Consumption, Investment, and Healthcare with Aging
CConsumption, Investment, and Healthcare with Aging ∗ Paolo Guasoni † Yu-Jui Huang ‡ Abstract
This paper solves the problem of optimal dynamic consumption, investment, and healthcarespending with isoelastic utility, when natural mortality grows exponentially to reflect Gompertz’law and investment opportunities are constant. Healthcare slows the natural growth of mortality,indirectly increasing utility from consumption through longer lifetimes. Optimal consumptionand healthcare imply an endogenous mortality law that is asymptotically exponential in the old-age limit, with lower growth rate than natural mortality. Healthcare spending steadily increaseswith age, both in absolute terms and relative to total spending. The optimal stochastic controlproblem reduces to a nonlinear ordinary differential equation with a unique solution, which hasan explicit expression in the old-age limit. The main results are obtained through a novel versionof Perron’s method.
JEL:
E21, I12
MSC (2010):
Keywords: healthcare, consumption-investment, Gompertz’ law, viscosity solutions, Perron’smethod. ∗ We thank for helpful comments seminar participants at Collegio Carlo Alberto, ETH Z¨urich, University ofLimerick, Alfred Renyi Institute, National Central University in Taiwan, the QMF conference at UTS Sydney, theCongress of the Bachelier Finance Society, and the University of Colorado at Boulder. † Boston University, Department of Mathematics and Statistics, 111 Cummington Mall, Boston, MA 02215,USA, and Dublin City University, School of Mathematical Sciences, Glasnevin, Dublin 9, Ireland, email: [email protected] . Partially supported by the ERC (278295), NSF (DMS-1412529), and SFI (16/SPP/3347and 16/IA/4443). ‡ Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA, email: [email protected] . Partially supported by NSF (DMS-1715439) and the University of Colorado (11003573). a r X i v : . [ q -f i n . M F ] J a n Introduction
The steady rise in both incomes and life expectancy over the past century alludes to tantalizing linksbetween healthcare, wealth, and mortality, which are the subject of heated debate. Understandingthe relative roles of healthcare, wealth, and medical progress in explaining longevity gains is asimportant as it is difficult (see Cutler et al. (2006) for a survey). Yet, with few exceptions, modelsof optimal consumption and investment have largely shied away from mortality and healthcare,leaving a wide gap between idealized theoretical settings and realistic empirical studies.Healthcare is different from consumption in typical goods and services: it often causes immediatepain, and is justified only by its expected effect in reversing or delaying the onset of disease and,ultimately, death. Healthcare does not directly generate utility, but, by reducing mortality risk, itextends the lifetime over which consumption yields utility.Mortality (the probability that someone alive today dies next year) displays an approximateexponential growth with age, an observation that has remained remarkably stable since its discoveryby Gompertz (1825), even as mortality has steadily declined at all age groups (Figure 1). A centralquestion is to which extent such decline can be ascribed to the availability and the optimal useof healthcare, and a satisfactory answer hinges on the predictions of a model in which healthcarechoices and their resulting mortality are endogenous. This is the goal of this paper.Our model focuses on a representative household that makes consumption, investment, andhealthcare spending decisions to maximize welfare. Consumption generates utility, healthcare re-duces the growth of mortality below its natural, constant rate of Gompertz’ law, while investmenthelps increasing wealth. Optimality is reached when their marginal values are the same.A careful representation of the impact of death is a critical issue in models with endogenousmortality: the ostensibly natural approach of expected lifetime utility leads to potential preferencefor death over life and violates the invariance of preferences to affine transformations. To avoid thispitfall, we assume that a death leaves the surviving household with a fraction of its previous wealth,but the same mortality rate. This representation is consistent with the interpretation of wealth aspresent value of current assets and future cash flows, and of the surviving household as a spouse inroughly the same age group. It also preserves affine invariance and unconditional preference of lifeover death.Equally important is the impact of healthcare on mortality growth – the efficacy function. Anunwise specification, in which sufficiently high spending can arrest and reverse aging, can leadto the implausible (and counterfactual) conclusion that early healthcare interventions can bringmortality to zero, leaving a household immortal albeit less wealthy. We exclude such dubiousoutcomes through two assumptions: first, the efficacy depends on healthcare spending relative towealth, thereby emphasizing the opportunity costs (such as foregone income) of healthcare andhealthy behaviors. Second, we posit that the same amount of healthcare is more effective whenhealth is worse, i.e., when mortality is higher.This paper contributes to the literature both mathematically and economically. On the tech-nical side, the model leads to an optimal control problem which features, in addition to the usualconsumption and investment, an arrival rate of jumps (deaths) that is partially controlled (byhealthcare) through a state variable (mortality). As such a control problem with jumps, it normallycorresponds to a Hamilton-Jacobi-Bellman (HJB) integro-differential equation. Yet, exploiting ascaling property of the value function, we are able to reduce the HJB equation to a nonlinear ordi-nary differential equation. Although such an equation does not admit explicit solutions, we provethat it has a single global solution through an unconventional version of Perron’s method.Perron’s method generally refers to the construction of a viscosity solution by interpositionbetween a supersolution and a subsolution, which are typically more tractable (cf. Janeˇcek and2
900 cohort1940 cohort
45 50 55 60 65 70 75110
Figure 1: Mortality rates (vertical axis, in logarithmic scale) at adults ages for the birth cohorts ofyears 1900 and 1940 (from top to bottom) in the United States (both males and females). Source:Berkeley Human Mortality Database.Sˆırbu (2012); Bayraktar and Zhang (2015)). By contrast, we apply Perron’s method not to asingle pair of super- and subsolutions, but to a collection thereof (Definition A.1). In additionto identifying the value function as the unique solution to the reduced nonlinear HJB equation(Theorem 3.1), this approach also delivers powerful estimates on the value function (Theorem 3.2),which in turn yield that Gompertz’ law holds asymptotically in the old-age limit. Such additionalinsights entail some challenges, such as verifying the subsolution property (Propositions A.2 andA.3): as the equation involved is of the first order, the regularizing property of a second-order termis not available to establish strict concavity, in contrast to standard viscosity applications.Proving that the solution to the differential equation is indeed the value function (verification)also requires new tools. Standard results do not apply, as they do not support controllable jumprates, and, in extant results that support them (Cohen and Elliott, 2015, chapters 20 and 21), jumprates depend on a control variable in a compact set, while in the present problem they depend ona state variable, and the control has unbounded support. The technical challenges of such anextension are overcome in Theorems B.1 and B.2, which establish a verification result that exploitsthe properties of the law of the candidate optimal process. On the economic side, the effect of (exogenous) mortality on household decisions has long beenrecognized. Yaari (1965) solves the problem of consumption and investment in a complete marketand finds that annuitization is optimal in the absence of a bequest motive. Vice versa, Richard(1975) proves that buying life insurance is optimal as a hedge against loss of future earnings. In theoptimal consumption framework, Rosen (1988) investigates the welfare changes from gains in life A related potential approach to dealing with controlled jumps is stochastic Perron’s method (a verification typeresult without smoothness required), as recently employed in Bayraktar and Li (2017). relative to health capital. By contrast, Hall and Jones (2007)emphasize the dual role of health in determining mortality and additively increasing utility fromconsumption.As this literature has evolved toward richer and more realistic models, Gompertz’ law of ex-ponential mortality growth, however, has remained a conspicuous absence. This paper makes thisfeature a central element: taking the mortality rate as sole state variable in addition to wealth,determined jointly by Gompertz’ law and healthcare choices, we investigate the mutual responsebetween healthcare spending and mortality growth, and the lower mortality rates that result.Several important implications are brought by our analysis. First, it identifies the marginalefficacy of optimal health spending as inversely proportional to the elasticity of consumption withrespect to mortality. Because in the model such elasticity increases with mortality, and healthcarespending has diminishing returns (decreasing marginal efficacy), it follows that health spendingrelative to wealth increases with age and mortality, converging to a maximum finite rate in theold-age limit. Second, health spending is nearly negligible in youth, but it rises rapidly with age,outpacing the growth in consumption and taking a larger share of total spending. At very oldages the trend reverses, as health spending rate stabilizes while consumption continues to risewith mortality. The latter effect, however, becomes visible only at ages that are not reached bymost individuals. Third, the model generates an endogenous mortality curve in which its naturalgrowth is reduced by healthcare. Importantly, endogenous mortality is also close to exponential,and asymptotically exponential in the old-age limit, thereby confirming the empirical observationthat Gompertz’ law has survived two centuries of medical progress. The reduction in mortalitygrowth depends on the efficacy of healthcare as well as the elasticity of intertemporal substitution(EIS), but not on other quantities.The rest of the paper is organized as follows: Section 2 first discusses in detail the assumptionson preferences, mortality, and healthcare, then provides rigorous definitions of the model and itsprobabilistic structure. Section 3 presents the main results in order of complexity, first in a baselinemodel with neither aging nor healthcare, then adding aging, and finally in the complete settingwith both aging and healthcare. Section 4 incorporates risky assets into the main model, whileSection 5 calibrates the main model and discusses the implications. Section 6 concludes.
The main model aims to understand optimal consumption and healthcare spending in relation tomortality, with a focus on a household seeking to maximize total welfare. Section 4 extends the4nalysis to include investment in risky assets.
A natural starting point are the familiar time-additive preferences, in which expected welfare hasthe representation E (cid:20)(cid:90) τ e − δt U ( X t c t ) dt (cid:21) , (2.1)where τ denotes the lifetime, U : R + → R is a utility function (i.e., increasing and concave), theparameter δ ≥ c t represents the rate of consumption per unit oftime, as a fraction of current wealth X t , interpreted as the household’s net worth, which includesthe present value of future income, not specified separately.The limit of this approach is that it tacitly reduces death to a change in preferences, assumingthat utility equals zero in the afterlife [ τ, ∞ ), and implying that death is preferable to negativeutility, as recognized by Shepard and Zeckhauser (1984), Rosen (1988), Bommier and Rochet(2006), Hall and Jones (2007), Bommier (2010).Yet, a more appealing approach than costless afterlife utility is to note that households haveconcrete bequest motives which center on the welfare of similar individuals. For example, uponhis death a man may leave behind a wife in a similar age group, hence with a similar mortalityrate. Such a household makes consumption and healthcare spending choices that account for thewelfare of both spouses over their lifetimes. Larger households face even more complex choices,which involve the lives of several people.Striking a balance between realism and tractability, suppose that a household experiences asequence of deaths at times ( τ n ) n ∈ N with 0 := τ < τ < · · · < τ n ↑ ∞ a.s. and that after eachdeath the surviving household members inherit a fraction ζ ∈ [0 ,
1] of wealth, while retaining thesame mortality. This assumption is clearly a simplification, as in reality households include only afew members, but this flaw is mitigated by the time-preference parameter δ , whereby the first fewlifetimes account for most of the expected utility.With this assumption, denoting by X t the household wealth at time t if no deaths have occurred,the actual household wealth after the n -th death is ζ n X t , and total welfare becomes E (cid:34) ∞ (cid:88) n =0 (cid:90) τ n +1 τ n e − δt U ( ζ n X t c t ) dt (cid:35) . (2.2)Furthermore, the discussion henceforth focuses on the isoelastic class U ( x ) = x − γ − γ < γ (cid:54) = 1 (2.3)which, in the absence of healthcare and mortality, generates consumption policies proportional towealth. Such a property is attractive because empirical consumption-wealth ratios do not displayany significant secular trend.In contrast to the lifetime-horizon approach described earlier, this model does not equate deathto a change in preferences, but rather to a loss for the surviving household, while leaving preferencesunchanged. The parameter ζ controls the severity of the loss: ζ = 1 implies immortality (commonin the literature as uncommon in reality), as the arrivals of τ n are inconsequential; at the otherextreme, with ζ = 0 death implies a total loss, after which only zero spending is possible, leadingto a constant utility rate of U (0). 5n general, ζ ∈ [0 ,
1] crudely summarizes the combined economic effects of death, which includeinheritance and estate taxes, loss of pensions and annuities, foregone future income, and a myriadof other actual or opportunity costs. (A loss of future cash flow is equivalent to a loss in wealth,assuming that the cash flow is replicable.) Although the model does not include explicitly life-insurance and annuity contracts, the parameter ζ can also be thought of as a measure of protectionof the household wealth against mortality losses, with full protection for ζ = 1 and no protectionfor ζ = 0.The model does not distinguish between the relative impact of household components withdifferent ages, as the focus of this paper is not on household structure but rather on the tradeoffbetween consumption and healthcare, which is now introduced. The household is homogeneous, in that all members share the same mortality M t starting at theinitial level M := m . In the absence of healthcare, mortality grows exponentially, consistentlywith the classical Gompertz (1825) law dM t = βM t dt. (2.4)Healthcare spending reduces mortality growth according to an efficacy function g : R + → R + of h t , the spending rate in healthcare as a fraction of household wealth: dM t = ( β − g ( h t )) M t dt. (2.5)The efficacy function g is assumed strictly increasing and concave, which reflects the diminishingreturns from increased health expenditure. In addition, g (0) = 0, which identifies β as the natural rate at which mortality grows in the absence of healthcare expenditures. Finally g is defined onlyon the positive real line, consistent with the interpretation of health investment as irreversible. The assumption that healthcare expenses affect mortality growth relative to wealth rather thanin absolute terms emphasizes the lost income and earning opportunities resulting from healthcareusage. For example, for households whose wealth is dominated by the value of future income, thelost income from healthcare usage is approximately proportional to wealth. In addition, means-tested subsidies and income taxes on health-insurance premiums effectively make the same medicalprocedures cheaper for poorer households, and the assumption of proportionality approximates thisdependence with a linear relation. Likewise, Chetty et al. (2016) recently find that life expectancyis significantly correlated with health behaviors but not with access to medical care. In reality thedeterminants of healthcare spending on mortality are complex (Cutler et al., 2006), and the relativeimportance of proportional and absolute components is largely an empirical question. The presentsimplification offers a plausible and parsimonious approximation that focuses on proportional costs.A related important reason to consider proportional costs is to avoid the unrealistic implicationthat wealth buys immortality. Indeed, if h t in (2.5) were to represent an absolute amount of healthexpenditures, wealthy individuals could effectively reduce mortality to zero through early healthexpenses while maintaining non-zero consumption. In reality, life expectancy in the top 1% incomepercentile is about five years higher than for median incomes (Chetty et al., 2016, Figure 2). See for example Grossman (1972), Ehrlich and Chuma (1990), Hall and Jones (2007). For example, Smith (1999, 2007) reports ill health as a leading cause of early retirement. In their words, geographical differences in life expectancy for individuals in the lowest income quartile weresignificantly correlated with health behaviors such as smoking [...], but were not significantly correlated with access tomedical care, physical environmental factors, income inequality, or labor market conditions. .3 Savings In the basic version of the model, the household leaves savings in a safe asset which earns a constantrate r , with no other financial or insurance contracts available. In particular, the household doesnot have access to life-insurance contracts that pay out in the event of death. This assumption isconsistent with the interpretation of wealth as inclusive of future income, which in practice can behedged only in rather limited amounts.With these assumptions, at each time t the household spends at rates c t in consumption and h t in healthcare, while earning a constant interest rate r on wealth. If N t denotes the number ofdeaths up to time t , regulated by the mortality dynamics in (2.5), household wealth Ξ t = X t ζ N t incorporating death losses evolves as: d Ξ t Ξ t = ( r − c t − h t ) dt − (1 − ζ ) dN t . (2.6)Note that the only source of randomness is the arrival of deaths, without which the model revertsto a deterministic consumption-investment problem, in which the optimal policy is to consume ata rate proportional to wealth.After the description and motivation of the main model provided here, the next section proceedswith the mathematical details required for the precise statement of the main result. The rigorous formulation of the model starts with the probability space (Ω , F , P ), which supportsa sequence { Z n } n ∈ N of independent, identically distributed random variables with an exponentiallaw P ( Z n > z ) = e − z for all z ≥ n ∈ N . (These random times are interpreted as mortality-adjusted times of death, as defined below.)Denote by L , +loc the collection of all nonnegative locally integrable functions f : R + → R + , andnote that L , +loc is metrizable, thus a Borel space. Define also L := (cid:110) { f n } n ∈ N (cid:12)(cid:12)(cid:12) f ∈ L , +loc , f n = f ( Z , ..., Z n ) for some Borel f : R n + → L , +loc (cid:111) , which represents the family of sequences of consumption-healthcare policies, with the policy afterthe n -th death depending possibly on the previous n times, in addition to calendar time.Consider a nondecreasing concave function g : R + → R + with g (0) = 0. For any ( t, m ) ∈ R and h ∈ L , +loc , let M t,m,h be the deterministic process defined by dM t,m,hs = M t,m,hs [ β − g ( h ( s ))] ds, M t,m,ht = m, (2.7)where β ≥ { Z n } n ∈ N . For any m ≥ { h n } ∈ L , construct recursively a sequence { τ n } n ∈ N ofrandom times as follows: first, set τ := 0 and m := m ; then, for each n ≥
0, define τ n +1 := inf (cid:26) t ≥ τ n (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) tτ n M τ n ,m n ,h n s ds ≥ Z n +1 (cid:27) , m n +1 := M τ n ,m n ,h n τ n +1 . (2.8)Now introduce the counting process { N t } t ≥ : N t := n for t ∈ [ τ n , τ n +1 ) , (2.9)7nd observe from the construction of { τ n } n ∈ N that P ( N t = n | Z , ..., Z n ) = P ( t ∈ [ τ n , τ n +1 ) | Z , ..., Z n ) = exp (cid:18) − (cid:90) tτ n M τ n ,m n ,h n s ds (cid:19) { t ≥ τ n } , (2.10)which means that the mortality rate of τ n +1 at time t is precisely M τ n ,m n ,h n t , as required.Consider now the collection of processes: A := (cid:40) c t = ∞ (cid:88) n =0 c n ( t )1 { τ n ≤ t<τ n +1 } , h t = ∞ (cid:88) n =0 h n ( t )1 { τ n ≤ t<τ n +1 } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { c n } , { h n } ∈ L (cid:41) . (2.11)The construction of A is understood as follows: first, use { h n } ∈ L to construct { τ n } as in (2.8);then, use { c n } , { h n } ∈ L to define the processes c t and h t . Then, the process M t,m,h is defined asin (2.7) for any h as in (2.11). With initial wealth x ≥ m ≥ t ≥
0, an household at each time s ≥ t chooses the rates of spending in consumption ( c s ≥
0) and healthcare ( h s ≥ r , household wealth X t,x,c,hs before mortality losses evolves as dX t,x,c,hs = X t,x,c,hs [ r − ( c s + h s )] ds, X t,x,c,ht = x. (2.12)The consumption and healthcare policies ( c, h ) ∈ A describe planned expenditures depending oncalendar time and past and current events, as follows. At time 0, the household chooses determin-istic policies c ( t ) and h ( t ), and mortality M ,m,h evolves accordingly as in (2.7). Upon the firstdeath at time τ , the surviving household carries on with wealth ζX ,x,c ,h τ and mortality M ,m,h τ ,switching to the deterministic policies c ( t ) and h ( t ). In general, if n deaths have occurred bytime t , the spending policies are c n ( t ) and h n ( t ).Healthcare makes mortality partially endogenous, and its effect is summarized by the efficacyfunction g , with which the houehold reduces the growth of mortality M ,m,h by selecting appropriate { h n } ∈ L . In the absence of healthcare (i.e. g ≡ M ,mt = me βt follows Gompertz’ law withparameter β . The household’s objective at time 0 is to maximize expected utility from intertemporalconsumption V ( x, m ) := sup ( c,h ) ∈A E (cid:20)(cid:90) ∞ e − δt U (cid:16) c t ζ N t X ,x,c,ht (cid:17) dt (cid:21) , (2.13)where the utility function U is of the isoelastic form in (2.3). The optimization problem considered in this paper departs from the classical consumption-investmentproblem in two aspects: aging , whereby natural mortality follows Gompertz’ law, and healthcare ,which slows down mortality growth.To understand the separate effects of each aspect, henceforth we present the main results inorder of complexity: First (Section 3.1), with neither aging nor healthcare – a minor variation ofthe classical setting. Second (Section 3.2), with aging but without healthcare – a partially newsetting that provides a reference for the general model with both aging and healthcare (Section3.3). 8 .1 Neither Aging nor Healthcare ( β = 0 and g ≡ ) When the mortality rate is constant ( β = 0), and healthcare is unavailable ( g ≡ forever young . The arrival times of death { τ n } n ∈ N are simply τ = 0 , τ n +1 = inf { t ≥ τ n | ( t − τ n ) · m ≥ Z n +1 } ∀ n ≥ , where m ≥ N defined in (2.9) is a Poissonprocess with intensity m ≥
0, and the collection of controls A in (2.11) reduces to C := (cid:40) c t = ∞ (cid:88) n =0 c n ( t )1 { τ n ≤ t<τ n +1 } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { c n } ∈ L (cid:41) . (3.1)The value function in (2.13) is then V ( x, m ) = sup c ∈C E (cid:20)(cid:90) ∞ e − δt U (cid:16) c t ζ N t X ,x,ct (cid:17) dt (cid:21) . (3.2)The next proposition describes the optimal policy in this basic setting: Proposition 3.1.
Let m ≥ satisfy δ + (1 − ζ − γ ) m − (1 − γ ) r > . (3.3) Then, for all x ≥ , V ( x, m ) = x − γ − γ c ( m ) − γ , where c ( m ) := δ + (1 − ζ − γ ) mγ + (cid:18) − γ (cid:19) r. (3.4) Furthermore, ˆ c t := c ( m ) , for all t ≥ , is an optimal control of (3.2) .Proof. See Section A.1.First, note that the parametric restriction in (3.3) is a well-posedness condition, which requiresthat the time-preference rate is large enough to prevent consumption from being deferred indefi-nitely. In fact, this restriction is equivalent to a positive consumption rate (3.4).The main message of this proposition is that, in the absence of both aging and healthcare, theoptimal consumption rate is a constant proportion of wealth, resulting from a weighted sum of theinterest rate r and of the discount rate δ +(1 − ζ − γ ) m , which captures the effects of time-preference δ and of mortality m , weighted for its impact via ζ . In particular, for a total loss ( ζ = 0), mortalityadds one-to-one to time-preference δ , and therefore it is equivalent to a higher δ , as in Yaari (1965).Importantly, higher mortality implies a higher consumption rate for EIS 1 /γ >
1, while theopposite holds for 1 /γ <
1. This dependence is explained in terms of the usual income andsubstitution effects in response of negative wealth shocks. On one hand, higher mortality ratespurs the household to consume before wealth is reduced by deaths (substitution effect). On theother hand, mortality shocks mean less future consumption, which in turn encourages savings toalleviate the consumption shock (income effect). Either of these countervailing effects prevailsabove or below γ = 1. At this threshold, which corresponds to logarithmic utility, the two effectsperfecly offset each other, and the consumption rate reduces to the time preference δ , regardless ofmortality m , its impact ζ , and the safe rate r . 9 .2 Aging without Healthcare ( g ≡ ) The next conceptual step is to add aging to the optimization problem, assuming that mortalitygrows according to Gompertz’ law ( β > g ≡ m ≥ M t = me βt for all t ≥
0. The times of death { τ n } n ∈ N then become τ = 0 , τ n +1 = inf (cid:26) t ≥ τ n (cid:12)(cid:12)(cid:12)(cid:12) ( e βt − e βτ n ) mβ ≥ Z n +1 (cid:27) ∀ n ≥ , (3.5)while the set of controls A in (2.11) again reduces to C in (3.1), and the value function in (2.13) tothe form (3.2).The next proposition describes the effect of aging on the optimal consumption-savings problem: Proposition 3.2.
Assume either one of the two conditions: (i) γ, ζ ∈ (0 , and δ + ( γ − r > ;(ii) γ, ζ > . Then, for any ( x, m ) ∈ R , V ( x, m ) = x − γ − γ u ( m ) − γ , where u ( m ) := (cid:20) β (cid:90) ∞ e − (1 − ζ − γ ) myβγ ( y + 1) − (cid:16) δ +( γ − rβγ (cid:17) dy (cid:21) − > is a strictly increasing function on (0 , ∞ ) satisfying(a) u (0) = c (0) = δ +( γ − rγ > , lim m →∞ ( u ( m ) − ( c ( m ) + β )) = 0 , and c ( m ) < u ( m ) < c ( m ) + β for all m ∈ (0 , ∞ ) . (3.7) (b) u (cid:48) (0+) = ∞ , u (cid:48) ( ∞ ) = − ζ − γ γ .Furthermore, ˆ c t := u ( me βt ) , for all t ≥ , is an optimal control of (3.2) .Proof. See Section A.2.Although this setting is known in the actuarial literature (see for example Huang et al. (2012)),the above result presents a few novel aspects, starting from the more complicated parametricrestrictions for well-posedness, which require that either γ, ζ ≤ γ, ζ >
1. In fact, the growingmortality rate generates a further motive for indefinite deferral of consumption, and a resulting ill-posed problem. Indeed, if γ > ζ <
1, with a growing mortality rate the household anticipatesso much future misery that it would attempt to reduce current consumption to zero. Indeed, onlya hypothetically positive mortality shock ( ζ >
1) would lead to a well-posed problem. Such a case ζ >
1, which would correspond to a life-insurance policy above the value of future income, is notrealistic and not pursued further: it is only discussed here to point out the source of ill-posednessfor γ > /γ > γ < β is the maximal increase in the consumption-wealth ratio resulting from aging, compared to another household with the same mortality butwithout aging. Of course, the increase in consumption rates results from the dominant substitutioneffect, which responds to higher mortality with earlier consumption.10 gingForever YoungForever Young + β Immortal (%) C on s u m p t i on - W ea l t h R a t i o ( % ) Figure 2: Consumption-wealth ratios (vertical, in percent) against instantaneous mortality rate(horizontal, in percent), with zero, constant, and exponentially growing mortality (solid, frombottom to top). The dashed line represents the asymptotic linear consumption rate with high,exponentially-growing mortality, which is also an upper bound at any mortality rate. Parametersas in Section 4.Finally, part (b) in the Proposition establishes that the consumption rate with respect to mor-tality increases very steeply near immortality ( m = 0), while becoming asymptotically linear in theold age limit ( m = ∞ ), and reaching the same slope − ζ − γ γ as in the case without aging.Figure 3.2 summarizes the attributes of the settings discussed so far: the flat bottom lineidentifies the constant consumption rate for a household with zero mortality. The line intersecting at m = 0 describes the consumption of a “forever young” household, for which mortality remains fixedat m over time. The curve above this line, which also intersects at m = 0, plots the consumptionrate of an aging household, which is higher in view of the substitution effect. The top dashedline, parallel to the constant mortality line, describes the asymptotic consumption rate of the aginghousehold for large m , which is also an upper bound.Importantly, these simplified settings provide a range in which the consumption rate of the fullmodel should lie. As healthcare curbs mortality growth, implied consumption rate should lie belowthe solution with aging but without healthcare, in anticipation of slower mortality growth. At thesame time, consumption should be higher than with a constant mortality, at least if healthcarecannot reverse aging, as it does not in reality. The full model incorporates both aging, with mortality increasing naturally with Gompertz’ law( β > g ≥ g that satisfies the well-posedness condition (3.9) below, which stipulates that even arbitrarily high amounts of healthcarecannot arrest or reverse mortality growth, in addition to the following regularity conditions: Assumption 3.1.
Let g : R + → R + be twice-differentiable with g (0) = 0 , g (cid:48) ( h ) > and g (cid:48)(cid:48) ( h ) < for h > , and satisfy the Inada condition: g (cid:48) (0+) = ∞ and g (cid:48) ( ∞ ) = 0 . (3.8)The restrictions on the other parameter values ensure well-posedness by excluding indefinitedeferral of consumption. Theorem 3.1.
Let Assumption 3.1 hold, and let < γ < and ¯ c := δγ + (cid:0) − γ (cid:1) r > . If g (cid:18) I (cid:18) − γγ (cid:19)(cid:19) < β with I := ( g (cid:48) ) − , (3.9) then the value function in (2.13) satisfies V ( x, m ) = x − γ − γ u ∗ ( m ) − γ where u ∗ : R + → R + is theunique nonnegative, strictly increasing solution to the equation L u ( m ) := u ( m ) − c ( m ) u ( m ) + mu (cid:48) ( m ) (cid:32) sup h ≥ (cid:26) g ( h ) − − γγ u ( m ) mu (cid:48) ( m ) h (cid:27) − β (cid:33) = 0 . (3.10) Furthermore, u ∗ is strictly concave, and (ˆ c, ˆ h ) defined by ˆ c t := u ∗ ( M t ) and ˆ h t := I (cid:18) − γγ u ∗ ( M t ) M t · ( u ∗ ) (cid:48) ( M t ) (cid:19) , for all t ≥ , (3.11) optimizes (2.13) .Proof. This result is a consequence of Proposition A.5 and Corollary A.1 below.This result identifies the optimal consumption policy as the solution of (3.10), a first-order,nonlinear ODE, in which the effect of healthcare is captured by the last nonlinear term. Such asolution does not have a closed-form expression even in relatively simple settings, such as the onediscussed next in Corollary 3.1, but it is nonetheless straightforward to calculate numerically, andso are its quantitative implications.A delicate point is that the solution to equation (3.10) is uniquely identified without any ad-ditional boundary conditions, because the equation has only one increasing solution defined for all m ∈ R + , while all others explode for finite m or start decreasing for m large enough. Note thatnatural boundary condition u (0) = c (0) holds for any local solution in [0 , ε ), and therefore doesnot identify the one defined for all m ∈ R + .Also, the above result establishes the optimality condition for healthcare expenditure in (3.11),whereby the marginal efficacy of optimal healthcare is inversely proportional to mu (cid:48) ( m ) /u ( m ), theelasticity of consumption with respect to mortality, where the constant of proportionality dependson preferences.The next results provides a deeper insight on the impact of healthcare on the growth rate ofmortality: 12 heorem 3.2. Let Assumption 3.1 and condition (3.9) hold, and let < γ < , δγ + (cid:0) − γ (cid:1) r > .Define β g := β − sup h ≥ (cid:26) g ( h ) − − γγ h (cid:27) ∈ (0 , β ) . (3.12) As (3.6) defines u ( m ) , define u g ( m ) analogously with β g in place of β . Then, for any m > , u g ( m ) ≤ u ∗ ( m ) ≤ min { u ( m ) , c ( m ) + β g } (3.13) and lim m →∞ ( c ( m ) − u ∗ ( m )) = β g . (3.14) Proof.
See Section A.3.The message of this result is that even if the exact effect of healthcare on consumption iscomplicated, it does admit simple upper and lower bounds (Figure 3). The lower bound is the oneobtained from u g ( m ), the consumption rate in a model in which healthcare is not available, butmortality grows at the lower rate β g < β . Indeed, the household would gladly give up access tohealthcare in exchange for such a lower rate of mortality growth.Upper bounds are consumption rates under the same mortality growth rate β but with no accessto healthcare (i.e. u ( m )), and for a forever young household, augmented by the adjusted growthrate β g (i.e. c ( m ) + β g ). The former estimate is more accurate at younger ages, while the latterat older ages. In all cases, the minimum consumption rate, i.e. the lower bound u g ( m ), yields thesharpest estimate.The next result specializes the analysis to a concrete model, assuming an efficacy function ofisoelastic type, thereby enabling comparative statics and parameter estimation. Corollary 3.1.
Let < γ < , ¯ c := δγ + (cid:0) − γ (cid:1) r > , and g : R + → R + of the form g ( z ) := a z q q , for some a > and q ∈ (0 , . If a − q q (cid:16) − γγ (cid:17) − q − q < β , then V ( x, m ) = x − γ − γ u ∗ ( m ) − γ for all ( x, m ) ∈ R , where u ∗ : R + → R + is the unique nonnegative, strictly increasing solution to the equation u ( m ) − c ( m ) u ( m ) − βmu (cid:48) ( m ) + 1 − qq a − q (cid:18) − γγ u ( m ) (cid:19) − q − q (cid:0) mu (cid:48) ( m ) (cid:1) − q = 0 , (3.15) Furthermore, (ˆ c, ˆ h ) defined by ˆ c t := u ∗ ( M t ) and ˆ h t := a − q (cid:18) − γγ u ∗ ( M t ) M t · ( u ∗ ) (cid:48) ( M t ) (cid:19) − − q , for all t ≥ , is an optimal control of (2.13) . With this result at hand, we investigate in Section 5 the model’s implications for the optimalpolicies and their resulting endogenous mortality.13 u c + β g u * (%) C on s u m p t i on - W ea l t h R a t i o ( % ) Figure 3: Consumption-wealth ratios (vertical axis, in percent) against instantaneous mortalityrates (horizontal). Solid curves correspond to aging households with natural mortality growth β ,with (center) and without (top) healthcare, and with adjusted mortality growth β g (bottom). Thedashed line represents the asymptotic linear consumption ratio, which is also an upper bound. The main model in the paper assumes that households’ savings are confined to a safe investment.This section discusses extending the model to include risky assets. The main question is to whatextent healthcare and endogenous mortality are sensitive to risky investments and – conversely –whether portfolio allocation is sensitive to the mortality rate. Theorem 4.1 below argues that thepresence of risky assets with constant investment opportunities is equivalent to an increase in thesafe rate, and that the resulting optimal portfolio is independent of mortality (cf. equations (4.10)and (4.11) below). Thus, the main results in the paper remain valid with the addition of riskyassets, up to reinterpreting the safe rate parameter as an equivalent safe rate that accounts foradditional investment opportunities.Specifically, consider a risky asset S satisfying the dynamics dS t = S t ( µ + r ) dt + S t σdW t , t ≥ , (4.1)where µ ∈ R and σ > W is a standard Brownian motion independent of { Z n } n ∈ N . The independence assumption is appropriate for most individuals, as death is unlikelyto affect, or result from, price changes of publicly traded securities. Denoting by π t the fraction ofwealth that the household invests in the risky asset at time t , when no deaths have occurred thewealth process is dX t X t = [(1 − π t ) r − c t − h t ] dt + π t dS t S t = [ r + µπ t − c t − h t ] dt + σπ t dW t , t ≥ . (4.2)14o properly define the value function and derive the associated HJB equation, our probabilityspace (Ω , F , P ) in Section 2.4 needs to be enlarged to account for the additional Brownian motion W . Specifically, let (Ω , F , P ) be the probability space supporting the i.i.d. exponential randomvariables { Z n } n ∈ N , as specified in Section 2.4. Let (Ω , F , P ) be another probability space thatsupports the Brownian motion W . Then, we take (Ω , F , P ) to be the product probability space of(Ω , F , P ) and (Ω , F , P ), endowed with the filtration {F t } t ≥ generated by { Z n } n ∈ N and W .We denote by E , E , and E the expectations taken under P , P , and P , respectively.Between two consecutive death times, c t and h t are no longer deterministic as in (2.11), becausethey may depend on the evolution of S in (4.1) (or, the Brownian motion W ). More precisely, let L (Ω ) denote the collection of processes f : R + × Ω → R such that E [ (cid:82) t f ( s ) ds ] < ∞ for all t ≥
0. Also consider L , +loc (Ω ), the subspace of L (Ω ) containing nonnegative processes. Define L (cid:48) := (cid:8) { f n } n ≥ (cid:12)(cid:12) f ∈ L (Ω ) , f n = f ( Z , ..., Z n ) for some Borel f : R n + → L (Ω ) (cid:9) . We also consider L (cid:48) + , defined as L (cid:48) with L (Ω ) replaced by L , +loc (Ω ). Now, for any { h n } ∈ L (cid:48) + ,the death times in (2.8) can be formulated as follows: for each ω = ( ω , ω ) ∈ Ω, τ n +1 ( ω ) := inf (cid:26) t ≥ τ n ( ω ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) tτ n M τ n ,m n ,h n s ( ω ) ds ≥ Z n +1 ( ω ) (cid:27) , m n +1 := M τ n ,m n ,h n τ n +1 ( ω ) . Thus, (2.10) can be rewritten as, for any fixed ω ∈ Ω , P ( t ∈ [ τ n ( · , ω ) , τ n +1 ( · , ω )) | Z , ..., Z n ) ( ω )= exp (cid:32) − (cid:90) tτ n ( ω ,ω ) M τ n ,m n ,h n s ( ω ) ds (cid:33) { t ≥ τ n } ( ω , ω ) , ∀ n ≥ . (4.3)Similarly to (2.11), the collection A (cid:48) of admissible controls contains all processes ( c t , h t , π t ) where c t = ∞ (cid:88) n =0 c n ( t )1 { τ n ≤ t<τ n +1 } , h t = ∞ (cid:88) n =0 h n ( t )1 { τ n ≤ t<τ n +1 } , π t = ∞ (cid:88) n =0 π n ( t )1 { τ n ≤ t<τ n +1 } , (4.4)with { c n } , { h n } ∈ L (cid:48) + and { π n } ∈ L (cid:48) .Now, define the value function as V ( x, m ) := sup c,h,π E (cid:20)(cid:90) ∞ e − δt U ( ζ N t X t c t ) dt (cid:21) = sup c,h,π ∞ (cid:88) n =0 E (cid:20)(cid:90) τ n +1 τ n e − δt U ( ζ n X t c t ) dt (cid:21) . (4.5) In the following, we will first derive heuristically the HJB equation for V ( x, m ) and the candidateoptimal policies. These heuristic guesses turn out to be truly optimal, as verified in Theorem 4.1.By the definition of V ( x, m ) in (4.5), V ( x, m ) = sup c,h,π E (cid:20)(cid:90) τ e − δt U ( c t X t ) dt + e − δτ (cid:90) ∞ τ e − δ ( t − τ ) U ( ζ N t c t X t ) dt (cid:21) = sup c,h,π E (cid:20)(cid:90) τ e − δt U ( c t X t ) dt + e − δτ V ( ζX τ , M τ ) (cid:21) , P [ τ ( · , ω ) > t ] = e − (cid:82) t M ,m,h s ( ω ) ds from (4.3), the above equationyields V ( x, m ) = sup c,h,π E (cid:20)(cid:90) ∞ e − (cid:82) t M s ds e − δt U ( c t X t ) dt + (cid:90) ∞ M t e − (cid:82) t M s ds e − δt V ( ζX t , M t ) dt (cid:21) = sup c,h,π E (cid:20)(cid:90) ∞ e − (cid:82) t ( δ + M s ) ds [ U ( c t X t ) + M t V ( ζX t , M t )] dt (cid:21) . This shows that the value function can be viewed alternatively as an infinite-horizon problemwith running payoff U ( c t X t ) + M t V ( ζX t , M t ) and discount rate δ + M t . Suppose that a dynamicprogramming principle holds for this alternative formulation, i.e. V ( x, m ) = sup c,h,π E (cid:20) (cid:90) T e − (cid:82) t ( δ + M s ) ds [ U ( c t X t ) + M t V ( ζX t , M t )] dt + e − (cid:82) T ( δ + M s ) ds V ( X T , M T ) (cid:21) , ∀ T > . (4.6)Then, in view of d (cid:16) e − (cid:82) t ( δ + M s ) ds V ( X t , M t ) (cid:17) = e − (cid:82) t ( δ + M s ) ds (cid:20) − ( M t + δ ) V ( X t , M t ) + V x ( X t , M t ) dX t + V m ( X t , M t ) dM t + 12 V xx ( X t , M t )( dX t ) (cid:21) , (4.6) implies that for all T > c,h,π E (cid:20) (cid:90) T e − (cid:82) t ( δ + M s ) ds [ U ( c t X t ) + M t V ( ζX t , M t ) + K ( X t , M t , c t , h t , π t )] dt (cid:21) , where K ( x, m, c, h, π ) := − ( δ + m ) V ( x, m ) + [ r + µπ − c − h ] xV x ( x, m )+ ( β − g ( h )) mV m ( x, m ) + 12 σ π x V xx ( x, m ) , thereby leading to the HJB equation0 = sup c ≥ { U ( cx ) − cxV x ( x, m ) } + mV ( ζx, m ) − ( δ + m ) V ( x, m ) + rxV x ( x, m )+ sup π ∈ R (cid:26) µπxV x ( x, m ) + 12 σ π x V xx ( x, m ) (cid:27) (4.7)+ βmV m ( x, m ) + sup h ≥ {− g ( h ) mV m ( x, m ) − hxV x ( x, m ) } . Assuming heuristically that V xx < V m <
0, the above equation suggests the followingcandidate optimal policies (ˆ c, ˆ π, ˆ h ) from the first-order conditions:ˆ c = V x ( x, m ) − /γ x , ˆ π = − µσ x V x ( x, m ) V xx ( x, m ) , ˆ h = ( g (cid:48) ) − (cid:18) − xV x ( x, m ) mV m ( x, m ) (cid:19) , (4.8)16hich in turn imply that (4.7) can be simplified as0 = γ − γ ( V x ( x, m )) γ − γ + mV ( ζx, m ) − ( δ + m ) V ( x, m ) + rxV x ( x, m ) + βmV m ( x, m ) − (cid:16) µσ (cid:17) V x ( x, m ) V xx ( x, m ) − mV m ( x, m ) sup h ≥ (cid:26) g ( h ) + hxV x ( x, m ) mV m ( x, m ) (cid:27) . (4.9)Using the ansatz V ( x, m ) = x − γ − γ u ( m ) − γ , the above equation reduces to0 = u ( m ) − ¯ c u ( m ) − βmu (cid:48) ( m ) + mu (cid:48) ( m ) sup h ≥ (cid:26) g ( h ) − − γγ u ( m ) mu (cid:48) ( m ) h (cid:27) , (4.10)where ¯ c ( m ) := δ + (1 − ζ − γ ) mγ + (cid:18) − γ (cid:19) (cid:18) r + 12 γ (cid:16) µσ (cid:17) (cid:19) . (4.11)The next result shows that the heuristic derivation above does lead us to truly optimal strategies. Theorem 4.1.
Including the risky asset S as in (4.1) , Proposition 3.1, Proposition 3.2, and The-orem 3.1 still hold true, with the interest rate r replaced by r + γ (cid:0) µσ (cid:1) . In particular, the optimalconsumption and health spending are as specified therein, with c replaced by ¯ c in (4.11) , while theoptimal portfolio is ˆ π t ≡ µγσ , t ≥ . Proof.
Comparing (3.4) and (4.11), note that ¯ c differs from c , and thus equation (4.10) differsfrom (3.15), only in the interest rate: r is now raised to r + σ (cid:0) µσ (cid:1) . It follows that we can re-stateall the results in Section 3, with the raised interest rate, by using appropriate verification resultsTheorem B.2 and Proposition B.2, extensions of Theorem B.1 and Proposition B.1 to account forthe risky asset S . The resulting optimal ratios in consumption and health spending are the sameas in Section 3, while the optimal ratio in investment is given by (4.8):ˆ π t = − µσ X t V x ( X t , M t ) V xx ( X t , M t ) = − µσ X t X − γt u ( M t ) − γ ( − γ ) X − γ − t u ( M t ) − γ = µγσ . This section discusses the implications of the isoelastic model in Corollary 3.1 for optimal policies,with the model’s parameters calibrated to the values r = 1%, δ = 1%, β = 7 . γ = 0 . ζ = 50%, a = 0 . q = 0 .
46, and m = 0 . r = 1% approximates the long-term average realrate on Treasury bills reported by (Beeler and Campbell, 2012), and our time preference δ = 1%is also consistent with their estimates, while γ = 0 .
67 corresponds to the estimates obtained byHarrison et al. (2007) in field experiments. The conventional value of ζ = 50% implies that thehousehold loses half of its wealth with each death, and describes a household in which future income(including pensions and annuities) represents a high proportion of the net worth.The values of the mortality natural growth β is estimated from mortality data for the US cohortborn in 1900, assuming no healthcare available. Holding these estimates constant, the healthcareparameters a and q appearing in the efficacy function are calibrated by matching the endogenousmortality curve with mortality data for the US cohort born in 1940.17 onsumptionHealthcare
40 50 60 70 800.20.5125 Age ( years ) S pend i ng - W ea l t h R a t i o s ( % )
40 50 60 70 8010121416 Age ( years ) H ea l t h c a r e - S pend i ng R a t i o ( % ) Figure 4: Left: Consumption- and healthcare-wealth ratios (vertical) at adult ages (horizontal).Right: Healthcare, as a fraction of total spending (vertical) at adult ages (horizontal).
Empirical studies thoroughly confirm the familiar observation that health spending increases withage, raising the natural question of whether the model’s predictions satisfy this basic property.The left panel in Figure 4 offers an affirmative answer. At age 40, annual healthcare spending isjust above 0.2% of wealth, but it rises quickly to almost 1% at age 80. Such increase is broadlyconsistent with the results of Hartman et al. (2008), who report at age 85 and older health spendingbetween 5.7 and 6.9 times as large as at general working-ages.In the model, healthcare is unattractive in the young age, as mortality is unlikely, and its poten-tial losses lead to low optimal healthcare spending. As mortality grows exponentially, healthcarespending increases accordingly, at a rate that is not far from mortality growth.The right panel in Figure 4 compares health spending to total spending, including consumption.Although the increases in mortality implies higher spending rate in both categories, the effect isvery pronounced for healthcare, as its weight increases from less than 10% at age 40 to almost 20%at age 80.
A central question for the model at hand is to what extent it can account for the secular decreasein mortality rates. Figure 5 attempts to evaluate the model performance under two simplifyingassumptions. First, suppose that the cohort with birth in 1900 essentially had no access to health-care, therefore that its mortality simply grows exponentially with Gompertz’ law at the naturalrate β . Second, suppose that the 1940 cohort had full access to healthcare, whence its mortalityrate follows the endogenous growth implied by the model.Both assumptions are clearly crude approximations, as healthcare did exist in 1900, thoughit was certainly less advanced and available than forty years later. Also, being forty-year old in1940, the 1900 cohort clearly had some access to healthcare. Yet, as mortality rates in the UnitedStates are available from 1933, data on the 1900 cohort begins with 33-year olds. Similarly, at thetime of writing mortality rates for 75-year-olds are not available for younger cohorts than 1940.Data availability aside, much of the rise in employer-sponsored insurance in the United Statesdeveloped in the wake of wage controls enacted during World War II, which makes the 1940 cohorta reasonable choice for the our purposes.Keeping in mind these limitations of the model and the data, Figure 5 shows that the calibratedparameters are able to largely explain the decline in mortality between the two cohorts as the18
900 cohort without healthcare1940 cohort with healthcare
40 50 60 70 800.10.20.51.02.05.010.0 Age H years L M o r t a lit y H % L Figure 5: Empirical (dots) and model-implied (lines) mortality rates at adult ages for the birthcohorts of 1900 and 1940result of health spending. In particular, it attests the ability of the model to reproduce declinesin mortality rates that are close to the ones observed historically, and that are consistent with theplausible levels of health spending described above.
The role of healthcare in reducing mortality rather than increasing current utility calls for a differenttreatment from other forms of consumption. At the same time, the exponential growth of mortalityis a leading driver of health spending. This paper combines these features in a model of optimalchoice of consumption and health spending, in which savings accrue a constant interest rate.The model captures a few stylized facts on mortality and health spending. Healthcare leads toan endogenously determined mortality curve that is close to exponential, albeit with a lower growthrate, consistently with the observed secular decline in mortality. Healthcare spending continues toincrease at adult ages, outpacing consumption growth, and therefore also its share of total spendingincreases.
A Proofs of Main Results
In this section, we prove the main results in Section 3 by verification arguments, relying on TheoremB.1 and Proposition B.1. Given f : R + → R , consider its Legendre transform (cid:101) f ( y ) := sup x ≥ { f ( x ) − xy } for y ∈ R + . A heuristic derivation as in Section 4.1 shows that the Hamilton-Jacobi equation19ssociated with V ( x, m ) in (2.13) is (cid:101) U ( w x ( x, m )) + mw ( ζx, m ) − ( δ + m ) w ( x, m )+ rxw x ( x, m ) + βmw m ( x, m ) + sup h ≥ {− mw m ( x, m ) g ( h ) − hxw x ( x, m ) } = 0 . (A.1)This is simply (4.9) without the second-order term, contributed by the added risky asset in Section 4. A.1 Neither Aging nor Healthcare ( β = 0 and g ≡ ) Recall the setup in Section 3.1. Since V ( x, m ) is nondecreasing in x by definition, the supremumin the last term of (A.1) vanishes, leading to the equation (cid:101) U ( w x ( x, m )) + mw ( ζx, m ) − ( δ + m ) w ( x, m ) + rxw x ( x, m ) = 0 . (A.2)If V is of the form V ( x, m ) = x − γ − γ v ( m ). Then, the above equation reduces to v ( m ) − γ − c ( m ) v ( m ) = 0 , where c ( m ) is defined as in (3.4). By setting v ( m ) = u ( m ) − γ , we obtain from the above equation u ( m ) − c ( m ) u ( m ) = 0 , (A.3)whence u ( m ) = c ( m ). We then prove Proposition 3.1 by verification. Proof of Proposition 3.1.
Set w ( x, m ) := x − γ − γ c ( m ) − γ . Note that (3.3) implies c ( m ) > γ > γ (cid:54) = 1. Case I: < γ <
1. By Theorem B.1, it suffices to verify (B.2) and (B.3) under current context.For any x ≥ c ∈ C , and n ∈ N , since 0 < γ <
1, we have0 ≤ E (cid:104) e − ( δ + m )( t − τ n ) w (cid:16) ζ n X ,x,ct , m (cid:17) (cid:12)(cid:12)(cid:12) Z , ..., Z n (cid:105) ≤ e − ( δ + m )( t − τ n ) ( ζ n X ,x,cτ n ) − γ − γ e (1 − γ ) r ( t − τ n ) c ( m ) − γ → t → ∞ , where the convergence follows from (3.3). This already verifies (B.2). On the other hand, since τ n is the sum of n independent, identically distributed exponential random variables with mean 1 /m ,0 ≤ E (cid:104) e − δτ n w (cid:0) ζ n X ,x,cτ n , m (cid:1)(cid:105) ≤ ζ (1 − γ ) n x − γ − γ c ( m ) − γ E (cid:104) e − δτ n e (1 − γ ) rτ n (cid:105) = ζ (1 − γ ) n x − γ − γ c ( m ) − γ (cid:90) ∞ e − ( δ +( γ − r ) t m n e − mt t n − ( n − dt = (cid:0) mζ (1 − γ ) (cid:1) n ( n − x − γ − γ c ( m ) − γ (cid:90) ∞ e − ( δ + m +( γ − r ) t t n − dt = (cid:32) mζ (1 − γ ) δ + m + ( γ − r (cid:33) n x − γ − γ c ( m ) − γ , (A.4)where the third equality requires δ + m + ( γ − r >
0, which is true under (3.3). Finally, notingthat (3.3) implies mζ (1 − γ ) δ + m +( γ − r ∈ (0 , E (cid:2) e − δτ mn w (cid:0) ζ n X ,x,cτ mn , m (cid:1)(cid:3) → n → ∞ ,which verifies (B.3). 20 ase II: γ >
1. By Proposition B.1, it suffices to establish (B.17)-(B.19) and show thatˆ c t ≡ c ( m ) satisfies (B.2) and (B.3). For any ε >
0, since g ≡
0, the sequence { τ εn } n ≥ constructedin Appendix B coincides with { τ n } n ≥ . The counting process N ε in (B.15) is therefore the sameas N in (2.9). It follows that (B.19) trivially holds under current context. Given x ≥ c ∈ C , and ε >
0, consider ( c ε ) t := c t X ,x,ct X ,x,ct + εe rt ∀ t ≥ . (A.5)By construction, X ,x + ε,c ε t = X ,x,c ε + εe rt for all t ≥
0. This, together with γ >
1, implies that forany n ∈ N ,0 ≥ E (cid:104) e − ( δ + m )( t − τ n ) w (cid:16) ζ n X ,x + ε,c ε t , m (cid:17) (cid:12)(cid:12)(cid:12) Z , ..., Z n (cid:105) ≥ e − ( δ + m )( t − τ n ) ( ζ n ε ) − γ e (1 − γ ) rt − γ c ( m ) − γ . Since the right hand side converges to 0 a.s. as t → ∞ , the above inequality in particular impliesthat (B.17) holds. On the other hand, by a calculation similar to (A.4),0 ≥ E (cid:104) e − δτ n w (cid:0) ζ n X ,x + ε,c ε τ n , m (cid:1)(cid:105) ≥ ζ (1 − γ ) n ε − γ − γ c ( m ) − γ E (cid:104) e − δτ n e (1 − γ ) rτ n (cid:105) = (cid:32) mζ (1 − γ ) δ + m + ( γ − r (cid:33) n ε − γ − γ c ( m ) − γ . Since (3.3) again guarantees that mζ (1 − γ ) δ + m +( γ − r ∈ (0 , E [ e − δτ n w ( ζ n X ,x,cτ n , m )] → n → ∞ , which verifies (B.18). Now, with ˆ c t ≡ c ( m ), for any n ∈ N ,0 ≥ e − ( δ + m )( t − τ n ) w ( X ,x, ˆ ct , m ) = ( X ,x, ˆ cτ n ) − γ − γ c ( m ) − γ e − [ δ + m +( γ − r − c ( m ))]( t − τ n ) , if t > τ n . Observing that (3.3) implies δ + m + ( γ − r − c ( m )) = c ( m ) + mζ − γ > , (A.6)we conclude that e − ( δ + m )( t − τ n ) w ( X ,x, ˆ ct , m ) → t → ∞ . This shows that ˆ c satisfies (B.2).Thanks to (A.6), a calculation similar to (A.4) yields0 ≥ E (cid:104) e − δτ n w (cid:16) ζ n X ,x, ˆ cτ n , m (cid:17)(cid:105) = ζ (1 − γ ) n x − γ − γ c ( m ) − γ E (cid:104) e − δτ n e (1 − γ )( r − c ( m )) τ n (cid:105) = (cid:18) mζ − γ c ( m ) + mζ − γ (cid:19) n x − γ − γ c ( m ) − γ → n → ∞ , which shows that ˆ c satisfies (B.3). A.2 Aging without Healthcare ( g ≡ ) Recall the setup in Section 3.2. The Hamilton-Jacobi equation (A.1) associated with the valuefunction V ( x, m ) now becomes (cid:101) U ( w x ( x, m )) + mw ( ζx, m ) − ( δ + m ) w ( x, m ) + rxw x ( x, m ) + βmw m ( x, m ) = 0 . (A.7)21f V is of the form V ( x, m ) = x − γ − γ v ( m ), the above equation turns into v ( m ) − γ − c ( m ) v ( m ) + βmγ v (cid:48) ( m ) = 0 , where c ( m ) is given by (3.4). Setting v ( m ) = u ( m ) − γ , we obtain from the above equation u ( m ) − c ( m ) u ( m ) − βmu (cid:48) ( m ) = 0 , (A.8)which admits the general solution u ( m ) = βe − (1 − ζ − γ ) mβγ (cid:20) Cβm δ +( γ − rβγ + (cid:90) ∞ e − (1 − ζ − γ ) muβγ u − (cid:16) δ +( γ − rβγ (cid:17) du (cid:21) − , where C ∈ R is a constant to be determined. Proposition 3.2 states that taking C = 0, which turns u ( m ) into u ( m ) in (3.6), leads to our value function V ( x, m ). In the following, we separate theproof of Proposition 3.2 into two parts. Lemma A.1.
Under the assumptions of Proposition 3.2, u defined in (3.6) is a strictly increasingfunction on (0 , ∞ ) satisfying (a) and (b) in Proposition 3.2.Proof. The definition of u in (3.6) directly implies that u is strictly increasing, u (0) = δ +( γ − rγ ,and u (cid:48) (0+) = ∞ . Since u solves (A.8), for any m ∈ (0 , ∞ ), u ( m ) − c ( m ) = βmu (cid:48) ( m ) u ( m ) > , where the inequality follows from u being positive and strictly increasing. On the other hand,using y + 1 < e y for y >
0, (3.6) yields u ( m ) < β (cid:20)(cid:90) ∞ exp (cid:26) − (cid:18) (1 − ζ − γ ) mβγ + 1 + δ + ( γ − rβγ (cid:19) y (cid:27) dy (cid:21) − = β (cid:20) δ + (1 − ζ − γ ) m + ( γ − rβγ + 1 (cid:21) = c ( m ) + β. Finally, Taylor’s expansion of u ( m ) at infinity shows u ( m ) = c ( m ) + β + O (1 /m ) . This implies u ( m ) − ( c ( m ) + β ) → m → ∞ , andlim m →∞ u (cid:48) ( m ) = lim m →∞ u ( m ) m = 1 − ζ − γ γ . Proof of Proposition 3.2.
Properties (a) and (b) are established in Lemma A.1. Here, we prove therest of the claims in Proposition 3.2. Set w ( x, m ) := x − γ − γ u ( m ) − γ . By Lemma A.1, it remains toshow that V ( x, m ) = w ( x, m ) and ˆ c t := u ( me βt ), t ≥
0, is an optimal control of (3.2). First, weobserve that ˆ c is an element of C . Indeed, for any compact subset K of R + , thanks to u ≤ c + β in Lemma A.1, (cid:90) K ˆ c t dt ≤ (cid:90) K δ + (1 − ζ − γ ) me βt + ( γ − rγ + β dt < ∞ . (A.9)22ow we deal with two cases separately. Case I: condition (i) holds. By Theorem B.1, it suffices to verify (B.2) and (B.3) undercurrent context. For any ( x, m ) ∈ R , c ∈ C , and n ∈ N , by using γ ∈ (0 ,
1) and X ,x,ct ≤ X ,x,cτ n exp ( r ( t − τ n )) on the set { t ≥ τ n } ,0 ≤ E (cid:20) exp (cid:18) − (cid:90) tτ n ( δ + me βs ) ds (cid:19) w (cid:16) ζ n X ,x,ct , me βt (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) Z , ..., Z n (cid:21) ≤ e − ( δ + m )( t − τ n ) ( ζ n X ,x,cτ n ) − γ − γ e (1 − γ ) r ( t − τ n ) u ( me βt ) − γ → t → ∞ , where the convergence follows from δ + ( γ − r > u being a nondecreasing function bydefinition. This in particular implies (B.2). On the other hand,0 ≤ E (cid:104) e − δτ n w (cid:16) ζ n X ,x,cτ n , me βτ n (cid:17)(cid:105) ≤ ζ (1 − γ ) n x − γ − γ E (cid:104) e − δτ n e (1 − γ ) rτ n u ( me βτ n ) − γ (cid:105) ≤ ζ (1 − γ ) n x − γ − γ u ( m ) − γ E (cid:104) e − ( δ +( γ − r ) τ n (cid:105) ≤ ζ (1 − γ ) n x − γ − γ u ( m ) − γ → n → ∞ , where the fourth inequality follows from δ + ( γ − r > γ, ζ ∈ (0 , Case II: condition (ii) holds. By Proposition B.1, it suffices to establish (B.17)-(B.19) and showthat ˆ c t := u ( me βt ) satisfies (B.2) and (B.3). For any ε >
0, since g ≡
0, the sequence { τ εn } n ≥ constructed in Appendix B coincides with { τ n } n ≥ . The counting process N ε in (B.15) is thereforethe same as N in (2.9). Thus, (B.19) trivially holds under current context. Given ( x, m ) ∈ R , c ∈ C , and ε >
0, consider the consumption policy c ε as in (A.5), and the associated property X ,x + ε,c ε t = X ,x,ct + εe rt for all t ≥
0. We then deduce from γ > u being an nondecreasingfunction that for any n ∈ N ,0 ≥ E (cid:20) exp (cid:18) − (cid:90) tτ n ( δ + me βs ) ds (cid:19) w (cid:16) ζ n X ,x + ε,c ε t , me βt (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) Z , ..., Z n (cid:21) ≥ e − ( δ + m )( t − τ n ) ( ζ n ε ) − γ e (1 − γ ) rt − γ u ( m ) − γ → t → ∞ , which in particular implies (B.17). Since γ > δ + ( γ − r >
0, we have0 ≥ E (cid:104) e − δτ n w (cid:16) ζ n X ,x + ε,c ε τ n , me βτ n (cid:17)(cid:105) ≥ ζ (1 − γ ) n ε − γ − γ u ( m ) − γ E [ e − ( δ +( γ − r ) τ n ] ≥ ζ (1 − γ ) n ε − γ − γ u ( m ) − γ → n → ∞ , where the convergence follows from γ, ζ >
1. This verifies (B.18). Now, for any n ∈ N , applyingˆ c t := u ( me βt ), t ≥
0, yields0 ≥ E (cid:20) exp (cid:18) − (cid:90) tτ n ( δ + me βs ) ds (cid:19) w (cid:16) X ,x, ˆ ct , me βt (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) Z , ..., Z n (cid:21) ≥ e − [ δ +( γ − r ]( t − τ n ) exp (cid:18) − (cid:90) tτ n (cid:104) me βs − ( γ − u ( me βs ) (cid:105) ds (cid:19) ( X ,x, ˆ cτ n ) − γ − γ u ( m ) − γ (A.10)23n the set { t ≥ τ n } . By direct calculation, u ( m ) defined in (3.6) can be expressed as u ( m ) = β e − m (1 − ζ − γ ) βγ (cid:16) m (1 − ζ − γ ) βγ (cid:17) − δ +( γ − rβγ Γ (cid:16) − δ +( γ − rβγ , m (1 − ζ − γ ) βγ (cid:17) , where Γ( s, z ) := (cid:82) ∞ z t s − e − t dt is the upper incomplete gamma function. Recalling the property Γ( s,z ) e − z z s − → z → ∞ , it follows thatlim m →∞ mγu ( m ) = 11 − ζ − γ > , (A.11)whence me βs > γu ( me βs ) for s large enough. This, together with u being a positive nonincreasingfunction, shows that (cid:82) tτ n (cid:2) me βs − ( γ − u ( me βs ) (cid:3) ds → ∞ a.s. as t → ∞ . We then conclude from(A.10) that ˆ c satisfies (B.2). It remains to show that ˆ c satisfies (B.3). Observe that0 ≥ E (cid:104) e − δτ n w (cid:16) ζ n X ,x, ˆ cτ n , me βτ n (cid:17)(cid:105) ≥ ζ (1 − γ ) n x − γ − γ u ( m ) − γ E (cid:20) e − ( δ +( γ − r ) τ n exp (cid:18)(cid:90) τ n ( γ − u ( me βs ) ds (cid:19)(cid:21) ≥ ζ (1 − γ ) n x − γ − γ u ( m ) − γ (cid:90) ∞ P ( τ n ∈ dt ) exp (cid:18)(cid:90) t ( γ − u ( me βs ) ds (cid:19) dt, (A.12)where the third line above follows from e − ( δ +( γ − r ) τ n ≤ γ >
1. Note that for each n ∈ N , (cid:80) ni =1 Z i has a gamma distribution with the law P ( (cid:80) ni =1 Z i ≤ z ) = n ) Γ( n, z ), where Γ( s ) := (cid:82) ∞ t s − e − t dt is the gamma function and Γ( s, z ) := (cid:82) z t s − e − t dt is the lower incomplete gammafunction. We then observe from (3.5) that P ( τ n ≤ t ) = P (cid:32) mβ ( e βt − ≥ n (cid:88) i =1 Z i (cid:33) = 1Γ( n ) Γ (cid:18) n, mβ ( e βt − (cid:19) ∀ t ≥ . It follows that P ( τ n ∈ dt ) = ddt P ( τ n ≤ t ) = 1Γ( n ) (cid:18) mβ (cid:19) n ( e βt − n − e − mβ ( e βt − βe βt ∀ t ≥ . (A.13)Also, for any α ∈ (cid:0) (1 − /γ )(1 − ζ − γ ) , (cid:1) , (A.14)(A.11) yields mγu ( m ) > α − ζ − γ for m large enough. Thus, there exists t ∗ > − ζ − γ αγ me βt > u ( me βt ) , for t ≥ t ∗ . (A.15)24y setting C ( t ∗ ) := exp (cid:0) (cid:82) t ∗ ( γ − u ( me βs ) ds (cid:1) , we obtain from (A.12) that0 ≥ E (cid:104) e − δτ n w (cid:16) ζ n X ,x, ˆ cτ n , me βτ n (cid:17)(cid:105) ≥ ζ (1 − γ ) n x − γ − γ u ( m ) − γ Γ( n ) (cid:18) mβ (cid:19) n (cid:20) C ( t ∗ ) (cid:90) t ∗ ( e βt − n − e − mβ ( e βt − βe βt dt + (cid:90) ∞ t ∗ ( e βt − n − e − mβ ( e βt − βe βt e (1 − γ )(1 − ζ − γ ) mαβ ( e βt − dt (cid:21) ≥ ζ (1 − γ ) n x − γ − γ u ( m ) − γ Γ( n ) (cid:18) mβ (cid:19) n (cid:20) C ( t ∗ ) (cid:90) ∞ y n − e − mβ y dy + (cid:90) ∞ y n − e − (cid:104) − α (1 − γ )(1 − ζ − γ ) (cid:105) mβ y dy (cid:21) = ζ (1 − γ ) n x − γ − γ u ( m ) − γ (cid:34) C ( t ∗ ) + (cid:18) − α (cid:18) − γ (cid:19) (cid:0) − ζ − γ (cid:1)(cid:19) − n (cid:35) , (A.16)where the second line follows form (A.13) and (A.15), the fourth line is due to the change of variable y = e βt −
1, and the last equality holds when 1 − α (cid:0) − γ (cid:1)(cid:0) − ζ − γ (cid:1) >
0, which is true under(A.14). Noting that (A.14) implies 1 − α (cid:0) − γ (cid:1)(cid:0) − ζ − γ (cid:1) > − (1 − ζ − γ ) = ζ − γ , we concludefrom (A.16) that E (cid:2) e − δτ n w (cid:0) ζ n X ,x, ˆ cτ n , me βτ n (cid:1)(cid:3) → n → ∞ , i.e. ˆ c satisfies (B.3). A.3 Aging with Healthcare
Recall the setup in Section 3.3 where mortality increases naturally according to Gompertz’ law( β > g : R + → R + is not constantly 0) toslow down the mortality growth. For the rest of this section, let Assumption 3.1 hold, and denoteby I : R + → R + the inverse function of g (cid:48) , and note that I is strictly decreasing.If the value function (2.13) is of the form V ( x, m ) = x − γ − γ v ( m ), then (A.1) yields v ( m ) − γ − c ( m ) v ( m ) + βmγ v (cid:48) ( m ) + 1 − γγ sup h ≥ (cid:26) − v (cid:48) ( m )1 − γ (cid:18) mg ( h ) + h (1 − γ ) v ( m ) v (cid:48) ( m ) (cid:19)(cid:27) = 0 , where c ( m ) is given by (3.4). By setting v ( m ) = u ( m ) − γ and assuming that u (cid:48) ( m ) ≥
0, the aboveequation becomes u ( m ) − c ( m ) u ( m ) − βmu (cid:48) ( m ) + mu (cid:48) ( m ) sup h ≥ (cid:110) g ( h ) − − γγ u ( m ) mu (cid:48) ( m ) h (cid:111) = 0 , if 0 < γ < ,mu (cid:48) ( m ) inf h ≥ (cid:110) g ( h ) − − γγ u ( m ) mu (cid:48) ( m ) h (cid:111) = 0 , if γ > . (A.17)Since g is a nondecreasing function with g (0) = 0, the infimum above equals 0. That is, when γ > < γ < L u ( m ) = 0 as in (3.10). In the following, we will employ Perron’s method to construct solutions to(3.10), under the assumption that ¯ c := δγ + (cid:18) − γ (cid:19) r > . (A.18) Definition A.1.
Let Π be the collection of ( p, q ) , where p, q : R + → R are continuous and satisfy(i) c ≤ p ≤ q ≤ c + β on (0 , ∞ ) . ii) p and q are strictly increasing and concave.(iii) p (resp. q ) is a viscosity subsolution (resp. supersolution) to (3.10) on (0 , ∞ ) .For any ( p, q ) ∈ Π , let S ( p, q ) denote the collection of continuous f : R + → R such that1. p ≤ f ≤ q on (0 , ∞ ) .2. f is strictly increasing and concave.3. f is a viscosity supersolution to (3.10) on (0 , ∞ ) . Remark A.1.
Under (A.18) and (3.9) , Π (cid:54) = ∅ . Indeed, c + β is a supersolution to (A.8) , andthus a supersolution to (3.10) . Specifically, for any m > , L ( c + β )( m ) ≥ ( c ( m ) + β ) β − βm (cid:18) − ζ − γ γ (cid:19) = β ¯ c + β > . (A.19) On the other hand, c is a subsolution to (3.10) : for any m > L c ( m ) = am (cid:32) sup h ≥ (cid:26) g ( h ) − − γγ ¯ cam h (cid:27) − β (cid:33) = am (cid:20) g (cid:18) I (cid:18) − γγ (cid:104) cam (cid:105)(cid:19)(cid:19) − − γγ (cid:104) cam (cid:105) I (cid:18) − γγ (cid:104) cam (cid:105)(cid:19) − β (cid:21) < , (A.20) where a := − ζ − γ γ and the inequality follows from (3.9) . Thus, Π contains at least ( c , c + β ) .Also note that for each ( p, q ) ∈ Π , S ( p, q ) (cid:54) = ∅ because by construction q ∈ S ( p, q ) . We first present a basic result for strictly increasing, concave functions f which are bounded by c and c + β . Note that the concavity of f implies f (cid:48) ( ∞ ) := lim m →∞ f (cid:48) ( m − ) is well-defined. Lemma A.2.
Assume < γ < and (A.18) . For any nonnegative, strictly increasing, and concave f : R + → R , we have f ( m ) mf (cid:48) ( m − ) ≥ for all m ∈ (0 , ∞ ) . If f additionally satisfies c ≤ f ≤ c + β on (0 , ∞ ) , then f (cid:48) ( ∞ ) = − ζ − γ γ and f ( m ) mf (cid:48) ( m − ) → as m → ∞ .Proof. Since f is strictly increasing with f (0) ≥
0, the concavity of f implies f ( m ) m ≥ f (cid:48) ( m − ) > f ( m ) mf (cid:48) ( m − ) ≥ m ∈ (0 , ∞ ). Suppose f is additional bounded by c and c + β .Since c is a linear function with slope − ζ − γ γ , if f (cid:48) ( ∞ ) (cid:54) = − ζ − γ γ , then f ( m ) / ∈ [ c ( m ) , c ( m ) + β ]for m large enough, a contradiction. Moreover, we deduce from c ( m ) ≤ f ( m ) ≤ c ( m ) + β and f (cid:48) ( ∞ ) = − ζ − γ γ > δ + ( γ − r (1 − ζ − γ ) m ≤ f ( m ) mf (cid:48) ( ∞ ) ≤ δ + ( γ − r + βγ (1 − ζ − γ ) m , m > . This implies f ( m ) mf (cid:48) ( ∞ ) → m → ∞ .The next result shows that c + α is a supersolution to (3.10) on (0 , ∞ ) for α large enough. Lemma A.3.
Assume < γ < , (A.18) , and (3.9) . For any α ∈ [0 , β ] , c + α is a supersolutionto (3.10) on (0 , ∞ ) if and only if α ∈ [ β g , β ] , where β g is defined in (3.12) . Specifically, α ∈ [ β g , β ] = ⇒ L ( c + α )( m ) > for all m > α ∈ [0 , β g ) = ⇒ L ( c + α )( m ) → −∞ as m → ∞ . roof. For any a, b >
0, consider the function θ ( m ) := L ( am + b ) = ( am + b ) (cid:20)(cid:18) a − − ζ − γ γ (cid:19) m + ( b − ¯ c ) (cid:21) + am ( (cid:96) ( m ) − β ) , (A.21)where (cid:96) ( m ) := sup h ≥ (cid:26) g ( h ) − − γγ (cid:20) bam (cid:21) h (cid:27) = g (cid:18) I (cid:18) − γγ (cid:20) bam (cid:21)(cid:19)(cid:19) − − γγ (cid:20) bam (cid:21) I (cid:18) − γγ (cid:20) bam (cid:21)(cid:19) . (A.22)By direct calculation, (cid:96) (cid:48) ( m ) = 1 − γγ bam I (cid:18) − γγ (cid:20) bam (cid:21)(cid:19) , (A.23) θ (cid:48) ( m ) = 2 a (cid:18) a − − ζ − γ γ (cid:19) m + a ( b − ¯ c ) + b (cid:18) a − − ζ − γ γ (cid:19) + a (cid:18) (cid:96) ( m ) − β + 1 − γγ bam I (cid:18) − γγ (cid:20) bam (cid:21)(cid:19)(cid:19) , (A.24) θ (cid:48)(cid:48) ( m ) = 2 a (cid:18) a − − ζ − γ γ (cid:19) − (cid:18) − γγ (cid:19) b am I (cid:48) (cid:18) − γγ (cid:20) bam (cid:21)(cid:19) > , (A.25)where the positivity follows from I (cid:48) ( y ) = ddy ( g (cid:48) ) − ( y ) = g (cid:48)(cid:48) (( g (cid:48) ) − ( y )) = g (cid:48)(cid:48) ( I ( y )) and g (cid:48)(cid:48) < α ∈ [0 , β ], c ( m ) + α = am + b with a = − ζ − γ γ and b = ¯ c + α . Then (A.21) reduces to θ ( m ) = am [ α − ( β − (cid:96) ( m ))] + αb. (A.26)Observe from (A.22) that β − (cid:96) ( m ) → β g as m → ∞ . It follows thatlim m →∞ L ( c + α )( m ) = lim m →∞ θ ( m ) = (cid:40) −∞ , if α ∈ [0 , β g );+ ∞ , if α ∈ ( β g , β ] . (A.27)This already shows that if α ∈ [0 , β g ), c + α cannot be a supersolution to (3.10) on (0 , ∞ ).It remains to show that if α ∈ [ β g , β ], c + α satisfies L ( c + α )( m ) > m >
0. This istrue for α = β , as explained in Remark A.1. For any α ∈ ( β g , β ), using a = − ζ − γ γ and b = ¯ c + α under current setting and (A.22), (A.24) becomes θ (cid:48) ( m ) = a (cid:20) α − β + g (cid:18) I (cid:18) − γγ (cid:20) bam (cid:21)(cid:19)(cid:19) − − γγ I (cid:18) − γγ (cid:20) bam (cid:21)(cid:19)(cid:21) , (A.28)which implies θ (cid:48) ( m ) → α − β < m ↓
0. This, together with lim m →∞ θ ( m ) = ∞ in (A.27)and θ (cid:48)(cid:48) ( · ) > θ must attain a global minimum at some m ∗ ∈ (0 , ∞ ). Using θ (cid:48) ( m ∗ ) = 0, we obtain from (A.28) that α − − γγ I (cid:18) − γγ (cid:20) bam ∗ (cid:21)(cid:19) = β − g (cid:18) I (cid:18) − γγ (cid:20) bam ∗ (cid:21)(cid:19)(cid:19) . (A.29)27he global minimum can then be computed as θ ( m ∗ ) = am ∗ [ α − β + (cid:96) ( m )] + αb = am ∗ (cid:20) α − β + g (cid:18) I (cid:18) − γγ (cid:20) bam ∗ (cid:21)(cid:19)(cid:19) − − γγ (cid:20) bam ∗ (cid:21) I (cid:18) − γγ (cid:20) bam ∗ (cid:21)(cid:19) (cid:21) + αb = b (cid:20) α − − γγ I (cid:18) − γγ (cid:20) bam ∗ (cid:21)(cid:19)(cid:21) = b (cid:20) β − g (cid:18) I (cid:18) − γγ (cid:20) bam ∗ (cid:21)(cid:19)(cid:19)(cid:21) > , where the second equality comes from (A.22), the third and fourth equalities follow from (A.29),and the final inequality is due to (3.9). We thus conclude that for any α ∈ ( β g , β ), L ( c + α )( m ) = θ ( m ) ≥ θ ( m ∗ ) > m ∈ (0 , ∞ ). Finally, for α = β g , since c + β g is the pointwise infimum ofthe supersolutions c + α , α ∈ ( β g , β ], it must also be a supersolution, thanks to (Crandall et al.,1992, Lemma 4.2). Observe from (A.28) that lim m ↑∞ θ (cid:48) ( m ) = a [ β g − β g ] = 0. This, together with θ (cid:48)(cid:48) > c + β g being a supersolution, shows that θ ( m ) must be strictly decreasing on(0 , ∞ ) with lim m ↑∞ θ ( m ) ≥
0. This already implies L ( c + β g )( m ) = θ ( m ) > m > Lemma A.4.
Assume < γ < , (A.18) , and (3.9) . For any ( p, q ) ∈ Π , p < c + β g on R + , with β g defined in (3.12) .Proof. By contradiction, suppose “ p < c + β g on R + ” does not hold. If there exists m > p ( m ) = c ( m ) + β g for all m ≥ m , then the subsolution property of p is violated, thanksto Lemma A.3. Thus, it remains to deal with the second case: there exists m > p ( m ) > c ( m ) + β g for m > m .Since p (cid:48) ( ∞ ) = − ζ − γ γ (by Lemma A.2 (i)), there must exist α ∈ ( β g , β ] such that p ≤ c + α and( c ( m ) + α ) − p ( m ) ↓ m → ∞ . Consider the collection of functions { c + α : α ∈ ( α + β g , α ) } .For each α ∈ ( α + β g , α ), we let θ α ( m ) := L ( c + α )( m ), and recall the formula of θ α in (A.26). Itshows that θ α ( m ) m = a [ α − ( β − (cid:96) ( m ))] + α (¯ c + α ) m → a [ α − β g ] as m → ∞ , where a := − ζ − γ γ . Moreover, in view of (A.22) with b replaced by ¯ c + α , the above convergenceis uniform in α ∈ ( α + β g , α ). That is, for any δ >
0, there exists M ( δ ) > m ≥ M ( ε ) and α ∈ ( α + β g , α ), | θ α ( m ) m − a [ α − β g ] | < δ . Taking δ := a ( α − β g )4 , we get for any m > M ( δ ) and α ∈ ( α + β g , α ), θ α ( m ) m > a [ α − β g ] − δ > a (cid:20) α + β g − β g (cid:21) − δ = a (cid:20) α − β g (cid:21) . (A.30)Fix ε ∈ (0 , a ( α − β g )4 β ). We can take α ∈ ( α + β g , α ) large enough such that c + α intersects p at28 ∗ > M ( δ ) and a < p (cid:48) ( m ∗ ) < a + ε . It follows that L ( p )( m ∗ ) = p ( m ∗ )( p ( m ∗ ) − c ( m ∗ )) + m ∗ p (cid:48) ( m ∗ ) (cid:32) sup h ≥ (cid:26) g ( h ) − − γγ p ( m ∗ ) m ∗ p (cid:48) ( m ∗ ) h (cid:27) − β (cid:33) = α ( c ( m ∗ ) + α ) + sup h ≥ (cid:26) g ( h ) m ∗ p (cid:48) ( m ∗ ) − − γγ p ( m ∗ ) h (cid:27) − βm ∗ p (cid:48) ( m ∗ ) ≥ α ( c ( m ∗ ) + α ) + sup h ≥ (cid:26) g ( h ) m ∗ a − − γγ ( c ( m ∗ ) + α ) h (cid:27) − βm ∗ ( a + ε )= θ α ( m ∗ ) − βm ∗ ε = m ∗ (cid:18) θ α ( m ∗ ) m ∗ − βε (cid:19) > , where the third equality follows from θ α ( m ∗ ) = L ( c + α )( m ∗ ), and the last inequality is due to(A.30) and the choice of ε . This implies that p cannot be a subsolution to (3.10) on (0 , ∞ ), acontradiction.Following Perron’s method, we introduce, for each ( p, q ) ∈ Π, the function u ∗ p,q ( m ) := inf f ∈S ( p,q ) f ( m ) , m ≥ . (A.31) Proposition A.1 (Supersolution Property) . Assume < γ < and (A.18) . For any ( p, q ) ∈ Π , u ∗ p,q ∈ S ( p, q ) .Proof. As a pointwise infimum of concave nondecreasing functions bounded by c and c + β , u ∗ p,q is by definition concave, nondecreasing, and bounded by c and c + β . The concavity of u ∗ p,q yieldsthe desired continuity. Then, by (Crandall et al., 1992, Lemma 4.2), u ∗ p,q , being continuous and apointwise infimum of viscosity supersolution, is again a viscosity supersolution. It remains to showthat u ∗ p,q is strictly increasing. Suppose to the contrary that u ∗ p,q ≡ κ > m ∗ ∈ (0 , ∞ ). The concavity of u ∗ p,q then implies that u ∗ p,q ≡ κ on [ m ∗ , ∞ ). It follows that L u ∗ p,q ( m ) = κ ( κ − c ( m )) < m large enough. This contradicts the supersolution property of u ∗ p,q . Proposition A.2 (Subsolution Property) . Assume < γ < and (A.18) . Fix ( p, q ) ∈ Π . Suppose u ∗ p,q is strictly concave at m ∈ (0 , ∞ ) in the following sense:for any m , m ∈ (0 , ∞ ) and λ ∈ (0 , such that m = λm + (1 − λ ) m ,u ∗ p,q ( m ) > λu ∗ p,q ( m ) + (1 − λ ) u ∗ p,q ( m ) . (A.32) Then, u ∗ p,q is a viscosity subsolution to (3.10) at m .Proof. If u ∗ p,q is strictly concave at m ∈ (0 , ∞ ) as defined above, there are three possibilities:(i) ( u ∗ p,q ) (cid:48) ( m − ) (cid:54) = ( u ∗ p,q ) (cid:48) ( m +); (ii) ( u ∗ p,q ) (cid:48) ( m − ) = ( u ∗ p,q ) (cid:48) ( m +), and u ∗ is strictly concave onthe interval [ m − κ, m + κ ] for some κ >
0; (iii) ( u ∗ p,q ) (cid:48) ( m − ) = ( u ∗ p,q ) (cid:48) ( m +), and there exists κ , κ > u ∗ p,q is linear on [ m − κ , m ] and strictly concave on [ m , m + κ ], or strictlyconcave on [ m − κ , m ] and linear on [ m , m + κ ].We assume, by contradiction, that there exists a test function ψ ∈ C ((0 , ∞ )) such that 0 =( u ∗ p,q − ψ )( m ) > ( u ∗ p,q − ψ )( m ) for all m ∈ (0 , ∞ ) \ { m } and L ψ ( m ) >
0. For the cases (i) and(ii), we can assume without loss of generality that ψ is strictly increasing and concave on (0 , ∞ ).Take δ > L ψ ( m ) > m ∈ ( m − δ, m + δ ). Then, for small29nough ε >
0, one can take 0 < δ ≤ δ such that for each 0 < η ≤ ε , L ( ψ − η )( m ) > m ∈ ( m − δ , m + δ ). Consider the function u η ( m ) := (cid:40) min { u ∗ p,q ( m ) , ψ ( m ) − η } , for m ∈ [ m − δ , m + δ ] ,u ∗ p,q ( m ) , for m / ∈ [ m − δ , m + δ ] . (A.33)When η is small enough, u η by construction is a concave, strictly increasing viscosity supersolutionto (3.10) on (0 , ∞ ), and u ∗ p,q − η ≤ u η ≤ u ∗ p,q . That is, u η ∈ S ( p, q ) as η is small enough. However,by definition u η < u ∗ p,q in some small neighborhood of m , which contradicts the definition of u ∗ p,q .Now we deal with the case (iii). Set a := ( u ∗ p,q ) (cid:48) ( m − ) = ( u ∗ p,q ) (cid:48) ( m +). In view of (3.10), to getthe desired subsolution property, it suffices to prove( u ∗ p,q ) ( m ) − c ( m ) u ∗ p,q ( m ) + am (cid:32) sup h ≥ (cid:26) g ( h ) − − γγ u ∗ p,q ( m ) am h (cid:27) − β (cid:33) ≤ . (A.34)We assume, without loss of generality, that u ∗ p,q is linear on [ m − κ , m ] and strictly concave on[ m , m + κ ]. Take { (cid:96) n } n ∈ N in ( m , m + κ ] such that (cid:96) n ↓ m and u ∗ p,q is differentiable at (cid:96) n .Then, the subsolution property we established above under case (ii) implies that L u ∗ p,q ( (cid:96) n ) ≤ n ∈ N . Observe that the map m (cid:55)→ sup h ≥ (cid:26) g ( h ) − − γγ u ∗ p,q ( m ) m ( u ∗ p,q ) (cid:48) ( m ) h (cid:27) is continuous around m , (A.35)thanks to g being strictly concave and nondecreasing with g (cid:48) ( ∞ ) = 0. As n → ∞ , L u ∗ p,q ( (cid:96) n ) ≤ u ∗ p,q and (A.35).We next establish the strict concavity of u ∗ p,q . Recall that I denotes the inverse function of g (cid:48) . Proposition A.3 (Strict Concavity) . Assume < γ < , (A.18) , and (3.9) . For any ( p, q ) ∈ Π , u ∗ p,q is strictly concave on (0 , ∞ ) .Proof. Assume, by contradiction, that u ∗ p,q is linear, i.e. u ∗ p,q ( m ) = am + b , on some interval of R + . Since u ∗ p,q ∈ S ( p, q ), we deduce from Lemma A.2 that a ≥ − ζ − γ γ and b ∈ [¯ c, ¯ c + β ]. Recall θ ( m ) := L ( am + b ) in (A.21). • Case I: a = − ζ − γ γ and b ∈ [¯ c, ¯ c + β g ). Then u ∗ p,q ( m ) = am + b = c ( m ) + α for m large enough,where α := b − ¯ c ∈ [0 , β g ). By Lemma A.3, lim m ↑∞ L ( u ∗ p,q )( m ) = lim m ↑∞ L ( c + α )( m ) = −∞ .This contradicts the supermartingale property of u ∗ p,q . • Case II: a = − ζ − γ γ and b ∈ [¯ c + β g , ¯ c + β ]. – Case II-1: u ∗ p,q ( m ) = am + b for all m ≥
0, with b ∈ (¯ c + β g , ¯ c + β ].Let us write u ∗ p,q ( m ) = c ( m ) + α , with α := b − ¯ c ∈ ( β g , β ]. For any ¯ α ∈ ( β g , α ), Lemmas A.3and A.4 imply that c + ¯ α belongs to S ( p, q ) and is strictly less than u ∗ p,q , which contradictsthe definition of u ∗ p,q . – Case II-2: u ∗ p,q ( m ) = am + (¯ c + β g ) for all m ≥ m ↓ θ ( m ) = b ( b − ¯ c ) > m ↓ θ (cid:48) ( m ) = b − ¯ c − β <
0. Thus, we can take m ∗ > θ ( m ∗ ) > θ (cid:48) ( m ∗ ) < θ ( m ∗ ) and θ (cid:48) ( m ∗ ) on a, b in (A.21) and (A.24), thereexists δ > a, b are replaced by ¯ a ∈ ( a, a + δ ) and ¯ b ∈ ( b − δ, b ),30 ( m ∗ ) > θ (cid:48) ( m ∗ ) < a ∈ ( a, a + δ ) and ¯ b ∈ ( b − δ, b ) such that¯ am ∗ + ¯ b = u ∗ p,q ( m ∗ ) and ¯ am + ¯ b > p ( m ) for m ∈ (0 , m ∗ ] (this is doable thanks to Lemma A.4).For clarity, let ¯ θ and ¯ θ (cid:48) denote θ and θ (cid:48) with a, b replaced by ¯ a, ¯ b . Now, we deduce fromlim m ↓ ¯ θ ( m ) = ¯ b (¯ b − ¯ c ) > θ ( m ∗ ) >
0, ¯ θ (cid:48) ( m ∗ ) <
0, and¯ θ (cid:48)(cid:48) ( m ) > m > θ ( m ) > m ∈ (0 , m ∗ ). Consider the function ψ ( m ) := ¯ am + ¯ b . By definition L ψ ( m ) = ¯ θ ( m ) > m ∈ (0 , m ∗ ). Thus, ψ ∧ u ∗ p,q belongs to S ( p, q ) and is strictly less than u ∗ p,q for m ∈ (0 , m ∗ ). This contradicts the definition of u ∗ p,q . – Case II-3: There exists m > u ∗ p,q ( m ) = am + b for all m ≥ m , and u ∗ p,q isstrictly concave at m in the sense of (A.32).By Proposition A.2, u ∗ p,q is a viscosity subsolution to (3.10) at m . Take ψ ( m ) := am + b , m ∈ (0 , ∞ ), as a test function of u ∗ p,q at m . The subsolution property of u ∗ p,q yields L ψ ( m ) ≤ ψ ( m ) = c ( m ) + α with α := b − ¯ c ∈ [ β g , β ]. Thus, by Lemma A.3, L ψ ( m ) > m >
0, a contradiction. • Case III: a > − ζ − γ γ and b = ¯ c . Then there exists m > u ∗ p,q ( m ) = am + b for m ∈ [0 , m ] and u ∗ p,q is strictly concave at m in the sense of (A.32). By Proposition A.2, u ∗ p,q isa viscosity subsolution to (3.10) at m . For all m ∈ (0 , ∞ ), define η ( m ) := (cid:18) a + bm (cid:19) (cid:20)(cid:18) a − − ζ − γ γ (cid:19) m + ( b − ¯ c ) (cid:21) + a ( (cid:96) ( m ) − β ) , with (cid:96) as in (A.22). Note that θ ( m ) = mη ( m ). By direct calculation and (A.23), η (cid:48) ( m ) = a (cid:18) a − − ζ − γ γ (cid:19) − bm (cid:20) ( b − ¯ c ) − − γγ I (cid:18) − γγ (cid:20) bam (cid:21)(cid:19)(cid:21) . (A.36)Since we currently have b = ¯ c , η (cid:48) ( m ) > m ∈ (0 , ∞ ). Now, take ψ ( m ) := am + b , m ∈ (0 , ∞ ), as a test function of u ∗ p,q at m . Then the subsolution property of u ∗ p,q implies0 ≥ L ψ ( m ) = θ ( m ) = m η ( m ). We therefore have η ( m ) < m ∈ (0 , m ). Thesupersolution property of u ∗ p,q , however, entails 0 ≤ L u ∗ p,q ( m ) = θ ( m ) = mη ( m ) for all m ∈ (0 , m ), a contradiction. • Case IV: a > − ζ − γ γ and b ∈ (¯ c, ¯ c + β ). – Case IV-1: There exists m > u ∗ p,q ( m ) = am + b for m ∈ [0 , m ], and u ∗ p,q isstrictly concave at m in the sense of (A.32).We first show that p (0) has to be strictly less than u ∗ p,q (0). If p (0) = u ∗ p,q (0), then lim m ↓ p (cid:48) ( m ) ≤ a ; otherwise, p ( m ) > u ∗ p,q ( m ) for m > u ∗ p,q ∈ S ( p, q ). Bythe concavity of p , we can take a real sequence { (cid:96) n } such that (cid:96) n ↓ p is differentiableat (cid:96) n . The subsolution property of p then implies L p ( (cid:96) n ) ≤ n ∈ N . As n → ∞ , weget p (0)( p (0) − ¯ c ) ≤
0, thanks to the finiteness of lim m ↓ p (cid:48) ( m ). This shows that p (0) < ¯ c , acontradiction to p ≥ c .By Proposition A.2, u ∗ p,q is a viscosity subsolution to (3.10) at m . Take ψ ( m ) := am + b , m ∈ (0 , ∞ ), as a test function of u ∗ p,q at m . Then the subsolution property of u ∗ p,q implies 0 ≥L ψ ( m ) = θ ( m ). Observe from (A.21) that lim m ↓ θ ( m ) = b ( b − ¯ c ) >
0. If lim m ↓ θ (cid:48) ( m ) ≥ θ (cid:48)(cid:48) > , ∞ ) (by (A.25)) implies that θ ( m ) > θ (0) > m >
0, whichcontradicts θ ( m ) ≤
0. If lim m ↓ θ (cid:48) ( m ) <
0, then we can follow the argument in Case II-2.Take 0 < m ∗ < m small enough such that θ ( m ∗ ) > θ (cid:48) ( m ∗ ) <
0. By the continuousdependence of θ ( m ∗ ) and θ (cid:48) ( m ∗ ) on a, b , there exists δ > a, b are replaced31y ¯ a ∈ ( a, a + δ ) and ¯ b ∈ ( b − δ, b ), θ ( m ∗ ) > θ (cid:48) ( m ∗ ) < a ∈ ( a, a + δ ) and ¯ b ∈ ( b − δ, b ) such that ¯ am ∗ +¯ b = u ∗ p,q ( m ∗ ) and ¯ am +¯ b > p ( m ) for m ∈ (0 , m ∗ ](this is doable thanks to p (0) < u ∗ p,q (0)). For clarity, let ¯ θ and ¯ θ (cid:48) denote θ and θ (cid:48) with a, b replaced by ¯ a, ¯ b . Now, we deduce from lim m ↓ ¯ θ ( m ) >
0, ¯ θ ( m ∗ ) >
0, ¯ θ (cid:48) ( m ∗ ) <
0, and ¯ θ (cid:48)(cid:48) > , ∞ ) that ¯ θ ( m ) > m ∈ (0 , m ∗ ). Consider the function φ ( m ) := ¯ am + ¯ b . By definition L φ ( m ) = ¯ θ ( m ) > m ∈ (0 , m ∗ ). Thus, φ ∧ u ∗ p,q belongs to S ( p, q ) and is strictly less than u ∗ p,q for m ∈ (0 , m ∗ ). This contradicts the definition of u ∗ p,q . – Case IV-2: There exist m , m ∈ (0 , ∞ ) such that u ∗ p,q ( m ) = am + b for m ∈ [ m , m ], and u ∗ p,q is strictly concave at m and m in the sense of (A.32).By Proposition A.2, u ∗ p,q is a viscosity subsolution to (3.10) at both m and m . Now, take ψ ( m ) := am + b , m ∈ (0 , ∞ ), as a test function of u ∗ p,q at m and m . Then the subsolutionproperty of u ∗ p,q implies 0 ≥ L ψ ( m ) = θ ( m ) and 0 ≥ L ψ ( m ) = θ ( m ). Since θ (cid:48)(cid:48) > , ∞ ) (by (A.25)), we must have θ ( m ) < m ∈ ( m , m ). The supersolutionproperty of u ∗ p,q , however, entails 0 ≤ L ψ ( m ) = θ ( m ) for all m ∈ ( m , m ), a contradiction. Proposition A.4 (Regularity) . Assume < γ < , (A.18) , and (3.9) . For any ( p, q ) ∈ Π , u ∗ p,q isa strictly concave classical solution to (3.10) on (0 , ∞ ) .Proof. For any ( p, q ) ∈ Π, Propositions A.1, A.2, and A.3 immediately imply that u ∗ p,q is astrictly concave viscosity solution to (3.10) on (0 , ∞ ). It remains to show that u ∗ p,q is differen-tiable everywhere on (0 , ∞ ). Assume, by contradiction, that there exists m ∈ (0 , ∞ ) such that a := ( u ∗ ) (cid:48) ( m +) < ( u ∗ ) (cid:48) ( m − ) =: b . Take { k n } n ∈ N and { (cid:96) n } n ∈ N in (0 , ∞ ) such that k n ↑ m , (cid:96) n ↓ m , and u ∗ p,q is differentiable at k n and (cid:96) n for all n ∈ N . By the viscosity solution property of u ∗ p,q , L u ∗ ( k n ) = L u ∗ ( (cid:96) n ) = 0, for all n ∈ N . As n → ∞ , we getsup h ≥ (cid:26) ( g ( h ) − β ) a − − γγ u ∗ p,q ( m ) m h (cid:27) = sup h ≥ (cid:26) ( g ( h ) − β ) b − − γγ u ∗ p,q ( m ) m h (cid:27) , which implies that a = b , a contradiction. Proposition A.5 (Verification) . Assume < γ < , (A.18) , and (3.9) . If u : R + → R + is anonnegative, strictly increasing, and concave classical solution to (3.10) on (0 , ∞ ) , then V ( x, m ) = x − γ − γ u ( m ) − γ for all ( x, m ) ∈ R . Furthermore, (ˆ c, ˆ h ) defined by ˆ c t := u ( M t ) and ˆ h t := I (cid:18) − γγ u ( M t ) M t · ( u ) (cid:48) ( M t ) (cid:19) , for all t ≥ , is an optimal control of (2.13) .Proof. Set w ( x, m ) := x − γ − γ u ( m ) − γ . In view of Theorem B.1, it suffices to show that (B.2) and (B.3)hold, and (ˆ c, ˆ h ) belongs to A . Since u is nonnegative, strictly increasing, and concave, Lemma A.2implies that ˆ h t ≤ I ( − γγ ) for all t ≥
0. Moreover, there exist a, b > u ( m ) < am + b forall m ≥
0. It follows that for any compact subset K ⊂ R + , (cid:90) K ˆ c t dt ≤ (cid:90) K aM t + b dt ≤ (cid:90) K ame βt + b dt < ∞ . c, ˆ h ) ∈ A .Under (3.9), u being a classical solution to (3.10) implies u ( m ) − u ( m ) c ( m ) ≥
0, and thus u ( m ) ≥ c ( m ) for all m ∈ (0 , ∞ ). Now, for any ( x, m ) ∈ R , ( c, h ) ∈ A , and n ∈ N , by using0 < γ < X ,x,c,ht ≤ X ,x,c,hτ n exp ( r ( t − τ n )), we have0 ≤ E (cid:20) exp (cid:18) − (cid:90) tτ n ( δ + M ,m,hs ) ds (cid:19) w (cid:16) ζ n X ,x,c,ht , M ,m,ht (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) Z , ..., Z n (cid:21) ≤ e − δ ( t − τ n ) ( X ,x,c,hτ n ) − γ − γ e (1 − γ ) r ( t − τ n ) E [ u ( M ,m,ht ) − γ | Z , ..., Z n ] ≤ e − ( δ +( γ − r )( t − τ n ) ( X ,x,c,hτ n ) − γ − γ (¯ c ) − γ → t → ∞ , where the second line is due to M ,m,ht ≥ u beingstrictly increasing with u ( m ) > u (0) ≥ c (0) = ¯ c >
0, and the convergence is a consequence of δ + ( γ − r = γ ¯ c >
0. This in particular implies (B.2). On the other hand, for each n ∈ N ,0 ≤ E (cid:104) e − δτ n w (cid:16) ζ n X ,x,c,hτ n , M ,m,hτ n (cid:17)(cid:105) ≤ ζ (1 − γ ) n x − γ − γ E (cid:104) e − δτ n e (1 − γ ) rτ n u ( M ,m,hτ n ) − γ (cid:105) ≤ ζ (1 − γ ) n x − γ − γ (¯ c ) − γ E (cid:104) e − ( δ +( γ − r ) τ n (cid:105) ≤ ζ (1 − γ ) n x − γ − γ (¯ c ) − γ → t → ∞ , where the last inequality is due to δ + ( γ − r = γ ¯ c >
0. The shows that (B.3) is also satisfied.Proposition A.5, together with Propositions A.4 and A.1, leads to:
Corollary A.1.
Assume < γ < , (A.18) , and (3.9) . Then u ∗ p,q is independent of the choice of ( p, q ) ∈ Π , and it is the unique nonnegative, strictly increasing, and concave classical solution to (3.10) on (0 , ∞ ) . Remark A.2.
Proposition A.5 and Corollary A.1 yield Theorem 3.1.
In the following, we will simply denote by u ∗ the function u ∗ p,q for any ( p, q ) ∈ Π. Corollary A.2 (Strict Concavity of u ) . Assume < γ < and (A.18) . Then u , defined in (3.6) , is strictly concave on (0 , ∞ ) .Proof. With g ≡
0, the equation (3.10) reduces to (A.8), and we can repeat the same argumentsin this section (with much simpler proofs) to show that the strictly concave u ∗ constructed underPerron’s method coincides with u .Now, we are ready to prove Theorem 3.2 Proof of Theorem 3.2.
First, observe that β g = β − g (cid:0) I (cid:0) − γγ (cid:1)(cid:1) + − γγ I (cid:0) − γγ (cid:1) . Then (3.9) impliesthat β g >
0. Since u is a solution to (A.8) and u (cid:48) ( m ) ≥
0, it is a supersolution to (3.10). This,together with Lemma A.1, Corollary A.2, and Remark A.1, shows that ( c , u ) ∈ Π. It follows that u ∗ = u ∗ c ,u ≤ u . Similarly, ( c , c + β g ) ∈ Π by Lemma A.3, which implies u ∗ = u ∗ c ,c + β g ≤ c + β .This already yields u ∗ ≤ min { u , c + β g } . On the other hand, thanks again to Lemma A.1 andCorollary A.2 with β replaced by β g , u g is nonnegative, strictly increasing, concave, and bounded33rom below and above by c and c + β g respectively. Then, Lemma A.2 implies u g ( m ) m ( u g ) (cid:48) ( m ) ≥ m >
0. It follows that β − sup h ≥ (cid:26) g ( h ) − − γγ u g ( m ) m ( u g ) (cid:48) ( m ) h (cid:27) ≥ β g , ∀ m > . Since u g is by construction a solution to (A.8) with β replaced by β g , the above inequality gives0 = ( u g ( m )) − u g ( m ) c ( m ) − β g m ( u g ) (cid:48) ( m ) ≥ ( u g ( m )) − u g ( m ) c ( m ) + m ( u g ) (cid:48) ( m ) (cid:32) sup h ≥ (cid:26) g ( h ) − − γγ u g ( m ) m ( u g ) (cid:48) ( m ) h (cid:27) − β (cid:33) = L u g ( m ) , for all m >
0. This shows that ( u g , c + β g ) ∈ Π, and thus u ∗ = u ∗ u g ,c + β g ≥ u g . B Verification
In this section, we provide a general verification theorem for the value function V ( x, m ) in (2.13).Given ( c, h ) ∈ A , we introduce, for each n ∈ N , the truncated policies ( c ( n ) , h ( n ) ) ∈ A : c ( n ) t := (cid:32) n − (cid:88) k =0 c k ( t )1 { τ k ≤ t<τ k +1 } (cid:33) + c n ( t )1 { t ≥ τ n } , h ( n ) t := (cid:32) n − (cid:88) k =0 h k ( t )1 { τ k ≤ t<τ k +1 } (cid:33) + h n ( t )1 { t ≥ τ n } . (B.1) Theorem B.1.
Let w ∈ C , ( R + × R + ) satisfy (A.1) . Suppose for any ( x, m ) ∈ R and ( c, h ) ∈ A , lim t →∞ E (cid:20) exp (cid:18) − (cid:90) tτ n ( δ + M ,m,h ( n ) s ) ds (cid:19) · w (cid:16) ζ n X ,x,c ( n ) ,h ( n ) t , M ,m,h ( n ) t (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) Z , ..., Z n (cid:21) = 0 ∀ n ≥ , (B.2)lim n →∞ E (cid:104) e − δτ n w (cid:16) ζ n X ,x,c ( n ) ,h ( n ) τ n , M ,m,h ( n ) τ n (cid:17)(cid:105) = 0 . (B.3) (i) w ( x, m ) ≥ V ( x, m ) on R + × R + .(ii) Suppose there exist two measurable functions ¯ c , ¯ h : R → R + such that ¯ c ( x, m ) and ¯ h ( x, m ) are maximizers of sup c ≥ { U ( cx ) − cxw x ( x, m ) } and sup h ≥ {− w m ( x, m ) g ( h ) − hxw x ( x, m ) } , respectively, for all ( x, m ) ∈ R . Let ¯ X , ¯ M , ¯ N denote the solutions to dX s = X s [ r − (¯ c ( X s , M s ) + ¯ h ( X s , M s ))] ds X = x,dM s = M s (cid:2) β − g (¯ h ( ζ N s X s , M s )) (cid:3) ds M = m,N s = ∞ (cid:88) k =0 k { T k ≤ t 0. Letting t → ∞ and in view of (B.2), w ( x, m ) ≥ (cid:90) ∞ e − (cid:82) s ( δ + M ,m,h ν ) dν U (cid:16) c ( s ) X ,x,c ,h s (cid:17) ds + (cid:90) ∞ e − (cid:82) s ( δ + M ,m,h ν ) dν M ,m,h s w ( ζX ,x,c ,h s , M ,m,h s ) ds. (B.7)Thanks to Fubini’s theorem and (2.10), observe that E (cid:20) (cid:90) τ e − δt U (cid:16) c t ζ N t X ,x,c,ht (cid:17) dt (cid:21) = E (cid:20)(cid:90) ∞ { τ >t } e − δt U (cid:16) c ( t ) X ,x,c ,h t (cid:17) dt (cid:21) = (cid:90) ∞ e − (cid:82) t ( δ + M ,m,h ν ) dν U (cid:16) c ( t ) X ,x,c ,h t (cid:17) dt, (B.8) E (cid:20) e − δτ w (cid:16) ζ X ,x,c,hτ , M ,m,hτ (cid:17) (cid:21) = E (cid:104) e − δτ w (cid:16) ζX ,x,c ,h τ , M ,m ,h τ (cid:17)(cid:105) = (cid:90) ∞ e − (cid:82) t ( δ + M ,m,h ν ) dν M ,m,h t w (cid:16) ζX ,x,c ,h t , M ,m,h t (cid:17) dt, (B.9)whence (B.5) holds true for n = 1 in view of (B.7)-(B.9). Now, suppose (B.5) holds true for n = k > 1. That is, w ( x, m ) ≥ E (cid:20) (cid:90) τ k e − δt U (cid:16) c t ζ N t X ,x,c,ht (cid:17) dt (cid:21) + E (cid:104) e − δτ k w (cid:16) ζ k X ,x,c,hτ k , M ,m,hτ k (cid:17)(cid:105) . (B.10)By writing x k = ζ k X ,x,c,hτ k and m k = M ,m,hτ k , we get (B.6) with (0 , x, m, c , h ) replaced by( τ k , x k , m k , c k , h k ). This, together with (B.2), gives w (cid:16) ζ k X ,x,c,hτ k , M ,m,hτ k (cid:17) = w ( x k , m k ) ≥ E (cid:20) (cid:90) ∞ τ k exp (cid:18) − (cid:90) tτ k ( δ + M τ k ,m k ,h k ν ) dν (cid:19) U (cid:16) c k ( t ) X τ k ,x k ,c k ,h k t (cid:17) dt (cid:12)(cid:12)(cid:12)(cid:12) Z , ..., Z k (cid:21) (B.11)+ E (cid:20)(cid:90) ∞ τ k exp (cid:18) − (cid:90) tτ k ( δ + M τ k ,m k ,h k ν ) dν (cid:19) M τ k ,m k ,h k t w (cid:16) ζX τ k ,x k ,c k ,h k t , M τ k ,m k ,h k t (cid:17) dt (cid:12)(cid:12)(cid:12)(cid:12) Z , ..., Z k (cid:21) . E (cid:104) e − δτ k w (cid:16) ζ k X ,x,c,hτ k , M ,m,hτ k (cid:17)(cid:105) ≥ E (cid:20) (cid:90) τ k +1 τ k e − δt U (cid:16) c t ζ N t X ,x,c,ht (cid:17) dt (cid:21) + E (cid:104) e − δτ k +1 w (cid:16) ζ k +1 X ,x,c,hτ k +1 , M ,m,hτ k +1 (cid:17)(cid:105) . (B.12)This, together with (B.10), shows that w ( x, m ) ≥ E (cid:20) (cid:90) τ k +1 e − δt U (cid:16) c t ζ N t X ,x,c,ht (cid:17) dt (cid:21) + E (cid:20) e − δτ k +1 w (cid:18) ζ k +1 X ,x,c,hτ k +1 , M ,m,hτ k +1 (cid:19)(cid:21) . The claim (B.5) therefore holds by induction. Letting n → ∞ in (B.5), by the monotone conver-gence theorem and (B.3), w ( x, m ) ≥ E (cid:2) (cid:82) ∞ e − δt U (cid:0) c t ζ N t X ,x,c,ht (cid:1) dt (cid:3) for all ( c, h ) ∈ A . Taking thesupremum over ( c, h ) ∈ A ( m ) leads to w ( x, m ) ≥ V ( x, m ).(ii) With ( c, h ) = (ˆ c, ˆ h ), the inequality (B.6) turns into an equality, whence (B.5) holds withequality. Sending n → ∞ , the monotone convergence theorem and (B.3) imply that w ( x, m ) = E (cid:20) (cid:90) ∞ e − δt U (cid:16) ˆ c t ζ N t X ,x, ˆ c, ˆ ht (cid:17) dt (cid:21) ≤ V ( x, m ) . This, together with part (i), shows that w ( x, m ) = V ( x, m ) and (ˆ c, ˆ h ) is an optimal control.Theorem B.1 can be extended to include the risky asset S in (4.1). Recall the setup in Section 4,especially A (cid:48) in (4.4) and the value function V in (4.5). For any ( c, h, π ) ∈ A (cid:48) , we can also considerthe truncated version ( c ( n ) , h ( n ) , π ( n ) ) ∈ A (cid:48) defined as in (B.1). Theorem B.2. Let w ∈ C , ( R + × R + ) satisfy (4.7) . For any ( x, m ) ∈ R and ( c, h, π ) ∈ A (cid:48) , (B.2) and (B.3) hold, with X ,x,c ( n ) ,h ( n ) replaced by X ,x,c ( n ) ,h ( n ) ,π ( n ) and E [ · | Z , ..., Z n ] by E [ · | F τ n ] .(i) w ( x, m ) ≥ V ( x, m ) on R + × R + .(ii) Suppose there exist measurable functions ¯ c , ¯ h , ¯ π : R → R + , with ¯ c and ¯ h as described inTheorem B.1 (ii) and ¯ π ( x, m ) being the maximizer of sup π ∈ R (cid:26) πµxw x ( x, m ) + 12 σ π x w xx ( x, m ) (cid:27) , ∀ ( x, m ) ∈ R . Let ¯ X , ¯ M , ¯ N denote the solutions to dX s = X s [ r + µ ¯ π ( X s , M s ) − (¯ c ( X s , M s ) + ¯ h ( X s , M s ))] ds + σX s ¯ π ( X s , M s ) dW s , X = x ; dM s = M s (cid:2) β − g (¯ h ( ζ N s X s , M s )) (cid:3) ds, M = m ; N s = ∞ (cid:88) k =0 k { T k ≤ t 1, i.e. (B.10) is true. As w is asolution to (4.7) and in view of (B.2), by Itˆo’s formula (B.11) holds, with E [ · | Z , ..., Z n ] replacedby E [ · | F τ n ]. By Fubini’s theorem and (4.3) as above, (B.12) follows. This, together with (B.10),implies that (B.5) holds for n = k + 1. Thus, (B.5) follows by induction. Letting n → ∞ in (B.5)and recalling (B.3), the same argument as at the end of the proof of Theorem B.1 (i) yields that w ( x, m ) ≥ V ( x, m ).(ii) This follows from the same argument as in the proof of Theorem B.1 (ii).In the sequel we relax the conditions in Theorem B.1. To this end, for any ( x, m ) ∈ R and( c, h ) ∈ A , suppose that the household is given additional wealth ε > x ≥ ε > t ≥ 0, the amount it spends in consumption (resp. healthcare)is c t (resp. h t ) multiplied by the standard account balance. The rates of spending in consumptionand healthcare therefore become( c ε ) t = c t X ,x,c,ht X ,x,c,ht + εe rt , ( h ε ) t = h t X ,x,c,ht X ,x,c,ht + εe rt ∀ t ≥ . (B.13)This new process h ε of spending rate in healthcare, different from h , changes the moments of deaths.More precisely, starting from time 0, the household takes h ε ( t ) := h ( t ) X ,x,c ,h t X ,x,c ,h t + εe rt (B.14)as instantaneous spending rates in healthcare. As in (2.8), the time of the first death is defined as τ ε := inf (cid:26) t ≥ (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) t M ,m,h ε s ds ≥ Z (cid:27) ≤ τ . Starting from time τ ε , the household takes h ε ( t ) := h ( t ) X ,x,c ,h t X ,x,c ,h t + εe rt { t<τ } + h ( t ) X ,x,c (1) ,h (1) t X ,x,c (1) ,h (1) t + εe rt { t ≥ τ } as instantaneous spending rates in healthcare. Set m ε := M ,m,h ε τ ε , the time of the second death isdefined as in (2.8) by τ ε := inf (cid:40) t ≥ τ ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) tτ ε M τ ε ,m ε ,h ε s ds ≥ Z (cid:41) ≤ τ . In general, for each n ∈ N , the household, starting from time τ εn , takes h εn ( t ) := n − (cid:88) k =0 h k ( t ) X ,x,c ( k ) ,h ( k ) t X ,x,c ( k ) ,h ( k ) t + εe rt { τ k ≤ t<τ k +1 } + h n ( t ) X ,x,c ( n ) ,h ( n ) t X ,x,c ( n ) ,h ( n ) t + εe rt { t ≥ τ n } 37s instantaneous spending rates in healthcare. Set m εn := M τ εn − ,m εn − ,h εn − τ εn , the ( n + 1) th deathmoment is defined as in (2.8) by τ εn +1 := inf (cid:40) t ≥ τ εn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) tτ εn M τ εn ,m εn ,h εn s ds ≥ Z n +1 (cid:41) ≤ τ n +1 . As in (2.9), we can introduce the counting process N εt := n for t ∈ [ τ εn , τ εn +1 ) . (B.15)Similarly, define for each n ∈ N , c εn ( t ) := n − (cid:88) k =0 c k ( t ) X ,x,c ( k ) ,h ( k ) t X ,x,c ( k ) ,h ( k ) t + εe rt { τ k ≤ t<τ k +1 } + c n ( t ) X ,x,c ( n ) ,h ( n ) t X ,x,c ( n ) ,h ( n ) t + εe rt { t ≥ τ n } . Observe that { c εn } , { h εn } ∈ L , and it can be checked that( c ε )( t ) = ∞ (cid:88) k =0 c εn ( t )1 { τ εk ≤ t<τ εk +1 } , ( h ε )( t ) = ∞ (cid:88) k =0 h εn ( t )1 { τ εk ≤ t<τ εk +1 } . This in particular shows that ( c ε , h ε ) ∈ A . In view of (B.13), we have the identity X ,x + ε,c ε ,h ε t = X ,x,c,ht + εe rt . (B.16) Proposition B.1. Let w ∈ C , ( R + × R + ) satisfy (A.1) . Suppose for any ( x, m ) ∈ R , ( c, h ) ∈ A ,and ε > , lim t →∞ E (cid:34) exp (cid:32) − (cid:90) tτ εn ( δ + M ,m,h ( n ) ε s ) ds (cid:33) w (cid:18) ζ n X ,x + ε,c ( n ) ε ,h ( n ) ε t , M ,m,h ( n ) ε t (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z , ..., Z n (cid:35) = 0 ∀ n ≥ , (B.17)lim n →∞ E (cid:104) e − δτ εn w (cid:16) ζ n X ,x + ε,c ε ,h ε τ εn , M ,m,h ε τ εn (cid:17)(cid:105) = 0 , (B.18)lim ε → E (cid:20)(cid:90) ∞ e − δt U (cid:16) c t ζ N εt X ,x,c,ht (cid:17) dt (cid:21) = E (cid:20)(cid:90) ∞ e − δt U (cid:16) c t ζ N t X ,x,c,ht (cid:17) dt (cid:21) . (B.19) (i) w ( x, m ) ≥ V ( x, m ) on R + × R + .(ii) Suppose the measurable functions ¯ h and ¯ c specified in Theorem B.1 (ii) exist, so that we candefine (ˆ c, ˆ h ) as in (B.4) . If (ˆ c, ˆ h ) ∈ A and satisfies (B.2) and (B.3) , then (ˆ c, ˆ h ) is an optimalcontrol to the problem (2.13) , and w ( x, m ) = V ( x, m ) on R + × R + .Proof. We carry out the same arguments as in Theorem B.1. With the aid of (B.17), we obtain(B.5), with ( x, c, h, τ n , N t ) replaced by ( x + ε, c ε , h ε , τ εn , N εt ). Letting n → ∞ and using (B.18), w ( x + ε, m ) ≥ E (cid:20)(cid:90) ∞ e − δt U (cid:16) ( c ε ) t ζ N εt X ,x + ε,c ε ,h ε t (cid:17) dt (cid:21) = E (cid:20)(cid:90) ∞ e − δt U (cid:16) c t ζ N εt X ,x,c,ht (cid:17) dt (cid:21) , where the equality follows from (B.16) and the definition of c ε in (B.13). Sending ε → 0, we obtainfrom (B.19) that w ( x, m ) ≥ E (cid:2) (cid:82) ∞ e − δt U (cid:0) c t ζ N t X ,x,c,ht (cid:1) dt (cid:3) . Taking the supremum over ( c, h ) ∈ A leads to w ( x, m ) ≥ V ( x, m ). The proof of (ii) is the same as Theorem B.1 (ii).38heorem B.2 can also be relaxed in a similar fashion. Proposition B.2. Let w ∈ C , ( R + × R + ) satisfy (4.7) . For any ( x, m ) ∈ R , ( c, h, π ) ∈ A (cid:48) , and ε > , suppose (B.17) , (B.18) , and (B.19) hold, with X ,x,c ( n ) ε ,h ( n ) ε replaced by X ,x,c ( n ) ε ,h ( n ) ε ,π ( n ) ε and E [ · | Z , ..., Z n ] by E [ · | F τ n ] .(i) w ( x, m ) ≥ V ( x, m ) on R + × R + .(ii) Suppose the measurable functions (¯ c, ¯ h, ¯ π ) specified in Theorem B.2 (ii) exist, so that wecan define (ˆ c, ˆ h, ˆ π ) therein. If (ˆ c, ˆ h, ˆ π ) ∈ A (cid:48) and satisfies (B.2) and (B.3) as specified inTheorem B.2, then (ˆ c, ˆ h, ˆ π ) is an optimal control to the problem (4.5) , and w ( x, m ) = V ( x, m ) on R + × R + . References Bayraktar, E. and Li, J. (2017), ‘On the controller-stopper problems with controlled jumps’, toappear in Applied Mathematics and Optimization , available at https://arxiv.org/abs/1609.03954.Bayraktar, E. and Zhang, Y. (2015), ‘Minimizing the probability of lifetime ruin under ambiguityaversion’, SIAM J. Control Optim. (1), 58–90.Beeler, J. and Campbell, J. Y. (2012), ‘The long-run risks model and aggregate asset prices: Anempirical assessment’, Critical Finance Review (1), 141–182.Bommier, A. (2010), ‘Portfolio choice under uncertain lifetime’, Journal of Public Economic Theory (1), 57–73.Bommier, A. and Rochet, J.-C. (2006), ‘Risk aversion and planning horizons’, Journal of the Eu-ropean Economic Association (4), 708–734.Chetty, R., Stepner, M., Abraham, S., Lin, S., Scuderi, B., Turner, N., Bergeron, A. and Cutler,D. (2016), ‘The association between income and life expectancy in the united states, 2001-2014’, JAMA (16), 1750–1766.Cohen, S. N. and Elliott, R. J. (2015), Stochastic calculus and applications , Probability and itsApplications, second edn, Springer, Cham.Crandall, M. G., Ishii, H. and Lions, P.-L. (1992), ‘User’s guide to viscosity solutions of secondorder partial differential equations’, Bulletin of the American Mathematical Society (1), 1–67.Cutler, D., Deaton, A. and Lleras-Muney, A. (2006), ‘The determinants of mortality’, The Journalof Economic Perspectives (3), 97–120.Ehrlich, I. (2000), ‘Uncertain lifetime, life protection, and the value of life saving’, Journal of healtheconomics (3), 341–367.Ehrlich, I. and Chuma, H. (1990), ‘A model of the demand for longevity and the value of lifeextension’, Journal of Political economy (4), 761–782.Gompertz, B. (1825), ‘On the nature of the function expressive of the law of human mortality, andon a new mode of determining the value of life contingencies’, Philosophical transactions of theRoyal Society of London , 513–583. 39rossman, M. (1972), ‘On the concept of health capital and the demand for health’, Journal ofPolitical economy (2), 223–255.Hall, R. E. and Jones, C. I. (2007), ‘The value of life and the rise in health spending’, The QuarterlyJournal of Economics (1), 39–72.Harrison, G. W., Lau, M. I. and Rutstr¨om, E. E. (2007), ‘Estimating risk attitudes in Denmark:A field experiment’, The Scandinavian Journal of Economics (2), 341–368.Hartman, M., Catlin, A., Lassman, D., Cylus, J. and Heffler, S. (2008), ‘US health spending byage, selected years through 2004’, Health Affairs (1), w1–w12.Huang, H., Milevsky, M. A. and Salisbury, T. S. (2012), ‘Optimal retirement consumption with astochastic force of mortality’, Insurance: Mathematics and Economics (2), 282–291.Hugonnier, J., Pelgrin, F. and St-Amour, P. (2013), ‘Health and (other) asset holdings’, The Reviewof Economic Studies (2), 663–710.Janeˇcek, K. and Sˆırbu, M. (2012), ‘Optimal investment with high-watermark performance fee’, SIAM J. Control Optim. (2), 790–819.Richard, S. F. (1975), ‘Optimal consumption, portfolio and life insurance rules for an uncertainlived individual in a continuous time model’, Journal of Financial Economics (2), 187–203.Rosen, S. (1988), ‘The value of changes in life expectancy’, Journal of Risk and uncertainty (3), 285–304.Shepard, D. S. and Zeckhauser, R. J. (1984), ‘Survival versus consumption’, Management Science (4), 423–439.Smith, J. P. (1999), ‘Healthy bodies and thick wallets: the dual relation between health andeconomic status’, The journal of economic perspectives: a journal of the American EconomicAssociation (2), 144–166.Smith, J. P. (2007), ‘The impact of socioeconomic status on health over the life-course’, The Journalof Human Resources (4), 739–764.Yaari, M. E. (1965), ‘Uncertain lifetime, life insurance, and the theory of the consumer’, The Reviewof Economic Studies (2), 137–150.Yogo, M. (2016), ‘Portfolio choice in retirement: Health risk and the demand for annuities, housing,and risky assets’, Journal of Monetary Economics80