question set 9 - Microeconomic Theory ‘ Jeffrey Carpenter...

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Unformatted text preview: Microeconomic Theory ‘ Jeffrey Carpenter Econ 255 ‘ Warner 305F Problem set number 9 jpc©middleburyedu Due: Friday December 4th (in my mailbox) X3241 CHAPTER 28 — GAME THEORY (1) (2) (3) (4) Cournot-Nash 1: Suppose there are two identical firms that face market demand, p=100—(y1+y2) and the cost structure, c(y)=10yi for i=1,2. ‘ ‘ a. Set up a normal (or strategic) form game in which the two firms can either produce the Cournot quantity or half the monopoly quantity (the entries should be profits). b. What is the Nash equilibrium of this normal form, game? Cournot-Nash 2: Suppose that the firms in (1) are no longer completely symmetric. Suppose the cost function for firm two changes to c(y):5y2 and firm one is the same. a. Set up a normal (or strategic) form game in which the two firms can either produce the Cournot quantity or half the monopoly quantity (the entries should be profits. b. What is the Nash equilibrium of this normal form game? Battle of the Sexes: Henry and Audrey like doing things together; however, there preferences are not identical. Henry prefers ballet while Audrey likes all-star wrestling. The following normal form game represents their preferences. Henry Ballet Wrestling Audrey Benet _—_ Wrestling -m- 1,3: - a. Find the pure strategy Nash Equilibria. b. Is there a mixed strategy equilibrium? Rock, Paper, Scissors: Remember “rock smashes scissors, paper.” Imagine the payoff to “winning” is +1 and the payoff to losing is -1. Then the normal for ” “paper covers rock,” and “scissors cuts _ of RPS is just: Player 2 Rock Paper _ Scissors Rock . —1 ,1 . 1 ,-1 Player 1 Paper Scissors a. Find the pure strategy Nash Equilibria. b. Is there a mixed strategy equilibrium? CHAPTER 17 — AUCTIONS (5) (6) (7) An Easy Second Price Auction: Suppose that there are two bidders in a second price auction, that bidder one’s value is v1=$5, that bidder two’s value is v2=$4, and that all bids have to be in multiples of $2. Draw the normal form and find any Nash equilibria. . A Harder Second Price Auction: Assume that each of “n” bidders has a private value. of vi i=1...n for the good being auctioned, that each bidder bids bi, and that the winner only has to pay bje the second highest bid. Prove that bidding one’s value is weakly dominant. A First Price Auction: Assume that there are 10 people participating in a first price sealed bid auction and that it is common knowledge that private values are distributed independently and uniformly on the [0,10] interval. Derive the optimal bid, b*, of a person whose private value is$5. 77 (m) .- [we - (39439)] 30 .— (Macs ‘ nod and = 700 V :5; ’77, (C) “1/1) :[ [cu _ (39.: 23,57] 30"”[3‘8 01/4/25" 300 =5 CC};- ’7 (#4?) () :[figa- (315221r332?'{"0(22€ I06V.7r~225' r .75" 3;; 7104/1, "72) 506:; -(72.5r3}.5‘3327:5;fi‘(”5\, 4/22?.5’-—225’: W ad, '6 I) 1/1/03; fl‘r""w7‘*""t '3’ SW i! 6% 4‘li 8") ankfdrflo: {7 w. to? 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