Franny is the manager of a used car lot. Manny is one of Franny's sales associates. Franny wants Manny to sell lots of used cars to maximize the profits for her lot, but isn't sure how best to structure Manny's contract in order to induce his best effort. Manny can either provide maximum effort or minimum effort, which are each stochastically linked to the used car lot's profits according to the following table showing the probability of low, medium or high profits for the lot given Manny's effort level: Probability of Medium Manny's Effort Probability of Low Profits Probability of High Profits Profits Maximum Effort 33.33% 33.33% 33.33% Minimum Effort 66.66% 33.33% 0% Providing maximum effort costs Manny 20 utils, while providing minimum effort only costs him 18 utils, so Manny will need to be compensated in order to provide the additional effort. Unfortunately, Franny manages several lots and does not get to observe Manny's effort level directly. She can only observe the profit level of the lot he is working on. Suppose Franny offers a standard wage (w) each month with a bonus (bm) if the lot earns medium profits and a bonus (bh ) if the lot earns high profits. Suppose Manny has a stochastic utility reference level (varying according to the probabilities in the table above) given by: u (ac) act ( x - r ) , ifrza x - 1(r - x), ifr > xA. Write down Manny's expected utility if he chooses to exert minimum effort B. Write down Manny's expected utility if he chooses to exert maximum effort C. Manny tells Franny that she does not need to worry about a fancy contract to motivate his effort level. He promises to work for free (w=0) until the end of the month, at which point he would expect his efforts to be sufciently compensated via a bonus. It should be clear that for Manny to exert maximum effort it must be the case that doing so brings him greater utility than doing nothing at all. This is known as the participation constraint among economists working in the eld of mechanism design. In this case, Manny can only achieve high prots if he provides maximum effort, but even with maximum effort those high prots are not certain. What is the smallest bonus must Franny offer to pay Manny in the event of high prots in order to get him to exert maximum effort? Assume for now that Manny is risk-neutral - i.e. A : 1 D. How would your answer change if Manny were loss-averse and A = 2 ? How big of a bonus would Franny need to offer Manny to induce his maximum effort? E. Suppose Franny looks at the bonus gure in part (D) and decides it is impossibly high. She instead wants to offer smaller bonus (9 = bm = bh) for medium or high prots and a xed wage (w). How large would that wage need to be to encourage maximum effort from Manny? To put the question another way, what wage level satises the incentive-compatibility constraint? Remember from your intermediate micro class that the incentive compatibility constraint requires the expected utility associated with the maximum effort level to equal or exceed the expected utility associated with the minimum effort level? F. Imagine now that Manny is overcondent in his abilities as a sales associate. Specically, he believes that the relationship between his effort and the used car lot's earnings are actually those given in this table: Probab'l't of Med' m Manny's Effort Probability of Low Prots Prots l l y I\" Probability of High Prots Maximum Effort 0% 33.33% 66.66% Minimum Effort 66.66% 33.33% 0% Return to the assumption that Manny is risk-neutral - i.e. A = 1 and is willing to work only for a bonus. What is the smallest bonus must Franny offer to pay Manny in the event of high prots in order to get him to exert maximum effort? How does this answer compare to the bonus Franny needed to offer the reasonably condent Manny previously considered in part (C)? G. Finally, show that if Manny is loss-averse and A : 2 Franny will nd it more expensive to employ Manny. Why is this the case