If he becomes an attorney, he will make M 900/yr if he becomes a partner in a Wall Street firm, but only M 25/yr if he fails to make partner. The probability of his becoming a partner is 0.2. Smith is an infallible judge of legal talent. After a brief interview, he can state with certainty whether John will become a partner. What is the most John would be willing to pay for this information? (Set up the relevant equation. You don’t need to solve it.)

*18. In the preceding problem, assuming that the interview is costless for Smith to conduct, is he getting the highest possible expected income for himself by charging John the same fee regardless of the outcome of the interview?

*19. There are two groups of equal size, each with a utility function given by where M 100 is the initial wealth level for every individual. Each member of group 1 faces a loss of 36 with probability 0.5. Each member of group 2 faces the same loss with probability 0.1.

a. What is the most a member of each group would be willing to pay to insure against this loss?

b. In part (a), if it is impossible for outsiders to discover which individuals belong to which group, will it be practical for members of group 2 to insure against this loss in a competitive insurance market? (For simplicity, you may assume that insurance companies charge only enough in premiums to cover their expected benefit payments.) Explain.

c. Now suppose that the insurance companies in part

(b) have an imperfect test for identifying which persons belong to which group. If the test says that a person belongs to a particular group, the probability that he really does belong to that group is x  1.0. How large must x be in order to alter your answer to part (b)?
*20. There are two groups, each with a utility function given by where M 144 is the initial wealth level for every individual. Each member of group 1 faces a loss of 44 with probability 0.5. Each member of group 2 faces the same loss with probability 0.1.

a. What is the most a member of each group would be willing to pay to insure against this loss?

b. If it is impossible for outsiders to discover which individuals belong to which group, how large a share of the potential client pool can the members of group 1 be before it becomes impossible for a private company with a zero-profit constraint to provide insurance for the members of group 2? (For simplicity, you may assume that insurance companies charge only enough in premiums to cover their expected benefit payments and that people will always buy insurance when its price is equal to or below their reservation price.) Explain.
*21. Given a choice between A (a sure win of 100) and B (an 80 percent chance to win 150 and a 20 percent chance to win 0), Smith picks A. But when he is given a choice between C (a 50 percent chance to win 100 and a 50 percent chance to win 0) and D (a 40 percent chance to win 150 and a 60 percent chance to win 0), he picks D. Show that Smith’s choices are inconsistent with expected utility maximization.