Social welfare function
Encyclopedia
In economics
, a social welfare function is a real-valued function
that ranks conceivable social states (alternative complete descriptions of the society) from lowest to highest. Inputs of the function include any variables considered to affect the economic welfare
of a society (Sen, 1970, p. 33). In using welfare measures of persons in the society as inputs, the social welfare function is individualistic
in form. One use of a social welfare function is to represent
prospective patterns of collective choice as to alternative social states.
The social welfare function is analogous to an indifference-curve
map 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. One point of a social welfare function is to determine how close the analogy is to an ordinal utility function for an individual with at least minimal restrictions suggested by welfare economics
. Kenneth Arrow
proved a more basic point
for a set of seemingly reasonable conditions.
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
and Pigou, Pareto
and Barone
, and Lerner. The function was real-valued and differentiable. 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
used in producing different commodities, including labor.
Necessary general conditions are that at the maximum value of the function:
Bergson showed how welfare economics
could describe a standard of economic efficiency despite dispensing with interpersonally-comparable cardinal utility
, the hypothesizaton 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
, 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
(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
, p. 221) himself stressed the flexibility of the social welfare function to characterize any one ethical belief, Pareto-bound or not, consistent with:
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 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.
(1963
) 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 functions) 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
). Arrow finds that nothing of behavioral significance is lost by dropping the requirement of social orderings that are real-valued (and thus cardinal) in favor of orderings, which are merely complete and transitive, such as a standard indifference-curve 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
and Jeremy Bentham
on), Arrow proved the General Possibility Theorem
that it is impossible to have a social welfare function that satisfies a certain set of "apparently reasonable" conditions.
and per capita income for the society. The rest of this article adopts the latter definition.
The form of the social welfare function is intended to express a statement of objectives of a society. For example, take this example of a social welfare function:
where
is social welfare and 
is the income of individual i among n in the society. In this case, maximising the social welfare function means maximising the total income of the people in the society, without regard to how incomes are distributed in society. Alternatively, consider the Max-Min utility function (based on the philosophical work of John Rawls
):
Here, the social welfare of society is taken to be related to the income of the poorest person in the society, and maximising welfare would mean maximising the income of the poorest person without regard for the incomes of the others.
These two social welfare functions express very different views about how a society would need to be organised in order to maximise welfare, with the first emphasizing total incomes and the second emphasising the needs of the poorest. 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
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, 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
, 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
, 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 an 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.
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, a social welfare function is a real-valued function
Function of a real variable
In mathematics, a function of a real variable is a mathematical function whose domain is the real line. More loosely, a function of a real variable is sometimes taken to mean any function whose domain is a subset of the real line....
that ranks conceivable social states (alternative complete descriptions of the society) from lowest to highest. Inputs of the function include any variables considered to affect the economic welfare
Economic welfare
Economic welfare broadly refers to the level of prosperity and living standards in an individual or group of persons. In the field of economics, it specifically refers to utility gained through the achievement of material goods and services...
of a society (Sen, 1970, p. 33). In using welfare measures of persons in the society as inputs, the social welfare function is individualistic
Methodological individualism
Methodological individualism is the theory that social phenomena can only be accurately explained by showing how they result from the intentional states that motivate the individual actors. The idea has been used to criticize historicism, structural functionalism, and the roles of social class,...
in form. One use of a social welfare function is to represent
Mathematical problem
A mathematical problem is a problem that is amenable to being represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more abstract nature, such as Hilbert's...
prospective patterns of collective choice as to alternative social states.
The social welfare function is analogous to an indifference-curve
Indifference curve
In microeconomic theory, an indifference curve is a graph showing different bundles of goods between which a consumer is indifferent. That is, at each point on the curve, the consumer has no preference for one bundle over another. One can equivalently refer to each point on the indifference curve...
map 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. One point of a social welfare function is to determine how close the analogy is to an ordinal utility function for an individual with at least minimal restrictions suggested by welfare economics
Welfare economics
Welfare economics is a branch of economics that uses microeconomic techniques to evaluate economic well-being, especially relative to competitive general equilibrium within an economy as to economic efficiency and the resulting income distribution associated with it...
. Kenneth Arrow
Kenneth Arrow
Kenneth Joseph Arrow is an American economist and joint winner of the Nobel Memorial Prize in Economics with John Hicks in 1972. To date, he is the youngest person to have received this award, at 51....
proved a more basic point
Arrow's impossibility theorem
In social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives , no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a...
for a set of seemingly reasonable conditions.
Bergson–Samuelson social welfare function
In a 1938 article Abram BergsonAbram Bergson
Abram Bergson , born Abram Burk, was an American economist. He was born in New York City....
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
Alfred Marshall
Alfred Marshall was an Englishman and one of the most influential economists of his time. His book, Principles of Economics , was the dominant economic textbook in England for many years...
and Pigou, Pareto
Vilfredo Pareto
Vilfredo Federico Damaso Pareto , born Wilfried Fritz Pareto, was an Italian engineer, sociologist, economist, political scientist and philosopher. He made several important contributions to economics, particularly in the study of income distribution and in the analysis of individuals' choices....
and Barone
Enrico Barone
Enrico Barone was a soldier, military historian, and economist.Barone studied the classics and mathematics before becoming an army officer. He taught military history for eight years from 1894 at the Officers' Training School. There he wrote a series of influential historical military works...
, and Lerner. The function was real-valued and differentiable. 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
Factors of production
In economics, factors of production means inputs and finished goods means output. Input determines the quantity of output i.e. output depends upon input. Input is the starting point and output is the end point of production process and such input-output relationship is called a production function...
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
Welfare economics is a branch of economics that uses microeconomic techniques to evaluate economic well-being, especially relative to competitive general equilibrium within an economy as to economic efficiency and the resulting income distribution associated with it...
could describe a standard of economic efficiency despite dispensing with interpersonally-comparable cardinal utility
Cardinal utility
In economics, cardinal utility refers to a property of mathematical indices that preserve preference orderings uniquely up to positive linear transformations...
, the hypothesizaton of which may merely conceal value judgments, and purely subjective ones at that.
| Earlier neoclassical welfare theory Welfare economics Welfare economics is a branch of economics that uses microeconomic techniques to evaluate economic well-being, especially relative to competitive general equilibrium within an economy as to economic efficiency and the resulting income distribution associated with it... , heir to the classical utilitarianism Utilitarianism Utilitarianism is an ethical theory holding that the proper course of action is the one that maximizes the overall "happiness", by whatever means necessary. It is thus a form of consequentialism, meaning that the moral worth of an action is determined only by its resulting outcome, and that one can... of Bentham Jeremy Bentham Jeremy Bentham was an English jurist, philosopher, and legal and social reformer. He became a leading theorist in Anglo-American philosophy of law, and a political radical whose ideas influenced the development of welfarism... , had not infrequently treated the Law of Diminishing Marginal Utility as implying interpersonally comparable utility, a necessary condition to achieve the goal of maximizing total utility of the society. Irrespective of such comparability, income or wealth is measurable, and it was commonly inferred that redistributing income from a rich person to a poor person tends to increase total utility (however measured) in the society.* But Lionel Robbins (1935 An Essay on the Nature and Significance of Economic Science Lionel Robbins' Essay sought to define more precisely economics as a science and to derive substantive implications. Analysis is relative to "accepted solutions of particular problems" based on best modern practice as referenced, especially including the works of Philip Wicksteed, Ludwig von... , ch. VI) argued that how or how much utilities, as mental events, would have changed relative to each other is not measurable by any empirical test. Nor are they inferable from the shapes of standard indifference curves. Hence, the advantage of being able to dispense with interpersonal comparability of utility without abstaining from welfare theory.
|
Auxiliary specifications enable comparison of different social states by each member of society in preference satisfaction. These help define Pareto efficiency
Pareto efficiency
Pareto efficiency, or Pareto optimality, is a concept in economics with applications in engineering and social sciences. The term is named after Vilfredo Pareto, an Italian economist who used the concept in his studies of economic efficiency and income distribution.Given an initial allocation of...
, 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 Samuelson
Paul Anthony Samuelson was an American economist, and the first American to win the Nobel Memorial Prize in Economic Sciences. The Swedish Royal Academies stated, when awarding the prize, that he "has done more than any other contemporary economist to raise the level of scientific analysis in...
(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
Foundations of Economic Analysis
Foundations of Economic Analysis is a book by Paul A. Samuelson published in 1947 by Harvard University Press. It sought to demonstrate a common mathematical structure underlying multiple branches of economics from two basic principles: maximizing behavior of agents and stability of equilibrium...
, 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 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.
For a two-person society, there is a graphical depiction of such welfare maximization at the first figure of Bergson–Samuelson social welfare functions. Relative to consumer theory for an individual as to two commodities consumed, there are the following parallels:
|
Arrow social welfare function (constitution)
Kenneth ArrowKenneth Arrow
Kenneth Joseph Arrow is an American economist and joint winner of the Nobel Memorial Prize in Economics with John Hicks in 1972. To date, he is the youngest person to have received this award, at 51....
(1963
Social Choice and Individual Values
Kenneth Arrow's monograph Social Choice and Individual Values and a theorem within it created modern social choice theory, a rigorous melding of social ethics and voting theory with an economic flavor...
) 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 functions) 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
Ceteris paribus
or is a Latin phrase, literally translated as "with other things the same," or "all other things being equal or held constant." It is an example of an ablative absolute and is commonly rendered in English as "all other things being equal." A prediction, or a statement about causal or logical...
). Arrow finds that nothing of behavioral significance is lost by dropping the requirement of social orderings that are real-valued (and thus cardinal) in favor of orderings, which are merely complete and transitive, such as a standard indifference-curve map
Consumer theory
Consumer choice is a theory of microeconomics that relates preferences for consumption goods and services to consumption expenditures and ultimately to consumer demand curves. The link between personal preferences, consumption, and the demand curve is one of the most closely studied relations in...
. 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
Production possibility frontier
In economics, a production–possibility frontier , sometimes called a production–possibility curve or product transformation curve, is a graph that compares the production rates of two commodities that use the same fixed total of the factors of production...
. 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
Invisible hand
In economics, invisible hand or invisible hand of the market is the term economists use to describe the self-regulating nature of the marketplace. This is a metaphor first coined by the economist Adam Smith...
and Jeremy Bentham
Jeremy Bentham
Jeremy Bentham was an English jurist, philosopher, and legal and social reformer. He became a leading theorist in Anglo-American philosophy of law, and a political radical whose ideas influenced the development of welfarism...
on), Arrow proved the General Possibility Theorem
Arrow's impossibility theorem
In social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives , no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a...
that it is impossible to have a social welfare function that satisfies a certain set of "apparently reasonable" conditions.
Cardinal social welfare functions
In the above contexts, a social welfare function provides a kind of social preference based on only individual utility functions, whereas in others it includes cardinal measures of social welfare not aggregated from individual utility functions. Examples of such measures are life expectancyLife expectancy
Life expectancy is the expected number of years of life remaining at a given age. It is denoted by ex, which means the average number of subsequent years of life for someone now aged x, according to a particular mortality experience...
and per capita income for the society. The rest of this article adopts the latter definition.
The form of the social welfare function is intended to express a statement of objectives of a society. For example, take this example of a social welfare function:
where
is social welfare and is the income of individual i among n in the society. In this case, maximising the social welfare function means maximising the total income of the people in the society, without regard to how incomes are distributed in society. Alternatively, consider the Max-Min utility function (based on the philosophical work of John RawlsJohn Rawls
John Bordley Rawls was an American philosopher and a leading figure in moral and political philosophy. He held the James Bryant Conant University Professorship at Harvard University....
):
Here, the social welfare of society is taken to be related to the income of the poorest person in the society, and maximising welfare would mean maximising the income of the poorest person without regard for the incomes of the others.
These two social welfare functions express very different views about how a society would need to be organised in order to maximise welfare, with the first emphasizing total incomes and the second emphasising the needs of the poorest. 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 Sen
Amartya Sen, CH is an Indian economist who was awarded the 1998 Nobel Prize in Economic Sciences for his contributions to welfare economics and social choice theory, and for his interest in the problems of society's poorest members...
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, a relative inequality measure. James E. Foster (1996) proposed to use one of AtkinsonAnthony Barnes Atkinson
Sir Anthony Barnes "Tony" Atkinson, FBA, is a British economist and has been a Senior Research Fellow of Nuffield College, Oxford since 2005.-Career:Atkinson served as Warden of Nuffield College from 1994 to 2005...
's Indexes, which is an entropy measure. Due to the relation between Atkinsons entropy measure and the Theil index
Theil index
The Theil index is a statistic used to measure economic inequality. It has also been used to measure the lack of racial diversity. The basic Theil index TT is the same as redundancy in information theory which is the maximum possible entropy of the data minus the observed entropy. It is a special...
, 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
Median
In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...
, 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 an 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
Theil index
The Theil index is a statistic used to measure economic inequality. It has also been used to measure the lack of racial diversity. The basic Theil index TT is the same as redundancy in information theory which is the maximum possible entropy of the data minus the observed entropy. It is a special...
provides further information about how this index is used in order to compute welfare functions.
See also
- Aggregation problemAggregation problemAn aggregate in economics is a summary measure describing a market or economy. The aggregation problem refers to the difficulty of treating an empirical or theoretical aggregate as if it reacted like a less-aggregated measure, say, about behavior of an individual agent as described in general...
- Arrow's impossibility theoremArrow's impossibility theoremIn social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives , no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a...
- Community indifference curveCommunity indifference curveA community indifference curve is an illustration of different combinations of commodity quantities that would bring a whole community the same level of utility. The model can be used to describe any community, such as a town or an entire nation...
- Distribution (economics)Distribution (economics)Distribution in economics refers to the way total output, income, or wealth is distributed among individuals or among the factors of production .. In general theory and the national income and product accounts, each unit of output corresponds to a unit of income...
- Economic welfareEconomic welfareEconomic welfare broadly refers to the level of prosperity and living standards in an individual or group of persons. In the field of economics, it specifically refers to utility gained through the achievement of material goods and services...
- Extended sympathyExtended sympathyExtended sympathy in welfare economics refers to interpersonal value judgments of the form that social state x for person A is ranked better than, worse than, or as good as social state y for person B...
- Justice (economics)Justice (economics)Justice in economics is a subcategory of welfare economics with models frequently representing the ethical-social requirements of a given theory. That theory may or may not elicit acceptance...
- Liberal paradoxLiberal paradoxThe liberal paradox is a logical paradox advanced by Amartya Sen, building on the work of Kenneth Arrow and his impossibility theorem, which showed that within a system of menu-independent social choice, it is impossible to have both a commitment to "Minimal Liberty", which was defined as the...
- Social choice theorySocial choice theorySocial choice theory is a theoretical framework for measuring individual interests, values, or welfares as an aggregate towards collective decision. A non-theoretical example of a collective decision is passing a set of laws under a constitution. Social choice theory dates from Condorcet's...
- Welfare economicsWelfare economicsWelfare economics is a branch of economics that uses microeconomic techniques to evaluate economic well-being, especially relative to competitive general equilibrium within an economy as to economic efficiency and the resulting income distribution associated with it...
- Production-possibility frontier