Question

Suppose mark currently has  $80,000 in wealth. Also suppose that there is a 1% chance that his house will be destroyed and cost $70,000 to repair (reducing your income to $10,000). Finally, suppose that his utility function is U = M , where M is income.

1. What is the expected value of this situation? What is the expected utility?
2. Would mark be willing to pay $500 to purchase an insurance policy that fully insures him against his loss? (Full insurance implies that the insurance company will compensate you for the cost of rebuilding your house.) Solve numerically and show with a graph.
3. What is the highest price that mark would be willing to pay for this insurance policy?

Answer 1

A) expected value = 0.99(80,000) + 0.01(80,000-70,000)= 79,300.

Expected utility = 0.99 (value of M without accident) + 0.01( value of M with accident)

= 0.99(80,000)+0.01(10,000) = 79300.

B) when mark buys insurance that fully insures him against the loss then his expected utility is :

= 0.99( M​​​​​​without accident - 500) + 0.01( M​​​​​​with accident -500) = 0.99 (80,000 - 500) + 0.01(80,000 - 500) = 80,000-500 = 79,500

Hence, comparing this with above part , his utility without insurance is lesser therefore he will buy insurance as he will be better off.

as can be seen from the graph, the utility is higher when he buys insurance ( 79,500) than when he doesn't buy one (79,300) .

C) The highest price that Mark would be willing to pay for this policy be x .

His expected utility is : 0.99(80,000 - x) + 0.1(80,000 - x)

This should be equal to utility without insurance ;

Hence 80,000-x = 79300 ==> x = 700.

Hence he can pay an insurance premium of max 700 .