Competitive Provision of Health Insurance: Consider the challenge of providing health insurance to a population with different probabilities of getting sick.
A. Suppose that, as in our car insurance example, there are two consumer types—consumers of type 1 that are likely to get sick, and consumers of type 2 that are relatively healthy. Let x represent the level of health insurance, with x = 0 implying no insurance and higher levels of x indicating increasingly generous health insurance benefits. Assume that each consumer type has linear demand curves (equal to marginal willingness to pay), with d1 representing the demand curve for a single consumer of type 1 and d2 representing the demand curve for a single consumer of type 2. Suppose further that the marginal cost of providing additional health coverage to an individual is constant, with MC1 >MC2.
(a) For simplicity, suppose throughout that d1 and d2 have the same slope. Suppose further, unless otherwise stated, that d1 has higher intercept than d2. Do you think it is reasonable to assume that type 1 has higher demand for insurance?
(b) Begin by drawing a graph with d1, d2, MC1 and MC2 assuming that the vertical intercepts of both demand curves lie above MC1. Indicate the efficient level of insurance 1 and 2 for the two types.
(c) Suppose the industry offers any level of x at price p = MC1. Illustrate on your graph the consumer surplus that type 1 individuals will get if this were the only way to buy insurance and they buy there optimal policy A. How much consumer surplus will type 2 individuals get?
(d) Next, suppose you want to offer an additional insurance contract B that earns zero profit if bought only by type 2 consumers, that is preferred by type 2 individuals to A and that makes type 1 consumers just as well off as they are under the options from part (c). Identify B in your graph.
(e) Suppose for a moment that it is an equilibrium for the industry to offer only contracts A and B (and suppose that the actual B is just slightly to the left of the B you identified in part (d)). True or False: While insurance companies do not know what type consumers are when they walk into the insurance office to buy a policy, the companies will know what type of consumer they made a contract with after the consumer leaves.
(f) In order for this to be an equilibrium, it must be the case that it is not possible for an insurance company to offer a “pooling price” that makes at least zero profit while attracting both type 1 and 2 consumers. (Such a policy has a single price p∗ that lies betweenMC1 and MC2.) Note that the demand curves graphed thus far were for only one individual of each type. What additional information would you have to know in order to know whether the zero-profit price p∗ would attract both types?
(g) True or False: The greater the fraction of consumers that are of type 1, the less likely it is that such a “pooling price” exists.
(h) Suppose that no such pooling price exists. Assuming that health insurance firms cannot observe the health conditions of their customers, would it be a competitive equilibrium for the industry to offer contracts A and B? Would this be a pooling or a separating equilibrium?
(i) Would you still be able to identify a contract B that satisfies the conditions in (d) if d1 = d2? What if d1 < d2?
B. Part A of this exercise attempts to formalize a key intuition we covered in section B of the text with a different type of model for insurance.
(a) Rather than starting our analysis by distinguishing between marginal costs of different types, our model from section B starts by specifying the probabilities θ and δ that type 1 and type 2 individuals will find themselves in the “bad state” that they are insuring against. Mapping this to our model from part A of this exercise, with type 1 and 2 defined as in part A, what is the relationship between δ and θ?
(b) To fit the story with the model from section B, we can assume that what matters about bad health shocks is only the impact they have on consumption — and that tastes are state independent. (We will relax this assumption in exercise 22.8). Suppose we can, for both types, write tastes over risky gambles as von-Neumann Morgenstern expected utility functions that employ the same function u(y) as “utility of consumption” (with consumption denoted y). Write out the expected utility functions for the two types.
(c) Does the fact that we can use the same u(y) to express expected utilities for both types imply that the two types have the same tastes over risky gambles— and thus the same demand for insurance?
(d) If insurance companies could tell who is what type, they would (in a competitive equilibrium) simply charge a price equal to each type’s marginal cost. How is this captured in the model developed in section B of the text?
(e) In the separating equilibrium we identified in part A, we had insurance companies providing the contract A that is efficient for type 1 individuals—but providing an inefficient contract B to type 2. Draw the model from section B of the text and illustrate the same A and B contracts. How are they exactly analogous to what we derived in part A?
(f) In part Awe also investigated the possibility of a potential pooling price—or pooling contract —breaking the separating equilibrium in which A and B are offered. Illustrate in the different model here how the same factors are at play in determining whether such a pooling price or contract exists.
(g) Evaluate again the True/False statement in part A(g).