We have now discussed problems of adverse selection and problems of moral hazard. In this problem, you may tn; your hand at a situation with both at the same time. If you find this problem too hard, wait until the end of the next chapter, when you should find it ven; easy. Consider a situation just like the salesman problem in section 16.2, with one additional fillip. Specifically, the salesman will be of either high or low ability. To be exact: You are going to hire a salesman to try to make a particular sale for you. You are risk neutral in all this. He is risk averse, with von Neumann-Morgenstern utility function y'x - a, where x is the amount you pay him, and a is an index of the level of effort he exerts. He is either of good ability or of bad ability, each with probability .5. In parts (a) and (b) you will know his level of ability; in later parts you will not. He will exert either a high level of effort or a low, and you will be unable to observe which. (Note: Good and bad pertain to ability, and high and low pertain to effort.) Regardless of his level of ability, he has reservation utility level 5. And if he puts in
a high level of effort, a in his utility function is 1; a = 0 for a low level of effort.
He either will make the sale or he won't. The sale, gross of yo'ur payment to him, will net you $100 ifhe makes it. The chances that he makes the sale depend both on his level of ability and on his level of effort. Specifically:
If he is good and gives high effort, he makes the sale with probability .9. If he is good and gives low effort, he makes the sale with probability .6. If he is bad and gives high effort, he makes the sale with probability .5. If he is bad and gives low effort, he makes the sale with probability .3. (This is a world in which ability counts more than effort.)
Although it doesn't make much sense in the context I have outlined, as sume that you are only allowed to deal with one salesman. You may offer him a contract, which he will either accept or reject; if he rejects, you net zero. Any time that the salesman is indifferent, he will elect the option that is more favorable to you. (That is, you don't have to worry about ties.)
(a) Suppose you are able to observe his ability, and you find that the salesman you are dealing with has bad ability. What, from your point of view, is the optimal contract to offer him? Will he give high or low effort under this contract? What will be your profits?
(b) Suppose you are able to observe his ability, and you find that the salesman you are dealing with has good ability. What, from your point of view, is the optimal contract to offer to him? Will he give high or low effort under this contract? What will be your profits?
(c) Now suppose that you are unable to observe the ability of the salesman you are dealing with. If you offer him a choice from the contracts you identified in parts (a) and (b), what will be the outcome for you? Why is this? Be very clear and explicit as to what you expect to happen in this case.
(d) You are unable to observe the ability of the salesman that you are dealing with, and you may either offer him a pair of contracts from which he can chodse, or a single contract, that he may elect to take or not. (If you offer a pair of contracts, he can refuse employment altogether as well, although remember that when he is indifferent, he is presumed to do what is in your interests.) I want you to produce an offer to him that does better than offering him the pair of contracts from parts (a) and (b). Say what the contract(s) is (are), and what will be the outcome of your offer. You should produce the best contract(s) you can -ideally, you should devise the optimal contract(s) to offer him, and you should prove that this is the optimal offering.