Question

Consider Mac who has current wealth (M) of $100,000. Mac faces a 5% chance of losing her automobile, which is currently worth $25,000. Suppose her utility function is M^1/2.

What is the expected value of the car?

What is Mac’s expected utility if she does not buy insurance?

How much would Marcy be willing to pay to insure this car?

Answer 1

There are two states of the world :-

1. Mac loses the automobile with probability as given 0.05
2. Mac does not lose the automobile with probability (1 - 0.05) = 0.95

Now, Incase of losing the automobile Mac will be just left with his Initial wealth = $100,000

and Incase of Mac not losing his automobile will have the wealth as his initial wealth plus the current worth of the automobile = $100,000 + $25,000 = $125,000

Now,

a) Expected Value (EV) :

So, Expected Value of Car = (0.05 * $0) + (0.95 * $25000)

= $0 + $23750

= $23,750

Expected Value of wealth = (0.05 * $100,000) + (0.95 * $125,000)

= $5000 + $118750

= $123,750

b) Now, As Utility function (u(M)) = M^{​​​​​​1/2}

Expected Utility (EU) :

So, Expected Utility without insurance = [0.05 * u($100,000)] + [0.95 * u($125,000)]

= [ 0.05 * 100,000^1/2] + [ 0.95 * 125,000^1/2]

= [ 0.05 * $316.23] + [ 0.95 * $353.55]

= $15.8115 + $335.8725

= $351.684

c) Now,

Mac would be willing to pay an Insurance Premium for insuring the car

Insurance Premium = Expected Wealth - Certainty Equivalent(CE) of wealth

As, u(Certainty Equivalent) = Expected Utility

So, CE^{​​​​​1/2} = 351.684

So, CE = 351.684²

So, CE = $123,681.63

So, ​​​As Expected wealth calculated earlier = $123,750

So, Insurance Premium = $123,750 - $123,681.63

= $68.37

So, Marcy would be willing to pay $68.37 for insuring the car.