Alfred is a risk-averse person with $100 in monetary wealth and owns a house worth $300, for total wealth of $400. The probability that his house is destroyed by fire (equivalent to a loss of $300) is pne = 0.5. If he exerts an effort level e = 0.3 to keep his house safe, the probability falls to pe = 0.2. His utility function is: U = w0.5 – e where e is effort level exerted (zero in the case of no effort and 0.3 in the case of effort). a. In the absence of insurance, does Alfred exert effort to lower the probability of fire? HINT: Calculate and compare the expected utility i) with effort, and ii) without effort. If effort is exerted, then the effort cost is paid regardless of whether or not a fire occurs. b. Alfred is considering buying fire insurance. The insurance agent explains that a home owner’s insurance policy would require paying a premium α and would repay the value of the house in the event of fire, minus a deductible “D”. [A deductible is an amount of money that the company does not reimburse in the event of a loss.] The company explains that, without a deductible, insured clients do not take adequate precautions (effort) to reduce the possibility of fire. Unfortunately, these actions are not observed by the company and monitoring their insurees’ behavior would be extremely costly. Hence, the insurance company cannot require effort as part of the policy. Verify that the company is correct that, without a deductible, insurees would not take precautions (spend effort) against the possibility of fire. (If there’s no deductible, the company would reimburse the entire value of the house in the event of a fire)? Answer the question comparing expected utility under the two action choices (e = 0 or e = 0.3).
Y5
Alfred is a risk-averse person with $100 in monetary wealth and owns a house worth $300, for total wealth of $400. The probability that his house is destroyed by fire (equivalent to a loss of $300) is pne = 0.5. If he exerts an effort level e = 0.3 to keep his house safe, the probability falls to pe = 0.2. His utility function is: U = w0.5 – e where e is effort level exerted (zero in the case of no effort and 0.3 in the case of effort).
a. In the absence of insurance, does Alfred exert effort to lower the probability of fire?
HINT: Calculate and compare the expected utility i) with effort, and ii) without effort. If effort is exerted, then the effort cost is paid regardless of whether or not a fire occurs.
b. Alfred is considering buying fire insurance. The insurance agent explains that a home owner’s insurance policy would require paying a premium α and would repay the value of the house in the event of fire, minus a deductible “D”. [A deductible is an amount of money that the company does not reimburse in the event of a loss.] The company explains that, without a deductible, insured clients do not take adequate precautions (effort) to reduce the possibility of fire. Unfortunately, these actions are not observed by the company and monitoring their insurees’ behavior would be extremely costly. Hence, the insurance company cannot require effort as part of the policy.
Verify that the company is correct that, without a deductible, insurees would not take precautions (spend effort) against the possibility of fire. (If there’s no deductible, the company would reimburse the entire value of the house in the event of a fire)? Answer the question comparing expected utility under the two action choices (e = 0 or e = 0.3).
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