1. Consider a simple coin-flipping game. If a head appears, the individual playing the game receives a payoff of $16. If a tail appears, the individual receives a payoff of $4. The individual's initial wealth is $100.
a. What price to play the game would make it actuarially fair?
b. If the individual's utility function is U(X) = X1/2, would the individual pay that price to play the game? X represents wealth.
c. Suppose the individual is confronted with two options: play the game at a cost of $10, or buy insurance that guarantees the individual the expected value of the game. What is the maximum premium this individual will pay for the insurance?
2. Consider an individual with a current wealth of $100,000 who faces the prospect of a 25% chance of losing $20,000 through theft of her car during the next year. If the person's utility function is U(X) = ln(X), where X is wealth:
a. calculate expected utility without insurance,
b. calculate the actuarially fair premium for full insurance,
c. calculate expected utility with full insurance at the actuarially fair premium, and
d. calculate the maximum amount the individual would pay for full insurance. Moral Hazard
3. Suppose the person in Question 2 could purchase and install and antitheft device that costs $1950 and reduces the probability of car theft from 25% to 1%.
a. How does the expected gain from installing the device compare to the cost?
b. In the absence of insurance, how does expected utility without the device compare to the expected utility from purchasing and installing the device? Use the same utility function, U(X) = ln(X).
c. Suppose the person can purchase full insurance at a price of $5,200. Assume that the insurance company makes no effort to monitor installation of antitheft devices. Will she purchase the insurance? Will she purchase the antitheft device?
d. Suppose that the insurance company can monitor installation of antitheft devices, but at a cost of $10 per customer. It then charges an insurance premium of $3,210 for those with devices - $3,000 for the expected loss, $200 for administrative overhead, and $10 for monitoring costs. If the person purchases this insurance policy, what will her utility be? How does this compare to expected utility when buying the device without insurance and to buying an unmonitored policy?
4. Continue with the values from Question 2. Suppose that half of all drivers have antitheft devices already installed and the other half do not. The cost of installation is no longer a consideration for drivers. Assume that there are no administrative or monitoring costs for the insurance company, that the probability of loss for those without the devices is 25% and 15% for those with the devices.
a. Show that if the insurance company could separate drivers without the devices and offer full insurance at the actuarially fair premium of $5,000, the higher risk drivers would prefer insurance to no insurance.
b. Show that if the insurance company could separate drivers with the devices and offer full insurance at the actuarially fair premium of $3,000, the lower risk drivers would prefer insurance to no insurance.
c. If the insurance company cannot separate high risk from low risk drivers, what is the actuarially fair premium when both risk classes are combined?
d. Would high risk drivers purchase insurance at that premium?
e. Would low risk drivers purchase insurance at that premium?