In exercise 23.3, we analyzed the case of a monopoly health insurance provider. We now extend the analysis to second-degree price discrimination, with x again denoting the degree of health insurance coverage.
A: Consider the same set-up as in part A of exercise 23.3 and assume there is an equal number of type 1 and type 2 consumers.
(a) Begin again by drawing a graph with the individual demands for the two types, d1 and d2, as well as the marginal costs. Indicate the efficient levels of health insurance x1∗ and x2∗ for the two types.
(b) Under second degree price discrimination, the monopolist does not know who is what type. What two packages of insurance level x and price P (that can have a per-unit price plus a fixed charge) will the monopolist offer? (Hint: You can assume that, if consumers are indifferent between two packages, they each buy the one intended for them.)
(c) Is the outcome efficient? Are consumers likely to prefer it to other monopoly pricing strategies?
(d) Suppose next that the demand from type 1 consumers is greater than the demand from type 2 consumers, with d1 intersecting MC1 to the right of where d2 intersects MC2. Would anything fundamental change for a first-degree or third-degree price discriminating monopolist?
(e) Illustrate how a second-degree price discriminating monopolist would now structure the two health insurance packages to maximize profit. Might relatively healthy individuals no longer be offered health insurance?
(f) True or False: Under second-degree price discrimination, the most likely to not buy any health insurance are the relatively healthy and the relatively young.
B: Consider again the set-up in part B of exercise 23.3. Suppose that a fraction γ of the population is of type 1, with the remainder (1−γ) of type 2. In analyzing second degree price discrimination, let the total payment Pn made by type n be in the form of a two-part tariff pn = Fn +pn xn.
(a) Begin by assuming that the monopolist will set p2 = (a) p¯ and p1 =MC1 = 8. Express the level of insurance x2 for type 2 consumers as a function of p. Then express consumer surplus for type 2 consumers as a function of p and denote it CS2 (p).
(b)Why would a second-degree price discriminating monopolist set F2 equal to CS+ (p) once she has figured out what p should be? What would the payment P2 (p) made by type 2 consumers to the monopolist be under p and F2 (p)?
(c) Suppose MC2 < p < MC1. For p in that range, what is the largest possible F1 that the monopolist can charge to type 1 consumers if she sets p1 =MC1 = 8.
(d) Suppose instead thatMC1 < p < 10. What would now be the largest possible F1 that is consistent with type 1 consumers not buying the type 2 insurance (assuming still that p1 =MC1 = 8?
(e) Given that the fraction of type 1 consumers is γ (and the fraction of type 2 consumers is (1−γ)), what is the expected profit E (π (p¯)) per customer from setting p2 = p when MC2 < p¯ (f) For both cases—i.e. for MC2 < p¯ < MC1 and whenMC1 < p¯ < 10—set up the optimization problem the second degree price discriminating monopolist solves to determine p¯. Then solve for p in terms of γ.
(g) Determine the value for p¯ when γ = 0. Does your answer make intuitive sense? What about when γ = 0.1, when γ = 0.2, when γ= 0.25? True or False: As the fraction of type 1 consumers increases, health insurance coverage for type 2 consumers falls.
(h) At what value for γ will type 2 consumers no longer buy insurance? If we interpret the difference in types as a difference in incomes (as outlined in the appendix), can you determine which form of price discrimination is best for low income consumers?