Consider the insurance market where each consumer has initial wealth w, and may suffer a loss of L if an accident occurs. Now, suppose there are two types of risks in the in- surance market, low and high. The low-risk consumers' accident prob- ability is L, while the high-risk ones' is H, where 0 < L < H < 1. Let the proportion of low-risk consumers be (0,1). Let each consumer's von Neumann-Morganstern utility function be defined by u(y) = (w y)2.

There are many insurance companies in the market, so that the price is competitively set. Only full-coverage insurance contracts are offered. In other words, if a consumer purchases the contract, then she is fully reimbursed for the loss when an accident occurs.

(a) Can there be more than one price in the market? Show your reasoning.

(b) Now, what are the equilibrium prices in equilibrium? Who pur- chase insurance in equilibrium? (Hint: consider any price p. Which consumers want to purchase insurance at this price? Would it be profitable for insurance companies to offer insurance con- tracts at this price?)