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ECON 401
Hartman
Autumn 2010
ANSWERS FOR PROBLEM SET VII
1.
Substitute for
t
I
to see that we want to maximize
1
0
() [
(
1)]
(1
)
ttt
t
t
t
t
pfK
q K
K
Z
r
222
3
2
111
2
1
000
1
0
2
()
[
(
(
(
)
[
)
]
...
1(
1
)
K
K
K
rr
1
1
1
[
(
1
)]
(
)[
(
1
)
]
...
)
)
t
t
t
t
t
t
t
tt
K
p
fK qK
K
. Now,
0
K
was
determined in the past, but we are free to choose all future
K
s starting with
1
K
.
For any of those
future
K
s, say
t
K
, the first order condition is
1
1
() (
1 )
0
)
)
t
t
t
qp
f
K
q
Z
Kr
r
.
This
condition can be rewritten as
1
(
1 ) (
t
pf K
q
r q
.
The user cost of capital for period
t
is
11
1
)
)
(
)
t
t
t
t
qr
q
q
r
q
q
q
; the first term after the equality is foregone interest,
the second term is the cost of depreciation over the period, and the last term is the net capital gain
which is subtracted.
2.a.
Substitute for
I
to see that we want to maximize
1/3
0
[3
(
)
]
rt
ep
K
q
KK
d
t
.
If we let
1/3
(, , )
[ 3
(
rt
GtKK
e
p K
qK
K
, then
2/3
/(
)
rt
GKe p
K
q
,
/
rt
GK eq
, and
rt
rt
dG
re
q
e
q
dt
K
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This note was uploaded on 04/04/2011 for the course ECON 401 taught by Professor Staff during the Spring '08 term at University of Washington.
 Spring '08
 Staff
 Macroeconomics

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