Question

Smith just bought a house for $250,000. Earthquake insurance, which would pay $250,000 in the event of a major earthquake, is available for $25,000. Smith estimates that the probability of a major earthquake in the coming year is 10 percent, and that in the event of such a quake, the property would be worth nothing. The utility (U) that Smith gets from income (I) is given as follows:

U(I) = I0.5. (Smith’s utility is the square root of her income.

Should Smith buy the insurance?

A) Yes.

B) No.

C) Smith is indifferent.

D) We need more information on Smith's attitude toward risk.

Answer 1

The correct answer is (A) Yes

He should buy insurance if expected utility if he bought insurance is greater than Expected utility if he didn't purchase.

Expected Utility(EU) = P₁U(I₁) + P₂U(I₂)

P₁ = Probability that he will will have income(or wealth) = I₁ and P₂ = Probability if he have Income(or wealth) I₂

If he buys insurance then P₁ = Probability that earthquake occurs = 10% = 0.10 , I₁ = Wealth after earthquake occurs = 250,000 - 25000 = 225,000 , P₂ = Probability that earthquake will not occurs = 90% = 0.90 , I₂ = Wealth if earthquake not occurs = 225,000. (Note that wealth before and after insurance is same because he has already incurred 25000 for insurance and after insurance whole wealth is recoverable).

Hence If he buys insurance, Expected Utility(EU) = 0.10U(225000) + 0.9U(225000) = 0.10(225000)^0.5 + 0.9(225000)^0.5 = (225000)^0.5 = 474.34

If he didn't buys insurance then P₁ = Probability that earthquake occurs = 10% = 0.10 , I₁ = Wealth after earthquake occurs=0, P₂ = Probability that earthquake will not occurs = 90% = 0.90 , I₂ = Wealth if earthquake not occurs = 250,000.

Hence If he buys insurance, Expected Utility(EU) = 0.10U(0) + 0.9U(250000) = 0 + 0.9(250000)^0.5 = 0.9*(250000)^0.5 = 450

So, EU if he buys insurance is greater than If he didn't purchased the insurance

Hence he should buy the insurance.

Hence, the correct answer is (A) Yes