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. Contents 1 Bergson-Samuelson social welfare function 2 Arrow social welfare function (constitution) 3 Cardinal social welfare functions 4 See also 5 References Bergson-Samuelson social welfare function

In a 1938 article Abram Bergson 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: 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 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. Earlier neoclassical welfare theory heir to the classical utilitarianism of Bentham 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 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. A practical qualification to this was any reduction in output from the transfer.

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: 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: The respective hypothetical utilities of the two persons in two-dimensional utility space is analogous to respective quantities of commodities for the two-dimensional commodity space of the indifference-curve surface The Welfare function is analogous to the indifference-curve map The Possibility function is analogous to the budget constraint Two-person welfare maximization at the tangency of the highest Welfare function curve on the Possibility function is analogous to tangency of the highest indifference curve on the budget constraint. Arrow social welfare function (constitution) Kenneth Arrow (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. 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 expectancy 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 W is social welfare and Yi 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 (1 G) where G 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. See also Aggregation problem Arrow's impossibility theorem Community indifference curve Distribution (economics) Extended sympathy Justice (economics) Liberal paradox Social choice theory Welfare economics Production-possibility frontier References Kenneth J. Arrow 1951 2nd ed. 1963 Social Choice and Individual Values ISBN 0-300-01364-7 Abram Bergson (Burk)"A Reformulation of Certain Aspects of Welfare Economics" Quarterly Journal of Economics 52(2) February 1938 310-34 James E. Foster and Amartya Sen 1996 On Economic Inequality expanded edition with annexe ISBN 0-19-828193-5). John C. Harsanyi 1987 interpersonal utility comparisons" The New Palgrave: A Dictionary of Economics v. 2 955-58 Prasanta K. Pattanaik (2008). "social welfare function." The New Palgrave Dictionary of Economics 2nd Edition Abstract. Jan de Van Graaff 1957 "Theoretical Welfare Economics" 1957 Cambridge UK: Cambridge University Press. Lionel Robbins 1935 2nd ed.. An Essay on the Nature and Significance of Economic Science ch. VI 1938 "Interpersonal Comparisons of Utility: A Comment" Economic Journal 43(4) 635-41 Paul A. Samuelson 1947 Enlarged ed. 1983 Foundations of Economic Analysis pp. xxi-xxiv & ch. VIII "Welfare Economics" ISBN 0-674-31301-1 1977. "Reaffirming the Existence of 'Reasonable' Bergson-Samuelson Social Welfare Functions" Economica N.S. 44(173) pp. 81-88. Reprinted in (1986) The Collected Scientific Papers of Paul A. Samuelson pp. 47-54. 1981. "Bergsonian Welfare Economics" in S. Rosefielde (ed.) Economic Welfare and the Economics of Soviet Socialism: Essays in Honor of Abram Bergson Cambridge University Press Cambridge pp. 223-66. Reprinted in (1986) The Collected Scientific Papers of Paul A. Samuelson pp. 3-46. Sen Amartya K. (1963). "Distribution Transitivity and Little's Welfare Criteria" Economic Journal 73(292) p. 771-78. 1970 1984 Collective Choice and Social Welfare (description) ch. 3 "Collective Rationality." ISBN 0-444-85127-5 (1982). Choice Welfare and Measurement MIT Press. Description and scroll to chapter-preview links. Kotaro Suzumura (1980). "On Distributional Value Judgments and Piecemeal Welfare Criteria" Economica 47(186) pp. 125-39. 1987 social welfare function" The New Palgrave: A Dictionary of Economics v. 4 418-20 Bergson-Samuelson social welfare functions in Paretian welfare economics from the New School.