Eco 601E: Advanced Microeconomics II (Spring, 2nd, 2011) Final Exam: July 27

1. True or False (15 points)

Answer whether each of the following statements is true (T) or false (F). You do NOT need to explain the reason.

(a) Pure-strategy Nash equilibrium may NOT exist even if the game is …nite. (b) There MAY exist a subgame perfect Nash equilibrium which is not a Nash

equilibrium.

(c) Nash bargaining solution ALWAYS requires two players to divide the surplus equally.

(d) The perfect Bayesian equilibrium puts NO restriction on beliefs at information sets that are not reached in equilibrium.

(e) In the simple moral hazard problem we studied in class, the optimal wage (= s( )) is NOT necessarily increasing in outcome (= x).

2. Static Game (20 points)

Consider a static game played by 2 players, described by the following payo¤ matrix in which is a parameter (some real number).

1 n 2 L R

U ;1 0; 0

D 0; 0 1 ;

(a) Is there a dominant strategy for each player? (b) Derive pure-strategy Nash equilibria.

(c) Now asuume 0 < < 1. Let p be the probability of choosing U and q be the probability of choosing L. Then, draw the best reply curve for each player in the …gure whose x-axis is p and y-axis is q. Indicate both pure and mixed- strategy Nash equilibria in your …gure.

Hint: Your answers in (a) and (b) may change depending on the value of . 3. Game Tree (30 points)

Consider dynamic games expressed by the game trees (i) (ii) in next page.

(a) How many information sets (that contain two or more decision node) does each game have?

(b) How many subgames (except for the entire game) does each game have? (c) Find all pure-strategy subame perfect Nash equilibria for each game.

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4. Repeated Game (25 points)

In each period, two …rms produce an identical good. The inverse demand curve for the good is P = 25 Q, where Q is the total quantity produced by the two …rms. Each …rm has a constant marginal cost 1 of producing the good.

(a) Find the Cournot Nash equilibrium of this game. What quantity would each

…rm produce? What would be the market price? What would be the pro…ts of each …rm?

(b) Suppose the …rms form a cartel: each …rm produces the same output and maximizes their joint pro…ts. What quantity would each …rm produce? What would be the market price? What would be the pro…ts of each …rm?

(c) Suppose that one …rm produces the cartel output (your answer in (b)). Sup- pose that the other …rm maximizes pro…ts knowing that the other …rm would produce cartel output. How much would it produce? What would be the market price? What would be the pro…ts of each …rm?

(d) Now suppose that the …rms interact inde…nitely through time. They discount future pro…ts at a discount factor . For what value of is there an equilibrium where …rms follow the “trigger strategies”, i.e., they produce cartel output as long as the other …rm has always produced cartel output and otherwise they produce Cournot Nash output?

5. Bayesian Game (20 points)

There are 10 envelopes and each of them contains a number 1 through 10. That is, one envelope contains 1, another envelope contains 2, and so on; these numbers cannot be observable from outside. Suppose there are two individuals. Each of them randomly receives one envelope and observes the number inside of her/his own envelope. Then, they are given an option to exchange the envelope to the other person; exchange occurs if and only if both individuals wish to exchange. Finally, individuals receive prize ($) equal to the number, i.e., she receives $X if the number is X. Assume that both individuals are risk-neutral so that they maximize expected value of prizes.

(a) Consider the above situation as a Bayesian game. Then, what is the individ- ual’s strategy?

(b) If an individual receives the envelope 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 equi- librium?

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