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.

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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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Hint: Note that every stage game above is a prisoner’**s** dilemma. You can focus on the trigger strategy, i.e., players choose a stage game Nash equilibrium (D; D) as a punishment whenever someone has once deviated from (C; C).
4. Auction (15 points)

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R i (or % i ) An individual i’**s** preference relation on X (an binary relation satisfying completeness and transitivity. → Let P i (or ≻i) and I i (or ∼i) be the associated relations of strict individual preference and indifference, respectively.

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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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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.

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(a) Derive each partner’**s** payo¤ function.
(b) Derive each partner’**s** best reply function and graphically draw them in a …gure. (Taking m in the horizontal axis and n in the vertical axis.)
(c) Is this game strategic complementarity, strategic substitution, or neither of them? Explain why.

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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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For a singleton information set, i.e., x = h(x), the player’**s** belief puts probability one on the single decision node.
(**2**) Given their beliefs, the players’ strategies must be
sequentially rational. That is, at each information set, the action taken by the player must be optimal given

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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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is strictly increasing in its first two arguments and strictly decreasing in b. Thus by increasing pollution, the firm can produce more output (or use less input). The consumer has a concave utility function U (y 1 , y **2** , b ) that is also increasing in its first
two arguments and decreasing in b.

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Constant Absolute Risk Aversion
Def We say that preference relation % exhibits invariance to
wealth if (x + p 1 ) % (x + p **2** ) is true or false independent of x.
Thm If u is a vNM continuous utility function representing preferences that are monotonic and exhibit both risk aversion and invariance to wealth, then u must be exponential,

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How to Measure Welfare Change | 厚生**の**変化をどうはかるか？
When the economic environment or market outcome changes, a consumer may be made better off ( 改善 ) or worse off ( 悪化 ). Economists often want to measure how consumers are affected by these changes, and have developed several tools for the assessment of welfare ( 厚生 ).

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

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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.

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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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