A monopolist sells health insurance to two types of consumers, "1" and "2". Type "1" is less healthy than type "2", such that the marginal cost of providing health services for type "1" is greater than for type "2". Suppose that _i() = (_i )_i is the type "i" demand for medical insurance. Interpret _i as the amount of medical insurance coverage the individual wants to purchase, with _i = 0 being no coverage and _i = 1 being full coverage. Let ₁ = 20 and ₂ = 8, for both types of consumers, and ₁ =7 and ₂ = 5, be the marginal costs of serving both types.

a) Determine the optimal level (without loss of well-being) of insurance for each type. b) Suppose the monopolist cannot distinguish between the two types of consumers, and can only charge a single price per unit of insurance sold. What price will it charge, knowing that there are consumers of type 1 and (1 ) of type 2, with < 0.5. Hint: write the monopolist's expected profit function and maximize it. c) Suppose now that the monopolist is able to discriminate price in the first degree. How much insurance is each type of consumer going to? consume? How much will each one pay? d) Now the monopolist only achieves third degree price discrimination. How much insurance will each type of consumer consume? How much will each one pay?