K.-T. Hua
coverage and both decisions of retirement and intergenerational transfers, in-
cluding fertility and education investment.
The production of the homogenous output
Y
t
uses physical capital
K
t
(
i.e.
, de-
preciates fully after one period) and effective labor
H
t
as inputs in the conven-
tionally neoclassical fashion (Mizuno and Yakita [9]). Under perfect competition
assumption, the real wage rate
w
t
and the gross rate of return on physical capital
r
t
are
(1)
(2)
where
represents physical capital per effective labor,
A
> 0 total
factor productivity and
the share of
K
t
to
Y
t
.
There are a large number of homogenous working parents, school-aged young
children, and retired grandparents in this model economy. Since our main focus
is on the population aging in relatively advanced economies, the survival rate is
100% from children to parents and is
from parents to grandparents.
A parent of generation t decides his current consumption
c
t
, his retirement con-
sumption
d
t+1
, number of identical children
n
t
,
children’s human capital level
h
t+1
, and the amount of time in working when old
. His expected life-
time utility is
( )
1 12 1 3 1
ln ln ln ln 1
t t t tt t
u c p d nh p l
σσ σ
++ +
=+ + +−
(3)
where
and
denote the tastes toward the retirement consumption,
children, and leisure time when old, respectively. A child of generation
t
+ 1
ob-
tains education investment
e
t
from his parent and spends all his childhood in
learning. He accumulates human capital according to
.
B
denotes
the productivity of learning process and
the elasticity of human cap-
ital accumulation with respect to education investment. The conventional as-
sumptions of diminishing returns to scale in
e
t
and constant returns to scale in
both et and
h
t
are applied here.
A working parent with 1 unit of time endowment considers sacrificing
δ
frac-
tion of time raising each child, earning labor wage at
w
t
, paying social security
tax at
, saving for his own old age consumption, and investing child-
ren’s education. His budget constraints are
(
)( )
11
t t t tt t tt
S n wh c en
τδ
=− − −−
(4)
( )
( )
1
1 11 1 1
1
1
tt
t tt t t t
rs
d wlh M
p
τ
+
+ ++ + +
+
= + −+
(5)
where
denotes the amount of social security benefit and working parents
participate in the actuarially fair type of saving.
This working parent maximizes Equation (3) subject to (4) and (5). His op-
timal decisions are
( )
( ) (
)
( )(
)
1 11 1
1 12 3
11 1
11
t t tt t t t t
t
t
r whp whpM
c
rp p
ττ
σσ σ
+ ++ +
+
+ − ++ +
=
+ + ++
(6)
DOI: 10.4236/tel.2018.815214 3483 Theoretical Economics Letters
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