07 Jun Question Econ 106P: Problem Set 1 Yingju Ma Aug 11, 2015
Question
Econ 106P: Problem Set 1
Yingju Ma
Aug 11, 2015
(Due on Thursday, Aug 20, 10:45am.)
No Late Submission Accepted.
You are encouraged to work as teams with 2-3 members, but please submit your own solution.
On the top of your answer sheet please show your name, UID, and whom you work with.
1. Two-Part Tari
Consider a monopoly rm and two consumers 1 and 2. The consumers demand functions are
P1 = 12
q1 and P2 = 10
q2 and the monopolys cost function is C(q) = 2q. Assume that the
monopoly uses a two-part tari pricing scheme.
A. What is the monopolys prot-maximizing quantity if it only faces consumer 1? What is the
resulting prot?
B. Assume that the monopoly faces both consumers. What is its prot-maximizing two-part tari
scheme if it needs to ensure participation (i.e. both consumers will buy something)? What is
the resulting prot?
C. Assume now that the monopoly faces 5 consumers with the same demand function as consumer
1, and one consumer with the same demand function as consumer 2. Would the monopoly
rather exclude the second type of consumer (i.e. set a pricing scheme such that the second
type would not buy anything)?
2. Joint Cost Production
Suppose a monopoly rm produce two nal products with same raw materials, the same as the
example showed in Lecture 2. Lets assume product 1 becomes more popular due to a change of
consumers taste. Then the demand for product 1 is lager than before, say P1 = 300
demand of product 2 does not change, P2 = 100
q1 . The
q2 . Firm has the same cost function: C0 = 30q0 ,
C1 = 10q1 , C2 = 4q2 .
A. Solve for rms optimal production plan.
B. Find the production plan that maximize the social welfare.
1
C. Compare the rms solution you nd in Part A with the solution we have in the slides, try to
interpret the dierence of these two results, with economic intuition.
3. Vertical Market Separation with Competitive Manufacturer
Consider a vertically separated market with the distributor as a monopoly in the retail market, but
with the manufacturer as a competitive rm. The demand function of the retail market is P = 20 q,
and the manufacturers cost function is C(q) = 4q.
A. What is the demand function faced by the manufacturer?
B. What is the wholesale price?
C. How many units would the distributor purchase from the manufacturer?
4. Pirates of the Caribbean
100 pirates need to divide 5 gold coins among themselves. They agree to the following rule:
Starting with pirate 1, each pirate proposes a plan, then the 100 pirates take a vote (for/against)
If at least half of the (remaining) pirates vote for the plan, then the gold coins are divided
accordingly; otherwise, the proposer is executed
We assume that
Each pirate values his life most.
Being alive, he prefers more gold coins than less.
Being alive and having the same number of gold coins, he prefers to watch others die.
A. If it were pirate 100s turn to propose, what is his best action? Given this, what is pirate 99s
best proposal? How about pirate?
B. Which pirate is the rst such that, if it were his turn to propose, there is no way of staying alive?
C. Let n be the answer of question 2. Is there a way such that pirate n
1 can stay alive?
D. How many pirates would die in the end?
5. Cournot Competition with Entry Cost
Consider a market with demand function P = 20
Q. There are 30 potential competing rms with
identical cost functions C(q) = q, but to enter the market they each need to pay an entry cost of 2.
In particular, each rm rst decides whether it will pay the cost to enter the market; no rm can
see others decisions in this phase. Then the rms who enter the market play Cournot competition.
A. If there are two rms in the market, what are their prots after taking into account the entry
cost?
2
B. Would it be protable for a third rm to pay the cost to enter the market?
C. How many rms would be in the market in the SPNE?
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