Sam is suing someone in court for $10,000. The probability that Sam will lose the case is 1/h where h is the number of hours that Sam's attorney works on the case

The lawyer charges $500 per hour if he is to be paid hourly, or he requests 20% of the settlement if he is to be paid on a contingency basis. Assuming both Sam and the attorney are risk-neutral wealth maximizers, is either contract efficient?

For a per hour contract, first determine the expected value of the settlement as a function of the hours worked. E(X) = 10,000 ? (1 - (1/h)) = 10,000-10,000h-1. For Sam, the marginal benefit is 10,000h-2 and his marginal cost is $500. Equating the two, Sam's expected outcome is maximized at h = 4.472 hours. The attorney's income is 500 ? h. He wishes to bill (hopefully work) as many hours as possible. This can (will) exceed 5. Under the contingency arrangement, Sam does not incur a per hour marginal cost. His marginal benefit is 8,000h-2 thus, he wants the attorney to work on the case until the expected marginal cost is zero; that is, h goes to infinity. The attorney incurs a $500 opportunity cost for each hour spent on the case but enjoys only 20% of the marginal expected benefit--his expected marginal benefit is 2,000h-2. Equating his marginal benefit to his marginal cost means he will stop work after two hours. Thus, both plans are inefficient in production. One plan has the attorney billing too many hours; the other has the attorney working not enough.