ECON 401
Hartman
Autumn 2010
SECOND EXAMINATION
with answers
1.
Consider a Solow-Swan model with labor augmenting technical change.
The savings rate,
s
, is
fixed.
The production function is
1/2
1/2
8(
)
YKA
L
where
Y
is output,
K
is the capital stock,
L
is the labor input, and
A
is the level of the labor augmenting technical change parameter.
Capital accumulates according to
K
I
K
where
I
is gross investment and
is the
depreciation rate.
The population grows at the constant proportional rate
n
, and each person
provides one unit of labor services so that
/
LL n
.
Labor augmenting technical change
occurs at the rate
g
so that
/
AA g
.
There is no government or international trade, and
therefore we must have
YCI
.
a.
Let
/ (
)
kKA
L
denote capital per unit of effective labor, and let
/
kd
t
.
Derive
the equation that gives
k
in terms of
k
and the parameters of the model.
(To receive
full credit, you must show the steps required to derive the equation.)
Answer:
Note first that
1/2
1/2
1/2
88
YK
A
L
y
k
AL
AL
AL
.
Since saving must equal
investment in this model, we must have Y
C
sY
K
K
which, after dividing
through by AL, becomes
sY
K
K
AL
AL
AL
or
1/2
8
K
sk
k
AL
.
Now,
2
()
KAL
LA
AL K
K
L
A
K
K
kn
g
k
AL
AL
L
A AL
AL
so that
K
g
k
AL
,
and therefore
1/2
)
k n gg
k
so that
1/2
)
ks
k
ng k
.
b.
Suppose that
0.03
n
,
0.12
,
0.05
g
, and
0.25
s
.
What is the steady state value
of
/(
)
L
?
What is output per unit of effective labor in the steady state?
What is
consumption per unit of effective labor in the steady state?
Answer:
Let
0
k
and substitute for n, g,
, and s to see that the steady state value of
k satisfies
1/2
20
.
2
0
kk
.
It follows that the steady state value of k is
*2
2
(2 / 0.2)
(10)
100
k
.
Output per unit of effective labor is
**
1
/
2
1
/
2
8(
)
8(100)
80
yk
and consumption per unit of effective labor is
*
1
/
2
1
/
2
(1
)
(.75)8(
)
(.75)8(100)
60
cs
y
k
.
c.
Does output per person grow in the long run?
If so, at what rate?
Answer:
Yes, Y/L does grow in the long run.
It grows at the same rate as A, which is
0.05 or 5% per period.
2.
Suppose the production function is
1/2 1/2
8
L
where
Y
is aggregate output,
K
is the capital
stock, and
L
is the input of labor.
Capital evolves according to
K
I
K
where
I
denotes
gross investment and
0.08
.
The population grows at the constant proportional rate of 2%
per period, and each person provides one unit of labor services so that
/
0.02
LL
.
There is
no government or international trade, and therefore we must have
.
Denote aggregate
consumption by
C
, and consumption per person by
/
cCL
.
There is no technical change.