ECO3011 Intermediate Microeconomic Theory: Problem Set 3
zezzzz
Instructor: Jieshuang He
Total points: 100
Due Date: Mar 19 (Tuesday) 5pm
Please show detailed steps for ALL the questions.
Academic Integrity: Copying is NOT allowed. If your answers are from collaborated work with other
classmates, please cite properly. Identical answers without citation are considered as cheating. For more
details, please refer to the course syllabus.
1. (15 pts) The government of Metropolis is considering building a bridge. There is just one size, and it
will either be built or not be built. It will cost 1 million if built, and nothing if not built. Assume
quasi-linear preferences for the local citizens. Local econometricians have estimated a market demand
curve for the bridge, given by X(p) = 1, 000, 000 200, 000p, where X(p) is the aggregate time of
usage of the bridge and p is the price charged by the government for use of the bridge, per unit time.
Building the bridge is socially worthwhile if and only if the net social benefit is non-negative.
(a) (4 pts) Should the bridge be built? Explain.
(b) (3 pts) If the government decided to charge the usage fee to just cover the cost of building the
bridge, what is the minimum value of p should be charged? Under the price, calculate the net
social benefit. Save your final results to 3 decimal points.
(c) (4 pts) Find the price that would maximize government revenue from the bridge, pX(p). If the
government chooses the price that maximizes revenue, what is the net social benefit?
(d) (4 pts) Find a formula for the net social benefit. At what price will the net social benefit be
maximized?
2. (20 pts) Lolita, an intelligent and charming Holstein cow, consumes only two goods, cow feed (made
of ground corn and oats) and hay. Her preferences are represented by the utility function U (x, y) =
x x2 /2 + y, where x is her consumption of cow feed and y is her consumption of hay. Lolita has been
instructed in the mysteries of budgets and optimization and always maximizes her utility subject to
her budget constraint. Lolita has an income of m that she is allowed to spend as she wishes on cow
feed and hay. The price of hay is always 1, and the price of cow feed will be denoted by p, where
0 < p  1.
(a) (5 pts) What type is Lolita’s utility function? Solve for Lolita’s inverse demand function for cow
feed.
(b) (3 pts) How much hay does Lolita choose?
(c) (4 pts) Suppose that Lolita’s daily income is 3 and that the price of feed is 0.50. What bundle
does she buy? What bundle would she buy if the price of cow feed rose to 1?
(d) (3 pts) How much money would Lolita be willing to pay to avoid having the price of cow feed rise
to 1?
(e) (3 pts) Suppose that the price of cow feed rose to 1. How much extra money would you have to
pay Lolita to make her as well-o↵ as she was at the old prices?
(f) (2 pts) At the price 0.50 and income 3, how much (net) consumer’s surplus is Lolita getting?
3. (24 pts) Suppose that electric dog polishers is demanded by dog breeders and pet owners. The demand
function of dog breeders for electric dog polishers is qb = max{200 p, 0}, and the demand function
of pet owners for electric dog polishers is qo = max{90 4p, 0}.
1
(a) (4 pts) At price p, what is the price elasticity of dog breeders’ demand for electric dog polishers?
What is the price elasticity of pet owners’ demand?
(b) (2 pts) At what price is the dog breeders’ elasticity equal to -1? At what price is the pet owners’
elasticity equal to -1?
(c) (6 pts) Draw in a graph the dog breeders’ demand curve as blue solid line, the pet owners’ demand
curve in red ink, and the market demand curve as a dashed line in pencil.
(d) (5 pts) Find a nonzero price at which there is positive total demand for dog polishers and at which
there is a kink in the demand curve. What is the market demand function for prices below the
kink? What is the market demand function for prices above the kink?
(e) (7 pts) Where on the market demand curve is the price elasticity equal to -1? At what price will
the revenue from the sale of electric dog polishers be maximized? If the goal of the sellers is to
maximize revenue, will electric dog polishers be sold to breeders only, to pet owners only, or to
both?
4. (28 pts) Governments in many cities around the world regulate the housing rent market by imposing
a rent control policy. In essence, this policy puts a price ceiling (below the equilibrium price) on the
amount of rent that landlords can charge in the apartment buildings a↵ected by the policy. Assume
for simplicity that income e↵ects do not play a significant role in the analysis of the housing market
throughout the following analyses.
(a) (2 pts) Draw a supply and demand graph with apartments on the horizontal axis and rents (i.e.
the monthly price of apartments) on the vertical. Illustrate the equilibrium under the rent control
policy.
(b) (2 pts) Suppose that the city government can easily identify those who get the most surplus from
getting an apartment. In the event of excess demand for apartments, the city then assigns the
right to live (at the rent-controlled price) in these apartments to those who get the most consumer
surplus. Illustrate the resulting consumer and producer surplus as well as the deadweight loss
from the policy in the above graph you draw.
(c) (3 pts) Now, suppose the city government cannot easily identify how much consumer surplus any
individual gets — and therefore cannot match people to apartments as in (b). So instead, the
mayor develops a “pay-to-play” system under which only those who pay monthly bribes to the
city will get to “play” in a rent controlled apartment. Assuming the mayor sets the required bribe
at just the right level to get all apartments rented out, illustrate the size of the monthly bribe in
the above graph you draw. Explain.
(d) (4 pts) Will the identity of those who live in rent-controlled apartments be di↵erent in (c) than
in (b)? Will consumer or producer surplus be di↵erent? What about deadweight loss?
(e) (3 pts) Next, suppose that the way rent-controlled apartments are allocated is through a lottery.
Whoever wants to rent a rent-controlled apartment can enter his/her name in the lottery, and the
mayor picks randomly as many names as there are apartments. Suppose the winners can sell their
right to live in a rent-controlled apartment to anyone who agrees to buy that right at whatever
price they can agree on. Who do you think will end up living in the rent-controlled apartments
(compared to who lived there under the previous policies)? Explain.
(f) (6 pts) The winners in the lottery in part (e) in essence become the suppliers of “rights” to
rent-controlled apartments while those that did not win in the lottery become the demanders.
Imagine that selling your right to an apartment means agreeing to give up your right to occupy
the apartment in exchange for a monthly check q. Draw a supply and demand graph in this
market for “apartment rights” and relate the equilibrium point to your previous graph of the
apartment market. Explain.
(g) (5 pts) What will be the equilibrium monthly price q ⇤ of a “right” to live in one of these apartments
compared to the bribe charged in (c)? What will be the deadweight loss in your original graph
of the apartment market? How does your answer change if lottery winners are not allowed to sell
their rights?
(h) (3 pts) Finally, suppose that instead the apartments are allocated by having people wait in line.
Who will get the apartments and what will deadweight loss be now? (Assume that everyone has
the same value of time.)
2
5. (13 pts) Suppose guns are traded in a perfectly competitive market. The supply and demand functions
are given by
D(p) = 40 p,
S(p) = 10 + p,
where p is the price in dollars.
(a) (5 pts) Solve for the equilibrium price and quantity sold for guns. Suppose that the government
decides to restrict the industry to selling only 20 guns. At what price would 20 guns be demanded?
How many guns would suppliers supply at that price? At what price would the suppliers supply
only 20 guns?
(b) (3 pts) The government wants to make sure that only 20 guns are bought, but it doesn’t want the
firms in the industry to receive more than the minimum price that it would take to have them
supply 20 guns. One way to do this is for the government to issue 20 ration coupons. Then in
order to buy a gun, a consumer would need to present a ration coupon along with the necessary
amount of money to pay for the good. If the ration coupons were freely bought and sold on the
open market, what would be the equilibrium price of these coupons?
(c) (5 pts) Draw a graph to show the area that represents the deadweight loss from restricting the
supply of guns to 20. How much is this expressed in dollars?
3