1 Eco 600E Advanced Microeconomics II
Term: Spring (2nd), 2009 Lecturer: Yosuke Yasuda
Problem Set 2 Due in class on July 23
1. Question 1 (10 points)
Consider the following two persons 3 x 3 game.
1 / 2 X Y Z
A 5,5 8,4 0,0
B 4,8 7,7 1,9
C 0,0 9,1 0,0
a) Find all the pure‐strategy Nash equilibria of this game.
b) Consider the two‐period repeated game in which the above stage game will be played twice. Suppose the payoffs are simply the sum of the payoffs in each stage game. Then, is there a subgame perfect Nash equilibrium that can achieve (B, Y) in the first period? If so, describe the equilibrium. If not, explain why.
2. Question 2 (25 points)
Two firms produce an identical good. The inverse demand curve for the good is P = 121
‐ X, where X is the total quantity produced by the two firms. Each firm has a constant marginal cost 1 of producing the good.
a) Suppose each firm i produces and sells xi units of the good. Write down an expression for firm i’s profit (as a function of the output of each firm).
b) Suppose that firms compete as quantity setting duopolists. Find the Cournot Nash equilibrium of this game. What quantities will they produce, what is the market price and how much profit does each firm earn?
c) Suppose that firm 1 decides how much to produce first; firm 2 chooses only after observing firm 1's choice. Find the subgame perfect Nash equilibrium (also called
“Stackelberg equilibrium”) of this game. What quantities will they produce, what is the market price and how much profit does each firm earn?
d) Suppose the firms formed a cartel: each firm produced the same output and
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maximized their joint profits. What quantity would each firm produce? What would be the market price? What would be the profit of each firm?
e) Now suppose that the firms interact indefinitely through time. The inverse demand curve in each period is given by P = 121 ‐ X, and firms discount future profits at a discount factor δ. For what value of δ is there an equilibrium where firms follow the
“trigger strategies” discussed in class, i.e., they produce cartel output as long as the other firm has always produced cartel output and otherwise they produce Cournot Nash output?
3. Question 3 (15 points)
Each of two individuals receives a ticket on which there is an integer from 1 to 10 indicating the size of a prize ($) she may receive. Assume the payoff of receiving the prize $X is X. The individuals’ tickets are assigned randomly and independently; the probability of an individual receiving each possible number is 0.1. Each individual is given the option of exchanging her prize for the other individual’s prize; the individuals are given this option simultaneously. If both individuals wish to exchange, then the prizes are exchanged; otherwise each individual receives her own prize. Each individual’s objective is to maximize her expected monetary payoff.
a) Consider the above situation as a Bayesian game. Then, what are the individuals’ strategies?
b) If an individual receives the ticket with $10, will she have an incentive to exchange or not? Explain why.
c) Solve for the Bayesian Nash equilibrium. Can the exchange happen in equilibrium?
4. Question 5 (20 points)
Suppose a government auctions one block of radio spectrum to two risk neutral mobile phone companies, i = 1,2. The companies submit bids simultaneously, and the company with higher bid receives a spectrum block. The loser pays nothing while the winner pays a weighted average of the two bids:
θb + (1 ‐ θ)b’
where b is the winner’s bid, b’ is the loser’s bid, and θ is some constant satisfying 0 ≤ θ
≤ 1. (In case of ties, each company wins with equal probability.) Assume the valuation of the spectrum block for each company is independently and uniformly distributed between 0 and 1.
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a) Solve a Bayesian Nash equilibrium.
Hint: You can assume the equilibrium strategy is symmetric and linear, i.e., bi = αvi for i = 1,2.
b) Show that bi = vi is always (weakly) better than bi = 0.5vi for i when θ=0.
5. Question 5 (30 points)
Consider a game between two friends, Amy and Brenda. Amy wants Brenda to give her a ride to the mall. Brenda has no interest in going to the mall unless her favorite shoes are on sale (S) at the large department store there. Amy likes these shoes as well, but she wants to go to the mall even if the shoes are not on sale (N). Only Amy subscribes to the newspaper, which carries a daily advertisement of the department store. The advertisement lists all items that are on sale, so Amy learns whether or not the shoes are on sale. Amy can prove whether or not the shoes are on sale by showing the newspaper to Brenda. But this is costly for Amy, because she will have to take the newspaper away from her sister, who will yell her later for doing so.
In this game, nature first decides whether or not the shoes are on sale, and this information is made known to Amy. That is, Amy observes whether nature chose S or N. Nature chooses S with probability p and N with probability 1 ‐ p. Then Amy decides whether or not to take the newspaper to Brenda (T or D). If she takes the newspaper to Brenda, then it reveals to Brenda whether the shoes are on sale. In any case, Brenda must then decide whether to take Amy to the mall (Y) or to forget it (F). If the shoes are on sale, then going to the mall is worth 1 unit of utility to Brenda and 3 to Amy. If the shoes are not on sale, then traveling to the mall is worth 1 to Amy and ‐1 to Brenda. Both players obtain 0 utility when they do not go to the mall. Amy's personal cost of taking the newspaper to Brenda is 2 units of utility, which is subtracted from her other utility amounts.
a) Draw the game tree of this game.
b) Does this game have a separating perfect Bayesian Nash equilibrium? If so, fully describe it.
c) Does this game have a pooling perfect Bayesian Nash equilibrium? If so, fully describe it.
Hint: A separating equilibrium means that Amy takes different strategies in S and N, while she chooses the same strategy in a pooling equilibrium. Your answer in (c) might depend on the value p.
