Discussion Paper No. 506

THE SENSITIVITY ANALYSISOF THE OPTIMAL LENGTH OF LIFE
〜 THE NUMERICAL APPROACH 〜

Ken Tabata
andYasushi Ohkusa

April 2000

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

The sensitivity analysis of the optimal length of life*)~ the numerical approach ~5thdraft 2000 /04/04

Ken Tabata
Graduate School of Economics, Osaka University
Yasushi Ohkusa
Institute of Social and Economic Research, Osaka University

Abstract
This paper solves the pure consumption Grossman model numerically following dynamic
programming method, and undertakes sensitivity analysis of the optimal length of life. This is thefirst analysis that treats the determination of optimal length of life explicitly with numerical analysis.
In numerical analysis, it is possible to treat the free terminal time problem directly, althoughprevious analytical analyses have to consider a sequence of the modified fixed terminal time
problem to treat the free terminal time problem. From the simulation results, decreases in the healthcapital depreciation rate or increases in the relative preference for health and the time discount rate
make the individual live longer. Furthermore, longer lives significantly change the optimal
allocation of consumption and the features of the optimal consumption path.

JEL classification: C61; C63; I10; I12
Keywords
Grossman Model, Optimal Endogenous Length of Life, The Sensitivity Analysis of the Optimal
Length of Life

Correspondence: Yasushi Ohkusa, Osaka University, Institute for Social and Economic
Research, 6‑1 Mihogaoka, Ibaraki, Osaka, Japan
tel: +81‑6‑6879‑8566 fax: +81‑6‑6878‑2766 e‑mail: ohkusa@iser.osaka‑u.ac.jp

ⅠIntroduction
The number of the countries, which subsides the retired people or elderly through public sector,
providing them with public pensions or public health care insurance1), has been increasing over thelast few decades. Public pensions have grown, and now amount to one tenth of the gross domestic
product of OECD countries. Moreover, in the United States, more than one third of total federal
expenditures is devoted to income support and health care assistance for the old. As a result, thesepublic assistance programs comprise a significant share of the income of retired people. The
generosity of this system and the increase in subsidy per head is partly responsible for the growth ofthe longevity of retired people in OECD countries by between one and three months per cohort in
the last decade (OECD health statistics in 1997)2). However, at the same time, the increase in thelongevity of the retired and of the subsidy per head has brought about a financial crisis in substantial
public old age assistance programmes. In order to move to a more efficient system, it is essential toanalyze the relationship between the public system and individual behavior affecting longevity,
using the model that treats the longevity of the retired endogenously3).
Recently, Ehlrich and Chuma (1990), Ried (1998, 1999) and Grossman (1998) developed theGrossman model that was proposed by Grossman (1972a) in relation to endogenous length of life.
Grossman (1972a) is the standard model that formalizes individual inter-temporal health investment
behavior as an individual dynamic optimization problem. These studies use the analytical methodand discuss the determination of the optimal length of life in a strictly delicate manner.
However, the studies suffer from certain problems or shortcomings concerning the sensitivity
analysis of optimal longevity. Ehlrich and Chuma (1990) uses the continuous time model and shows
how parameter changes affect individual optimal length of life, using the method of comparative
dynamics proposed by Oniki (1972). However, Eisenring (1999) questions the plausibility ofapplying the Oniki’s method to Ehlrich and Chuma (1990) model. Much more restrictive
assumptions must be necessary, at least for the clear results presented in Ehlrich and Chuma (1990),
although these are not explicitly mentioned in their paper.
While Ried (1998) uses the discrete time model and applies the comparative dynamic method,
he focuses only on the case where marginal parameter changes are too small to alter the individual’s
optimal longevity. He can not show how changes in exogenous parameters affect the individual’s
optimal longevity, because the application of the comparative dynamic method to the discrete timemodel is impossible in the case where a parameter change causes the Lagrange multiplier to jump.
As expounded in Ried (1999), sufficiently large non-marginal parameter changes alter individual
longevity or terminal period and so causes this jump. However, as Grossman (1998) stresses, it isvery important to consider the case where individual optimal loongevity responds elastically tochange in exogenous parameters, i.e., the case of sufficiently large non-marginal change inparameters. The analytical method appears to suffer from certain limitations in dealing with this2

range of parameter changes in the discrete model.
Thus, the objective of this paper is to extend Ried(1999)’s contribution to the case of
sufficiently large non-marginal parameter changes. To overcome the limitation of the analyticalmethod, this paper solves the pure consumption Grossman model numerically, following the methodof dynamic programming and undertakes the sensitivity analysis of optimal longevity.
The pure consumption Grossman model is a simplified version of the Grossman model. In the
Grossman model, the marginal benefit of health capital consists of both monetary and psychic
benefits. Monetary benefit are produced by the increase in working time stemming from longer
healthy time. Psychic benefit are associated with these longer times of good health itself. The pureconsumption Grossman model only focuses on psychic benefit and assumes that retired individuals
are the targets for analysis4). The pure consumption Grossman model is therefore appropriate for the
analysis of the retired individual’s health investment. Since the object of further analysis is mainly
the longevity behavior of the retired old, we use the pure consumption model.
The merit of the numerical approach is as follows. First, it can explicitly show how exogenous
parameter changes affect the optimal longevity. Second, although it is heavily related to the former,
the algorithm for searching optimal longevity is different from the standard analytical approach.
It is well known that there are several methods for searching the optimal solution. Roughly
there are two types of strategy. The simpler method computes the objective and constraint functionsat several points and selects the feasible point that yields the greatest value. Judd (1998) calls this the‘comparison method’. The comparison method uses no derivative information. Moreover it is very
powerful because it can establish a global maximum, supposing that grid search proceeds intensively
and extensively over the feasible space. The second methods uses the information of the gradients orcurvature of objective and constraints such as first order conditions (FOCs). Usually, the second
method of establishing the optimum is more efficient in terms of calculation time than the first
method if the optimization problem has a normal structure. However, the derivative information canestablish only the local maximum. So, if the structure of the optimal problem is unusual, the former
strategy is safer and better than the latter, as it excludes the local maximum.
Ried(1998,9) and Grossman(1998) propose the following method of establishing optimal
longevity. They characterize the problem of individuals choosing their longevity as the ‘free terminal
time problem’. That is, the individual can actually choose the terminal period optimally. However, inthe discrete time Grossman model, it seems impossible to treat the free terminal time problem
directly with the analytical method. The authors thus use a step-by-step strategy. They consider the
fixed terminal time problem with exogenous death constraints on the terminal health capital. This
constraint forces the individual to die exogenously at terminal period, and individuals plan theiroptimal behavior in the right of this constraint. In addition, they consider the modified fixed terminal
time problem. The modified fixed terminal time problem has no terminal health constraints. Thus3

individuals may die endogenously at the terminal period, or have sufficiently high health capital atterminal period. They then define that free terminal time problem as the modified fixed terminal time
problem with endogenous death, which achieves the highest inter-temporal utility. Thus, they
actually consider the sequence of the modified fixed terminal time problem with different terminal
periods. They then solve the FOCs of each problem and evaluate each inter-temporal utility byreference to the solution satisfying each FOCs. Finally, they compare each utility level and choose
the terminal period with the highest inter-temporal utility as the optimal length of life.
However, because the numerical approach adopted in this paper can treat the free terminal time
problem directly, we do not have to follow the step by step strategy. Moreover, due to the numericalmethod, we do not have to use the information about FOC, curvature of the objectives andconstraints, for solving free terminal time problem. In stead, we follow the comparison method. We
directly evaluate the inter-temporal-utility of the free terminal time problem, and then choose theoptimal path of health capital and financial assets and so does optimal length of life. Technically, inorder to treat the free terminal time problem directly, the planning period must be sufficiently large.
Intuitively, the planning period in the numerical dynamic programming method does not necessarilyequal the individual optimal length of life, although in previous analytical analysis the terminal
period become the individual optimal length of life.
It is impossible and pointless to determine which method is better for establishing the optimal
length of life, as the latter approach seems analytically impossible. However, it may be safer andbetter to use our algorithm, which assures the global maximum, because the optimal problem has an
unusual structure such that length of life is determined endogenously. Moreover, it enables us tounder take sensitivity analysis of optimal longevity5).
To our knowledge, Picone et al.(1998) is the only analysis that solves the Grossman model
analytically. It introduces uncertainty into the pure consumption Grossman model, and solves themodel using the dynamic programming method6). The authors analyze how the degree of uncertainty
affects individual heath investment and saving behavior. Their analysis should be regarded highly, as
there are few theoretical applied analyses of the Grossman model. However, because of theirdifferent concerns, they assume that the death rarely occurs endogenously. This paper, therefore,
extends Piconne et al.(1998) to the study of endogenous optimal length of life. Eisenring (1999)
points out that whether individual length of life is determined endogenously may influence thecharacteristics of an individual’s optimal health investment path. Thus the endogenity of life may
influence the optimal path7). However, the objective of this paper is to demonstrate clearly themechanism for determining optimal length of life. Thus it does not introduce any uncertainty into themodel, and uses a simpler model than that of Picone et al.(1998). Hence it is impossible todirectlycompare our results with those of Picone et al.(1998). The introduction of uncertainty only induces
earlier death only exogenously. Uncertainty is not, therefore, essential to determining optimal length4

of life.
This paper is organized as follows. Section II presents the model, and section III explain the
dynamic programming procedure and the method of determining the optimal length of life. Section
IV describes the parameter value setting, sectionⅤ and Ⅵ present and discuss the simulation
results, and section Ⅶ summarizes the results and suggests further extensions.

Ⅱ The Model
This section presents the model that describes the optimal behavior of the retired individual. As
individual has already retired, he/she has no labor income. In addition, uncertainty is not kept out of
the model in order to demonstrate clearly the mechanism for determining endogenous death. The
individual therefore chooses levels of consumption and health investment that maximize utility inretirement under several constraints, and builds up the optimal level of health capital and financial
assets. More specifically, the model has a two-dimensional state space consisting of financial assets
and health capital. The range of health capital H t is defined from H to H and Wt is also definedfrom 0 to W . In addition, individual is assumed to have no bequest motive: that is, it is optimal forthe individual to exhaust all his or her financial assets during a lifetime. The individual therfore
solves the following problem

Maxt =τ ∑t =1 β t u t (C t , H t ) ,τ(1)τ,{Ct , M t }t =1

s.t.Wt +1 = (1 + r )Wt − C t − M t ,H t +1 = M t + (1 − δ ) H t .

(2)

C t > 0, M t ≥ 0 1 ≤ t < τ

(4)

(3)

where β is the time discount rate, C t is current consumption, M t is the health investment andδ is the health capital depreciation rate. In addition, τ expresses the individual length of life
(retirement periods). Thus from eq(1) the individual gains utility from consumption and healthcapital, and future utility is discounted by β . Eq(2) expresses the motion equation for financial
assets and implies the individual’s financial budget constraints. Eq (3) expresses the motion equationof health capital and implies the health investment function. This model concerns with the bangbang control problem heavily discussed by Ehlrich and Chuma(1990) and thus uses the linear healthinvestment function8) The depreciation of health capital δ is constant over time, and is different
from the related studies. Suppose current health investment is 0, health capital decreases constantly
through δ . Moreover, from eq(4), C t must be positive and M t must be non-negative during
lifetime periods9). In actual numerical analysis, the utility function is specified as5αut ( H t , C t ) = H t Ct

1− αand 0 ≤ α ≤ 1 10). In this specification, α represents the preference forhealth capital. If α is higher, the preference for health capital is higher than for consumption.
Finally, individual optimal length of life (retirement periods) in this model is defined as

τ = min{t | H t < H }(5)Thus individual optimal length of life is determined when the optimal health capital becomes lower
than the minimum level. For convenience, when the individual is dead, the set of health capital andfinancial asset is assumed as

D = {H t < H, Wt ≤ 0 | ∀t > τ}In addition, the utility ut in dead time and after become11)

u t = 0 for t > τ

(6)

(7)

Ⅲ Solution procedure for the Model
This paper principally follows the method suggested by Bertsekas(1976). As is well known,
dynamic programming consists of a backward process and a forward process. In this algorithm, theplanning period, T , is set and the value function at T is defined as 0 by terminal condition. Notethat T is different from τ , length of life. Given the value function at T , the value function at

T − 1 is calculated using the optimization problem at T − 1 . The value function for each proceeding
period from T − 1 to the initial period is calculated in a similar manner in T − 1 steps. Then, in theforward process, the optimal problem is solved again from initial period to T − 1 , given initial value
and the obtained value function in the backward process. As a result, the optimal values of statevariables and control variables are determined sequentially. This model treats the free terminal time
problem directly. Technically, in order to show the endogenous death in our algorithm, T must belarge enough relative to τ for all the people with different initial endowments to die endogenouly.
Thus, the appropriate size of T is determined according to the range of initial health capital andfinancial assets. Under a sufficiently large T , individual optimal length of life τ is determinedthrough the optimal change in the feasibility set.
Ⅲ-1 Backward Solving
For the sake of clarity in the explanation below, we define the set of health capital and financial
asset in t as

[

][ ]

ψ = H , H × 0, W

(8)

The backward process is expressed as follows. By terminal condition, the value function in6T , VT (H T , WT ), is defined as

VT (H T , WT ) = 0(9)Thus eq(9) indicates that all the values of VT (H T , WT ) at any combination of H T and WT in ψ ,are evaluated as 0 by terminal condition. Moreover, when the certain combination of H t and Wt
is the element of D or death status at t , the value of Vt (H t , Wt ) of this H t and Wt is also

Vt (H t , Wt ) = 0(10)

The optimal problem in T -1, using the value function in T , is expressed as follows
α1− αmax H T −1 CT −1

cT −1, M T −1

+ β VT ( H T , WT )(11)

s.t.CT −1 = (1 + r )WT −1 − WT − M T −1

(12)

M T −1 = H T − (1 − δ ) H T −1

(13)

M T −1 ≥ 0, CT −1 > 0(14)

Thus the value function in T -1is evaluated as

{

α

1− α

VT −1 ( H T −1 ,WT −1 ) = max H T −1 CT −1
cT −1, M T −1}

+ βVT ( H T ,WT )(15)s.tCT −1 = (1 + r )WT −1 − WT − M T −1
M T −1 = H T − (1 − δ) H T −1
M T −1 ≥ 0, CT −1 > 0

(16)
(17)
(18)

By substituting eq(16) and eq(17) into eq(15), we obtain

VT −1 ( H T −1 ,WT −1 ) =

{

}

α

max H T −1 ((1 + r )WT −1 − WT − H T + (1 − δ) H T −1 )1−α + βVT ( H T , WT )H T ,WT(19).

s.t.H T − (1 − δ) H T −1 ≥ 0, (1 + r )WT −1 − WT − H T + (1 − δ) H T −1 > 0
As a result VT −1 (H T −1 ,WT −1 ) is defined over all combinations of H T −1 and WT −1 in ψ . Eq(19)expresses the evaluation of the optimal mapping from given H T −1 and WT −1 to optimal combinationof H T and WT . Optimal H T and WT is chosen among the feasible combination of H T and WT ,which satisfies eq(16) and eq(17), over the potential set of ψ ∪ D . Note that D is also included in
the feasible combination in T .
For example, the set of H T −1 and WT −1 over ψ , which satisfies the condition thatH − (1 − δ) H T −1 ≥ (1 + r )WT −1 , is defined as
BT −1 = {H T −1,WT −1 ∈ ψ | H − (1 − δ) H T −1 ≥ (1 + r )WT −1 }(20)

From eq(17), H − (1 − δ) H T −1 expresses health investment in T -1 which is necessary tomaintain H in T . From the model’s assumption, in case that H is not maintained, the individual
dies and the utility level in t is 0. Thus, as a result of the optimal problem in T -1, when the
combination of H T −1 and WT −1 satisfies BT −1 , optimal consumption and health investment in T -1

7

becomeCT −1 = (1 + r )WT −1
M T −1 = 0(21)

Thus, WT become 0 and H T become lower than H . This means that the set of D is chosen
through

optimalbehavior

following

VT −1 (H T −1 , WT −1 ) is redefined as follows

thetheoretical

model

in

section Ⅱ .Therefore,αVT −1 ( H T −1 ,WT −1 ) = H T −1 ((1 + r )WT −1 )1−α(22)Intuitively, eq(22) expresses the value of VT −1 (H T −1 , WT −1 ) at some combinations of H T −1 andWT −1 over ψ T −1 which choose the death status endogenously in T .
However, in T , all the values of VT (H T , WT ) at any combination of H T and WT in ψ , areevaluated as 0 by terminal condition. Thus, the value function in T evaluates the element of both ψand D as 0. The terminal condition also indicates that the utility at T cannot be obtained by the
choice of any combination of H T and WT in ψ . This terminal condition is exogenous constraints,
which is not shown in the theoretical model presented in sectionⅡ.This exogenous condition isadded in order to apply the numerical dynamic programming method to solve the model. However
this terminal condition actually influences the optimal choice process at T -1.
From eq(20), ψ \ BT −1 is defined asψ \ BT −1 = {(H T −1,WT −1 )∈ ψ T −1 | H − (1 − δ) H T −1 < (1 + r )WT −1 }(23)

Because of the terminal condition, given any combination of H T −1 and WT −1 in ψ \ BT −1 ,consumption and health investment in T -1 becomeCT −1 = (1 + r )WT −1
M T −1 = 0(24)

Therefore, VT −1 (H T −1 , WT −1 ) is also redefined asαVT −1 ( H T −1 ,WT −1 ) = H T −1 ((1 + r )WT −1 )1−α(25)

Due to eq(24), given any combination of H T −1 and WT −1 in ψ \ BT −1 , WT become 0. On
the other hand, from eq(17) and eq(24), H T becomesH = (1 − δ) H
if H T −1 ≥ (1 − δ) −1 H
T −1
 T
(26)

−1
H T ≤ H
if H T −1 < (1 − δ) H
Thus, supposing H T −1 is large enough, H T can never fall below  . In short, the value ofVT −1 (H T −1 , WT −1 ) at some combinations of H T −1 and WT −1 in ψ \ BT −1 is potentially biased by8the terminal condition, which is the exogenous constraint and is not assumed in the free terminal
time problem presented in sectionⅡ. Because the terminal condition induces the choice as eq(24).
Moreover, from eq(24), WT become 0 even any given H T −1 and WT −1 in ψ , and thus cannot
maintain positive consumption and non-negative health investment.. From the clarity of the
expalnation, the choice that WT =0 due to termianl condition through eq(24) is defined as
exogenous death. Thus eq(25) express the value function of H T −1 and WT −1 which chooses
exogenous death in T .
Similarly, the value function in T -2 is evaluated through the optimal problem in T -2.

{αVT − 2 ( H T − 2 ,WT − 2 ) = max H T − 2 CT − 2
cT −  , M T − 

1− α

}

+ βVT −1 ( H T −1 ,WT −1 )(27)

s.tCT − 2 = (1 + r )WT − 2 − WT −1 − M T − 2
M T − 2 = H T −1 − (1 − δ) H T − 2
M T − 2 ≥ 0, CT − 2 > 0

(28)
(29)
(30)

By substituting eq (28) and eq(29) into eq(27)VT − 2 ( H T − 2 , WT − 2 ) ={} (31)

αmax H T − 2 ((1 + r )WT − 2 − WT −1 − H T −1 + (1 − δ) H T − 2 )1− α + βVT −1 ( H T −1 , WT −1 )H T −1,WT −1

s.t.H T −1 − (1 − δ) H T − 2 ≥ 0, (1 + r )WT − 2 − WT −1 − H T −1 + (1 − δ) H T − 2 > 0
Similar to the value function in T -1, eq(31) expresses the evaluation of the optimal mapping fromgiven H T − 2 and WT −2 in ψ to optimal combination of H T −1 and WT −1 . Especially, given somecombination of H T − 2 and WT −2 in ψ , the element of D is chosen in T -1 as the optimal choice,VT −1 (H T −1 , WT −1 ) become 0. Thus, as a result of optimal problem in T -2, following equations are

satisfied.

CT − 2 = (1 + r )WT − 2M T − 2 = 0(31)

Thus, eq(31) is redefined asαVT − 2 ( H T − 2 ,WT − 2 ) = H T − 2 ((1 + r )WT − 2 )1−α(32)

Intuitively, eq(32) expresses the value of VT − 2 ( H T − 2 , WT − 2 ) at H T − and WT − over ψ whichchooses the death status endogenously in T -1. In addition, although exogenous death may not

T -2 directly, it influences it indirectly throughVT −1 ( H T −1 ,WT −1 ) . Eq(32) includes VT −1 ( H T −1 , WT −1 ) , which evaluates the optimal mapping fromgiven H T −1 and WT −1 selected in the optimal problem in T -1 to optimal combination of H T
and WT . Thus if the value of VT −1 ( H T −1 , WT −1 ) is not biased by terminal condition, eq(32)influence the optimal problem in

evaluates the optimal path of all health capital and financial asset of all H T − 2 and WT −2 in ψ9from T -2 to T . Conversely, if the value of VT −1 ( H T −1 , WT −1 ) is biased by terminal condition, thevalue of VT −  ( H T − , WT −  ) is also biased.
From the above discussion, supposing the value of VT −1 ( H T −1 , WT −1 ) is obtained without thebias of the terminal condition or exogenous death, VT −  ( H T − , WT −  ) can determine the optimal
path of health capital and financial asset of all combination of H T − 2 and WT −2 in ψ from T -2
to T . Of course, this expresses the optimal path in the free terminal time problem presented in
sectionⅡ. By performing a similar calculation from T -2 backwards to the initial period, the valuefunction in initial period, V1 (H 1 ,W1 ) , is calculated. This, V1 (H 1 ,W1 ) can determine the optimal
path of health capital and financial asset of all combination of  1 and W1 in ψ from initial to T .

In order to avoid the effect of the terminal condition on the value of V1 (H 1 ,W1 ) at any

combination of  1 and W1 in ψ , the optimal path of health capital and financial asset, which

V1 (H 1 ,W1 ) determines, have to be the element of D before T endogenously. More specifically, it
is appropriate for the optimal health capital and financial assets path for any combination of H 1
and W1 in ψ to be at least the element of BT −1 in T -1. In short, they choose the element of D in
T . Supposing this condition is satisfied, the value of V1 (H 1 ,W1 ) at any combination of H 1 andW1 in ψ can be evaluated without any bias from the terminal condition upon the sequence of the
optimal choice. To satisfy this condition, T must be large enough not to affect the value of

V1 (H 1 ,W1 ) at all combination of  1 and W1 in ψ . In practice, it is confirmed by performing thesensitivity analysis concerning T to evaluate the effect of V1 (H 1 ,W1 ) , whether this condition issatisfied or not.
From the aspect of the optimal longevity, this backward process is loosely translated asfollows12). Let Vt (H t , Wt ) as 0. This means individual dies at t . In this case, intuitively, thebackward procedure from t to initial period means the process of establishing initial health capitaland financial asset which achieve zero value function optimally in t . At the same time, it predicts
the optimal sequence of health capital and financial asset. This indicates that backward process canestablish the individual who dies optimally in t . Thus under the planning period of T , thebackward process can establish not only the individual who dies optimally in T , but also those whodies optimally in t before T . This indicates the backward process can establish the optimal
longevity of every one in initial period, so long as T is set at a sufficiently large value according to
the range of initial endowments.
Ⅲ-2 Forward Simulation
Given the value of Vt ( H t , Wt ) for all t from the above procedure, the individual whose
initial state variables are H 1 and W1 solves the following problem to show explicitly the optimal
path of health capital and financial assets:

10

{

}

V1 ( H 1 , W1 ) = max H 1 ((1 + r )W1 − W2 − H 2 + (1 − δ) H 1 )1−α + βV2 ( H 2 , W2 )H 2 ,W2αs.t.{H 2 ,W2 }∈ Λ2(34)

where Λ2 shows the feasible set of health capital and financial assets in the second period.

Λ 2 consists of survival, Ω 2 and death, D . Thus Λ2 is defined as

Λ2 = Ω 2 ∪ D(35)

The individual chooses optimal health capital and financial assets in the second period to maximize
the utility in retirement periods. The set of survival in the second period is expressed as(36)
Ω2 = {max[H , (1 − δ) H 1 ] ≤ H 2 < (1 + r )W1 + (1 − δ) H 1 ,0 ≤ W2 < (1 + r )W1 }where (1 − δ) H 1 expresses health capital in the second period with no health investment at the first
period. Supposing (1 - δ)H1 is below H , it is not included in Ω 2 . (1 + r )W1 + (1 − δ) H 1
expresses the health capital in the second period, if all the interest and principle is spent initially.
However, it is also not included in Ω 2 , because consumption has to be positive for the individual tosurvive. In addition, financial assets in the second period are 0 , if the individual exhausts all initialfinancial assets and interest through initial consumption and health investment. On the other hand,the financial assets in the second period are (1 + r )W1 , with no initial health investment andconsumption. However, it is also not included in Ω 2 because of the strictly positive constraints for
consumption. If it is infeasible to choice an element of Ω 2 , the individual dies and choices anelement of D . In the case of D , the value function in the second period is expressed as

V2 ( H 2 , W 2 ) = 0(37)

Therefore, the distinction between death and survival is expressed as the differences in the size ofthe feasibility set.
Under Λ2 , optimal health capital and financial asset in the second period are determined
simultaneously as follows{H 2 ( H 1 ,W1 ), W2 ( H 1 ,W1 )}arg max( H 1 α ((1 + r )W1 − W2 − H 2 + (1 − δ) H 1 )1− α + β V2 ( H 2 , W2 )) H 2 ,W2

if Λ2 \ D ≠ φ
=
D


if Λ2 \ D = φ(38)

In the case that Λ2 \ D = φ , individual cannot choose an element of Ω 2 and dies before or at 2.
Thus health capital and financial assets are defined as an element of D .
Moreover, in the case of survival, Λ2 \ D ≠ φ , consumption and health investment at the first
period are expressed as11
C1 ( H 1 ,W1 ) = (1 + r )W1 − W2 ( H 1 ,W1 ) − H 2 ( H 1 ,W1 ) + (1 − δ) H 1
M 1 ( H 1 ,W1 ) = H 2 ( H 1 ,W1 ) − (1 - δ) H 1
if Λ2 \ D ≠ φ


(39)

Contrary when individual dies just at 2nd period, these are expressed asC1 ( H 1 , W1 ) = (1 + r)W1

M 1 ( H 1 , W1 ) = 0



(40)

if Λ2 \ D = φ

In this case, individual consumes all the initial financial assets and interest income and neverundertakes health investment.
Suppose an individual with certain initial endowments satisfies the condition that Λ2 \ D = φ ,and T is defined appropriately to satisfy 2 < T . In this case, the individual’s optimal length of life,τ = 2 , is determined endogenously. Moreover, in the actual calculation process, this individual
technically continues to choose an element of D and gains 0 utility during the period from 2 to T ,as if he or she lived after period 2.
A similar process is performed from the second to T -1 period, and the sequence of optimal
health capital and financial assets can be calculated. In addition, consumption and health investment
are also determined optimally. The development of Λ t depends heavily upon initial endowments. It
means that the optimal length of life and the optimal path differs by the difference in initialendowments. Therefore, technically T must be large enough to satisfy the condition that allindividual with different initial health capital and financial assets die endogenously and satisfy
that τ < T .
Generally, the optimal problem in t is presented as

{

}

α

Vt ( H t ,Wt ) = max H t ((1 + r )Wt − Wt +1 − H t +1 + (1 − δ) H t )1− α + β Vt +1 ( H t +1 ,Wt +1 )H t +1,Wt +1

s.t.{H t +1 ,Wt +1 }∈ Λ t +1

(41)

where Λ t +1 consists of the case of survival Ω t +1 and death D , and is thus expressed asΛt +1 = Ω t +1 ∪ D .
(42)
The set of survival Ω t +1 is defined as(43)
Ω t +1 = {max[H , (1 − δ) H t ] ≤ H t +1 < (1 + r )Wt + (1 − δ) H t ,0 ≤ Wt +1 < (1 + r )Wt }Then, suppose Λt +1 \ D = φ , the value function in t + 1 is defined as(44)
Vt +1 ( H t +1 , Wt +1 ) = 0
Under Λ t +1 , optimal health capital and financial assets are determined simultaneously as
follows

12

{H t +1 ( H t ,Wt ), Wt +1 ( H t ,Wt )}arg max( H t α ((1 + r )Wt − Wt +1 − H t +1 + (1 − δ) H t )1−α + βVt +1 ( H t +1 , Wt +1 )) H t +1 ,Wt +1
if Λt +1 \ D ≠ φ
=
D


if Λt +1 \ D = φ(45)

When Λ t +1 \ D = φ , the individual dies before or at t + 1 , and health capital and financial assets
are defined as D . Moreover,{H t ,Wt }∈ D , {H s ,Ws }s≥τ ∈ D technically means that theindividual continues to choose the element of D after death. We can, therefore, treat dead people’s
behavior as well as that of survivors following eq.(44). In this sense, death and survival are continual
phenomena.
Moreover, when the individual survives in t + 1 , consumption and health investment in t are
expressed asC t ( H t ,Wt ) = (1 + r )Wt − Wt +1 ( H t ,Wt ) − H t +1 ( H t ,Wt ) + (1 − δ) H t
(46)
M t ( H t ,Wt ) = H t +1 ( H t ,Wt ) − (1 - δ) H tif Λt +1 \ D ≠ φConversely, when individual dies in just t + 1 , consumption and health investment in t are
expressed asC t ( H t , Wt ) = (1 + r)Wt
(47)
 M t ( H t , Wt ) = 0
if Λt +1 \ D ≠ φ and Bt

where Bt is defined in eq(20). When H − (1 − δ) H t is more than (1 + r)Wt , the individual
cannot maintain minimal health due to budget constraints. In this case, the individual consumes hisor her entire financial assets and interest in t , and never undertakes health investment. Finally,
when the individual dies before the t + 1 period, both consumption and health investment in t
should be 0, That isC t ( H t , Wt ) = 0
 M t ( H t , Wt ) = 0



(48)

if Λs \ D ≠ φ( s < t + 1)

Therefore, the dead individual chooses the element of D during the period of {t | τ ≤ t ≤ T }from eq(44), and gains 0 utility. Since the development in Λ t +1 depends heavily upon initialendowments, T must be set up appropriately according to the range of initial endowments so that all13

initial individuals can satisfy the condition that Λt \ Dt = φ before T . If T is not large enoughfor all people to die endogenously, some individuals who survive in T die exogenously by theterminal condition that VT (H T , WT ) = 0 . In the case of exogenous death13), the individual plans hisor her optimal behavior by regarding the 0 value function as given. Thus, the terminal condition
induces this individual to exhaust his or her financial assets more rapidly so that he or she has nofinancial asset at T as shown in eq(24). This path may not coincide with the optimal path when

T is sufficient large and the individual experiences endogenous death.

Ⅳ Parameter Values and Numerical Evaluation
This model may be relatively simpler than that of Picone et al.(1998) for the following reasons.
First, as one of the objectives of this model is to show the mechanism of the determination of
optimal length of life clearly and perform the sensitivity analysis for some parameters overreasonable range, thus the simpler model is appropriate for this purpose. Secondly, there are noadequate empirical studies for the parameter used in the Grossman model. The model consists of thefollowing parameters defined above: health capital depreciation rate ( δ ); the preference parameterfor health capital ( α ); time discount rate ( β ); and interest rate ( r ). However, there exist no
adequate empirical studies of the possible values of α and δ . Grossman (1972a) is the only
investigation with estimates, but the estimation method is not so appropriate. Picone et al. (1998) set

α as 0.614) and made δ a function of age15) in their benchmark case following Grossman (1972a).
However, the plausibility of these values is not well established, and much more precise empiricalstudies are necessary.
In the benchmark case, α is 0.7, δ is 0.3, β is 0.9 and r is 0. Moreover, H is 0.5, W is0, H is 99.5 and W is 99 and 100 equal-scaled grid16). Thus the health capital of entire periods(from initial to T ) are divided into 100 different equally-levels from 0.5 to 99.5. Financial asset arealso divided into 100 equally spaced from 0 to 99. This implies that there exist 10000 combinations
over every period. Especially initial 10000 combination is interpreted as 10 000 people withdifferent initial combinations who plans their optimal life after the retirement. In other words, we
assume the society that retired-people distributes uniformly over heath capital and financial asset. In
addition, this paper does not consider the range that the financial assets become negative. It is
impossible to return resources, even if the retired people borrow from financial market. Because thethey only gain the interest income from positive financial asset
The assumption that δ is the increasing function of age is unnecessary complexity for the finite
length of life in the pure consumption Grossman model. Thus we set δ as constant value of 0.3 andrelatively high for the clarity of endogenous optimal length of life. Moreover, we set r as 0, since.
the value of r heavily affects the characteristics of optimal health investment path under a linear

14

health investment function. As the purpose of this paper is to demonstrate the determination of
optimal length of life, we do not treat this matter in depth. We then set r as 0, and discuss the
optimal length of life under a linear health investment function. While the assumption that retiredpeople distributed uniformly over heath capital and financial asset is unrealistic, it is unnecessary to
use a more realistic distribution for our purposes as a benchmark case. Nevertheless, it is easy toobtain the result from a realistic distribution only by weighting the results from a uniform
distribution17).
Finally, this paper undertakes sensitivity analysis in order to assure everyone’s endogenous
death and to confirm the robustness of the model. First, we discuss the value of T . If T is ofsufficiently large value, everyone in initial period dies endogenously and satisfies the condition thatΛ t \ D = φ before T . Thus optimal length of life is determined endogenously. Theoretically, it ispreferable to set T at a sufficiently large and nearly infinite value, since an individual with anylevel of initial health capital and financial assets can die endogenously. However, too large value forT takes too much time and is difficult in practice. Thus, we attempt sensitivity analysis concerning
T from 5 to 30 for every period, given the parameters held constant at the benchmark case, and thenestablish the appropriate value of T 18). The appropriate value of T is determined when the valueof V1 (H 1 ,W1 ) over ψ is not influenced by the length of T . Second, after fixing T at 25, weundertake sensitivity analysis concerning δ , α and β respectively. The range of δ is from 0.5
to 0.1, of α from 0.5 to 0.9, and of β from 0.5 to 0.9. The size of the steps is 0.1.

ⅤSensitivity analysis concerning T and endogenous death

This section tries to find the appropriate value of T under which the value of V1 (H 1 ,W1 ) at

any combination of  1 and W1 in ψ is not influenced by the terminal condition. This indicatesthat all initial people die endogenously before T. The endogenous death of all initial people isassured by sensitivity analysis of V1 (H 1 ,W1 ) at all elements in ψ concerning T. If the value of

V1 (H 1 ,W1 ) over ψ is not influenced by the length of T and converges to constant, all
V1 (H 1 ,W1 ) over ψ are not biased by terminal condition at all. Under such V1 (H 1 ,W1 ), theoptimal path, determined by V1 (H 1 ,W1 ), follows the free terminal time problem presented in

sectionⅡ.
In order to demonstrate the characteristics of the model, we first look at the typical individualwith a certain combination of  1 and W1 in ψ and find the T under which the value of

V1 (H 1 ,W1 ) at given  1 and W1 converge to the constant. In other words, we try to find the T that is

sufficiently large for this individual to die endogenously. Then we move to the discussion of thevalue of T under which the value of V1 (H 1 ,W1 ) at any combination of  1 and W1 over ψ
becomes constant, or all initial people die endogenously.

15

Figure 1 presents the relationship between the value of V1 (H 1 ,W1 ) of the typical individual
and T. This typical individual has 60th level of  1 and W1 , so the value of each is 59.5 and 59
respectively. When T is under 11, the value of V1 (H 1 ,W1 ) increases as T increases. However,when T is over 11, it converges to a constant. Therefore, if T is over 11, the value of V1 (H 1 ,W1 )of this individual is not influenced by the terminal condition. Figure 2 shows the relationshipbetween this individual’s length of life and T to explain the intuitive meanings of Figure1. When T
is less than 11, this individual length of life τ is equal to T . This indicates that the individual dies
exogenously by the terminal condition. However when T is more than 11, the individual’s length oflife τ become less than T by choosing an element of D endogenously before T . This indicatesthat the individual dies endogenously.
Figure 3 to Figure 6 show the relationship between the financial assets, consumption, healthinvestment , and health capital path of this individual and T. The solid line represents the path when

T is 5. Similarly, dash, dash-dot, and dot lines present the case of 7, 9,11 and 15 respectively. As
shown explicitly in these figures, dash-dot and dot lines have exactly the same lengths and values at
each period. Therefore, if T is more than 11, individual optimal paths of financial assets,
consumption and health capital converge on the same line. Moreover, the path of these variables is
co0nsistent with the predicted results in the section Ⅲ. For example, as T gets shorter, healthinvestment and especially consumption in the initial period increase, as shown in Figure 4 and 5. As
a result, from Figure 3, financial asset decrease more rapidly as T gets shorter, and fall to nearly 0
at each T . It is consistent with the predicted result in eq(24) because of the inducement effect of theterminal condition of exogenous death. The differences in the paths of the solid line and the dot line
apparently express the effect of the terminal condition on the optimal path. Moreover, the obtained
health capital path is also consistent with the predicted result in eq (26). From Figure 7, supposing

H T −1 is sufficiently large, H T remains to be positive.

The results presented above show that the value of V1 (H 1 ,W1 ) of this typical individual

becomes constant under a sufficiently large T . This means that optimal length of life is determined

endogenously under a sufficiently large T . In addition, the value of V1 (H 1 ,W1 ) of this typicalindividual indicates the welfare level of his or her life in retirement 19). Thus, when the individual

dies exogenously by the terminal condition, individual welfare falls below the level in the case ofendogenous death. This is because the terminal condition induces the individual to consume thefinancial assets more rapidly in order to exhaust them completely at T . However, neither does
health capital, since individual gains utility from health capital, and thus has an incentive to maintainhealth through the initial health investment. Moreover, the results of converged welfare under a large

T can be translated as follows. As the individual can choose the longevity he or she wants, theindividual determines it so as to maximize lifetime welfare. Therefore, even if T gets larger,
individual optimal length of life does not alter, and neither does individual welfare.

16

Next, we move to the discussion of the value of T under which the value of V1 (H 1 ,W1 ) at any
combination of  1 and W1 in ψ becomes constant or all initial people die endogenously. Figure 7

presents the relationship between the average value of V1 (H 1 ,W1 ) over all the combination of  1

and W1 in ψ and T . As T becomes larger, the average value of V1 (H 1 ,W1 ) increases

monotonically. Then, when T is over 13, it converges on the constant Therefore, supposing T isover 13, the value of V1 (H 1 ,W1 ) at any combination of  1 and W1 in ψ is not influenced by
the terminal condition. As a result, as explained abpve, all initial people die endogenously if T is
more than 13. To confirm this results, Figure 8 shows the relationship between the social average
longevity and T . Social average longevity represents average retirement periods of 10000 retiredpeople at the initial period. From Figure 8, as T lengthens from 5 to 30, the numbers of retired
people who die exogenously by the terminal condition decrease, and thus social average longevity
also lengthens linearly. Then, when T is more than 13, social average longevity converges on theconstant. Therefore, all retired people die endogenously, when T is more than 13.
From these results, if T is more than 13, all retired people die endogenously in the wholesociety, and thus everyone determines his or her longevity so as to maximize welfare during
retirement periods. Therefore, the difference in T does not alter either social average longevity or
welfare. As a result, when T is 25, we can confirm that all retired-people die endogenously in any
parameter value of the following sensitivity analysis.

Ⅵ The sensitivity analysis of the optimal path
This section undertakes sesitivity analysis for various parameters. Since T is large enough,length of life is determined endogenously. First, the optimal length of life and the optimal path at thebenchmark case are presented. Then we undertake sensitivity analysis for δ , α and β
sequentially.

Ⅵ-1 The benchmark caseThis sub-section describes the cumurative death rate in the whole society and the optimal pathin the benchmark case. At the benchmark, α is 0.7, δ is 0.3, β is 0.9 and r is 0. We also set T
at 25. For the clarity of description, this paper presents figures for the individuals whose initial levelsof health capital and financial asset are 20th, 40th, 60th, 80th, and 100th, respectively. Moreover the
initial 60th case is mainly referred to in words.
The cumurative death rate at t in the society is defined as the ratio of the cumulative numberof those who die until t period to the retired people in initial period. In the benchmark case, as
shown in Figure 9, everyone has died by the 13th period, and thus the average length of life becomes
about 9.4 periods. Figure 10 shows the individual optimal health investment path. Almost all health17

investment is undertaken at the initial period. In the case of 60th initial stocks, initial healthinvestment occupies the 26.9% of initial financial assets, becoming nearly 0 during the remainder ofretirement. As a result, as shown in Figure 11, optimal health capital decreases exponentiallythrough δ after the second period. Similarly, from Figure 12, optimal finantial assets decrease
exponentially. This means that people use up financial assets to finance health investment andconsumption. Optimal consumption also decreases, although more gradually than financial assets, as
shown in Figure 13.
In the above figures, endogenity of longevity is confirmed by the fact that the optimal path ofhealth capital becomes 0 before the 25th period. Principally, the time discount rate and the linear
health investment function explain the features in the dynamics of the endogenous variables. Owing
to the time discount rate, an individual prefers to maintain higher health capital and to consume
much in earlier periods. Since the health investment function is linear, the current health investment
additively increases the health capital of the next period if the constant health capital depreciation
effect is ignored. This means that the marginal effect of health investment on health capital does notchange due to the level of the current health capital. Therefore health investment is mostly
undertaken in the initial period; thereafter it becomes nearly 0, and thus health capital decreases
exponentially through δ after second period. This feature of health investment, which becomes 0
in certain period, is consistent with the Ehlrich and Chuma (1990) which emphasized the point of thebang-bang control problem under the linear health investment function. Consumption also decreases
because of the time discount rate, but more gradually, partly because of the specification of theutility function. As a result, financial assets also decrease exponentially. These simulation results arealmost consistent with the results at Picone et al. (1998). However, as Picone et al. (1998) assume ahigh negative value of disutility for death so as to avoid endogenous death- moreover, their
specification of health capital depreciation and utility functions also differs- there are somedifferences in the characteristics of optimal health capital and consumption path20).
Ⅵ-2 Changes in the health capital depreciation rate ( δ )This subsection first analyses how the depreciation for health capital, δ , affect the optimal
path of the typical individual when changing from 0.5 to 0.1 by the step of 0.1. Other parameters are
the same as in the benchmark case. This individual has the 60th level of initial health capital andfinantial assets. This subsection then discusses how the death rate and average longevity in the wholesociety alter due to the same change in δ .
The relationship between optimal longevity and δ is shown in Figure 14. As δ falls, optimal
length of life increases, and thus the longevity when δ = 0.1 is twice as long as when δ =0.5. Thistendency is consistent with the analytical results of Ehlrich and Chuma (1990). Health investment at
the initial period also declines, as in Figure 15. Although health investment decreases, health capital

18

increases at every period due to the direct effect of δ , as shown in Figure 16. Similarly, financial
assets also increase, as shown in Figure 17. This means that the individual runs down financial asset
more slowly. This affects the level of consumption at earlier periods, and so the consumption-age
profile becomes flatter as shown in Figure 18. For example, when δ is 0.5, consumption decreases
by 83.3 % from the initial to period 4, but when δ is 0.1, it changes only by 36.6 % over the same
duration.
Figure 19 and 20 show how the death rate and the average longevity in the whole society
respond to the same change in δ . The death rate decreases over all retirement periods due to thedecrease in δ . Thus, although all retired people die by period 10 when δ =0.5, they survive until
period 20 when δ =0.1. Therefore, average longevity increases as δ decreases, and thus average
longevity when δ =0.1 is nearly twice as long as when δ =0.5. These results show explicitly that
longevity responds elastically to the change in parameters. The decrease in δ increases individual
health investment in previous analysis, such as Ehlrich and Chuma (1990) and Eisenring (1999).
However, our results suggest that health investment in the initial period declines. One of the reasonsof this somewhat counter-intuitive result is the use of the pure consumption model, whereas Ehlrich
and Chuma (1990) use the full Grossman model. Indeed, the analysis of Grossman (1972b) also
indicates that the sign of the response of health investment is not clear in the pure consumption
model. This result is explained intuitively as follows. Under the linear health investment function
such that H t +1 = M t + (1 − δ ) H t , a lower value of δ can affect health capital in the next period,
not only through health investment in the present period but also through the smaller direct
depreciation effect of the health capital H t . If this direct effect sufficiently increases health capitalat the next period, the individual may be able to achieve more utility through less current healthinvestment. The decrease in δ also affects the consumption pattern through longevity. Due to thedirect effect of δ , health capital is maintained more than H . Therefore, it is optimal for theindividual to save financial assets for the longer remaining period of life, and so consumption
decreases in earlier periods. In addition, longer optimal life changes the optimal inter-temporal
allocation of consumption, and so the features of the optimal consumption path. Since endowment
stays constant and investment is not affected significantly by changes in δ , a longer life results inthe smaller financial assets per lifetime period.
Ⅵ-3 Changes in the Relative Preference for Health Capital ( α )This subsection analyses how the parameter of the preference for H t , α , affect the optimal
path of the same individual when changing from 0.5 to 0.9. As shown in Figure 21, optimal
longevity lengthens as α become larger. However, it is not affected when α is more than 0.721).
Figure 22 shows that health investment at initial period increases. For example, lifetime healthinvestment when α =0.9 is 21 times as large as when α =0.5. Thus, health capital also increases, as

19

shown in Figure 23. On the other hand, financial assets are run down more rapidly at earlier periods
and more slowly at later periods, as shown in Figure 24. For example, when α is 0.5, financial
assets decrease by 87 % from period 2 to period 6, while when α is 0.9, financial assets change
only by 66.7 % over the same duration. Consumption also decreases over a lifetime, as shown in
Figure 25. In particular, consumption stays at a low level for almost every period if α =0.9. In
addition, the consumption-age profile becomes flatter when α is higher.
Figure 26 and Figure 27 present the influence of a change in α on the death rate and theaverage longevity in the whole society. As shown in Figure 26, an increase in α makes the death
rate lower over entire lifetimes. However, the marginal effect on the death rate is smaller than δ .
When α is 0.5, all retired old people die by period 12, whereas when α is 0.9, they survive until
period 14. The marginal effect on average longevity is also smaller than δ , and thus average
longevity grows only 20 % due to the increase in α from 0.5 to 0.9. The increase in α means thatthe individual has a relative preference for health capital over his or her entire retirement periods.
Thus, the individual undertakes larger initial health investment in order to increase health capitalover his or her entire lifetime. Hence, the death rate at each period declines and longer average
longevity is achieved. In order to finance the increase in health investment, the individual not only
decreases consumption but also exhausts the financial assets more rapidly. However, at the sametime individuals plan a longer life due to the increase in health capital, and so it is not optimal toexhaust the financial asset completely over their remaining lifetimes. The speed of the decrease infinancial assets therefore falls in later periods. Since the longer optimal length of life changes the
optimal inter-temporal allocation of consumption, the slope of the consumption-age profile becomesflatter.
Ⅵ-4 Changes in the time discount rate ( β )Finally, we show how change in the time discount rate, β , affects the optimal path of thesame typical individual when changing from 0.5 to 0.9. Surely other paprameters are held constant at
the benchmark. Figure 28 shows how the optimal length of life differs due to the change in β . The
optimal length of life lengthens as β increases. As a result, lonvevity when β = 0.9 is about 2.2
times as long as when β =0.5. Moreover, health investment continues for slightly longer periods
and so does health capital (Figure 29 and 30) as β become larger22). Similarly, financial assets
increase at every period, as shown in Figure 31. This means that individuals exhaust financial assets
more slowly. Therefore, Figure 32 shows how the consumption-age profile becomes flat. When β
is 0.75, for example, consumption changes by 86.9 % from the first period to the forth. On the other
hand, when β is 0.95, it changes by only 60 % over the same duration.
Figure 33 and 34 show the effect of β on the death rate and average longevity in the wholesociety. As shown in Figure 33, the death rate during retirement decreases monotonically as β20increases. For example, all retired people live until period 7 when β is 0.5, whereas they survive
until 13 periods when β is 0.9. As a result, average longevity lengthens as β increases, and soaverage longevity when β =0.9 is about 1.9 times as long as when β =0.5. The increase in the time
discount rate means that the individual places more value on a longer retirement priods. Therefore,
health capital and health investment increases so as to maintain the health capital above H for
longer periods; conversely, individuals try to save their assets by reducing consumption at earlier
periods. As a result, death rate during retirement decreases and average longevity increases.

Ⅶ Concluding Remarks
This paper solves the pure consumption Grossman model numerically by using the dynamic
programming method, and undertakes sensitivity analysis of the optimal length of life. Grossman(1998) and Ried (1999) have discussed the determination of optimal length of life analytically.
However, in the discrete time model, these authors cannot determine how the parameters affect the
optimal length of life. Although Picone et al.(1998) is the rare analyses try to extend the Grossman
model numerically, it does not discuss the determination of optimal length of life. This paper,
hterefore, focuses on the issue of the optimal length of life and extends the Picone et al.(1998) in thisdirection. To our knowledge, this is the first analysis that treats the optimal length of life withnumerical analysis.
This paper shows the numerical algorithm that individual length of life is determined
endogenously under a sufficiently large planning period, T . To confirm the availability of thismethod, this paper undertakes sensitivity analysis concerning T to search appropriate value of T .
From the simulation results, it is proved that all people’s value of the value function of initial period
converge under sufficiently large T . In short, the longevity of all retired people in the wholesociety is determined endogenously. After fixing T at 25, we undertake sensitivity analysis withseveral parameters, health capital depreciation, relative preference for health capital, and the time
discount rate. Simulation results show that a longer optimal length of life results from decreases in
the health capital depreciation rate or increases in the relative preference for health and in time
discount rate. Furthermore, longer lives significantly change the optimal allocation of consumption
and the features of the optimal consumption path.
By making the individual length of life endogenous in the numerical method, applied analysis
of longevity becomes easier in the Grossman model than the analytical method. For example,
Philipson and Becker (1998) analyzes how pension affects the retired people's behavior for longevity.
It claims that public pension will have the effect of inducing more health investment and longer lives
than private insurance, although it deteriorates the welfare level due to the distortion of the lifetime
resource allocation. The author calls it moral hazard effect in annuities. Moreover, they discuss how

21this longevity inducement affects the premiums of annuities and health care insurance. However,their model is inappropriate for rigorous analysis for longevity. Thus our model enables us toinvestigate this longevity effect more carefully supposing annuity and health care insurance are
introduced. Also, it is also important to check how the endogeneity of the length of life itself affectthe optimal path of health capital and health investment. In addition, there are very few analyses
which try to extend the Grossman model by introducing insurance and annuities23). Therefore, It isalso essential to show how the existence or the characteristic of health insurance and annuity affects
the individual health investment behavior. Nevertheless, we may be able to consider the optimal
insurance or annuity under the Grossman model. Finally in order to extend numerical applied
analysis of Grossman model, the reliability of parameter value is of great importance. Therefore,
more reliable empirical analysis for Grossman model is also significant theme for the future
research.

22

Footnote＊)Acknowledgement
We benefited from comments by Professors Christoph Eisenring, Kouichi Futagami, Fumio Ohtake,Yoshihiko Seoka, Makoto Mori, Tetsuya Nakajima, Masako Ii and Akihisa Shibata. In addition, I
would like to thank Professors Walter Ried for sending us the latest version of his paper. Needless tosay, any remaining errors are ours.
1)According to Philipson and Becker (1998), the number of the countries that provide the retired
people income transfer or health care assistance program through public sector has increased
fivefold. ( from 33 to 155) since 1940.
2)Zweifel and Breyer (1997) emphasize the close relationship between longer average longevity andthe amount of subsidy per retired person. As the rate of the retired increase in the population due tothe improvement of average longevity, retired people become more influential in the politicaldecision-making process. The public sector is therefore induced to expend excess resources on the
medical technology preferred by the retired, or on higher subsidies through old-age assistanceprograms. As a result, average longevity improves further and reinforces the political power of theretired. Zweifel and Breyer (1997) call this phenomenon Sisyphus syndrome.
3)Philipson and Becker(1998)is the first analysis which analyzes this issue formally. However the
method and the model used seems to be inappropriate for rigorous analysis.
4)On the contrary, the model which focuses on the wage through healthy time is called ‘the pureinvestment Grossman model’. Thus it mostly targets young individuals.
5)We have tried to use the terminology of the existing literature. However, as our method of
establishing optimal length of life is totally different from the analytical studies, it is sometimes
difficult to describe our ideas with the terminology of the existing literature. Thus, to avoid
misunderstanding, we mainly use terminology that is familiar in the literature of dynamic
programming.
6)The introduction of uncertainty into the Grossman model is discussed in Chang (1996), Dardanoni
and Wagstaff (1987), Liljas (1998) and Picone et al (1998)
7)Eisenring (1999) applies the comparative dynamic method to the fixed terminal time model andcompares the results with the analysis of Ehlrich and Chuma (1990), which applies the same method23

to the model with endogenous longevity. In Ehlrich and Chuma (1998), an increase in initial healthdoes not induce lower health investment, as the demand for longevity also increases. However in
Eisenring (1999) health investment decreases due to the increase in the initial health stock.
8)The bang -bang control problem means that health investment in each period becomes the corner
solution on the feasible set in the optimization problem due to the linear health investment function.
In this case, the maximum level of health investment is undertaken under constraints, when themarginal value of health capital happens to become larger than the marginal value of financial assets.
Ehlrich and Chuma (1999) discusses the possibility of the bang-bang control problem in Grossman(1972a), as it assumes that the health production function is constant return to scale.
9)If financial assets are completely exhausted by the one-period before, the individual cannot
consume a positive amount in current period and thus dies in this period. Therefore, endogenous
death occur even if the health capital at one-period before is high enough to keep the current healthcapital above the minimum without health investment in one-period before. Needless to say, inalmost all cases, it may not be optimal.
10) Certainly, the specification of the utility function influences longevity and the optimal path. Ried
(1999) argues the possibility of the non-existence of the solution, if H t is near H and the utility
function satisfies the Inada condition. We can also employ the utility function such as.

u ( H t , Ct ) = ( H t − H ) α Ct

1− αalternatively. However, the obtained results do not change
αsignificantly. Thus, this paper presents only the results obtained from H t C t

1− α. The non-existence

issue proposed by Ried (1999) may occur only under the fixed terminal time problem, because nonexistence means that individuals cannot survive, even though they have to survive until the terminal
period under the fixed terminal time problem. In other words, it is obviously not optimal to survive
for such long fixed periods. As our simulation method treats the free terminal time problem directly,
this problem does not matter, though it is one of the important issues in the existing analytical
literature.
11) As referred in Bleinchrodt and Quiggin (1999), 0 utility assumption in dead time is normal in
health economics or health evaluation. However, there is no positive reason for it. If utility in deadtime has different values, the difference in the utility level in death and survival may have a different
jump. As a result, it may significantly alter the optimal path and optimal longevity.

2412) This intuitive explanation may be incorrect mathematically. As we explained earlier, the valuefunction in t just evaluates the optimal mapping from all grid points of t to the feasible set in

t + 1 , using the information of the value function from t + 1 to T .
13) This does not imply that the optimal problem with a shorter T is equal to the fixed terminal time

problem in the existing literature. Some persons die by terminal condition that VT (H T , WT ) = 0 ordies exogenously at  . However, there is the case that H T ≥ H as a result of optimal choice, as
shown in eq (26). This apparently contradicts the definition of the fixed terminal time problem.
Needless to say, this may be the modified fixed terminal time problem14) α is set as 0.7 in our paper, because the grid of state variables used in this paper is coarse.
Supposing we use the lower value, the response of health investment to the change in the parametercannot be shown clearly in the figure. This is mostly due to the insufficient precision of our grid.
However, the usage of more precise grid setting consumes too much time . Therefore, this paper uses

α , which is 0.1 point higher than Picone et al.(1998).
15) Picone et al.(1998) specify health capital depreciation as the increasing function of age such asδ = 1 − 0.012 exp(0.021(32 + t )) , following Grossman (1972b). As a result, δ changes from
0.124 to 0.152. Moreover, in order to investigate how individual attitudes against uncertainty affect[C H ] − 1 . Then
individual optimal behavior, the utility function is specified asγ
t1− γ 1− σt

1− σγ − σ( γ − 1) express the Arrow-Pratt measure of individual attitudes for relative risk aversion.
16) As it is plausible to suppose that some retired people have no financial assets, this paper sets thelowest financial assets level at 0. For the finite length of life, H must be positive. If an individual
does not make any health investment in his or her life, health capital decreases exponentially througheq(3), but health capital never reaches 0 within finite periods.
17)As there is no interaction between people in the initial period, the optimal path of the certain
value of the initial endowment does not change, even if the initial distribution is different from theuniform distribution.
18) This sensitivity analysis is undertaken to find the appropriate value of T . Therefore, this isdifferent from the sequence of the modified-fixed terminal time problem for searching optimal
longevity in the existing literature. Our method can evaluate the free terminal time problem directly
under appropriate T and thus there is no need of the sequence of the problem with different

25planning periods to find optimal longevity.
19) This individual welfare exactly equals to the inter-temporal utility in the existing literature due tothe well-known characteristics of the value function.
20) In the footnote5, Picone et al.(1998) assume utility in death ( H t ≤ 0.001 ) as

5*[(0.0001)]1−σ − 1 . As σ is 0.9 at the benchmark case, it indicates that death is assumed have a1− σlarge negative value.
21)This result shows the one of shortcomings of our numerical analysis. If the grid of the statevariables is not so fine, the subtle difference in individual length of life is approximated to a same
number. However, the usage of the more precise grid takes too much time for computation.
Therefore, we cannot help using the rather rough grid.
22)This figure also shows the one of the shortcomings of our analysis. As referred before, the grid of
the state variables in our analysis is rough and thus the subtle difference in health investment andhealth capital in each period is approximated a same number. Thus the image of the figure may
change slightly when we use the more precise grid but it does not change the essence of our results.
23) Liljas(1998) is probably the only analysis which introduces the insurance into the Grossman
model

26

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