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Problem Set 5: Due on July 5 Advanced Microeconomics II (Spring, 2nd, 2012) 1. Question 1 (4 points) 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]. 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.

elimination of strictly dominated strategies can never be selected (with positive probability) in a mixed-strategy Nash equilibrium.[r]

  A tree starts with the initial node and ends at.. terminal nodes where payoffs are specified..[r]

Prisoners’ Dilemma: Analysis
   ( Silent , Silent ) looks mutually beneficial outcomes, though    Playing Confess is optimal regardless of other player’s choice!    Acting optimally ( Confess , Confess ) rends up realizing!!

Prisoners’ Dilemma: Analysis (3)
  (Silent, Silent) looks mutually beneficial outcomes, though    Playing Confess is optimal regardless of other player’s choice!   Acting optimally ( Confess , Confess ) rends up realizing!!

payoff) while M gives 1 irrespective of player 1’s strategy.   Therefore, M is eliminated by mixing L and R .   After eliminating M , we can further eliminate D (step 2) and L (step 3), eventually picks up ( U , R ) as a unique outcome.

   Both the Bertrand and Cournot models are particular cases of a more general model of oligopoly competition where firms choose prices and quantities (or capacities.).   Ber[r]

Review of Lecture 5    Indifference property in mixed strategy NE.   If a player chooses more than one strategy with positive probability, she must be indifferent among such pure strategies: choosing any of them generate same expected payoff.

   If the stage game has a unique NE, then for any T , the finitely repeated game has a unique SPNE: the NE of the stage game is played in every stage irrespective of the histor[r]

3(a - e)/4, is greater than aggregate quantity in the Nash equilib- rium of the Cournot game, 2(a - e)/3, so the market-clearing price is lower in the Stackelberg game.. Thus, i[r]

   If the stage game has a unique NE, then for any T , the finitely repeated game has a unique SPNE: the NE of the stage game is played in every stage irrespective of the histor[r]

  Exist exactly one for ANY exchange problem.   Always Pareto efficient and individually rational[r]

  Paul Romer (1955-, 内生的成長理論) → 学界から消えた！？   Ben Bernanke (1953-, マクロ、金融) → FRB議長を辞めたのは好材料？   Douglas Diamond (1953-, 銀行取付) → 金融は無い？   清滝信宏 (1955-, マクロ、金融) → まだ早い

3(a - e)/4, is greater than aggregate quantity in the Nash equilib- rium of the Cournot game, 2(a - e)/3, so the market-clearing price is lower in the Stackelberg game.. Thus, i[r]

5. Mixed Strategy (20 points) Consider a patent race game in which a “weak” firm is given an endowment of 4 and a “strong” firm is given an endowment of 5, and any integral amount of the endowment could be invested in a project. That is, the weak firm has five pure strategies (invest 0, 1, 2, 3 or 4) and the strong firm has six (0, 1, 2, 3, 4 or 5). The winner of the patent race receives the return of 10. Both players are instructed that whichever player invests the most will win the race and if there is a tie, both lose: neither gets the return of 10.

(b) If consumer’s choice satis…es the weak axiom of revealed preferences, we can always construct a utility function which is consistent with such choice behav- iour. (c) If a consumer problem has a solution, then it must be unique whenever the consumer’s preference relation is convex.

(c) Solve for the total saving S by all types who save and the total borrowing B.. by all types who borrow.[r]

    5. Bayesian Nash Equilibrium (12 points)  There are three different bills, $5, $10, and $20. Two individuals randomly receive one  bill each. The (ex ante) probability of an individual receiving each bill is therefore 1/3.  Each  individual  knows  only  her  own  bill,  and  is  simultaneously  given  the  option  of  exchanging her bill for the other individual’s bill. The bills will be exchanged if and only  if  both  individuals  wish  to  do  so;  otherwise  no  exchange  occurs.  That  is,  each  individuals can choose either exchange (E) or not (N), and exchange occurs only when  both  choose  E.  We  assume  that  individuals’  objective  is  to  maximize  their  expected  monetary payoff ($). 

Find (all) pure‐strategy Nash equilibrium if it exists.  iii.[r]

e z . The prices of the three goods are given by (p, q, 1) and the consumer’s wealth is given by ω. (a) Formulate the utility maximization problem of this consumer. (b) Note that this consumer’s preference can be expressed in the form of U (x, y, z) = V (x, y) + z. Derive V (x, y).