] E {V (W, H, t + ) H}, (8) where we have set ρ m (H j ) ρ + λ m (H j ), the endogenous discount rate. Hence, an unhealthy agent has a shorter expected life horizon and is tantamount to a more impatient individual. 2 Medicare coverage for young disabled and Medicaid for poor households are abstracted from for tractability reasons. 9

11 Optimality Letting subscripts denote partial derivatives, the first-order and Envelope conditions reveal that the optimal allocation is characterized by: U c = {t<t } exp [ ρ m (H)] E {U c H} R f, (9) U c OOP I (I, t) = {t<t } exp [ ρ m (H)] E {V H H} I g I, () w(t, x) = U l + Ig l U c I g OOP I (I, t), () I where P I, OOP I (I, t) = P I [ X (t)( ψ)] if P I I < D otherwise. (2) The marginal value of health solves the recursion: V H = {t<t } exp [ ρ m (H)] ( ρ m HE {V H} + E H {V H} +E {V H H} [( δ(t)) + I g H ] φ(t)e {V Hɛ s H}) (3) The Euler condition (9) equalizes the marginal utility cost of foregone current consumption when savings are increased to the expected discounted marginal utility benefit of future wealth times the rate of return on the safe asset. It is entirely standard except for the presence of endogenous discounting. In particular, a healthy agent may increase savings in order to accommodate a longer planning horizon, just as would a more patient individual. The Euler equation () equates the current marginal utility cost of out-ofpocket health expenditures to the expected future marginal benefit of the additional health procured by investment. Equation () is a static optimality condition that equates the marginal cost of leisure (i.e. after-tax wages) to its marginal benefit (i.e. direct marginal rate of substitution between leisure and consumption plus the marginal reduction in out-of-pocket expenditures made possible by leisure).

12 As equation (2) and Figure make clear, the marginal OOP cost of health expenditures is kinked at the deductible for insured agents, and encourages them to spend more than D. Moreover, Medicare implies that cost is age-dependent as young uninsured agents become covered at age 65, encouraging them to postpone health expenditures. Also, the insured status in (2) also affects the leisure health expenditures tradeoff in (), by making self-insurance through leisure relatively more costly for insured agents. Again, the age-dependent effects for young uninsured individuals becoming covered by Medicare are to be expected. Finally, the Envelope condition (3) decomposes the marginal value of health as the reduction in mortality risk times the continuation utility, plus the marginal value of morbidity risk reduction, plus the expected future marginal value of undepreciated health stock. Observe that the marginal value of health will decline when the depreciation rates δ(t), φ(t) become large (e.g. as t T ). Finite lives plus increasing depreciation then make it optimal for the agent to run-down his health capital as certain death approaches. 3 Numerical methods Numerous factors contribute to render analytical solutions to the model challenging at best and intractable at worst. First, the marginal cost of out-of-pocket health expenditures is kinked at the deductible (see Figure ). Second, the presence of Medicare induces discrete jumps in the cost of insurance depending on age. Third, autonomous time variation is also found in both depreciation rates δ(t), φ(t), as well as in wages w(t). Last, and most importantly, discounting is not constant, but endogenous in the health status. Alternative solution methods in endogenous mortality problems include the analytical approximations advocated by Hugonnier et al. (22). This approach considers closedform solutions to a simpler problem with exogenous discounting upon which, a first-order expansion is performed to approximate the more complex setup. Unfortunately, the

13 problem considered here is quite different, as it involves labour-leisure choices, as well as insurance for health expenditures, rather than for health shocks. The consequence is that approximation methods cannot readily be applied to our framework, and that numerical alternatives must be invoked. These numerical approaches can be split in two parts: iteration and simulation. 3. Iterative methods The iterative step consists in solving the model numerically by backward induction via a Value Function Iteration approach. This involves discretizing the state space which comprises the insurance plans, as well as the health and wealth statuses. First, we consider 5 exogenous insurance plans corresponding to No (N) and Private (P) insurance when young ( t < 65), and No, Private or Medicare (M) when old (t 65), and denoted x = (x y, x o ) X = {PM, PP, PN, NM, NN}. The descriptions and corresponding parametric restrictions are outlined in Table. Plans PM and PP encompass the full insurance cases with, and without Medicare. Plan PN captures the effects of employment-provided insurance which is terminated at retirement, whereas plans NN and NM illustrate the effects of market failures leading to exclusion from health insurance. 3 This classification allows for a convenient identification of the marginal effects of (i) young agents insurance status conditional on the elders insurance status (by contrasting PM vs NM and PN vs NN), as well as those of the (ii) elders insurance status conditional on young insurance status. For the latter, the marginal effect of private insurance is obtained by contrasting PP vs PN, whereas that of Medicare obtains from the PM vs PN and NM vs NN comparisons. Finally, the pure budget constraint effects of Medicare are isolated from the coverage effects by imposing full insurance, and computing the difference between plans PM vs PP. 3 See French and Jones (2) for an empirical analysis of the effects of employer-provided health insurance on savings and retirement. Plan NP is arguably of limited empirical relevance, and is abstracted from. 2

14 Second, denote by Z = (H, W ) Z, the discretized state space of dimension K Z, ɛ = (ɛ s, ɛ m ) {, } 2, the health shocks, and by Q = (C, I, l) Q, the discretized control space of dimension K Q. After setting K Z, K Q and maximal longevity T, 4 we solve the following over t = T, T,..., : V (Z, t, x) = max {Q Q} U(Q) + {t<t } exp[ ρ m (Z)]E {V (Z, t +, x) Z} s.t. Z = Z (Z, Q, ɛ, t, x) where linear interpolation is used to calculate the next-period value function. In order to account for the long reach of childhood effects (e.g. Smith, 999), the model is solved for all periods (i.e. t =,... T ), rather than only until a sufficient degree of convergence has been obtained. The output we recover consists of the age- and plan-specific allocations, and welfare: {Q(Z, t, x), V (Z, t, x)} T t=, Z Z, x X (4) that will be used in the simulation phase. 3.2 Simulation methods The iteration phase just described is performed over a pre-determined exogenous state space Z. In order to compute the optimal solutions along the optimal path, it is necessary to simulate the model forward by using the allocation (4) in conjunction with the laws of motion for Z in (2) and (5), as well as the shocks ɛ generated from the endogenous intensities in (). Specifically, the Monte-Carlo setup is as follows: 4 Specifically, we use a K Z = 3 3 grid for the state space, a K Q = grid for the control space Q, and T = years maximal longevity. Robustness to using a finer grid (i.e. K Z = 5 5, and K Q = 5 5 5) is verified and confirmed. Note that the well-known curse of dimensionality is particularly important in our setup as the state has to be augmented by the years horizon, as well as the 5 insurance plans in X. 3

15 . We define the adult population of agents, also indexed by Z Z. The population is initialized using the state, optimal rules and value function at age 6: Z Z t = Z Q Z t = Q(Z Z t, 6, x) (5) V Z t = V (Z Z t, 6, x) 2. For each agent Z Z, and for each subsequent year t = 7, 8,... T, the shocks are generated, the state is updated, and the optimal path is simulated forward using a spline interpolation of the results obtained in the iterative phase (4): ɛ Z t Z Z t Q Z t V Z t {, } 2 λ(z Z t ) = Z (Z Z t, Q Z t, ɛ Z t, t, x) = Z Z t (x) = interpn(q(z, t, x), Z Z t ) = Q Z t (x) = interpn(v (Z, t, x), Zt Z ) = Vt Z (x) (6) 3. The initialization (5) and simulation (6) are repeated over n =, 2,..., N populations. Several issues are worth mentioning. First, the output we recover: { {Q Z t,n(x), Vt,n(x), Z Zt,n(x) } } Z N, Z Z, x X, t=6 n= is the one along the optimal path, i.e. with all endogeneities accounted for. Second, the sample moments can be calculated either (i) across agents Z, age t, and replication n to obtain the population statistics, or (ii) across agents Z, and replication n to obtain the life cycle statistics. Third, the curse of dimensionality is again present, with (K Z = 3 3 grid points) (T = 84 years) (dim(x) = 5 insurance plans) (N = 5 replications) =.89 8 entries 6 series of interest. Fourth, the choice of the discretized state space Z is non trivial. Specifically, if chosen too tightly in step, the 4

16 interpolation used for step 2 will be imprecise, since it will be calculated for values outside the grid, i.e. not used to compute the optimal rules in the iteration. If chosen too wide, then the grid will be too coarse, again leading to imprecision in computing the allocations along the optimal path. As the computing time explodes rapidly with grid size, refining the grid is not a feasible option. Finally, death is of course an absorbing state, so that special care must be taken to ensure that dead agents remain so within one replication n of the Monte-Carlo. Related to that point, the statistics must be computed using survivors only to account for survivorship bias. Indeed, the simulated allocation of surviving elders must reflect the observed allocation of agents who are likely in better health in an endogenous mortality settings. Towards that aim, for each agent Z, we thus resort to: 5 { Q Z t (x), V Z t (x), Z Z t (x) } = NaN, t T Z m. The sample moments are then computed by excluding NaN values from the calculations (i.e. with nanmean, nanmedian in Matlab). Moreover, this strategy allows us to compute the life expectation and the survival rate across all replications: L(x) = [ T t ( S t (x), where St,n(x) Z = exp λ m H Z s,n (x) )], t= s= and S t (x) is the survival rate integrated across agents and replications. In what follows, both the life expectation L(x), and the age-specific survivor rate S(x, t) will be used to assess the realism of the numerical results. 3.3 Functional forms The model is now parameterized in order to numerically solve the optimal rules. Specifically, we consider decreasing convex intensities and a CRS gross investment functions, as 5 In practice, ɛ Z m,t(x) = max { ɛ Z m,t(x), ɛ Z m,t (x) } is invoked to ensure absorbing mortality. 5

17 well as a CES utility: λ m (H) = λ m + λ m H ξm, (7) λ s (H) = λ s2 λ s2 λ s + λ s H ξs, λ s < λ s2, (8) I g (H, I, l) = AI η I l η l H η I η l, η I, η l [, ] (9) d(t) = d exp(d t), d {δ, φ}. (2) U(C, l) = [ β C C γ + β l l γ] γ, β C, β l [, ], (2) Equations (7) and (8) both encompass limits to self-insurance as the intensities are bounded below by λ m, and λ s. Morbidity risk is also bounded above by λ s2 to avoid spiralling optimal paths where health falls, inducing more sickness, and further depreciation and certain subsequent sickness and death. The Cobb-Douglas technology (9) ensures diminishing returns to both expenditures and leisure inputs for gross investment. It also entails a long reach of health history such that the agent cannot solve all current health problems by investing large amounts. Moreover, the depreciation rates in (2) are both increasing and convex in age, implying less pressing health issues for younger individuals. Finally, the Constant Elasticity of Substitution (CES) specification of preferences (2) allows for direct positive value of leisure that supplements its health-enhancing ability. 6 Unlike CRRA preferences, CES are unconditionally non-negative and therefore guarantee preference for life over death. The calibration of the model s parameters is presented in Table 2. The parameters are chosen by using the data as much as possible. First, known parameters are calibrated to their data values. Second, the free parameters are selected either from the literature, or so as to minimize the distance between the data and simulated cross-sectional, population, and life cycle moments. As can be observed from Figure 2, our calibration is consistent with accelerating stochastic and deterministic health depreciation rates after middle age 6 Consistent with the model, health has no direct utilitarian value aside from its positive effects on health production as well as morbidity and mortality risks reduction. The results presented below are also qualitatively robust to including a direct preference for health U(C, l, H) in the CES function (2). 6

18 (Panel a), 7 and an increasing wage profile which peaks around age 6, and falls after retirement (Panel b). 4 Results This section describes the predicted optimal policies, welfare, and derived variables of the model. Following a brief discussion of the output obtained from the iterative phase in Section 4., we present the results obtained from the simulation phase in Section 4.2, starting first with the population moments followed by the life cycle properties. 4. Iterative results Figure 3 displays the optimal allocations, as well as the welfare functions of the predetermined health and wealth state. For that purpose, we compute the median values between ages 2 98, under plan PM, at each point of the state grid, and across the simulation output. The optimal consumption share of wealth (Panel A) is decreasing in wealth, whereas leisure (Panel B) is increasing in wealth, as expected (e.g. Scholz and Seshadri, 22, Fig. 2, for wealth effects on healthy leisure). Both are increasing in health up to a certain level after which both are decreasing. This non-monotonicity can be explained by the opposing effects of the reductions in morbidity and mortality risks as health improves. On the one hand, lower sickness risks entails reductions in expected OOP expenditures that can warrant more current consumption and/or leisure. On the other hand, a lower risk of dying implies that more savings must be set aside to cover a longer expected horizon, justifying lower consumption and/or leisure. Our results suggest that at low health levels, the morbidity risk effects dominate, whereas the mortality effects dominate at sufficiently high levels of health. The health investment function in Panel C is monotone increasing in wealth. It increases in health for the very poor and unhealthy, consistent with the intuition that 7 The calibration of the depreciation rates in (2) is chosen to ensure non-negative health upon realization of a health shock, i.e. δ(t) + φ(t), t. 7

19 it may become optimal to cut health expenses when death is near certain, disposable resources are scant, and health deteriorates further. Investment is otherwise decreasing in health, consistent with findings that the rich and unhealthy spend more on preventive and curative health care (e.g Smith, 999; Wu, 23; Barros et al., 28). Finally welfare in Panel D is clearly monotone increasing in both wealth and in health, as expected from the discussion of Envelope condition (3). Observe that concavity is more pronounced with respect to health, as can be anticipated from the diminishing returns in the selfinsurance technology (7) and (8), and in the gross investment function (9). The previous results are obtained over a given state space, and by integrating timevarying optimal policies over the age domain. In what follows we calculate the agedependent policies along the simulated optimal path, thereby fully endogenizing the evolution in the health and wealth statuses and reinstating the life cycle properties. 4.2 Simulation results 4.2. Population moments In order to compute the population statistics for our predicted variables, we simulate the model for adults over ages 6 98, and then calculate the medians for the surviving agents, and for the five health insurance plans (PM, PN, PP, NN and NM). The medians are contrasted with observed moments obtained from data discussed in Appendix C. The close correspondence between observed and predicted moments in Table 3 confirms that the model does quite well in capturing the population features of the data. Consumption, investment, health and expected longevity are accurately reproduced, while the other variables are reasonably close given the caveats associated with the model and/or data. Indeed, the model s OOP variable includes all health expenditures, i.e. both preventive and curative expenses. By not taking into account limitations on the types of expenses being reimbursed by the insurer, it likely overstates the actual out-ofpocket expenditure. Moreover, observed healthy leisure activities are approximated by 8

20 the average amount of time spent sleeping (7.6 hours per day) and leisure and sports (2.6 hours per day), which could overestimate the actual time spent improving health. Finally, the observed wealth measures is net worth and includes real estate which is abstracted from in the model. We postpone the discussion of the allocations differences across the insurance plans until the next section. Suffice it to say that they do lead to significant differences in decisions and ex-post statuses, notably a lower health and shorter longevity for uninsured elders, and a higher welfare for plans involving Medicare. The population results obtained thus far integrate the simulated survivors data across all age and do not reveal how the optimal life cycle is affected. In particular, they remain silent on how the agent s intertemporal substitution is determined by the insurance scheme. We turn to these issues next Life-cycle properties The simulated life cycles are presented in Figures 4. For that purpose, we compute the median values at each age across the health and wealth statuses, and across the simulation output. To facilitate the discussion, the simulated and observed (when available) levels are reported in Panels a, whereas the median of the differences across insurance statuses are reported in Panels b, with the marginal effects of young (resp. old) agents insurance statuses in b.a B (resp. b.c F). Health status and survival The simulated health statuses in Figure 4.a predict an optimal decline that is consistent with those observed in the data (e.g. Scholz and Seshadri, 22; Van Kippersluis et al., 29). The health paths start diverging after age 3 for young insurees, and after retirement for old insurees, who both have better health than the uninsured. The health values for plans NM and PN display opposite paths after age 6, with NM catching up with other insured elders, and PN catching down with NN once health insurance entitlement terminates. 9

21 Our results in Panels b confirm that insurees are healthier, with more cross-generational effects when uninsured at other periods. In particular, the differences in health for insured young agents (Panels b.a, B) peak at retirement, and fall thereafter, with more longlasting effects when the elders status is uninsurance. The effects of elders insurance statuses on their health are more important in magnitude. Old insurees remain permanently healthier after age 3, when the young status is uninsured (Panel b.d), and after 65, when the young status is insured (Panels b.d F), with health gains peaking at 75 in both cases. We observe the same marginal effects whether insurance is provided through Medicare (b.e), or private insurers (b.f). Similarly, the budget constraint effects are minor in Panel b.c. The predicted optimal decline in the health status induces a corresponding simulated fall in the survival rates in Figure 5.a. The decline is again consistent with those in the data, even though the concavity is somewhat under-estimated, possibly because of our assumption of age-independent death intensity parameters in (7). In Panels b, we observe that the young agents status has no consequence for survival (b.a and B), and that no differences in survival based on elders insurance status are found before retirement (b.d F). Afterwards, old insurees enjoy better survival rates. Again, both the differences between Medicare and privately provided insurance, and the budget constraint effects of Medicare in Panel b.c are insignificant. Health expenditures and healthy leisure The simulated health dynamics both induce and result from investment and healthy leisure decisions over the life cycle. First, in Figure 6.a, the predicted upward trend of investment accurately matches the one observed in the data, with insurees clearly consuming more health services once health problems begin to accelerate at middle age and/or entitlement begins. Second, in Panels b, we find no evidence of cross-generational effects in investment. In particular, the insured young agents invest more around middle age, but not at other periods of their lives, with effects being independent of elders statuses (Panels b.a, B). In comparison, 2

22 the status effects on health investment for elders are more important in magnitude and increase after 65, but are not apparent before that (Panels b.d F). We thus find no indication of pre-retirement postponement of expenses until coverage begins. Both the differences between Medicare and private insurance (b.e and F), and budget constraint effects of Medicare (B.C) are again insignificant. Second, the out-of-pocket expenditures in Figure 7.a and insurance effects in Panels b both display corresponding patterns, and are consistent with the intuition that health insurance leads to an increase in utilization, but a decrease in costs of health resources. They also confirm the absence of cross-generational effects. Indeed, the OOP s are lower at middle age for insured young agents in Panels b.a, B and for insured elders after 65, in Panels b.d F, but not at other periods of their lives. Panels b.d F again show no clear distinction between PP and PM, independence from young agents status, and confirm the weakness of Medicare budget constraint effects (b.c). Third, the leisure paths in Figure 8.a display strong similarities with observed patterns; they are initially low, followed by a sharp increase when wages fall after age 65 (see Figure 2, Panel b), consistent with observed behavior (e.g. Rust and Phelan, 997). The post-retirement match is admittedly less convincing, with predicted healthy leisure falling in order to induce the corresponding optimal fall in health, whereas observed leisure does not. This suggests that our assumption that healthy leisure is an age-independent share of total leisure is possibly excessive. Our results in Panels b are consistent with insurance allowing more inter-temporal substitution induced by movements in wages. Indeed young insurees reduce leisure when both wages and health expenditures are high around middle age, and increase it more than for the uninsured when wages fall after retirement, especially if uninsured when old (b.b). The elders status effects on leisure in Panels b.d F are weaker, yet also consistent with a reduction at middle age, followed by more leisure afterwards for insurees, except in the last periods of life where uninsured elders mitigate exploding health care costs and slow down the fall in health through more healthy leisure, compared to insurees. 2

23 Wealth, consumption and welfare The simulated and observed wealth dynamics in Figure 9.a again coincide, with asset accumulation when young, a peak occurring around retirement, followed by optimal dis-savings. The predicted optimal fall in wealth is admittedly faster than the observed one after age 75, possibly reflecting the absence of bequest motives, and of housing wealth in our modeling strategy. 8 The uninsured young (resp. old) agents clearly prefer a flatter asset accumulation (resp. dissipation) when young (resp. old). Panel b results are consistent with insurance reducing the need to build up ex-ante precautionary wealth balances, whereby lower exposure to OOP risks results in higher ex-post wealth. In particular being insured when young relaxes the need to build up wealth reserves for health problems at middle age. Lower OOP expenditures (Figure 7) combined with lower leisure at middle age both contribute to higher ex-post wealth after 6. Similarly, a lower exposure to OOP expenses for insured elders induces lower post-retirement precautionary wealth up to the last periods of life (b.d F). Once again, Medicare vs private differences and budget constraint effects in Panels b.c F are minimal. Corresponding movements in the consumption share of wealth can be found in Figure. Again adequate match with the data is apparent with falling shares after late 3 s. Note again that the faster dis-savings towards the end of life predicted by the model is not matched in the data with observed behavior maintaining high wealth balances (i.e. low consumption shares) until death. Our results in Panels b point towards the conflicting influences on consumption of lower disposable resources due to premia and Medicare taxes, but also lower financial exposure to OOP risks. Young insurees in Panels b.a, B have lower consumption shares between age 2-3 due to the insurance premia, and higher shares between age 3-45 due to the lesser requirement of building up wealth balances against middle age health problems. For the same reasons, old insurees in b.d F display similar paths displaced forward in time, with lower shares at middle age, and higher shares after retirement. Whereas Medicare-private differences are insignificant in 8 See also De Nardi et al. (2, 29); Dynan et al. (24) for evidence of slow asset decumulation in old age. 22

24 b.e and F, the budget constraint effects in b.c are clearly apparent with lower shares between age 3-5 due to Medicare taxes. Finally, the combination of single-peaked wealth, falling health and survival rates implies that welfare in Figure.a is also increasing up to retirement, and falling thereafter. The plots reveal that non-insurance for the young agents is optimal up to the middle age when wealth and wages are low, and health problems are scant (see also Panels b.a, and B), and suboptimal afterwards, consistent with the non-insurance data. 9 Conversely, non-insurance for the elders is always suboptimal (see panels b.c F). The optimality of the Medicare programs is also clearly apparent with plans PM and NM yielding the highest welfare after middle age Life cycle effects of Medicare Our results have a number of implications for the life cycle responses to Medicare. First, our findings are consistent with Medicare and private insurance being close substitutes. Indeed, the comparisons of Panels b.e (PM vs PN) and b.f (PP vs PN) reveal very similar marginal effects of the insurance status for elders, whether that insurance is provided through Medicare, or through private markets. Second, the pure budget constraint effects of Medicare, as isolated in Panels b.c (PM vs PP), was found to be moderate, indicating that Medicare affects the life cycle mostly through its effect on coverage to otherwise uninsured elders (see also McClellan and Skinner, 29, for a similar conclusion from a different perspective). Taxes on labor revenues tend to reduce disposable income when young. However, the subsidy of health insurance premia after 65 is tantamount to a lump-sum transfer, and increases wealth after retirement. The modest net effects indicate that current taxes and future subsidies apparently offset one another, thereby casting some doubt on the hypothesis that Medicare entitlement is net financial worth. 9 The percentage of people without health insurance falls from 3.4% for ages to 5.7% for ages (National Center for Health Statistics, 2, Tab. 4). 23

25 Third, the joint (i.e. budget constraint plus coverage of uninsured elders) effect of Medicare on health outcomes is much more significant, and is consistent with those outlined by the empirical literature. In particular, the model reproduces better health and survival (Lichtenberg, 22; Khwaja, 2, 27; Finkelstein and McKnight, 28; Card et al., 29; Scholz and Seshadri, 22), as well as more investment after age 65 (Lichtenberg, 22; Khwaja, 27; Finkelstein, 27; Card et al., 29), yet lower OOP s (Khwaja, 27; Finkelstein and McKnight, 28; Scholz and Seshadri, 22; De Nardi et al., 2), and lower precautionary wealth (De Nardi et al., 2, 29; Scholz and Seshadri, 22), and more leisure (Currie and Madrian, 999; French, 25). The comparisons of Panels b.d (NM vs NN) and b.e (PM vs PN) shows that the insurance status when young has a modest impact on how Medicare affects these post-entitlement health outcomes. Fourth, any pre-entitlement joint effect of Medicare concerns mainly the financial variables, and conversely depends on the insurance status of the younger agents. In particular, Medicare entitlement warrants more inter-temporal substitution in hours worked for the uninsured young agents in Panels b.d who prefer to decrease healthy leisure when wages are at their highest and shift the response to eventual resulting health problems to later when wages are lower and Medicare coverage begins. Young insurees in Panel b.e are less responsive to such inter-temporal substitution. Moreover, they do not need to build up high wealth balances as those who lose their health insurance at entitlement. Consequently, wealth is lower after middle age. Such intertemporal substitution is absent regarding health expenditures, thereby casting doubt on the significance of the piling up hypothesis. Finally, our findings confirm that exclusion from health insurance market becomes very detrimental at middle age, but not for younger adults who may still prefer to remain uninsured when wealth is low, the health stock is high and health problems are scant. In contrast, health insurance for elders is always optimal. Universal eligibility of insurance, whether via Medicare or private markets, might therefore not be Pareto improving. Also, Medicare was found to be welfare improving at the individual level, with the effects on 24