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# ECONS 424 - STRATEGY AND GAME THEORY HOMEWORK #4 - ANSWER KEY Exercise 2 Chapter 16 Watson Solving by backward induction: We start from the second...

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1 E CON S 424 – S TRATEGY AND G AME T HEORY H OMEWORK #4– A NSWER K EY Exercise 2 ­ Chapter 16 Watson Solving by backward induction: 1. We start from the second stage of the game where both firms compete in prices. Since market demand is Q = a ­ p , then products are homogeneous, and in addition, we are told in the exercise that the firm setting the lowest price gets all the market. Hence, we are in a Bertrand game of price competition, and we know from class that the equilibrium price firms set is P 1 = P 2 = 0. Importantly, note that prices are not functions of the expenditure on advertising that firm 1 makes during the first period. 2. Since this is the case, firm 1 knows that by spending more money on advertising it will not increase the profits during the second period. As a consequence, a = 0 during the first period. Therefore, the subgame perfect equilibrium is a = 0 during the first stage and P 1 = P 2 = 0 during the second stage.
2 Exercise 8 ­ Chapter 16 Watson a. Without payoffs, the extensive form is as follows [Note that we are using dashed lines to denote that firm 2 chooses q 2 without observing firm 1’s output q 1 . Similarly, firm 3 chooses q 3 without observing firm 1 and firm 2’s output, q 1 and q 2 , respectively.]: Solving by backward induction, we must first find the output level of every possible entry/no entry scenario. By doing so, we will be able to find the profits resulting from every possible entry/no entry scenario, and then we will be ready to compare firms’ profits from entering and not entering: 1. We first solve firms’ output in the subgame that starts after firm 1 and 2 enter. [In the figure, this subgame is the upper part, where firms are selecting q 1 , q2and q 3 ] This is just a Cournot game of quantity competition with three firms competing with each other by simultaneously selecting output. Hence, ݍ ൌݍ ൌݍ ൌ3 . a. PROFITS: In this case, note that the profits of every firm in this Cournot oligopoly game with three firms are: ሺ12 െ ܳሻݍ ൌ ሺ12 െ ݍ െݍ െݍ ሻݍ ൌ ሺ12െ3െ3െ3ሻכ3ൌ3כ3ൌ9 .
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1 E CON S 424 S TRATEGY AND G AME T HEORY M IDTERM E XAM #2 (T AKE -H OME E XAM ) D UE DATE : T UESDAY , M ARCH 26 TH 2013, IN CLASS . Instructions: Write your answers to each exercise in a different page. Show all your work, and be as clear as possible in your answer. You can work in groups, but each student must submit a copy of his/her exam. The due date of this take-home exam is Tuesday, March 26 th , in class. I strongly recommend you to work a few exercises every day, rather than trying to solve all exercises in one day. Late submission will be subject to significant grade reduction.
2 Exercise 1 Consider the entry-exit two-stage game represented in the figure below in which firm A is the incumbent firm that faces potential entrant firm B. In stage I, firm B decides whether to enter into A's market or whether to stay out. The cost of entry is denoted by ε . In stage II, the established firm, firm A, decides whether to stay in the market or exit. The game tree reveals that firm A can recover some of its sunk entry cost by selling its capital for the price φ , where 0 φ ε ≤≤ . Solve the two problems: a) Compute the subgame-perfect equilibrium strategies of firms B and A assuming that 60 ε < . Prove your answer. Answer the above question assuming that 60 100 φ ε < ≤< Exercise 2 Return to the game with two neighbors creating positive externalities on each other that you solve in Homework #4 (Exercise #3, see the exercise and its answer key on the course website) . Continue to suppose that player i's average benefit per hour of work on landscaping is 10 2 j i l l + Continue to suppose that player 2’s opportunity cost of an hour of landscaping work is 4. Suppose that player l’s opportunity cost is either 3 or 5 with equal probability (1/2), and that this cost is player 1’s private information (player 2 does not observe this information). a) Find player 1’s best response function when his opportunity cost is 3. Denote it as l L 1 (l 2 ). b) Find player 1’s best response function when his opportunity cost is 5. Denote it as l H 1 (l 2 ). c) Find player 2’s best response function. Denote it as l L 2 (l L 1, l H 1 ). d) Solve for the Bayesian-Nash equilibrium. Exercise 3 Consider two consumers (1, 2), each with income M to allocate between two goods. Good 1 provides 1 unit of consumption to its purchaser and α units of consumption to the other consumer, where 01 α . Each consumer i, i = 1,2, has the utility function 12 log( ) ij i Ux x = + where 11 1 ii j x yy α =+ is his consumption of good 1 (that takes into account the amount of good 1 purchased by individual i and j ), and 2 i x is his consumption of good 2.
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