Civil Engineering Reference
In-Depth Information
economy, there are infinite homogeneous consumers whose total measure is 1.
Representative consumer chooses the per capita consumption and per capita labor
l
t
(
s
t
)
to maximize this discounted utility
∞
t
=
0
s
t
ρ
t
s
t
s
t
s
t
π
t
(
)
u
(
c
t
(
)
,
l
t
(
))
N
t
.
Budget constraint consumers faced
s
t
s
t
s
t
s
t
s
t
s
t
s
t
s
t
−
1
s
t
C
t
(
)+[
1
+
τ
xt
(
)]
X
t
(
)=[
1
−
τ
lt
(
)]
ω
t
(
)
L
t
(
)+
r
t
(
)
K
t
(
)+
T
t
(
)
.
Capital transfer equation
s
t
s
t
−
1
s
t
(
)=(
−
δ
)
(
)+
(
)
.
K
t
+
1
1
K
t
X
t
s
t
s
t
s
t
s
t
−
1
(
)
,
(
)
,
(
)
,
(
)
C
t
respectively represent total consumption in the econ-
omy, total investment, the total sum payment-type tax, and the total capital stock.
ω
X
t
T
t
K
t
s
t
s
t
(
)
,
(
)
,
ρ
,
δ
,
,
g
represent wage rate, rental rate of capital, discount rate, the
depreciation rate of capital, the total population, and total population growth.
The enterprise's profit maximization problem
r
t
N
t
t
s
t
s
t
−
1
s
t
s
t
s
t
−
1
s
t
s
t
(
)
(
(
)
,
(
))
−
(
)
(
)
−
ω
(
)
(
)
.
max
A
t
F
K
t
Z
t
L
t
r
t
K
t
L
t
s
t
s
t
t
{
K
t
(
)
,
L
t
(
)
}
s
t
s
t
(
)
,
(
)
Among which
Y
t
G
t
represents the total social output and exogenous de-
mand.
2.1
Model Solution
2.1.1
Smooth Conversion of the Model
Because of exogenous technological progress and population growth, the model
variables will grow unboundedly. So we define variables without trend:
c
t
=
C
t
/
(
Z
t
N
t
)
,
x
t
=
X
t
/
(
Z
t
N
t
)
,
k
t
=
K
t
/
(
Z
t
N
t
)
,
t
t
=
T
t
/
(
Z
t
N
t
)
,
g
t
=
G
t
/
(
Z
t
N
t
)
,
y
t
=
Y
t
/
(
Z
t
N
t
)
.
The capital transfer equation can be described as follows using variables without
trend:
(
1
+
g
n
)(
1
+
g
z
)
k
t
+
1
=(
1
−
δ
)
k
t
+
x
t
.
The consumer's objective function can be changed as
∞
t
=
0
s
t
β
t
s
t
s
t
s
t
max
π
t
(
)
U
(
c
t
(
)
,
l
t
(
))
.
(1)
Budget constraint consumers faced changes to
s
t
s
t
s
t
s
t
s
t
s
t
s
t
s
t
−
1
s
t
(
)+[
+
τ
(
)]
(
)=[
−
τ
lt
(
)]
ω
(
)
(
)+
(
)
(
)+
(
)
.
c
t
1
x
t
1
L
t
r
t
k
t
t
t
xt
t
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