W6: Agency and Payment Type

Assume that the patient’s net benefit function is, \(NB(x) = B(x) - p_{d}x\). Further assume that \(B(x)=16x-2x^{2}\) 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, \(n(NB)=NB\). 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, \(B(x) - p_{d}x = 16x-2x^{2}\). 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, \(\pi = n(NB) \left[R + (p_{s}-c) \right]\). Given the functional forms provided in this question, we can rewrite this as, \(\pi = \left(16x - 2x^{2}\right) \left(R + (p_{s}-c)x\right)\).

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, \(\frac{\mathrm{d}\pi}{\mathrm{d}x} = (16 - 4x)(R+ (p_{s}-c)x) + (p_{s}-c)(16x-2x^{2})=0\).

Setting \(R=0\) and \(p_{s}-c=1\), this reduces to \((16-4x)x + 16x - 2x^{2} = 16x-4x^{2} + 16x - 2x^{2} = 32x-6x^{2}=0\). 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 \((16-4x)(1+x) + 16x - 2x^{2} = 6x^{2} - 28x - 16=0\). 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 \((16-4x) = 0\), 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, \(n(NB) = a*NB\). 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.