Problem 2 - Adverse Selection: Suppose there are two equal-sized groups of customers in the market: high-risk and low-risk. If a full policy is offered to high-risk individuals, the expected coverage is $200. If it is offered to low-risk individuals, the expected coverage is $170.

a) Assume that insurance companies know which individuals are high-risk and low-risk. Also, assume that the insurance market is in a long-term competitive equilibrium. What is the premium of the policy sold to each type of individual? (Hint: if the market is in long term equilibrium, it means that insurance companies' profit is zero. Remember that this implies that the insurance policy is fair.)

For a full policy, the high-risk customers are willing to pay a maximum of $212, and the low-risk

customers are willing to pay a maximum of $180.

b) What is the benefit (surplus) of a full policy for each type of customer? Will high-risk customers accept the policy? Will low-risk customers accept the policy?

Assume that insurance companies cannot distinguish between different customers (although cus- tomers may differentiate themselves through self-selection). However, insurance companies know the proportion of high and low-risk individuals in the market.

c) Assuming that the only policy to be offered offers full coverage, at what premium will it be sold? What is the effect of the adverse selection problem on social welfare?

Some insurance companies are considering an alternative policy that pays only a fraction of the loss (therefore, the policy is not full). This policy is less attractive to customers than the first one. High-risk people are willing to pay a maximum of $130, while low-risk people are willing to pay a maximum of $120. The expected coverage of this policy for high-risk customers is $125 and for low-risk customers $119.

d) Show that offering a partial policy at a premium of $119 acts as a screening device. If the insurance companies offer both policies (full policy at a premium of $200 and partial policy at a premium of $119), they separate the risk types into two groups.

e) What is the average benefit of the insurance policy per high-risk person they purchase? What is the average benefit of the insurance policy per low-risk person they purchase? What is the average benefit per person?

f) If the government provided insurance free of charge to all individuals, then what would be the benefit per person? What would be the expected cost to the government (expected coverage per person)? What would be society's surplus?

g) Use your answers to parts e) and f) to provide an argument for the state provision of insurance.