Suppose there are two types of workers, high ability and low ability and and equal numbers of each in your labor market, but you can't observe each workers ability before you hire him. Suppose each worker can choose an effort level 3 which is any number greater than 0.

High workers output q=2e.

Low workers output q=e

All workers have a cost of effort function equal to c(e) = e^2

Their utility from a wafe of w and an effort level of e are U(w,e) = w-e^2

It is standard practive to pay workers per unit of output so you need to decide on a per-unit rate to offer, denoted b.

Thus Wage is w=bq

The market price for your good is 5 and your cost of raw materials is 1 per unit. So

Profit = 5q-q-w=4q-w. Assume no fixed costs.

Question 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?

Question 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?

Question c. Given you don't know the workers ability which value of b maximizes your profits? What are your maximized expected profits?

Question d. How much would you pay for a screening that let you know the workers ability?