22.3 In exercise 22.2, we showed how an efficient equilibrium with a complete set of insurance markets can be reestablished with truthful signaling of information by consumers. We now illustrate that signaling might not always accomplish this.

A. Begin by once again assuming the same set-up as in exercise 22.1. Suppose that it costs c to truthfully reveal who you are and 0.25c more for each level of exaggeration; that is, for a C student, it costs c to reveal that he is a C student, 1.25c to falsely signal that he is a B student, and 1.5c to falsely signal that he is an A student.

a. Begin by assuming that insurance companies are pricing A-, B-, C-, and D-insurance competitively under the assumption that the signals they receive are truthful. Would any student wish to send false signals in this case?

b. * Could A-insurance be sold in equilibrium (where premiums have to end up at zero-profit rates given who is buying insurance)? (Hint: Illustrate what happens to surplus for students as premiums adjust to reach the zero-profit level.)

c. * Could B-insurance be sold in equilibrium? What about C- and D-insurance?

d. * Based on your answers to

(b) and (c), can you explain why the equilibrium in this case is to have only D-insurance sold, and bought by both D and F students? Is it efficient?