As r and w are given and positive and l should not be zero, unless the
t
Ž .
lifetime income constraint is satisfied, h must be zero so as to satisfy Eq. 28 .
t
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 s V y T y TH y
t t
t
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 health
t
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. This
t 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 h w TL , h g 0,1 ,
29
Ž .
Ž .
t t
t t
which includes an insurance for the time loss due to illness. Introducing this type of insurance, the Hamiltonian is formulated 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 1 y h V y f H
e q m I M ,TH
y d H ,Q H
d t
Ž . Ž
. Ž
. Ž
.
. Ž
.
t t
t t
t t
t t
t t
t t
30
Ž .
Using the expression TL s V y f H ,
31
Ž .
t t
t
5
This insurance also affects the individual’s workable time TW through the time constraint. In the
t
short run, an increase in time input for health investment TH will reduce the workable time TW since
t t
the increase in TH will not increase health capital H immediately. Needless to say, in the long run,
t t
the effect will be indefinite due to the increase in healthy time f H .
t t
the first-order conditions are defined by E J
Eu EZ
EZ
t t
yu t yr t
s E e
y lq e
s 0, 32
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, 33
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,
34
Ž .
t t
EM EM
EM
t t
t
E J EI
EI
t t
yr t
s ylp e
q m s 0,
35
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 1 y h e q m E d q
H .
Ž .
˙
t t
t t
t t
t
ž
EH Eh
EH
t t
t
36
Ž .
Ž .
Eq. 34 implies m s lp e
yr t
. 37
Ž .
t t
Since l is constant, we obtain m
p
˙ ˙
t t
s y r .
38
Ž .
m p
t t
Ž .
and so Eq. 36 can be rearranged as m
Eu 1
Ed
˙
t t
yu t yr t
y s yf
E e
q lw 1 y h e q E d q
H .
Ž .
t t
t t
t
ž
m Eh
m EH
t t
t t
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 Ž
. ryu t
f E
e q w 1 y h
s r q E d q H
p y p .
Ž .
˙
t t
t t
t t
t
ž ž
l Eh
EH
t t
40
Ž .
Obviously, under this alternative insurance, these first order conditions can be satisfied simultaneously. Hence, the existence of an optimal solution is assured.
Of course, the key point of this alternative insurance is that it does not create a distortion between time input TH and medical expenditure M . It has an effect
t t
only through H . Needless to say, changes in H would affect all control
t t
variables, including TH and M , but its effects are along the optimal policy
t t
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
