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Economics 102, Winter, 2010
Midterm Exam # 2 practice questions
Professor Lee Ohanian
(1) Show that in the Solow model in which population grows at rate n, that
the Golden Rule requires that the marginal product of capital be equal to the
sum of the depreciation rate and the growth rate of the population. Why is the
golden rule MPK higher for a higher population growth rate?
K
t
+1
=
I
t
+ (1
)
K
t
in percapita terms we have
K
t
+1
L
t
=
I
t
L
t
+ (1
)
K
t
L
t
This equation can be rewritte as:
K
t
+1
L
t
+1
L
t
+1
L
t
=
I
t
L
t
+ (1
)
K
t
L
t
in the new variables is:
k
t
+1
(1 +
n
) =
i
t
+ (1
)
k
t
In the steadystate (we drop the time subscripsts)
k
(
+
n
) =
i
Investment percapita be written as output minus consumption percapita
k
(
+
n
) =
y
c
Hence
c
=
y
k
(
+
n
)
The optimal condition for the golden rule is when we derive respect to capital
@c
@k
=
@y
@k
(
+
n
)
If we make the derivative equal to zero (which is the neccesary condition for a
maximum)
@y
@k
(
+
n
) = 0
MPK
= (
+
n
)
1
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View Full DocumentWhen
n
is higher, goldenMPK is higher due to shrinking capital stock
per worker, because of higher amount of investment necessary to provide higher
number of new workers with capital.
k
is spreaded thinly among a larger number
of workers
(2) Show that in the Solow model in which there is constant growth in worker
e¢ ciency that a steady state exists for the variable
Y=
(
NE
)
;
and that once the
steady state is reached that per capita output grows at rate
g:
Start with the production function
Y
=
K
(
NE
)
1
Divide through
NE
Y
NE
=
K
NE
±
NE
NE
±
1
y
=
k
where
y
=
Y
NE
and
k
=
K
NE
:
The law of motion of capital can be reexpressed as:
k
=
i
(
+
n
+
g
)
k
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 Spring '08
 Serra
 Economics

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