In
welfare economics
Welfare economics is a branch of economics that uses microeconomic
Microeconomics is a branch of mainstream economics
Mainstream economics is the body of knowledge, theories, and models of economics, as taught by universities worldwide, tha ...
, a social welfare function is a
function
Function or functionality may refer to:
Computing
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* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
that ranks social states (alternative complete descriptions of the society) as less desirable, more desirable, or
for every possible pair of social states. Inputs of the function include any variables considered to affect the
economic welfare The welfare definition of economics is an attempt by Alfred Marshall
Alfred Marshall (26 July 1842 – 13 July 1924) was an English economist, who was one of the most influential economists of his time. His book, '' Principles of Economics'' ...
of a society. In using welfare measures of persons in the society as inputs, the social welfare function is
individualistic
Individualism is the Ethics, moral stance, political philosophy, ideology and social outlook that emphasizes the intrinsic worth of the individual. Individualists promote the exercise of one's goals and desires and to value independence and self ...
in form. One use of a social welfare function is to
represent
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* Represent (Fat Joe album), ''Represent'' (Fat Joe album), 1993
* ''Represent'', an album by DJ Magic Mike, 1994 ...
prospective patterns of collective choice as to alternative social states. The social welfare function provides the government with a simple guideline for achieving the optimal distribution of income.
The social welfare function is analogous to the
consumer theory
The theory of consumer choice is the branch of microeconomics
Microeconomics is a branch of mainstream economics
Mainstream economics is the body of knowledge, theories, and models of economics, as taught by universities worldwide, that ar ...
of
–
budget constraint
In economics
Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behavi ...

tangency for an individual, except that the social welfare function is a mapping of individual preferences or judgments of everyone in the society as to collective choices, which apply to all, whatever individual preferences are for (variable) constraints on factors of production. One point of a social welfare function is to determine how close the analogy is to an ordinal
utility function
As a topic of economics
Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumpti ...
for an individual with at least minimal restrictions suggested by welfare economics, including constraints on the number of
factors of production
In economics
Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behavio ...
.
There are two major distinct but related types of social welfare functions:
* A Bergson–Samuelson social welfare function considers welfare for a ''given set'' of individual preferences or welfare rankings.
* An Arrow social welfare function considers welfare across ''different possible sets'' of individual preferences or welfare rankings and seemingly reasonable axioms that constrain the function.
[Prasanta K. Pattanaik, 2008. "social welfare function," '']The New Palgrave Dictionary of Economics
''The New Palgrave Dictionary of Economics'' (2018), 3rd ed., is an twenty-volume reference work on economics
Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, ...
'', 2nd Edition
Abstract.
/ref>
Bergson–Samuelson social welfare function
In a 1938 article, Abram Bergson
Abram Bergson (April 21, 1914 in New York City
New York, often called New York City to distinguish it from New York State
New York is a state
State may refer to:
Arts, entertainment, and media Literature
* ''State Magazine'', a mont ...
introduced the ''social welfare function''. The object was "to state in precise form the value judgments required for the derivation of the conditions of maximum economic welfare" set out by earlier writers, including Marshall
Marshall may refer to:
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and , and Barone, and Lerner Lerner is a German and Jewish family name. Its literal meaning can be either "student" or "scholar". It may refer to:
Organizations
* Lerner Enterprises, a real estate company
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* Lerner Publishing Group, a publisher of childr ...

. The function was real-valued and differentiable
In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...

. It was specified to describe the society as a whole. Arguments of the function included the quantities of different commodities produced and consumed and of resources
Resource refers to all the materials available in our environment which help us to satisfy our needs and wants. Resources can broadly be classified upon their availability — they are classified into renewable
A renewable resource, also know ...
used in producing different commodities, including labor.
Necessary general conditions are that at the maximum value of the function:
* The marginal "dollar's worth" of welfare is equal for each individual and for each commodity
* The marginal "diswelfare" of each "dollar's worth" of labor is equal for each commodity produced of each labor supplier
* The marginal "dollar" cost of each unit of resources is equal to the marginal value productivity for each commodity.
Bergson showed how welfare economics
Welfare economics is a branch of economics that uses microeconomic
Microeconomics is a branch of mainstream economics
Mainstream economics is the body of knowledge, theories, and models of economics, as taught by universities worldwide, tha ...
could describe a standard of economic efficiency despite dispensing with ''interpersonally-comparable'' cardinal utility
In economics, a cardinal utility function or scale is a utility index that preserves Preference (economics), preference orderings uniquely up to positive affine transformations. Two utility indices are related by an affine transformation if for the ...
, the hypothesization of which may merely conceal value judgments, and purely subjective ones at that.
Auxiliary specifications enable comparison of different social states by each member of society in preference satisfaction. These help define ''Pareto efficiency
Pareto efficiency or Pareto optimality is a situation where no individual or preference criterion can be better off without making at least one individual or preference criterion worse off or without any loss thereof. The concept is named after V ...
'', which holds if all alternatives have been exhausted to put at least one person in a more preferred position with no one put in a less preferred position. Bergson described an "economic welfare increase" (later called a ''Pareto improvement'') as at least one individual moving to a more preferred position with everyone else indifferent. The social welfare function could then be specified in a ''substantively'' individualistic sense to derive Pareto efficiency (optimality). Paul Samuelson
Paul may refer to:
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The Eastern Orthodox Church, officially the Orthodox Catholic Church, is the List ...

(2004, p. 26) notes that Bergson's function "could derive Pareto optimality conditions as ''necessary'' but not sufficient for defining interpersonal normative equity." Still, Pareto efficiency could also characterize ''one'' dimension of a particular social welfare function with distribution of commodities among individuals characterizing ''another'' dimension. As Bergson noted, a welfare improvement from the social welfare function could come from the "position of some individuals" improving at the expense of others. That social welfare function could then be described as characterizing an equity dimension.
Samuelson (1947
It was the first year of the Cold War
The Cold War was a period of geopolitical tension between the United States and the Soviet Union and their respective allies, the Western Bloc and the Eastern Bloc, which began following World W ...
, p. 221) himself stressed the flexibility of the social welfare function to characterize ''any'' one ethical belief, Pareto-bound or not, consistent with:
* a complete and transitive ranking (an ethically "better", "worse", or "indifferent" ranking) of all social alternatives and
* one set out of an infinity of welfare indices and cardinal indicators to characterize the belief.
He also presented a lucid verbal and mathematical exposition of the social welfare function (1947, pp. 219–49) with minimal use of Lagrangean multipliers and without the difficult notation of differentials used by Bergson throughout. As Samuelson (1983, p. xxii) notes, Bergson clarified how production and consumption efficiency conditions are distinct from the interpersonal ethical values of the social welfare function.
Samuelson further sharpened that distinction by specifying the ''Welfare function'' and the ''Possibility function'' (1947, pp. 243–49). Each has as arguments
In logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents stat ...
the set of utility functions for everyone in the society. Each can (and commonly does) incorporate Pareto efficiency. The Possibility function also depends on technology and resource restraints. It is written in implicit form, reflecting the ''feasible'' locus of utility combinations imposed by the restraints and allowed by Pareto efficiency. At a given point on the Possibility function, if the utility of all but one person is determined, the remaining person's utility is determined. The Welfare function ranks different hypothetical ''sets'' of utility for everyone in the society from ethically lowest on up (with ties permitted), that is, it makes interpersonal comparisons of utility. Welfare maximization then consists of maximizing the Welfare function subject to the Possibility function as a constraint. The same welfare maximization conditions emerge as in Bergson's analysis.
Arrow social welfare function (constitution)
Kenneth Arrow
Kenneth Joseph Arrow (23 August 1921 – 21 February 2017) was an American economist, mathematician, writer, and political theorist
{{unreferenced, date=June 2015
A political theorist is someone who engages in constructing or evaluating politica ...
(1963
Events January
* January 1
January 1 or 1 January is the first day of the year in the . There are 364 days remaining until the end of the year (365 in s). This day is known as since the day marks the beginning of the year. __TOC__ ...
) generalizes the analysis. Along earlier lines, his version of a social welfare function, also called a 'constitution', maps a set of individual orderings (ordinal utility function In economics, an ordinal utility function is a function representing the Preference (economics), preferences of an agent on an ordinal scale. Ordinal utility theory claims that it is only meaningful to ask which option is better than the other, but ...
s) for everyone in the society to a social ordering, a rule for ranking alternative social states (say passing an enforceable law or not, ceteris paribus
' or ' () is a Latin phrase meaning "other things equal"; English translations of the phrase include "all other things being equal" or "other things held constant" or "all else unchanged". A prediction or a statement about a ontic, causal, epist ...
). Arrow finds that nothing of behavioral significance is lost by dropping the requirement of social orderings that are ''real-valued'' (and thus cardinal
Cardinal or The Cardinal may refer to:
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) in favor of orderings, which are merely ''complete'' and ''transitive'', such as a standard indifference curve
In economics
Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behavi ...

map. The earlier analysis mapped any set of individual orderings to ''one'' social ordering, whatever it was. This social ordering selected the top-ranked ''feasible'' alternative from the economic environment as to resource constraints. Arrow proposed to examine mapping different sets of individual orderings to possibly different social orderings. Here the social ordering would depend on the set of individual orderings, rather than being ''imposed'' (invariant to them). Stunningly (relative to a course of theory from Adam Smith
Adam Smith ( 1723 – 17 July 1790) was a Scottish economist, philosopher as well as a moral philosopher
Ethics or moral philosophy is a branch of philosophy that "involves systematizing, defending, and recommending concepts of right and ...

and Jeremy Bentham
Jeremy Bentham (; 15 February 1748 Old_Style_and_New_Style_dates">O.S._4_February_1747.html" ;"title="Old_Style_and_New_Style_dates.html" ;"title="nowiki/>Old Style and New Style dates">O.S. 4 February 1747">Old_Style_and_New_Style_dates.htm ...

on), Arrow proved the '' general impossibility theorem'' which says that it is impossible to have a social welfare function that satisfies a certain set of "apparently reasonable" conditions.
Cardinal social welfare functions
A cardinal social welfare function is a function that takes as input numeric representations of individual utilities (also known as cardinal utility
In economics, a cardinal utility function or scale is a utility index that preserves Preference (economics), preference orderings uniquely up to positive affine transformations. Two utility indices are related by an affine transformation if for the ...
), and returns as output a numeric representation of the collective welfare. The underlying assumption is that individuals utilities can be put on a common scale and compared. Examples of such measures can be:
* life expectancy
Life expectancy is a statistical measure of the average time an organism is expected to live, based on the year of its birth, its current age, and other demographic
Demography (from prefix ''demo-'' from Ancient Greek
Ancien ...

,
* per capita income.
For the purposes of this section, income is adopted as the measurement of utility.
The form of the social welfare function is intended to express a statement of objectives of a society.
The utilitarian
Utilitarianism is a family of normative
Normative generally means relating to an evaluative standard. Normativity is the phenomenon in human societies of designating some actions or outcomes as good or desirable or permissible and others as ba ...
or Benthamite
Jeremy Bentham (; 15 February 1748 ld Style and New Style dates, O.S. 4 February 1747
LD may refer to:
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*Lorraine "L.D." Delacorte, a character on the TV series ''Degrassi
''Degrassi'' is a Canadian teen drama
In film
A film, also called a movie, motion picture or moving pi ...
– 6 June 1832) was an English philosopher, jurist, and social reformer regarded as the founder of modern utilitarianism.
Bentham defined as the "fundamen ...
social welfare function measures social welfare as the total or sum of individual incomes:
:
where is social welfare and is the income of individual among individuals in society. In this case, maximizing the social welfare means maximizing the total income of the people in the society, without regard to how incomes are distributed in society. It does not distinguish between an income transfer from rich to poor and vice versa. If an income transfer from the poor to the rich results in a bigger increase in the utility of the rich than the decrease in the utility of the poor, the society is expected to accept such a transfer, because the total utility of the society has increased as a whole. Alternatively, society's welfare can also be measured under this function by taking the average of individual incomes:
:
In contrast, the max-min or Rawlsian social welfare function (based on the philosophical work of John Rawls
John Bordley Rawls (; February 21, 1921 – November 24, 2002) was an American moral
A moral (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was ori ...
) measures the social welfare of society on the basis of the welfare of the least well-off individual member of society:
:
Here maximizing societal welfare would mean maximizing the income of the poorest person in society without regard for the income of other individuals.
These two social welfare functions express very different views about how a society would need to be organised in order to maximize welfare, with the first emphasizing total incomes and the second emphasizing the needs of the worst-off. The max-min welfare function can be seen as reflecting an extreme form of uncertainty aversion on the part of society as a whole, since it is concerned only with the worst conditions that a member of society could face.
Amartya Sen
Amartya Kumar Sen (; born 3 November 1933) is an Indian economist and philosopher, who since 1972 has taught and worked in the United Kingdom
The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom ( ...

proposed a welfare function in 1973:
:
The average per capita income of a measured group (e.g. nation) is multiplied with where is the Gini index
In economics
Economics () is a social science
Social science is the Branches of science, branch of science devoted to the study of society, societies and the Social relation, relationships among individuals within those so ...
, a relative inequality measure. James E. Foster (1996) proposed to use one of Atkinson's Indexes, which is an entropy measure. Due to the relation between Atkinsons entropy measure and the Theil index #REDIRECT Theil index
The Theil index is a statistic primarily used to measure economic inequality and other economic phenomena, though it has also been used to measure racial segregation.
The first presentation of this method of measuring inequal ...
, Foster's welfare function also can be computed directly using the Theil-L Index.
:
The value yielded by this function has a concrete meaning. There are several possible incomes which could be earned by a ''person'', who randomly is selected from a population with an unequal distribution of incomes. This welfare function marks the income, which a randomly selected person is most likely to have. Similar to the median
In statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data
Data (; ) are individual facts, statistics, or items of information, often numeric. In a m ...

, this income will be smaller than the average per capita income.
:
Here the Theil-T index is applied. The inverse value yielded by this function has a concrete meaning as well. There are several possible incomes to which a ''Euro'' may belong, which is randomly picked from the sum of all unequally distributed incomes. This welfare function marks the income, which a randomly selected Euro most likely belongs to. The inverse value of that function will be larger than the average per capita income.
The article on the Theil index provides further information about how this index is used in order to compute welfare functions.
Axioms of cardinal welfarism
Suppose we are given a preference relation The term preference relation is used to refer to orderings that describe human preferences for one thing over an other.
* In mathematics, preferences may be modeled as a weak ordering or a semiorder, two different types of binary relation. One speci ...
''R'' on utility profiles. ''R'' is a weak total order
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
on utility profiles—it can tell us, given any two utility profiles, if they are indifferent or one of them is better than the other. A reasonable preference ordering should satisfy several axioms:
1. Monotonicity, i.e., if the utility of an individual increases while all other utilities remain equal, ''R'' should strictly prefer the second profile. E.g., it should prefer the profile (1,4,4,5) to (1,2,4,5). This is related to Pareto optimality
Pareto efficiency or Pareto optimality is a situation where no individual or preference criterion can be better off without making at least one individual or preference criterion worse off or without any loss thereof. The concept is named after V ...

.
2. Symmetry, i.e., ''R'' should be indifferent to permutation of the numbers in the utility profile. E.g., it should be indifferent between (1,4,4,5) and (5,4,1,4).
3. Continuity: for every profile ''v'', the set of profiles weakly better than ''v'' and the set of profiles weakly worse than ''v'' are closed set
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
s.
4. Independence of unconcerned agents, i.e., ''R'' should be independent of individuals whose utilities have not changed. E.g., if ''R'' prefers (2,2,4) to (1,3,4), then it also prefers (2,2,9) to (1,3,9); the utility of agent 3 should not affect the comparison between two utility profiles of agents 1 and 2. This property can also be called locality or separability. It allows us to treat allocation problems in a local way, and separate them from the allocation in the rest of society.
Every preference relation with properties 1–4 can be represented as by a function ''W'' which is a sum of the form:
:
where ''w'' is a continuous increasing function.
It is also reasonable to require:
5. Independence of common scale, i.e., the relation between two utility profiles does not change if both of them are multiplied by the same scalar (e.g., the relation does not depend on whether we measure the income in cents, dollars or thousands).
If the preference relation has properties 1–5, then the function ''w'' belongs to the following one-parameter family:
* for ,
* for ,
* for .
This family has some familiar members:
* The limit when is the ''leximin'' ordering;
* For we get the Nash bargaining solution—maximizing the product of utilities;
* For we get the utilitarian
Utilitarianism is a family of normative
Normative generally means relating to an evaluative standard. Normativity is the phenomenon in human societies of designating some actions or outcomes as good or desirable or permissible and others as ba ...
welfare function—maximizing the sum of utilities;
* The limit when is the ''leximax'' ordering.
If, in addition, we require:
6. the Pigou–Dalton principle,
then the parameter ''p'', in the above family, must be at most 1.
See also
* Aggregation problem
An ''aggregate'' in economics
Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuse ...
* Gorman polar form Gorman polar form is a functional form for indirect utility function__NOTOC__
In economics, a consumer's indirect utility function
v(p, w) gives the consumer's maximal attainable utility when faced with a vector p of goods prices and an amount of in ...
* Arrow's impossibility theorem#REDIRECT Arrow's impossibility theorem#REDIRECT Arrow's impossibility theorem
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