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A lawyer works as an agent for an injured plaintiff. By working harder, the lawyer can increasethe payoff to her plaintiff. Assume there is no randomness; the lawyer can guarantee any payoff ofp she wants. But guaranteeing a higher payoff means the lawyer has to work hard, which is costly.Specifically, suppose that if the lawyer wants to guarantee a payoff p then she has to suffer a costof c(p) = p2/2. For all of the following questions you must show your work; correct answers alonewill get you zero credit.

Problem 4.1 Suppose the plaintiff pays the lawyer a fixed fraction q of her payoff p (where 0 ≤q ≤ 1). For any p, the lawyer gets total profit of qp - c(p) and the plaintiff gets (1 - q)p. Whatpayoff p will the lawyer choose? What will be the resulting profits of the two parties?

Problem 4.2 Which q is best for the plaintiff?Problem 4.3 Now suppose the lawyer takes as payment p - a, where a ≤ p. What will be thelawyer's effort and the two parties' payoffs?

Problem 4.4 Which a is best for the plaintiff, subject to the constraint that the lawyer would neveraccept a payment scheme that gives her negative profit?

Problem 4.5 What does the plaintiff prefer: his best q in Problem 4.2 (proportional payment), orhis best a in Problem 4.4 (fixed payment)?

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