### Cost-shifting vs cross-subsidization

**Matthew Martin**3/26/2015 08:34:00 AM

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I think it might help here to distinguish two separate things: the "cost-shifting" hypothesis can be either a values claim or a causal claim. As Frakt repeatedly noted in his NYT post as well as on TIE, there's plenty of evidence that providers do cross-subsidize by charging different prices to different types of customers. As a

*values statement*, you could certainly interpret this as meaning that higher margin patients, such as those with private insurance, "bear" a larger share of the provider's fixed costs than lower margin Medicare patients, because they represent a larger share of the provider's revenue. But that's extremely different than the

*causal claim*that lower payment rates for Medicare cause providers to increase prices to private insurers.

The causal claim version of the cost-shifting hypothesis is not supported by either economic theory or empirical evidence. Consider the theory: we'll let [$]P_m[$] denote the rates paid to Medicare and [$]P_i[$] be the rates to private insurers, while [$]m[$] is the number of Medicare patients and [$]y[$] is the number of privately insured patients that the hospital treats. The function [$]c\left(m+y\right)[$] gives the hospital's total costs of treating all their patients. We'll assume that the government controls [$]P_m[$] and sets it below market rates so that [$]P_m \lt P_i[$]. Demand for healthcare depends in part on price, so [$]y[$] is actually a function of [$]P_i,[$] and we'll denote this demand function as [$]y=f\left(P_i\right)[$]. We can, of course, write a similar function for Medicare but since the government sets [$]P_m[$], we'll leave that implicit for our purposes. The hospital chooses [$]P_i[$] and [$]m[$]--the number of medicare patients it will accept--to maximize profits [$]\pi[$] given by [$$]\max_{P_i,m}\pi=P_mm+P_if\left(P_i\right)-c\left(m+f\left(P_i\right)\right).[$$] The first order conditions are

\begin{align}

P_i:&~f\left(P_i\right)+P_if'\left(P_i\right) =c'\left(m+f\left(P_i\right)\right)f'\left(P_i\right) \\

m:&~P_m=c'\left(m+f\left(P_i\right)\right)

\end{align}

where [$]f'\left(P_i\right)[$] is the first derivative of [$]f\left(P_i\right)[$] while [$]c'\left(m+f\left(P_i\right)\right)[$], the first derivative of [$]c\left(m+f\left(P_i\right)\right)[$], is the marginal cost function. Combining the two FOCs we get

\begin{equation}

f\left(P_i\right)=\left(P_m-P_i\right)f'\left(P_i\right) \label{intuit}

\end{equation}

Totally differentiating, we get

\begin{equation}

\frac{\partial P_i}{\partial P_m}=\frac{f'\left(P_i\right)}{2f'\left(P_i\right)-f''\left(P_i\right)\left(P_m-P_i\right)}

\end{equation}

Ok, for a standard demand curve, [$]f'\left(P_i\right)<0[$] and we postulated that [$]P_m-P_i<0[$]. Off the top of my head I don't recall any theorems about the second derivative of demand [$]f''\left(P_i\right)[$] but I will argue it is negative, because we typically think of people wanting infinite--or at least disproportionately large quantities--of something when its price is zero, which doesn't happen if the second derivative is positive (assuming twice continuous differentiability, etc). Hence, [$]\frac{\partial P_i}{\partial P_m}>0[$] which means this theory predicts that a decrease in Medicare payment rates will actually cause hospitals to decrease prices to private insurers as well. (Did I do all the math right? Some one please check me.)

We can intuit this result by examining equation \eqref{intuit} and recalling that [$]y=f\left(P_i\right)[$]. A decrease in [$]P_m[$] increases the right-hand side of the equation, implying an increase in [$]y[$], and we know the only way to increase [$]y[$] is by decreasing the price [$]P_i[$]. That is, hospitals seek to offset lost revenues from Medicare patients by serving fewer Medicare patients and enticing more privately insured patients with lower prices.

Austin Frakt has already gone through the evidence, and in fact confirms exactly what this little theory predicted: lower Medicare rates decrease rather than increase prices to private insurers.