The Acme Auto Insurance Company is risk neutral and seeks to offer insurance at its
actuarially fair price. All drivers in the community have the same initial income, $400, and know that if they are involved in an auto accident, their income falls to $0.
All drivers have the same von Neumann-Morgenstern utility index: U(Y) = Y½
There are two groups of drivers in the community. One half are safe drivers and face a 25% chance of being in the accident. The others are risky drivers and face a 75% chance of being in an accident.
a) If the insurance company can identify who is a safe driver and who is a risky driver and charge them different prices, what premium will members of each group pay for insurance? How much insurance will a representative driver in each group buy? What level of income will each type of driver enjoy in the two states of the world?
b) Now assume that while each individual knows whether she is a safe or risky driver, the insurance company does not. Acme only knows that overall there is a 50% chance of an individual being in an accident, so it charges $0.50 per $1 of coverage to everyone. Solve for the optimal decision for each type of driver. You may assume that there is no limit on the amount of insurance an individual can buy. Is the insurance company in long run equilibrium at this price? [Hint: what is Acme’s expected profit per driver?] Explain briefly what is going on in this case.
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