dynamicStackelbergAns

dynamicStackelbergAns - A Dynamic Oligopoly Game Consider...

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A Dynamic Oligopoly Game Consider the following Stackelberg game with two f rm sandtwos ta te so f nature. In the high state, the inverse demand function is given by p H = 120 q 1 q 2 where p H is the price and q i is the quantity supplied by f rm i ( i =1 , 2 ). In the low state, the inverse demand function is given by p L =80 q 1 q 2 The probability of each state is 1 2 . Assume that production costs are zero. The timing of moves is the following. Firm 1 moves f rst and selects its quantity. Then f rm 2 chooses its quantity after observing f rm 1’s quantity. That is, q 2 can be a function of q 1 . (a) Suppose f rm 2 can observe the true state of nature before choosing its quantity, but f rm 1 cannot. Calculate the unique subgame perfect Bayesian Nash equilibrium. Be careful to fully specify f rm 2’s strategy. (b) Suppose f rm 1 can observe the true state of nature before choosing its quantity, but f rm 2 cannot. Calculate one weak perfect Bayesian equilibrium (WPBE) of this game. Remember to specify beliefs as well as strategies. (c) Without doing any calculations, carefully explain the intuition for why there are many WPBE of the game in part (b). Answers (a) Since f rm 1 does not know the state, we have a subgame starting when f rm 2 observes the state and f rm 1’s quantity. Given q 1 , f rm 2’s pro f tinthe high state is given by π H 2 = (120 q 1 q H 2 ) q H 2 . (1) Di f erentiating (1) with respect to q H 2 , and setting the expression equal to zero, we have f rm 2’s optimal strategy for the subgame, q H 2 =60 q 1 2 . (2) A similar calculation for the low state yields q L 2 =40 q 1 2 . (3) Thus, (2) and (3) constitute f rm 2’s subgame perfect equilibrium strategy.
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dynamicStackelbergAns - A Dynamic Oligopoly Game Consider...

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