Social welfare function

In economics a social welfare function can be defined as a real-valued function that ranks conceivable social states (alternative complete descriptions of the society) from lowest on up as to welfare of the society. Inputs of the function include any variables considered to affect welfare of the society (Sen, 1970, p. 33). In using welfare measures of persons in the society as inputs, the social welfare function is individualistic in form. Ostensive 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.

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: 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.
 * 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.

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.
 * 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.

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.

Cardinal social welfare functions again
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. Motivation for such a measure is in its appeal, whether to officials, advisors, or voters.

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:

$$W = Y_1 + Y_2 + \cdots + Y_n$$

where $$W$$ is social welfare and $$Y_i$$ 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):

$$W = \min(Y_1, Y_2, \cdots, Y_n)$$

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 risk 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:

$$W = Income \times (1-Inequality)$$

$$Income$$ is the average per capita income of a measured group (e.g. nation). $$Inequality$$ is the relative inequality of the income distribution within that group. Here Sen used the Gini Index. James E. Foster (1996) proposed to use one of Atkinson's Indexes, which is an entropy measure. Foster's welfare function also can be computed directly using the Theil Index.