Worksheet 6: Physician agency and capitated payments

Assume that the patient’s net benefit function is, $N B (x) = B (x) - p_{d} x$ . Further assume that $B (x) = 16 x - 2 x^{2}$ and that the patient is fully insured such that $p_{d} = 0$ (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, $n (N B) = N B$ . The physician receives a fixed payment, $R$ , per patient, along with a margin of $p_{s} - c$ for each unit of care, $x$ . The same amount of care $x$ is provided to each patient.

Question 1:

What is the patient’s optimal amount of care?

If they could, the patient would choose $x$ to maximize their net benefit, $B (x) - p_{d} x = 16 x - 2 x^{2}$ . This net benefit is optimized at $x = 4$ .

Question 2:

Write out the physician’s profit function.

The physician’s profit is simply their number of patients times the revenue per patient, $π = n (N B) [R + (p_{s} - c)]$ . Given the functional forms provided in this question, we can rewrite this as, $π = (16 x - 2 x^{2}) (R + (p_{s} - c) x)$ .

Question 3:

Find the physician’s optimal amount of care if $R = 0$ and $p_{s} - c = 1$

To find the physician’s optimum, we must first take the derivative of the profit function and set this equal to 0, $\frac{d π}{d x} = (16 - 4 x) (R + (p_{s} - c) x) + (p_{s} - c) (16 x - 2 x^{2}) = 0$ .

Setting $R = 0$ and $p_{s} - c = 1$ , this reduces to $(16 - 4 x) x + 16 x - 2 x^{2} = 16 x - 4 x^{2} + 16 x - 2 x^{2} = 32 x - 6 x^{2} = 0$ . Solving for $x$ yields $x = 5.33$ . This is larger than the patient’s optimal, $x = 4$ , 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 $R = 1$ and $p_{s} - c = 1$ .

Plugging these values into the same first order condition, we get $(16 - 4 x) (1 + x) + 16 x - 2 x^{2} = 6 x^{2} - 28 x - 16 = 0$ . We can solve this with the quadratic equation, in which case $x = \frac{28 \pm \sqrt{(} 1168)}{12} \approx (- 0.515, 5.181)$ . Since we can’t provide negative care, we’ll take the positive root so that $x \approx 5.181$ . While these numbers are small, the important part is that we’ve introduced a capitated payment $R = 1$ 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.

Question 5:

Find the physician’s optimal if $R = 1$ and $p_{s} - c = 0$ .

Again substituting these values into he first order condition, we get $(16 - 4 x) = 0$ , or $x = 4$ . So now the physician and patient optimal are the same value. Note that this is not a general solution…it only arises due to the linear patient function, $n (N B) = a * N B$ . For nonlinear functions, we won’t generally be able to easily obtain the same optimal values from the physician and patient’s perspective. Instead, $R > 0$ will yield some amount of undertreatment as we saw in the general case in class.