Assume that the patient’s net benefit function is, . Further assume that and that the patient is fully insured such that (i.e., the insurer pays the full price of care, and the patient pays nothing). Finally, assume that the physician’s number of patients in their practice is governed by, . The physician receives a fixed payment, , per patient, along with a margin of for each unit of care, . The same amount of care is provided to each patient.
Question 1:
What is the patient’s optimal amount of care?
If they could, the patient would choose to maximize their net benefit, . This net benefit is optimized at .
Question 2:
Write out the physician’s profit function.
The physician’s profit is simply their number of patients times the revenue per patient, . Given the functional forms provided in this question, we can rewrite this as, .
Question 3:
Find the physician’s optimal amount of care if and
To find the physician’s optimum, we must first take the derivative of the profit function and set this equal to 0, .
Setting and , this reduces to . Solving for yields . This is larger than the patient’s optimal, , which should be expected since there is a positive FFS payment in this setup and no capitated payment.
Question 4:
Find the physician’s optimal amount of care if and .
Plugging these values into the same first order condition, we get . We can solve this with the quadratic equation, in which case . Since we can’t provide negative care, we’ll take the positive root so that . While these numbers are small, the important part is that we’ve introduced a capitated payment and managed to bring “overtreatment” down from 1.333 to 1.181. In other words, the inclusion of the capitated payment brought us closer to the patient’s optimal amount of care.