Worksheet 8: Pricing in a Two-price Market

Consider the firm’s inverse demand curve in the private insurance market, $d = 16 - q$ , and costs, $c (q) = 5 + q^{2}$ . Assume that there exists a public insurer that pays a fixed price of $\bar{p} = 10$ .

Question 1

How many private patients will the provider serve?

The practice will serve private insurance patients until the marginal revenue from those patients falls below the marginal revenue of a public patient. In this case, the marginal revenue of a public patient is $\bar{p} = 10$ . The Marginal revenue of a private practice is $16 - 2 q$ . These are equal at $q = 3$ . So the practice will see 3 private insurance patients.

Note, we should check first that the practice will see any public patients. The way to do this is to make sure that the marginal revenue from a public patient is above the marginal revenue of $M R = M C$ in the private market. In this case, $M R = M C$ at $q = 4$ and $M R = 8$ . This is below $\bar{p} = 10$ , so the practice will servce some portion of the public market whenever $\bar{p} > 8$ .

Question 2

How many public patients?

The practice will see patients to the point where $M R = \bar{p} = M C$ , which holds for $2 q = 10$ , or $q = 5$ . Combined with our answer in part 1, the practice will see 3 private insurance patients and 2 public insurance patients.

Question 3

What if $\bar{p}$ drops to $9.

In this case, the practice will again see total patients to the point where $M C = \bar{p}$ . This holds for $2 q = 9$ , or $q = 4.5$ . The practice will switch to the public market at $M R = \bar{p}$ . This holds for $16 - 2 q = 9$ , or $q = 3.5$ . So the drop in the fixed payment rate will lead to an increase in the number of private insurance patients seen (from 3 to 3.5), and a decrease in the number of public patients seen (from 2 to 1). The total number of patients seen in this case also decreases (from 5 to 4.5).