4. Adverse SelectionConsider the Rothschild and Stiglitz (1976) insurance model under asymmetric information.Suppose that insurance companies offer price-quantity contracts. There are two types ofagents with type i = H or L. The initial wealth for all agents is W. An agent of type i hasprobability pi of losing an amount L when the bad event happens. All agents have the sameutility function. Let W = 24, L = 16, u(x) = 2√x, pL =12, and pH =34.(Hint: you will need to use the quadratic formula: x =-b±√b2-4ac2a).

(a) Compute the marginal rates of substitution for the two types.

(b) Compute the wealth in the good and bad states, i.e. Wg and Wb, for each type in theseparating equilibrium.

5. Moral HazardAn individual has initial wealth of $80,000 and faces a potential loss of $36,000. The probabilityof loss depends on the amount of effort the individual puts into trying to avoid it.If the individual puts a high level of effort, then the probability of loss is 5%, while if sheexerts low effort, the probability is 15%. The individual's utility is u(x) = √x if low effortand u(x) = √x - 1 if high effort.(a) If the individual remains uninsured, what level of effort will be chosen, i.e. low or high?(b) If the individual is offered full insurance with a premium of $2,250, what level of effortwill be chosen, i.e. low or high?(c) What will be the insurance company's expected profit from the full insurance contractwith premium $2,250? (Hint: you must consider the probability of the effort that theindividual selects from part (b))