A Difficulty in the Concept of Social Welfare
Author(s): Kenneth J. Arrow
Source: Journal of Political Economy, Vol. 58, No. 4 (Aug., 1950), pp. 328-346
Published by: The University of Chicago Press
Stable URL: http://www.jstor.org/stable/1828886
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A DIFFICULTY IN THE CONCEPT OF SOCIAL WELFARE
KENNETH J. ARROW'
Stanford University
cial choice in any given situation, e.g.,
I. INTRODUCTION
a religious code.
IN A capitalist democracy there are
The last two methods of social choice,
essentially two methods by which
dictatorship and convention, have in
social choices can be made: voting,
their formal structure a certain definite-
typically used to make "political" de-
ness absent from voting or the market
cisions, and the market mechanism,
mechanism. In an ideal dictatorship,
typically used to make "economic" de-
there is but one will involved in choice;
cisions. In the emerging democracies
in an ideal society ruled by convention,
with mixed economic systems, Great
there is but the divine will or perhaps,
Britain, France, and Scandinavia, the
by assumption, a common will of all
same two modes of making social
individuals concerning social decisions,
choices prevail, though more scope is
so that in either case no conflict of indi-
given to the method of voting and to
vidual wills is involved. The methods
decisions based directly or indirectly
of voting and of the market, on the
on it and less to the rule of the price
other hand, are methods of amalga-
mechanism. Elsewhere in the world,
mating the tastes of many individuals
and even in smaller social units within
in the making of social choices. The
the democracies, the social decisions
methods of dictatorship and convention
are sometimes made by single individare, or can be, rational in the sense
uals or small groups and sometimes
that any individual can be rational in
(more and more rarely in this modern
his choice. Can such consistency be
world) by a widely encompassing set
attributed to collective modes of choice,
of traditional rules for making the so-
where the wills of many people; are in-
volved?
'This paper is based on research carried on at
It should be emphasized here that
the RAND Corporation, a project of the United
States Air Force, and at the Cowles Commission
the present study is concerned only
for Research in Economics and is part of a longer
with the formal aspects of the foregoing
study, "Social Choice and Individual Values," to
question. That is, we ask if it is for-
be published by John Wiley & Sons as a Cowles
Commission monograph. A version was read at the
mally possible to construct a procedure
December, 1948, meeting of the Econometric So-
for passing from a set of known indi-
ciety. I am indebted to A. Kaplan, University of
vidual tastes to a pattern of social deCalifornia at Los Angeles, and J. W. T. Youngs,
University of Indiana, for guidance in formulating
cision-making, the procedure in ques-
the problem, and to A. Bergson and A. G. Hart,
tion being required to satisfy certain
Columbia University, and T. C. Koopmans, Cowles
natural conditions. An illustration of
Commission and the University of Chicago, who
have read the manuscript and made valuable com-
the problem is the following well-known
ments on both the presentation and the meaning.
"paradox of voting." Suppose there is a
Needless to say, any error or opacity remaining is
community consisting of three voters
the responsibility of the author.
328
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THE CONCEPT OF SOCIAL WELFARE 329
and this community must choose among
three alternative modes of social action
(e.g., disarmament, cold war, or hot
war). It is expected that choices of this
imply rational behavior on the part of
the community and which will be satis-
factory in other ways?3
If we adopt the traditional identifica-
type have to be made repeatedly, but
tion of rationality with maximization
sometimes not all of the three alterna-
of some sort, then the problem of
tives will be available. In analogy with
achieving a social maximum derived
the usual utility analysis of the indi-
vidual consumer under conditions of
constant wants and variable price-in-
from individual desires is precisely the
problem which has been central to the
field of welfare economics.4 However,
come situations, rational behavior on
the search for a clear definition of opti-
the part of the community would mean
mum social welfare has been plagued
that the community orders the three
by the difficulties of interpersonal com-
alternatives according to its collective
parisons. The emphasis, as is well
preferences once for all and then
known, has shifted to a weaker defini-
chooses in any given case that alter-
tion of optimum, namely, the determi-
native among those actually available
nation of all social states such that no
which stands highest on this list. A
individual can be made better off with-
natural way of arriving at the collecout making someone worse off. As Pro-
tive preference scale would be to say
fessors Bergson, Lange, and Samuelson
that one alternative is preferred to
have argued, though, the weaker defini-
another if a majority of the community
tion cannot be used as a guide to social
prefer the first alternative to the secpolicy; the second type of welfare eco-
ond, i.e., would choose the first over the
nomics is only important as a prelimi-
second if those were the only two alter-
2 It may be added that the method of decision
natives. Let A, B. and C be the three
sketched above is essentially that used in deliber-
alternatives, and i, 2, and 3 the three
ative bodies, where a whole range of alternatives
individuals. Suppose individual i preusually comes up for decision in the form of suc-
fers A to B and B to C (and therefore
cessive pairwise comparisons. The phenomenon de-
scribed in the text can be seen in a pure form in
A to C), individual 2 prefers B to C
the disposition of the proposals before recent Con-
and C to A (and therefore B to A), and
gresses for federal aid to state education, the three
individual 3 prefers C to A and A to B
alternatives being no federal aid, federal aid to pub-
lic schools only, and federal aid to both public
(and therefore C to B). Then a major-
and parochial schools.
ity prefers A to B, and a majority pre-
3 The problem of collective rationality has been
fers B to C. We may therefore say that
discussed by Professor Frank H. Knight, but chiefly
the community prefers A to B and B to
in terms of the socio-psychological prerequisites;
see "The Planful Act: The Possibilities and LimitaC. If the community is to be regarded
tions of Collective Rationality," in Freedom and
as behaving rationally, we are forced to
Reform (New York: Harper & Bros., I947), pp.
say that A is preferred to C. But, in
fact, a majority of the community pre-
335-69, esp. pp. 346-65).
'See P. A. Samuelson, Foundations of Economic
Analysis (Cambridge, Mass.: Harvard University
fers C to A." So the method just out-
Press, I94.7), chap. viii; A. Bergson (Burk), "A
lined for passing from individual to col-
Reformulation of Certain Aspects of Welfare Eco-
lective tastes fails to satisfy the condi-
nomics," Quarterly Journal of Economics, L1I
(2938), 3IO-34; 0. Lange, "The Foundations of
tion of rationality as we ordinarily un-
Welfare Economics," Econonzetrica, X (2942),
derstand it. Can we find other methods
225-28; M. W. Reder, Studies in the Theory of
of aggregating individual tastes which
Welfare Economics (New York, I947), chaps. i-v.
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330 KENNETH J. ARROW
nary to the determination of a genuine
compensation is not actually paid.
social maximum in the full sense. E.g.,
Apart from the ethical difficulties in the
under the usual assumptions, if there is
an excise tax imposed on one commod-
ity in the initial situation, it can be
argued that the removal of the tax ac-
companied by a suitable redistribution
acceptance of this principle, there is a
formal difficulty which was pointed out
by Professor Scitovszky:7 it is possible
that simultaneously x should be pre-
ferred to y and y be preferred to x. Just
of income and direct tax burdens will
as in the case of majority voting, this
improve the position of all individuals
method of aggregating individual pref-
in the society. But there are, in general,
erences may lead to a pattern of social
many redistributions which will accom-
choice which is not a linear ordering of
plish this end, and society must have
the social alternatives. Note that in
some criterion for choosing among them
both cases the paradox need not occur;
before it can make any change at all.
all that is said is that there are prefer-
Further, there is no reason for confining
ence patterns which, if held by the indi-
the range of possible social actions to
vidual members of the society, will give
those which will injure no one as com-
rise to an inconsistent pattern of social
pared with the initial situation, unless
the status quo is to be sanctified on
ethical grounds. All we can really say is
choice. Unless the trouble-breeding in-
dividual preference patterns can be
ruled out by a priori assumption, both
that society ought to abolish the excise
majority voting and the compensation
tax and make some redistribution of in-
principle must be regarded as unsatis-
come and tax burdens; but this is no
factory techniques for the determina-
prescription for action unless there is
tion of social preferences.
some principle by which society can
make its choice among attainable in-
come distributions, i.e., a social indif-
ference map.
The aim of the present paper is to
show that these difficulties are general.
For any method of deriving social
choices by aggregating individual pref-
Voting can be regarded as a method
of arriving at social choices derived
from the preferences of individuals.
Another such method of more specifi-
cally economic content is the compensa-
erence patterns which satisfies certain
natural conditions, it is possible to find
individual preference patterns which
give rise to a social choice pattern
which is not a linear ordering. In partic-
tion principle, as proposed by Mr. Kal-
ular, this is very likely to be the case if,
dor:5 in a choice between two alterna-
as is frequently assumed, each indi-
tive economic states x and y, if there is
vidual's preferences among social states
a method of paying compensations un-
are derived purely from his personal
der state x such that everybody can be
consumption-leisure-saving situation in
made better off in the state resulting
each.8 It is assumed that individuals act
from making the compensations under
rationally, in the sense that their be-
x than they are in state y, then x should
6 See W. J. Baumol, "Community Indifference,"
be chosen in preference to y, even if the
Review of Economic Studies, XIV (I946-47),
44-48.
5 N. Kaldor, "Welfare Propositions of Economics
'T. Scitovszky, "A Note on Welfare Proposi-
and Interpersonal Comparisons of Utility," Eco-
tions in Economics," Review of Economic Studies,
nomic Journal, XLIX (I939), 549-652; see also
IX (1942), 77-88.
J. R. Hicks, "The Foundations of Welfare Eco-
8 See, e.g., Samuelson, op. cit., pp. 222-24; Bergnomics," Economic Journal, XLIX (I939), 698-
,oi and 71I-I2.
son, op. cit., pp. 3i8-20; Lange, op. Cit., p. 2i6.
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THE CONCEPT OF SOCIAL WELFARE 331
havior in alternative situations can be
described by an indifference map. It is
further assumed that utility is not
measurable in any sense relevant to
preferred to x. The decisions made for
different pairs are assumed to be consist-
ent with one another, so that, for ex-
ample, if x is preferred to y and y to z,
welfare economics, so that the tastes of
then x is preferred to z; similarly, if x
an individual are completely described
is indifferent to y and y to z, then x is
by a suitable preference pattern or in-
indifferent to z. Having this ordering of
difference map.
all possible alternatives, the chooser is
now confronted with a particular op-
II. DEFINITIONS AND NOTATION
portunity set S. If there is one alterna-
I. A NOTATION FOR PREFERENCES AND CHOICE
tive in S which is preferred to all others
in S, then the chooser selects that one
In this paper I shall be interested in
alternative.9
the description of preference patterns
Preference and indifference are relaboth for the individual and for society.
tions between alternatives. Instead of
It will be found convenient to represent
working with two relations, it will be
preference by a notation not customar-
slightly more convenient to use a single
ily employed in economics, though fa-
relation, "preferred or indifferent." The
miliar in mathematics and particularly
statement, "x is preferred or indifferent
in symbolic logic. We assume that there
to y," will be symbolized by xRy. The
is a basic set of alternatives which
letter R, by itself, will be the name of
could conceivably be presented to the
the relation and will stand for a knowlchooser. In the theory of consumers'
edge of all pairs such that xRy. From
choice, each alternative would be a
commodity bundle; in the theory of the
our previous discussion, we have, for
any pair of alternatives x and y, either
firm, each alternative would be a com-
that x is preferred to y or y to x or that
plete decision on all inputs and outputs;
the two are indifferent. That is, we have
in welfare economics, each alternative
assumed that any two alternatives are
would be a distribution of commodities
comparable. But this assumption may
and labor requirements. These alterna-
be written symbolically,
tives are mutually exclusive; they are
denoted by small letters, x, y, z.... On
Axiom I: For all x and y, either xRy or yRx.
any given occasion the chooser has
Note that Axiom I is presumed to hold
available to him a subset S of all pos-
when x _ y, as well as when x is dis-
sible alternatives, and he is required to
tinct from y, for we ordinarily say that
choose one out of this set. The set S is
x is indifferent to itself for any x, and
a generalization of the well-known op-
this implies xRx. Note also that the
portunity curve; thus, in the theory of
9 It may be that there is a subset of alternatives
consumer's choice under perfect compe-
in S, such that the alternatives in the subset are
tition, it would be the budget plane.
each preferred to every alternative not in the subset,
It is assumed further that the choice is
while the alternatives in the subset are indifferent
to one another. This case would be one in which the
made in this way: Before knowing the
highest indifference curve which has a point in com-
set S. the chooser considers in turn all
mon with a given opportunity curve has at least
possible pairs of alternatives, say x and
two points in common with it (the well-known case
of multiple maxima). In this case, the best thing to
y, and for each pair he makes one and
say is that the choice made in S is the whole subset;
only one of three decisions: x is prethe first case discussed is one in which the subset
ferred to y, x is indifferent to y, or y is
in question, the choice, contains a single element.
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332 KENNETH J. ARROW
word "or" in the statement of Axiom I
different to y." It is clear that P and I,
does not exclude the possibility of both
so defined, correspond to the ordinary
xRy and yRx. That word merely as-
notions of preference and indifference,
serts that at least one of the two events
respectively.
must occur; both may.
Lenima: a) For all x, xRx.
The property mentioned above of
b) If xPy, then xRy.
c) If xPy and yPz, then xPz.
consistency in the preferences as be-
d) If xIy and yIz, then xIz.
tween different pairs of alternatives
e) For all x and y, either xRy or
may be stated more precisely, as fol-
yPx.
lows: if x is preferred or indifferent to
I) If xPy and yRz, then xPz.
y and y is preferred or indifferent to z,
All these statements are intuitively self-
then x must be either preferred or in-
evident from the interpretations placed
different to z. In symbols,
on the symbols.
Axioin II: For all x, y, and z, xRy and yRz
For clarity, we shall avoid the use of
imply xRz.
the terms "preference scale" or "pref-
A relation satisfying both Axiom I and
Axiom II is termed a weak ordering or
sometimes simply an ordering. It is
clear that a relation having these two
properties taken together does create
erence pattern" when referring to R,
since we wish to avoid confusion with
the concept of preference proper, de-
noted by P. We shall refer to R as an
"'ordering relation" or "weak ordering
a ranking of the various alternatives.
relation" or, more simply, as an "order-
The adjective "weak" refers to the fact
ing" or "weak ordering." The term
that the ordering does not exclude in-
difference, i.e., Axioms I and II do not
exclude the possibility that for some
distinct x and y, both xRy and yRx.
It might be held that the two axioms
in question do not completely charac-
terize the concept of a preference pat-
"preference relation" will refer to the
relation P.
Suppose that we know the choice
which would be made from any given
pair of alternatives; i.e., given two
alternatives x and y from which the
chooser must select, we know whether
he would take x or y or remain indiftern. For example, we ordinarily feel
ferent between them. Since choosing x
that not only the relation R but also
from the pair x, y implies that x is prethe relations of (strict) preference and
ferred to y, and similarly with a choice
of indifference satisfy Axiom II. It can
of y, a knowledge of the choice which
be shown that, by defining preference
would be made from any two given
and indifference suitably in terms of R,
alternatives implies a knowledge of the
it will follow that all the usually de-
full preference scale; from earlier re-
sired properties of preference patterns
marks this, in turn, implies a knowl-
obtain.
edge of the choice which would be
Definition I: xPy is defined to mean not yRx.
made from any set of alternatives actu-
ally available. Hence, one of the conse-
The statement "xPy" is read, "x is pre-
quences of the assumption of rational
ferred to y."
behavior is that the choice from any
Definition : xIy means xRy and yRx.
collection of alternatives can be deter-
The statement "xly" is read, "x is in-
mined by a knowledge of the choices
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THE CONCEPT OF SOCIAL WELFARE 333
analysis if the generality were re-
which would be made from pairs of
stricted by a prior knowledge of the
alternatives.
nature of individual orderings of social
2. THE ORDERING OF SOCIAL STATES
states. This problem will be touched on
In the present study the objects of
choice are social states. The most pre-
again.
In general, then, there will be a dif-
cise definition of a social state would be
ference between the ordering of social
a complete description of the amount
states according to the direct consump-
of each type of commodity in the hands
tion of the individual and the ordering
of each individual, the amount of labor
when the individual adds his general
to be applied by each individual, the
standards of equity (or perhaps his
amount of each productive resource in-
standards of pecuniary emulation)."
vested in each type of productive activ-
We may refer to the former ordering
ity, and the amounts of various types
as reflecting the tastes of the individual
of collective activity such as municipal
and the latter as reflecting his values.
services, diplomacy and its continuation
The distinction between the two is
by other means, and the erection of
by no means clear cut. An individual
statues to famous men. It is assumed
with aesthetic feelings certainly derives
that each individual in the community
pleasure from his neighbor's having a
has a definite ordering of all conceiva-
well-tended lawn. Under the system of
ble social states in terms of their de-
a free market, such feelings play no
sirability to him. It need not be as-
direct part in social choice; yet, psy-
sumed here that an individual's atti-
chologically, they differ only slightly
tude toward different social states is de-
fromen the pleasure in one's own lawn.
termined exclusively by the commodity
Intuitively, of course, we feel that not
bundles which accrue to his lot under
all the possible preferences which an
each. The individual may order all so-
individual might have ought to count;
cial states by whatever standards he
his preferences for matters which are
deems relevant. A member of Veblen's
leisure class might order the states sole-
ly on the criterion of his relative in-
"none of his business" should be ir-
relevant. Without challenging this view,
I should like to emphasize that the de-
come standing in each; a believer in the
cision as to which preferences are rele-
equality of man might order them in
vant and which are not is itself a value
accordance with some measure of in-
judgment and cannot be settled on an
come equality. Indeed, since, as men-
a priori basis. From a formal point of
tioned above, some of the components
view, one cannot distinguish between
of the social state, considered as a vec-
an individual's dislike of having his
tor, are collective activities, purely indi-
grounds ruined by factory smoke and
vidualistic assumptions are useless in
analyzing such problems as the division
of the national income between public
and private expenditure. The present
notation permits perfect generality in
this respect. Needless to say, this gen-
his extreme distaste for the existence
of heathenism in Central Africa. There
are probably not a few individuals in
this country who would regard the for-
mer feeling as irrelevant for social pol-
icy and the latter as relevant, though
erality is not without its price. More
10 This distinction has been stressed to the author
information would be available for
by M. Friedman, University of Chicago.
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KENNETH J. ARROW
the majority would probably reverse
the judgment. I merely wish to empha-
size here that we must look at the en-
tire system of values, including values
be denoted by P and I, respectively,
primes or seconds being attached when
they are attached to the relation R,
respectively.
Throughout this analysis, it will be
about values, in seeking for a truly
general theory of social welfare.
It is the ordering according to values
which takes into account all the desires
of the individual, including the highly
important socializing desires, and which
is primarily relevant for the achieve-
ment of a social maximum. The mar-
ket mechanism, however, takes into ac-
assumed that individuals are rational.,
by which is meant that the ordering re-
lations Ri satisfy Axioms I and II. The
problem will be to construct an order-
ing relation for society as a whole which
is also to reflect rational choice-making,
so that R also will be assumed to satis-
fy Axioms I and II.
count only the ordering according to
III. TIHE SOCIAL WELFARE FUNCTION
tastes. This distinction is the analogue,
I. FORMAL STATEMENT OF TIlE PROBLEM
on the side of consumption, of the di-
OF SOCIAL CHOICE
vergence between social and private
costs in production which has been deI shall largely restate Bergson's for-
veloped by Professor Pigou."
mulation of the problem of making
As for notation, let RI be the orderwelfare judgments"2 in the terminology
ing relation for alternative social states
here adopted. The various arguments
from the standpoint of individual i.
of his social welfare function are the
Sometimes, when several different orcomponents of what I have here termed
dering relations are being considered
the "social state," so that essentially
for the same individual, the symbols
he is describing the process of assign-
will be distinguished by adding a supering a numerical social utility to each
script. Corresponding to the ordering
social state, the aim of society then
relation Ri, we have the (strict) prefbeing described by saying it seeks to
erence relation Pi and the indifference
maximize the social utility or social
relation Ii,. If the symbol for the orderwelfare subject to whatever technologi-
ing has a prime or second attached
cal or resource constraints are relevant,
(thus, R', R'), then the corresponding
or, put otherwise, it chooses the social
symbols for preference and indifference
state yielding the highest possible so-
will have the prime or second attached,
cial welfare within the environment.
respectively.
As with any type of behavior described
Similarly, society as a whole will be
by maximization, the measurability of
considered provisionally to have a sosocial welfare need not be assumed; all
cial ordering relation for alternative so-
that matters is the existence of a social
cial states, which will be designated by
ordering satisfying Axioms I and II.
R, sometimes with a prime or second.
As before, all that is needed to define
Social preference and indifference will
such an ordering is to know the rela-
'" A. C. Pigou, The Economics of Welfare
tive ranking of each pair of alterna-
(London: Macmillan & Co., I920), Part II, chap.
vi. For the analogy see Samuelson, Op. Cit., p. 224;
tives.
Reder, op. cit., pp. 64-67; G. Tintner, "A Note on
The relative ranking of a fixed pair
Welfare Economics," Econometrica, XIV (1946),
12 Bergson, op. cit.
69-78.
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THE CONCEPT OF SOCIAL WELFARE 335
of alternative social states will vary, in
otherwise, the whole social ordering re-
general, with changes in the values of
lation R is to be determined by the
at least some individuals; to assume
individual ordering relations for social
that the ranking does not change with
states, Rl,..., Rn. We do not exclude
any changes in individual values is to
here the possibility that some or all
assume, with traditional social philoso-
phy of the Platonic realist variety, that
of the choices between pairs of social
states made by society might be inde-
there exists an objective social good
pendent of the preferences of certain
defined independently of individual de-
particular individuals, just as a func-
sires. This social good, it was frequent-
ly held, could be best apprehended by
tion of several variables might be inde-
pendent of some of them.
the methods of philosophic inquiry.
Definition 3: By a "social welfare function"
Such a philosophy could be and was
will be meant a process or rule which, for each
used to justify government by elite,
set of individual orderings RI, . . . , R,, for al-
ternative social states (one ordering for each
secular or religious, although the con-
individual), states a corresponding social ordernection is not a necessary one.
ing of alternative social states, R.
To the nominalist temperament of
As a matter of notation, we shall let
the modern period the assumption of
R be the social ordering corresponding
the existence of the social ideal in some
to the set of individual orderings R1,
Platonic realm of being was meaning-
. .. , Rn, the correspondence being that
less. The utilitarian philosophy of Jere-
established by a given social welfare
my Bentham and his followers sought
function; if primes or seconds are
instead to ground the social good on the
added to the symbols for the individual
good of individuals. The hedonist psy-
chology associated with utilitarian phi-
orderings, primes or seconds will be
added to the symbol for the correlosophy was further used to imply that
sponding social ordering.
each individual's good was identical
with his desires. Hence, the social good
was in some sense to be a composite
of the desires of individuals. A view-
point of this type serves as a justifi-
cation of both political democracy and
laissez faire economics or at least an
economic system involving free choice
of goods by consumers and of occupa-
There is some difference between the
concept of social welfare function used
here and that employed by Bergson.
The individual orderings which enter
as arguments into the social welfare
function as defined here refer to the
values of individuals rather than to
their tastes. Bergson supposes indi-
vidual values to be such as to yield a
tions by workers.
The hedonist psychology finds its ex-
pression here in the assumption that
individuals' behavior is expressed by
individual ordering relations Ri. Utili-
tarian philosophy is expressed by say-
ing for each pair of social states that
the choice depends on the ordering re-
lations of all individuals, i.e., depends
on R1, . . . , R,,, where n is the number
of individuals in the community. Put
social value judgment leading to a par-
ticular rule for determining the allo-
cation of productive resources and the
distribution of leisure and final prod-
ucts in accordance with individual
tastes. In effect, the social welfare
function described here is a method
of choosing which social welfare func-
tion of the Bergson type will be ap-
plicable, though of course I do not ex-
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336 KENNETH J. ARROW
cdude the possibility that the social
choice actually arrived at will not be
the proof will be considerably simpli-
fied by considering only two.
consistent with the particular value
judgments formulated by Bergson. But
in the formal aspect the difference be-
tween the two definitions of social wel-
fare function is not too important. In
Bergson's treatment the tastes of indi-
viduals (each for his own consump-
tion) are represented by utility func-
tions, i.e., essentially by ordering re-
lations; hence, the Bergson social wel-
fare function is also a rule for assign-
ing to each set of individual orderings a
social ordering of social states. Further,
as already indicated, no sharp line can
be drawn between tastes and values.
A special type of social welfare func-
We shall not ask, in general, that the
social welfare function be defined for
every logically possible set of individual
orderings. On a priori grounds we may
suppose it known that preferences for
alternative social states are formed
only in a limited set of ways, and the
social welfare function need only be
defined for individual orderings formed
in those ways. For example, we may
suppose (and will later on) that each
individual orders social alternatives ac-
cording to his own personal consump-
tion under each (the purely individu-
alistic case). Then the social welfare
function need be defined only for those
tion would be one which assigns the
sets of individual orderings which are
same social ordering for every set of
admissible, in the sense of being con-
individual orderings. In this case, of
sistent with our a priori assumptions
course, social choices are completely
about the empirical possibilities.
independent of individual tastes, and
Condition I: The social welfare function is de-
we are back in the Platonic case.
fined for every admissible pair of individual or-
derings, RI, R,.
For simplicity of exposition, it will
be assumed that the society under
Condition i, it should be emphasized,
study contains only two individuals
is a restriction on the form of the social
and that the total number of alterna-
welfare function, since we are requiring
tives which are conceivable is three.
that for some sufficiently wide range of
Since the results to be obtained are
sets of individual orderings, the social
negative, the latter restriction is not a
welfare function give rise to a true so-
real one; if it turns out to be impossicial ordering.
ble to construct a social welfare func-
2. POSITIVE ASSOCIATION OF SOCIAL AND
tion which will define a social ordering
INDIVIDUAL VALUES
of three alternatives, it will a fortiori
be impossible to define one which will
order more alternatives. The restric-
Since we are trying to describe so-
cial "welfare" and not some sort of
tion to two individuals may be more
"illfare," we must assume that the so-
serious; it is conceivable that there
cial welfare function is such that the
may be suitable social welfare func-
social ordering responds positively to
tions which can be defined for three
individuals but not for two, for ex-
ample. In fact, this is not so, and the
alterations in individual values or at
least not negatively. Hence, we may
state the following condition:
results stated in this paper hold for
Condition 2: If an alternative social state x
any number of individuals. However,
rises or does not fall in the ordering of each
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THE CONCEPT OF SOCIAL WELFARE 337
individual without any other change in those
consider two sets of individual order-
orderings and if x was preferred to another alings such that, for each individual, his
ternative y before the change in individual
ordering of those particular alternatives
orderings, then x is still preferred to y.
under consideration is the same each
time, then we require that the choice
3. THE INDEPENDENCE OF IRRELEVANT
ALTERNATIVES
made by society be the same if indi-
vidual values are given by the first set
Just as for a single individual, the
of orderings as if they are given by the
choice made by society from any given
second.
set of alternatives should be independ-
Condition 3: Let RI, R2, and RI, R' be two
ent of the very existence of alterna-
sets of individual orderings. If, for both inditives outside the given set. For exam-
viduals i and for all x and y in a given set of al-
ple, suppose an election system has
ternatives S, xRjy if and only if xR'y, then the
been devised whereby each individual
social choice made from S is the same whether
lists all the candidates in order of his
the individual orderings are RI, R2, or RI, R'.
(Independence of irrelevant alternatives.)
preference, and then, by a preassigned
procedure, the winning candidate is
The reasonableness of this condition
derived from these lists. (All actual
can be seen by consideration of the
election procedures are of this type,
possible results in a method of choice
although in most the entire list is not
which does not satisfy Condition 3, the
required for the choice.) Suppose an
rank-order method of voting frequent-
election is held, with a certain number
ly used in clubs.'3 With a finite num-
of candidates in the field, each indi-
ber of candidates, let each individual
vidual filing his list of preferences, and
rank all his candidates, i.e., designate
then one of the candidates dies. Surely,
his first-choice candidate, second-choice
the social choice should be made by tak-
candidate, etc. Let preassigned weights
ing each of the individual's preference
be given first, second, etc., choices, the
lists, blotting out completely the dead
higher weight to the higher choice, and
candidate's name, and considering only
then let the candidate with the high-
the orderings of the remaining names in
est weighted sum of votes be elected.
going through the procedure of deter-
In particular, suppose there are three
mining the winner. That is, the choice
voters and four candidates, x, y, z, and
to be made among the set of surviving
w). Let the weights for first, second,
candidates should be independent of
third, and fourth choices be 4, 3, 2, and
the preferences of individuals for the
I, respectively. Suppose that individu-
nonsurviving candidates. To assume
als i and 2 rank the candidates in the
otherwise would be to make the result
order x, y, z, and w, while individual 3
of the election dependent on the obvi-
ranks them in the order z, w, x, and y.
ously accidental circumstance of wheth-
Under the given electoral system, x is
er a candidate died before or after the
chosen. Then, certainly, if y is deleted
date of polling. Therefore, we may re-
from the ranks of the candidates, the
quire of our social welfare function that
the choice made by society from a given
set of alternatives depend only on the
system applied to the remaining can-
didates should yield the same result,
orderings of individuals among those al"This example was suggested by a discussion
ternatives. Alternatively stated, if we
with G. E. Forsythe, National Bureau of Standards.
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338 KENNETH J. ARROW
especially since, in this case, y is in-
to be such as to prevent us, by its very
ferior to x according to the tastes of
definition, from expressing a preference
every individual; but, if y is in fact
for some given alternative over another.
deleted, the indicated electoral system
Definition 4: A social welfare function will be
would yield a tie between x and z.
said to be imposed if for some pair of distinct
The condition of the independence
alternatives x and y, xRy for any set of individu-
al orderings RI, R2, where R is the social orderof irrelevant alternatives implies that
ing corresponding to RI, R2.
in a generalized sense all methods of
social choice are of the type of voting.
In other words, when the social wel-
If S is the set consisting of the two
fare function is imposed, there is some
alternatives x and y, Condition 3 tells
pair of alternatives x and y such that
us that the choice between x and y is
the community can never express a
determined solely by the preferences of
preference for y over x no matter what
the members of the community as be-
the tastes of both individuals are, in-
tween x and y. That is, if we know
deed even if both individuals prefer y
which members of the community pre-
to x; some preferences are taboo. (Note
fer x to y, which are indifferent, and
that, by Definition i, asserting that
which prefer y to x, then we know what
xRy holds for all sets of individual
choice the community makes. Knowing
orderings is equivalent to asserting that
the social choices made in pairwise com-
yPx never holds.) We certainly wish
parisons in turn determines the entire
to require of our social welfare func-
social ordering and therewith the social
tion the condition that it not be im-
choice made from any set of alter-
posed in the sense of Definition 4; we
natives. Condition 2 guarantees that
certainly wish all choices to be possible
voting for a certain alternative has the
if unanimously desired by the group.
usual effect of making surer that that
Condition 4: The social welfare function is
alternative will be adopted.
not to be imposed.
Condition i says, in effect, that, as
Condition 4 is stronger than need be
the set of alternatives varies and indi-
for the present argument. Some devidual orderings remain fixed, the dif-
cisions, as between given pairs of alter-
ferent choices made shall bear a cer-
natives, may be assumed to be imposed.
tain type of consistent relation to one
All that is required really is that there
another. Conditions 2 and 3, on the
be a set S of three alternatives such
other hand, suppose a fixed set of alter-
that the choice between any pair is not
natives and say that for certain par-
constrained in advance by the social
ticular types of variation in individual
welfare function.
values, the various choices made have
It should also be noted that Condia certain type of consistency.
tion 4 excludes the Platonic case dis-
4. THE CONDITION OF CITIZENS'
cussed in section i of Part III above.
SOVEREIGNTY
It expresses fully the idea that all so-
We certainly wish to assume that the
individuals in our society be free to
cial choices are determined by individ-
ual desires. In conjunction with Con-
choose, by varying their values, among
dition 2 (which insures that the deter-
the alternatives available. That is, we
mination is in the direction of agreeing
do not wish our social welfare function
with individual desires), Condition 4
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THE CONCEPT OF SOCIAL WELFARE 339
Condition 5: The social welfare function is
expresses the same idea as Professor
not to be dictatorial (nondictatorship).
Bergson's Fundamental Value Propo-
sitions of Individual Preference, which
We have now imposed five apparent-
state that of two alternatives between
ly reasonable conditions on the con-
which all individuals but one are indif-
struction of a social welfare function.
ferent, the community will prefer one
over the other or be indifferent between
the two according as the one individual
prefers one over the other or is indif-
These conditions are of course value
judgments and could be called into
question; taken together, they express
the doctrines of citizens' sovereignty
ferent between the two.'4 Conditions 2
and rationality in a very general form,
and 4 together correspond to the usual
with the citizens being allowed to have
concept of consumers' sovereignty;
a wide range of values. The question
since we are here referring to values
is that of constructing a social order-
rather than to tastes, we might refer
ing of all conceivable alternative social
to them as expressing the idea of citi-
zens' sovereignty.
states from any given set of individual
orderings of those social states, the
method of construction being in accord-
5. THE CONDITION OF NONDICTATORSHIP
ance with the value judgments of citi-
A second form of social choice not of
zens' sovereignty and rationality as ex-
a collective character is the choice by
pressed in Conditions I-5.
dictatorship. In its pure form this
IV. THE POSSIBILITY THEOREM FOR
means that social choices are to be
SOCIAL WELFARE FUNCTIONS
based solely on the preferences of one
I. THE RANGE OF POSSIBLE INDIVIDUAL
man. That is, whenever the dictator
ORDERINGS
prefers x to y, so does society. If the
For simplicity we shall impose on the
dictator is indifferent between x and
individual preference scales two cony, presumably he will then leave the
ditions which in fact have almost inchoice up to some or all of the other
variably been assumed in works on
members of society.
welfare economics: (I) each individ-
Definition 5: A social welfare function is said
ual's comparison of two alternative soto be "dictatorial" if there exists an individual i
such that for all x and y, xPiy implies xPy re-
cial states depends only on the comn-
gardless of the orderings of all individuals other
modities that he receives (and labor
than i, where P is the social preference relation
that he expends) in the two states, i.e.,
corresponding to those orderings.
he is indifferent as between any two
Since we are interested in the consocial states in which his own consump-
struction of collective methods of social
tion-leisure-saving situations are the
choice, we wish to exclude dictatorial
same or at least indifferent to him;
social welfare functions.
(2) in comparing two personal situ-
" Bergson, op. cit., pp. 3 I8-20. The Fundamental
ations in one of which he receives at
Value Propositions of Individual Preference are
least as much of each commodity (in-
not, strictly speaking, implied by Conditions 2 and
cluding leisure and saving as commodi-
4 (in conjunction with Conditions i and 2), al-
though something very similar to them is so im-
ties) and more of at least one com-
plied; see Consequence I in Part IV, section 2
modity than in the other, the individual
below. A slightly stronger form of Condition 2
will prefer the first situation. Suppose
than that stated here would suffice to yield the
that among the possible alternatives
desired implication.
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340 KENNETH J. ARROW
Let the individuals be designated as i
there were three, none of which gave
and 2, and let R1 and R2 be the order-
any individual at least as much of both
ings by i and 2, respectively, of the
commodities as any other. For example,
alternatives x, y, z. Let P1 and P2 be
suppose that there are two individuals
the corresponding preference relations;
and a total of ten units of each of two
e.g., x'Piy' means that individual i
commodities. Consider three alternative
strictly prefers x' to y'.
distributions described by the accom-
panying tabulation. The individualistic
Consequence I: If x'Py' and xTP2y', then
x'Py'.
INDIVIDUAL I INDIVIDUAL 2
I.e., if both prefer x' to y', then so-
ciety must prefer x' to y'.
ALTERNATIVE
PROOF.-By Condition 4 there are
Com- Com- Com- Com-
modity i modify 2 modity i modity 2
orderings R' and R', for individuals
i and 2, respectively, such that, in the
I.......... 5 I 5 9
2 ......... 4 2 6 8
corresponding social preference, x'P'y'.
3 .3 3 7 7
Form R"' from R' by raising x', if
need be, to the top, while leaving the
restrictions imposed do not tell us any-
relative positions of the other two alter-
thing about the way either individual
natives alone; form R"' from R~ in the
orders these alternatives. Under the
same way. Since all we have done is
individualistic assumptions there is no
raise alternative x' in everyone's es-
a priori reason to suppose that the two
teem, while leaving the others alone,
individuals will not order the alterna-
x' should still be preferred to y' by so-
tives in any given way. In the sense of
ciety in accordance with Condition 2,
Part III, section i , above, all individual
so that x'P"y'. But, by construction,
orderings of the three alternatives are
both individuals prefer x' to y' in the
admissible. Condition i therefore re-
orderings R', R"', and society prefers
quires that the social welfare function
x' to y'. Since, by Condition 3, the so-
be defined for all pairs of individual
cial choice between x' and y' depends
orderings, R1, R2.
only on the individual orderings of
those two alternatives, it follows that
2. THE POSSIBILITY THEOREM
whenever both individuals prefer x' to
Some consequences will be drawn
y', regardless of the rank of the third
from Conditions I-5 for the present
alternative, society will prefer x' to y',
case of a social welfare function for
which is the statement to be proved.
two individuals and three alternatives.
Consequence 2: Suppose that for some x' and
It will be shown that the supposition
y', whenever x'PIY' and y'P2x', x'Py'. Then, for
that there is a social welfare function
that x' and y', whenever x'Py', x'Py'.
satisfying those conditions leads to a
I.e., if in a given choice, the will of
contradiction.
Let x, y, and z be the three alterna-
tives among which choice is to be made,
individual i prevails against the oppo-
sition of 2, then individual i's views
e.g., three possible distributions of com-
will certainly prevail if 2 is indifferent
modities. Let x' and y' be variable sym-
or if he agrees with i.
bols which represent possible alterna-
tives, i.e., range over the values x, y, z.
PROOF.-Let R1 be an ordering in
which x'Ply', R2 be any ordering. Let
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THE CONCEPT OF SOCIAL WELFARE 341
Suppose individual i prefers x to y
RI be the same ordering as R1, while
and y to z, while individual 2 prefers
RI is derived from R2 by depressing
x' to the bottom while leaving the rela-
y to z and z to x. Individual 2 then pre-
tive positions of the other two alter-
fers y to x. By (i) society prefers x to
natives unchanged. By construction,
y. Also, both prefer y to z; by Conse-
x'PIy', y'P'x'. By hypothesis, x'P'y',
quence i, society prefers y to z. Since
where P' is the social preference rela-
society prefers x to y and y to z, it must
tion derived from the individual order-
prefer x to z. Therefore, we have ex-
ings R', R'. Now the only difference
hibited orderings R1, R2 such that xPiz,
between R', R. and R1, R2 is that x'
zP2x, but xPz. Since the social choice
is raised in the scale of individual 2 in
between x and z depends only on the
the latter as compared with the former.
individual preferences for x and z,
Hence, by Condition 2 (interchanging
whenever xPIz and ZP2x, xPz . (2)
the R.'s and the R"s) it follows from
Now suppose R1 is the ordering y, x,
x'p'y' that x'Py'. I.e., whenever R1, R2
z, and R2 the ordering z, y, x. By Conare such that x'Piy', then x'Py'.
sequence i, yPx; by (2) xPz, so that
Consequence 3: If x'PIy' and y'Px', then
yPz. By the same reasoning as before,
x'Iy'.
whenever yP.z and zP2y, yPz . (3)
I.e., if the two individuals have exact-
ly opposing interests on the choice beIf R1 is the ordering y, z, x, and R2
tween two given alternatives, then sothe ordering z, x, y, it follows from
ciety will be indifferent between the
Consequence i and (3) that zPx and
alternatives.
yPz, so that yPx. Hence,
PROOF.-Suppose the consequence is
whenever yPIx and xP2y, yPx. (4)
false. Then, for some orderings R1 and
If R1 is the ordering z, y, x, and R2
R2 and for some pair of alternatives x'
the ordering x, z, y, then from Conseand y', we would have x'Piy', y'P2x',
quence i and (4), zPy and yPx, so that
but not x'Iy'. In that case, either x'Py'
zPx.
or y'Px'. We will suppose x'Py' and
show that this supposition leads to a
Whenever zPIx and xP,2z, zPx. (5)
contradiction; the same reasoning
If R1 is the ordering z, x, y, and R2
would show that the assumption y'Px'
x, y, Z, then, using (5), zPx and xPy,
also leads to a contradiction.
so that zPy.
Without loss of generality it can be
assumed that x' is the alternative x, y'
Whenever zPy and yP2z, zPy. (6)
y. Then we have, for the particular
From (i) it follows from Conse-
orderings in question, XPly, yP2x, and
quence 2 that whenever xPiy, xPy.
xPy. Since the social choice between
Similarly, from ( i ) to (6) it follows that
x and y depends, by Condition 3, only
for any pair of alternatives x', y', when-
on the individual choices as between
ever x'P1y', then x'Py'. That is, by
x and y, we must have
Definition 5, individual i would be a
whenever xPIy and yP2x, xPy. (I)
dictator. This is prohibited by Con-
It will be shown that (i) leads to a con-
tradiction.
dition 5, so that (i) must be false.
Therefore, Consequence 3 is proved.
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342 KENNETH J. ARROW
the market mechanism does not create
Now suppose individual i has the
a rational social choice.
ordering x, y, z, while individual 2 has
the ordering z, x, y. By Consequence i,
V. SOME IMPLICATIONS FOR THE
xPy. (7)
FORMATION OF SOCIAL WEL-
FARE JUDGMENTS
Since yPiz, ZP2y, it follows from Conse-
quence 3 that
I. INTERPRETATION OF THE POSSIBILITY
THEOREM
ylz. (8)
The interpretation of the Possibility
From (7) and (8), xPz. But, also xPiz,
Theorem is given by examination of the
ZP2x, which implies xlz by Consequence
meaning of Conditions I-5. In particu3. It cannot be that x is both preferred
lar, it is required that the social orderand indifferent to z. Hence the assump-
ing be formed from individual ordertion that there is a social welfare func-
ings and that the social decision be-
tion compatible with Conditions I-5
tween two alternatives be independent
has led to a contradiction.
of the desires of individuals involving
Put another way, if we assume that
any alternatives other than the given
our social welfare function satisfies
two (Conditions I and 3). These conConditions 2-3 and we further sup-
ditions taken together serve to exclude
pose that Condition i holds, then either
interpersonal comparison of social utilCondition 4 or Condition 5 must be
ity either by some form of direct meas-
violated. Condition 4 states that the so-
urement or by comparison with other
cial welfare function is not imposed;
alternative social states. Therefore, the
Condition 5 states that it is not dicta-
Possibility Theorem can be restated as
torial.
follows:
Possibility Thleorem.-If there are
If we exclude the possibility of inter-
at least three alternatives among which
personal comparisons of utility, then
the members of the society are free to
the only methods of passing from indi-
order in any way, then every social
vidual tastes to social preferences which
welfare function satisfying Conditions
will be satisfactory and which will be
2 and 3 and yielding a social ordering
defined for a wide range of sets of indi-
satisfying Axioms I and II must be
vidual orderings are either imposed or
either imposed or dictatorial.15 The
dictatorial.
Possibility Theorem shows that, if no
The word "satisfactory" in the fore-
prior assumptions are made about the
going statement means that the social
nature of individual orderings, there
welfare function does not reflect indi-
is no method of voting which will re-
viduals' desires negatively (Condition
move the paradox of voting discussed in
2) and that the resultant social tastes
Part I, neither plurality voting nor any
shall be represented by an ordering
scheme of proportional representation,
having the usual properties of ration-
no matter how complicated. Similarly,
ality ascribed to individual orderings
'5 The negative outcome expressed in this theorem
(Condition I and Axioms I and II).
is strongly reminiscent of the intransitivity of the
concept of domination in the theory of multiperson
In view of the interpretations placed
games; see John von Neumann and Oskar Morgenon the conditions for a social welfare
stern, Theory of Gamles and Economnic Behavior
function in Part III above, we can also
(2d ed.; Princeton University Press, I947), pp.
phrase the result this way: If con38-39.
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THE CONCEPT OF SOCIAL WELFARE 343
impose on individual tastes are those
sumers' values can be represented by a
implied by the individualistic assump-
wide range of individual orderings, the
tions, then, as we have seen, there is no
doctrine of voters' sovereignty is incom-
patible with that of collective ration-
satisfactory social welfare function pos-
ality.
sible when there is more than one com-
modity. Since, as we have seen, the only
If we wish to make social welfare
judgments which depend on all indi-
purpose of the determination of the
vidual values, i.e., are not imposed or
maximal states is as a preliminary to
the study of social welfare functions,
dictatorial, then we must relax some
the customary study of maximal states
of the conditions imposed. It will con-
tinue to be maintained that there is
under individualistic assumptions is
pointless. There is, however, a qualifi-
no meaningful interpersonal compari-
son of utilities and that the conditions
cation which should be added. It is con-
wrapped up in the word "satisfactory"
ceivable that, if further restrictions are
are to be accepted.16 The only condition
added to the individualistic ones, a so-
that remains to be eliminated is the one
cial welfare function will be possible.
stating that the method of forming a
Any state which is maximal under the
social ordering would work properly
combination of individualistic and other
restrictions will certainly be maximal
for a wide range of sets of individual
if only individualistic restrictions are
orderings. That is, it must be supposed
imposed on the individual orderings.
that it is known in advance that the
individual orderings R1, . . . , R. for
Hence, if the proper handling of the
social actions satisfy certain conditions
social welfare problem is deemed to be
more restrictive than those hitherto
the imposition of further restrictions
in addition to the individualistic ones,
introduced.
then the social maximum in any given
2. A REFLECTION ON THE NEW WELFARE
situation will be one of the maximal ele-
ECONOMICS
ments under the combined restrictions
As noted in Part I, the so-called "new
welfare economics" has concentrated on
and hence one of the maximal elements
under individualistic conditions. It is
the determination of the totality of so-
therefore not excluded that the current
cial states which have the property that
new welfare economics will be of some
any change which benefits one individ-
ual injures another-"maximal states"
in Lange's terminology. In particular,
this problem has usually been analyzed
under the assumption that individual
use in restricting the range in which we
must look for the social maximum.
The failure of purely individualistic
assumptions to lead to a well-defined
social welfare function means, in effect,
desires for social alternatives are
that there must be a divergence be-
formed in the individualistic way de-
tween social and private benefits if we
scribed above in Part IV, section i. But
are to be able to discuss a social opti-
if the only restrictions that we wish to
mum. Part of each individual's value
system must be a scheme of socio-
16 The only part of the last-named conditions
ethical norms, the realization of which
that seems to me to be at all in dispute is the as-
sumption of rationality. The consequences of
cannot, by their nature, be achieved
dropping this assumption are so radical that it
through atomistic market behavior.
seems worth while to explore the consequences of
These norms, further, must be suffimaintaining it.
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34-4 KENNETH J. ARROW
outputs s and t, respectively, and ap-
ciently similar among the members of
portionments s' and t', respectively, to
the society to avoid the difficulties out-
the given individual. If s> t, but s'<t',
lined above.
then society prefers x to y, while the
3. A ONE-COMMODITY WORLD
individual prefers y to x.
The insufficiency of the individualis-
The qualitative nature of the differ-
tic hypotheses to permit the formation
ence between the single- and multicom-
of a social welfare function, as devel-
modity cases makes any welfare argu-
oped in the previous sections, hinged
ments based on an implicit assumption
on the assumption that there was more
of a single commodity dubious in its
than one commodity involved. An in-
applicability to real situations. The fun-
vestigation of the one-commodity case
damental difficulty is that, in a world
may be of interest to bring out more
of more than one commodity, there is
clearly the issues involved.
no unequivocal meaning to comparing
In a one-commodity world, if we
total production in any two social states
make assumptions i and 2 of Part IV,
save in terms of some standard of value
section i, there is for any given indi-
to make the different commodities comvidual only one possible ordering of the
mensurable; and, usually such a stand-
social states. He orders various social
ard of value must depend on the distri-
states solely according to the amount
bution of income. In other words, there
of the one commodity he gets under
is no meaning to total output independ-
each. In such a situation the individual
ent of distribution, i.e., of ethical judg-
orderings are not variables; Conditions
ments.
2, 3, and 4 become irrelevant, since
they relate to the variation in the social
4. DISTRIBUTIONAL ETHICS COMBINED
WITH INDIVIDUALISM
ordering corresponding to certain speci-
fied types of changes in the individual
We may examine briefly a set of
orderings. Condition 5 (nondictator-
assumptions about individual values
ship) becomes a much weaker restric-
which seem to be common to those
tion, though not completely irrelevant.
who feel that the new welfare econom-
Any specification of a social ordering
ics is applicable in a fairly direct way
which does not coincide completely
with the ordering of any one individual
will be a social welfare function com-
patible with all the conditions. For ex-
to the solution of specific economic
problems. It is assumed that there are
(i) an accepted (let us say, unani-
mously accepted) value judgment that
ample, for each fixed total output, we
if everybody is better off (more pre-
might set up arbitrarily an ordering of
cisely, if everybody is at least as well
the various distributions; then order
any two social states with different total
off and one person better off) in one
social state than another according to
outputs in accordance with the total
his tastes, then the first social state is
output, any two social states with the
preferred to the second; and (2) a uni-
same total output according to the arbi-
versally accepted ordering of different
trary ordering. This sets up a genuine
possible welfare distributions in any
weak ordering which does not coincide
given situation. The latter value judg-
with the ordering of any one individual.
ment usually takes an egalitarian form.
For let x and y be two states with total
This ethical schema is quite explicit
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THE CONCEPT OF SOCIAL WELFARE 345
in the work of Bergson; the second
value judgment is contained in his
Propositions of Relative Shares.17 The
same set of ethics underlies the com-
pensation principle of Professors Kal-
dor and Hicks. More recently, some
proposals made by Professors Johnson
and Modigliani for meeting the prob-
lem of the increased cost of food due to
European demand seem to have been
individual will vary with variations in
the tastes of all. Our previous argu-
ments as to the nonexistence of social
welfare functions were based on the
diversity of values; do they carry over
to this particular kind of unanimity?
The actual distribution of welfare
dictated by the second value judgment
cannot be stated simply in money
terms. As Professor Samuelson points
based on value judgments i and 2
out, such a value judgment is not con-
above.'8 To prevent the inequitable
sistent with any well-defined social or-
shift in real income to farmers, it was
proposed that there should be imposed
dering of alternative social states.'9 The
distribution of real income, for a given
an excise tax on food, accompanied by a
environment, must vary with individual
per capita subsidy to consumers. Under
tastes. Thus, for a given set of individ-
the assumption that the supply of agriual tastes (as represented by the order-
cultural goods is completely inelastic,
the tax would be absorbed by the farm-
ing relations of all individuals, each for
his own consumption) and a given en-
ers while the subsidy would have no
vironment, there is a given distribution
substitution effects at the margin, so
of purchasing power (somehow de-
that the marginal rate of substitution
fined); then exchange under perfectly
for any pair of commodities would be
competitive conditions proceeds until
the same for all consumers and hence
an optimum distribution is reached.
the first value judgment would be ful-
The given distribution of real income
filled. The taxes and subsidies perform
and the individual tastes uniquely de-
a purely distributive function and can
termine the final outcome, which is a
be so arranged as to restore the status
social state. Therefore, the given ethical
quo ante as near as may be, though
system is a rule which selects a social
actually the payment of a per capita
state as the choice from a given collec-
subsidy implies a certain equalizing
tion of alternative distributions of
effect.
goods as a function of the tastes of all
The value judgments are assumed
individuals. If, for a given set of tastes,
here to hold for any individual. Note
the range of social alternatives varies,
that even to state these judgments we
we expect that the choices will be con-
must distinguish sharply between valsistent in the sense that the choice func-
ues and tastes (see Part II, sec. 2). All
tion is derivable from a social weak or-
individuals are assumed to have the
dering of all social states. Thus, the
same values at any given instant of
time, but the values held by any one
ethical scheme discussed in this section,
which we may term the "Bergson social
welfare function," has the form of a
7 Bergson, op. cit.
rule assigning a social ordering to each
"8D. G. Johnson, "The High Cost of Food-a
Suggested Solution," Journal of Political Economy,
possible set of individual orderings rep-
LVI (I948), 54-57; Modigliani's proposals are con-
resenting tastes. Mathematically, the
tained in a press release of the Institute of World
Affairs, New York, October, 1948.
' Samuelson, Op. Cit., p. 225.
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346 KENNETH J. ARROW
Bergson social welfare function has,
then, the same form as the social wel-
fare function we have already dis-
ful than the actual practice of individ-
ualism in permitting the formation of
social welfare judgments.
It must of course be recognized that
cussed; though, of course, the inter-
pretation is somewhat different, in that
the meaning of Conditions 2-5 has
the individual orderings represent
changed. The previous arguments for
tastes rather than values and that the
their validity assumed that the individ-
whole function is the end product of
ual orderings represented values rather
certain values assumed to be unani-
than tastes. It seems obvious that Con-
mously held rather than a method of
ditions 2, 4, and 5 have the same intrin-
reconciling divergent value systems. If
sic desirability under either interpreta-
the range of tastes is not restricted by
tion. Condition 3 is perhaps more
a priori considerations (except that
doubtful. Suppose there are just two
they must be truly tastes, i.e., refer
commodities, bread and wine. A distri-
only to an individual's own consump-
bution, deemed equitable by all, is
tion, however that may be defined),
arranged, with the wine-lovers getting
then, indeed, the Bergson social welfare
more wine and less bread than the ab-
function is mathematically isomorphic
stainers get. Suppose now that all the
to the social welfare function under in-
wine is destroyed. Are the wine-lovers
dividualistic assumptions. Hence the
entitled, because of that fact, to more
Possibility Theorem is applicable here;
we cannot construct a Bergson social
welfare function, i.e., cannot satisfy
value judgments i and 2, which will
satisfy Conditions 2-5 and which will
than an equal share of bread? The an-
swer is, of course, a value judgment.
My own feeling is that tastes for un-
attainable alternatives should have
nothing to do with the decision among
yield a true social ordering for every set
the attainable ones; desires in conflict
of individual tastes. Essentially, the
with reality are not entitled to consid-
two value judgments amount to erect-
eration, so that Condition 3, reinter-
ing individualistic behavior into a value
preted in terms of tastes rather than of
judgment. It is not surprising, then,
that such ethics can be no more success-
values, is a valid value judgment, to me
at least.
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