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Eco 601E: Advanced Microeconomics II (Fall, 2nd, 2014)

Midterm Exam: December 22

1. True or False (20 points)

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

(a) If an agent is risk averse, her risk premium is ALWAYS positive.

(b) When every player has a (strictly) dominant strategy, the strategy profile that consists of each player’s dominant strategy MUST be a Nash equilibrium. (c) If there are two Nash equilibria in pure-strategy, they can ALWAYS be Pareto

ranked, i.e., one equilibrium Pareto dominates the other.

(d) The third-degree price discrimination NEVER increases the total surplus. (e) If a game has a mixed-strategy Nash equilibrium, there CANNOT be other

equilibrium in pure-strategy.

2. Expected Utility (20 points)

Suppose a decision maker is on a game show, and he has equal probability of winning

$3, $30, $300. His vNM utility function is

u(x) = 1 3x

2,

where x is the dollar amount he wins.

(a) Calculate the expected value of the lottery. (b) Calculate his expected utility from the lottery.

(c) Is he risk averse, risk neutral, or risk loving? Explain.

(d) Derive his certainty equivalent of the lottery. Is it smaller or larger than the expected value (your answer in (a))?

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3. Nash equilibrium (15 points)

Find all pure strategy Nash equilibria in each of the following games.

(a)

1  2 L R

U 100, 1 0, 1 D 100 , 0 0, 0

(b)

1  2 L M R

U 200, 0 50, 1 2, 2

D 100, 100 100, 50 1, 1

(c)

1  2 L M R

U −2, −2 −1, 0 4, 0

M 0, 0 2, 2 3, 1

D 0, 0 2, 4 3, 3

4. Games with Continuous Strategies (25 points)

You and your n − 1 roommates (n ≧ 2) each have five hours of free time that could be used to clean your apartment. You all dislike cleaning, but you all like having a clean apartment: each person i’s payoff is the total hours spent (by everyone) cleaning, minus a number c (> 0) times the hours spent individually cleaning. That is,

ui(s1, s2, . . . , sn) = Xn

j=1

sj − csi.

Assume everyone chooses simultaneously how much time to spend cleaning.

(a) Find a strictly dominant strategy for each player. Does it depend on the parameters c or n? Explain.

(b) Find the Nash equilibrium if c < 1. (c) Find the Nash equilibrium if c > 1.

(d) Is the equilibrium outcome in (b) or (c) Pareto efficient? Explain in each case. (e) Now suppose that the cleaning cost is changed from csi to 1

2s

2i for everyone. Find the Nash equilibrium.

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5. Mixed Strategy (20 points)

Consider a patent race game in which a “weak” firm is given an endowment of 4 and a “strong” firm is given an endowment of 5, and any integral amount of the endowment could be invested in a project. That is, the weak firm has five pure strategies (invest 0, 1, 2, 3 or 4) and the strong firm has six (0, 1, 2, 3, 4 or 5). The winner of the patent race receives the return of 10. Both players are instructed that whichever player invests the most will win the race and if there is a tie, both lose: neither gets the return of 10.

(a) Show that there is no pure-strategy equilibrium in this game.

(b) Is there any strictly dominated strategy? If yes, describe which strategy is dominated by which strategy. If no, briefly explain the reason.

(c) Derive the mixed-strategy Nash equilibrium.

Hint: You may first take the iterated elimination of strictly dominated strate- gies, and reduce the set of strategies that each player would select (with posi- tive probability) in a mixed strategy equilibrium.

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