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1. (a) Suppose a manager and a worker interact as follows. The manager decides whether to hire or not hire the worker. If the manager does not hire the worker, then the game ends. When hired, the worker chooses to exert either high effort or low effort. On observing the workerís effort, the manager chooses to retain or fire the worker. In this game, does "not hire" describe a strategy for the manager? Explain.
(b) Find the pure strategy Nashequilibria in the following games:
2. Consider the following Cournot quantity-setting game. Suppose there are n > 2 firms and that all firms i = 1, .., n simultaneously and independently choose output yi ∈ [0; 1): Aggregate production is:
Let p(y) = a - y be the inverse market demand function, where a > 0: Assume no fixed costs and that the firms have the same marginal cost c < a:
(a) Describe the normal-form of this game.
(b) Find the symmetric Nash equilibrium.
(c) What happens to the equilibrium price as n approaches infinity? Explain.
3. Let there be two firms that compete by setting quantities sequentially. Output is homogenous and the inverse market demand isp (y) = 12 - y: Assume that both firms have zero cost, and that their objective is to maximize profits.
(a) Draw the extensive form of this game.
(b) Verify that the following strategy constitutes a Nash equilibrium:
(c) Explain why the Nash equilibrium above is not a subgame perfect Nash equilib- rium.
4. Consider the following simultaneous-move game of asymmetric information played by two firms. First, nature selects a number x 2 f4; 8g with equal probability. Firm one observes x; but firm two does not. Firm two believes that x is either 4 or 8 with equal probability. Once firm one has observed x; the two firms compete by simultaneously setting quantities y1 and y2 ; respectively. Assume that costs are zero. The payoffs to firmi = 1; 2 with i ≠ j are:
πi (x; yi ; yj ) = (x - yi - yj ) yi :
That is, x is a measure of how strong demand for the good is.
(a) Find the firms'best reply functions.
(b) What is the Bayesian Nash equilibrium (BNE)?
(c) Verify that having information about x gives firm one an advantage over firm two.