# Social welfare function

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. 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 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.

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.

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:

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

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):

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

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::$W_mathrm\left\{Gini\right\} = overline\left\{ ext\left\{Income cdot left\left( 1-G ight\right)$ The average per capita income of a measured group (e.g. nation) is multiplied with $\left(1-G\right)$m 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.

:$W_mathrm\left\{Theil-L\right\} = overline\left\{ ext\left\{Income cdot mathrm\left\{e\right\}^\left\{-T_L\right\}$

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 inequal 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.

:$W^\left\{-1\right\}_mathrm\left\{Theil-T\right\} = overline\left\{ ext\left\{Income cdot mathrm\left\{e\right\}^\left\{T_T\right\}$

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 inequally 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.

* Aggregation problem
* Arrow's impossibility theorem
* Distribution (economics)
* Extended sympathy
* Justice (economics)
* Social choice theory
* Welfare economics

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 Jounal 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," "", v. 2, 955-58
* Prasanta K. Pattanaik (2008). "social welfare function." "The New Palgrave Dictionary of Economics", 2nd Edition [http://www.dictionaryofeconomics.com/article?id=pde2008_S000172&q=socia%20welfare&topicid=&result_number=1 Abstract.]
* 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
* ____, 2004, " [http://www.nap.edu/readingroom/books/biomems/abergson.pdf Abram Bergson 1914-2003: A Biographical Memoir] "
* Amartya K. Sen, 1970 [1984] , "Collective Choice and Social Welfare", ch. 3, "Collective Rationality" ISBN 0-444-85127-5
* Kotaro Suzumura, 1987, “social welfare function," "The New Palgrave: A Dictionary of Economics", v. 4, 418-20
* [http://cepa.newschool.edu/het/essays/paretian/paretosocial.htm#swf Bergson-Samuelson social welfare functions] in Paretian welfare economics from the New School.

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