Concept Architecture
Concept
Theoretically, the Social Welfare Function is a formal mathematical function that aggregates the welfare or utility of individuals into a single measure of social welfare. Originating in welfare economics and social choice theory, it provides a framework for evaluating alternative allocations of resources according to explicit social value judgements. In health economics, the social welfare function is used to evaluate healthcare policies that involve trade-offs between efficiency and equity by combining individual health outcomes, utilities or wellbeing into a measure of collective welfare.
Mathematically, a social welfare function maps a vector of individual utilities into a scalar measure of social welfare. Different functional forms embody different ethical principles. A utilitarian social welfare function sums individual utilities, whereas isoelastic and Atkinson-type functions incorporate inequality aversion by assigning greater weight to improvements experienced by individuals with lower levels of welfare. The mathematical framework therefore allows explicit modelling of societal preferences regarding distributive justice.
In practice, social welfare functions are estimated by combining individual health outcomes, utilities or quality-adjusted life years with societal preference weights derived from normative assumptions or empirical preference studies. They are implemented in distributional cost-effectiveness analysis, equity-weighted economic evaluation, healthcare priority setting and resource allocation models to compare alternative policies while accounting for both total health gain and its distribution across populations.
Purpose
Used to aggregate individual welfare into a measure of societal welfare, evaluate healthcare policies under alternative equity objectives and support resource allocation decisions that balance efficiency with distributive fairness.
Mathematical Formulae
Primary Formula
General form:
W = f(U?, U?, ?, U?)
where:
- W = social welfare
- U? = utility or welfare of individual i
Supporting Formulae
Utilitarian social welfare function:
W = ????� U?
Isoelastic social welfare function:
W = ????� U?^(1?�) / (1?�),?� ? 1
where:
- � = inequality aversion parameter
Related Mathematical Methods
- Social choice theory
- Welfare economics
- Distributional cost-effectiveness analysis
- Equity-weighted cost-effectiveness analysis
- Atkinson social welfare analysis
- Optimisation under equity constraints
Example
A health authority compares two allocation strategies.
Strategy A
- Patient 1: 8 QALYs
- Patient 2: 8 QALYs
- Patient 3: 8 QALYs
Utilitarian social welfare:
W = 8 + 8 + 8 = 24
Strategy B
- Patient 1: 14 QALYs
- Patient 2: 7 QALYs
- Patient 3: 3 QALYs
Utilitarian social welfare:
W = 14 + 7 + 3 = 24
Although both strategies produce identical total welfare under the utilitarian function, an inequality-averse social welfare function would assign a higher welfare value to Strategy A because health outcomes are distributed more equally.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUM | =SUM(B2:B101) | Calculates a utilitarian social welfare function by summing individual utilities or QALYs. |
| POWER | =POWER(B2,1-$E$1)/(1-$E$1) | Computes individual contributions to an isoelastic social welfare function. |
| SUMPRODUCT | =SUMPRODUCT(B2:B101,C2:C101) | Calculates weighted social welfare using equity weights. |
| Solver | Solver Optimisation | Identifies the resource allocation that maximises social welfare subject to healthcare budget constraints. |
VBA (Optional)
Automate calculation of alternative social welfare functions using different inequality aversion parameters and compare healthcare allocation scenarios.
Sources
- Bergson A. A Reformulation of Certain Aspects of Welfare Economics. Quarterly Journal of Economics. 1938.
- Samuelson PA. Foundations of Economic Analysis. Harvard University Press.
- Atkinson AB. On the Measurement of Inequality. Journal of Economic Theory. 1970.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
- Asaria M, Griffin S, Cookson R. Distributional Cost-Effectiveness Analysis: A Tutorial. Medical Decision Making.
Related Concepts (2)
Library
Publications
1
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 mathematical representation of how individual wellbeing combines into overall social welfare, embodying explicit judgements about equality versus total welfare.
Source: Bergson 1938
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: 26 Sep 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-EE-WE-022
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