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Unformatted text preview: Problem Set 4 (Week of May 9) ECOS3012- Strategic Behavior, Semester 1, 2011 1. Work through all the end of chapter problems in the chapter “Backward Induction and Subgame Perfection” in Watson’s text. 2. (Stackelberg Competition) In a certain market with two firms, the price is P = 10- Q if the total output is Q ≤ 10 and equals 0 otherwise. The firms compete by choosing how much to produce in the following manner. Firm 1 is the leader and chooses her production first. Firm 2, the follower observes how much Firm 1 has produced and then selects her production. The marginal cost of production for both firms is zero. (a) Assuming that either firm can only choose her production to be only 2, 4 or 6, i. Draw the extensive form game. How many subgames are there? Ans. The extensive form is shown below. I will let you fill in the payoffs corresponding to the terminal nodes. There are four subgames. f1 f2 a l b m c h l f2 d l e m f h m f2 g l h m i h h ii. What is the subgame perfect equilibrium? Ans. I will let you solve for this via backward induction after filling in the payoffs for the terminal nodes. Also for the next question. iii. Give a Nash equilibrium of this game that is not a subgame perfect equilibrium. (b) Now assume that either firm can choose any quantity continuously between 0 and 10. Write down the subgame perfect equilibrium. What quantity is produced? If a firm could choose, would it prefer to be a leader or a follower? p1 p2 q 1 q 2 (10- q 1- q 2 ) q 1 , (10- q 1- q 2 ) q 2 Ans. When f1 plays a quantity 0 ≤ q 1 ≤ 10, then f2 achieves the highest payoff by playing a best response to q 1 , which is given by q 2 := R 2 ( q 1 ) = (10- q 1 ) / 2. (Recall, this is just the reaction function of f2.) Therefore, if f1 were to choose a strategy q 1 , then f2 would respond with R 2 ( q 1 ) leading to a profit of (10- q 1- R 2 ( q 1 )) × q 1 = (10- q 1 ) × q 1 / 2 which achives a maximum at q * 1 = 5. So the Subgame Perfect Equilibrium is: f1 plays q * 1 and f2 plays R 2 . In equilibrium, payoff of f1 is 12 . 5 and that of f2 is (10- q * 1- R 2 ( q * 1 )) × R 2 ( q * 1 ) = 6 . 25. Hence, a firm would prefer to be a leader. 3. (Bargaining in Bilateral Monopoly.) A is a monopolist that uses an special input that only B produces to create a unique product. It turns out that if B offers input whose quality is q , which costs her c ( q ), A is able to sell her unique creation for R ( q ) where 1 c ( · ) is a strictly increasing cost convex function and R ( · ) is a strictly increasing concave revenue function. Initially, take R ( q ) = √ q and c ( q ) = q 2 / 2....
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This note was uploaded on 08/20/2011 for the course ECON 101 taught by Professor Etw during the Spring '11 term at UniversitÃ di Bologna.
- Spring '11