In class last Thursday, while I explained the logic behind the so-called Rothschild-Stiglitz model (also presented in the last couple of paragraphs of my “Adverse Selection – a definition, some examples, and some solutions” posting and in pp. 22-23 of my “Moral Hazard and Adverse Selection” lecture note), I didn’t quite get around to providing a numerical example, so here it is.
Suppose that insurance clients are identical in all respects except for accident risks. Specifically, all such clients have initial wealth of $125, U(W) = W^.5 (i.e., square root utility), and exposed to a loss of $100 when an accident occurs. There are only two possible states – accident and no accident. However, some clients are high risk (i.e., probability of accident is p(h) = .75, whereas other clients are low risk; i.e., p(l) = .25.
The insurer’s problem is that he/she cannot determine which clients are high risk and which clients are low risk; consequently, the insurer decides to implement the Rothschild-Stiglitz model so as to induce self selection by clients. Specifically, a high risk contract called Policy A is offered which provides 100% coverage for $75, and a low risk contract called Policy B is offered which provides 10% coverage for $2.50. The following spreadsheet shows that high risk clients will self-select into Policy A, thus earning 0 profit for the insurer, whereas low risk clients will self-select into Policy B, thus earning 0 profit for the insurer:
Here, note that high risk and low risk clients prefer policies A and B respectively. Certainly the insurer would like for low risk clients to select Policy A (since this would provide profit to the insurer of $50 per client), but low risk clients rationally select Policy B since it maximizes their expected utilities. Of course, the objective here is to make sure that Policy B is not attractive to high risk clients by limiting the level of coverage offered under this contract, and indeed, Policy A (which offers 100% coverage at a price that is actuarially fair to high risk clients) is preferred. Since Policy B offers 10% coverage at a price that is actuarially fair to low risk clients, this high risk clients prefer Policy A; however, if the coinsurance rate on Policy B were, say, 20%, then high risk clients would prefer Policy B and there would be adverse selection (involving a per policy loss due to high risk clients opting into Policy B) of $10 per high risk client.