University of Minnesota
Department of Economics
Econ 3102: Intermediate Macroeconomics
Problem Set 1 Answer Key
1
Problems (65 points)
Exercise 1 (10 points)
Consider the utility function
U
(
C, ‘
) =
γlogC
+ (1

γ
)
log‘
.
(a) Derive the marginal utility of consumption. Is it decreasing in C?
Answer:
The marginal utility of consumption is:
U
C
(
C, ‘
) =
γ
C
(b) Derive the marginal utility of leisure. Is it decreasing in leisure?
Answer:
The marginal utility of leisure is:
U
‘
(
C, ‘
) =
1

γ
‘
(c) Does this utility function satisfy the Inada Conditions? Show your answer.
Answer:
Yes. Given the marginal utilities, it is straightforward to verify that
U
C
(0
, ‘
)
=
+
∞
U
‘
(
C,
0)
=
+
∞
,
Exercise 2 (15 points)
Consider a representative consumer whose preferences over consumption and leisure are given by the utility
function
U
(
C, l
) =
C
1
/
3
l
2
/
3
. Assume that she has a total of
h
= 18 hours which she can use for leisure or
she can work for the wage rate
w
= 6. Finally, assume that she enjoys a dividend income
π
= 36 and has to
pay taxes
T
= 24.
Determine the optimal consumption bundle (
C
*
, ‘
*
).
Answer:
You can do this in a variety of ways; here I’ll use the MRS
=
w
trick.
Since the MRS equals
the ratio of the marginal utilities, we get that
MU
‘
/MU
C
=
w
; given that the marginal utilities are just the
partial derivatives of the utility function, we rearrange and get
U
‘
(
C, ‘
)

wU
C
(
C, ‘
) = 0
or
2
3
C
1
/
3
‘

1
/
3
= 6
1
3
C

2
/
3
‘
2
/
3
.
(1.1)
Multiply across by
‘

1
/
3
C

2
/
3
to get
C
= 3
‘.
(1.2)
Now plug (2) in the budget constraint:
3
‘
= 6(18

‘
) + 36

24
.
(1.3)
Working this out, we get
‘
*
= 13
.
33
. Since
C
= 3
‘
, we get
C
*
= (3)(13
.
33) = 40
.
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 Summer '10
 ECON
 Economics, Macroeconomics, Utility, optimal consumption

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