Social welfare function

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.^[1] 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.^[2]

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 societal preference based on only individual utility functions, whereas in others it also outputs cardinal measures of social welfare not aggregated from individual utility functions. Examples of such measures can be life expectancy and per capita income for the society. For the purposes of this article, income is adopted as the measurement of utility in this section.

The form of the social welfare function is intended to express a statement of objectives of a society. Consider the Utilitarian or Benthamite social welfare function which measures social welfare as the total or sum of individual incomes:

W = \sum_{i=1}^n Y_i

where $W$ is social welfare and $Y_i$ is the income of individual $i$ among $n$ individuals in society. In this case, maximizing the social welfare means maximizing the total income of the people in the society, without regard to how incomes are distributed in society. It does not distinguish between an income transfer from rich to poor and vice versa. If an income transfer from the poor to the rich results in a bigger increase in the utility of the rich than the decrease in the utility of the poor, the society is expected to accept such a transfer, because the total utility of the society has increased as a whole. Alternatively, society's welfare can also be measured under this function by taking the average of individual incomes:

W = \frac{1}{n}\sum_{i=1}^n Y_i = \overline{Y}

In contrast, the Max-Min or Rawlsian social welfare function (based on the philosophical work of John Rawls) measures the social welfare of society on the basis of the welfare of the least well-off individual member of society:

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

Here maximizing societal welfare would mean maximizing the income of the poorest person in society without regard for the income of other individuals.

These two social welfare functions express very different views about how a society would need to be organised in order to maximize welfare, with the first emphasizing total incomes and the second emphasizing the needs of the worst-off. 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:

W_\mathrm{Gini} = \overline{Y} \left( 1-G \right)

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.

W_\mathrm{Theil-L} = \overline{Y} \mathrm{e}^{-T_L}

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.

W^{-1}_\mathrm{Theil-T} = \overline{Y} \mathrm{e}^{T_T}

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.

Notes

↑ Amartya K. Sen, 1970 [1984], Collective Choice and Social Welfare, ch. 3, "Collective Rationality." p. 33, and ch. 3*, "Social Welfare Functions." Description.
↑ Prasanta K. Pattanaik, 2008. "social welfare function," The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.

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
Bergson–Samuelson social welfare functions in Paretian welfare economics from the New School.
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
Pattanaik, Prasanta K. (2008), "social welfare function (definition)", in Durlauf, Steven N.; Blume, Lawrence E., The new Palgrave dictionary of economics (2nd ed.), Basingstoke, Hampshire New York: Palgrave Macmillan, ISBN 9780333786765.

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