Homework # 2 - ISyE 6230 – Economic Decision Analysis II – Spring 2008Due Thurs Feb 7Grading: Up to 10 points per problem for completing each problem; the remaining 50 points will come from grading one randomly selected problem for correctness. 1. Suppose company A is a monopolist supplier of aircraft engines. Company B is considering entering the market. Both the companies have to invest in capital investment that is difficult to divest if they are to serve the market. Therefore, both firms need to choose between several investment options. As expected, their payoffs depend on both their actions. Company A can choose to invest big or small. Company B after observing A’s action can either decide to stay out of the market or enter the market by making either a large investment or a minor investment. If A chooses to invest big, it will make a profit of 80 if B stays out, and will incur a loss of 30 if B enters the market. If A chooses to invest small, it will make a profit of 30 if B stays out, incurs a loss of 50 if B enters with large investment, or incurs a loss of 20 if B enters with minor investment. If A chooses to invest big, B will make a profit of 0 if it stays out, incurs a loss of 40 if it makes large investment, or makes a profit of 20 if it makes a minor investment. However, if A chooses to invest small, B will make a profit of 0, 60, 30 if it stays out, invests large or invests minor respectively. a. Draw the extensive form of this game.b. What are the information sets in this game? What are the subgames?c. Is this a game of perfect or imperfect information, and why?d. Identify and Nash equilibria of this game.2. Make the following modification in problem 1: Assume that company A realizes that it has an option of not investing in capital, and run it’s existing plant more efficiently by changing the management. This will lead to a profit of 10 to A no matter what B decides to do. However, B will make a profit of 0 if it stays out, profit of 150 if it makes a large investment, or profit of 75 if it makes a minor investment. Company A chooses whether or not to invest. Company B knows whether A has invested or not but will not know if A chose to invest big or small when B must decide whether to enter the market and how much to invest. All other payoffs remain the same as in problem 1 other than what has been specifically changed. a. Draw the extensive form of this game.b. What are the information sets in this game? What are the subgames?c. Is this a game of perfect or imperfect information, and why?d. Identify and Nash equilibria of this game.3. Suppose that the two companies in (1) and (2) also supply railway engines. Current demand is characterized by )1000(501QP where P is the price of the 1good (railway engines) and Q is the quantity. A company’s cost function is2)(2QQC .a. Suppose that A cannot supply the market due to some legal issues. What isthe best quantity for B to supply?b. What is the price B demands?c. Assume that A has successfully resolved it’s legal issues and is able to supply the market. Calculate the Cournot equilibrium output for each firm.d. What is the equilibrium price?e. Find the Stackelberg quantity leadership equilibrium for the game where Bis the leader. Calculate the equilibrium output for each firm. What is the equilibrium price?4. Three oligopolists operate in a market with inverse demand given byQaQP )(, where 321qqqQ and iq is the quantity produced by firm i. Each firm has a constant marginal cost of production, c, and no fixed cost. The firms choose their quantities as follows: 1) firm 1 chooses01q; (2) firms 2 and 3 observe 1q and then simultaneously choose 2q and3q, respectively. a. What is the subgame-perfect outcome?b. Now, assume that all three firms above are Cournot oligopolists. Consider the infinitely repeated Cournot game. What is the lowest value of the discount factor such that these three firms can use trigger strategies to sustain the monopoly output level in a subgame-perfect Nash equilibrium. Show all your work.
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