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quadratic equation

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?

Solution Preview

a) If the firm knew the type of each individual and there is no problem of adverse selection, the contracts offered in equilibrium will be issued at fair premium and full loss coverage.
Since the amount of loss is $30, the loss coverage will be $30 i.e. the individual will receive $30 in case of loss.
The expected value of loss for a low risk individual = Probability of loss * Amount of loss = 1/5*30=$6
Contract for low risk individual will be ($6, $30)

The expected value of loss for a high risk individual = Probability of loss * Amount of loss = 1/3*30=$10
Contract for high risk individual will be ($10, $30)

b) When the firm does not know the type of individual, there will be problem of adverse selection where the individual ...

Solution Summary

Competitive Screening is denoted.

$2.19