Final answer:
Tom Cruiser maximizes his expected utility by considering buying insurance to cover the risk of his car being stolen. The exact amount of insurance (K) he should buy can be calculated by setting up an expected utility maximization problem with his utility function, considering the probabilities of the car being stolen and the cost of the insurance.
Step-by-step explanation:
In determining how much insurance Tom Cruiser will buy, we look at his expected utility with and without insurance. Tom's total wealth without the car is $100,000, and with the car, it's $200,000.
If his car is stolen there's a 50% chance he will have $100,000 in wealth and a 50% chance he will have $200,000 in wealth. So without insurance, his expected utility is:
- 0.5 * u(100,000) + 0.5 * u(200,000)
- 0.5 * ln(100,000) + 0.5 * ln(200,000)
- 0.5 * 11.5129 + 0.5 * 12.2061 = 11.8595
If Tom buys insurance he pays 0.6K for K amount of insurance. For K=100,000, if the car is stolen, the insurance pays out K, and he loses 0.6K due to the insurance cost regardless of theft. So the two possible wealth levels are 200,000 - 0.6K if the car is not stolen and 100,000 if the car is stolen. His expected utility with full insurance will be:
- 0.5 * u(200,000 - 0.6 * 100,000) + 0.5 * u(100,000)
- 0.5 * ln(140,000) + 0.5 * ln(100,000)
- 0.5 * 11.8494 + 0.5 * 11.5129 = 11.6812
However, Tom's objective is to maximize his expected utility. To find the optimal amount of insurance K he would buy, we need to set up and solve the following problem:
Maximize E[u(w)] = 0.5 * ln(100,000 + K - 0.6K) + 0.5 * ln(200,000 - 0.6K)
By taking the first derivative and setting it to zero, we can find the maximum expected utility for Tom. The calculation will require the use of calculus to derive the first-order condition and solve for K.