In principle, there is a good argument for paying Medicare Advantage (MA) plans more than what it would cost to cover a beneficiary under FFS Medicare: a spillover effect that causes per beneficiary FFS Medicare costs to be lower as MA enrollment grows. In recent years, the MA “overpayment” (for lack of a better term) was about 13%. That is, each beneficiary enrolled in an MA plan cost taxpayers 13% more than it would have if that beneficiary had selected the FFS option. However, at least one study suggests that as enrollment in Medicare HMOs increases, FFS costs per beneficiary decreases. Thus, the argument goes, MA overpayments may fully or partially pay for themselves.

The mechanisms for this offset effect are: (1) Medicare HMOs induce more efficient provider practice patterns that spillover into FFS; (2) Medicare HMOs draw enrollment out of Medicare supplements (wrap-around coverage that fills in FFS Medicare cost sharing and provides additional benefits). Supplements have been shown to increase FFS costs since reduced cost sharing increases moral hazard. Bundling coverage within one at-risk organization–an MA plan–both encourages that organization to control utilization and removes cost risk from FFS Medicare.

At least two publications I’ve read recently have made a qualitative argument that MA overpayments may not be as costly to taxpayers as they naively appear if one accounts for the offsetting effects just described (Book and Capretta 2010, Francis 2009). In support of this claim, they cite a 2008 Journal of Health Economics paper by Chernew, DeCicca, and Town, “Managed care and medical expenditures of Medicare beneficiaries.” In that paper the authors use a well-defended instrumental variables approach and 1994–2001 data to estimate the size of the Medicare HMO spillover. The point estimate of their preferred specification is that each percentage point increase in Medicare HMO market penetration leads to a one percent decrease in FFS costs.

That’s a huge effect! It’s tempting to conclude that Medicare HMOs can reduce overall Medicare costs even if they’re paid more than FFS costs. However, this conclusion is actually not supported by the findings of Chernew, DeCicca, and Town. In a series of e-mail exchanges, the authors of that paper convinced me of this fact. The math that shows it is a bit beyond what is customary for a typical blog post, but I’m going to go through it anyway because this is not a typical blog.

Let *f* be per beneficiary FFS cost, *p* be Medicare HMO market penetration (the proportion of beneficiaries enrolled in an HMO), *a* be the overpayment, and *θ *be the elasticity of FFS costs w.r.t. HMO market penetration (the spillover effect). I’ll emphasize that HMO market penetration is a function of the overpayment by writing *p = p(a)*. Though I won’t write *θ=θ(p)*, it is nevertheless true that the degree to which HMO market penetration affects FFS costs is not constant.

Medicare HMOs are paid *(1+a)f* for each beneficiary enrolled. Due to the spillover effect, a beneficiary enrolled in FFS Medicare costs taxpayers *(1-θp(a))f. *Also, by assumption, proportion *p(a) *of beneficiaries are enrolled in an HMO and proportion *(1-p(a)) *are enrolled in FFS. Consequently, an average beneficiary costs the program

*c(a)* = *p(a)(1+a)f + (1-p(a))(1-θp(a))f *.

What’s the optimal value overpayment, *a*, that minimizes program costs?

To find it, take the derivative of the program cost function, *c(a)*, w.r.t. *a*, set it to zero, and solve for *a*. I’ll omit the calculus and algebra and just assert that the answer is

*a = -p(a)(g(a)+2θ) + θ ,*

where *g(a) *is the reciprocal of the derivative of *p(a)* w.r.t. *a*. We can already sense that it is possible for the optimal *a *to be negative or zero. A lot depends on the functional form of *p(a)*. That makes sense. Think about it. If very few additional beneficiaries enroll in an HMO even as overpayments reach high levels, the program isn’t going to save a lot of money; it could even lose money. Of course, the optimal *a *also depends on the size of the spillover effect, *θ*.

Let’s simplify things to get some more intuition. I don’t believe for a second that *p(a)* is linear in *a* but assuming so will help get a grip on the basic concepts. So, let *p(a) = βa*. Plugging that into the expression for the optimal *a* above, we find that the optimal *a* is

*a = θ/(2 + 2βθ) *.

Now it is easy to see that the optimal *a* is non-negative if *β > -1/ θ (it is assumed that θ is non-positive). OK, maybe that’s not easy enough. So, let’s simplify further and plug in the value for θ found by Chernew, DeCicca, and Town θ = -1. With that, the optimal a is*

*a = 1/(2 β – 2). *

Clearly *a* is non-negative only if *β > 1*. That means that overpayments are justified if MA enrollment growth grows at least as fast. If a one percent overpayment does not at least lead to a once percent increase in MA market penetration, then MA payments won’t pay for themselves. But that’s reasoning from a faulty model. We know that *p* is not a linear function of *a*, which is what I assumed just for developing intuition. So, hopefully you get the idea.

Put another way, *β *must vary with *p*. Put yet another way, I expect that HMO overpayments to at least partially pay for themselves in some range of HMO enrollment, but not in others. At some point all the practice pattern changes have been wrung out of the system and there’s no more spillover juice left. Where’s the threshold? I don’t know. Nobody does.

Finally, it should be noted that much has changed in the Medicare Advantage program (formerly Medicare+Choice) since 2001, the last year of the study window, which warrants revisiting this whole spillover issue. The authors note this in their conclusion,

Given their substantial magnitude,we suspect additional large changes in penetration might translate into somewhat smaller effects. Second, our results do not apply to private FFS plans, which have benefited from generous payment and do not likely generate substantial spillovers. Third, our analysis predates the Medicare Modernization Act (MMA) of 2003. The MMA expanded the set of plans available to Medicare beneficiaries, added a prescription drug benefit, and enhanced the system of risk adjustment. Spillovers may differ in the post MMA era if these changes alter the way managed care plans behave or the set of people who selected Medicare managed care plans.

A shorter way to say this is that the spillover effect, *θ*, estimated by Chernew, DeCicca, and Town is at the margin of HMO market penetration observed in their data. It’s not the *average *effect, nor is it likely to apply at the margin that exists today. As I said above, *θ *varies with market penetration, *p. *It may not even be negative for all *p*.