Kenneth Arrow on healthcare economics: a 21st century appreciation

Nobel laureate Kenneth Arrow passed away on February 21, 2017. In a classic, fifty-year-old paper entitled Uncertainty and the Welfare Economics of Medical Care, Arrow discussed how:

“the operation of the medical-care industry and the efficacy with which it satisfies the needs of society differs from… a competitive model… If a competitive equilibrium exists at all, and if all commodities relevant to costs or utilities are in fact priced in the market, then the equilibrium is necessarily [Pareto] optimal” (emphasis added)

Note the implicit assumption that price reflects value, to which I’ll return. As Arrow elegantly explained, there are vast differences between the actual healthcare market and the competitive model, and, moreover, these differences arise from important features of the actual healthcare market.

Identifying the lack of realism of the competitive model in health care may lead to deeper understanding of the actual system. In essence this is what Arrow does. Although both medical care and our expectations have changed greatly, Arrow ’63 is still valid and worth reading today.

Here is Arrow’s summary of the differences between the healthcare market and typical competitive markets.

The nature of demand

Demand for medical services is irregular and unpredictable:

“Medical services, apart from preventive services, afford satisfaction only in the event of illness, a departure from the normal state of affairs… Illness is, thus, not only risky but a costly risk in itself, apart from the cost of medical care.”

Expected behavior of the physician

“It is at least claimed that treatment is dictated by objective needs of the case and not limited by financial considerations… Charity treatment in one form or another does exist because of this tradition about human rights to adequate medical care.”

Product uncertainty

“Recovery from disease is as unpredictable as its incidence…  Because medical knowledge is so complicated, the information possessed by the physician as to the consequences and possibilities of treatment is necessarily very much greater than that of the patient, or at least so it is believed by both parties.”

Supply conditions

Barriers to entry include licensing and other controls on quality (accreditation) and costs.

“One striking consequence of the control of quality is the restriction on the range offered… The declining ratio of physicians to total employees in the medical-care industry shows that substitution of less trained personnel, technicians and the like, is not prevented completely, but the central role of the highly trained physician is not affected at all.”

Pricing practices

There are no fixed prices:

“extensive price discrimination by income (with an extreme of zero prices for sufficiently indigent patients)… the apparent rigidity of so-called administered prices considerably understates the actual flexibility.”

Avik Roy observes in a critical National Review article that “Because patients don’t see the bill until after the non-refundable service has been consumed, and because patients are given little information about price and cost, patients and payors are rarely able to shop around for a medical service based on price and value.”

Medicine has seen major changes since Arrow’s 1963 paper. For example, the treatment of blocked coronary arteries has evolved from coronary bypass to angioplasty to early stents and finally drug-eluting stents. We have seen the advent of minimally invasive surgery, robotic surgery and catheter-based cardiac valve repair and replacement. We have seen drugs to treat hepatitis C and biologicals to treat arthritis and cancer. Many conditions have been transformed from acute to chronic but (at least temporarily) manageable. There are also divergent trends, such as increases in both natural childbirth and Caesarean sections.

In the last 50 years, medicine has become more powerful, but also significantly more complex and overall, more expensive. Intensive care units are a good example, both valuable therapeutically, but expensive to provide. At the same time, many treatments are both better (more valuable to the patient) and less expensive to provide; these range from root canal (frequently two visits to the dentist instead of four) to the significantly less invasive treatments for many cardiac rhythm abnormalities (radio-frequency ablation) and stents for coronary artery disease. The advent of epinephrine auto-injectors has been a lifesaver, but the cost of the Epi-Pen has increased significantly.

Can a competitive economic system appropriately and reasonably price such treatments and devices? Arrow argues that, if not, non-market social institutions will arise and address these challenges. Here is a deeper look.

Arrow’s first two points are still virtually axiomatic today: demand for medical services has become even more unpredictable with the continued growth of advanced, effective interventions and corresponding, appropriately increasing (in my opinion), patient expectations. Similarly, as medical care advances, we increasingly see medical care as a human right and in many cases, a societal obligation. We have come to expect treatment dictated by objective needs and not limited by financial considerations, not only from physicians but from a growing number of key players including pharmaceutical companies. To their credit, in many cases (AIDS comes to mind) pharmaceutical companies have responded by sharply reducing prices in the developing world.

Powerful chemotherapeutic and biologic drugs may have increased the uncertainty and asymmetry of information observed by Arrow, both in their effectiveness and in their side effects. In many cases one needs the language and mathematics of probability and statistics to evaluate, assess and describe their efficacy and utility. One needs an understanding of probability to determine when and how to use common preventive techniques, such as mammograms and PSA screening. Here is an example, paraphrased from Gigerenzer and Edwards (see also Strogatz). Women 40 to 50 years old, with no family history of breast cancer, are a low-risk population; the overall probability of breast cancer in this population is 0.8%. Assume that mammography has a sensitivity of 90% and a false positive rate of 7%.  A woman has a positive mammogram. What is the probability that she has breast cancer? Among 25 German doctors surveyed, 36% said 90% or more, 32% said 50-80%, and 32% said 10% or less. Most (95%) of United States doctors thought the probability was approximately 75%.  (See the links above for the answer, or see my next blog on the challenge of communicating probability).

Arrow’s information asymmetry remains, despite the growing availability of accessible medical information on the web, perhaps for good reasons such as the ability to effectively address the needs of sicker patients.

I would amend Arrow’s discussion of supply conditions to include a wide variety of cost barriers ranging from large fixed costs of ICUs to the costs of medical research. The high cost of basic medical services relative to per capita GDP in the the developing world represents a barrier as high as any faced in the developed world.  As Arrow notes, society has addressed this challenge through a variety of pricing mechanisms outside traditional competitive models. This may not, and in general will not achieve a Pareto optimum, but their wide endorsement by society does indeed suggest that these approaches achieve a more general optimum.

“I propose here the view that, when the market fails to achieve an optimal state, society will, to some extent at least, recognize the gap, and nonmarket social institutions will arise attempting to bridge it… But it is contended here that the special structural characteristics of the medical-care market are largely attempts to overcome the lack of optimality due to the nonmarketability of the bearing of suitable risks and the imperfect marketability of information. These compensatory institutional changes, with some reinforcement from usual profit motives, largely explain the observed noncompetitive behavior of the medical-care market, behavior which, in itself, interferes with optimality. The social adjustment towards optimality thus puts obstacles in its own path.”

It is this view which I find too limiting. I would suggest that society has at least implicitly concluded that price alone does not define value, and thus formed a broader definition of optimality, not simply Pareto optimality in a competitive market. Society is finding and supporting ways to overcome obstacles toward this broader sense of optimality.

The Bill & Melinda Gates Foundation vaccination project aims to reduce the number of children that die each year from preventable disease (currently around 1.5 million). The lifebox project, founded by Dr Atul Gawande, provides affordable, high quality pulse oximeters to the developing world and now seeks to address basic surgical safety in the developing world. Important advances also arise in the developing world; most recently, an easy to deliver, more effective oral cholera vaccine developed in Vietnam.

Arrow himself recognizes the limits of a traditional economic description of the medical care market in his concluding Postscript, arguing that “The logic and limitations of ideal competitive behavior under uncertainty force us to recognize the incomplete description of reality supplied by the impersonal price system.” I conclude more generally that prices not only do not necessarily represent value in medical care (as Arrow observed), but that the combination of uncertainty, externalities, high costs, divergent economies, and technological advance means that price alone cannot describe value in medical care. A broader more general theory of healthcare economics with a foundation standing on the shoulders of giants such as Kenneth Arrow, with perhaps a more general multi-dimensional Pareto optimum, might help us all better understand where we are and where we might go.



ICU triage: a challenge and an opportunity

In a well-publicized snapshot of the challenge of ICU triage, Chang and colleagues wrote:

Critical care services can be life-saving, but many patients admitted to intensive care units (ICUs) are too sick or, conversely, not sick enough to benefit. Intensive care unit overutilization can produce more costly and invasive care without improving outcomes.

Emphasis added. Hyder provides an interesting critique to which Chang and Shapiro respond. In this post, I shall consider over-utilization by those “not sick enough to benefit”: 23.4% of the 808 patients admitted to the UCLA Medical Center in the study by Chang et al. This over-utilization provides both a challenge and a win-win opportunity (better outcomes at lower cost) if we can meet the challenge.

In a forward-looking vision, which some may regard as optimistic, Anesi et al wrote:

In the year 2050 we will unambiguously reimburse healthcare based on value, and so there is good reason to suspect that we will have targeted and reduced many services that provide little or no benefit to patients…

It can be argued that ICU over-utilization, on average, provides no overall benefit, while significantly increasing costs. Gooch and Kahn observed that US spending on critical care represents nearly 3% of GDP, while:

In contrast, the United Kingdom spends only 0.1% of its gross domestic product on critical care services, with no evidence of worse patient outcomes and similar life expectancies as in the United States. Although there are many differences between these 2 countries, one significant difference is intensive care unit (ICU) bed supply. The United States has 25 ICU beds per 100 000 people, as compared with 5 per 100 000 in the United Kingdom. As a result, ICU case-mix differs substantially. In the United Kingdom, the majority of ICU patients are at high risk for death, whereas in the United States, many patients are admitted to the ICU for observation.

As observed by Halpern, these differences come at a significant cost in the US:

The number of intensive care unit (ICU) beds in the United States has continued to increase over the last 3 decades, as have ICU utilization rates and costs, and this despite the lack of any federal, regional, or critical care society mandates to justify these increases. Some experts believe that the increase in the number of ICU beds has led to inappropriate use of these beds by patients who are either too healthy or too sick to benefit from intensive care. This may in part explain the stable national ICU occupancy rate of approximately 68% between 1985 and 2010 and suggests that ICU utilization has simply risen to meet the increased number of beds.

Emphasis added. I shall consider here only ICU usage by patients too healthy to benefit. Although the economics behind reducing ICU over-utilization by “those not sick enough to benefit” appears simple, the underlying cause is in fact likely complex.


This one appears easy: lower costs and potentially better outcomes

At the same time, I recall several caveats, well known to health economists, but important in planning and communication:

  1. We expect ICUs to be available when needed, including for emergencies and disasters,
  2. ICUs have high fixed costs,
  3. Decision-making is critical: incremental costs of adding capacity become fixed costs in the future.

Chris Sampson recently reviewed a study aimed at overconsumption or misconsumption (a consequence of over-utilization). The authors of that paper suggest that “cultural change might be required to achieve significant shifts in clinical behaviour.” Chris laments that this study did not ‘dig deeper’; here we aim to dig deeper in one specific area: ICU triage for patients “not sick enough to benefit.” More questions than answers at this stage, but hopefully the questions will ultimately lead to answers.

I begin by stepping back: economic decisions frequently involve compromises in allocating scarce resources. Decisions in health economics are frequently no different. How scarce are ICU resources? What happens if they are less scarce? What are the costs? Increasing availability can frequently lead to increased utilization, a phenomenon called “demand elasticity”. For example, increasing expressway/motorway capacity “can lead to increased traffic as new drivers seize the opportunity to travel on the larger road”, and thus no reduction in travel time. Gooch and Kahn further note that:

The presence of demand elasticity in decisions regarding ICU care has major implications for health care delivery and financing. Primarily, this indicates it is possible to reduce the costs of US hospital care by constraining ICU bed supply, perhaps through certificate of need laws or other legislation.

I offer a highly simplified sketch of how ICU over-utilization by those “not sick enough to benefit” is one driver of a vicious cycle in ICU cost growth.


ICU over-utilization by patients “not sick enough to benefit” as a driver for ICU demand elasticity

Who (if anyone) is at fault for this ICU vicious cycle? Chang and Shapiro offer one suggestion:

For medical conditions where ICU care is frequently provided, but may not always be necessary, institutions that utilize ICUs more frequently are more likely to perform invasive procedures and have higher costs but have no improvement in hospital mortality. Hospitals had similar ICU utilization patterns across the 4 medical conditions, suggesting that systematic institutional factors may influence decisions to potentially overutilize ICU care.

Emphasis added. I note that demand elasticity is not in itself bad; it must simply be recognized, controlled and used appropriately. As part of a discussion in print on the role of cost considerations in medical decisions, Du and Kahn write:

Although we argue that costs should not be factored into medical decision-making in the ICU, this does not mean that we should not strive toward healthcare cost reduction in other ways. One strategy is to devise systems of care that prevent unnecessary or unwanted ICU admissions—given the small amount of ICU care that is due to discretionary spending, the only real way to reduce ICU costs is to prevent ICU admissions in the first place.

Du and Kahn also argue for careful cost-effectiveness analyses, such as that supported by NICE in the UK:

These programs limit use of treatments that are not cost-effective, taking cost decisions out of the hands of physicians and putting them where they belong: in the hands of society at large… We will achieve real ICU savings only by encouraging a society committed to system-based reforms.

Emphasis added. One can debate “taking cost decisions out of the hands of physicians”, though Guidet & Beale‘s and Capuzzo & Rhodes‘s argument for more physician awareness of cost might provide a good intermediate position in this debate.

Finally, increasing ICU supply (that is, ICU beds) in response to well-conceived increases in ICU demand is not in itself bad; ICU supply must be able to respond to demands imposed by disasters or other emergencies. We need to seek out novel ways to provide this capacity without incurring potentially unnecessary fixed costs, perhaps from region-wide stockpiling of supplies and equipment, and region-wide pools of on-call physicians and other ICU personnel. In summary, current health-related literature offers a wide-ranging discussion of the growing costs of intensive care; in my opinion: more questions than answers at this stage, but hopefully the questions will ultimately lead to answers.


“Doing the math” on the distribution of healthcare expenditures: a Pareto-like distribution is inevitable

Yesterday I explored one of the major challenges to affordable, universal health insurance, namely the high cost of providing care to the sickest patients. The extreme distribution of healthcare costs means that “Targeting the highest spenders represents the greatest opportunity to have a significant impact on overall spending”, an opportunity for insurance carriers  to reduce costs by risk selection, as well as for public policy. Here is a deeper look into the math behind the distribution of healthcare expenditures, using 2012 US data as a model.

One can fit a Pareto (power law, 80/20) distribution with scale coefficient \alpha – that is, prob(expenditure)\sim 1/expenditure^{\alpha+1} – to the data in several ways. For a Pareto distribution with scale coefficient \alpha, the per-capita expenditure at a given percentile from the top scales as 1/\%ile^{1/\alpha}. The first two of these approaches yield a scale coefficient 1/\%ile^{0.893}, with expenditures scaling as :

  1. Use the 80/20 rule modified to fit the data: the top 25% ranked by healthcare expenditures account for 86.7% of costs; thus \alpha=1.115.
  2. Use the ratio of mean to median expenditure, 5.05:1; thus \alpha=1.119.
    However, a graphical analysis finds that the data does not follow such a Pareto distribution, shown as a black dashed line in the following figure (representing a Pareto distribution with \alpha=1.117 and median expenditure $854, the actual median expenditure).
  3. Use data for the most expensive patients (10% through 30% percentiles from the top), for these patients, per-capita expenditure scales as 1/\%ile^{1.24}, (R^{2}=0.994), shown as a dashed red line in the figure above; thus \alpha=0.806.
  4. Use the fraction of total expenses paid by the most expensive patients. A comparison of the fraction of expenses paid by the most expensive 1%, 5% and 10% finds that this scales as x^{0.4228}, (R^{2}=0.987), shown as a dashed black line in the figure below. This scaling exponent is 1-1/\alpha; thus \alpha=1.733. (Scaling added to figure modified from Cohen, 2014)


Thus, there really is no typical patient. For discussion and implications, see Feyman, who called the empirical distribution of healthcare costs “worse than Pareto”. The Pareto-like (hyper-Pareto?) empirical distribution of expenditures presents a severe challenge to risk pooling through insurance without limiting the highest expenditures through risk selection (illegal!).

Pareto distributions differ sharply from normal distributions, with important consequences for payment models. For a Pareto-like distribution with \alpha\leq2 at large expenditures, the variance is not defined, and sample variance approaches infinity with increasing sample size. Therefore, unlike the case of distributions with finite variance, variability in the mean of a sample of size N does not decrease with N. This violates a standard requirement for insurance; that risk pooling over a large sample reduces variability in the mean expenditure, and thus, standard insurance models cannot effectively price health insurance when the highest per capita expenditures follow Pareto distributions.

Moreover, a Pareto-like distribution may be a natural consequence of advances in healthcare: our growing ability to manage multiple simultaneous chronic conditions, with consequent exponential growth in costs, while extending life expectancy, so that the probability of dying is not only not reduced, but may actually increase. In a mathematically limiting case, with no bound on healthcare costs, these dynamics yield a Pareto distribution.

In fact, if one extrapolates the power law for a broad range of the sickest patients (the 10th through 30th percentiles of expenditures from the top), obtaining a Pareto distribution with \alpha\leq1, even the mean is not defined and the sample mean approaches infinity with increasing sample size. The actual distribution of healthcare cost for the very sickest patients clearly falls below the empirical Pareto distribution with \alpha=0.806, such a distribution predicts a cost at the 1st percentile of $178,194, well above the average for the top 1% of $97,956. Deviations from this distribution for the very sickest patients may reflect current limits on healthcare and thus healthcare expenses. These limits may be relaxed with advances in healthcare, causing further growth in costs.

A Pareto-like distribution of healthcare costs is here to stay, and must be reflected in how we share the burden of healthcare and provide care to our sickest patients.