There are two groups of equal size: the clumsy and the graceful. Each group has a utility function given by = where = is the initial wealth level for every individual. Each member of the clumsy group faces accidentally damaging their own property and incurring a loss of 36 with probability 0.5. Each member of the graceful group 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 the graceful group 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 a test for identifying which persons belong to which group (they force them to make a video of themselves attempting ballet). The test is good but not perfect - if the test says that a person belongs to a particular group, the probability that he really does belong to that group is < . How large must be in order to alter your answer to part (b)?