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In welfare economics, a social welfare function is a function that ranks social states (alternative complete descriptions of the society) as less desirable, more desirable, or indifferent for every possible pair of social states. Inputs of the function include any variables considered to affect the economic welfare of a society.〔Amartya K. Sen, 1970 (), ''Collective Choice and Social Welfare'', ch. 3, "Collective Rationality." p. 33, and ch. 3 *, "Social Welfare Functions." (Description. )〕 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 the consumer theory of indifference-curve/budget constraint equilibrium 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 for an individual with at least minimal restrictions suggested by welfare economics, including constraints on the amount of factors of production. 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'', 2nd Edition. (Abstract. )〕 ==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. 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. ==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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Social welfare function」の詳細全文を読む スポンサード リンク
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