Intermediate Microeconomics, 2008
Problem Set No 6
due: Thursday, Jan 31;
Wed, Jan 30
Problems
1) We considered preferences represented by the following utility function in
the last homework:
U
(
q
1
; q
2
) =
q
1
q
2
.
Income is °xed at
Y
= 2
and the price of good two is °xed at
p
2
= 1
.
a) Suppose
p
1
increases.
Illustrate graphically how the change in demand
can be decomposed into a substitution e/ect and an income e/ect.
b) Suppose
p
1
increases from
1
4
to
1
2
. Derive the total change of the demand
for good one. Decompose that change into the change due to the substitution
e/ect and the change due to the income e/ect.
Note:
The total change in
demand is
D
1
°
1
2
; Y
±
°
D
1
°
1
4
; Y
±
at
Y
= 2
. The change due to the substitution
e/ect is
H
1
°
1
2
;
°
U
0
±
°
H
1
°
1
4
;
°
U
0
±
. The change due to the income e/ect is the
change in demand
D
1
°
1
2
; Y
±
when income changes from the hypothetical level
Y
=
E
°
1
2
;
°
U
0
±
to
Y
=
E
°
1
2
;
°
U
N
±
= 2
.
°
U
0
and
°
U
N
are the utility levels at the
old and the new price,
p
0
1
=
1
4
and
p
N
1
=
1
2
, respectively.
c) Suppose
p
1
increases from
1
4
to
4
. Again, decompose the change of demand
into the change that is due to the substitution e/ect and the change that is due
to the income e/ect.
d) Calculate the elasticity of demand at
p
1
=
1
4
. Decompose this elasticity
into the Hicks elasticity
"
°
and the weighted income elasticity
°±
.
3) Let
p
2
= 1
. Consider the following utility functions:
U
(
q
1
; q
2
)
=
min
f
q
1
; q
2
g
U
(
q
1
; q
2
)
=
q
1
+
q
2
U
(
q
1
; q
2
)
=
q
2
+
p
q
1
a) Illustrate the indi/erence curves graphically and illustrate how a price
change a/ects demand.
Carefully illustrating the indi/erence curves will be
helpful. You might want to check earlier problem sets when graphing the indif
ference curves.
b) Illustrate how a change in income changes demand.
1
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c) Derive the expenditure function
E
(
p
1
; U
)
and the Hicks demand curve
for good one,
H
1
(
p
1
; U
)
for
U
= 4
and for general
U
.
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 Winter '08
 KUHN
 Supply And Demand, Utility, tax revenues, Consumer price index, Hicks

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