**Introduction**

One of the key ideas in contemporary economic theory in general and law and economics in particular is the *social welfare function*.
Law students without a background in economics might be put off by the
fact that social welfare functions are expressed in mathematical
notation, but there is no reason to be intimidated. The basic ideas are
easily grasped and the mathematical notation can be mastered in just a
few minutes. This post provided an introduction to the idea of the
social welfare function for law students, especially first year law
students, with an interest in legal theory. Here we go!

**Background**

Normative EconomicsThe idea of a

social welfare functionis part of normative economics. There are several plausible formulations of normative economics, but almost all of normative economics begins with the fundamental idea of utility as a conception or measure of the good. Economists may disagree about the nature of utility, the relationship of utility to social welfare, and the role of welfare in public policy, but most (if not all) economists would assent to the abstract proposition that ceteris paribus more utility is a good thing. But this apparent agreement is at a very abstract and ambiguous level. There are many different ideas about what "utility" is.

Cardinal and Ordinal Interpretations of UtilityOne key divide is between cardinal and ordinal interpretations of utility. An

ordinal utility functionfor an individual consists of a rank ordering of possible states of affairs for that individual. An ordinal function tells us that individual i prefers possible world X to possible world Y, but it doesn't tell us whether X is much better than Y or only a little better.A

cardinal utility functionyields a real-number value for each possible state of affairs. If we assume that utility functions yield values expressed in units of utility or utiles, then individual's utility function might score possible world P at 80 utiles and possible world Q at 120 utiles. We might represent the utility function U of individual i for P and Q as follows:

Ui(P) = 80

Ui(Q) = 120

The distinction between cardinal and ordinal utilities is potentially important for utilitarianism, at least on certain interpretations. As a theory of evaluation, utilitarianism is the view that an action is the best action if and only if the action maximizes utility when compared with all possible alternative actions. For technical reasons, utilitarianism requires both cardinality and full interpersonal comparability. This point about utilitarianism is closely related to the history of welfare economics, the explicitly normative branch of economic theory.

Measurement ProblemsBoth cardinality and interpersonal comparability pose measurement problems for economists. Even in the case of a single individual, it is difficult to reliability measure cardinal utilities. Measurements that support interpersonal comparisons are even more difficult to justify, and cardinal interpersonal comparisons seem to require the analyst (the person making the comparison) to make a variety of controversial value judgments. Market prices won't do as a proxy for utility, for a variety of reasons including wealth effects. The challenge for welfare economics was to develop a methodology that yields robust evaluations but does not require the use of cardinal interpersonally comparable utilities.

ParetoThis is the point at which Pareto arrives on the scene. Suppose that all the information we have about individual utilities is ordinal and non-interpersonally comparable. In other words, each individual can rank order states of affairs, but we (the analysts) cannot compare the rank orderings across persons. The weak Pareto principle suggests that possible world (state of affairs) P is socially preferable to possible world (state of affairs) Q, if everyone's ordinal ranking of P is higher than their ranking of Q. Weak Pareto doesn't get us very far, because such unanimity of preferences among all persons is rare. The strong Pareto principle suggests that possible world (state of affairs) P is socially preferable to possible world (state of affairs) Q, if at least one person ranks P higher than Q and no one ranks Q higher than P. Unlike weak Pareto, strong Pareto does permit some relatively robust conclusions.

The New Welfare EconomicsThe so-called new welfare economics was based on the insight that market transactions without uncompensated negative externalities satisfy strong Pareto. If the only difference between state P and state Q is that in P, individuals i1 and i2 engage in an exchange (money for widgets, chickens for shoes) where both prefer the result of the exchange, then the exchange is Pareto efficient. A state of affairs where no further Pareto efficient moves (or trades) are possible is called Pareto optimal. The assumption about externalities is, of course, crucial. If there are negative externalities of any sort, then the trade is not Pareto efficient.

Weak Pareto and the Arrow Impossibility TheoremWeak Pareto plus ordinal utility information allows some social states (or possible worlds) to be ranked on the basis of everyone's preferences. A method for transforming individual utility information into such a social ranking is called a social welfare function. Kenneth Arrow's famous impossibility theorem demonstrates that it is impossible to construct a social utility function that can transform individual ordinal rankings into a social ranking in cases not covered by weak Pareto, if certain plausible assumptions are made. Arrow's theorem has spurred two lines of development in welfare economics. One line of development relaxes various assumptions that Arrow made; for example, we might relax Arrow's assumption that the social ranking must be transitive (if X is preferred to Y and Y is preferred to Z, then X must be preferred to Z). The other line of development considers the possibility of allowing information other than individual, noncomparable ordinal utilities. It is this second line of development that is relevant to the use of social welfare functions in contemporary law and economics.

**Social Welfare Functions**

Suppose that we allow full interpersonal comparability and cardinal utility information. This is sufficient to support what are called Bergson-Samuelson utility functions, which have the form:

W(x) = F (U1(x), U2(x), . . . UN(x))

Where

W(x) represents a real number social utility value for some state of affairs (or possible world) X,

F is some increasing function that yields a real number,

U1(x) is a cardinal, interpersonally comparable utility value yielded by some procedure for individual 1 for state of affairs X, and

N is the total number of individuals.

Bergson-Samuelson social welfare functions are named after Paul Samuelson and Avram Bergson.

**What Are the Plausible Social Welfare Functions?**

There are a variety of different possible functions that can be substituted for F. Here are some of the most important possibilities:

Classical-utilitarian SWF--We could substitute summation for F, and simply add the individual utility values; this is sometimes called a Benthamite or classical-utilitarian social welfare function famously associated with Jeremy Bentham. The classical utility social welfare function can be represented as follows:W(x)={U1(x) + U2(x) + U(3(x) . . . Un(x)}

Average-utilitarian SWF--The classical SWF adds the utilities. This raises some very interesting issues when the different states of the world (x or y) have different population sizes. When deciding whether to add additional individuals, the classical-utilitarian SWF says more is better until we reach the point where adding more actually reduces the overall level of utility. One way to avoid this implication is use the average level of utility instead of the sum, as in the following formula:W(s){[U1(x) + U2(x) + U(3(x) . . . Un(x)]/n}

In other words, we divide the sum of utilities by the number of individuals!

Bernoulli-Nash SWF--In the alternative, we could substitute the product function (¡Ç) and multiply individual utilities. This is sometimes called a Bernoulli-Nash social welfare function, which can be represented as follows:W(x)={U1(x) * U2(x) * U(3(x) . . . Un(x)}

Rather than adding individual utilities, we multiply them! And yes, the "Nash" in Bernoulli-Nash is John Nash of "A Beautiful Mind" fame.

**What About the Problem of Interpersonal Comparison?**

*Social welfare functions*
are much discussed in legal theory these days. One of the reasons for
the contemporary debate over social welfare functions is that this
approach has been championed by Louis Kaplow and Steven Shavell (both
of the Harvard Law School). Their book, *Welfare versus Fairness*, has put the welfarist approach to normative economics "front and center."

One of the interesting theoretical questons about SWFs concerns the problem of interpersonal comparison. How do we get the values to plug into U1(x), U2(x), and so forth. That is, how do we compare up with a way of putting my utility and your utility on the same scale. As I understand the state of play, this is not a topic on which economists agree. Some economists believe that there is no objective way of producing interpersonally comparable cardinal utility values. But some economists believe that a third-party (the legal analyst or the economist) can do the job of assigning values to individual utilities.

**Conclusion**

We've barely begun to scratch
the surface of the many interesting theoretical issues that attend the
use of social welfare functions in legal theory. Some of those issues
were explored in a prior *Legal Theory Lexicon* entry on Balancing Tests.
Even if you have absolutely no background in economics, there is no
reason to shy away from the debates about social welfare functions. The
notation, although at first intimidating, is actually very simple. The
foundational ideas, although sometimes articulated in the jargon of
economic theory, really go to fundamental questions in moral theory. I
hope this post has given you the tools to begin to discuss these ideas!

**Related Lexicon Entries**

Legal Theory Lexicon 008: Utilitarianism

Legal Theory Lexicon 060: Efficiency, Pareto, and Kaldor-Hicks

**Links**

Social welfare function, Wikipedia

**Bibliography**

Kenneth J. Arrow, Social Choice and Individual Values (2d ed. 1963)

Abram Bergson,

A Reformulation of Certain Aspects of Welfare Economics, 52 Quarterly Jounal of Economics 310 (1938).John C. Harsanyi,

Interpersonal Utility Comparisons, 2 The New Palgrave: A Dictionary of Economics, 955 (1987).Lawrence Solum, Public Legal Reason, 92 Virginia Law Review 1449 (2006).

Kotaro Suzumura, Social Welfare Function, 4 The New Palgrave: A Dictionary of Economics, v. 4, 418-20 (1987).

(This entry was last modified on August 30, 2009.)