Insurance versus risky or safe behavior

Consider a town with 1000 people. People can engage in either risky or safe behavior while driving. If they engage in risky behavior, they have a 10% chance of having an accident; however, they get a \$1000-equivalent bump in utility from risky driving. If they engage in safe driving, there is only a 1% chance that they have an accident, but they get no satisfaction from driving. Accidents cost \$20,000 on average.
a. Assume there is no car insurance available, and calculate the expected value of safe and risky driving for an individual. Will people engage in risky or safe driving? Calculate total utility in this town.
b. Suppose that there is car insurance available. The insurance company assumes that everyone drives safely. What price will they charge for insurance? Who will buy the insurance?
c. Calculate expected values for safe and risky driving again under these conditions. Does anyone have an incentive to change their behavior after buying insurance?
d. What do you expect the insurance company to do in response? What will be the resulting expected values of driving for each individual? Calculate total utility in this equilibrium and compare to your answer in (a).

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