Question

Suppose there is a \(50-50\) chance that an individual with logarithmic utility from wealth and with a current wealth of \(\$ 20,000\) will suffer a loss of \(\$ 10,000\) from a car accident. Insurance is competitively provided at actuarially fair rates. a. Compute the outcome if the individual buys full insurance. b. Compute the outcome if the individual buys only partial insurance covering half the loss. Show that the outcome in part (a) is preferred. c. Now suppose that individuals who buy the partial rather than the full insurance policy take more care when driving, reducing the damage from loss from \(\$ 10,000\) to \(\$ 7,000 .\) What would be the actuarially fair price of the partial policy? Does the individual now prefer the full or the partial policy?

Short answer

Consider both scenarios before and after the individual decides to drive more carefully. Answer: The individual prefers full insurance before deciding to drive more carefully. However, after driving more carefully and reducing the potential loss, the individual prefers the partial insurance policy.

Step-by-step solution

Calculate the insurance premium for full coverage

To calculate the actuarially fair insurance premium for full coverage, we need to know the probability of an accident happening and the associated loss. In this case, the probability is 50% and the loss would be $10,000. Therefore, the premium for full coverage would be: Premium = Probability x Loss = 0.5 x 10,000 = 5,000$

Calculate the wealth after purchasing full coverage insurance

If the individual purchases full insurance, they would pay the premium, which would decrease their wealth by $5,000. Therefore, the new wealth would be: Wealth after full insurance = Current wealth - Premium = 20,000 - 5,000 = 15,000$

Compute the utility of the outcome after purchasing full insurance

The individual's utility is given by the logarithm of their wealth. With full insurance and wealth of $15,000, the utility would be: Utility (full insurance) = ln(wealth after full insurance) = ln(15,000) b. Partial Insurance

Calculate the insurance premium for partial coverage

For partial coverage, the insurance would cover only half the loss. Therefore, the premium for partial coverage would be: Premium (partial) = Probability x (Loss / 2) = 0.5 x (10,000 / 2) = 2,500$

Calculate the wealth after purchasing partial coverage insurance in both scenarios (accident and no accident)

If the individual purchases partial insurance, their wealth would be affected in two scenarios: when an accident occurs and when it does not. Therefore, the new wealth in each scenario would be: Wealth (no accident) = 20,000 - 2,500 = 17,500$ Wealth (accident) = 20,000 - 2,500 - 5,000 = 12,500$ where 5,000$ is half the loss, covered by the individual.

Compute the expected utility of the outcome after purchasing partial insurance

The expected utility would be the average of the logarithm of the wealth in both scenarios, weighted by the probability of each scenario: Expected Utility (partial insurance) = 0.5 × ln(Wealth (no accident)) + 0.5 × ln(Wealth (accident)) = 0.5 × ln(17,500) + 0.5 × ln(12,500) Now, compare the utilities to determine which is preferred. Utility (full insurance) = ln(15,000) > ln(17,500) / 2 + ln(12,500) / 2 = Expected Utility (partial insurance) Since the utility with full insurance is greater than the expected utility with partial insurance, the individual prefers full insurance. c. Actuarially Fair Price of Partial Policy and Preference

Calculate the new premium for partial coverage after careful driving

An individual who buys partial insurance and drives more carefully reduces the damage from a loss to $7,000. The new premium for partial coverage would be: New Premium (partial) = Probability x (Reduced Loss / 2)= 0.5 x (7,000 / 2) = 1,750$

Calculate the wealth after purchasing partial coverage insurance in both scenarios (accident and no accident) with reduced loss

The new wealth in each scenario would be: Wealth (no accident) = 20,000 - 1,750 = 18,250$ Wealth (accident) = 20,000 - 1,750 - 3,500 = 14,750$ where 3,500$ is half the reduced loss, covered by the individual.

Compute the expected utility of the outcome after purchasing partial insurance with reduced damage

The expected utility would be the average of the logarithm of the wealth in both scenarios: Expected Utility (partial insurance, reduced damage) = 0.5 × ln(Wealth (no accident)) + 0.5 × ln(Wealth (accident)) = 0.5 × ln(18,250) + 0.5 × ln(14,750) Now, compare the utilities to determine which is preferred. Utility (full insurance) = ln(15,000) < Expected Utility (partial insurance, reduced damage) Since the utility with full insurance is now less than the expected utility with partial insurance and reduced damage, the individual now prefers the partial policy.