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Social welfare functions have occupied the center stage of welfare economics since the philosopher Jeremy Bentham (1748-1832) and other utilitarian theorists advocated that humans should seek the greatest total welfare, understood as the sum of individual utilities. The noted economists Abram Bergson (1914-2003) and Paul A. Samuelson (b. 1915) generalized the concept. Following the seminal work of the economic theorists Kenneth J. Arrow (b. 1921) and Amartya K. Sen (b. 1933), the theory of social choice has examined the various possible forms of such functions in detail.

In the most standard form, a social welfare function is a mapping that determines the level of social welfare as a function of individual levels of welfare. In fact, a numerical measure of social welfare is not really needed for the evaluation of social situations; the only relevant part of a social welfare function is how it ranks social situations from the best to the worst.

The theory of social choice analyzes two important features of a social welfare function. The first feature is its degree of aversion to inequalities between individual levels of welfare. For instance, the utilitarian function, computed as the sum of individual indices of welfare, is indifferent to inequalities because only the total matters. In contrast, the maximin function, which identifies social welfare with the welfare of the worst-off individuals in the population, displays an infinite aversion to inequalities since it gives absolute priority to the worst off. There are countless intermediate possibilities between these two extreme functions. The second important feature of a social welfare function is the information about individual indices that is used by the function. For instance, the utilitarian function focuses on individual gains and losses when it compares two situations: A situation is better than another if the sum of welfare gains when one moves to it is greater than the sum of losses. By contrast, the maximin function focuses on individual levels: A situation is better if the level of welfare of the worst-off individuals is greater. The link between the two features—inequality aversion and informational basis—has been thoroughly scrutinized by the theory of social choice. For surveys, see Walter Bossert and John A. Weymark’s “Utility in Social Choice” in the Handbook of Utility Theory (Barbera, Hammond, and Seidl 2004, pp. 1099-1178) and Claude D’Aspremont and Louis Grevers’ “Social Welfare Functionals and Interpersonal Comparability” in the first volume of the Handbook of Social Choice and Welfare (Arrow, Sen, and Suzumura 2002, pp. 459-542).

The link between the measurement of social welfare and the measurement of inequalities has also been clarified. The level of social welfare in a given situation can be decomposed into the positive contribution of total welfare and the negative contribution of inequalities. More precisely, it is generally possible to write social welfare as the difference of two terms. One is the sum of individual indices of welfare. The other, which comes as a deduction, is computed as the product of an inequality index and the sum of individual indices. In other words, one can typically say that inequalities reduce social welfare by a certain percentage, and this percentage itself is the value of an inequality index. Classical analyses of this link are provided by Anthony B. Atkinson in his essay “On the Measurement of Inequality” (1970) and Amartya K. Sen and James Foster in their book On Economic Inequality (1997).

A controversial issue has been the possibility of constructing social welfare functions on the sole basis of individual ordinal and noncomparable preferences over the social situations. The standard social welfare function relies on individual indices of welfare, but in its most general form it can simply define a ranking of social situations as a function of the preference rankings of these situations for each member of the population. This is important because there is much skepticism in economics about the possibility of making interpersonal comparisons of welfare in a consensual way. Bergson and Samuelson argued that the construction of a purely ordinal social welfare function was possible, but Arrow, in Social Choice and Individual Values (1951), provided an impossibility theorem and identified a deep problem in the idea of “aggregating” individual ordinal preferences. This problem has been questioned, however, because Arrow’s theorem involved restrictive conditions that do not appear compelling and are not even considered in important branches of welfare economics, such as the theory of fair allocation or cost-benefit analysis. It is now emerging that interesting social welfare functions can be based on individual ordinal and noncomparable preferences, in particular on the basis of the equity concepts developed by the theory of fair allocation. Overviews of the controversies can be found in Marc Fleurbaey and Peter Hammond’s contribution, “Interpersonally Comparable Utility,” to the Handbook of Utility Theory (Barbera, Hammond, and Seidl 2004, pp. 1179-1285) and in an article by Marc Fleurbaey and Philippe Mongin (2005).

Recent Developments

Recent developments have dealt with the refinements of social welfare functions to incorporate a variety of fairness concepts or to make it possible not only to compare social situations for a given population, but also to compare social situations across different populations. The latter is important to make judgments about the desirable size of the population, and deep dilemmas plague this issue. A social welfare function such that every addition of an individual with positive welfare is considered to improve the social situation will generate extreme populationist evalu-ations—for example, the judgment that a larger population always makes a situation better, provided the larger population is large enough, no matter how low individual welfare is in this situation. However, a social welfare function that is based on some notion of average individual welfare is likely to be Malthusian because any addition of an individual with less-than-average welfare is considered undesirable. The compromise proposed in Population Issues in Social-Choice Theory, Welfare Economics, and Ethics (Blackorby, Bossert, and Donaldson 2005) is to define a critical level such that the addition of an individual is good only if his welfare is above this threshold.

An important issue in fairness concepts has been the incorporation of considerations of freedom and individual responsibility, motivated by theories of justice such as that of John Rawls (1971). Social welfare functions have been developed that are based on the idea of measuring individual situations in terms of opportunities or resources rather than ultimate welfare. A variety of functions have been proposed, in particular because there are different possible interpretations of the implications of individual responsibility. In one interpretation, individual responsibility removes the need for redistribution, which has motivated the construction of social welfare functions that disregard individual characteristics for which the individuals are responsible and focus on the compensation of inequalities for which individuals are not responsible. In another interpretation, individual responsibility removes the need for inequality aversion in the evaluation. Along this vein, social welfare functions have been proposed that are of the utilitarian sort along the responsibility dimensions of individual characteristics and of the maximin sort along the non-responsibility dimensions. Surveys of this field can be found in John E. Roemer’s book Theories of Distributive Justice (1996).

Bibliography:

Arrow, Kenneth J. 1951. Social Choice and Individual Values. New York: Wiley.
Arrow, Kenneth J., Amartya K. Sen, and Kotaro Suzumura, eds. 2002. Handbook of Social Choice and Welfare. Vol. 1. Boston: Elsevier.
Atkinson, Anthony B. 1970. On the Measurement of Inequality.Journal of Economic Theory 2 (3): 244–263.
Barbera, Salvador, Peter J. Hammond, and Christian Seidl, eds. 2004. Handbook of Utility Theory. Vol. 2. Boston: Kluwer Academic Publishers.
Bergson, Abram. 1938. A Reformulation of Certain Aspects of Welfare Economics. Quarterly Journal of Economics 52 (2): 310–334.
Blackorby, Charles, Walter Bossert, and David Donaldson. 2005. Population Issues in Social-Choice Theory, Welfare Economics, and Ethics. Cambridge, U.K.: Cambridge University Press.
Fleurbaey, Marc, and Philippe Mongin. 2005. The News of the Death of Welfare Economics Is Greatly Exaggerated. Social Choice and Welfare 25 (2): 381–418.
Rawls, John. 1971. A Theory of Justice. Cambridge, MA: Harvard University Press.
Roemer, John E. 1996. Theories of Distributive Justice. Cambridge, MA: Harvard University Press.
Samuelson, Paul Anthony. 1947. Foundations of Economic Analysis. Cambridge, MA: Harvard University Press.
Sen, Amartya K. 1970. Collective Choice and Social Welfare. San Francisco: Holden-Day.
Sen, Amartya K., and James Foster. 1997. On Economic Inequality, rev. ed. Oxford: Clarendon Press.