Pareto Efficiency

Pareto Efficiency in Health Care Allocation When Every Gain Has a Loser

In the allocation of health care resources, Pareto efficiency appears mainly as a benchmark rather than a decision rule. An allocation meets it when no change to how a budget, staff or treatments are distributed could help one patient or group without harming another. The benchmark rules out plain waste, it frames Arrow's account of why markets for medical care fall short, and its narrowness is the reason cost-benefit analysis relies on the weaker Kaldor-Hicks compensation test and part of why cost-effectiveness analysis turns to maximising health. Almost every real decision in a health system, from funding a new medicine to closing a ward, creates some losers, so a strict Pareto test can approve very little. The criterion belongs to the wider framework of welfare economics.

The Pareto test applied to a health care allocation

The Pareto criterion compares allocations person by person. A move from one allocation to another is a Pareto improvement if at least one person gains and nobody loses, and an allocation is Pareto efficient (also called Pareto optimal) once no such move remains. Brouwer and colleagues, in their review of welfarism and extra-welfarism in health economics, describe the optimum as the point at which no reallocation could be made without reducing at least one person's utility, and note that there may be many such optima.

$$y \succ_P x \iff u_i(y) \ge u_i(x) \text{ for every } i \text{ and } u_j(y) > u_j(x) \text{ for at least one } j$$

Here $x$ is the current allocation, $y$ is a proposed feasible alternative, $u_i(\cdot)$ is the welfare of person $i$ under an allocation, $i$ runs over everyone affected and $j$ is any one affected person. The symbol $y \succ_P x$ reads "$y$ is Pareto superior to $x$". An allocation $x$ is Pareto efficient when no feasible $y$ satisfies this condition.

Two features matter here. First, the criterion is silent on distribution, and Brouwer and colleagues stress that choosing between the many optima is impossible with the Pareto criterion alone. Second, the measure being compared can change. In the welfarist tradition it is individual utility, but Brouwer and colleagues note that the Pareto criterion can be read as an aggregation rule and applied to non-utility characteristics such as health. Pareto efficiency can therefore be judged in QALYs as well as in utility, and an analysis should state which one it means.

Why medical care markets fall short of the Pareto benchmark

Arrow's 1963 paper on the welfare economics of medical care stated that a competitive equilibrium, with all commodities relevant to costs or utilities priced, is optimal in Pareto's sense. He argued that the special structural characteristics of the medical care market are attempts to overcome a lack of optimality caused by the nonmarketability of risk-bearing and the imperfect marketability of information, so that the market shows a failure to reach an optimal state in the sense of Pareto. The theorems are set out on the welfare economics page and the case for intervention on the market failure page.

Pareto reasoning about prices and insurance in health care

Arrow also applied the Pareto test directly to features of the medical care market. On price discrimination by income, he wrote that it is a source of nonoptimality, since in principle everyone would be better off if prices were made equal for all and the rich compensated the poor for the change in their relative positions. The size of the loss, he added, depends on the extent of discrimination and on each income group's price elasticity of demand.

His treatment of insurance is a second example. Where the insurer is risk-averse and its loading depends on the uncertainty of the risk, Arrow showed that the Pareto optimal policy involves some element of coinsurance, so that coverage of costs above the minimum limit (the deductible) is less than 100 per cent. His appendix credits this result to Borch. Here Pareto optimality is applied to a contract between two parties rather than to a whole economy. A policy is optimal when no other policy could raise the expected utility of the insured without lowering that of the insurer.

From actual to potential compensation in the Kaldor-Hicks test

Because almost every policy has losers, economists sought a test that could approve changes without interpersonal comparisons of utility. Feldman's dictionary entry on Kaldor-Hicks compensation explains that Kaldor and Hicks proposed in 1939 that a move is desirable if those who gain could in theory compensate those harmed and remain better off. The compensation need not be paid, and Feldman quotes Kaldor's view that whether to pay it is a political question. Feldman notes that the test is therefore sometimes called the potential Pareto improvement test.

In health care this test underpins cost-benefit analysis. Turner and colleagues, in an introduction to the main types of economic evaluation, describe cost-benefit analysis as resting on a welfarist foundation in which improvements are judged as potential Pareto improvements, with benefits valued by willingness to pay. They add that the criterion is almost always applied hypothetically, so some people win and others lose. Brouwer and colleagues go further, arguing that the potential Pareto criterion, which gives an equal weight to each unit of money gained or lost regardless of whose it is, makes explicit interpersonal comparisons of uncompensated gains and losses, and they suggest it may be viewed as a serious compromise of the Pareto principle.

Pareto efficiency and cost-effectiveness analysis

Cost-effectiveness analysis and cost-utility analysis usually rest on a different foundation. Turner and colleagues describe them as extra-welfarist, with the objective of maximising societal health measured as the sum of individual health, often in QALYs. Brouwer and colleagues explain that under extra-welfarism changes with uncompensated losers, even changes that would fail the potential Pareto criterion, may be judged improvements by appeal to additional ethical criteria, and that, by comparing people in terms of health, extra-welfarism avoids the policy paralysis that results from refusing to compare individual welfares and relying on the Pareto criterion.

The two criteria can disagree. Garber and Phelps, deriving a cost-effectiveness criterion from expected utility theory, concluded that, because the appropriate criterion varies with characteristics such as age, income and risk aversion, a uniform cost-effectiveness criterion applied to a heterogeneous population is unlikely to yield Pareto-optimal resource allocations. In a fixed budget the opportunity cost of funding one technology falls on other patients whose care is displaced, so a positive net health gain is not a Pareto improvement unless nobody loses. Read in extra-welfarist terms, a threshold rule follows the health-maximising objective, so approval under it does not show that a decision is Pareto superior.

Illustrative example of one budget and two patient groups

The numbers below are illustrative. A commissioner has 10 budget units to divide between a programme for group A and a programme for group B, and outcomes are judged in QALYs, the health reading of the Pareto test described above. Each unit spent on group A yields 2 QALYs, and each unit spent on group B yields 1 QALY.

1. Allocation X. Group A receives 4 x 2 = 8 QALYs and group B receives 4 x 1 = 4 QALYs, a total of 12. Two units buy care with no health benefit, so X is not Pareto efficient.
2. Move from X to Y. Moving one wasted unit to each programme gives A 5 x 2 = 10 QALYs (a gain of 2) and B 5 x 1 = 5 QALYs (a gain of 1). Both groups gain and neither loses, so this is a Pareto improvement.
3. Efficiency of Y. All 10 units now produce health, and any further unit moved to one group must be taken from the other, lowering its QALYs. Y is Pareto efficient.
4. Allocation Z. Giving all 10 units to A yields 10 x 2 = 20 QALYs for A and none for B. Z is also Pareto efficient, because B can only gain at A's expense.
5. Move from Y to Z. Group A gains 20 minus 10 = 10 QALYs and group B loses 5, so total health rises by 10 minus 5 = 5 QALYs. This is not a Pareto improvement, although a health-maximising rule would choose Z.

The Pareto test removes X but cannot choose between Y and Z, which give the same budget very different distributions. Choosing Z maximises total QALYs and choosing Y protects group B; this is the equity-efficiency trade-off. The example treats each group as if all its members gain or lose together; with real patients the test applies person by person, so a group-level gain can still hide individual losers.

A money version shows the compensation test. Suppose a service change benefits 500 people, each willing to pay GBP 300 (GBP 150,000 in total), and harms 200 people, each needing GBP 400 to be fully compensated (GBP 80,000). The net gain is 150,000 minus 80,000 = GBP 70,000, so the change passes the Kaldor-Hicks test, but without payment it is not a Pareto improvement. If GBP 80,000 is actually transferred, GBP 160 from each gainer, each gainer keeps 300 minus 160 = GBP 140 (GBP 70,000 in total) and each loser is fully compensated, so the change becomes an actual Pareto improvement.

Efficiency is not fairness

Pareto efficiency is consistent with any distribution of health or income that cannot be changed without a loss to someone. Brouwer and colleagues quote Sen's remark that an economy can be Pareto-optimal and still "perfectly disgusting", and his point that a state can be Pareto-optimal with some people in extreme misery so long as they cannot gain without cutting into the luxury of the rich. Allocation Z above is efficient in exactly this sense.

Arrow drew the matching conclusion in 1963. A change that makes everyone better off should be made, but he described that judgement as a very weak one, and being at a Pareto optimal allocation does not mean society should not move to another. Such a move hurts someone, and deciding whether the gains justify it requires interpersonal comparisons, which are value judgements. In health economics those comparisons are made openly through a social welfare function, equity weights or a decision maker's stated objective.

Common misunderstandings

Several errors recur in health economics writing about Pareto efficiency. Most come from reading more into the criterion than it contains, or from confusing it with other ideas that share the name.

• An aggregate gain is a Pareto improvement. A policy that raises total QALYs or passes a cost-benefit test can still leave some patients worse off. Only a change with no losers is a Pareto improvement.
• Potential compensation is compensation. The Kaldor-Hicks test asks whether gainers could compensate losers, not whether they do.
• Pareto efficient means best. There are many Pareto efficient allocations, and the criterion cannot rank them.
• Pareto efficiency and allocative efficiency are the same. Allocative efficiency in health economics usually means directing resources to the uses that maximise a stated objective, such as total health, which picks one point among the Pareto efficient allocations. Technical efficiency concerns producing an output with the fewest inputs.
• A Pareto frontier in modelling is a welfare judgement. In multi-objective optimisation the Pareto frontier is the set of options that cannot be improved on one objective without worsening another, such as health gain and cost. Its objectives are criteria, not the welfare of individual people.
• The 80/20 rule is Pareto efficiency. Pareto analysis in quality improvement, which ranks the few causes behind most problems, shares the name only.

Sources

• Arrow KJ. Uncertainty and the welfare economics of medical care. American Economic Review. 1963;53(5):941-973. https://www.aeaweb.org/aer/top20/53.5.941-973.pdf
• Brouwer WBF, Culyer AJ, van Exel NJA, Rutten FFH. Welfarism vs. extra-welfarism. Journal of Health Economics. 2008;27(2):325-338. doi:10.1016/j.jhealeco.2007.07.003. Author version: https://eprints.whiterose.ac.uk/id/eprint/3789/
• Feldman AM. Kaldor-Hicks compensation. In: Newman P, editor. The New Palgrave Dictionary of Economics and the Law, volume 2. London: Macmillan; 1998. p. 417-421.
• Garber AM, Phelps CE. Economic foundations of cost-effectiveness analysis. Journal of Health Economics. 1997;16(1):1-31. doi:10.1016/S0167-6296(96)00506-1
• Turner HC, Archer RA, Downey LE, Isaranuwatchai W, Chalkidou K, Jit M, Teerawattananon Y. An introduction to the main types of economic evaluations used for informing priority setting and resource allocation in healthcare: key features, uses, and limitations. Frontiers in Public Health. 2021;9:722927. doi:10.3389/fpubh.2021.722927