Problem Set 4: Due on June 21
Advanced Microeconomics II (Spring, 2nd, 2012)
1. Question 1 (6 points)
A monopolist faces two kinds of consumers: students and non-students. The demand function of students is q = x(100 − 2p). The demand function of non-students is given as q = y(100 − p). Assume that marginal cost of production is zero.
(a) First suppose that the monopolist must set a single price to sell to all con- sumers. What price would the monopolist charge? How much will students and non-students consume, respectively?
(b) Now suppose that the monopolist can charge different prices to students and non-students. What price would the monopolist charge in each market? How much will students and non-students consume, respectively?
(c) Compute the social welfare (total surplus) in (a) and (b). When does the single price regulation (in (a)) generate higher welfare than without it (in (b))? 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 (4 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.
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4. 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) Is there any mixed strategy Nash equilibrium in which every person decides to make a call with the same probability p? If yes, derive such p.
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