Directory UMM :Journals:Journal of Health Economics:Vol19.Issue5.Sep2000:

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Discussion

Correction note on ‘‘The demand for health

with uncertainty and insurance’’

Ken Tabata

a

, Yasushi Ohkusa

b,) a

Graduate School of Economics, Osaka UniÕersity, Japan b

Institute of Social and Economic Research, Osaka UniÕersity, 6-1 Mihogaoka, Ibaraki, Osaka, Japan

Received 9 July 1999; received in revised form 5 November 1999; accepted 25 January 2000

Abstract

w

This paper proves the non-existence of an optimal solution under the Liljas Liljas, B.

Ž1998 . The demand for health with uncertainty and insurance. J. Health Econ., 17,.

x

153–170 type of insurance. The reason for the non-existence is that the insurance induces the individual to increase his time input, relative to medical expenditure in the household production of health investment. Hence, it distorts the balance of inputs in the production of health investment. Moreover, it also distorts the household production for consumption goods through time constraints. Therefore, this paper proposes an alternative insurance that covers the time loss due to illness and has an optimal solution. q_{2000 Elsevier Science}

B.V. All rights reserved.

Keywords: Optimal solution; Medical insurance; Grossman model

1. Introduction

The Grossman model, originally developed by Grossman in 1972, is very valuable in order to analyze the individual lifetime health investment behavior. Unfortunately, there are few papers that introduce uncertainty and insurance into

AbbreÕiations: D13; D61; D80; I10; I12

)_{Corresponding author. Tel:q81-6-6879-8566; fax:q81-6-6878-2766.}

Ž .

E-mail address: ohkusa@iser.osaka-u.ac.jp Y. Ohkusa .

0167-6296r00r$ - see front matterq2000 Elsevier Science B.V. All rights reserved.

Ž .

1 _{Ž .}

the Grossman model in spite of their importance in the real world. Liljas 1998 is probably the first to do so, thus contributing significantly to the development of the Grossman model. Unfortunately, there is one crucial error in this paper. The

Ž .

optimal solution does not exist under the Liljas 1998 type of insurance.

This paper proves the non-existence of an optimal solution under the Liljas

Ž1998 type of insurance and proposes an alternative insurance that has an optimal.

solution. A more careful treatment on the Liljas’ model is necessary for the development of the Grossman model and health economics. This paper is orga-nized as follows. Section 2 reformulates the Liljas model so as to make the non-existence proof much clear. Section 3 proves the non-existence of an optimal

Ž .

solution under a Liljas 1998 type of insurance and Section 4 proposes an alternative type of insurance which has an optimal solution. Finally, Section 5 summarizes the results.

2. Solution to the optimization problem

Ž .

Although the original work by Liljas 1998 solves the model following

Ž .

Grossman 1972a,b , this paper uses the method of maximum principle to make the non-existence proof more tractable.2_{Almost all the notations and assumptions}

Ž .

in this paper are the same as in Liljas 1998 to facilitate comparisons. Thus, for a more precise explanation of the assumptions and notations in this model, see

Ž .

Appendix A of Liljas 1998 .

Ž .

In Liljas 1998 , uncertainty is introduced into health capital such as

HsHE_{qs ´,}

Ž .

₁

t t t

H_max

E

Ht s

H

H g H d H ,t

Ž

t

.

t

Ž .

2

Hmin

Since Hmin is assumed to be positive,3 then

yHE t

Pr

ž

´G

/

s1

Ž .

3

s_t

This means that H is never negative.t

1

Ž . Ž . Ž .

Dardanoni and Wagstaff 1987 , Selden 1993 , and Chang 1996 also introduced uncertainty into the Grossman model. However, these studies treated only one or two period model.

2

It is well known that the dynamic optimization problem can be solved through several methods

Žsee Kaimen and Schwartz, 1991 . Wagstaff 1986 also solves the Grossman model with the method. Ž .

of maximum principle.

3 _{Ž .}

Thus, the individual utility maximization problem under uncertainty is reformu-lated as

T Hmax _yut

Max.

H H

u

Ž

ftH , Z et t

.

g H d H d t ,

Ž

t

.

t

Ž .

4

0 H_min

T Hmax _{yr t}

s.t.R.s

H H

Ž

pt tIqq zt tqw TL et t

.

d H d t ,t

Ž .

5

0 H_min

˙

HtsItydt

Ž

H ,Qt t

.

H .t

Ž .

6

The Hamiltonian reads

H_max T

yut Js

H

H

u

Ž

ftH , Zt t

Ž

X ,Tt t

.

.

e d t

H_min 0

T

yr t

yl

H

Ž

pt tI M ,TH

Ž

t t

.

qq Zt t

Ž

X ,Tt t

.

qwt

Ž

VyftHt

.

.

e d t

0

qmt

Ž

I M ,THt

Ž

t t

.

ydt

Ž

H ,Qt t

.

Ht

.

g H d H

Ž

t

.

t

Ž .

7

T _yut

sE

H

u

Ž

ftH , Zt t

Ž

X ,Tt t

.

.

e d t

0

T

yr t

yl

H

Ž

pt tI M ,TH

Ž

t t

.

qq Zt t

Ž

X ,Tt t

.

qwt

Ž

VyftHt

.

.

e d t

0

qmt

Ž

I M ,THt

Ž

t t

.

ydt

Ž

H ,Qt t

.

Ht

.

wx

where E is defined on the health capital. This can be rearranged as

T

yut

Js

H

E u

Ž

ftH , Zt t

Ž

X ,Tt t

.

.

e yl p

Ž

t tI M ,TH

Ž

t t

.

qq Zt t

Ž

X ,Tt t

.

0

yr t

qwt

Ž

VyftHt

.

.

e qmt

Ž

I , M ,THt

Ž

t t

.

ydt

Ž

H ,Qt t

.

Ht

.

d t

Ž .

8 The first-order conditions of this problem are

EJ Eu EZt _yut EZt _{yr t}

sE e ylqt e s0,

Ž .

9

EXt EZt EXt EXt

EJ Eu EZt _yut EZt _{yr t}

sE e ylqt e s0,

Ž .

10

ETt EZt ETt ETt

EJ EI_t EI_t

yr t

s ylpt e qmt s0,

Ž .

11

EJ EIt _{yr t} EIt

s ylpt e qmt s0,

Ž .

12

ETHt ETHt ETHt

EJ Eu Ed_t

yut yr t

ym

˙

ts sft

ž

E e qlw et

/

ymtE dtq H .t

Ž .

13

EH_t Eh_t EH_t

Ž . Ž .

Since Z X ,Tt t t and I M ,THt t t are independent of H , they are not evaluated byt

Ž .

the expectation. From Eq. 11 ,

mslpeyr t_.

Ž .

₁₄

t t

Note that lis constant, since it is the multiplier for the individual life time budget constraint and not for the variable financial asset in each period.4 Thus, we obtain

m

˙

t p

˙

t

s yr ,

Ž .

15

m_t p_t

Ž .

and rearranging Eq. 13

m

˙

t Eu _{yut yr t} 1 Edt

s yft

ž

E e qlw et

/

qE dtq H .t

Ž .

16

m_t Eh_t m_t EH_t

Ž . Ž . Ž .

Substituting Eqs. 14 and 15 into Eq. 16 , the Euler equation becomes

1 Eu Ed_t

Žryu.t

ft

ž

_lE e qwt

/ ž

s rqE dtq Ht

/

ptyp

˙

t.

Ž .

17

Eht EHt

This is the same as the Euler equation under the uncertainty presented by Liljas

Ž1998 . Hence, the change in the resolution procedure does not alter the solution in.

any way.

3. Non-existence of the optimal solution

This section presents the nonexistence of an optimal solution when the Liljas

Ž1998 type of insurance is introduced. Liljas 1998 introduces the social insur-. Ž .

ance as

htw TW Ht

Ž

Ž

max

.

yTW H

Ž

t

.

.

,htg

Ž

0,1 .

.

Ž .

18

Ž .

In order to express TW as TW H , the author implicitly assumes the existence_{t t} of an optimal solution under this type of insurance. Indeed, if there is an optimal solution, then all endogenous variables should only be the function of H .t

4

Unfortunately, this assumption is not so obvious since the existence of an optimal

Ž .

solution is not proven in Liljas 1998 . Therefore, this insurance should be reformulated as

htw TWt

Ž

yTW ,t

.

htg

Ž

0,1 ,

.

Ž .

19 where TW is the maximum attainable working time. The Hamiltonian of the optimization problem with insurance reads as

T

yut

Js

H

E u

Ž

ftH , Zt t

Ž

X ,Tt t

.

.

e yl p

Ž

t tI M ,TH

Ž

t t

.

qq Zt t

Ž

X ,Tt t

.

0

yr t

qwt

Ž

VyftHt

.

qhtw TWt

Ž

y

Ž

ftHtyTtyTHt

.

.

.

e

qmt

Ž

I M ,THt

Ž

t t

.

ydt

Ž

H ,Qt t

.

Ht

.

d t

Ž .

20 Note that we use the relation

TWtsftHtyTtyTH .t

Ž .

21 The first-order conditions are

EJ Eu EZt _yut EZt _{yr t}

sE e ylqt e s0,

Ž .

22

EXt EZt EXt EXt

EJ Eu EZ_t EZ_t

yut yr t yr t

sE e ylqt e ylhtw et s0,

Ž .

23

ET_t EZ_t ET_t ET_t

EJ EIt _{yr t} EIt

s ylpt e qmt s0,

Ž .

24

EMt EMt EMt

EJ EIt _{yr t} EIt _{yr t}

s ylpt e qmt ylhtw et s0,

Ž .

25

ETHt ETHt ETHt

EJ

ym

˙

ts

Ž .

26

EH_t

Eu Ed_t

yut yr t

sft

ž

E e qlw et

/

ymtE dtq Ht

Eht EHt

ETWt _{yr t}

qlhtw Et e

EHt

For the existence of an optimal solution, all first-order conditions should be

Ž .

satisfied. From Eq. 24 , it follows that

mslpeyr t

,

Ž .

27

t t

Ž .

Substituting this expression into Eq. 25 , we obtain

lhw eyr t_s0.

Ž .

₂₈

As r and w are given and positive andt l should not be zero, unless the

Ž .

lifetime income constraint is satisfied, ht must be zero so as to satisfy Eq. 28 . This means that there is no insurance. Thus, there is no optimal solution under the

Ž .

Liljas 1998 type of insurance.

Ž .

Since Liljas 1998 implicitly assumed the existence of an optimal solution, the insurance proposed in the paper may be inadequate. Moreover, the reason for the non-optimal solution under this insurance is that it just covers the time input TH_t to produce gross health investment, through the relation of TW_tsVyT_tyTH_ty

TL . Thus, this insurance affects individuals by increasing their time input TH ,_{t t} relative to medical expenditure M . As a result, the balance of inputs for healtht

investment is substantially distorted. Moreover, this insurance also distorts the balance of inputs in the household production function for consumption goods through the time constraint.5

Ž . Ž .

This can easily be proved using Eqs. 22 and 23 . For the existence of the optimal solution, the insurance should not distort the balance of the medicare expenditure M and time inputs TH in the production of health investment. Thist t

indicates the insurance which only covers medical care expenditure also distorts the balance of inputs and has no optimal solution, even though this type of insurance is the most likely in actual world.

4. An alternative way to introduce insurance

This section proposes an alternative insurance to that of Liljas type, under which the first order conditions for optimization problem are satisfied simultane-ously. That is the insurance with an optimal solution. The proposed insurance is

htw TL ,t t htg

Ž

0,1 ,

.

Ž .

29

which includes an insurance for the time loss due to illness. Introducing this type of insurance, the Hamiltonian is formulated as

T _yut

Js

H

E u

Ž

ftH , Zt t

Ž

X ,Tt t

.

.

e yl p

Ž

t tI M ,TH

Ž

t t

.

qq Zt t

Ž

X ,Tt t

.

0

yr t

qw 1t

Ž

yht

. Ž

VyftHt

.

.

e qmt

Ž

I M ,THt

Ž

t t

.

ydt

Ž

H ,Qt t

.

Ht

.

d t 30

Ž .

Using the expression

TLtsVyftH ,t

Ž .

31

5

the first-order conditions are defined by

EJ Eu EZt _yut EZt _{yr t}

sE e ylqt e s0,

Ž .

32

EXt EZt EXt EXt

EJ Eu EZt _yut EZt _{yr t}

sE e ylqt e s0,

Ž .

33

ETt EZt ETt ETt

EJ EIt _{yr t} EIt

s ylpt e qmt s0,

Ž .

34

EMt EMt EMt

EJ EIt _{yr t} EIt

s ylpt e qmt s0,

Ž .

35

ETHt ETHt ETHt

EJ Eu Ed_t

yut yr t

ym

˙

ts sft

ž

E e qlw 1t

Ž

yht

.

e

/

qmtE dtq H .t

EH_t Eh_t EH_t

36

Ž .

Ž .

Eq. 34 implies

mslpeyr t_.

Ž .

₃₇

t t

Since l is constant, we obtain

m

˙

t p

˙

t

s yr .

Ž .

38

m_t p_t

Ž .

and so Eq. 36 can be rearranged as

m

˙

t Eu _{yut yr t} 1 Edt

y s yft

ž

E e qlw 1t

Ž

yht

.

e

/

qE dtq H .t

mt Eht mt EHt

39

Ž .

Ž . Ž . Ž .

Substituting Eqs. 37 and 38 into Eq. 39 , the Euler equation, obtained under this alternative insurance, is given by

1 Eu Ed_t

Žryu.t

ft

ž

E e qw 1t

Ž

yht

.

/ ž

s rqE dtq Ht

/

ptyp

˙

t.

l Eht EHt

40

Ž .

Obviously, under this alternative insurance, these first order conditions can be satisfied simultaneously. Hence, the existence of an optimal solution is assured.

only through H . Needless to say, changes in Ht t would affect all control variables, including THt and M , but its effects are along the optimal policyt

function for non-insurance model. Therefore, it does not distort the balance of inputs in both household production functions.

Unfortunately, this type of insurance has no foothold in the actual world. It suggests that insurers should insure against all time loss, including time loss in housekeeping due to illness. However, it is well known that making individuals express their real time loss is difficult due to the moral hazard based on asymmetric information. On the contrary, as it is shown, a more realistic insur-ance, which insures only medical care expenditure, does not have an optimal solution. Therefore, this may imply that an insurance with optimal solution is difficult to maintain in the real world.

5. Concluding remarks

This paper proves the non-existence of an optimal solution under the Liljas

Ž1998 type of insurance and proposes an alternative insurance with an optimal.

solution. The reason for the non-existence of such an optimal solution is that it creates a distortion between time input and medical expenditure in the health investment function. As a result, it distorts the balance of inputs in both household production functions for health investment and consumption goods. Hence, we propose an alternative insurance that does not distort the balance of inputs for health investment. A more careful treatment of the Liljas model is significant for the development of the Grossman model.

Acknowledgements

The authors would like to thank the anonymous referee for indicating a lot of mistakes and making valuable suggestion in an earlier version of this paper. Needless to say, any remaining errors are ours.

Appendix A

By introducing financial asset dynamics explicitly, the more familiar type of dynamic optimization problem can be formulated. Furthermore, through this formulation, it is easily shown that the condition where lshould be constant over

Ž .

The individual utility maximization problem is reformulated as

T

yut

Max.

H

E u

Ž

ftH , Zt t

.

e d t

Ž .

a1

0

˙

HtsE Itydt

Ž

H ,Qt t

.

Ht

Ž .

a2

˙

AtsE rAtqwtfHtyq Z X ,Tt

Ž

t t

.

yptI M ,TH

Ž

t t

.

Ž .

a3 where A is a financial asset. Then, the Hamiltonian is formulated ast

yut

˜

JsE u

Ž

ftH , Zt t

Ž

X ,Tt t

.

.

e qlt

Ž

rAtqwtftHtyq Zt t

Ž

X ,Tt t

.

ypt tI M ,TH

Ž

t t

.

.

qmt

Ž

I M ,THt

Ž

t t

.

ydt

Ž

H ,Qt t

.

Ht

.

Ž .

a4

˜

where lt is the costate variable on the transition equation for financial assets. Obviously, it expresses the individual budget constraint at each period. The first-order conditions are

EJ Eu EZ_t EZ_t

yut

˜

sE e yltqt s0.

Ž .

a5

EX_t EZ_t EX_t EX_t

EJ Eu EZt _yut EZt

˜

sE e yltqt s0.

Ž .

a6

ETt EZt ETt ETt

EJ EIt EIt

˜

s yl pt t qmt s0.

Ž .

a7

EMt EMt EMt

EJ EI_t EI_t

˜

s yl pt t qmt s0.

Ž .

a8

ETH_t ETH_t ETH_t

EJ

˙˜

˜

ylts sltr ,

Ž .

a9

EA_t

EJ Eu Ed_t

yut

˜

ym

˙

ts sE fte ymtE Htqdt qltwtft.

Ž

a10

.

EHt Eht EHt

Ž .

Eq. a7 implies

˜

mtsl pt t.

Ž

a11

.

and we can obtain

˙˜

m

˙

t p

˙

t lt

s q .

Ž

a12

.

˜

Ž .

On the other hand, Eq. a9 implies

˙˜

l_t

s yr ,

Ž

a13

.

˜

l_t

and so

˜

yr t

˜

ltse l0

Ž

a14

.

Ž .

and Eq. a12 can be reexpressed by

m

˙

t p

˙

t

s yr .

Ž

a15

.

m_t p_t

Ž .

As from Eq. a10

˜

m

˙

_t 1 Eu Ed_t l_t

yut

s y E fte qE Htqdt y wtft,

Ž

a16

.

m_t m_t Eh_t EH_t m_t

Ž . Ž . Ž . Ž .

substituting Eqs. a11 , a12 and a14 into Eq. a16 , we obtain

1 Eu Ed_t

Žryu.t

f

ž

˜

E e qwt

/

s

ž

rqE dtq Ht

/

ptyp

˙

t.

Ž

a17

.

Eh EH

l0 t t

˜

Ž . Ž .

Eq. a17 is the same as Eq. 17 , if l is l₀. It implies that l is just the marginal utility of initial wealth. Thus, it is constant by definition. Therefore, this condition is not essential in the model.

References

Chang, F.-R., 1996. Uncertainty and investment in health. J. Health Econ. 15, 369–376.

Dardanoni, V., Wagstaff, A., 1987. Uncertainty, inequalities in health and the demand for the health. J. Health Econ. 6, 283–290.

Grossman, M., 1972a. The Demand for Health: A Theoretical and Empirical Investigation. National Bureau of Economic Research, New York.

Grossman, M., 1972b. On the concept of health capital and the demand for health. J. Political Econ. 80, 223–255.

Kamien, M., Schwartz, L.N., 1991. Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management. North Holland.

Liljas, B., 1998. The demand for health with uncertainty and insurance. J. Health Econ. 17, 153–170. Selden, T.M., 1993. Uncertainty and health care spending by the poor: the human capital model

revisited. J. Health Econ. 12, 109–115.