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

1 さらに読み込む

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

14 さらに読み込む

(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)

2 さらに読み込む

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

2 さらに読み込む

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.

2 さらに読み込む

(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.

2 さらに読み込む

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.

13 さらに読み込む

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.

さらに見せる
3 さらに読み込む

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,

2 さらに読み込む

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.

13 さらに読み込む

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.

13 さらに読み込む

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**).

12 さらに読み込む

(c) Confirm that by choosing the tax t appropriately, the socially optimal level of pollution is produced.
(d) Add a second firm with a different production function. Now the consumers observe a pollution level b = b 1 + b **2** . Show that the social optimum can still

3 さらに読み込む

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

15 さらに読み込む

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

12 さらに読み込む

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

1 さらに読み込む

Dual Problem - Theory | 双対問題 - 理論 (3)
Thm Suppose the consumer’**s** preference is continuous, monotone and strictly convex. Then, we have the following relations between the Hicksian and Marshallian demand functions for p ≫ 0, ω ≥ 0 and u ∈ R, and i = 1, **2**, ..., n:

28 さらに読み込む

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

1 さらに読み込む

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.

2 さらに読み込む

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 .

15 さらに読み込む