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### Problem Set 3: Due on July 17

Advanced Microeconomics II (Spring, 2nd, 2013)

1. Question 1 (4 points) Consider the following labor market signaling game. There are two types of worker. Type 1 worker has a marginal value product of 1 and type 2 worker has a marginal value product of 2. The cost of signal z for type 1 is C1(z) = z and for type 2 is C2(z) = (1 − c)z. The worker is type 1 with probability p and 2 with probability 1 − p. There are two firms that play a Bertrand wage bidding game for the services of the worker, which simplifies wage determination: the equilibrium wage becomes the expected marginal value product of the worker. (a) Show that there is a separating PBE in which type 1 does not signal and type

2 chooses z ∈ [1,11

c].

(b) Show that the equilibrium payoff to the worker in the pooling equilibrium with no signaling is w(p) = 2 − p.

2. Question 2 (9 points) Consider a game between two friends, Amy and Brenda. Amy wants Brenda to give her a ride to the mall. Brenda has no interest in going to the mall unless her favorite shoes are on sale (S) at the large department store there. Amy likes these shoes as well, but she wants to go to the mall even if the shoes are not on sale (N ). Only Amy subscribes to the newspaper, which carries a daily advertisement of the department store. The advertisement lists all items that are on sale, so Amy learns whether or not the shoes are on sale. Amy can prove whether or not the shoes are on sale by showing the newspaper to Brenda. But this is costly for Amy, because she will have to take the newspaper away from her sister, who will yell her later for doing so.

In this game, the nature first decides whether or not the shoes are on sale, and this information is made known to Amy. That is, Amy observes whether the nature chose S or N . The nature chooses S with probability p and N with probability 1−p. Then Amy decides whether or not to take the newspaper to Brenda (T or D). If she takes the newspaper to Brenda, then it reveals to Brenda whether the shoes are on sale. In any case, Brenda must then decide whether to take Amy to the mall (Y ) or to forget it (F ). If the shoes are on sale, then going to the mall is worth 1 unit of utility to Brenda and 3 to Amy. If the shoes are not on sale, then traveling to the mall is worth 1 to Amy and −1 to Brenda. Both players obtain 0 utility when they do not go to the mall. Amy’s personal cost of taking the newspaper to Brenda is 2 units of utility, which is subtracted from her other utility amounts.

(a) Draw the game tree of this game.

(b) Does this game have a separating perfect Bayesian Nash equilibrium? If so, fully describe it.

(c) Does this game have a pooling perfect Bayesian Nash equilibrium? If so, fully describe it.

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Hint: A separating equilibrium means that Amy takes different strategies in S and N , while she chooses the same strategy in a pooling equilibrium. Your answer in (c) might depend on the value p.

3. Question 3 (4 points) Let F and G be distribution functions with support [0, ω], and let f and g be the corresponding probability density functions. The hazard rate of F is the function λ : [0, ω) → R+ defined by

λ(x) = f(x) 1 − F (x). Show that F (x) can be expressed by λ(x) as follows:

F(x) = 1 − exp (

x

0

λ(t)dt )

.

4. Question 4 (8 points)

A monopolist can produce a good in different qualities. The cost of producing a unit of quality s is s2. Consumers buy at most one unit and have utility function u(s|θ) = θs if they consume one unit of quality s and 0 if they do not consume. The monopolist decides on the quality and price that it is going to produce. Con- sumers observe qualities and prices and decide which quality to buy if at all.

(a) Characterize the first-best solution.

(b) Suppose that the seller cannot observe θ: θ ∈ {θL, θH} and Pr[θ = θL] = β with 0 < θL< θH. Set up the seller’s optimization problem under this asymmetric information structure.

Hint: By the taxation principle, you can focus on the pair of contracts: one is for the low type and the other is for the high type.

(c) Show that the low type consumers (θL) receive no rents under the second-best solution.

(d) Characterize the second-best solution. You can ignore possibility of a corner solution such that the low type consumers are excluded.

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