Monday, March 29, 2010

Who Dominates Health Care Costs?

There's nothing like a little bout with MRSA to make one pay a little more attention to the state of health care legislation. Two nights in the ER are definitely making me thankful for health insurance. Knowing that it wasn't going to cost me an arm and a leg to get antibiotics through an IV (in fact, it probably saved me the leg) definitely helped me to seek care early, rather than waiting for the infection to get truly life-threatening. And that probably saved in health care costs in the long run.

I've heard it argued many times by the other side that universal health care will drive up the cost of health care for everyone, because so-called "healthy people" will be paying, through their premiums, for the bills of the "unhealthy". Ignoring that:
  1. the above is a tautological statement about what insurance is
  2. people routinely go from the "healthy" group to the "unhealthy" group and back again
  3. we should maybe feel a moral obligation to care for the unhealthy
Yeah, ignoring that, I wanted to know if the underlying assumption was, in fact, true. Who dominates health care costs? Is it the small number of extremely sick people? Or is it the larger number of moderately sick people?

To answer that question, I went searching for the population distribution of health care costs. I found the following publication: Variations in Lifetime Healthcare Costs across a Population (Forget et al. 2008). To the left are reproduced Figs. 4 and 5.

Given all the hype, I was somewhat underwhelmed to see that these curves depict (with the exception of an excess at the lowest cost bin) a gamma distribution. This isn't surprising, because a gamma distribution is supposed to represent the sum of a bunch of exponentially-distributed random variables.

To find the contribution of people in each cost bin to the total health care cost of the population, we simply need to multiply the population of that bin (drawn from a gamma function) by the mean health care cost of that bin (a linearly increasing function). Setting the mode of the gamma distributon to $90k for females, and tweaking the k and theta parameters (I'll chi-by-eye it at k=4.5, theta=1.0) we get the following distributions of fractional population (black) and fractional total health care cost (red), as a function of lifetime healthcare cost:

So who dominates health care costs? Those just slightly above the mode, which is to say, the large number of people who are just a little sicker than most. And that really could be any of us, folks.


  1. Hi Aaron,
    Sorry to comment here on your Guitar Chord song page, but I couldn't find a way to contact you thru that page. I wanted to thank you for posting your renditions of the many songs. I am a 40 something year old mom, just now learning to strum away with my 12 year old daughter.....excellent quality time, for sure. As we are both beginners, we appreciate the chords over lyrics to learn to play familiar songs. beginners, it would really help us if you also included with each song, the suggested strum pattern. As absolute beginners, we have no idea background knowledge to draw on, and no guitar playing role model to turn to for suggestions as to strum patterns, so we are completely clueless as to what we should be strumming like. Please consider adding the suggested strum patterns to the many Songs/lyrics/chords that you have posted and shared. Thanks

  2. Aaron and Sarah, It's late here in Massachusetts, so I'm just going to write a
    few lines this time. You may know, it is not
    possible to leave a message on your guest page. YOUR MUSIC PAGES ARE A RARE GIFT!!
    The clear and "print friendly" manner in which
    you've entered them is so appreciated by me.
    The choice of songs, also. Sounds like you are busy people, and yet have enough time to
    do this..thank you!! I tried to read some of
    your blogs...holy cow, that's all I'll say for now. I have been playing guitar for
    approx 45 years... took it out of the closet
    a few months ago, after a couple years (maybe 10 or more)or collecting cobwebs. Your website has definately inspired me!!!! Love your website. Carol I have 4 grown kids and 10 to hear more about your mom

  3. A totally unrelated comment. If you are the owner of I found this page a few years ago and I seem to remember a tool for transposing the chords. It is a great selection of songs but I find I need to sing in a different key than you do. Your guestbook is no longer working, I wanted to ask there. Feel free to email me directly at and let me know if this was my imagination or not.

  4. Aaron, I don't agree that the following statement is a tautology of health insurance: "universal health care will drive up the cost of health care for everyone, because so-called "healthy people" will be paying, through their premiums, for the bills of the "unhealthy". If anything, it is a tautology of universal health care, which is very different from insurance. Insurance is the underwriting of risk, whereby parties pool together their funds to cover a small percentage of claimants. The percentage of claimants has to perforce be small, or the system is no longer insurance. Do the math: if an individual pays $200 a monnth in premiums and incurs a $100,000 medical expense, how many other payers have to not make a claim in order for the insurer to stay solvent? Quite a few.

    Health care, on the other hand, is the actual doctors, medications, treatments and tests provided by the medical industry. This is different from insurance.

    So "Universal Health Care" is not insurance at all, but a promise to give everyone in the population free (or cheap) medical care whether they pay premiums or not.

    I won't bore you with why the concept of "Universal Health Insurance" is a contradiction in terms. I just respectfully point out that in Universal Health Care only those who can afford the higher taxes and fees will be the ones paying for everyone else's use of the benefits. How moral is that?

    By the way, I also greatly appreciate your guitar charts.

    Kindest regards.