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Unformatted text preview: Economics 467 Spring 2010 Karl Dunz Problem Set 4 Answers 1. (a) A perfect Bayesian equilibrium for this game consists of giving values for the following: ∙ The probability a low cost monopoly chooses high capacity. Call this u1D44E u1D43F . ∙ The probability a high cost monopoly chooses high capacity. Call this u1D44E u1D43B . ∙ The probability the potential entrant chooses to enter if the monopoly has low capacity. Call this u1D44F u1D43F . ∙ The probability the potential entrant chooses to enter if the monopoly has high capacity. Call this u1D44F u1D43B . ∙ The probability the potential entrant thinks the monopoly is low cost if the monopoly has high capacity. Call this u1D707 u1D43B . ∙ The probability the potential entrant thinks the monopoly is low cost if the monopoly has low capacity. Call this u1D707 u1D43F . The extensive form for this game is given below. Nature low cost p 1p high cost Entrant Entrant Monopoly high capacity Enter1, 9 Out 0, 15 low capacity Enter 1, 8 Out 0, 12 Monopoly high capacity Enter 1, 3 Out 0, 8 low capacity Enter 1, 4 Out 0, 8 (b) When the monopoly has low capacity the potential entrant obtains 1 if it enters and 0 if it doesn’t. This is independent of whether the monopoly is high cost or low cost. Therefore, the potential entrant will always enter when it sees the monopoly has low capacity. So u1D44F u1D43F = 1. This means that a monopoly with low costs will receive 8 if it chooses low capacity. Such a monopoly will receive either 9 (if there is entry) or 15 (if there is no entry) if it chooses high capacity. Since 1 the monopoly’s payoff is higher with high capacity no matter what the potential entrant does, a low cost monopoly will always choose high capacity. So u1D44E u1D43F = 1. (c) If the potential entrant observes the monopoly having high capacity, staying out results in an expected payoff of 0 no matter what. The entrant’s expected payoff of entering is − u1D707 u1D43B +(1 − u1D707 u1D43B ). Therefore, the potential entrant will enter ( u1D44F u1D43B = 1) if u1D707 u1D43B < 1 / 2, stay out ( u1D44F u1D43B = 0) if u1D707 u1D43B > 1 / 2, and can choose any value of u1D44F u1D43B if u1D707 u1D43B = 1 / 2. (d) Given that we have already established that u1D44F u1D43F = 1, a highcost monopoly will obtain a payoff of 4 if it chooses low capacity and an expected payoff of 3 u1D44F u1D43B + 8(1 − u1D44F u1D43B ) = 8 − 5 u1D44F u1D43B if it chooses high capacity. These two payoffs are equal when u1D44F u1D43B = 4 / 5. Therefore, the highcost monopoly will choose high capacity (5....
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 Spring '10
 Dunz
 Game Theory, Monopoly, PBE, high capacity

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