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Practice Problems II Answers 1. The payoff matrix looks like this: Player 1 R P S R 0, 0 1, -1 -1, 1 Player 2 P -1, 1 0, 0 1, -1 S 1, -1 -1, 1 0, 0 I’ve used yellow and green highlighting (respectively) to show player 1’s and player 2’s best responses to each other. Since there is no cell in which both numbers are highlighted, there is no pure-strategy Nash Equilibria. In other words, we can’t find any combination of strategies such that neither player would wish to change his strategy. However, it turns out there is a mixed-strategy Nash Equilibrium of this game, which is for both players to randomize over R, P, and S with probabilities 1/3, 1/3, and 1/3. We did not learn in class how to calculate a mixed-strategy equilibrium, but if you’ve ever played this game, you probably understand and play this equilibrium intuitively. 2. The payoff matrix looks like this: Akeem Reach today Don’t reach Rojelio Reach today Don’t reach $50, $50 $100, $0 $0, $100 $75, $75 (If both reach today, they split the $100, and there’s nothing left tomorrow. If both don’t reach, the $100 grows to $150, which they split between them. If one reaches and the other doesn’t, the one who reaches gets the whole $100, the other gets nothing, and there’s nothing left for tomorrow.) As with the previous problem, highlighting shows the players’ best responses. {Reach today, Reach today} is the Nash Equilibrium of this game, since both payoffs are highlighted in that cell. This game is a Prisoners’ Dilemma, because both players have dominant strategies, and the resulting dominant strategy equilibrium yields payoffs that are worse for both players than they would have gotten if both had acted differently. 3. The payoff matrix looks like this: Hector Hard bargain Easy bargain Menelaus Hard bargain Easy bargain $0, $0 $500, $100 $100, $500 $300, $300 (If both drive a hard bargain, there’s no deal, so there’s zero change relative to the baseline. If both drive an easy bargain, Hector pays $700 for a $1000 gain, and Menelaus receives $700 for accepting a $400 loss, yielding net $300 each. If Hector drives an easy bargain and Menelaus a hard bargain, Hector pays $900 for a $1000 gain, and Menelaus receives $900 for a $400 loss, yielding net $100 and $500 respectively. Similar reasoning for a payment of $600 yields the numbers in the upper right cell.) The highlighted best responses show that there are two Nash Equilibria: {Easy, Hard} and {Hard, Easy}. This game is like the game of Chicken, because there are multiple equilibria, one preferred by one party and one preferred by the other, as well as a “disaster” outcome in which both parties lose out on possible gains. 4. (a) Under Cournot competition, firm 1 perceives its demand curve as P = [140 – 2q2] – 2q1. So firm 1’s marginal revenue is the same thing with twice the slope: MR = [140 – 2q2] – 4q1. Setting this equal to the MC of 20, we get: [140 – 2q2] – 4q1 = 20 4q1 = 120 – 2q2 qR1 = 30 – (1/2)q2 That’s firm 1’s reaction function. Since the firms are identical, firm 2’s reaction is similar: qR2 = 30 – (1/2)q1 The equilibrium occurs where the reaction functions cross each other. Solve the system of equations by substituting one into the other: q1 = 30 – (1/2)[30 – (1/2)q1] q1 = 15 + (1/4)q1 (3/4)q1 = 15 q*1 = 20 Since the firm’s are identical, q*2 = 20 as well, so Q* = 40, and P* = 140 – 2(40) = 60. (b) Under Bertrand competition, the firms undercut each other’s prices until their prices both equal the marginal cost: P*1 = P*2 = 20. Plugging this into the demand curve, we get 20 = 140 – 2Q, so 2Q = 120, so Q* = 60, which is split between the two firms. (c) Under Stackelberg competition, we use backward induction to find the solution. Once firm 1 has chosen its quantity, firm 2 will respond according to the reaction function we found earlier: qR2 = 30 – (1/2)q1. Predicting this, firm 1 perceives the following demand curve: P = 140 – 2q2 – 2q1 P = 140 – 2[30 – (1/2)q1] - 2q1 P = 80 – q1 Find the MR by doubling the slope, and set it equal to the MC: MR = 80 - 2q1 = 20 2q1 = 60 q*1 = 30 Plug this into firm 2’s reaction function to get firm 2’s quantity: q*2 = 30 – (1/2)(30) = 15 So the market quantity is Q* = 30 + 15 = 45, and the market price is P* = 140 – 2(45) = 50. 5. The calculations are similar to those above, so I’ll just give the answers: (a) Cournot: q*1 = q*2 = 60, Q* = 120, P* = 50 (b) Bertrand: P*1 = P*2 = 30, Q* = 180. (c) Stackelberg: q*1 = 90, q*2 = 45, Q* = 135, P* = 45 6. (a) Go through the same procedure as in the previous problems to find firm 1’s reaction function. But since the firms’ marginal costs differ, you need to calculate firm 2’s reaction function as well. They turn out to be: qR1 = 90 – (1/2)q2 qR2 = 75 – (1/2)q1 Then solve the system of equations by plugging one into the other. q1 = 90 – (1/2)[ 75 – (1/2)q1] q1 = 52.5 + (1/4)q1] (3/4)q1 = 52.5 q*1 = 70 Plugging this into firm 2’s reaction function, we get q*2 = 75 – (1/2)(70) = 40. So Q* = 70 + 40 = 110, and P* = 210 – 110 = 100. (b) Under Bertrand competition, the firms undercut each until firm 2 (the higher-cost firm) leaves the market. Firm 1 sets a price just under firm 2’s MC; P*1 = 59, and Q* = 151 (all of it sold by firm 1). If it is costly for firm 2 to re-enter the market, firm 1 could possibly raise its price higher after firm 2 leaves. 7. The calculations are similar to those above, so I’ll just give the answers: (a) Cournot: q*1 = 14, q*2 = 11, Q* = 25, P* = 45 (b) Bertrand: P*1 = 17, Q* = 36.5, and firm 2 leaves the market.