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
This section presents the nonexistence of an optimal solution when the Liljas Ž
. Ž
. 1998 type of insurance is introduced. Liljas 1998 introduces the social insur-
ance as h w TW H
y TW H , h g 0,1 .
18
Ž .
Ž .
Ž .
Ž .
Ž .
t t
max t
t
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
This condition is not essential for this model. By introducing financial assets in each period explicitly in this model, the meaning of this condition can be presented explicitly. See Appendix A for
this proof.
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
h w TWy TW , h g 0,1 , 19
Ž .
Ž .
Ž .
t t
t t
where TW is the maximum attainable working time. The Hamiltonian of the optimization problem with insurance reads 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 q h w TWy f H y T y TH e
Ž .
Ž .
Ž .
.
t t
t t
t t
t t
t
qm I M ,TH y d H ,Q
H d t
20
Ž .
Ž .
Ž .
Ž .
t t
t t
t t
t t
Note that we use the relation TW s f H y T y TH .
21
Ž .
t t
t t
t
The first-order conditions are E J
Eu EZ
EZ
t t
yu t yr t
s E e
y lq e
s 0, 22
Ž .
t
E X EZ
E X E X
t t
t t
E J Eu
EZ EZ
t t
yu t yr t
yr t
s E e
y lq e
y lh w e s 0,
23
Ž .
t t
t
ET EZ
ET ET
t t
t t
E J EI
EI
t t
yr t
s ylp e
q m s 0,
24
Ž .
t t
EM EM
EM
t t
t
E J EI
EI
t t
yr t yr t
s ylp e
q m y lh w e
s 0, 25
Ž .
t t
t t
ETH ETH
ETH
t t
t
E J ym s
26
Ž . ˙
t
EH
t
Eu Ed
t yu t
yr t
s f E
e q lw e
y m E d q H
t t
t t
t
ž
Eh EH
t t
ETW
t yr t
qlh w E e
t t
EH
t
For the existence of an optimal solution, all first-order conditions should be Ž
. satisfied. From Eq. 24 , it follows that
m s lp e
yr t
, 27
Ž .
t t
Ž .
Substituting this expression into Eq. 25 , we obtain lh w e
yr t
s 0. 28
Ž .
t t
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
