In Chapter 22, we worked with models in which high and low cost customers compete for insurance. Consider the level x of health insurance that consumers might choose to buy, with higher levels of x indicating more comprehensive insurance coverage.
A: Suppose that there are relatively unhealthy type 1 consumers and relatively healthy type 2 consumers. The marginal cost of providing additional insurance coverage is then MC1 and MC2, with MC1 >MC2. Unless otherwise stated, assume that d1 = d2 —i.e. the individual demand curves for x are the same for the two types. Also, suppose that the number of type 1 and type 2 consumers is the same, and some portion of each demand curve lies above MC1.
(a) Begin 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) Suppose the monopolist cannot tell consumers apart and can only charge a single price to both types. What price will it be and what level of insurance will each type purchase?
(c) How does your answer change if the monopolist can first-degree price discriminate?
(d)What if he can third-degree price discriminate?
(e) Suppose you worked for the Justice Department’s anti-trust division and you only cared about efficiency. Would you prosecute a first-degree price discriminating monopolist in the health insurance market? What if you cared only about consumer welfare?
(f) In the text we suggested that it is generally not possible without knowing the specifics of a case whether third degree price discrimination is more or less efficient than no price discrimination by a monopolist. For the specifics in this case, can you tell whether type 1 consumers are better off without this price-discrimination? What about consumer type 2?
(g) Would it improve average consumer surplus to prohibit the monopolist from third-degree price discriminating? Would it be more efficient?
B: Suppose next that we normalize the units of health insurance coverage such that the demand function is xn(p) = (θn −p)/θn for type n. You can interpret x = 0 as no insurance and x = 1 as full insurance. Let θ1 = 20 and θ2 = 10 for the two types of consumers, and let MC1 = 8 and MC2 = 6.
(a) Determine the efficient level of insurance for each consumer type.
(b) If a monopolist cannot tell who is what type and can only charge a single per-unit price for insurance, what will she do assuming there are γ type 1 consumers and (1−γ) type 2 consumers, with γ < 0.5?
(c) What would the monopoly price be if γ = 0? What if γ = 2/7? What is the highest that γ can be and still result in type 2 consumers buying insurance?
(d) Suppose that the monopolist first-degree price discriminates. How much insurance will each consumer type purchase? How much will each type pay for her coverage?
(e) How do your answers to (d) change if the monopolist third-degree price discriminates?
(f) Let the payment that individual n makes to the monopolist be given by Pn = Fn + pn + xn.
Express your answers to (c), (d) and (e) in terms of F1, F2, p1 and p2.
(g) Suppose γ = 0.5 — i.e. half of the population is type 1 and half is type 2. Can you rank the three scenarios in (c), (d) and (e) from most efficient to least efficient?
(h) Can you rank them in terms of their impact on consumer welfare for each type? What about in terms of population weighted average consumer welfare?