Legal Theory Lexicon 025: Social Welfare Functions
 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 Economics
The idea of a social welfare function is 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 Utility One key divide is between cardinal and ordinal interpretations of utility. An ordinal utility function for 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 function yields a realnumber 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
Measurement Problems Both 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 cardinal interpersonally comparable utilities.
Pareto This is the point at which Pareto arrives on the scene. Suppose that all the information we have about individual utilities is ordinal and noninterpersonally 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 Economics The socalled new welfare economics was based on the insight that market transactions without 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 Theorem Weak 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 utility 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.

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

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.
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:

Classicalutilitarian SWFWe could substitute summation for F, and simply add the individual utility values; this is sometimes called a Benthamite or classicalutilitarian 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)}

W(s){[U1(x) + U2(x) + U(3(x) . . . Un(x)]/n}
BernoulliNash SWFIn the alternative, we could substitute the product function (¡Ç) and multiply individual utilities. This is sometimes called a BernoulliNash social welfare function, which can be represented as follows:

W(x)={U1(x) * U2(x) * U(3(x) . . . Un(x)}
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 thirdparty (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!
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