W10: 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).