Economics 314: Suggested Solutions to HW 5
TA: Yan Li
March 10, 2009
1
Case 1, Handout 2
From the budget constraint,
c
1
=
°
0
(
°
1
°
c
0
)
:
U
(
c
0
; c
1
) =
±c
0
+ (1
°
±
)
c
1
;
0
±
±
±
1
Plugging in the
c
1
from the budget constraint, we get a function of only
c
0
given by:
f
(
c
0
) = [
±
°
(1
°
±
)
°
0
]
c
0
+ (1
°
±
)
°
0
°
1
1. [(i)]
2. If
±
°
(1
°
±
)
°
0
>
, then consuming more
c
0
increases utility. So we
would want to consume the maximum possible
c
0
;
which is obtained
when we plug in
c
1
= 0
in the budget constraint. This gives
c
0
=
°
0
:
Solution:
(
c
0
; c
1
) = (
°
1
;
0)
3. If
±
°
(1
°
±
)
°
0
<
, consuming more
c
0
decreases utility.
So
c
0
= 0
:
This gives
c
1
=
°
0
°
1
:
Solution
(
c
0
; c
1
) = (0
; °
0
°
1
)
:
4. If
±
°
(1
°
±
)
°
0
=
, then any
(
c
0
; c
1
)
is optimal subject to the budget
constraint.
1
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2
Case 2, Handout 2
f
(
c
0
; c
1
) =
±
ln
c
0
+ (1
°
±
) ln(
°
0
:
(
°
1
°
c
0
))
Here because of log functions, we can rule out corner solutions, so
c
0
is
optimal if
f
0
(
c
0
) = 0
:
f
0
(
c
0
) = 0
=
)
±
c
0
°
1
°
±
°
1
°
c
0
= 0
Solving for the equation gives:
c
0
=
±°
1
=
)
c
1
= (1
°
±
)
°
0
°
1
3
Case 3, Handout 2
The equation for
f
0
(
c
0
) = 0
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 Spring '07
 MBIEKOP
 Economics, Macroeconomics, Utility, $10, Yan Li

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