Problem_Set_8_Solutions - Kaushik Basu Spring 2008 Econ 367...

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Kaushik Basu Spring 2008 Econ 367 Game Theoretic Methods Problem Set 8 1. Two firms, producing the same good, face the following inverse demand function: ) ( 10 2 1 x x p + - = where x i is the output produced by firm i and p is price in dollars. Firm 2 can produce this good incurring zero cost. Firm 1 can also produce this good with no per-unit cost but must incur an initial sunk cost of k dollars in order to start production. Firm 1 chooses x 1 in period 1 and then firm 2 (after seeing 1's choice) chooses x 2 in period 2. Describe the subgame perfect equilibrium for cases where (a) k = 0, (b) k = 15 and (c) k = 20. Using backward induction, firm 2 in period 2 maximizes P*Q = [10-(x 1 +x 2 )]x 2 which solves to 5/2 – x 1 /2. Firm 1, now knowing that this is what firm two will choose, maximizes P*Q = [10-(x 1 +(5/2 – x 1 /2))]x 1 . Solving, x 1 =5, x 2 =5/2. This implies that profit for firm 1 is 12.5. So any sunk cost k greater than 12.5, firm 1 will not produce. So in both parts b & c, firm 1 does not produce and firm 2 produces the monopoly amount, 5. 2. Two firms face the following inverse demand function: ) ( 12 2 1 x x p + - = Each firm i's total cost of production is given by: . 2 , 1 , 3 = = i x C i i (a) How much will each firm produce in a Cournot equilibrium? (b)
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Problem_Set_8_Solutions - Kaushik Basu Spring 2008 Econ 367...

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