In exercise 22.2, we showed how an efficient equilibrium with a complete set of insurance markets can be re-established 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; i.e. 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?
(e) Now suppose that the value students attach to grades is different: They would be willing to pay as much as 4c to guarantee their usual grade and 0.9c more for each level of grade above that. Suppose further that the cost of telling the truth about yourself is still c but the cost of exaggerating is 0.1c for each level of exaggeration about the truth. How much surplus does each student type get from signaling that he is an A student if A-insurance is priced at 2c?
(f) Suppose that insurance companies believe that any applicant for B insurance is a random student from the population of B, C, D and F students; that any applicant for C insurance is a random student from the population of C, D and F students; and any applicant for D insurance is a random student from the population of D and F students. How would they competitively price B,C and D insurance?
(g) Suppose that, in addition, insurance companies do not sell insurance to students who did not send a signal as to what type they are. Under these assumptions, is it an equilibrium for everyone to signal that they are A students?
(h) There are two sources of inefficiency in this equilibrium. Can you distinguish between them?
B. In exercise 22.2B, we introduced a new “signaling technology” that restored the efficient allocation of insurance from an initially inefficient allocation in a self-selecting separating equilibrium. Suppose that insurance companies believe anyone who does not send a signal that he is a δ type must be a θ type.
(a) Suppose that c f is below the range you calculated in B (d) of exercise 22.2. Can you describe a pooling equilibrium in which both types fully insure and both types send a signal that they are δ types?
(b) In order for this to be equilibrium, why are the beliefs about what a non-signal would mean important? What would happen if companies believed that both types are equally likely not to signal?
(c) True or False: for equilibrium like the one you described in part (a) to be equilibrium, it matters what firms believe about events that never happen in equilibrium.