1. (Adverse Selection) Consider a labor market model with many identical firms hiring workers. The firms produce a homogeneous product with a constant-returns-to-scale technology and act as price takers (we normalize the price of the product to $1). A worker, if hired by a firm, can produce µ units of output per day, where µ differs across workers. There are three possible values of µ: µ1 = 10, µ2 = 20 and µ3 = 30. A worker with daily output µi is called a type i ∈ {1; 2; 3} worker. The fraction of each type of worker is 1=3. A worker can choose to work either at a firm or at home. If a type i worker chooses to work at home, he earns $ (0:7 × µi) per day. Assume that a type i worker chooses to work for a firm if and only if he can obtain a wage no less than $ (0:7 × µi) per day.

(a) Suppose that µ is publicly observable. Specify the competitive equilibrium of the labor market with complete information. Who will be employed in the equilibrium? Is the equilibrium outcome Pareto efficient?

(b) Suppose that µ is each worker’s private information, and the firms only know its distribution. Specify the competitive equilibrium of the labor market with asymmetric information. Who will be employed in the equilibrium? Is this equilibrium outcome Pareto efficient?

(c) Now suppose that every worker earns $15 per day if he chooses to work at home, regardless of his type. Hence, a worker chooses to work for a firm if and only if he can obtain a wage no less than $15 per day. Everything else in the model remains the same as above. As in part b, specify the competitive equilibrium of the labor market with asymmetric information. Who will be employed in the equilibrium? Is this equilibrium outcome Pareto efficient?

2. (Signaling) Prof. Li has an extra ticket to a concert. He wants to sell it, but prefers to sell it to a person who cares about people in need. The ticket is worth $1000 to whoever he sells it to. A potential buyer approaches him. Prof. Li cannot tell whether this potential buyer is a caring person or not, but believes the probability he is to be 1=2. If the potential buyer does care, then donation to the charity yields him some utility. More specifically, for every $1 he donates, he receives 50 cents worth of utility from being helpful for those in need. In effect, then, donating $1 costs him only 50 cents. On the other hand, for a person who does not care, donating $1 costs $1 since they do not get such utility from the donation. Prof. Li asks the potential buyer to donate, sees how much the person donates, and then sets a price for the ticket. If he concludes that the potential buyer is a caring person, he sets the price at $200. If he concludes that the potential buyer is not a caring person, he sets the price at $600. If he cannot tell, he sets the price at the expected value, $400. The potential buyer knows all of the above.

(a) Specify all the separating equilibria.

(b) Specify all the pooling equilibria. 1

3. (Signaling) There are three types of workers, type 1, 2, and 3, whose marginal products are $40, $80, and $100 respectively. The fraction of each type is 1/3. Employers observe only the worker’s level of education, not his type, and pay the expected marginal product given their observations. Workers only care about the salary they get minus the cost of the education. The cost of education for a type 1 is $20 per year, for a type 2 is $10 per year, and for a type 3 is $5 per year.

(a) Fully describe a pooling equilibrium in which all three types of workers obtain the same level of education.

(b) Is there a pooling equilibrium in which all three types of workers obtain 2 years of education? Explain.

(c) Fully describe a completely separating equilibrium in which three types of workers choose three different levels of education.

(d) Is there a completely separating equilibrium in which a type III worker obtains 3 years of education? Explain.

(e) Fully describe a partially separating equilibrium in which the workers of type I and II obtain no education, and a type III worker obtains some education.