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dynamicStackelbergAns

# dynamicStackelbergAns - A Dynamic Oligopoly Game Consider...

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A Dynamic Oligopoly Game Consider the following Stackelberg game with two fi rms and two states of 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 fi 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 fi rst and selects its quantity. Then fi rm 2 chooses its quantity after observing fi rm 1’s quantity. That is, q 2 can be a function of q 1 . (a) Suppose fi rm 2 can observe the true state of nature before choosing its quantity, but fi rm 1 cannot. Calculate the unique subgame perfect Bayesian Nash equilibrium. Be careful to fully specify fi rm 2’s strategy. (b) Suppose fi rm 1 can observe the true state of nature before choosing its quantity, but fi 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 fi rm 1 does not know the state, we have a subgame starting when fi rm 2 observes the state and fi rm 1’s quantity. Given q 1 , fi rm 2’s pro fi t in the high state is given by π H 2 = (120 q 1 q H 2 ) q H 2 . (1) Di ff erentiating (1) with respect to q H 2 , and setting the expression equal to zero, we have fi 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 fi rm 2’s subgame perfect equilibrium strategy.

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dynamicStackelbergAns - A Dynamic Oligopoly Game Consider...

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