ECON3014 : GAME THEORY Exercise 4 1. Consider a competitive labor market with a continuum of workers. Each worker is characterized by his productivity 2 fH; Lg. The proportion of workers with type H is . Before applying for a job, each worker chooses the level of his education e 0. The education is costly and has no eect on productivity. If a worker with productivity gets an education e and a wage ! his payois U(e; !; ) = ! ?? e2 : If a worker decides not to participate in the job market, he gets zero payo.

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ECON3014 : GAME THEORY Exercise 4 1. Consider a competitive labor market with a continuum of workers. Each worker is characterized by his productivity 2 fH; Lg. The proportion of workers with type H is . Before applying for a job, each worker chooses the level of his education e 0. The education is costly and has no eect on productivity. If a worker with productivity gets an education e and a wage ! his payois U(e; !; ) = ! ?? e2 : If a worker decides not to participate in the job market, he gets zero payo. a) Argue that in the competitive market, a wage paid to a worker equals to his expected productivity (conditional on all the information about this worker known to rms). b) What is the dierence between separating and pooling equilibria in this model? Find the separating equilibrium with the lowest possible education level for the high type. c)What is the largest level of education that can be chosen by workers with productivity H in equilibrium in this model? d) Find a pooling equilibrium in this model. e) Suppose the equilibrium that you found in b) is played. The government comes up with a bill that prohibits wage discrimination based on education (i.e. holding other publicly observable workers’ characteristics the same, rms can not oer dierent wages to people with dierent education levels). To pass the bill, the government holds a referendum in which only workers can vote. Assuming that voters vote sincerely, i.e. voters pick an alternative that gives them the highest payo, and that a simple majority is required for passing the bill, derive a necessary and sucient condition for passing this bill (the condition must be formulated in terms of fundamentals of the model) 2. Firm A (the acquirer”) is considering taking over rm T (the target”). It does not know rm T’s value; it believes that this value, when rm T is controlled by its own management, is at least $0 and at most $100, and assigns equal probability to each of the 101 dollar values in this range. Firm…

ECON3014 : GAME THEORY Exercise 4 1. Consider a competitive labor market with a continuum of workers. Each worker is characterized by his productivity 2 fH; Lg. The proportion of workers with type H is . Before applying for a job, each worker chooses the level of his education e 0. The education is costly and has no eect on productivity. If a worker with productivity gets an education e and a wage ! his payois U(e; !; ) = ! ?? e2 : If a worker decides not to participate in the job market, he gets zero payo. a) Argue that in the competitive market, a wage paid to a worker equals to his expected productivity (conditional on all the information about this worker known to rms). b) What is the dierence between separating and pooling equilibria in this model? Find the separating equilibrium with the lowest possible education level for the high type. c)What is the largest level of education that can be chosen by workers with productivity H in equilibrium in this model? d) Find a pooling equilibrium in this model. e) Suppose the equilibrium that you found in b) is played. The government comes up with a bill that prohibits wage discrimination based on education (i.e. holding other publicly observable workers’ characteristics the same, rms can not oer dierent wages to people with dierent education levels). To pass the bill, the government holds a referendum in which only workers can vote. Assuming that voters vote sincerely, i.e. voters pick an alternative that gives them the highest payo, and that a simple majority is required for passing the bill, derive a necessary and sucient condition for passing this bill (the condition must be formulated in terms of fundamentals of the model) 2. Firm A (the acquirer”) is considering taking over rm T (the target”). It does not know rm T’s value; it believes that this value, when rm T is controlled by its own management, is at least $0 and at most $100, and assigns equal probability to each of the 101 dollar values in this range. Firm…

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