Suppose that everything in the grade insurance market is as described in exercise 22.1. But instead of taking the asymmetric information as fixed, we will now ask what can happen if students can transmit information. Assume throughout that no insurance company will sell A insurance to students other than A students, B insurance to students other than B students, etc. whenever they know what type students are.
A. Suppose that a student can send an accurate “signal” to me about the type of student he is by expending effort that costs c. Furthermore, suppose that each student can signal that he is a better student than he actually is by expending additional effort c for each level above his true level. For instance, a “C student” can signal his true type by expending effort c but can falsely signal that he is a “B student” by expending effort 2c and that he is an “A student” by expending 3c.
(a) Suppose everyone sends truthful signals to insurance companies and that insurance companies know the signals to be truthful. What will be the prices of A-insurance, B-insurance, C-insurance and D-insurance?
(b) How much surplus does each student type get (taking into account the cost c of sending the truthful signal).
(c) Now investigate whether this “truth-telling” can be part of a real equilibrium. Could B students get more surplus by sending a costlier false signal? Could C, D or F students?
(d)Would the equilibrium be any different if it was costless to tell the truth but it costs c to exaggerate the truth by each level? (Assume F-students would be willing to pay 1.5c for getting an F just as other students are willing to pay 1.5c to get their usual grade.)
(e) Is the equilibrium in part (d) efficient? What about the equilibrium in part (c)? (Hint: Think about the marginal cost and marginal benefit of providing more insurance to any type.)
(f) Can you explain intuitively why signaling in this case addresses the problem faced by the insurance market?
B. In Section B of the text, we considered the case of insurance policies (b,p) in an environment where the “bad outcome” in the absence of insurance is x1 and the “good” outcome in the absence of insurance is x2. We further assumed two risk types: δ types that face the bad outcome with probability δ and θ types that face the bad outcome with probability θ —where θ > δ.
(a) Suppose that both types are risk averse and have state-independent tastes. Show that, under actuarially fair insurance contracts, they will choose the same benefit level b but will pay different insurance premiums.
(b) Suppose throughout the rest of the problem that insurance companies never sell more than full insurance; i.e. they never sell policies with b higher than what you determined in (a). In Section B we focused on self-selecting equilibria where insurance companies restrict the contracts they offer in order to get different types of consumers to self-select into different insurance policies. In Section A, as in part A of this question, we focused on explicit signals that consumers might be able to send to let insurance companies know what type they are. How much would a θ type be willing to pay to send a credible signal that he is a δ type if this will permit him access to the actuarially fair full insurance contract for δ types?
(c) Suppose for the rest of the problem that u(x) = lnx is a function that permits us to represent everyone’s tastes over gambles in the expected utility form. Let x1 = 10, x2 = 250, δ= 0.25 and θ = 0.5 as in the text. Suppose further that we are currently in a self-selecting equilibrium of the type that was discussed in the text (where not all actuarially fair policies are offered to δ types). How much would a δ type be willing to pay to send a credible signal to an insurance company to let them know he is in fact a δ type?
(d) Suppose we are currently in the separating equilibrium but a new way of signaling your type has just been discovered. Let ct be the cost of a signal that reveals your true type and let c f be the cost of sending a false signal that you are a different type. For what ranges of ct and c f will the efficient allocation of insurance in this market be restored through consumer signaling?
(e) Suppose ct and cf are within the ranges you specified in (d). Has efficiency been restored?