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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.

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]

Let w = (w 1 , w 2 , w 3 , w 4 ) ≫ 0 be factor prices and y be an (target) output. (a) Does the production function exhibit increasing, constant or decreasing returns to scale? Explain. (b) Calculate the conditional input demand function for factors 1 and 2. (c) Suppose w 3 >

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 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.

(d) Suppose that this game is played finitely many times, say T (≥ 2) times. De- rive the subgame perfect Nash equilibrium of such a finitely repeated game. Assume that payoff of each player is sum of each period payoff. (e) Now suppose that the game is played infinitely many times: payoff of each player is discounted sum of each period payoff with some discount factor δ ∈ (0, 1). Assume specifically that A = 16, c 1 = c 2 = 8. Then, derive the

Consider the following two-person dynamic game. In the …rst period, game A is played; after observing each player’s actions, they play game B in the second period. Assume that the payo¤s are simply the sum of the payo¤s of two games (i.e., there is no discounting).

(d) What is the Nash equilibrium of this game? 4. Mixed Strategy (15 points) Three …rms (1, 2 and 3) put three items on the market and can advertise these products either on morning (= M ) or evening TV (= E). A …rm advertises exactly once per day. If more than one …rm advertises at the same time, their pro…ts become 0. If exactly one …rm advertises in the morning, its pro…t is 1; if exactly one …rm advertises in the evening, its pro…t is 2. Firms must make their daily advertising decisions simultaneously.

5. Bayesian Game (20 points) There are 10 envelopes and each of them contains a number 1 through 10. That is, one envelope contains 1, another envelope contains 2, and so on; these numbers cannot be observable from outside. Suppose there are two individuals. Each of them randomly receives one envelope and observes the number inside of her/his own envelope. Then, they are given an option to exchange the envelope to the other person; exchange occurs if and only if both individuals wish to exchange. Finally, individuals receive prize ($) equal to the number, i.e., she receives $X if the number is X. Assume that both individuals are risk-neutral so that they maximize expected value of prizes.

Prisoners’ Dilemma: Analysis
   ( Silent , Silent ) looks mutually beneficial outcomes, though    Playing Confess is optimal regardless of other player’s choice!    Acting optimally ( Confess , Confess ) rends up realizing!!

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, p 1 and p 2 . The firms’ products are differentiated. After the prices are set, consumers demand 24 − p i +

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:

Suppose you are on the admission committee of the CRIPS, and must decide the minimum acceptance score of the entrance examination. There are two kinds of students, excellent and geniuses. All students would like to be admitted to the CRIPS as long as their expected ben- e…ts (in monetary term) are non-negative, but the object of the admis- sion committee is to accept only geniuses. It is presumably easier for geniuses to obtain high scores on the exam. In particular, suppose that the cost of obtaining a score of x out of 100 is $1200x for an excellent student and $1000x for a genius. The value to each student of being admitted to the CRIPS is $90,000. Then, what range of (minimum) exam scores would meet the admission committee’s objective?

That is, given that the incumbent’s information set is reached, choosing A is clearly optimal irrespective of her belief over nodes. → Choosing F looks non-credible. SPNE cannot eliminate this since there is no proper subgame. Weak perfect Bayesian equilibrium (explain in detail later) is enough to exclude (O, F ) in this case.

A belief about other players’ types is a conditional probability distribution of other players’ types given the player’s knowledge of her own type p i (t −i |t i ). When nature reveals t i to player i, she can compute the belief p i (t −i |t i ) using Bayes’ rule:

First-Price: General Model (1) Consider a first-price auction with n bidders in which all the conditions in the previous theorem are satisfied. Assume that bidders play a symmetric equilibrium, β(x). Given some bidding strategy b, a bidder’s expected payoff becomes

Proof Sketch (2): Existence of Pivotal Voter Lemma 3 (Existence of Pivotal Voter) There is a voter n ∗ = n(b) who is extremely pivotal in the sense that by changing his vote at some profile he can move b from the very bottom of the social ranking to the very top.

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

(c) Any finite game has at least one Nash equilibrium in pure strategies. 2. Expected Utility (16 points) Suppose that an individual can either exert effort or not. Her initial wealth is $100 and the cost of exerting effort is c. Her probability of facing a loss $75 (that is, her wealth becomes $25) is 1

A bargaining situation is described by a tuple hX, D, % 1 , % 2 i: X is a set of possible agreements: a set of possible consequences that the two players can jointly achieve. D ∈ X is the disagreement outcome: the event that occurs if the players fail to agree.