Problem Set 1: Due on June 20
Advanced Microeconomics II (Spring, 2nd, 2013)
1. Question 1 (3 points) Suppose a monopolist with constant marginal costs prac- tices third-degree price discrimination. Group A’s elasticity of demand is ϵA and Group B’s is ϵB. At the same level of demand in each group, ϵA> ϵB always holds. Then, which group will face a higher price? Explain.
Remark: The elasticity of demand depends on the amount of the good demanded. 2. Question 2 (6 points)
Two drivers are deciding how fast to drive their cars. Driver i chooses speed xi and gets utility ui(xi) from this choice; assume u′ >0 and u′′ <0. However, the faster they drive, the more likely they get involved in a mutual accident. Let p(x1, x2) be the probability of an accident, assumed to be increasing in each argument, and let ci >0 be the cost that the accident imposes on driver i.
(a) Show that each driver has an inventive to drive too fast from the social point of view (i.e., the maximization of total surplus).
(b) If driver i is fined an amount ti in the case of an accident, how large should ti
be to internalize the externality?
(c) Suppose now that driver i gets utility ui(xi) only if there is no accident. What is the appropriate fine in this case?
3. Question 3 (6 points) There are three people who consume public and private goods. The public good is denoted by x, and yi represents person i’s consumption of the private good (i = 1, 2, 3). The prices of both the public good and the private good are $1 per unit. The initial endowments of the private good are (ω1, ω2, ω3) = (10, 10, 10). The three people have the following utility functions:
u1(x, y1) = ln x + y1. u2(x, y2) = 2 ln x + y2. u3(x, y3) = 3 ln x + y3.
(a) Assume that the public good is purchased, privately and that person 3 is the first to go to the market and buy the public good. Assume he does not act strategically; he ignores persons 1 and 2 when he buys x, and thinks only of his own utility maximization problem. What is the outcome? How much of the public good does person 3 buy? How much do persons 1 and 2 buy? (b) Use the Samuelson optimality condition to find the Pareto optimal quantity
of the public good x∗.
(c) Describe the Lindahl equilibrium.
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4. Question 3 (6 points)
(a) Suppose every player has a strictly dominant strategy. Then, show that the strategy profile in which everyone takes this strictly dominant strategy be- comes a unique Nash equilibrium.
(b) Suppose every player has a weakly dominant strategy. Then, is the strategy profile in which everyone takes this weakly dominant strategy a unique Nash equilibrium? If yes, explain your reason. If not, construct the counter example. (c) Provide an example of static game (with infinitely many strategies) which does
not have any Nash equilibrium, including mixed strategy equilibrium. 5. Question 4 (4 points)
A crime is observed by a group of n people. Each person would like the police to be informed but prefers that someone else make the phone call. They choose either
“call” or “not” independently and simultaneously. A person receives 0 payoff if no one calls. If someone (including herself) makes a call, she receives v while making a call costs c. We assume v > c so that each person has an incentive to call if no one else will call.
(a) Derive all pure-strategy Nash equilibria.
(b) Derive a symmetric mixed strategy Nash equilibrium in which every person decides to make a call with the same probability p.
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