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Two players, 1 and 2, each own a house. Each player i values her own house at v i and this is private information. The value of player i’s house to the other player j(6= i) is 3 2 v i . The values v i are drawn independently from the interval [0, 1] with uniform distribution. Suppose players announce simultaneously whether they want to exchange (E) their house of not (N). If both players agree to an exchange, the exchange takes place. Otherwise no exchange occurs.

Explain. (b) Show that any risk averse decision maker whose preference satisfies indepen- dence axiom must prefer L 2 to L 3 . 3. Question 3 (4 points) Suppose a monopolist with constant marginal costs prac- tices third-degree price discrimination. Group A’s elasticity of demand is ǫ A and

Players 1 (proposer) and 2 (receiver) are bargaining over how to split the ice-cream of size 1. In the first stage, player 1 proposes a share {x, 1 − x} to player 2 where x ∈ [0, 1] is player 1’s own share. Player 2 can decide whether accept the offer or reject it. If player 2 accepts, then the game finishes and players get their shares. If player 2 rejects, the game move to the second stage, in which the size of the ice-cream becomes δ(∈ (0, 1)) of the original size due to melting. In the second stage, by flipping a coin, the ice-cream is randomly assigned to one of the players. Suppose each player maximizes expected size of the ice-cream that she can get. Derive a subgame perfect Nash equilibrium of this game.

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.

Problem Set 2: Posted on November 4 Advanced Microeconomics I (Fall, 1st, 2014) 1. Question 1 (7 points) A real-valued function f (x) is called homothetic if f (x) = g(h(x)) where g : R → R is a strictly increasing function and h is a real-valued function which is homo- geneous of degree 1. Suppose that preferences can be represented by a homothetic utility function. Then, prove the following statements.

Problem Set 2: Posted on November 18 Advanced Microeconomics I (Fall, 1st, 2013) 1. Question 1 (7 points) A real-valued function f (x) is called homothetic if f (x) = g(h(x)) where g : R → R is a strictly increasing function and h is a real-valued function which is homo- geneous of degree 1. Suppose that preferences can be represented by a homothetic utility function. Then, prove the following statements.

Problem Set 2: Due on May 10 Advanced Microeconomics I (Spring, 1st, 2012) 1. Question 1 (2 points) Suppose the production function f satisfies (i) f (0) = 0, (ii) increasing, (iii) con- tinuous, (iv) quasi-concave, and (v) constant returns to scale. Then, show that f must be a concave function of x.

A good is called normal (resp. inferior) if consumption of it increases (resp. declines) as income increases, holding prices constant.. Show the following claims.[r]

Problem Set 2: Due on May 14 Advanced Microeconomics I (Spring, 1st, 2013) 1. Question 1 (6 points) (a) Suppose the utility function is continuous and strictly increasing. Then, show that the associated indirect utility function v(p, ω) is quasi-convex in (p, ω). (b) Show that the (minimum) expenditure function e(p, u) is concave in p.

j + x j − x i x j , where x i is i’s effort and x j is the effort of the other player. Assume x 1 , x 2 ≥ 0. (a) Find the Nash equilibrium of this game. Is it Pareto efficient? (b) Suppose that the players interact over time, which we model with the infinitely repeated version of the game. Let δ denote the (common) discount factor of the players. Under what conditions can the players sustain some positive effort level k = x 1 = x 2 > 0 over time?

4. Question 4 (5 points) Consider a game of election with asymmetric information among voters. Whether candidate A or candidate B is elected depends on the votes of two citizens (denoted by 1 and 2). The economy may be in one of two states, α and β. The citizens agree that candidate A is best if the state is α and candidate B is best if the state is β. The payoff for each citizen is symmetric and given as follows: 1 if the best candidate wins, 0 if the other candidate wins, and 1/2 if the candidates tie. Suppose that citizen 1 is informed of the true state, whereas citizen 2 believes it is α with probability 0.9 and β with probability 0.1. Each citizen may either vote for candidate A, vote for candidate B, or not vote.

(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.

Axiomatic Approach (2) PAR (Pareto Efficiency) Suppose hU, di is a bargaining problem with v, v ′ ∈ U and v ′ i > v i for i = 1, 2. Then f (U, d) 6= v. The axioms SYM and PAR restrict the behavior of the solution on single bargaining problems, while INV and IIA require the solution to exhibit some consistency across bargaining problems.

1. True or False (9 points) Answer whether each of the following statements is true (T) or false (F). You do NOT need to explain the reason. (a) The backward induction solution coincides with the subgame perfect Nash equilibrium for ANY perfect information game.

(a) Pure-strategy Nash equilibrium may NOT exist even if the game is …nite. (b) There MAY exist a subgame perfect Nash equilibrium which is not a Nash equilibrium. (c) Nash bargaining solution ALWAYS requires two players to divide the surplus equally.

4. Auctions (30 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:

where u i (x, θ i ) is the money-equivalent value of alternative x ∈ X. This assumes the case of private values in which player i’s payoff does not depend directly on other players’ types. If it does, then it is called common values case. The outcome (of the mechanism) is described by

(a) Derive all pure strategy Nash equilibria. (b) Show that the following type of asymmetric Nash equilibria does NOT exist: One firm chooses pure strategy M , and other two firms use mixed strategies. (c) Derive a symmetric mixed strategy Nash equilibria. You may assume that

Second degree price discrimination (or nonlinear pricing) Prices differ depending on the number of units of the good bought, but not across consumers from the beginning. Each consumer faces the same price schedule, but it involves different prices for different amounts of the good purchased. Quantity discount and two part tariffs (in mobile phone, taxi, printers etc) are the obvious examples.

2. Duopoly (15 points) Consider a duopoly game in which two firms, denoted by Firm 1 and Firm 2, simultaneously and independently select their own prices, p 1 and p 2 , respectively. The firms’ products are differentiated. After the prices are set, consumers demand A − p i +