Kevin Corinth
University of Chicago
Economics 20000
Spring 2012
Problem Set 6
Equilibrium and Externalities
Due Monday, May 14 at 7:00pm
1. Suppose there is an economy with two consumers (A and B) and two goods (good 1
and good 2); each consumer is endowed with 5 units of good 1 and 5 units of good 2.
For each set of consumer preferences, what is the Pareto set (depict it in an Edgeworth
Box or write it mathematically)?
(a) For both consumers, the two goods are perfect complements (in 1 for 1 proportions).
(b) For both consumers, the two goods are perfect substitutes (in 1 for 1 proportions).
(c) uA (x1 , x2 ) = x1 x2 and uB (x1 , x2 ) = x1 x2 .
(d) uA (x1 , x2 ) = x1 x2 and uB (x1 , x2 ) = −x1 x2 .
(e) Each consumer strictly prefers the bundle (0, 0) to all other bundles, and is indifferent between all non-zero bundles.
2. Suppose there is an economy with two consumers, Anson and Brandon, and two
commodities, pounds of food, x1 , and gallons of water, x2 . Anson’s preferences
are represented by the utility function uA (x1 , x2 ) = x1 x2 , and his initial endowment is wA = (18, 4). Brandon’s preferences are represented by the utility function
uB (x1 , x2 ) = ln x1 + x2 , and his initial endowment is wB = (3, 6). Let p1 and p2 denote
the prices of food and water, respectively.
(a) What is the total amount of food and water available to trade in this economy?
(b) What is the value of Anson’s endowment? What is the value of Brandon’s endowment?
(c) Set up Anson and Brandon’s utility maximization problems.
(d) Find Anson and Brandon’s demands for food and water? Use x1A (p1 , p2 ) and
x2A (p1 , p2 ) to denote Anson’s demands, and use x1B (p1 , p2 ) and x2B (p1 , p2 ) to denote Brandon’s demands.
(e) Find a Walrasian equilibrium [Hint: remember only the relative prices matter, so
feel free to set p∗1 = 1].
(f) Draw an Edgeworth Box that depicts the Walrasian equilibrium. Include the
endowment allocation, the allocation actually consumed, the budget constraint
that depicts the equilibrium prices, and sketches of the indifference curves for each
person containing the endowment allocation, and sketches of the indifference curves
for each person containing the allocation actually consumed.
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(g) Verify that the consumed allocation under the Walrasian equilibrium is Pareto
efficient. [Hint: What has to be true about each person’s indifference curves at a
Pareto efficient allocation?] Which theorem tells us that this must be true?
3. Suppose the only thing each person in the United States cares about is how much
wealth s/he has, and each person always wants more wealth no matter how much s/he
already has.
(a) Is the endowment allocation Pareto efficient?
(b) What is the Pareto set (the set of all allocations which are Pareto efficient)?
(c) Should the government redistribute wealth?
4. (a) It is sometimes said that the job of the economist is to find a “free lunch,” that
is, to make some people better off while making nobody worse off. Which welfare
theorem works against the economist?
(b) Economists are sometimes consulted to find a way to create markets to achieve
socially desirable outcomes. Which welfare theorem supports this work?
5. Allison is chronically homeless, has been diagnosed with schizophrenia, and prefers to
live in Uptown (sleeping in the park and spending her days in the public library). Bill
also lives in Uptown, but he prefers not to have to see Allison. Allison’s preferences
over money, x1 , and hours per day she spends in Uptown, x2 , are represented by the
√
utility function uA (x1 , x2 ) = x1 + x2 . Bill’s preferences over money, y1 , and hours
per day Allison spends outside of Uptown, y2 , are represented by the utility function
√
uB (y1 , y2 ) = y1 + y2 . Current Uptown law gives Allison the right to spend as much
time in Uptown as she wants per 24 hour day. Allison has a daily income of $0 and
Bill has a daily income of $100.
(a) Draw an Edgeworth Box which represents this economy. Label the graph with
each consumer, all four axes, the endowment allocation and the indifference curves
for each consumer containing their endowment bundle.
(b) Is it Pareto efficient for Allison to spend all of her time in Uptown? Lightly shade
the region on your Edgeworth Box which includes all allocations which would make
both Allison and Bill better off than the endowment allocation.
(c) An economist suggests creating a market so that Bill doesn’t have to face Allison
in person when paying her off to not have to see her. What is the equilibrium price
per hour for Bill not having to see Allison? How many hours does Allison spend
in Uptown? How much money does Allison end up with? Draw the budget line
and equilibrium allocation on your Edgeworth Box from part (a).
(d) While Bill appreciates not having to face Allison to pay her off, paying a daily
price still makes him feel a bit uneasy. The economist proposes a solution - just
pass a law that forbids Allison to be in Uptown unless she pays an hourly fee to
spend time there. How much money should Bill donate to Allison per day that
would lead to the same allocation under the equilibrium you found above? What
will the hourly fee be?
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(e) Indicate the Pareto set on your Edgeworth Box from part (a). Should society
always strive to move to Pareto efficient allocations?
6. Think of a situation where there is an externality. Why do you think property rights
are not well-defined in this situation? Would society be better off if property rights
were well-defined? Explain.
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