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ECON 1110 S2: Intermediate Microeconomics
Spring 2014
Professor: Geoffroy de Clippel
Problem Set 5 - Solutions
1. (100 points) Allison’s utility function is u(x1 , x2 ) = 2x1 x2 . She earns income of $10.
The price of good 1 is p1 = $ 21 and the price of good 2 is p2 = $1. Both goods are
perfectly divisible so that it is possible to consume a fraction of each.
(a) (10 points) Find Allison’s optimal choice. How much utility is associated with
this bundle? Call this bundle A.
These are Coob-Douglas preferences and the optimal choice is an interior solution.
The optimal choice of x1 and x2 must satisfy:
i. M RS = − pp12
ii. m = p1 x1 + p2 x2
In this case M RS =
x2
x1
and
p1
p2
= 21 .
The optimal choice is the bundle (10, 5) and the utility level associated with this
bundle is u(10, 5) = 100
(b) (10 points) Draw the budget line, the optimal choice A and the indifference curve
that goes through A
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(c) (10 points) Suppose the price of good 1 goes up to $1. Find Allison’s new optimal
choice. Call this bundle B. What is the new utility level associated with B? Draw
the budget line, the optimal choice B and the indifference curve that goes through
B.
From conditions i and ii we have:
B=(5,5); u(5, 5) = 50
(d) (20 points) How much of an income increase has to occur for Allison in order to
make her original bundle A just affordable at the new prices?
old
old
Incomenew = pnew
1 x1 + p2 x2 = 15
∆Income = 15 − 10 = 5
(e) (20 points) If we give Allison the extra income from above, what would her new
optimal choice be under the new prices? Call this choice C. Draw the budget line,
the optimal choice C and the indifference curve that goes through C.
Again, from conditions i and ii we have:
C=(7.5,7.5); u(7.5, 7.5) = 112.5
(f) (15 points) Calculate the substitution effect on good 1.
A
xC
1 − x1 = 7.5 − 10 = −2.5
(g) (15 points) Calculate the income effect on good 1.
C
xB
1 − x1 = 5 − 7.5 = −2.5
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