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Answers to Homework #3
We have seen that when the population is growing, the equation of motion of capital
becomes:
(
29
(
29
k
n
k
f
s
k
⋅
+

⋅
=
∆
δ
,
where
s
is the saving rate,
n
is the rate of growth of population, and
is the depreciation
rate. Now suppose that the aggregate production function is given by:
(
29
3
/
2
3
/
1
,
L
K
L
K
F
Y
=
=
,
and let
y
denote output per worker, and
k
denote capital per worker.
a. (10 pts.)
Using algebra (show your steps
) and the aggregate production function, show
that the production function per worker in this case is given by:
(
29
3
/
1
k
k
f
y
=
=
Ans:
To get the production function per worker we need to divide the aggregate
production function with the number of workers. Doing so yields:
⇒
=
⇒
=
⇒
=
⇒
=
⇒
=


3
/
1
3
/
1
3
/
1
3
/
1
3
/
1
1
3
/
2
3
/
1
3
/
2
3
/
1
L
K
y
L
K
y
L
K
y
L
K
y
L
L
K
L
Y
3
/
1
k
y
=
⇒
b) (10 pts.)
Using algebra (show your steps
) and the equation of motion of capital, show
that the level of capital per worker at the steadystate in this case is given by:
2
/
3
*
+
=
n
s
k
Ans:
We know that at the steady state capital per worker is constant so that
0
=
∆
k
. This
is the
steadystate condition
. According to the equation of motion of capital, the change
in capital is the difference between actual investment and breakeven investment, so that:
(
29
(
29
k
n
k
f
s
k
⋅
+

⋅
=
∆
Using the equation of motion and the steadystate condition we get:
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29
(
29
(
29
⇒
⋅
+
=
⇒
⋅
+
=
⋅
⇒
⋅
+

⋅
=
3
/
1
3
/
1
3
/
1
*
*
*
*
*
*
0
k
k
n
s
k
n
k
s
k
n
k
s
δ
(
29
(
29
⇒
=
+
⇒
⋅
+
=
⇒
⋅
+
=
⇒

3
/
2
3
/
2
3
/
1
1
*
*
*
k
n
s
k
n
s
k
n
s
(
29
2
/
3
2
/
3
3
/
2
2
/
3
*
*
+
=
⇒
=
+
n
s
k
k
n
s
.
c. (10 pts.)
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 Spring '08
 Alexandrakis

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