All residents in Smallville work at the only local factory. They all earn the same annual wage W. There is some risk that a worker may suffer a minor injury on the job because the factory has a lot of dangerous equipment. If a worker is hurt on the job during the year, the worker suffers a financial loss of L (this includes doctor bills and other costs). There are three different types of residents. Type A residents are quite accident-prone, type B residents are moderately careful, and type C residents are extremely careful. The chance that a worker of type A has an accident during the year is 2/5, the chance that a worker of type B has an accident is 1/5, and the chance that a worker of type C has an accident is zero. a) Insurance companies offer to sell accident insurance to residents in this town. Assume that companies can tell what type each resident is. Describe the insurance contracts (the relationship between the insurance premium, a, paid by the worker and the net benefit, b, received by the worker in the case of an accident) that competitive firms will offer to workers of each type. [Hint: You are simply finding the zero-profit condition for insurance firms when dealing with each type of worker.] b) Each individual has an expected utility function that looks like the following: EU = p ln(YA) + (1−p) ln(YNA),
where p is the probability of an accident, YA is an individual's net income in case of an accident, and YNA is an individual's net income if no accident occurs. Assume that W is equal to 100 and L is equal to 50. If a person of type B is offered an actuarially fair insurance contract, derive the amount of insurance this person would choose to buy. Interpret your result. [Hint: Find the optimal amount of insurance by choosing either a or b to maximize a type B person's expected utility subject to the appropriate no-profit insurance constraint.] c) Would a type A person prefer full insurance at actuarial fair rates appropriate to their own risk or the insurance contract chosen by type B individuals in part b)? [Do not work this out algebraically. Derive your answer using a graph with net income in the accident state on one axis and net income in the no-accident state on the other.] Would an insurance company be willing to offer full insurance at rates appropriate to type B's risk if it could not tell type A and type B individuals apart? d) Explain how the government can help to improve problems of adverse selection like the one encountered in part c).