II. THE SOCIAL WELFARE FUNCTION
1. Bentham's Utilitarianism. To Bentham, the util-
ity of each individual was an objectively meaningful
magnitude; from the point of view of the community,
one man's utility is the same as another's, and therefore
it is the sum of the utilities of all individuals which
ought to determine social policy. Bentham is indeed
concerned strongly to argue that the actual measure-
ment of another's utility is apt to be very difficult, and
therefore it is best to let each individual decide as much
as possible for himself. In symbols, if U1,..., Un are
the utilities of the n individuals in the society, each
being affected by a social decision, the decision should
be made so as to make the sum, U1 + U2 +... + Un,
as large as possible. An expression of this form, which
defines a utility for social choices as a function of the
utilities of individuals, is usually termed a social welfare
function. Bentham's conclusion is really clearly enough
stated, but there are considerable gaps in the underly-
ing argument. The addition of utilities assumes an
objective or at least interpersonally valid common unit;
but no argument is given for the existence of one and
no procedure for determining it, except possibly the
view that the just noticeable difference is such a unit.
Even if the existence and meaningfulness of such a unit
is established, it is logically arbitrary to add the util-
ities instead of combining them in some other way.
The argument that all individuals should appear
alike in a social judgment leads only to the conclusion
that the social welfare function should be a symmetric
function of individual utilities, not that it should be
a sum.
The Bentham criterion was defended later by John
Stuart Mill, but his arguments bear mostly on the
propriety and meaning of basing social welfare judg-
ments on individual preferences and not at all on the
commensurability of different individuals' utilities or
on the form of social welfare function. Mill, like Henry
Sidgwick and others, considered the primary use of
Bentham's doctrines to be applicability to the legal
system of criminal justice; since the conclusions arrived
at were qualitative, not quantitative, in nature, vague-
ness on questions of measurability was not noticed.
After the spread of marginal utility theory, the
economist F. Y. Edgeworth expounded the notion of
utility much more systematically than Bentham had
done, with little originality in the foundations, though
with a great deal of depth in applications. In particular,
he applied the sum-of-utilities criterion to the choice
of taxation schemes. The implication is one of radical
egalitarianism, as indeed Bentham had already per-
ceived. If, as is usually assumed, the marginal utility
of money is decreasing, if all individuals have the same
utility function for money, and if a fixed sum of money
is to be distributed, then the sum of utilities is
maximized when money income is distributed equally.
(Here, “money” may be thought of as standing for all
types of desired goods.) Then the only argument against
complete equality of income is that any procedure to
accomplish it would also reduce total income, which
is the amount to be divided. The argument can be also
put this way; resources should be taken from the rich
and given to the poor, not because they are poorer
per se but because they place a higher value on a given
quantity of goods. If it were possible to differentiate
between equally wealthy individuals on the basis of
their sensitivities to income increments, it would be
proper to give more to the more sensitive.
Apart from Edgeworth, there was little interest in
applying the sum-of-utilities criterion to economic or
any other policy. Very possibly, the radically egali-
tarian implications were too unpalatable, as they
clearly were to Edgeworth. Subsequent work on “wel-
fare economics,” as the theory of economic policy is
usually known, tended to be very obscure on funda-
mentals (although very clarifying in other ways).
2. Ordinalist Views of the Social Welfare Function.
Pareto's rejection of cardinal utility rendered mean-
ingless a sum-of-utilities criterion. If utility for an
individual was not even measurable, one could hardly
proceed to adding utilities for different individuals.
Pareto recognized this problem.
First of all, he introduced a necessary condition for
social optimality, which has come to be known as
Pareto-optimality: a social decision is Pareto-optimal
if there is no alternative decision which could have
made everybody at least as well off and at least one
person better off. In this definition, each individual is
expressing a preference for one social alternative
against another, but no measurement of preference
intensity is required. Pareto-optimality is thus a purely
ordinal concept.
It is, however, a weak condition. It is possible to
compare two alternative social decisions only if there
is essential unanimity. To put the matter another way,
among any given set of alternatives there will usually
be many which would satisfy the definition. A mani-
festly unjust allocation, with vast wealth for a few and
poverty for many, will nevertheless be Pareto-optimal
if there is no way of improving the lot of the many
without injuring the few in some measure. Pareto
himself was very clear on this point.
Pareto-optimality is nevertheless a very useful con-
cept in clearing away a whole realm of possible deci-
sions which are not compatible with any reasonable
definition of social welfare. It might be argued that
every application of utilitarianism in practice, as to
law, has in fact used only the concept of Pareto-
optimality. In welfare economics, similarly, it has
turned also to be useful in characterizing sharply the
types of institutional arrangements which lead to
efficient solutions, making it possible to isolate the
debate on distributive problems which it cannot solve.
Pareto later (1913) went further. He suggested that
each individual in his judgments about social decisions
considers the effects on others as well as on himself.
The exposition is a bit obscure, but it appears to coin-
cide with that developed later and independently by
the economist, Abram Bergson (1938). Each individual
has his own evaluation of a social state, which is a
function of the utilities of all individuals: Wi(U1,...,
Un). Since the evaluation is done by a single individual,
this function has only ordinal significance. The Ui's
themselves may be thought of as an arbitrary numerical
scaling of the individuals' preferences; they also have
only ordinal significance, but this creates no conceptual
problem, since the choice of the social welfare func-
tion, Wi, for the ith individual, already takes account
of the particular numerical representation of individ-
uals' ordinal utilities.
Interpersonal comparisons of utility are indeed
made, but they are ethical judgments by an observer,
not factual judgments.
Pareto (but not Bergson) went one step further. The
“government” will form the social welfare function
which will guide it in its choices by a parallel
amalgamation of the social welfare functions of the
individuals, i.e., a function, V(W1,..., Wn). Pareto's
concept of a social welfare function remained un-
known, though the concept of Pareto-optimality be-
came widely known and influential beginning with the
1930's, as is clear in Bergson's work. The latter became
very influential and is accepted as a major landmark;
but in fact it has had little application.
Bergson accepted fully the ordinalist viewpoint, so
that the ethical judgments are always those of a single
individual. This approach loses, however, an important
feature of most thinking about social welfare, namely,
its impartiality among individuals, as stressed by
Bentham and given classic, if insufficiently precise,
expression in the categorical imperative of Kant. In
Bergson's theory, any individual's social welfare func-
tion may be what he wishes, and it is in no way
excluded that his own utility plays a disproportionate
role. Pareto, by his second-level social welfare function
for the government implicitly recognized the need for
social welfare judgments not tied to particular individ-
uals. But the ordinalist position seems to imply that
all preferences are acts of individuals, so that in fact
Pareto had no basis for the second level of judgment.
3. Conjoint Measurement and Additive Social Wel-
fare Functions. In the field of social choice, as in that
of individual choice, the methods of conjoint measure-
ment have led to cardinal utilities which are consistent
with the general operational spirit of ordinalism.
William S. Vickrey, in 1945, suggested that the von
Neumann-Morgenstern theory of utility for risk-
bearing was applicable to the Bergson social welfare
function. The criterion of impartiality was interpreted
to mean that the ethical judge should consider himself
equally likely to have any position in society. He then
would prefer one decision to another if the expected
utility of the first is higher. The utility function used
is his von Neumann-Morgenstern utility function, i.e.,
that utility function which explains his behavior in risk-
bearing. Since all positions are assumed to be equally
likely, the expected utility is the same as the average
utility of all individuals. In turn, making the average
utility as large as possible is equivalent to maximizing
the sum of utilities, so that Vickrey's very ingenious
argument is a resuscitation, in a way, of Benthamite
utilitarianism.
Though Vickrey's criterion is impartial with respect
to individual's positions, it is not impartial with respect
to their tastes; the maker of the social welfare judgment
is implicitly ascribing his own tastes to others. Further
it has the somewhat peculiar property that social
choices among decisions where there may be no
uncertainty are governed by attitudes towards risk-
bearing.
Fleming, in 1952, took another direction, which has
not been followed up but which is worthy of note.
Suppose that an ethical judge is capable of making
social welfare judgments for part of the society inde-
pendently of the remainder. More precisely, suppose
that for any social decision which changes the utilities
of some individuals but not of others, the judge can
specify his preferences without knowing the utility
level of those unaffected by the decision. Then it can
be shown that there are cardinal utility functions for
the individuals and a cardinal social welfare function,
such that, W = U1 +... + Un. W and U1,..., Un
are interval scales, but the units of measurement must
be common. Again there is additivity of utility, but
note now that the measurements for individual utility
and for social welfare are implied by the social welfare
preferences and do not serve as independent bases for
them.
Harsanyi in 1955 in effect synthesized the points of
view of Vickrey and of Fleming. His argument was
that each individual has a von Neumann-Morgenstern
utility function, expressing his attitude toward risk, and
society, if it were rational, must also have a von
Neumann-Morgenstern utility function. It is then easy
to demonstrate that society's utility function must be
a weighted sum of the individuals' utilities, i.e.,
W = a1U1 +... + anUn. Since each individual utility
is an interval scale, we can choose the units so that
all the coefficients ai, are 1. This result differs from
Vickrey's in that the utility function of the ith individ-
ual is used to evaluate his position, rather than the
utility function of the judge.
Distantly related to these analyses is the revival, by
W. E. Armstrong, and by Leo Goodman and Harry
Markowitz, of Bentham's use of the just noticeable
difference as an interpersonally valid unit of utility.
It has proved remarkably difficult to formulate theories
of this type without logical contradiction or at least
paradoxical implications.
So far all these results have led to a sum-of-utilities
form, though with varying interpretations. As remarked
earlier, the notion of impartiality requires symmetry
but not necessarily additivity. John Rawls in 1958
proposed an alternative form for the social welfare
criterion, to maximize the minimum utility in the
society. This formulation presupposes an ordinal inter-
personal comparison of utilities. He shares with
Vickrey and Harsanyi a hypothetical concept of an
original position in which no individual knows who
he is going to be in the society whose principles are
being formulated. However, he does not regard this
ignorance as being adequately formulated by equal
probabilities of different positions; in view of the
permanence of the (hypothetical) choice being made,
he argues that a more conservative criterion, such as
maximizing the minimum, is more appropriate than
maximizing the expected value.
