# Some observations concerning yesterday’s Rothschild-Stiglitz class problem

Yesterday, we considered the following problem in class:

• Assume that consumers are identical in all respects expect for their loss probabilities; some are high risk, and others are low risk.
• Members of the high-risk group have loss probability pH = 65%, whereas members of the low risk group have loss probability pL = 35%.
• Each consumer has initial wealth of \$100 and utility U(W)=W.5.
• There are only two possible states of the world, loss and no loss. If a loss occurs, then consumers lose their initial wealth of \$100.
• Insurance contract offerings include the following:
• Policy A provides full coverage for a price of \$65.
• Policy B provides full coverage for a price of \$45.50.
• Policy C provides 60% coverage for a price of \$39.
• Policy D provides 30% coverage for a price of \$13.65.

The objective here is to identify the set of contract offerings that would prevent adverse selection. If you consider the pricing of these four insurance contracts, Policy A involves full insurance that is actuarially fair for high-risk consumers. We know from the Bernoulli principle that these consumers would like to purchase this contract. The challenge is to identify contracts that are favorable for the low-risk consumers but not for the high-risk consumers. Clearly we would not want to offer contract B, since everyone would select this contract and we would lose \$19.50 on every high-risk consumer who purchased it (while breaking even on every low-risk consumer). High-risk consumers won’t want Policy C because it offers actuarially fair partial coverage, which provides lower expected utility than actuarially fair full coverage. However, low-risk consumers would be willing to purchase Policy C, so if A and C were offered, the insurer would break even on A and make \$18 in profit from low-risk consumers who purchase Policy C. Given a choice between being uninsured, buying Policy A, or buying Policy C, low-risk consumers would purchase Policy C since it would offer higher expected utility than the other alternatives. Policy D would also be an acceptable alternative; if high-risk consumers purchased this contract, the insurer would lose \$5.85 per high-risk consumer. However, if Policy A was also offered, none of the high-risk consumers would purchase Policy D. But low-risk consumers would prefer Policy D since it would offer higher expected utility than the other alternatives.

For what it’s worth, I have uploaded a spreadsheet consisting of expected utility calculations for this problem:

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