Economics 104A
Solution for Problem Set #4
Winter 2010
1. Let
u
(
x
1
, x
2
) = 2
√
x
1
+
x
2
be a consumer’s utility function.
Let the prices of
good 1 and good 2 and the consumer’s income be ¯
p
1
= 1, ¯
p
2
= 2, and ¯
m
= 100,
respectively.
a) What is the income effect caused by a quantity tax of
t
= 1 on good 1?
Explain your answer.
Answer:
Note first that the utility function implies
MRS
= 1
/
√
x
1
.
It
thus follows that the optimal quantity of good 1 at prices
p
1
and
p
2
is
determined by 1
/
√
x
1
=
p
1
/p
2
.
This shows that the optimal quantity of
good 1 is independent of the consumer’s income. Thus, the income effect of
a quantity tax of
t
= 1 on good 1 is 0. As mentioned above, the consumer’s
utility function implies that the MRS does not depend on good 2. Hence a
change in income only changes the optimal quantity of good 2 but not the
optimal quantity of good 1.
b) Use CV to determine how much in dollar terms the quantity tax hurts the
consumer.
Answer:
From the answer to part a, the optimal bundle at original prices
and income has quantity of good 1 equal to (
p
2
/p
1
)
2
= 4 and quantity of
good 2 equal to (
m/p
2
)

(
p
2
/p
1
) = 48. The utility level at this bundle is
52. Let
m
0
be the new income that the consumer would need to order to
achieve utility level 52 after the quantity tax has been imposed. Then,
m
0
satisfies
2
p
2
p
1
+
t
+
m
0
p
2

p
2
p
1
+
t
= 52
.
Solving the above equation, we get
m
0
= 102. Thus, CV =
m
0

m
= $2
.
c) Use EV to determine at most how much of the income would the consumer
be willing to give up to prevent the quantity tax, assuming he does not lie.
Answer:
After the quantity tax has been imposed, the optimal bundle for
the consumer has quantity of good 1 equal to (
p
2
/p
1
+
t
)
2
= 1 and quantity
1
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of good 2 equal to (
m/p
2
)

(
p
2
/p
+
t
) = 49. Thus the utility level at this
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 Winter '10
 Staff
 Supply And Demand, Utility, 5m, 7m, quantity tax, 8.75m

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