ECON 401
Hartman
Autumn 2010
SECOND EXAMINATION
with answers
1.
Consider a SolowSwan model with labor augmenting technical change.
The savings rate,
s
, is
fixed.
The production function is
1/2
1/2
8
(
)
Y
K
AL
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
/
L
L
n
.
Labor augmenting technical change
occurs at the rate
g
so that
/
A
A
g
.
There is no government or international trade, and
therefore we must have
Y
C
I
.
a.
Let
/ (
)
k
K
AL
denote capital per unit of effective labor, and let
/
k
dk
dt
.
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
8
8
Y
K
AL
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
s k
k
AL
.
Now,
2
(
)
(
)
(
)
KAL
LA
AL K
K
L
A
K
K
k
n
g k
AL
AL
L
A
AL
AL
so that
(
)
K
k
n
g k
AL
,
and therefore
1/2
8
(
)
s k
k
n
g g
k
so that
1/2
8
(
)
k
s k
n
g
k
.
b.
Suppose that
0.03
n
,
0.12
,
0.05
g
, and
0.25
s
.
What is the steady state value
of
/ (
)
k
K
AL
?
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
2
0.20
k
k
.
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
y
k
and consumption per unit of effective labor is
*
*
*
1/2
1/2
(1
)
(.75)8(
)
(.75)8(100)
60
c
s 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
Y
K
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
L
L
.
There is
no government or international trade, and therefore we must have
Y
C
I
.
Denote aggregate
consumption by
C
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Staff
 Macroeconomics, Steady State, Stock and flow

Click to edit the document details