Social welfare function (1938)

Introduced by American economist ABRAM BERGSON (1914-2003) as a rejection of the cardinal utility approach to welfare economics, social welfare function is understood as meaning the determination of a society’s taste for different economic states.

There are two approaches to the welfare function: first, that it is an imposed structure; second, that it devises a single constitutional/voting system which changes the rankings of the individual into a single society ranking.

Also see: impossibility theorem

A Bergson, ‘A Reformulation of Certain Aspects of Welfare Economics’, Quarterly Journal of Economics 52, 2 (February 1938) 310-34

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 hypothesization 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 impossibility theorem which says that it is impossible to have a social welfare function that satisfies a certain set of “apparently reasonable” conditions.

Cardinal social welfare functions

cardinal social welfare function is a function that takes as input numeric representations of individual utilities (also known as cardinal utility), and returns as output a numeric representation of the collective welfare. The underlying assumption is that individuals utilities can be put on a common scale and compared. Examples of such measures can be:

  • life expectancy,
  • per capita income.

For the purposes of this section, income is adopted as the measurement of utility.

The form of the social welfare function is intended to express a statement of objectives of a society.

The utilitarian or Benthamite social welfare function measures social welfare as the total or sum of individual incomes:

{\displaystyle W=\sum _{i=1}^{n}Y_{i}}

where {\displaystyle W} is social welfare and {\displaystyle Y_{i}} is the income of individual {\displaystyle i} among {\displaystyle 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:

{\displaystyle 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:

{\displaystyle 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:

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

The average per capita income of a measured group (e.g. nation) is multiplied with {\displaystyle (1-G)} where {\displaystyle 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.

{\displaystyle 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.

{\displaystyle W_{\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 a 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.

Axioms of cardinal welfarism

Suppose we are given a preference relation R on utility profiles. R is a weak total order on utility profiles—it can tell us, given any two utility profiles, if they are indifferent or one of them is better than the other. A reasonable preference ordering should satisfy several axioms:[4]:66–69

1. Monotonicity, i.e., if the utility of an individual increases while all other utilities remain equal, R should strictly prefer the second profile. E.g., it should prefer the profile (1,4,4,5) to (1,2,4,5). This is related to Pareto optimality.

2. Symmetry, i.e., R should be indifferent to permutation of the numbers in the utility profile. E.g., it should be indifferent between (1,4,4,5) and (5,4,1,4).

3. Continuity: for every profile v, the set of profiles weakly better than v and the set of profiles weakly worse than v are closed sets.

4. Independence of unconcerned agents, i.e., R should be independent of individuals whose utilities have not changed. E.g., if R prefers (2,2,4) to (1,3,4), then it also prefers (2,2,9) to (1,3,9); the utility of agent 3 should not affect the comparison between two utility profiles of agents 1 and 2. This property can also be called locality or separability. It allows us to treat allocation problems in a local way, and separate them from the allocation in the rest of society.

Every preference relation with properties 1-4 can be represented as by a function W which is a sum of the form:

{\displaystyle W(u_{1},\dots ,u_{n})=\sum _{i=1}^{n}w(u_{i})}

where w is a continuous increasing function.

It is also reasonable to require:

5. Independence of common scale, i.e., the relation between two utility profiles does not change if both of them are multiplied by the same scalar (e.g., the relation does not depend on whether we measure the income in cents, dollars or thousands).

If the preference relation has properties 1-5, then the function w belongs to the following one-parameter family:

  • {\displaystyle w_{p}(x)=x^{p}} for {\displaystyle p>0},
  • {\displaystyle w_{0}(x)=\ln(x)} for {\displaystyle p=0},
  • {\displaystyle w_{p}(x)=-x^{p}} for {\displaystyle p<0}.

This family has some familiar members:

  • The limit when {\displaystyle p\to -\infty } is the leximin ordering;
  • For {\displaystyle p=0} we get the Nash bargaining solution—maximizing the product of utilities;
  • For {\displaystyle p=1} we get the utilitarian welfare function—maximizing the sum of utilities;
  • The limit when {\displaystyle p\to \infty } is the leximax ordering.

If, in addition, we require:

6. the Pigou–Dalton principle,

then the parameter p, in the above family, must be at most 1.

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