Dynamic Model of Health Investment:

Theoretical and Empirical Analysis of the Influence of Risks over Time and Age on Medical Expenditures

Takanori IdaKyoto University, Faculty of Economics

Presented at PSAM5, 11/30/2000

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The purposes of this paper

(1) to set up the Grossman model with endogenous death

(2) to theoretically and empirically investigate the influence of two risk parameters on medical expenditures.

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Two Risk Parameters

The first risk:

the relationship between health stock and death risk: since death risk decreases over time owing to the progress in medical technology and public health, it will be a decreasing factor in medical expenditures.

The second risk:

the depreciation rate of health stock: since this increases over age, it will be an increasing factor in medical expenditures.

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Model Analysis

Definitions and Conditions(1) The hazard rate, which is the conditional probability of

dying in a short interval of time after having survived, is assumed to be a decreasing function of health stock.

(2) The time available for the individual is divided into healthy time and sick time, where healthy time is positively related to the health stock while sick time is negatively related.

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The individual can obtain two utilities from health:

(1) the direct utility of being healthy, which is brought by the healthy

state itself

(2) the indirect utility of being healthy, which one can get by working

and consuming.

The intertemporal utility, which shall be maximized by an agent, is an

expected value over lifetime that is discounted by the interest rate, r.

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three kinds of constraints.

(1) The income budget constraint.the most income that we would get if we worked for all available

time is equal to the sum of the expenditure on health investment,

the expenditure on consumption goods and the income foregone

owing to sickness.

(2) The health stock condition. the health investment equals the net increase in health stock plus its

depreciation.

(3) The initial condition of health stock.

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The Optimization Problem:

MAX 　 ∫ 0TExp(-rt)(1-Ft)Utdt

S.T. 　 Zt=wTHt-pIt, dHt/dt=It-tHt, H0=H(0)

a costate variable: t≡(t)

the current-value Hamiltonian: L≡(1-Ft)Ut+t(It-tHt).

The maximum principle :

∂L/∂It=-p(1-Ft)vt'+t=0 (1)

dHt/dt=It-tHt (2)

dt/dt=rt-LHt=(r+t)t+Ft(ut+vt)-(1-Ft)(ut'+wvt')THt' (3)

Exp(-rT)T=0, and H0=H(0).7

The Specification of Functional Forms

Pure investment model(1) vt=-Exp(-(wTHt-pIt)) : the utility of

consumption goods is a constant absolute-risk-aversion function.

(2) THt=LogHt, THt'=/Ht: the relationship between health stock and healthy time is logarithmic.

(3)t=/Ht: the health stock is in inverse proportion to the conditional probabilistic density of death.

(4) Tt=T, t=: the available time and depreciation rate are constant.

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By letting dHt/dt and dIt/dt zero, the simplified first order

conditions about the health investment and health stock

are obtained.

It=Ht (7)

It={(w+-1)-r}Ht/(w+)+1/p. (8)

Furthermore, equations (7) and (8) have an equilibrium,

i.e., (H*, I*)=((w+)/p(+r), (w+)/p(+r)).

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Comparative Statics

(1) the depreciation rate of health stock,

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(2) the risk rate of health stock on death,

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Empirical Analysis

The induced variable:

the real (1994) expenditure on medical care in a month

(Medical-Care, ¥1)

The explanatory variables

the real (1994) income in a month (INC, ¥1000),

the age-adjusted death rate in a selected year (Time-Risk, per

1000 population),

the death rate at a selected age (Age-Risk, per 10000

population),

the female dummy (D-Female),

the year dummy (D-Year)13

The data : THE NATIONAL SURVEY OF FAMILY

INCOME AND EXPENDITURE: ONE-PERSON

HOUSEHOLDS (1969/1974/1979/1984/1989/1994 ) and THE LIFE TABLES.

The estimation equation:

(Ni)1/2MCi=(Ni)1/2a+b(Ni)1/2INCi+c(Ni)1/2TRi+d(Ni)1/2ARi+e

(Ni)1/2D-Fi+(Ni)1/2∑fjD-Yi+(Ni)1/2ui.

The weight: the number of tabulated households (N).

The coefficients are a~f, and the disturbance term is u.

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Table 1: Estimation result (R2=0.650942)Variable Const. Income Time risk Age risk D-female D-1974 D-1979 D-1984 D-1989 D-1994

Coefficient -32520.1 4.809672 2622.65 8.72727 11155.1 3621.47 8041.61 10011.0 13073.6 15149.5Standard error 10300.3 2.789844 846.897 1.71919 2967.84 1181.25 2549.59 3128.96 3813.73 4107.37

t-statistic -3.15721 1.72399 3.09678 5.0764 3.75865 3.06578 3.15408 3.19945 3.42803 3.68836

Table 2: Increase in medical care per month over time (1969/1994)Medical care Income (¥1000) Female ratio Time risk Age risk D-1994

Average1969 837.7 127.8 0.45 10.5 18.2 0Average 1994 4637.9 276.4 0.60 5.1 64.7 1

Difference 3800.2 148.6 0.14 -5.4 46.5 1Coeff.*Diff. 3800.2 714.8 1598.2 -14068.0 405.7 15149.5Contribution 100% 19% 42% -370% 11% 399%

Table 3: Increase in medical care per month over age (under-50/ over-50 in 1994)Medical care Income (¥1000) Female ratio Time risk Age risk D-1994

Under -50 3709.4 308.8 0.36 5.9 7.7 1Over-50 5582.4 243.5 0.84 4.3 122.63 1

Difference 1873.0 -65.3 0.47 -1.6 114.9 0Coeff.*Diff. 1873.0 -314.1 5283.6 -4099.3 1002.7 0Contribution 100% -17% 282% -219% 54% 0% 15

The Estimation Result

Table 1 shows the estimation results. R2 is 0.65, and the t-values are almost significant.

Although the constant is negative, all other coefficients are positive as expected. This is consistent with our theoretical conclusion that the coefficients of two risk parameters are positive:

the death risk decreases as time passes (resulting in the decrease in medical expenditure),

while it increases with age (resulting in the increase in medical expenditure). See this Fig.

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The transition of medical expenditure over time

Table 2 : a comparison of the average figures of Japanese one-person-households between 1969 and 1994.

The percentage contribution to the difference in medical expenditures between 1969 and 1994, ¥3800.2,

The negative effect of the time risk (-370%) and the positive effect of the year dummy (399%) are particularly large,

in comparison with the other contributions of the increases in income (19%), age risk (11%), and female dummy (42%).

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The cross-sectional change

of medical expenditure in 1994

Table 3 : a comparison of the average figures of Japanese one-person-households between under-50 year olds and over-50 year olds.

The percentage contribution to the difference in medical expenditures between the under-50 and the over-50 group, ¥1873.0,

the negative effect of the time risk (-219%) and the positive effect of the female dummy (282%) are particularly large,

in comparison with the other contributions of the increases in income (-17%), and age risk (54%).

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ConclusionFirst, the dynamic model of health investment with

endogenous death was established and the influences of two risk parameters were investigated. The decrease in the time risk, which is interpreted to represent social risks such as medical technology and public health, reduced the expenditure on medical care. The increase in the age risk, which is interpreted to represent individual risks such as aging, increased the expenditure on medical care.

Second, the medical expenditures of Japanese one-person households from 1969 to 1994 were estimated. The influences of two risk parameters on the medical expenditures were confirmed as expected, and the sexual distinction and the institutional change were also important.

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