Proof of Pratt’**s** Theorem (1) Sketch of the Proof.
To establish (i) ⇔ (iii), it is enough to show that P is positively related to r. Let ε be a “small” random variable with expectation of zero, i.e., E(ε) = 0. The risk premium P (ε) (at initial wealth x) is defined by

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(b) Revenue equivalence theorem claims that the equilibrium bidding strategy under the …rst-price auction is ALWAYS identical to the one under the second- price auction.
(c) EVERY perfect Bayesian equilibrium is a weak perfect Bayesian equilibrium. **2**. Dynamic Game (14 points)

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

b + (1 )b 0
where b is the winner’**s** bid, b 0 is the loser’**s** bid, and is some constant
satisfying 0 1. (In case of ties, each company wins with equal probability.) Assume the valuation of the spectrum block for each company is independently and uniformly distributed between 0 and 1.

(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.

Arrow’**s** Requirements of the SWF (1)
Unrestricted Domain (UD) The domain of f must include all possible combinations of individual preference relations on X.
Weak Pareto Principle (WP) For any pair of alternatives x and y in X, if xP i y for all i, then xP y.

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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.

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3. Auction (9 points)
Consider a “common-value auction” with two players, where the value of the object being auctioned is identical for both players. Call this value V and suppose that V = v 1 + v **2** , where v i is independently and uniformly distributed between 0 and 1,

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 +

Rm Each of these utility functions measures the change in the player’**s** utility. If there is no trade, then there is no change in utility. It would make no difference to define, say, the seller’**s** utility to be p if there is trade at price p and v **s** if there is no trade.

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1 **2** .
A leader never becomes worse off since she could have achieved Cournot profit level in the Stackelberg game simply by choosing the Cournot output: a gain from commitment. A follower does become worse off although he has more information in the Stackelberg game than in the Cournot game, i.e., the rivals output.

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Rm Since every subgame of an infinitely repeated game is identical to the game as a whole, we have to consider only two types of subgames: (i) subgame in which all the outcomes of earlier stages have been (C1, C**2**), and (ii) subgames in which the outcome of at least one earlier stage differs from (C1, C**2**).

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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

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1 Nature draws a type t i for the Sender from a set of feasible
types T = {t1 , ..., t I} according to a probability distribution
p(ti), where p(ti) > 0 for every i and p(t 1 ) + · · · + p(tn) = 1.
**2** Sender observes ti and then chooses a message mj from a set

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Solve the following problems in Snyder and Nicholson (11th):. 1.[r]

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.

Solve the following problems in Snyder and Nicholson (11th):. 1.[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.

Open Set and Closed Set (**2**)
Boundary and interior
◮ A point x is called a boundary point of a set S in R n
if every ε-ball centered at x contains points in S as well as points not in S. The set of all boundary points of a set S is called boundary, and is denoted ∂S .

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Solve the following problems in Snyder and Nicholson (11th):. 1.[r]