Question

There are two types of drivers on the road today, Speed Racers with a 5% chance of anaccident per year, and Low Riders with a 1% change per year. There are an equal number of SpeedRacers and Low Riders. The cost of an accident is $12,000. Assume that there is free entry into theinsurance market, the insurers always know the relative shares of each type in the population, andfirms are risk-neutral. Assume that your income is $12,000 in the absence of an accident.

If,U(Y) =√Y, would be a premium of $100 for the cheaper policy that covers onefourth of the losses (pays out $3000) allow both contracts to be sold? (Remember, their income is$12,000 if the bad event does not happen.)

Answer 1

First we consider the case of no-insurance

Income in case of accident=12000-12000=0

Utility in case of accident=U(0)=0^0.5=0 utils

Income in case of no accident=12000

Utility in case of no accident=U(12000)=12000^0.5=109.5445 utils

Now take the case of insurance

Income in case of accident=12000-300-12000+3000=2700

Utility in case of accident=U(2700)=2700^0.5=51.9615 utils

Income in case of no accident=12000-200=$11700

Utility in case of no accident=U(11700)=11700^0.5=108.1665 utils

Consider the case of Speed Racers

Expected utility without insurance=0.05*U(0)+0.95*U(12000)

Expected utility without insurance=0.05*0+0.95*109.5445=104.0673 utils

Expected utility with insurance=0.05*U(2700)+0.95*U(11700)

Expected utility with insurance=0.05*51.9615+0.95*108.1665=105.3563 utils

Since expected utility is higher in case of insurance. Speed Racers will buy the insurance

Consider the case of Low Riders

Expected utility without insurance=0.01*U(0)+0.99*U(12000)

Expected utility without insurance=0.01*0+0.99*109.5445=108.4491 utils

Expected utility with insurance=0.01*U(2700)+0.99*U(11700)

Expected utility with insurance=0.01*51.9615+0.99*108.1665=107.6045 utils

Since expected lower is higher in case of insurance. Low Riders will not buy the insurance

Contracts will not be sold to both type drivers. Only Speed Racers will buy it.