Assignment:
Suppose you are a factory owner who needs to hire a worker to run a production line. You know there are two types of worker, high ability and low ability, and there are equal numbers of each in your labor market, but you cannot observe the ability of an individual worker before you hire him. Suppose that workers choose an effort level e, which can be any non-negative number, and which you cannot observe. With a high-ability worker, your production line's output will be q = 2e. With a low-ability worker, your production line's output will be q = e. You know that all workers have a cost of effort function equal to c(e) = e2, and their utility functions from a wage of w and an effort level of e are U(w,e) = w - e2. It is standard practice to pay workers per unit of output, so you need to decide on the per-unit rate to offer, denoted b.
Thus, the wage you will be paying is w = bq. The market price for your good is 5, and your cost of raw materials per unit is 1, so your profit will be π = 5q - q - w = 4q - w (assume no fixed costs for simplicity).
a. What is the optimal effort level for a low-ability worker receiving a per-unit wage of b in this scenario? What would the output of such a worker be?
b. What is the optimal effort level for a high-ability worker receiving a per-unit wage of b in this scenario? What would the output of such a worker be?
c. Given that you do not know the ability of the worker before you hire him, which value of b maximizes your expected profits? What are your maximized expected profits?
d. (extra credit) How much would you be willing to pay for a screening system that let you know worker ability before you make your hiring decision?