16. The country of Adventureland’s two citizens, Bill and Ted, both earn $1,000 per week working the same job at the same company. Bill and Ted each face some risk of being laid off due to bad market conditions next year, in which case they will have an income of only $250 from an alternative part -time job they would fall back
on. There is a 10% probability that Bill we be laid off, and a 30% probability that Ted will be laid off. Bill and Ted have the same utility function U ln(C).The government is considering providing some social unemployment insurance. In particular, they are considering two plans: The first would pay any worker who loses his job $100, and the second would pay any worker who loses his job $600. Both would be financed by collecting a tax from any worker who keeps his job. 

a. Under each plan, how high would the government have to set the tax so that it would not expect to lose money on the plan?
b. Assuming it sets the tax rate you found in part a, compute the well-being of Bill and Ted under each of the plans. How do Bill and Ted rank the three possibilities (the two policies and the status quo)? Explain the pattern you see in terms of redistribution and risk aversion.
c. Which plan is best if the society has a utilitarian social welfare function? A Rawlsian social welfare function?

(See Chapter 3 if you need a reminder about social welfare functions!)

The government of Arendelle is considering providing a social unemployment insurance program. In particular, they are considering two options: the first would pay any worker who loses their job $100/week for a year, and the second would pay $600/week for a year. Both would be financed by collecting a tax from workers that keep their (full-time) jobs for the first option, the tax would be $25/week, and for the second option, the tax would be $150/week.

Calculate the expected utility of both types of workers under each program, as well as in the absence of an unemployment insurance program. How do the workers rank the possibilities? Explain your answer in terms of redistribution and risk aversion.

Now assume that if laid-off workers receive unemployment benefits of $600/week, they will choose not to work part-time. How does this affect the expected utility of the workers under this program?

The kingdom of Arendelle has equal numbers of two types of workers, domestic servants and ice sellers, which both earn $1,000 per week. All workers face a risk of being laid off next year due to bad market conditions, in which case they will have an income of only $250/week from an alternative part-time job. Domestic servants have a 10% probability of being laid off, and ice sellers have a 30% probability. All have the same utility function: U = ln(C).

a. Suppose that workers would like to insure against being laid off. If insurers can offer different insurance contracts to each type of worker, what will be the actuarially fair premiums for full insurance for each group? (Assume that full insurance pays a benefit of $1000 per week so that part-time work is not necessary.)

b. Explain why a private market for unemployment insurance is unlikely to exist.

Consider a car owner who has an 80% chance of no accidents in a year. For simplicity, assume that there is a 10% probability that after the accident the car will need repairs costing $500, an 8% probability that the repairs will cost $5,000, and a 2% probability that the car will need to be replaced, which will cost

$15,000. Based on this information, the probability distribution, f(x), of the random variable, X, loss due to accident, is:

where the first column is the probability of the event (i.e. P(x=0) =0.8) and the second is the severity.

Assuming risk retention, calculate the object risk to the car owner. Show ALL your work.

Now consider an insurance company that will reimburse repair costs resulting from accidents for 100 car owners, each with the same probabilities and losses as in part a). Calculate the objective risk for the insurance company. How is this number compared to that of part a)? Explain. Show ALL your work.

Suppose that the insurance company provides insurance to the same 100 car owners but now it introduces a deductible of $500. The claim payment distribution for EACH policy would now be:

where