Eco 601E: Advanced Microeconomics II (Fall, 2nd, 2013)

Final Exam: January 28

1. Dynamic Game (24 points)

Consider the following two-person dynamic game. In the first period, game A is played; after observing each player’s actions, they play game B in the second period. Assume that the payoffs are simply the sum of the payoffs of two games (i.e., there is no discounting).

1 2 L R

U 3, 3 0, 4 D 4, 0 1, 1

### Game A

1 2 L^{′} R^{′}
U^{′} 4, 4 0, 0
D^{′} 0, 0 2, 2

### Game B

(a) Suppose above games are played separately, that is, each game is played only once. Then, derive all Nash equilibria for each game.

(b) Now consider the dynamic game. How many subgames (except for the entire game) does this game have?

(c) Is there a subgame perfect Nash equilibrium that can achieve (U, L) is the first period? If so, describe the equilibrium strategies. If not, explain why.

2. Duopoly (32 points)

Consider a duopoly game in which two firms, denoted by firm 1 and firm 2, simul-
taneously and independently select their own price, p_{1} and p_{2}. The firms’ products
are differentiated. After the prices are set, consumers demand 24 − pi + ^{p}^{j}

2 ^{units}
(i 6= j, i = 1, 2) of the good that firm i produces. Assume that each firm’s marginal
cost is 6, and the payoff for each firm is equal to the firm’s profit.

(a) Derive firm 1’s payoff function and the best reply function.

(b) Solve the pure-strategy Nash equilibrium of this game. How much profit does each firm earn?

(c) Now suppose that firms decide prices sequentially: firm 1 sets its price p_{1} first,
and firm 2 chooses price only after observing firm 1’s price. Find the subgame
perfect equilibrium of this game. How much profit does each firm earn?

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(d) Recall that any subgame perfect equilibrium must be a Nash equilibrium. Ex- plain why the Nash equilibrium you derive in (b) is different from the subgame perfect equilibrium in (c).

3. Repeated Game (24 points)

Find conditions on the discount factor under which cooperation (=(C, C)) can be supported in the infinitely repeated games with the following stage games.

1 2 C D

C 3, 3 0, 4
D _{4, 0 1, 1}

### Game 1

1 2 C D

C 3, 4 0, 7
D _{5, 0 1, 2}

### Game 2

1 2 C D

C 3, 2 0, 4
D _{7, 0 2, 1}

### Game 3

Hint: Note that every stage game above is a prisoner’s dilemma. You can focus on the trigger strategy, i.e., players choose a stage game Nash equilibrium (D, D) as a punishment whenever someone has once deviated from (C, C).

4. Auctions (30 points)

Suppose that the government auctions one block of radio spectrum to two risk neu- tral 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.

(a) Suppose that company 2 takes a linear strategy, b_{2} = αv_{2}. Then, derive the
probability such that company 1 wins as a function of b_{1}.

(b) Solve a Bayesian Nash equilibrium. You can assume that the equilibrium strategy is symmetric and linear, i.e., bi = αvi for i = 1, 2.

(c) Show that bi = vi (for i = 1, 2) is a weakly dominant strategy when θ = 0.

5. Social Choice (10 points)

Explain the Arrow’s impossibility theorem by words within 10 lines.

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