Does the insurance company manage to screen its customers with these contracts?

A monopoly insurance company provides accident insurance to two types of customers: low risk customers, for whom the probability of an accident is 0.25, and high risk customers, for whom the probability of an accident is 0.5. There is an equal number of both types of customers. Without insurance, each customerís wealth is 16 if there is no accident, but 0 if there is an accident. Customersí von Neumann Morgenstern utility of wealth is: u(w) = p w.The insurance company cannot identify the type of a customer when the customer applies for an insurance contract. Suppose the insurance company o§ers the following two contracts. The Örst contract o§ers a payout of 8 in case there is an accident, and requires customers to pay a premium of 7. The second contract o§ers a payout of 16 in case there is an accident, but requires customers to pay a premium of 10. (a) Determine for low risk customers and then also for high risk customers which, if any, of these contracts they will buy. (b) Does the insurance company manage to screen its customers with these contracts? (c) Calculate the insurance companyís expected proÖt if it o§ers these contracts.

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