Question

Cindy is an expected utility maximizing consumer who has an initial wealth of $160,000 and is subject to fire risk. There is a 5% chance of a major fire with a loss of $70,000 and a 5% chance of a disastrous fire with a loss of $120,000. Her utility function is U = W1/2. Cindy is offered an insurance policy costing $12,000 with a deductible provision, which requires that she pay the first $7,620 on any fire loss (that is, she is paid the amount of the loss, less $7,620, in case of a fire). (Hint: To see how the deductible works, suppose Cindy suffered a loss due to fire. Her wealth, if she bought insurance, in that state of the world would be equal to the following: W = $160,000 - Loss + Claim - Premium With full insurance, the claim is equal to the loss. But with a deductible, her claim is equal to the loss *minus* the deductible (i.e. Claim = Loss - Deductible)

How many states of the world are there in this example?

What is Cindy’s expected utility if she buys the insurance? (Round your answer to two decimal points)

What is Cindy’s expected wealth if she buys the insurance?

What is the most Cindy would be willing to pay for this insurance policy? (Hint: Use Solver to get the answer to this part. You have to find the premium that gives Cindy the exact same level of expected utility as she receives when she does not buy the insurance policy as calculated in question #3.) (Round your answer to the nearest integer)

Answer 1

There are states of world first one being the possibility that major fire occur, second being the possibility that disastrous fire occur and last possibility that no fire will occur.

Expected Utility if buy insurance =EU(insurance)

EU(insurance)= (wealth with no fire and insurance)^1/2 + (wealth with major fire and insurance)^1/2 + (wealth with disastrous fire and insurance) ^1/2

= [(0.90)(160000-0+0-12000)]^1/2 + [(0.05)(160000-70000+70000-7620-12000)]^1/2 + [(0.05)(160000-120000+120000-7620-12000)]^1/2 .............equation 1

=532.53

Expected Wealth after insurance = EW(insurance)

EW(insurance)= wealth with no fire and insurance + wealth with major fire and insurance + wealth with disastrous fire and insurance

= (160000-0+0-12000)(0.9) + (160000-70000+70000-7620-12000)(0.05) + (160000-120000+120000-7620-12000)(0.05)

= 148000*0.9 + 140380*0.05 + 140380*0.05

=133200 + 140380*0.10

=133200 + 14038

=145238

Expected utility without insurance = EU( without insurance)

EU(without insurance) = utility with no fire no insurance + utility with major fire no insurance + utility with disastrous fire and no insurance

=[(0.9)(160000)]^1/2 + [(0.05)(160000-70000)]^1/2 + [(0.05)(160000-120000)]^1/2

=491.28

Now for the amount cendy would be willing to pay for the insurance policy, we have to change 12000 to variable p in the equation 1

We get EU(insurance)= [(0.9)(160000-0+0-p)]^1/2 + [(0.05)(160000-70000+70000-7620-p)] ^1/2+ [(0.05)(160000-120000+120000-7620-p)]^1/2

Now, EU(no insurance) = EU(insurance),

Onsolving for p we will get the amount cindy would be willing to pay i.e approximately 27046.72.