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Social Welfare Function

A social welfare function (SWF) combines individual health or wellbeing into overall social welfare, setting how equality is weighed against the total.

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Last reviewed byDr Darrin Baines

Concept Architecture

Social Welfare Function Forms in Health Economics

A social welfare function ranks whole distributions of health or wellbeing, so that an analyst can say which of two allocations society should prefer when they differ in who gains as well as in how much is gained. In health economics it is the device that shows the familiar rule of maximising total QALYs to be one value judgement among several, and that turns concern for the worse off into an explicit parameter. The Atkinson index and distributional cost-effectiveness analysis have their own pages.

From Bergson's welfare function to health

Bergson's 1938 paper set out to state precisely the value judgements needed to derive conditions for maximum economic welfare. He wrote economic welfare as a function of every individual's consumption and work and showed that, once individual preferences are respected, the function takes the form E=E[S1,…,Sn]E = E[S^1, \ldots, S^n], where EE is economic welfare and SiS^i is the indifference function of individual ii. Brouwer, Culyer, van Exel and Rutten describe how this Bergson-Samuelson function lets analysts choose among distributions on a welfare frontier, provided an explicit normative choice is made about distribution, the matter on which the Pareto criterion is silent.

In general form, for nn people, the function is written as follows.

W=W(h1,h2,…,hn)W = W(h_1, h_2, \ldots, h_n)

where WW is social welfare and hih_i is the health, or in welfarist versions the utility, of person ii. If WW rises whenever one person's hih_i rises and nobody's falls, the function respects Pareto efficiency, and every Pareto improvement raises welfare. Health-related functions use health, often quality-adjusted life expectancy, in place of utility; Dolan and Tsuchiya note that health is more readily compared between people than utility is. The wider framework is on the welfare economics page.

Three forms and the parameter that links them

Each form gives a different answer to how much a unit of health is worth when it goes to someone who already has a lot of it. The utilitarian form adds health up, so a unit counts the same for everyone. The Rawlsian, or maximin, form judges a distribution by its worst-off member. The isoelastic form, which Atkinson derived from the requirement of constant relative inequality aversion, lies between them and is written as follows.

W=∑i=1nhi 1−ε1−εfor ε≠1,W=∑i=1nln⁡hifor ε=1W = \sum_{i=1}^{n} \frac{h_i^{\,1-\varepsilon}}{1-\varepsilon} \quad \text{for} \ \varepsilon \neq 1, \qquad W = \sum_{i=1}^{n} \ln h_i \quad \text{for} \ \varepsilon = 1

where WW is social welfare, hih_i is the health of person ii, nn is the number of people and ε\varepsilon is the inequality aversion parameter, which is zero or positive. Atkinson showed that ε=0\varepsilon = 0 gives a linear function that ranks distributions solely by their total, and that as ε\varepsilon tends to infinity the function becomes the minimum, taking account only of the lowest group. Differentiating gives the weight on a small gain to person ii as hi−εh_i^{-\varepsilon}, so a gain to someone with half the health of another counts 2 to the power ε\varepsilon times as much.

FormSocial welfareWeight on a small gainHealth economics reading
Utilitarian∑hi\sum h_iEqual for allMaximise total QALYs
Isoelastic∑hi1−ε/(1−ε)\sum h_i^{1-\varepsilon}/(1-\varepsilon)Proportional to hi−εh_i^{-\varepsilon}Weighted priority to the less healthy
Rawlsian (maximin)min⁡ihi\min_i h_iAll weight on the worst offImprove the least healthy group only

Health-related welfare functions and their parameters

Wagstaff's 1991 paper argued that a common theme in criticisms of QALYs was concern about inequality, which the method then pursued by QALY advocates would never capture, and suggested a method that incorporated both kinds of concern. Asaria, Griffin and Cookson describe his iso-elastic function as the one most commonly used in the health inequalities literature. For two groups it can be written as follows.

W=[αha−r+(1−α)hb−r]−1/rW = \left[ \alpha h_a^{-r} + (1-\alpha) h_b^{-r} \right]^{-1/r}

where WW is social welfare, hah_a and hbh_b are the health of groups aa and bb, α\alpha is the relative weight on group aa and rr sets the curvature of the welfare contours. Dolan and Tsuchiya state that r=−1r = -1 gives an inequality-neutral utilitarian function and that rr approaching infinity gives the Rawlsian maximin. Setting α=0.5\alpha = 0.5 for two equal-sized groups and comparing the formulas shows that the function is an increasing transformation of the isoelastic form with ε=1+r\varepsilon = 1 + r, so the two families rank distributions identically.

Williams used this function to turn the fair innings argument, the view that everyone is entitled to a normal span of health, into numbers. As Asaria and colleagues describe it, he set α\alpha at 0.5, took a range of values of rr and fixed an equally distributed equivalent of 70 QALYs as the fair innings; the slopes of the welfare contour at each group's current health gave equity weights for use in cost-effectiveness analysis. Dolan and Tsuchiya used α\alpha for a different purpose, to represent individual responsibility. In their 2003 discussion paper on a York interview study, the median respondent was indifferent between saving 100 people who had not cared for their health and 45 who had, which implies that, with equal health, a gain to the first group carries about half the weight.

Maximising QALYs as the utilitarian special case

Standard cost-utility analysis already embodies a social welfare function. Cookson and colleagues describe population health gain as the unweighted sum of individual gains, resting on the judgement that "a QALY is a QALY". That is the utilitarian form, ε=0\varepsilon = 0 or r=−1r = -1. The NICE manual (PMG36) justifies cost-effectiveness analysis by NICE's focus on maximising health gains from a fixed NHS and personal social services budget, and in its reference case the committee regards all QALYs as being of equal weight, except in specific circumstances, such as the QALY weightings for severity.

Under this function a gain to a group with a short life expectancy counts the same as one to a group with a long one, so choosing cost-utility analysis with unweighted QALYs is itself a distributional choice. The equity-efficiency trade-off page discusses when the two objectives conflict.

How inequality aversion is elicited

The parameter is a social value judgement, so it comes from outside the clinical evidence; surveys of the public are one source. In an online survey with 244 usable responses, Robson and colleagues found a median Atkinson ε\varepsilon of 10.95 for England, which at current quality-adjusted life expectancy weights a small gain to the poorest fifth between six and seven times as highly as one to the richest fifth. Estimates depend on the elicitation method; the Atkinson index page covers the survey design and other studies.

Worked example ranking three allocations as inequality aversion rises

The figures are illustrative. A population has three equal-sized groups with quality-adjusted life expectancy (QALE) of 60, 68 and 76 years. Three uses of one budget are compared, net of the health displaced elsewhere (opportunity cost). P spreads gains across all groups, T targets the two less healthy groups, and R targets the least healthy group with services that displace care used by the other two.

OptionGroup 1 QALEGroup 2 QALEGroup 3 QALESum of group gains
P (population-wide)60.368.576.71.5
T (targeted)61.068.376.01.3
R (worst off first)61.267.675.60.4

Step 1 (utilitarian). Total QALE is 60.3 + 68.5 + 76.7 = 205.5 for P, 61.0 + 68.3 + 76.0 = 205.3 for T and 61.2 + 67.6 + 75.6 = 204.4 for R. P ranks first, as a conventional cost-utility analysis would find.

Step 2 (Rawlsian). The least healthy group has 60.3 under P, 61.0 under T and 61.2 under R, so R ranks first and P last.

Step 3 (isoelastic, moderate aversion). With ε=2\varepsilon = 2 the function becomes W=−∑1/hiW = -\sum 1/h_i, shown here multiplied by 1,000. For P the reciprocals of 60.3, 68.5 and 76.7 give 0.0165837 + 0.0145985 + 0.0130378 = 0.0442201, so W = -44.2201. For T, 0.0163934 + 0.0146413 + 0.0131579 = 0.0441926, so W = -44.1926, and for R, 0.0163399 + 0.0147929 + 0.0132275 = 0.0443603, so W = -44.3603. T has the highest welfare, so the isoelastic function picks neither extreme.

Step 4 (the range of aversion). Solving numerically for ties, P is preferred for ε\varepsilon below about 1.23, T between about 1.23 and 13.0, and R above about 13.0. At the English median of 10.95, T still ranks first, narrowly ahead of R.

Moving from P to T gives up 0.2 years of QALE across the three groups for a gain of 0.7 years to the least healthy group; R gives up a further 0.9 years for another 0.2. The switching values show a committee the range of inequality aversion over which each option is preferred.

Using a welfare function in evaluation and HTA

In the distributional cost-effectiveness framework the function is the last step. The cost-effectiveness tutorial by Asaria, Griffin and Cookson first applies dominance rules, which rank options with minimal restriction on the form of the welfare function, and specifies the function more fully only for options those rules cannot separate. Equity-weighted analysis is the related approach of attaching weights to QALYs directly.

NICE limits the role of such functions in technology appraisal. PMG36 states that distributional cost-effectiveness analysis should only be used as supporting evidence, and NICE's support document on health inequalities says that health inequality aversion weights should not be applied to QALYs, including through mathematical functions that use inequality aversion parameters.

Limitations and misreadings of social welfare functions

Every form imposes structure. Atkinson called the symmetric, additively separable form restrictive, and the isoelastic family builds in relative inequality aversion. The Kolm function instead reflects absolute inequality, and the distributional cost-effectiveness tutorial treats the choice as a further value judgement, suggesting a range of measures when there is no clear choice. Equity weights derived from a fair innings function have to be updated as the health distribution changes, and the inequality aversion and responsibility parameters in the Dolan and Tsuchiya formulation are difficult to identify separately.

Three misreadings recur. The first is that QALY maximisation avoids distributional judgements, when it applies the utilitarian function. The second is that the maximin form reproduces Rawls's theory; Brouwer and colleagues note that Rawls explicitly excluded health and health care from his list of primary goods, so the health version is an analogy. The third treats an elicited value such as 10.95 as an agreed standard, when it is one survey estimate and another English study using the same basic questionnaire implied a much higher one.

Sources

Functions & Formulae (5)

W(h_i, theta) = W

Maps the health of n people, or of n groups of equal size, and the parameters theta that state a distributional value judgement to a single social welfare value W, so that allocations can be ranked when they differ in who gains as well as in how much is gained. When W rises whenever one person's health rises and nobody's falls, every Pareto improvement raises welfare. The package gives Atkinson's isoelastic additive form for epsilon other than 1, its logarithmic form at epsilon of 1, the maximin limit, the two-group CES form of Wagstaff and the slope of its social indifference curve. The equally distributed equivalent (EDE) health that gives the same W is live as HE-FM-ATK-002 and HE-FM-ATK-003, the Atkinson index as HE-FM-ATK-001, the relative weight on a small gain as HE-FM-ATK-004 and equity-weighted QALY totals as HE-FM-EQWA-001, and HE-FM-WE-004 holds the linear two-group welfare with a switching weight; none is redrafted here. The health forgone for equity and the inequality aversion at which two options swap rank are left to the Equity-Efficiency Trade-off and Distributional Cost-Effectiveness packages. Notation follows the Social Welfare Function article.

  • Isoelastic Atkinson health social welfare of three equal-sized groups for epsilon other than 1

    W = h_1^(1 - epsilon) / (1 - epsilon) + h_2^(1 - epsilon) / (1 - epsilon) + h_3^(1 - epsilon) / (1 - epsilon)

    Adds h_i^(1-epsilon)/(1-epsilon) over the groups, the welfare function Atkinson derived from constant relative inequality aversion with its two constants set to 0 and 1, which leaves every ranking unchanged. The line is written for the three groups of the article's example, and with n people or groups the sum has n terms. At epsilon of 0 W is total health, the utilitarian rule of maximising total QALYs; as epsilon rises the weight on a small gain to group i, h_i^(-epsilon), shifts priority to the less healthy (HE-FM-ATK-004), and the ranking approaches the maximin rule of HE-FM-SWF-003. Solving W = n h_EDE^(1-epsilon)/(1-epsilon) for h_EDE gives the EDE health of HE-FM-ATK-002, so W and EDE health rank allocations identically, and W has meaning only in comparisons made at the same epsilon.

  • Logarithmic health social welfare of three equal-sized groups at epsilon of 1

    W = log(h_1) + log(h_2) + log(h_3)

    At epsilon of 1 the isoelastic form of HE-FM-SWF-001 divides by zero, and Atkinson's welfare function becomes the natural logarithm, so W adds the logarithms of the groups' health. Once the constant n/(1-epsilon), which does not affect rankings, is taken away, HE-FM-SWF-001 tends to this form as epsilon tends to 1, so rankings change smoothly with epsilon through 1. The weight on a small gain to group i is 1/h_i, so a gain of 1% counts the same for every group. Dividing W by n and taking the exponential gives the geometric-mean EDE health of HE-FM-ATK-003. The function log is the natural logarithm.

  • Maximin health social welfare of three groups as the Rawlsian limit

    W = min(h_1, min(h_2, h_3))

    Sets social welfare equal to the health of the least healthy group, the Rawlsian or maximin form. Atkinson gives it as the limit of the isoelastic family as epsilon tends to infinity (HE-FM-SWF-001), and Dolan and Tsuchiya as the limit of the two-group CES form as r approaches infinity (HE-FM-SWF-004). Two allocations with the same minimum have the same W and are ranked as tied however the other groups fare, so a gain to a healthier group alone leaves W unchanged. The nested min written here returns the smallest of the three values.

  • Wagstaff two-group CES health social welfare with a group weight and curvature

    W = (alpha * h_a^(-r) + (1 - alpha) * h_b^(-r))^(-1 / r)

    Combines the health of two groups through a constant elasticity of substitution (CES) function, the iso-elastic form of Wagstaff 1991 that Asaria, Griffin and Cookson describe as the most commonly used in the health inequalities literature. The weight alpha pivots the welfare contours around the line of equal health and r sets their curvature. At r = -1 W is the weighted sum alpha h_a + (1-alpha) h_b, the inequality-neutral utilitarian case, and as r approaches infinity W approaches the smaller of the two health values, the maximin of HE-FM-SWF-003. With alpha = 0.5, W is the EDE health of HE-FM-ATK-002 for two groups with epsilon = 1 + r, so the isoelastic and CES families rank distributions identically. Unlike the weighted two-group social welfare of HE-FM-WE-004, whose weight on a group's QALYs is fixed, the weight here changes with the health gap (HE-FM-SWF-005).

  • Marginal rate of substitution and contour slope of the two-group CES health welfare

    MRS = (alpha / (1 - alpha)) * (h_b / h_a)^(1 + r); slope = -MRS

    Divides the marginal welfare of health in group a by that in group b. The result is the health of group b that could be given up for one more unit to group a with W unchanged, and the social indifference curve through the point, drawn with h_a on the horizontal axis, has slope minus MRS. Dolan and Tsuchiya read the MRS as the equity weight on a marginal gain to group a relative to group b. At equal health it reduces to alpha/(1-alpha) for any r, which is how they identify alpha from survey choices, and with alpha = 0.5 it equals the Atkinson relative weight of HE-FM-ATK-004 with epsilon = 1 + r when group a is the less healthy, so the factor alpha/(1-alpha) is what this formula adds. Williams' fair innings equity weights come from these slopes at current health on the contour whose EDE is 70 QALYs.

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Library

Publications

3
  • Report

    The social welfare function and individual responsibility: some theoretical issues and empirical evidence from health — Dolan P, Tsuchiya A, HEDS Discussion Paper 03/01, August 2003; revised version published in Journal of Health Economics, Vol. 28, No. 1, pp. 210-220 (2009) ed., 2003 (University of Sheffield)

    Discussion paper introducing individual responsibility into the health-related social welfare function, cited for its York interview study in which the median respondent was indifferent between saving 100 lives in one group and 45 in another, giving a weight of about 0.45 on a health gain to someone who had not cared for their health.

    PDFView source →
  • Book

    Measuring health inequality in the context of cost-effectiveness analysis — Asaria M, Griffin S, Cookson R, In: Health and Inequality, Research on Economic Inequality, Volume 21, pp. 491-507 ed., 2013 (Emerald Group Publishing)

    Book chapter on measuring health inequality within cost-effectiveness analysis, cited for describing social welfare as a function of the distribution of quality-adjusted life expectancy, for the Wagstaff iso-elastic form as the most commonly used function and for the Williams fair innings calibration around an equally distributed equivalent of 70 QALYs.

    BookView source →
  • Book

    The Economics of Health and Health Care — Folland, Goodman, Stano & Danagoulian, 9th Edition ed., 2024 (Routledge)

    The market-leading general health economics textbook, giving comprehensive coverage of health economics through core economic themes and balancing theory, empirical evidence and public policy. The ninth edition adds chapters on health disparities and pandemic economics.

    BookView source →

Frequently Asked Questions (6)

  • What is a social welfare function?

    A social welfare function (SWF) combines individual health or wellbeing into overall social welfare, setting how equality is weighed against the total.

    Source: Asaria et al. 2013

  • What does a social welfare function take as its inputs?

    A social welfare function takes the wellbeing of each member of society as its inputs and returns a single ranking of social states as its output. The individual levels of wellbeing are the arguments, and the form of the function specifies how they are weighted and combined, whether added equally, weighted toward the worse off, or judged only by the minimum. Choosing the form is choosing a set of distributional values. Bergson (1938) introduced this formal representation.

    Source: Bergson 1938

  • How does a social welfare function represent value judgements?

    A social welfare function represents value judgements through its form, which determines how individual welfare is weighted and combined. A function that simply sums utilities treats each person's welfare equally regardless of their position, embodying indifference to distribution; a function that gives greater weight to the worse-off embodies a preference for equality. The curvature and weights of the function encode how much equality is valued relative to total welfare, so choosing a function is choosing a distributive judgement.

    Source: Bergson 1938

  • What is a Bergson-Samuelson social welfare function?

    A Bergson-Samuelson social welfare function is a function that ranks social states according to the welfare of individuals, expressing society's values about how individual wellbeing should be aggregated. Introduced by Bergson and developed by Samuelson, it does not prescribe a particular form but provides a general framework in which distributive value judgements are made explicit. It allows welfare economics to separate the positive analysis of how policies affect individuals from the normative choice of how to weigh those effects.

    Source: Bergson 1938

  • How do different social welfare functions differ?

    Different social welfare functions differ in how they weigh individual welfare and distribution. The utilitarian function sums utilities, indifferent to how they are distributed. Egalitarian functions give more weight to the worse-off, and the Rawlsian, or maximin, function attends only to the least well-off. Between these lie functions with varying degrees of aversion to inequality. The choice among them reflects different views about the value of equality relative to total welfare, and it determines how policies affecting distribution are ranked.

    Source: Bergson 1938

  • How is a social welfare function used in health economics?

    In health economics, a social welfare function provides a framework for weighing total health gain against its distribution, making explicit whether and how much priority to give the worse-off. Standard cost-effectiveness analysis implicitly uses a function that maximises total health, treating each QALY equally; incorporating equity concerns amounts to adopting a function that weights health gains to disadvantaged groups more heavily. The concept thus clarifies the value judgements behind priority-setting and allows efficiency and equity to be combined in a single evaluation.

    Source: Bergson 1938

Trust Record

Verified by Dr Darrin Baines

British health economist

Professional identity: darrinbaines.org

Verification date: 27 Oct 2026

Content version: 1.0.2

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