ECON 401
Hartman
Winter 2009
SECOND EXAMINATION
1.
Consider an economy with the production function
1/2
6
YN
K
where
Y
,
N
, and
K
are aggregate output,
the aggregate labor input, and the aggregate capital input, respectively.
Each person in the economy
provides one unit of labor services per period, and therefore
N
also represents the number of people in this
economy.
Assume that the population grows at the constant proportional rate
n
so that
/
NN n
, and that
capital depreciates at the rate
.
There is no government, and the savings rate,
s
, is given exogenously.
a.
Let
/
kKN
denote capital per person.
Find the function giving output per person in terms of
k
;
i
.
e
., find the function
( )
y
fk
where
/
y YN
is output per person.
[
Answer:
1/2
6
y
k
.]
b.
Develop the expression giving
/
kd
t
in terms of
k
,
s
,
, and
n
for this economy.
[
Answer:
1/2
() (
)
6
(
)
k
s
f
knk
s
k
nk
.
]
c.
If
0.1
s
,
0.15
, and
0.05
n
, what is the steady state value of
k
?
[
Answer:
9
ss
k
.
]
d.
Suppose again that
0.15
and
0.05
n
.
What is the golden rule value of
k
?
[
Answer:
225
GR
k
.
]
e.
What savings rate,
s
, gives rise to the steady state value of
k
corresponding to the golden rule?
[
Answer:
0.5
s
.
]
2.
Consider the problem of maximizing
0
()
bt
eu
c
d
t
where
c
is the control variable, and the state variable,
k
,
evolves according to
/2
t k
c k
where
is a constant and
kk
0
when
t
0
.
a.
Find the differential equation relating
/
cd
t
to
k
and
c
that must be satisfied for an optimum.
(That is, find the Euler equation for this problem.
You may use the calculus of variations, or the
regular Hamiltonian, or the current value Hamiltonian.)
[
Answer:
1/2
(
)
ucb
k
c
uc
.
]
b.
What is the transversality condition for this problem?
(You may ignore the constraint
( )
0
kt
.)
[
Answer:
lim
( )
0
bt
t
euc
]
c.
Suppose that
c
() l
n
where ln denotes the natural logarithm and assume that
1/10
b
.
Use the answer to part (a) to find the values of
k
and
c
in the steady state corresponding to an
optimal trajectory.
Explain why this steady state satisfies the transversality condition.
[
Answer:
25
ss
k
and
7.5
ss
c
.
Since
ss
cc
and
() 1
/
c
,
lim
( )
(1/ 7.5)lim
0
bt
bt
tt
e
.
]
3.
In class we derived the investment demand function of a firm which faces adjustment costs for investment
and has a production function with constant returns to scale.
This problem uses a phase diagram to work
out the qualitative implications of adjustment costs when the production function has decreasing returns to
scale.
For simplicity assume that capital is the only input to production.