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?

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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:

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Three firms (1, **2** and 3) put three items on the market and can advertise these products either on morning (= M ) or evening TV (= E). A firm advertises exactly once per day. If more than one firm advertises at the same time, their profits become 0. If exactly one firm advertises in the morning, its profit is 1; if exactly one firm advertises in the evening, its profit is **2**. Firms must make their daily advertising decisions simultaneously.

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However, it is difficult to assess how reasonable some axioms are without having in mind a specific bargaining procedure. In particular, IIA and PAR are hard to defend in the abstract. Unless we can find a sensible strategic model that has an equilibrium corresponding to the Nash solution, the appeal of Nash’**s** axioms is in doubt.

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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

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

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(a) Find a Bayesian Nash equilibrium of the game in pure strategies in which each player i accepts an exchange if and only if the value v i does not exceed some
threshold θ i
(b) How would your answer to (a) change if the value of player i’**s** house to the other player j becomes 5

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

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

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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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(e) The social welfare function introduced by Arrow is to derive social UTILITY by adding up individual utilities.
**2**. Externalities (25 points) Consider a one-consumer, one-firm economy (or equiv- alently an economy with many identical consumers and firms.) There are two private commodities. The firm also produces a level of pollution b. The produc- tion set of the firm is the convex set γ = {(y 1 , y **2** , b | G(y 1 , y **2** , b ) ≤ 0)}, where G

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(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.

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mechanism to make social choices should not depend on society’**s** members holding any particular sorts of views. WP is very straightforward, and one that economists, at least, are quite comfortable with. It says society should prefer x to y if every single member of society prefers x to y.

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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

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object for each buyer is independently and uniformly distributed between 0 and 1. (a) Suppose that buyer **2** takes a linear strategy, b **2** = v **2** . Then, derive the
probability such that buyer 1 wins as a function of b 1 .
(b) Solve a Bayesian Nash equilibrium.

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Similarly, player **2** must be indi¤erent amongst choosing X and Y , which implies 4q + 6(1 q) = 7(1 q)
, 5q = 1 , q = 1=5.
Thus, the mixed-strategy equilibirum is that player 1 takes A with probability 1=5 (and B with probability 4=5) and player **2** takes X with probability 3=4 (and Y with probability 1=4).

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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,

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

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

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