# トップPDF PS2 2 最近の更新履歴 yyasuda's website ### PS2 2 最近の更新履歴 yyasuda's website

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) ### Lec2 13 最近の更新履歴 yyasuda's website

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 ### Final2 08 最近の更新履歴 yyasuda's website

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. ### Midterm2 10 最近の更新履歴 yyasuda's website

(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. ### Final2 11 最近の更新履歴 yyasuda's website

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, ### Lec2 12 最近の更新履歴 yyasuda's website

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, C2), and (ii) subgames in which the outcome of at least one earlier stage differs from (C1, C2).

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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. ### Lec2 最近の更新履歴 yyasuda's website

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,

13 さらに読み込む ### Micro2 最近の更新履歴 yyasuda's website

How to Measure Welfare Change | 厚生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] ### PS2 最近の更新履歴 yyasuda's website

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. ### Lec2 最近の更新履歴 yyasuda's website

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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