1 20 points There are two drivers in the town of Radiator Sp

1. (20 points) There are two drivers in the town of Radiator Springs, McQueen and Sally. McQueen has a 5% chance of causing an accident; Sally has a 1% chance. The cost of an accident is $12,000.

a. (10 pts) Suppose an insurance company knows each driver’s type. What premium would the insurance company charge each driver?

b. Now suppose that there is asymmetric information, so that the insurance company does not know for sure each driver’s type. Would insurance be sold if:

i. (5 pts) Both drivers self-reported their type to the insurance company?

ii. (5 pts) No information at all is known about individual driver’s types? If you are uncertain as to whether insurance would be sold at all, explain why.

2. (25 points) Hammidou Diallo will be chosen in the second round of the NBA draft. His utility function is U= ln(C) , where C is the dollar amount of consumption. His salary as a second-round pick will be $562,493, the league minimum. He is considering purchasing personal injury insurance from Lloyd’s of London to insure his knees. There is a 2% chance that he will have a seasonending knee injury, in which case his contract stipulates he will only be paid $100,000.

a. (5 pts) What is Diallo’s expected utility?

b. (10 pts) Calculate the actuarially fair premium for the optimal amount of coverage he should buy.

c. (5 pts) What is his expected utility if he purchases the optimal amount of coverage?

d. (5 pts) What is the most he would be willing to pay for coverage?

3. (30 points) Fairyland has two citizens, Cosmo and Pixie. Each has the same utility function, U= ln(C) . Both earn $1000 in their full-time jobs, but both face some risk of being laid off due to a potential shortage of fairy dust, in which case they will only earn $250 in an alternate part-time job. There is a 10% chance that Cosmo will be laid off and a 30% chance Pixie will be laid off. The Governor of Fairyland, Jorgen von Strangle, is considering providing unemployment insurance. In particular, Jorgen is considering two plans: the first would pay any worker who loses his job $100; the second would pay any worker who loses his job $600. Both plans would be financed by collecting a tax from any worker who keeps his job.

a. (10 pts) Under each plan, how high does Jorgen have to set the tax so that the government does not lose money on the plan?

b. (10 pts) For the tax from part a, compute the well-being of Cosmo and Pixie under each plan.

c. (10 pts) For three possible policies (the two plans and the status quo of no plan), how do Cosmo and Pixie rank the policies, respectively, in terms of their own well-being.

Solution

ans2. U= ln(C) , Income = $562,493 , P(loss) = 0.02 , loss = $462,493

a. expected utility = P(no loss) ln(cons when no loss) + P(loss) ln(cons when loss)

= 0.8 ln(562493) + 0.2 ln (100000)

= 12.8946922678

b. actuarially fair premium = P(loss) * loss = 0.2 * 462493 = $92498.6

c. expected utility when buys insurance = P(loss) ln ( cons. when loss + premium payment) + P(no loss) ln(cons when no loss+ premium)

= 0.2 * ln ( 562493 - 462493 + 462493 - 92498.6) + 0.8 * ln( 562493 -92498.6)

= 13.0604760587

d. expected utility when no insurance = expected utility when insurance

12.8946922678 =  0.2 * ln ( 562493 - 462493 + 462493 - x) + 0.8 * ln( 562493 -x) where x= premium amount

=> 12.8946922678 = ln ( 562493 - x)

=> e^{12.8946922678} = 562493 - x

=> x = 562493 - e^{12.8946922678} = 562493 - 398193.07026 = $164299.92974

hence the max amount that he is willing to pay as premium is much more than the acturially fair amount ($164299.92974 > $92498.6 ) . this is because he is a risk averse individual .