The amount of output produced by each worker at a firm depends upon the worker's ability. Specifically, each worker produces 1 + θ units of goods each day, where θ represents the worker's ability and varies between workers. The abilities of the firm's current workers are uniformly distributed on the interval [0, 1]. Workers know their own abilities, but the firm cannot observe an individual worker's ability or his contribution to output

a) What is the average daily output of the firm's workers?

b) The firm reduces its workforce by 20% by offering the workers a "buy-out" (i.e., financial compensation for voluntarily leaving the firm). Workers with higher abilities are more confident of finding equally good employment elsewhere, and hence the 20% of the workforce with the highest ability leave the firm. What is the average daily output of the firm's workers after the buy-out?

c) Suppose that the firm randomly selects 20% of its workforce to be laid off. What is the average daily output of the firm's workers after the lay-offs?

d) Suppose that the firm offers a buy-out in order to reduce its workforce by 20%, but that its buy-out is so generous that 40% of the workforce accepts it. The firm reduces its workforce by 20% by giving the buy-out to half of the workers who wanted it and retaining the remainder of the workers. The workers given the buy-out are randomly selected from the workers who asked for it.

What is the average daily output of the firm's workers after the buy-out?