Adverse selection and Insurance

There are two states G and B, where G is the no-loss state and B the loss state. The customer has initial wealth W0 = 4, and will suffer loss (damage) D = 3 if state B occurs. The customer is an expected utility maximizer with a logarithmic utility-of-wealth function, so if his final wealth levels (consumption) in the two states are WG and WB, and the probability of loss is , his expected utility is

E(U) = (1)ln(WG)+ln(WB)

The insurance company is risk-neutral and wants to maximize expected profits, but perfect competition in the insurance market keeps its expected profit equal to zero. Insurance contracts will be characterized by the amounts of wealth (WG,WB) that the customer will end up with in the two states.

That is, if X is the premium and X the coverage, then WG = W0 X and WB = W0 D X + X.

Customers are of two types. The high risk type has probability of loss equal to 0.5; the low risk type has probability of loss equal to 0.25. The customers type is private information. You are asked to consider a separating equilibrium.

a. What is the insurance companys zero expected profit condition and what does this mean for the prices (of one euro of insurance claim) for each customer type?

b. Given the price implied by the zero-profit condition of the company, what is the budget constraint for each customer type?

c. What insurance contract intended for the high risk type will be offered in the equilib- rium? What is the resulting expected utility?

d. What insurance contract intended for the low risk type will be offered in the equilib- rium? What is the resulting expected utility? What sure amount of wealth would yield the same utility to this customer?

e. Compare the low risk type customers expected utility and the equivalent sure wealth amounts in three situations: first where he gets no insurance at all, second where he gets the separating equilibrium contract, and third, the hypothetical ideal optimum where type is observable, and the customer is given full insurance at the statistically fair price for the low type.