Adam has wealth of $40,000 as long as his business does not burn down. However, there is a 50% probability that his business will burn down, causing a loss of $30,000 and leaving him with $10,000 of wealth. Adam's utility function is given by U = W^0.5, where W is wealth. Suppose Adam purchases insurance to cover the potential $30,000 loss. If the insurance premium is actuarially fair, Adam's utility with insurance is:

Question

Quizplus · Accepted Answer

Without insurance, Adam's expected utility is:

EU1 = 0.5(WS1U1P10.5S1S1P1) + 0.5(WS2U2P10.5S2S2P0) 
= 0.5(40,0001(0.5)1(1)) + 0.5(10,0001(0.5)0(0)) 
= 10,000√40

With insurance, Adam's expected utility is:

EU2 = WS1U1P1 = (40,000 - p)1(0.5)1(1) 
= (40,000 - p)√2

To find the actuarially fair premium, we set EU2 equal to the expected value of the insurance payment, which is 0.5(30,000) = 15,000:

(40,000 - p)√2 = 15,000 
p = 28,284

So, Adam pays $28,284 for insurance. His expected utility with insurance is:

EU3 = WS1U1P1 - p = (40,000 - 28,284)1(0.5)1(1) - 28,284 = 15,811.53

Therefore, the best choice is C, which is Adam's utility with insurance.

Adam Has Wealth of $40,000 as Long as His Business