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 $16-2q$. 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 $MR=MC$ in the private market. In this case, $MR=MC$ at $q=4$ and $MR=8$. 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 $MR=\bar{p}=MC$, which holds for $2q=10$, 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 $MC=\bar{p}$. This holds for $2q=9$, or $q=4.5$. The practice will switch to the public market at $MR=\bar{p}$. This holds for $16-2q=9$, 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).