Econ 3102: Intermediate Macroeconomics
Homework #1 Answer Key
1
Problems (65 points)
Exercise 1 (10 points)
First note that the utility function is unde°ned when
C
= 0
or
‘
= 0
, so we need only focus on the cases
in which
C >
0
and
‘ >
0
.
(a) Derive the marginal utility of consumption. Is it decreasing in
C
?
Answer: The marginal utility of consumption is:
U
C
(
C; ‘
)
=
dU
(
C; ‘
)
dC
=
°
C
>
0
U
CC
(
C; ‘
)
=
dU
C
(
C; ‘
)
dC
=
°
°
°
C
2
±
<
0
Since
U
CC
is strictly less than zero, we know that marginal utility is a decreasing function in
C
.
(b) Derive the marginal utility of leisure. Is it decreasing in leisure?
Answer: The marginal utility of leisure is:
U
‘
(
C; ‘
)
=
dU
(
C; ‘
)
d‘
=
1
°
°
‘
>
0
U
‘‘
(
C; ‘
)
=
dU
‘
(
C; ‘
)
d‘
=
°
²
1
°
°
‘
2
³
<
0
Since
U
‘‘
is strictly less than zero, we know that marginal utility is a decreasing function in
‘
.
(c) Does this utility function satisfy the Inada conditions? Show your answer.
Answer: Yes. Given the marginal utilities, it is straightforward to verify that
lim
C
!
0
U
C
(
C; ‘
)
=
lim
‘
!
0
U
‘
(
C; ‘
) = +
1
lim
C
!
+
1
U
C
(
C; ‘
)
=
lim
‘
!
+
1
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; ‘
) =
C
1
=
3
‘
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
±
= 36
and has to
pay taxes
T
= 24
.
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 Summer '10
 ECON
 Economics, Macroeconomics, Utility, optimal consumption

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