Eco 601E: Advanced Microeconomics II (Spring, 2nd, 2013)
Final Exam: July 18
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) A pure-strategy Nash equilibrium ALWAYS exists when the game is finite. (b) ANY sequential equilibrium is a perfect Bayesian equilibrium.
(c) The situation of asymmetric information is called “hidden action” if the agents who have private information move earlier than the agents who do not. (d) It could be possible that a Nash bargaining solution is NOT Pareto efficient.
(e) The social welfare function introduced by Arrow is to derive social UTILITY by adding up individual utilities.
2. Externalities (25 points) Consider a one-consumer, one-firm economy (or equiv- alently an economy with many identical consumers and firms.) There are two private commodities. The firm also produces a level of pollution b. The produc- tion set of the firm is the convex set γ = {(y1, y2, b | G(y1, y2, b) ≤ 0)}, where G is strictly increasing in its first two arguments and strictly decreasing in b. Thus by increasing pollution, the firm can produce more output (or use less input). The consumer has a concave utility function U (y1, y2, b) that is also increasing in its first two arguments and decreasing in b.
Hint: Note that a set of (y1, y2, b) such that G(·) = 0 yields a production frontier. (a) Write down the first-order conditions for the socially optimal production plan
and pollution level.
(b) Assume there is a pollution tax t on each unit of pollution produced and the firm and consumers are price takers. Then, write down the first-order conditions for profit maximization and utility maximization.
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(c) Confirm that by choosing the tax t appropriately, the socially optimal level of pollution is produced.
(d) Add a second firm with a different production function. Now the consumers observe a pollution level b = b1+ b2. Show that the social optimum can still be achieved by the imposition of a tax.
(e) Let b∗ be the socially optimal pollution level. Suppose the government an- nounces that it will sell b∗ “rights” to pollute. Will the tax rate emerge as the equilibrium market-clearing price?
3. Static Game (20 points)
Consider a static game played by 2 players, described by the following payoff matrix in which θ is a parameter (some real number).
1 \ 2 L R
U θ, θ 0, 0
D 0, 0 1 − θ, 1 − θ
(a) Is there a dominant strategy for each player? (b) Derive all pure-strategy Nash equilibria.
(c) Derive all mixed-strategy Nash equilibria.
(d) Now assume 0 < θ < 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 figure whose x-axis is p and y-axis is q. Indicate both pure and mixed- strategy Nash equilibria in your figure.
Hint: Your answers in (a) – (c) may change depending on the value of θ.
4. Duopoly (20 points)
Consider a duopoly game in which two firms, denoted by firm 1 and firm 2, simul- taneously and independently select their own price, p1 and p2. The firms’ products are differentiated. After the prices are set, consumers demand 24 − pi + p2j 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. 2
(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 p1 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? (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).
5. Auctions (20 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, b2 = αv2. Then, derive the probability such that company 1 wins as a function of b1.
(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.
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