The New Palgrave Dictionary of Economics

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Social Welfare Function

  • Prasanta K. Pattanaik
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DOI: https://doi.org/10.1057/978-1-349-95121-5_1348-2

Abstract

The article deals with the related, though distinct, notions of a social welfare function due to A. Bergson and P. Samuelson on the one hand and K.J. Arrow on the other. After introducing the two formal concepts, it gives a brief outline of Arrow’s well-known impossibility theorem, considers some alternative intuitive interpretations of the notion of a social welfare function, and discusses the informational bases of social welfare judgements.

Keywords

Aggregation Arrow’s theorem Bergson–Samuelson social welfare function Cardinal utility Independence of irrelevant alternatives Interpersonal comparison of utilities Majority rule Non-utility information Pareto principle Preferences Rights Sen, A. Social choice Social preferences Social welfare function Voting paradoxes 

JEL Classifications

D6 

The concept of a social welfare function is central in welfare economics, the branch of economics that explores the implications of various ethical criteria for deciding what promotes social welfare, what public policies the society should choose, and so on. It was first introduced by Bergson (1938), and was subsequently elaborated by Samuelson (1947). A related, though not identical, notion with the same name was introduced by Arrow (1951); as we shall see below, Arrow’s interpretation was different from Bergson’s earlier concept. The technical literature on the social welfare function is large. The following discussion will, however, focus mainly on conceptual issues relating to the social welfare function.

The Basic Notation

Consider a given society consisting of individuals 1, 2, …, n, with X a given set of alternative social states. For our purpose, a social state can be thought of as a complete description of all ‘relevant’ aspects of the state of affairs prevailing in society, but one can also use more limited interpretations of a social state. An element of X can be thought of as a vector, each component of the vector representing some particular feature of the state of society. The elements of X will be denoted by x, y, z, and so forth. Each individual i is assumed to have a preference ordering (‘at least as good as’) defined over X (an ordering over X is defined to be a binary relation T over X such that T is reflexive (for all x in X, xTx), connected (for all distinct x and y in X, xTy or yTx), and transitive (for all x, y, and z in X, if xTy and yTz, then xTz)). Such a preference ordering will be denoted by R i , \( {R}_i^{\prime }, \) and so on. An n-tuple of preference orderings over X, with the preference ordering of each individual figuring exactly once in the n-tuple, will be called a preference profile. For all social states x and y, and for every individual i, [xP i y if and only if xR i y and not yR i x] and [xI i y if and only if xR i y and yR i x]. Intuitively, P i denotes the strict preference relation for individual i and I i denotes the indifference relation for individual i. R, R′, and so forth will denote social weak preference relations (‘socially at least as good as’) over X. Thus, xRy will denote that x is at least as good as y for the society. Given a social weak preference relation R, one can define P (‘socially better than’) and I (‘socially indifferent to’) in terms of R in the same way as P i and I i are defined in terms of R i .

The Bergson–Samuelson Social Welfare Function

The Bergson–Samuelson social welfare function (SWF (BS)) is a function W that specifies exactly one real number, W(x), for each social state x in X. The intended interpretation is this: for all x and y in X, W(x) ≥ W(y)denotes that x offers at least as high a level of social welfare as y, and similarly for W(x)> W(y) and W(x) = W(y) . W(x) , W(y), and so on are thus ordinal indices of social welfare attached to social states. What determines the form of the function? For example, what determines whether x should be assigned a higher number than y or y should be assigned a higher number than x? This, as both Bergson (1938) and Samuelson (1947) pointed out, will depend on the value judgements that we use. It is not surprising that very little can be done with the mathematical notion of an SWF (BS) at this level of generality. Even then, the abstract notion, by itself, makes one thing clear: if one wants to say anything specific about social welfare, one must introduce explicit value judgements.

Specific conclusions emerge from the Bergson–Samuelson social welfare function as one introduces additional value judgements that restrict the form of the function. Thus, assuming that each individual has an (ordinal) utility function U i defined over X, Samuelson considers an ‘individualistic’ form for SWF (BS) where the social welfare indices attached to social states depend exclusively on the individual utilities, so that one can write the SWF (BS) as
$$ W= F\left({U}^1,\dots {U}^n\right) $$
(1)

Assuming further that social states are simply allocations (a specification of the quantity of each commodity figuring in each consumer’s consumption bundle and the quantity of each commodity figuring in each producer’s production plan), F is increasing in U 1, U 2, …, and U n , and each consumer’s utility depends only on her consumption bundle, Samuelson (1947, pp. 229–49) derived conditions for the maximization of social welfare subject to the relevant resource constraints and technological constraints of the society.

Arrow’s Social Welfare Function

Arrow (1951) introduced a somewhat different concept of a social welfare function (SWF (A)). An SWF (A) is a functional rule G, which, for every possible preference profile, (R 1,…,R n ), belonging to a non-empty class of preference profiles, defines exactly one ordering R over X. Thus, we write
$$ R= G\left({R}_1,\dots, {R}_n\right). $$
(2)

The intuitive interpretation of R figuring in this definition is that it represents the society’s weak preference relation over social states, R being constrained to be an ordering. Thus, an SWF (A) gives us a (unique) social ordering of all social states once the individuals’ orderings over the social states are given.

What is the relationship between the notion of an SWF (BS) and that of an SWF (A)? Suppose an SWF (BS) has the form given by (1), and suppose the utility functions U i (i = 1,2,…,n) are ordinal so that they have no more significance beyond just the orderings that they, respectively, represent (recall that the Bergson–Samuelson social welfare indices are also ordinal). Then noting that, for every profile of real valued utility functions over X, we have Bergson–Samuelson social welfare indices for social states, such that the ordering implied by the social welfare indices does not change with a change in the profile of utility functions so long as the orderings implied by the utility functions do not change (see Samuelson 1947, p. 228), we would have a unique social ordering over X for every profile of individual orderings over X and, hence, an Arrow-type social welfare function. Thus, underlying an SWF (BS) as given in (1), there is always an SWF (A). (The converse is not necessarily true since the social ordering R of Arrow may not be representable by a real valued function over X.)

Arrow’s Impossibility Theorem

If Arrow did not impose any restrictions on an SWF (A), then the definition, by itself, would be of no more substantive interest than just the definition of an SWF (BS) without any specific assumptions about the properties of the SWF (BS). Arrow, however, proceeded to introduce specific restrictions regarding the form of an SWF (A). He postulated that an SWF (A), G, must satisfy the following properties.
  • Universal Domain (UD): the domain of G must include every possible preference profile.

  • Weak Pareto Criterion (WP): for every profile of individual orderings, (R 1,…,R n ), in the domain of G and for all x and y in X, if xP i y for every individual i, then xPy.

    The weak Pareto criterion, which is a weak version of the familiar Pareto principle, just says that, if every individual strictly prefers social state x to social state y, then the social ordering must rank x higher than y.

  • Independence of Irrelevant Alternatives (IIA): Consider any two profiles of preference orderings, (R 1,…,R n ) and \( \left({R}_1^{\prime },\dots, {R}_n^{\prime}\right) \), in the domain of G and any two social states x and y. If, for every individual i, [xR i y if and only if \( x{R}_i^{\prime } y \)] and [yR i x if and only if \( y{R}_i^{\prime } x \)], then [xRy if and only if xR y] and [yRx if and only if yR x] where the social orderings R and R’ correspond to the preference profiles (R 1,…,R n ) and \( \left({R}_1^{\prime },\dots, {R}_n^{\prime}\right) \), respectively.

    IIA requires that, if the individual orderings change but everybody’s ranking of a pair of social states remains unchanged, then the social ranking of those two social states must remain unchanged though the social ranking over other pairs of social states may change.

  • Non-dictatorship (ND): there does not exist any individual k such that for all social states x and y and for every profile of individual orderings (R 1,…,R n ) in the domain of G, if xP k y, then xPy.

    ND just says that there should not be any individual such that, whenever she strictly prefers any social state x to any other social state y, x must rank higher than y in the social ordering, irrespective of other people’s preferences.

Arrow’s (1951) famous impossibility theorem tells us that, if there are at least three social states in X, then there does not exist any SWF (A) that simultaneously satisfies WP, IIA, and ND. The result has the flavour of a paradox since, prima facie, the properties postulated by Arrow for his social welfare function seem plausible. It may be useful to consider two examples to illustrate how Arrow’s theorem ‘works’. Consider first the simple majority rule (SMR) which says that, for every preference profile, (R 1,…,R n ), and for all x and y in X, xRy if and only if the number of individuals who consider x to be at least as good as y is greater than or equal to the number of individuals who consider y to be at least as good as x. While the SMR satisfies WP, IIA, and ND, it does not yield a social ordering for every preference profile. Thus, if we have three individuals, 1, 2, and 3, and three alternatives, x, y, and z, then, for the preference profile such that (xP 1 y & yP 1 z & xP 1 z), (yP 2 z & zP 2 x & yP 2 x), and (zP 3 x & xP 3 y & zP 3 y), the SMR gives us (xPy & yPz & zPx) which is not an ordering (this, in fact is the well known ‘voting paradox’). Let us take a second example, the Borda rule, which for every preference profile specifies the social ordering as follows. On the assumption that X has m elements, if an individual places a social state x in the first position in his preference ordering, then x gets m points from him; if an individual places x in the second position in his preference ordering, then x gets m −1 points from him; and so on. (In stating the Borda rule, I have ignored the case where an individual may be indifferent between two social states. For a complete specification of the Borda rule, however, the assignment of points in such cases needs to be specified.) A social state a is considered to be socially at least as good as a social state b under the Borda rule if and only if the sum of all the points received by a from all individuals is greater than or equal to the corresponding sum for b. The Borda rule satisfies all the conditions of Arrow excepting IIA. To see that it violates IIA, let us consider the case where we have two individuals (1 and 2), three alternatives (x, y, and z), and two preference profiles, (R 1, R 2) and \( \left({R}_{\;1}^{\prime },{R}_{\;2}^{\prime}\right) \) as follows: xP 1 y & yP 1 z & xP 1 z; zP 2 y & yP 2 x & zP 2 x;\( x{P}_1^{\prime } y\& y{P}_1^{\prime } z\& x{P}_1^{\prime } z; \) and\( z{P}_2^{\prime } x\& x{P}_2^{\prime } y\& z{P}_2^{\prime } y \). Given the profile (R 1, R 2), each of x and z receives a total of four points, and, hence, we have xIz, but, given the preference profile \( \left({R}_{\;1}^{\prime },{R}_{\;2}^{\prime}\right), \) x receives a total of five points while z receives a total of four points, and hence, we have xP’z. This violates IIA since the social ranking of x and z changes when we go from (R 1, R 2) to \( \left({R}_{\;1}^{\prime },{R}_{\;2}^{\prime}\right), \), though the ranking of x and z has remained the same for each individual.

An impossibility theorem such as Arrow’s (1951) compels us to think what has gone ‘wrong’, which of the requirements are unreasonable, and which of the restrictions need to be discarded or modified to provide a way out of the impasse. In this brief article, I shall not explore these questions, which have been discussed in great detail in the large literature that followed Arrow (1951). Instead, I turn to some basic issues about how one is to interpret the notion of the social welfare function itself.

Alternative Intuitive Interpretations

Some important questions have been raised by a number of economists (see, for example, Bergson 1954; Little 1952; Sen 1977a) about the interpretation of a social ranking of social states, such as the social ordering yielded by Arrow’s social welfare function and the ordering implied by the welfare indices given to us by a Bergson–Samuelson social welfare function.

It has been claimed by both Little (1952) and Bergson (1954) that the social ordering R figuring in Arrow’s definition of a social welfare function is the result of an aggregation procedure or ‘constitution’ that aggregates a given profile of individual preference orderings reflecting the individuals’ judgements or opinions. In contrast, as Bergson pointed out, the welfare indices that come from his social welfare function were intended to reflect a given individual’s personal value judgements about what was good for the society (in a somewhat similar fashion, Sen 1977a, makes a distinction between committee decision and social welfare judgements). Arrow (1963) agreed that he did intend his social welfare function to be a constitution or a rule for aggregating people’s opinions, but he claimed that such aggregation was, indeed, the central issue of welfare economics.

An example may be helpful in clarifying the distinction. Suppose someone, say, individual i, says that a complete ban on smoking in all public places is better than a prohibition of smoking only in a few designated public places. Suppose, when asked to give the reason why he thinks so, he gives us as the reason the fact that 99% of the population in the society have the opinion that a complete ban on smoking will be better for society. This may be a good enough reason if i’s original statement is a statement about how society should rank the two policies for the purpose of social action, given the existing opinions or judgements of the people. It is, however, possible that individual i made his original statement as his personal judgement about what would promote society’s welfare. In that case, we would find his response to the request for justification a little strange: we would feel that he should give ‘independent reasons’ for his statement rather than referring to the judgements of other people. Typically, in aggregating people’s opinions or judgements through a ‘constitution’ or a committee decision procedure, we do not look into the basis of people’s opinions; we take them as given and simply try to find out what will be a reasonable way of reconciling different opinions. In contrast, in forming our personal social welfare judgements, typically we do not aggregate individuals’ opinions or judgements (in fact, we often question the basis of other people’s judgements when they do not conform to our ethical beliefs), though we do take into account other people’s well-being. Also, in our personal judgements about social welfare, we often compare the welfare losses or gains of one person with those of another, while, in aggregating opinions or judgements, we rarely take into account the strength with which one individual favours x over y with the strength with which another person favours y over x (see Sen 1977a, p. 159).

Arrow (1963) sees the basic purpose of welfare economics as that of analysing procedures for aggregating individual opinions so as to arrive at social decisions. Therefore, he interprets the social ordering that results from such aggregation as the basis of social action, the aggregation procedure being his social welfare function. Nevertheless, as he pointed out, ordering R in the definition of an SWF (A) could also be interpreted as reflecting the social welfare judgement of a given individual, say, i. In that case, the SWF (A) would reflect the rule (s) by which i derived his personal social welfare judgement regarding social states, given the preferences of the individuals in the society. If, however, we adopt this interpretation of the SWF (A), then it will be appropriate to interpret the preference orderings that constitute the arguments of the SWF (A) as reflecting the individuals’ welfares rather than their value judgements or opinions since it is not clear why a person would use other people’s judgements to form his own social welfare judgement.

Both the interpretations of a social welfare function would seem to be important for welfare economics conceived in a broad fashion. In some ways, the analysis of personal judgements about social welfare and the aggregation of the opinions of the individuals in society correspond to two distinct phases that can often be discerned in a democratic process. The first stage is the stage of deliberation where people engage in ethical debates about each other’s personal social welfare judgements. The second stage is the stage of voting or aggregation of people’s judgements, where people’s opinions or judgements, as they emerge from the debates and deliberations of the first stage, are taken as given, and attention is focused on arriving at a ‘reasonable compromise’ on the basis of these judgements (see Little 1952; Pattanaik 2005).

The Informational Basis of Social Welfare Judgements

Arrow’s analytical structure does not permit us to consider certain types of information, which, intuitively, we often regard as important for forming our social welfare judgements. I note two such informational constraints.

Cardinal Utility and Interpersonal Comparisons of Utilities

The SWF (A) defines social ordering as a function of the individuals’ preference orderings over X. Thus, social ordering does not use any cardinal feature of individual welfare, and interpersonal comparisons either of the levels of individual welfare or of individual welfare gains and losses does not play any role in the determination of Arrow’s social ordering. The same is also true of the ‘individualistic’ Bergson–Samuelson social welfare function (see (1)), given Samuelson’s assumption that the individual utility functions are all ordinal. Such complete eschewal of cardinal notions of individual welfare and all interpersonal comparisons of individual welfares goes counter to our intuition when one interprets the Arrow’s social ordering as reflecting someone’s social welfare judgement rather than simply as the result of aggregating opinions through a ‘constitution’. As I have noted earlier, in forming our social welfare judgements, we typically take into account the welfare of individuals. In doing so, we also often resort to interpersonal comparisons of individuals’ welfare levels or changes in their welfares. The SWF (A) would not allow us to do this. For example, consider two allocations x = (98, 2) and y = (97, 3) of 100 units of some desired resource between two individuals, 1 and 2. Suppose someone wants to say that a move from x to y, involving a redistribution in favour of 2 will improve social welfare because, at x, 1 has a higher level of utility than 2, and, even after redistribution, 1’s welfare continues to be higher than 2’s welfare. This justification cannot be given in the Arrow framework since the framework does not permit such interpersonal comparison of welfare levels. Nor can the person justify his social welfare comparison of x and y in the framework by saying that individual 1’s utility loss from the switch from x to y is outweighed by 2’s gain of utility since Arrow’s framework permits neither cardinal individual utilities nor interpersonal comparisons of utility differences. In the literature that followed Arrow (1951), a series of important contributions (see, for example, Harsanyi 1955; Sen 1970b, 1977b, 1979; d’Aspremont and Gevers 1977; Gevers 1979) have explored social welfare judgements based on much richer utility information incorporating cardinal and interpersonally comparable individual utilities, and have demonstrated that Arrow- type impossibility results often lose their bite in this expanded analytical structure.

Non-utility Information

Sen (1977b) demonstrated that, though the definition of an SWF (A), by itself, does not rule out the possibility of using non-utility information, such as the information contained in the description of social states, in making social welfare judgements, a somewhat stronger version of WP, together with IIA, does rule out that possibility. The stronger version requires that, for all social states x and y and for every preference profile (R 1,…,R n ), [if x I i y for all individuals i, then x I y], and, further, [if xR i y for all individuals i and xP i y for some individual i, then xPy].

Individual rights based on the notion that an individual should be able to make free choices in affairs relating to his or her private life is an important example of an ethical value based on non-utility information. The concept of an individual’s private life, which John Stuart Mill (1859) considered so important, cannot, however, be articulated in terms of individual utilities alone. While i’s religion may cause just as much disutility for his neighbours as his playing loud music in early hours of the morning, Mill (1859) would have considered i’s religion, but not his playing loud music in early hours of the morning, to be an aspect of i’s private life. Sen (1970a) investigated the implications of granting individuals the right to make free choices with respect to their private affairs irrespective of how others feel about their choices. In his celebrated result on the impossibility of the Paretian liberal, Sen (1970a) demonstrated that respect for such individual rights clashes sharply with WP, even if one discards IIA and replaces the Arrow requirement that the social weak preference relation R be an ordering by the much weaker requirement that R be reflexive and connected and P be acyclic (P is said to be acyclic if and only if there do not exist x 1,x 2,. . .,x n in X such that x 1 Px 2 & x 2 Px 3 & & x n − 1 Px n & x n Px 1). While Sen departed radically from the Arrow format by introducing individual rights that have non-utility information as their basis, he still retained one basic feature of the Arrow format. His analysis was in terms of a social weak preference relation specified by a function of individual orderings over the social states. Sen’s formulation of an individual’s right to make free choices in his own private life was introduced as a restriction on this function, the restriction being contingent on the individual’s preferences over social states that differed only with respect to some features of his private life. An alternative version of Sen’s theorem uses the notion of social choice rather than social preference. The point under consideration applies to that version as well. Given any set of feasible social states, social choice from that set is still a function of individual preference orderings over social states, and the rights of an individual are formulated as restrictions on social choices, the restrictions being contingent on the individual’s preferences over certain social states.

Several subsequent writers (see, for example, Sugden 1985; Gaertner et al. 1992), who argued for a formulation of individual rights in terms of game forms, had to abandon Sen’s format altogether, given their conception of an individual’s right as the individual’s freedom to choose any of the actions or strategies permissible under the right rather than in terms of constraints imposed on social weak preferences (or social choice) by the individual’s preferences over certain types of social states. (See, however, Pattanaik and Suzumura 1996, for an attempt to put the problem of social choice of a rights structure, viewed as a game form, back in the framework of an Arrow-type social welfare function.)

To conclude, it will perhaps be fair to say that, while the concept of a social welfare function has been a powerful analytical tool for investigating implications of value judgements relating to social welfare, the individualistic version (see (1)) of the Bergson–Samuelson formulation, as well as Arrow’s formulation of the concept, had certain limitations and that some important developments in welfare economics have their origin in attempts to overcome those limitations.

See Also

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Authors and Affiliations

  • Prasanta K. Pattanaik
    • 1
  1. 1.