Utility Possibility Frontier

Utility Possibility Frontier in Welfare Economics and Health

A utility possibility frontier is a boundary drawn in utility space, with the utility of one person or group on each axis, that marks the best combinations an economy can reach once its resources and technology are fixed. Every point on it is Pareto efficient, so a move along the frontier helps one party only by harming another. Health economists use the idea to keep two questions apart, whether an allocation of a health budget or an insurance contract wastes gains that were available, and which of the efficient outcomes society should prefer. Both questions are where welfarist cost-benefit analysis and the extra-welfarist health maximisation behind cost-effectiveness analysis part company.

How fixed resources map onto utility space

Welfare economics builds the frontier in steps. Fixed resources and technology define a production possibility frontier for outputs such as hospital care and other goods. Each output bundle can then be shared between people in many ways, and each sharing gives a pair of utility levels. Culyer and Evans (1996) describe the fundamental theorems of welfare economics as linking initial endowments, preferences and technology to potential outcomes, reached through transformation, exchange and consumption, in multi-dimensional utility space. The outer edge of that set is the frontier, and the Edgeworth box and its contract curve show the exchange step.

For two people the frontier can be written as the highest utility the second person can reach for each level guaranteed to the first.

$$\bar{U}_2(u_1) = \max_{x \in X} \{ U_2(x) \text{ such that } U_1(x) \ge u_1 \}$$

where $\bar{U}_2(u_1)$ is the frontier, $x$ is an allocation of goods and services, $X$ is the set of allocations that resources and technology make feasible, $U_i(x)$ is the utility of person $i$ under allocation $x$, and $u_1$ is the utility level promised to person 1.

Culyer and Evans describe the frontier as the potentially infinite set of Pareto-efficient points, where no one's utility can be raised by reallocating production or consumption without reducing someone else's. A point inside the frontier wastes resources, because a Pareto improvement is available. A point beyond it cannot be reached with current resources. The frontier is therefore a Pareto frontier whose objectives are individual utilities.

Arrow's frontier for a patient and an insurer

Arrow (1963) gives a health example in his paper on the welfare economics of medical care. Each possible insurance policy against a medical loss gives an expected utility to the insured person and another to the insurer. Arrow plots every such pair and shows that, when both parties have diminishing marginal utility of wealth, the boundary of this expected-utility-possibility set is convex to the northeast. It follows, he shows, that any Pareto-optimal policy can be found by maximising a weighted sum of the two expected utilities.

$$\alpha \, U[Y(X)] + \beta \, V[Z(X)]$$

where $U$ is the insured person's utility function, $V$ is the insurer's, $Y(X)$ and $Z(X)$ are their final wealth after a loss of size $X$, and $\alpha$ and $\beta$ are non-negative weights, at least one of them positive. Arrow maximises this for each size of loss.

Arrow's Proposition 2 states that if both parties are risk averse and the only costs are the losses covered, any non-trivial Pareto-optimal policy pays part, but not all, of each extra unit of loss, which is coinsurance. Different ratios of the weights pick different points on the same frontier, settling how the gains from pooling risk are split between patient and insurer. That split is a distributional matter, separate from whether the contract is efficient.

The welfare theorems and why health care misses the frontier

Arrow's First Optimality Theorem puts a competitive equilibrium, with every relevant commodity priced, on the frontier, and his Second, with no increasing returns and some minor conditions, makes every frontier point a competitive equilibrium for some distribution of purchasing power. He warns that where the actual market differs significantly from the competitive model, separating allocation from distribution becomes impossible in most cases.

Culyer and Evans list what is needed for a competitive economy to reach its frontier, including full information, the absence of externalities and public goods, and, under uncertainty, a complete set of contingent claims markets free of adverse selection and moral hazard. They report that many health economists, the authors included, struggle to see that list matching the health services sector in any country. For health policy the frontier is therefore a benchmark for allocative efficiency, not a description of where a health system actually is.

Choosing a point with a social welfare function

The frontier says nothing about which efficient point is best. Culyer and Evans stress that the theorems tell nothing about the goodness of different points on the frontier relative to each other, or even to some points off it. Arrow adds that any move between two optimal allocations must hurt someone, but society can still want it if the gains to some are judged to outweigh the injury to others, and such interpersonal comparisons are value judgements.

Brouwer, Culyer, van Exel and Rutten (2008) explain that the Bergson and Samuelson social welfare function lets analysts select a preferred distribution on the welfare frontier, provided an explicit normative choice about distribution is made. Fleurbaey's entry in the Stanford Encyclopedia of Philosophy writes this function in general form.

$$W = W(U_1(x), \ldots, U_n(x))$$

where $W$ is social welfare, $x$ is the social state and $U_i(x)$ is the utility of person $i$ in that state, for $n$ people.

The chosen point is where the highest attainable welfare contour touches the frontier. Sum-utilitarianism has straight-line contours and picks the point with the largest total. The maximin criterion, which Fleurbaey links to Rawls's difference principle, puts absolute priority on the worst-off and picks the most equal efficient point. Fleurbaey adds that social choice theory has shown how to build a continuum of social welfare functions between those two. HM Treasury's Green Book (2026) lets UK appraisals weight costs and benefits by income group, with welfare weights based on an estimated elasticity of marginal utility of consumption of 1.25, and requires weighted estimates to be shown alongside unweighted ones. Equity-weighted analysis applies weights to health gains in a similar way.

Worked example of one budget, two groups and three welfare rules

All numbers here are illustrative. A health board shares a fixed programme budget between group A and group B, where group B has a severe chronic condition and gains less utility from each pound. With a share $s$ of the budget going to group A, utility indices are as follows.

$$u_A = 100\sqrt{s}, \qquad u_B = 60\sqrt{1 - s}$$

where $u_A$ and $u_B$ are the utility indices of the two groups, $s$ is the share of the budget given to group A and $(1 - s)$ is the share given to group B. The square roots build in diminishing returns to spending.

Squaring and adding removes $s$ and gives the frontier.

$$\frac{u_A^2}{100^2} + \frac{u_B^2}{60^2} = 1$$

where 100 and 60 are the indices each group reaches with the whole budget.

1. The current position. The board now achieves 60 for group A and 25 for group B, because some spending is duplicated. The left-hand side of the frontier equation is 0.36 plus 0.1736, or 0.5336, below 1, so the position lies inside the frontier.

2. Utilitarian rule. Maximising the total sets the marginal gains equal, so 50 divided by the square root of $s$ equals 30 divided by the square root of $(1 - s)$. This gives $s$ = 25 / 34 = 0.7353. Group A then has 100 × 0.8575 = 85.75 and group B has 60 × 0.5145 = 30.87, a total of 116.62.

3. Maximin rule. The worst-off group does best when the two indices are equal, so 10,000 × $s$ equals 3,600 × $(1 - s)$ and $s$ = 3,600 / 13,600 = 0.2647. Each group then has 100 × 0.5145 = 51.45, and the total falls to 102.90.

4. Weighted rule. With an illustrative weight of 2 on group B, the condition becomes 50 divided by the square root of $s$ equals 60 divided by the square root of $(1 - s)$, so $s$ = 25 / 61 = 0.4098. Group A has 100 × 0.6402 = 64.02, group B has 60 × 0.7682 = 46.09, and weighted welfare is 64.02 + 2 × 46.09 = 156.20.

Each rule picks a different efficient point, and the maximin point is not even a Pareto improvement on the current position. The frontier corner where group A takes the whole budget scores 100 and 0. A utilitarian ranks that corner above the current position, because 100 is more than 85, while maximin and the weighted rule rank the current position higher, at 25 against 0 and 110 against 100. An efficient point need not be socially better than an inefficient one, as Culyer and Evans argue.

Health frontiers and the extra-welfarist reading

Extra-welfarist analysis keeps the geometry but changes what goes on the axes. Brouwer and colleagues note that the Pareto rule is an aggregation and decision rule that could be applied to health, and that in health care the evaluative space has focused on health, often measured in QALYs, because QALYs measure health and not because they capture utility. They add that an outcome which is optimal under welfarism may be judged suboptimal if health is badly distributed.

In a European Observatory book, Thomas and Chalkidou follow Wagstaff's 1991 paper and draw a utility possibilities frontier for two uses of health care resources, one that extends life and one that improves functional ability. A societal welfare line at 45 degrees, which assumes that a health outcome is worth the same whoever receives it, touches the frontier at the allocatively efficient mix. That tangency is the unweighted health-maximising rule. Equity weights that value health by who receives it would tilt that line, and the total health then given up is the equity-efficiency trade-off.

Utility frontiers and production frontiers compared

Several frontiers in health economics share a shape but plot different things. The production possibility frontier plots outputs, or health for two patient groups, while the utility possibility frontier plots the utilities people derive from those outputs, so one output bundle can sit behind many utility points. A frontier drawn for one fixed bundle of outputs can never lie outside the frontier drawn over every bundle the production possibility frontier allows. The general Pareto frontier is the parent idea for all of these.

Limitations and common misreadings of the frontier

The commonest error is to treat any point on the frontier as better than any point inside it. Culyer and Evans note that the classic Paretian framework says no such thing, and the worked example shows a weighted rule preferring an inefficient position to an efficient corner. A related error comes from language. Culyer and Evans warn that terms such as efficiency and optimality sound normative, and are often given the weight they carry in ordinary speech.

A second limitation is that the frontier depends on preferences that are never fully observed. Fleurbaey notes that Bergson and Samuelson shared the view that only ordinal, non-comparable information about individual preferences should be retained. On that view the curvature of the frontier changes with the arbitrary scaling of each person's utility, and only the order of points along each axis carries meaning. The utilitarian and weighted rules in the worked example need cardinal, comparable indices.

A third limitation concerns comparisons between policies. When two policies give frontiers that cross, neither set of possibilities contains the other, and a hypothetical compensation test can give conflicting answers. Fleurbaey reports that the same compensation criterion can declare A better than B and B better than A, and that Scitovsky's 1941 proposal to combine the Kaldor and Hicks tests does not prevent intransitive judgements. That reversal is known as the Scitovsky paradox.

Sources

• Arrow KJ. 1963. Uncertainty and the welfare economics of medical care. American Economic Review 53(5): 941-973. https://www.aeaweb.org/aer/top20/53.5.941-973.pdf
• Brouwer WBF, Culyer AJ, van Exel NJA, Rutten FFH. 2008. Welfarism vs. extra-welfarism. Journal of Health Economics 27(2): 325-338. https://doi.org/10.1016/j.jhealeco.2007.07.003 (author copy: https://eprints.whiterose.ac.uk/id/eprint/3789/)
• Culyer AJ, Evans RG. 1996. Mark Pauly on welfare economics: normative rabbits from positive hats. Journal of Health Economics 15: 243-251. Abridged reprint in Cookson R, Claxton K (eds). 2012. The Humble Economist: Tony Culyer on Health, Health Care and Social Decision Making. York: University of York and Office of Health Economics. https://www.york.ac.uk/media/che/documents/books/The%20Humble%20Economist%20corrected%20PDF%20version.pdf
• Fleurbaey M. 2026. Normative economics and economic justice. Stanford Encyclopedia of Philosophy, first published 28 May 2004, substantive revision 6 May 2026. https://plato.stanford.edu/entries/economic-justice/
• HM Treasury. 2026. The Green Book (2026). Updated 28 October 2026. https://www.gov.uk/government/publications/the-green-book--2/the-green-book-2026
• Thomas R, Chalkidou K. 2016. Cost-effectiveness analysis. In Cylus J, Papanicolas I, Smith PC (eds). Health System Efficiency: How to Make Measurement Matter for Policy and Management. Copenhagen: European Observatory on Health Systems and Policies, Health Policy Series 46. https://iris.who.int/server/api/core/bitstreams/2794c497-bcca-4a89-8161-b65f1b5cb998/content