Ž .
Journal of Health Economics 19 2000 811–820 www.elsevier.nlrlocatereconbase
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.
Ž .
1998 . 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: ohkusaiser.osaka-u.ac.jp Y. Ohkusa . 0167-6296r00r - see front matter q 2000 Elsevier Science B.V. All rights reserved.
Ž .
PII: S 0 1 6 7 - 6 2 9 6 0 0 0 0 0 3 6 - 9
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 Ž
. 1998 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 H s H
E
q s ´ , 1
Ž .
t t
t
H
max
E
H s H g H d H ,
2
Ž .
Ž .
H
t t
t t
H
min
Since H is assumed to be positive,
3
then
min
yH
E t
Pr ´ G s 1
3
Ž .
ž
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 Ž
. Ž
. see Kaimen and Schwartz, 1991 . Wagstaff 1986 also solves the Grossman model with the method
of maximum principle.
3
Ž . It is shown from Eq. 3 that the initial health capital must be more than zero for the health capital
during their lifetime to be more than zero.
Thus, the individual utility maximization problem under uncertainty is reformu- lated as
T H
max
yu t
Max. u f H , Z e
g H d H d t , 4
Ž .
Ž .
Ž .
H H
t t
t t
t
H
min
T H
max
yr t
s.t.R.s p I q q z q w TL e
d H d t , 5
Ž .
Ž .
H H
t t t
t t
t t
H
min
˙
H s I y d H ,Q H .
6
Ž .
Ž .
t t
t t
t t
The Hamiltonian reads
H T
max
yu t
J s u f H , Z
X ,T e
d t
Ž .
Ž .
H H
t t
t t
t
H
min
T
yr t
yl p I M ,TH
q q Z X ,T q w V y f H
e d t
Ž .
Ž .
Ž .
Ž .
H
t t t
t t
t t
t t
t t
qm I M ,TH y d H ,Q
H g H d H
7
Ž .
Ž .
Ž .
Ž .
Ž .
t t
t t
t t
t t
t t
T
yu t
s E u f H , Z
X ,T e
d t
Ž .
Ž .
H
t t
t t
t
T
yr t
yl p I M ,TH
q q Z X ,T q w V y f H
e d t
Ž .
Ž .
Ž .
Ž .
H
t t t
t t
t t
t t
t t
qm I M ,TH y d H ,Q
H
Ž .
Ž .
Ž .
t t
t t
t t
t t
wx where E
is defined on the health capital. This can be rearranged as
T
yu t
J s E u f H , Z
X ,T e
y l p I M ,TH q q Z
X ,T
Ž .
Ž .
Ž .
Ž .
Ž
H
t t
t t
t t t
t t
t t
t t
yr t
qw V y f H e
q m I , M ,TH y d H ,Q
H d t
8
Ž .
Ž .
Ž .
Ž .
. Ž
.
t t
t t
t t
t t
t t
t
The first-order conditions of this problem are E J
Eu EZ
EZ
t t
yu t yr t
s E e
y lq e
s 0, 9
Ž .
t
E X EZ
E X E X
t t
t t
E J Eu
EZ EZ
t t
yu t yr t
s E e
y lq e
s 0, 10
Ž .
t
ET EZ
ET ET
t t
t t
E J EI
EI
t t
yr t
s ylp e
q m s 0,
11
Ž .
t t
EM EM
EM
t t
t
E J EI
EI
t t
yr t
s ylp e
q m s 0,
12
Ž .
t t
ETH ETH
ETH
t t
t
E J Eu
Ed
t yu t
yr t
ym s s f
E e
q lw e y m E d q
H . 13
Ž . ˙
t t
t t
t t
ž
EH Eh
EH
t t
t
Ž .
Ž .
Since Z X ,T and I M ,TH
are independent of H , they are not evaluated by
t t
t t
t t
t
Ž .
the expectation. From Eq. 11 , m s lp e
yr t
. 14
Ž .
t t
Note that l is 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
p
˙ ˙
t t
s y r ,
15
Ž .
m p
t t
Ž .
and rearranging Eq. 13 m
Eu 1
Ed
˙
t t
yu t yr t
s yf E
e q lw e
q E d q H .
16
Ž .
t t
t t
ž
m Eh
m EH
t t
t t
Ž .
Ž .
Ž .
Substituting Eqs. 14 and 15 into Eq. 16 , the Euler equation becomes 1
Eu Ed
t Ž
. ryu t
f E
e q w
s r q E d q H
p y p . 17
Ž . ˙
t t
t t
t t
ž ž
l Eh
EH
t t
This is the same as the Euler equation under the uncertainty presented by Liljas Ž
. 1998 . Hence, the change in the resolution procedure does not alter the solution in
any way.
3. Non-existence of the optimal solution
