Editing Social welfare function (section)

==Ordinal welfare==
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In a 1938 article, [[Abram Bergson]] introduced the term ''social welfare function,'' with the intention "to state in precise form the value judgments required for the derivation of the conditions of maximum economic welfare." The function was real-valued and [[Differentiable function|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 [[factors of production|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 "dis-welfare" 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 argued that [[welfare economics]] had described a standard of economic efficiency despite dispensing with ''interpersonally-comparable'' [[cardinal utility]], the hypothesization of which may merely conceal value judgments, and purely subjective ones at that.

Earlier neoclassical welfare theory, heir to the classical [[utilitarianism]] of [[Jeremy Bentham|Bentham]], often treated the [[law of diminishing marginal utility]] as implying interpersonally comparable utility. 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 ([[An Essay on the Nature and Significance of Economic Science|1935]], ch. VI) argued that how or how much utilities, as mental events, change relative to each other is not measurable by any empirical test, making them [[unfalsifiable]]. Robbins therefore rejected such as incompatible with his own philosophical [[behaviorism]].

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.&nbsp;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 ([[Foundations of Economic Analysis|1947]], p.&nbsp;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.
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.&nbsp;243–49). Each has as [[Function (mathematics)#The vocabulary of functions|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.
{| style="border:1px solid #999; text-align:leftcenter; margin: auto;" cellspacing="20"
| For a two-person society, there is a graphical depiction of such welfare maximization at the first figure of [https://web.archive.org/web/20060215000915/http://cepa.newschool.edu/het/essays/paretian/paretosocial.htm#swf 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.
|}
[[Kenneth Arrow]]'s [[Social Choice and Individual Values|1963 book]] demonstrated the problems with such an approach, though he would not immediately realize this. Along earlier lines, Arrow's version of a social welfare function, also called a 'constitution', maps a set of individual orderings ([[ordinal utility function]]s) for everyone in society to a social ordering, which ranks alternative social states (such as which of several candidates should be elected).

Arrow found that contrary to the assertions of [[Lionel Robbins]] and other [[Behaviorism|behaviorists]], dropping the requirement of real-valued (and thus [[cardinal utility|cardinal]]) social orderings makes [[Decision theory|rational]] or [[Coherence (philosophical gambling strategy)|coherent]] behavior at the social level impossible. This result is now known as [[Arrow's impossibility theorem]]''.'' Arrow's theorem shows that it is impossible for an ordinal social welfare function to satisfy a standard axiom of [[Decision theory|rational behavior]], called [[independence of irrelevant alternatives]]. This axiom says that changing the value of one outcome should not affect choices that do not involve this outcome. For example, if a customer buys apples because he prefers them to blueberries, telling them that cherries are on sale should not make them buy blueberries instead of apples.

[[John Harsanyi]] later strengthened this result by [[Harsanyi's utilitarian theorem|showing]] that if societies must make decisions under [[uncertainty]], the unique social welfare function satisfying [[Coherence (philosophical gambling strategy)|coherence]] and [[Pareto efficiency]] is the [[utilitarian rule]].