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Consider the following game of incomplete information played by a worker (W) and a firm (F). M' c' N WH High (p) E e M C 6,10 0,4

Consider the following game of incomplete information played by a worker (W) and a firm (F). M' c' N WH High (p) E e M C 6,10 0,4

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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5. Consider the following game of incomplete information played by a worker (W) and a firm (F). 10, 10 4,4 10,0 4,4 M' C' M' C' (r) i (1-r) N WH High (p) Low (1-p) E N' WL E' M M с 6, 10 0.4 3,0 -3.4 The worker has private information about her level of ability. With probability p she is a high-ability type (H) and with probability 1 - p she is a low-ability type (L). After observing her own type, the worker decides whether to obtain a costly education (E) or not (N); think of E as getting a degree. The firm observes the worker's education but the firm does not observe the worker's quality type. The firm then decides whether to employ the worker in an important managerial job (M) or in a much less important job (C). The payoffs are represented in the above extensive form. (a) Is there a separating perfect Bayesian equilibrium in which the high- ability type worker obtains an education and the low-ability type does not? If yes, fully describe such an equilibrium. If no, prove there is not such an equilibrium. (b) Are there any pooling perfect Bayesian equilibria in which both types of workers take the same actions (pure strategies) regarding education? If yes, fully describe all such equilibria. If these depend on p, describe how. If no, prove there is not such an equilibrium. (c) If there are equilibria in both (a) and (b), which does the firm prefer? Explain. (d) If there are equilibria in both (a) and (b), which does the low-ability worker prefer? Explain. (e) If there are equilibria in both (a) and (b), which does the high-ability worker prefer? Explain. (f) In light of your answers to (c), (d), and (e), is it possible to rank the pooling and signaling/separating equilibria from a social welfare perspective? Explain.
Transcribed Image Text:5. Consider the following game of incomplete information played by a worker (W) and a firm (F). 10, 10 4,4 10,0 4,4 M' C' M' C' (r) i (1-r) N WH High (p) Low (1-p) E N' WL E' M M с 6, 10 0.4 3,0 -3.4 The worker has private information about her level of ability. With probability p she is a high-ability type (H) and with probability 1 - p she is a low-ability type (L). After observing her own type, the worker decides whether to obtain a costly education (E) or not (N); think of E as getting a degree. The firm observes the worker's education but the firm does not observe the worker's quality type. The firm then decides whether to employ the worker in an important managerial job (M) or in a much less important job (C). The payoffs are represented in the above extensive form. (a) Is there a separating perfect Bayesian equilibrium in which the high- ability type worker obtains an education and the low-ability type does not? If yes, fully describe such an equilibrium. If no, prove there is not such an equilibrium. (b) Are there any pooling perfect Bayesian equilibria in which both types of workers take the same actions (pure strategies) regarding education? If yes, fully describe all such equilibria. If these depend on p, describe how. If no, prove there is not such an equilibrium. (c) If there are equilibria in both (a) and (b), which does the firm prefer? Explain. (d) If there are equilibria in both (a) and (b), which does the low-ability worker prefer? Explain. (e) If there are equilibria in both (a) and (b), which does the high-ability worker prefer? Explain. (f) In light of your answers to (c), (d), and (e), is it possible to rank the pooling and signaling/separating equilibria from a social welfare perspective? Explain.
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