Assignment:
The utility of each individual is u(w) = 80w - w^2, where 0 ≤ w ≤ 40 is wealth. The initial wealth is $40. The individuals may suffer a loss of $30. There are two types of individuals. Either an individual has low risk of loss, in which case the probability of loss is 1/5 , or high risk, in which case the probability of loss is 1/3 . The individuals know own type. Two firms simultaneously offer insurance contracts of the form (P,B), where P > 0 is the premium the individuals pays for insurance and B > 0 is the bonus paid to the individuals in case of loss. The firms know only that the proportion 3/4 of the individuals has high risk and proportion 1/4 has low risk.
The individuals can either accept one of the contracts or reject all. If an individual accepts, the individual's payoff is the expected utility from the contract. If an individual rejects, the payoff is the expected utility from the no insurance situation. For a firm the payoff is the expected profit from an accepted contract, or zero if no contracts are accepted.
(a) If the firms knew the type of each individual, what contract(s) would they offer in equilibrium?
(b) When the firms know only that the proportion 3/4 of the individuals has high risk and proportion 1/4 has low risk, what contract(s) are offered in equilibrium?
Provide complete and step by step solution for the question and show calculations and use formulas.