Assume again the basic set-up from exercise 22.1.
A. We will now investigate the role of firm screens as opposed to consumer signals.
(a) Suppose that an insurance company can screen students. More precisely, suppose an insurance company can, for a fee of c, obtain a student’s transcript and thus know what type a student is. If insurance companies will only sell insurance of type i to students who have been screened as type i , what would be the equilibrium insurance premium for each insurance assuming perfect competition (and no recurring fixed costs)?
(b)Would each insurance type be offered and bought in equilibrium?
(c) How high would the cost of obtaining transcripts have to be in order for the insurance market to collapse?
(d) In the case of signaling, we had to consider the possibility of “pooling equilibria” in which the same insurance is sold to different types of students who care sufficiently for the higher grade to each be willing to pay the zero-profit premium as well as, for some, to pay the cost of falsely signaling their type. Suppose it is illegal to charge different prices to different customers for the same insurance product. If insurance companies can screen for the relevant information, could it ever be the case — assuming that individuals care sufficiently much about higher grades — that several types will get the same insurance? (Hint: Suppose an insurance company attempted to price a policy such that several types would get positive surplus by buying this policy. Does another insurance company have an incentive to compete some of the potential customers for that policy away?)
(e) Does the separating equilibrium that results from screening of customers depend on how many of each different type are in the class—and what exactly the curve is that is imposed in the class?
(f) Suppose we currently have a market in which a large number of insurers sell the different insurance types at the zero-profit price after screening customers to make sure insurance of type i is only sold to type i . Now suppose a new insurance company enters the market and devises “B insurance for C students”. Will the new company succeed in finding customers? (g) Would your answer to (f) change if students are willing to pay 1.5c to insure their usual grade and c (rather than 0.5c) for each grade above the usual?
(h) True or False: When insurance companies screen, the same insurance policy will never be sold to different student types at the same price, but it may be the case that students of different types will insure for the same grade.
B. Now consider the introduction of screening into the self-selection separating equilibrium of Section B of the text. As in the text, suppose that consumption in the absence of insurance is 10 in the bad state and 250 in the good state and that δ types have a probability of 0.25 of reaching the bad state while θ types have a probability of 0.5 of reaching that state. Suppose further that individuals are risk averse and their tastes are state-independent.
(a) Instead of graphing b on the horizontal and p on the vertical axis, begin by graphing x2 — i.e. consumption in the good state—on the horizontal and x1 —i.e. consumption in the bad state—on the vertical. Indicate with an “endowment” point E where consumption would be in the absence of insurance.
(b) Illustrate the actuarially fair insurance contracts for the two types of consumers—and indicate the two insurance policies that are offered in a self-selection separating equilibrium.
(c) Suppose a “screening industry” — i.e. an industry of firms that can identify what type an insurance applicant is for a cost of k per applicant — emerges. If an insurance firm gives applicants the option of paying k (as an application fee) to enable the company to pay a screening firm for this information, would θ types pay it?
(d) What is the highest that k can be in order for δ types to agree to pay the fee. Illustrate this in your graph.
(e) The applicant’s decision of whether or not to pay the fee is really a decision of whether to send a signal. How is this different from the type of signal we analyzes in exercise 22.3? In particular, why does θ’s signaling behavior matter in exercise 22.3 but not here?
(f) Suppose that instead of asking applicants to pay the screening fee, the insurance company paid to get the information from the screening firms for all applicants before determining the terms of the insurance contract they offered. Will the highest that k can be to change the self-selection separating equilibrium differ from what you concluded in part (d)?
(g) Will the insurance allocation be efficient if the screening industry ends up selling information to insurance firms?