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Unformatted text preview: Department of Economics
Columbia University W3211
Fall 2011 S O L U T I O N S T O P r ob l e m Se t 9
R E C I T A T I O N Q U E S T I O NS O N L Y
I n t e r m e d i a t e M i c r o e co n o m i cs
P r of . Se y h a n E r d e n A r k on a c F ol l ow i ng q u est i ons w i l l not b e g r a d e d , t h e y a r e f o r you to p r a c t i c e a n d w i l l b e d i sc usse d a t t h e
r e c i t a t ion :
Ch.14 question 4
question 7 1. Ch 13 question 4
Assume you’re Lori. If Max works, your best strategy is to give no bonus (your payoff is 3), rather than give a bonus (your payoff is 1). If Max loafs, again, your best strategy is to give no bonus (your payoff is 0), rather than give a bonus (your payoff is 1). Hence “No Bonus” is Lori’s dominant strategy. Now assume you’re Max. If Lori offers you a bonus, your best strategy is to loaf because the payoff is 3, rather than 2 when you work. If Lori gives no bonus, again, your best strategy is to loaf (payoff of 0 vs. payoff of 1 if you work). This means that “Loaf” is Max’s dominant strategy. Combining two dominant strategies together suggests that the pair No Bonus – Loaf is the Nash equilibrium outcome of this static game. 2. Ch 13 question 5 a. There are two Nash equilibria (the off diagonals). If either firm produces 20 while the other produces 10, neither player has an incentive to change strategies given the strategy of the other player. b. If Firm 1 can choose first, it will commit to selling 20 units, and Firm 2 sells 10 units. If Firm 1 were to choose 10 units, Firm 2 would choose to produce 20 units, reducing Firm 1’s payoff by $10. c. If Firm 2 can choose first, it will sell 10 units, and Firm 1 will sell 20. If Firm 2 were to produce 20 units, Firm 1 would produce only 10, reducing Firm 2’s payoff by $5. 3. There are no purestrategy Nash equilibria in this game. In each cell, one of the players always would prefer to switch, given the move of the other. 4. The incumbent must compare the twoperiod profits under two scenarios. In the first scenario, the incumbent overproduces in the first period in order to reduce marginal cost in the second period, knowing that it will be a monopolist in the second period. In the second scenario, the incumbent produces the profitmaximizing output level in the first period, resulting in duopoly profits (with higher marginal cost) in the second period. In the first game tree below, the monopolist overproduces in the first period, resulting in total profits of $1000, which exceeds the profits with profitmaximizing production in the first period. In the second game tree, profits are greater if the incumbent does not overproduce in the first period. Figure 14.1 5. a. b.
c. d. The pure strategy Nash equilibrium in this game is for both Warner Bros. and the T3 producer to release their movies on July 4. Given that the T3 producer releases its movie on July 4, the best choice for Warner Bros. is to release its movie on July 4. On the other hand, given that Warner Bros. releases its movie on July 4, the best choice for the T3 producer is to release its movie on July 4. The release on July 18 by the T3 producer and on July 4 by Warner Bros. maximizes joint profit. Note that 90 30 120 is the greatest sum of profits. The maximum Warner Bros. is willing to pay is the difference between its profit when both movies are released on July 4th, and its payoff when it has bought the release of T3 (released on July 18) and Matrix (released on July 4), i.e., 90 50 40. The profit the T3 producer earns if it does not sell its right of release is 50. Therefore the minimum price the T3 producer accepts for the sale of its right of release is 50. Since the maximum that Warner Bros. is willing to pay (40) is less than the minimum that the T3 producer is willing to accept (50), there is no mutually beneficial price at which trade can take place. Warner Bros. will release its movie on July 4th and T3 on July 18. 6. With only one firm, the deadweight loss is equal to the deadweight loss of a monopoly; that is, (243 147) (192 96)/2 (243 147) (243 147)/2 4608 (see Figure 14.2(a) in the text). With three firms, the deadweight loss is (195 147) (195 147)/2 1152, decreasing by 75%. 7. See Figure 14.2. The graph shows response curves for the two airlines. The airline with the lower marginal cost produces more: Southwest transports Q1 passengers and US Airways transports Q2 passengers. This result is shown algebraically in Solved Problem 14.1. Figure 14.2 8. The best response function of Firm 2 is a m2 bq1
q2 . 2b
Figure 14.3 shows bestresponse functions for Firm 1 and Firm 2. If marginal costs of the two firms are equal, then NashCournot outputs of the firms are also equal (equilibrium e1 ). Increasing marginal cost of the second firm shifts its bestresponse function inward, which leads to lower output of Firm 2 (equilibrium e2 ). q2
b a m2
x m2 m e1
e2 m2 m x am
Figure 14.3 a m2
b q1 10 . ...
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