Question

Problem #3 Jan's wealth amounts to $100,000. His car, worth $20,000, is exposed to the risk of being stolen and the probability of theft actually taking place is 25%. Jan's expected utility function takes the form U(W) - In(W), where W-wealth. a) Find Jan's expected utility (EU) in the situation when he does not buy car theft insurance. b) Find the fair insurance premium assuming that the insurance company does not bear any administrative costs. c) Find the maximum amount that Jan would be willing to pay for car theft insurance Jan can also install a car alarm, which costs him $1,750. Installing this alarm decreases the probability of the car being stolen from 25% to 15%. d) If Jan decides not to purchase insurance, will he buy and install the car alarm? e) What is the maximum amount that the insurance company can charge Jan for insurance if it requires that Jan purchases and installs the car alarm and verification whether Jan fulfilled this requirement is costless (full compensation, i.e. of $20,000, is guaranteed in the case of theft)? f) What can the insurance company do when it is not able to verify whether Jan installed the car alarm?

Answer 1

Answer:

1)

Jan's wealth W=$100000

Jan'e wealth if car is stolen= 100000-20000=$80000

Probability of theft=25%

Expected utility E(u)=25%*ln(80000)+(1-25%)*ln(100000)=11.457 Eq 1

2) Fair insurance premium = 25% * amount of loss=25%*20000=$5000

3) Given expected utility of jan E(U)= 11.457

for utility of 11.46 wealth would be W=e^(11.457)=$94560.96

So Jane maximum ready to pay for insurance = 100000-94560.96=$5439

4) if Jane installs the alarm then her wealth would be =100000-1750=$98250

Jan'e wealth if car is stolen= 98250-20000=$78250

Probability of theft=15%

Expected utility E(u)=15%*ln(78250)+(1-15%)*ln(98250)=11.461 Eq 2

From equation 1 and 2 it is clear that after installing alarm Jan'e expected utility will increase so she will install the alarm.