-4 DIFFICULTY IN THE CONCEPT OF SOCIAL WELFARE

KENNETH J. ARROW^ Stanford University

I. INTRODUCTION cia1 choice in any given situation, e.g., N A capitalist democracy there are a religious code. essentially two methods by which The last two methods of social choice, I social choices can be made: , dictatorship and convention, have in typically used to make "political" de- their formal structure a certain definite- cisions, and the market mechanism, ness absent from voting or the market typically used to make "economic" de- mechanism. In an ideal dictatorship, cisions. In the emerging democracies there is but one will involved in choice; with mixed economic systems, Great in an ideal society ruled by convention, Britain, France, and Scandinavia, the there is but the divine will or perhaps, same two modes of making social by assumption, a common will of all choices prevail, though more scope is individuals concerning social decisions, given to the method of voting and to so that in either case no conflict of indi- decisions based directly or indirectly vidual wills is involved. The methods on it and less to the rule of the price of voting and of the market, on the mechanism. Elsewhere in the world, other hand, are methods of amalga- 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 individ- are, 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 f choice, of traditional rules for making the so- where the wills of many peopk are in- 'This paper is based on research carried on at volved? the RAND- Corporation, a project of the United It should be emphasized here that States Air Force, and at the Cowles Commission the present study is concerned for Research in Economics and is part of a longer studv. "Social Choice and Individual Values." to with the formal aspects of the foregoing be published by John Wiley & Sons as a Cowles question. That is, we ask if it is for- Commission monograph. A version was read at the mally to construct a procedure December, 1948, meeting of the Econometric So- ciety. I am indebted to A. Kaplan, University of passing a set indi-

California at Los An~eles,and 1. W. T. Younss.-, - . vidual tastes to- a Latt tern~ of~ social~ de- University of Indiana, for guidance in formulating cision-making, the procedure in ques- the problem, and to A. Bergson and A. G. Hart, , and T. C. Koopmans, Cowles tion being required to certain

Commission and the Universitv of Chicago.u, who natural conditions. An illustration of have read the manuscript and made valuable com- the problem is the following well-known ments on both the presentation and the meaning. Needless to say, any error or opacity remaining is " of voting'" Suppose there is a the responsibility of the author. community consisting of three voters THE COKCEPT OF SOCIAL ITELFARE 329 and this con~munitymust choose among iinply rational behavior on the pa~tof three alternative modes of social action the com~llunityand which will be satis- (e.g., disarmament, cold war, or hot factory in other ways?" war). It is expected that choices of this 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 wiil be available. In analogy with achieving a social maximum derived the usual utility analysis of the indi- from individual desires is precisely tile vidual consumer under conditions of problem which has been central to the constant wants and variable price-in- field of .~owever, come situations, rational behavior on the search for a clear definition of opti- the part of tile connmunity would mean mum social welfare has been plagued that the con~lllunityorders the three by the difficulties of interpersoial 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 anlong those actually available nation of all social states such that no which stands highest on this list. A individual can be made better olf with- natural way of arriving at the collec- out making someone worse OH. As Pro- tive preference scale would be to say fessors Bergson, Lange, and Sai~~uelson that one alternative is preferred to have argued, though, the weaker defini- another if a majority of the corninurlity tion cannot be used as a guide to social prefer the first alternative to the sec- policy; the second type of welfare eco- ond, i.e., would ciloose the first over the nomics is only important as a preii~ni- second if those were the only two alter- natives. Let "1, B, and C be the three 'It may be added that the method of decision sketched above is essentially tint used in deiiber- I, 2, alternati~res,and and 3 the three ative bodies, where a whole ranse of ~:!ternati\-es individuals. Suppose individual I pre- usually comes up for decision in the form of suc- ftrs -4 to rF; and B to C (and therefore cessive pairnise comparisons. The phenomenon de- scribed in the test can be seen in a pure for~ilin A to C), individual 2 prefers 13 to C the disposition of the proposals before recent Con- and C to '1 (ai:d therefore B to A), and gresses for federal aid to state education. the three individual 3 prefers C to A a~ldA to B alternatives being no federai aid, federal aid to pub- (and therefore C to 23). Then a major- lic schools only, and federal aid lo both public and parochial schools. ity prefers il to R, and a majority pre- The problem of collective rationality has been fcrs B to C. lye may therefore say that discussed by Professor Frank H. Knight, but chieily the comm~~nityprefers A to B and B to in terms of the socio-psychological prerecjuisiies: C. If the community is to be regarded see "The Planful Act: The Possibilities and Limita- tioils of Collective Rationality," in Frcrdoilz and as behaving rationally, we are forced to R~forl7z(New York: Harper & Bros., 194;), pp. say that il is preferred to C. But, in 335-69, esp. PP. 346-65). fact, a majority of the community pre- 'See P. -4. Samuelzon, For~nrlatiotzsoj Eco~zoi~zZ'c fers C to A.2 So the method just out- Analysis (Cambridge, Mass.: Press, rgd~),chap. viii; .4. Bergson (Burk), ".4 lined for passing from individual to col- Reiormulation of Certain .\specis of LVeliare Eco- lective tastes fails to satisfy the condi- nomics," Quarterly Joz~r;zal of Econo~rlics, LI1 tion of rationality as we ordinarily un- (~gjS),310-34; 0. Lange, "The Foundations of derstand it. Can we find other methods Welfare Economics," Econo:?z~trica,X (1942)) 21 j-28; M. W. Reder. Sttldies iw the Theory of of aggregating individual tastes which Welfare Economics (New York, 1947)) chaps. i-v. nary to the determination of a genuine conzpensation 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 acceptance of this prin~iple,~there is a an excise tax imposed on one commod- formal difficulty which was pointed out ity in the initial situation, it can be by Professor Scito~szky:~it is possible argued that the removal of the tax ac- that simultaneously x should be pre- corllpanied by a suitable redistribution 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 choice. Unless the trouble-breeding in- the status quo is to be sanctified on dividual preference patterns can be ethical grounds. All we can really say is 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 The aim of the present paper is to make its choice among attainable in- show that these difficulties are general. come distributions, i.e., a social indif- For any method of deriving social ference map. choice; by aggrega'ciilg iildividLia1 pref- 1-oting can be regarded as a method erence patterns which satisfies certain of arriving at social choices derived natural conditions, it is possible to find from the preferences of individuals. individual preference patterns which Another such method of more specifi- give rise to a social choice pattern cally econonlic content is the compensa- 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:' 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.s It is assumed that individuals act from making the compensations under rationally, in the sense that their be- .T than they are in state y, then x should 'See W. J. Baumol, "Community Indifference," be chosen in preference to y, even if the Revierti of Econonzic Studies, XIV (1946-471, W. Kaldor, "We!fare Propositions of Economics 41-48. and Interpersonal Comparisons of Utility," Eco- 'T. Scitovszky, ",4 Note on Welfare Proposi- notflit Journal, XLIX (19391, j4g-652; see also tions in Economics," Revim of Econonzic Studies, J. R. Hicks, "The Foundations of UTelfare Eco- 1x (1942), 77-88. nomics," Econonzic Journal, XLIX (19391, 698- See, e.g., Samuelson, op. cit., pp. 222-24; Berg- ;OI and 711-12. son, op. cit., pp. 318-20; Lange, op. cit., p. 216. havior in alternative situations can be preferred to x. The decisions made for described by an indifference map. It is differentpairs are assumed to be consist- further assumed that utility is not ent with one another, so that, for ex- measurable in any sense relevant to ample, if x is preferred to y and y to z, weliare 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 dil'iere~lcemap. all possible alternatives, the chooser is now confronted with a particular op- 11. DEFINITIONS AND NOTATION portunity set S. If there is one alterna- I. il hO1ATIO.U FOR PKEFCKCNCcS AND CHOICE tive in S which is preferred to all others In this paper I shall be interested in in S, then the chooser selects that one the description of preference patterns alternati~e.~ both for the individual and for society. Preference and indifierence are rela- It will be found convenient to represent tions between alternatives. Instead of preference by a notation not customar- working with two relations, it will be ily empioyed in economics, though fa- slightly more convenient to use a single miiiar in mathematics and particularly relation, "preferred or indifferent." The in symbolic logic. We assume that there statement, "x is preferred or indifferent is a basic set of alternatives which to y," will be symbolized by xRy. The could conceivably be presented to the letter R, by itself, will be the name of chooser. In the theory of consumers' the relation and will stand for a knowl- choice, each alternative would be a edge of all pairs such that xRy. From commodity bundle; in the theory of the our previous discussion, we have, for firm, each alternative would be a conl- any pair of alternatives x and y, either plete decision on all inputs and outputs; that x is preferred to y or y to x or that in welfare econon~ics,each alternative the two are indifferent. That is, we have would be a distribution of commodities assumed that any two alternatives are and labor requirements. These alterna- comparable. But this assunlption may tives are inutually exclusive; they are be written symbolically, denoted by small letters, x, y, z. . . . On Axio~rzI: For all x and y, either xRy or yRx. any given occasion the chooser has available to him a subset S of all pos- Xote that Axiom I is presumed to hold sible alternatives, and he is required to when x = y, as well as when x is dis- choose one out of this set. The set S is tinct from y, for we ordinarily say that a generaiization of the well-known op- x is indifferent to itself for any x, and portunity curve; thus, in the theory of this i~zlplies xRx. Kote also that the consumer's choice under perfect compe- 'It may be that there is a subset of alternatiies in S, such that the alternatives in the subset are tition, it would be the budget plane. each preferred to exery alternative not in the subset. It is assunled further that the choice is while the alternati~esin the subset are indifferent made in this way: Before knowing the to one another. This case would be one in which the hichest 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 aiternatives, say x and two points in common with it (the well-known case y, and for each pair he makes one and of multiple maxima). In this case, the best thing to say is that the choice made in S is the whole subset; only one of three decisions: x is pre- the first case discussed is one in which the subset ferred to y, x is indifferent to y, or y is in questioi~,the choice, contains a single element. word '.or7' 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 xKy 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. Leiiunu: a) For ail x, xRx. The property mentioned above of b) If xPy, then xRy. consistency in the preferences as be- c) If xPy and yPz, then xPz. tween different pairs of alternatives d) If xIy and ylz, then xIz. e) For a11 x and y, either xRy or inay be stated more precisely, as fol- yPx. lows: if x is preferred or indifferent to J) If xPy and yRz, then xPz. y and y is preferred or indifferent to z, ,411 these statements are intuitively self- then x must be either preferred or in- evident from the interpretations placed diiierent to z. In syn~bols, on the symbols. drroin 11: For all x, y, and z, xRy and yKz For clarity, we shall avoid the use of imply xRz. the terms "preference scale" or "pref- X relation satisfying both Axiom I and erence pattern" when referring to £2, Xxionl I1 is termed a weak ordering or since we wish to avoid coilfusion with sonletinies simply an ordering. It is the concept of preference proper, de- clear that a relation having these two noted by P. We shall refer to R as an properties taken together does create "ordering relation7' or "weak ordering a ranking of the various alternatives. relation7'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- ,'preference relation7' will refer to the difference, i.e., Axioms I and I1 do not relation P. exclude the possibility that for some Suppose that we know the choice distinct x and y, both xRy and yRx. which would bc made from any given It might be held that the two axioms pair of alternatives; i.e., given two in question do not completely cliarac- alternatives x and y from which the terize the concept of a preference pat- chooser must select, we know whether tern. For example, we ordinarily feel he would take x or y or remain indif- that not only the relation R but also ferent between them. Since choosing x the relations of (strict) preference and from the pair x, y imp!ies that x is pre- of indifference satisfy Axiom 11. It can ferred to y, and similarly ~~itha choice be shown that, by defining preference of y, a kno.vvledge of the choice which would be made froin any two given a;ld 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 Dcfi~utiol~I: XI'^ is defined to mean not yRx. made from any set of alternatives actu- The statenlent "xPy" is read, "x is pre- ally available. Hence, one of the conse- ferred to y." quences of the assumption of rational behavior is that the choice froill any DcJinitio?l2: sly means xRy and yRx. col!ection of alternatives can be deter- The statement "xly" is read, "x is in- mined by a knowledge of the choices which would be made fro111 pairs of analysis if the generality were re- alternatives. stricted by a prior linowledge of the nature of individual orderings of social 2. THE ORDLRISG OF SOC! \L STATES states. This problem will be touched on In the present study the objects of again. choice are social states. Tile most pre- In general, then, there will be a dif- cise definition of a social state would be ference between the ordering of social a co~npletedescription 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) .I0 vested in each type of productive activ- Ke 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 vtrlz~es. services, diplomacy ar~dits cor:tinuation The distinction be:ween the two is by other means, and the erection of by no means clear cut. An individual statues to famous man. It is assumed with aesthetic feelings certzinly derives that each individual in the comn~rnity pleasure from his neighbor's havi~lga has a definite ordering of all co:~ceiva- well-tended la~vn.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 socizl choice; yet, psy- sumed here that an individual's atti- chologically, they differ only slightly tude toward different social states is de- fro:n the pleasure in one's own lawn. termined exclusive!y by the cominodity I:.,tu:tively, of course, we feel that not bundles which accrue to his lot under ail the possible preferences which an each. The individual may order all so- individual might have ought to count; cial states by ~vhateverstandards he his preferences for matters which are deems relevant. .A member of Veblen's "none oi his business" should be ir- leisure class might order the states sole- releva!~t.\ri*ithout challenging this view, ly on the criterion of his relative in- 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 Inan might order them in vant ar?d rrhicli are not is itself a value accordance with some measure of in- judgment and cancot 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 his extreme distaste for the existence analyzing such problenls as the division of heathecism in Central Africa. There of the national income between public are proSably not a few individuals in and private expenditi~re.The present this country who mould regard the for- notation permits perfect generality in mer feeling as irrelevant for social pol- this respect. Seedless to say, this gen- icy and the latter as relevant, though erality is not without its price. Illore This distinction has been stressed to the author information would be available for by M. Friedman, University of Chicago. J. ARROW the majority would probably reverse be denoted by P and I, respectively, the judgment. I merely wish to empha- primes or seconds being attached when size here that we must look at the en- they are attached to the relation R, tire system of values, including values respectively. about values, in seeking for a truly Throughout this analysis, it will be general theory of social welfare. assumed that individuals are rational, It is the ordering according to values by which is meant that the ordering re- which takes into account all the desires lations R, satisfy Axioms I and 11. The of the individual, including the highly problem will be to construct an order- important socializing desires, and which ing relation for society as a whole which is primarily relevant for the achieve- is also to reflect rational choice-making, ment of a social maximum. The mar- so that R also will be assumed to satis- ket mechanism, however, takes into ac- fy ,Axioms I and 11. count only the ordering according to tastes. This distinction is the analogue, 111. TIIE on the side of consumption, of the di- I. FORhlAL STATELTEXT OF TIIE PROBLEM vergence between social and private OF SOCIiL CHOICE costs in production which has been de- I shall largely restate Bergson's for- veloped by Professor Pigou." nlulation of the problern of making .Is for notation, let R, be the order- welfare judgments1' 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 or- components 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 super- ing a numerical social utility to each script. Corresponding to the ordering social state, the aim of society then relation R,,we have the (strict) pref- being described by saying it seeks to erence relation P, and the indifference maximize the social utility or social relation I,. If the symbol for the order- welfare subject to whatever technologi- ing has a prime or second attached cal or resource constraints are relevant, (thus, R:,RJ, 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. t\s with any type of behavior described Similarly, society as a whole will be by maximization, the measurability of considered provisionally to have a so- social 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 11. R, sometimes with a prime or second. -4s before, all that is needed to define Social preference and indifference will such an ordering is to know the rela- ''A. C. Pigou, The Econo~nic~of D'alfav (London: Mactnillan & Co. rgzo), Palt 11, chap tive ranking of each pair of alterna- \i For the analogy see Samuelson, op. fit, p 224 ; tives. Reder, op cit., pp. 64-67; G Tintner, "A Kote on The relative ranking of a fixed pair Welfare Economics," Econonzatvicn, XIV ( 1946), 69-78, '' Bctgson. op. cit. THE COXCEPT OF SOCIAL 'CI'ELFARE 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 individuaI ordering relations for socia1 that the ranking does not change with states, XI,. . . , R,. We do not exclude any changes in individual values is to here the possibility that some or all assume, with traditional social philoso- of the choices between pairs of social phy of the Platonic realist variety, that 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- tion of several variables might be inde- ly held, couid be best apprehended by pendent of some of them. tile meihods of philosophic inquiry. Dcfi~zitiott3: By a "social welfarc 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 indiviclual ordcrings R,,. . . , R, for al- secular or religious, although the con- ternative social states (one ordering for each nection is not a necessary one. individual), states a corresponding social order- To the nominalist temperament of ing of alternative social states, R. the modern period the assun~ptionof As a matter of notation, we shall let the existence of the social ideal in soille R be the social ordering corresponding Platonic realm of being was meaning- to the set of individual orderings RI, less. The utilitarian philosophy of Jere- . . . , X,,the correspondence being that my Bentham and his followers sought established by a given social welfare instead to ground the social good on the function; if primes or seconds are good of individuals. The hedonist psy- added to the syn~bolsfor the individual chology associated with utilitarian phi- orderings, primes or seconds will be losophy was further used to imply that added to the symbol for the corre- each individual's good was identical sponding social ordering. with his desires. Hence, the social good There is some difference between the was in some sense to be a composite concept of social welfare function used of the desires of individuals. A view- here and that employed by Bergson. point of this type serves as a justifi- The individual orderings which enter cation of both political democracy and as arguments into the social welfare laissez laire economics or at least an function as defined here refer to the economic system involvi~gfree choice values of individuals rather than to of goods by consumers and of occupa- their tastes. Bergson supposes indi- tions by workers. vidual values to be such as to yield a The hedonist psychology finds its ex- social value judgment leading to a par- pression here in the assumption that ticular rule for determining the allo- individuals' behavior is expressed by cation of productive resources and the individual ordering relations R,.Utili- distribution of leisure and final prod- tarian philosophy is expressed by say- ucts in accordance with individual ing for each pair of social states that tastes. In effect, the social welfare the choice depends on the ordering re- function described here is a method lations of all individuals, i.e., depends of choosing which social welfare func- on RI,. . . , I?,,, where n is the number tion of the Bergson type will be ap- of individuals in the community. Put plicable, though of course I do not ex- 336 KENNETH J. ARROW clude the possibility that the social the proof will be considerably simpli- choice actually arrived at will not be fied by considering only two. consistent with the particular value We shall not ask, in general, that the judgments formulated by Bergson. But social welfare function be defined for in the formal aspect the difference be- every logically possible set of individual tween the two definitions of social wel- orderings. On a priori grounds we may fare function is not too important. In suppose it known that preferences for Bergson's treatment the tastes of indi- alternative social states are formed viduals (each for his own consump- only in a limited set of ways, and the tion) are represented by utility func- social welfare function need only be tions, i.e., essentially by ordering re- defined for individual orderings formed lations; hence, the Bergson social wel- in those ways. For example, we may fare function is also a rule for assign- suppose (and will later on) that each ing to each set of individual orderings a individual orders social alternatives ac- social ordering of social states. Further, cording to his own personal consump- as already indicated, no sharp line can tion under each (the purely individu- be drawn between tastes and values. alistic case). Then the social welfare A special type of social welfare func- 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 Conditiort I: The social welfare function is de- we are back in the Platonic case. fined for every admissible pair of individual or- For simplicity of exposition, it will dering~,RI,R,. 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 impossi- cial ordering. ble to construct a social welfare func- tion which will define a social ordering 2. POSITIVE ASSOCIATION OF SOCIAL AND of three alternatives, it will a fortiori INDIVIDUAL VALUES be impossible to define one which will Since we are trying to describe so- order more alternatives. The restric- cial "welfare7' 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 alterations in individual values or at individuals but not for two, for ex- least not negatively. Hence, we may ample. In fact, this is not so, and the state the following condition : results stated in this paper hold for Cofzditiolz 2: If an alternative social state x any number of individuals. However, rises or does not fall in the ordering of each THE CONCEPT OF SOCIAL WELFARE 337 individual without any other change in those consider two sets of individual order- orderings and if x ~vaspreferred to another al- ing~such that, for each individual, his ternative y berore the change in iiidividual ordering of those particular alternatives orderings. then zis still preferred to y. under consideration is the same each

3 TIIE INDtPEKDESCE OF IRREJAE\'AST time, then we require that the choice ALTERNATIL'ES made by society be the same if indi- Just as for a single individual, the vidual values are given by the first set choice made by society from any given of orderings as if they are given by the set of alternatives should be independ- second. ent of the very existence of alterna- Condifion 3: Let R,, R,, and R:, R: he two tives outside the given set. For exam- sets of individual orderings. If, for both indi- viduals i and for all z and in a given s~tof al- ple, suppose an e!ection system has ternatives S, xRiy if and only if XR:?),then the been devised whereby each individual social choice made from S is the sanle whether lists all the candidates in order of his the indivitiual orderings are R,, R,, or R:, R:. preference, and then, by a preassigned (Indepentlcnce of irrelevant alternatives.) procedure, the winning candidate is The reasonableness of this condition derived from these lists. (-411 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.13 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 ~~eightedsum 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 surviviilg a. Let the weights for first, second, candidates should be independent of third, acd fourth choices be 4, 3, 2, and the preferences of individuals for the nonsurviving candidates. To assume I, respectively. Suppose that individu- otherwise mould be to make the result als I and 2 rank the candidates in the of the election dependent on the obvi- order r,y, z, and w,while individual 3 ously accidental circumstance of wheth- ranks them in the order z, w,x, and y. er a candidate died before or after the TJnder the given , x is date of polling. Therefore, we may re- chosen. Then, certainly, if y is deleted quire of our social welfare function that from the ranks of the candidates, the the choice made by socicty from a given system applied to the remaining can- set of alternatives depend only on the didates should yield the same result, orderi~gsof individuals among those al- ?:I This examplc was sucrgcstrd by a discussion teriiatives. -Alternatively stated. if we with G. E. Forsythc, National Bureau ol Stanria!.tls. 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 DcJxition 4: X social wclfarc function will he would yield a tie between x and z. said to bc inaposed if for some pair of distinct The condition of the independence alternatives x and v, .rRy for any set of individu- of irrelevant alternatives implies that al orderings R,, R,, wllere R is tlie social order- in a generalized sense all methods of ing corresponding to RI,R,. 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 indi:yiduals 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; 117e 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 alternative will be adopted. Colzdition 4: The social welfare function is not to be imposed. Condition I says, in effect, that, as the set of alternatives varies and indi- Condition 4 is stronger than need be vidual orderings remain fixed, the dif- for the present argument. Some de- ferent choices made shall bear a cer- cisions, as between given pairs of alter- tain type of consistent relation to one natives, may be assumed to be imposed. another. Conditions 2 and 3, on the All that is required really is that there other hand, suppose a fixed set of alter- be a set S of three alternatives such natives and say that for certain par- that the choice between any pair is not ticular types of variation in individual constrained in advance by the social values, the various choices made have welfare function. a certain type of consistency. It should also be noted that Condi- tion 4 excludes the Platonic case dis- 4. THE CONDITION OF CITIZENS' cussed in section I of Part I11 above. SOVEREIGNTY It expresses fully the idea that all so- We certainly wish to assume that the cial choices are determined by individ- individuals in our society be free to 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 'IIJE COSCEPT OF SOCIAL IVELFARE 339 expresses the same idea as Professor Co~zditio?~5: The social welfare functio~lis R ~ Fundamental~ value~ proPo-~ not~ to be dictatorial~ (nondictatorship).' ~ sitions ol 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 These conditions are of course value over the other or be indifferent between judgments and could be called into the two according as the one individual question; taken together, they express prefers one over the other or is indif- the doctrines of citizens' sovereignty ferent between the two.'Vonditions 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 questioil 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- states from any given set of individual zens' sovereignty. orderings of those social states, the method of construction being in accord- j. THE CONDITION OF N0XDICT.ATORSHIP 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 1-5. dictatorship. In its pure form this means that social choices are to be IV. THE POSSIBILITY THEOREM FOR based solely on the preferences of one SOCIAL WELFARE FUNCTIONS man. That is, whenever the dictator I. THE RANGE OF POSSIBLE IXDIVIDUAL prefers x to y, so does society. If the ORDERINGS dictator is indifferent between x: and For simplicity we shall impose on the y, presumably he will then leave the individual preference scales two con- choice up to some or all of the other ditions which in fact have alnlost in- members of society. variably been assunled in worlis oil Defiilitioll 5: :Isocial ~vclfarefunction is said welfare economics: (I) each iadivid- to I,~utlictatorial~if an individual i ual's comparison of two alterrative so- such that for all x and y, x~,?implies XP~re- cial states depends only on the ~0111- gardless of the ortlerings 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., correspondi~lgto those orderings. he is indifferent as between any two Since we are interested in the con- social states in which his own consump- structior. of collective methods of social tion-leisure-savi~~gsituations are the choice, KC wish to excicde dictatorial same or at least indifferent to him; social welfare functions. (2) in comparing two personal situ- Bergson, op. [it., pp. 31s-20. The Fundamental ations in one of which he receives at Value Propositions of Individual Preference are least as of each commodity (in- not, strictly speaking, implied by Conditions z 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 corn- plied; see Consequence I in Part IV, section 2 rnOdity than in the other, the individual below. A slightly stronger form of Condition 2 than that stated here would suffice to yield the will prefer the first situation. Suppose desired implication. that among the possible alternatives 340 KENNETH J. ARROW there were three, none of which gave Let the individuals be designated as I any individual at least as much of both and 2, and let RI and R2 be the order- commodities as any other. For example, ing~by I and 2, respectively, of the suppose that there are two individuals alternatives x, y, z. Let PI and P2 be and a total of ten units of each of two the corresponding preference relations; commodities. Consider three alternative e.g., x'Plyf means that individual I distributions described by the accom- strictly prefers x' to y'. panying tabulation. The individualistic Consequence I: If xlP,y'and x'P,yl,then x1Py'. I.e., if both prefer x' to y', then so- ciety must prefer x' to y'. ALTERNATIVE Com- Com- PROOF.-By Condition 4 there are modity I modity 2 modity I I modity 2 orderings Ri and R4, for individuals I...... 5 I and 2, respectively, such that, in the z...... 4 corresponding social preference, xfP'y'. 3...... 3 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 111, section I, above, all individual so that xfP"y'. But, by construction, orderings of the three alternatives are both individu4ls 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, RI, Rz. only on the individual orderings of

2. THE POSSIBILITY THEOREM those two alternatives, it follows that whenever both individuals prefer x' to Some consequences will be drawn y', regardless of the rank of the third from Conditions 1-5 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. It will be shown that the supposition Consequence z: Suppose that for some x' and that there is a social welfare function y', whenever x1P,y' and y'P,xl,x'Pyl. Then, for x' y', xlP,y',xlPy'. satisfying those conditions leads to a that and whenever contradiction. 1.e.) if in a given choice, the will of Let x, y, and z be the three alterna- individual I prevails against the oppo- tives among which choice is to be made, sition of 2, then individual 1'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- PROOF.-Let RI be an ordering in tives, i.e., range over the values x, y, z. which xfPly', Ra be any ordering. Let THE COSCEPT OF SOCIAL WELFARE 34 I

be the same ordering as K1, while Suppose individual I prefers x to y is derived froin by depressing and y to z, nhile individual 2 prefers to th~bottom while leaving the rela- y to z and z to x. Individual 2 then pre- tive positior~sof the other two alter- fers y to x. By (I) society prefers x to natives unchanged. By construction, y. -41~0,both prefer y to z; by Conse- xfpiy', yf2':x'. By hypothesis, x'P'y', quence I, society prefers y to z. Since where Y' is the social preference rela- society prefers x to y and y to z, it must tion derived froin the individual order- prefer x to z. Therefore, we have ex- ings R:, R:. iYow the only difference hibited orderings RI, Ka such that xP~z, between R:, R: and R1, R2 is that x' zPzx, 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, Eence, by Condition 2 (interchanging h hen ever xP,z and aP,x,xPz . the Rl's and the Ri's) it follows from (2) x'2-"yf that x'Py'. I.e., whenever R1, R2 Now suppose XI is the ordering y, x, are such that x'P~y', then x'Py'. z, and Ra the ordering z, y, x. By Con- sequence I, yPx; by (2) xPz, so that yPz. By the same reasoning as before, I.e., if the two individuals have exact- whenever yP,z and zP,y,yPz . (3) ly opposing interests on the choice be- If K1 is the ordering y, z, x, and R:! tween two given alternatives, then so- the 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 yP,x and xP,g,yPx . false. 'l'hcn, for sonle orderings RI and (4) 1% and for son~epair of alternatives x' If XI is the ordering z, y, x, and R. and y', we would have x'Ply', yrP:!x', the ordering x, z, y, then from Conse- but not x'ly'. In that case, either xrPy' quence I and (4), zPy and yPx, so that or y'Pxr. IYe will suppose xrPy' and ZPX. show that this supposition leads to a IYheilevcr zP,n and z,P~B,Z~X . (5) contradiction; the same reasoning would show that the assumption yrPn-' If RI is the ordering z, n-, y, and R2 also leads to a contradiction. x, y, z, then, using (s), zPx and xPy, 11-ithout loss of generality it can be so that zPy. assumed that x' is the alternative n-, y' = y. Then we have, for the particular orderings in question, xPly, yPax, and From (I) it follows froin Conse- xPy. Since the social choice between quence 2 that whenever xPly, n-Py. n: and y depends, by Condition 3, only Similarly, from (I) to (6) it follows that on the individual choices as between for any pair of alternatives x', y', when- x and y, we must have ever n-'Ply', then x'Py'. That is, by Definition 5, individual I would be a xY,y yP2x,rYy whenever and . (1) dictator. This is prohibited by Con- It will be shown that (I) leads to a con- dition 5, so that (I) must be false. tradiction. Therefore, Consequence 3 is proved. 342 KENNETH J. ARROW

Sow suppose individual I has the the market mechanism does not create ordering x, y, z, while individual 2 has a rational social choice. the ordering z, x, y. By Consequence I, V. SOME IhIPLICATIONS FOR THE XI'\' . (7) FORJIATION OF SOCIAL TVEL- Since yPlz, zP-y, it fo1lov;s from Conse- FARE JUDGMENTS yuence 3 that I. INTERPRETATIOK OF THE I'OSSIEILITY ~12. (9 THEORETvI From( 7) (81,%pz.BU~,also xplz, The interpretation of the Possibility zl-'2x,which illlplies x~zby Consequence Theorem is given by examination of the 3. 1t cannot be that is both preferred meaning Of Conditions 1-5. Ill particu- and indiiierent to z. H~~~~the assump- lar, it is required that the social order- tion that there is a social welfare func- ing be formed from individual order- tion compatible with conditions I-5 ings and that the social decision be- has led to a contradiction. tween two alternatives be independent 1kt anotller lvay, if u-e assume that of the desires of individuals involving our social function satisfies any alternatives other than the given ConditioGs 2-3 and we further sup- tkvO (Conditions 1 and 3). These con- pose that Condition I holds, then either ditions taken together serve to exclude conditior! or condition must be interpersonal comparison of social util- violated. eonditioll states that the so- ity either by some form of direct meas- cia] function is not imposed; urement or by conlparison with other colldition states that it is not dicta- alternative social states. Therefore, the lorial. Possibiiity Theorem can be restated as Possioility T1~eorem.-If there are at least thrse alternatives among which If we exclzlde the possibility of i91tel'- the members of the society are free to 3@"o"al comPi1risons of utility, then order in any Ivay, then every social the 0oirly methods of passiag /?'om indi- function satisfying conditions aidual tastes to social preferences which and and a social ordering will be satisfactory and which will be satisfying ~~i~~~ I and 11 must be defined for a wide runge of sets of indi- eitiler imposed or dictatorial.ls ~l~~ niduol orderings are either imposed or Possibility Theorem shows that, if no prior assumptions are made about the The word "satisfactory" in the fore- nature of individual orderings, there going statement means that the social is no method of voting lv.ich will re- welfare function does not reflect indi- move the paradox of voting discussed ill viduals' desires negatively (Condition Part I, neither pluraiity voting nor any 2) and that the resultant social tastes schen~eof proportional representation, be represented an ordering no matter horv complicated. Similarly, having the properties Of ration- ality ascribed to individual orderings 13 The negati1.e outcomc expressed in this theorem is strongly reminiscent of the intransitivity of the ('Ondition I and Axioms I and 'I). concept of domination in the theory of multiperson In view of the interpretations placed games; sce John vcn Keumann and Oslcar Morgen- On the conditions for a social welfare stern, Theory oJ Gallies and Econoiizic B~ilavior (zd ed.; Princeton Uni\.ersity Press, 1g3;), pp. function in EJart 111 above, we can also 38-39. phrase the result this way: If con- TIIE CONCEPT OF SOCI.iL WELFARE 343 sumers' values can be represented by a impose on individual tastes are those wide range of individual orderings, the implied by the individualistic assump- doctrine of voters' sovereignty is incom- tions, then, as we have seen, there is no patible with that of collective ration- satisfactory social welfare function pos- ality. sible when there is more than one com- If we wish to make social welfare modity. Since, as we have seen, the only judgments which depend 011 all indi- purpose of the determination of the vidual vaiues, i.e., are not imposed or maximal states is as a preliminary to dictatorial, then we must relax some the study of social welfare functions, of the conditions imposed. It will con- the customary study of maximal states tinue to be maintained that there is under individualistic assumptions is no meaningful interpersonal compari- pointless. There is, however, a qualifi- son of utilities and that the cond:t'ions cation which should be added. It is con- v:rapped up in the word "satisfzctory" ceivable that, if further restrictions are are to be accepted.'" 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 neth hod of forming a Any state which is maximal under the social ordering would work properly combination of individualistic and other for a wide range of sets of individual restrictions will certainly be maximal orderings. That is, it must be supposed if only individualistic restrictions are that it is known in advance that the imposed on the individual orderings. individual orderings RI,. . . , 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 introduced. in addition to the individualistic ones, then the social maximum in any given 2. A REFLECTION OK THE NETT TTTLFARE situation will be one of the maximal ele- ECOKORIICS ments under the combined restrictions ,4s noted in Part I,tl:e so-called "new and hence one of the niaxiinal elements G-elfareeconomics" has concentrated on under individualistic conditions. It is the deter~ilinationof the totality of so- therefore not excluded that the current cial states which have the property that new welfare econonlics \vill be of some any charlze v;hich benefits one individ- use in restricting the range in which we ual injures another-"ma:iimal states7' must look for the social maximum. in Lange's tern~inology. In particular, The failure of purely individualistic this problenl has usually been analyzed assumptions to lead to a well-defined under the assumption that individual 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 Fart I\.', 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 schen~eof socio- 'The only part of the last-named conditions that scenls to me to he at all in dispute is the as- ethical norms, the realization of which sumption of rationality. The consequences of cannot, by their nature, be achieved dropping this assumption are so radical that it seems worth while to explore the consecluences of through atomistic market behavior. nlaintaining it. These norms, further, must be suffi- cicntly similar anlong the members of outputs s and t, respectively, and ap- the society to avoid the difficulties out- portioninents s' and t', respectively, to lined above. the given individual. Ir' s > t, but s'

l7 Bergson, op. cit. welfare function," has the form of a

le D. G. Johnson, "The High Cost of Food-a rule assigning a social ordering to each Suggested Solution," Journal of Political Economy, possible set of individual orderings rep- LVI (rgqS), j4-j7 ; Modigliani's proposals are con- tained in a press release of the Institute of World resenting tastes. &fathematically, the Affairs, New York, October, 1948. Is Samuelson, op. cit., p. 225. Bergson social welfare function has, ful than the actual practice of individ- then, the same form as the social wel- ualism in permitting the formation of fare function we have already dis- social welfare judgments. cussed; though, of course, the inter- It must of course be recognized that 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 con~modities,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 socizl welfare more wine and less bread than the ab- function is mathen~aticallyisoniorphic stainers get. Suppose now that all the to the social welfare function under in- wine is destroyed. Are the wine-lovers dividualistic assu:nptions. Hence the entitled, because of that fact, to more Possibility Theorem is applicable here; than an equal share of bread? The an- we cannot construct a Bergson social swer is, of course, a value judgment. welfare function, i.e., cannot satisfy lly own feeling is that tastes for un- value judgments I and 2, which will attainable alternatives should have satisfy Conditions 2-5 and which will cothing 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 beha~iorinto a value preted in terms of tastes rather than of judgment. It is not surprising, then, values, is a valid value judgment, to me that such ethics can be no more success- at least. http://www.jstor.org

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You have printed the following article: A Difficulty in the Concept of Social Welfare Kenneth J. Arrow The Journal of Political Economy, Vol. 58, No. 4. (Aug., 1950), pp. 328-346. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28195008%2958%3A4%3C328%3AADITCO%3E2.0.CO%3B2-R

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4 A Reformulation of Certain Aspects of Welfare Economics Abram Bergson The Quarterly Journal of Economics, Vol. 52, No. 2. (Feb., 1938), pp. 310-334. Stable URL: http://links.jstor.org/sici?sici=0033-5533%28193802%2952%3A2%3C310%3AAROCAO%3E2.0.CO%3B2-%23

5 Welfare Propositions of Economics and Interpersonal Comparisons of Utility Nicholas Kaldor The Economic Journal, Vol. 49, No. 195. (Sep., 1939), pp. 549-552. Stable URL: http://links.jstor.org/sici?sici=0013-0133%28193909%2949%3A195%3C549%3AWPOEAI%3E2.0.CO%3B2-B

5 The Foundations of Welfare Economics J. R. Hicks The Economic Journal, Vol. 49, No. 196. (Dec., 1939), pp. 696-712. Stable URL: http://links.jstor.org/sici?sici=0013-0133%28193912%2949%3A196%3C696%3ATFOWE%3E2.0.CO%3B2-M

6 Community Indifference William J. Baumol The Review of Economic Studies, Vol. 14, No. 1. (1946 - 1947), pp. 44-48. Stable URL: http://links.jstor.org/sici?sici=0034-6527%281946%2F1947%2914%3A1%3C44%3ACI%3E2.0.CO%3B2-K

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7 A Reconsideration of the Theory of Tariffs T. De Scitovszky The Review of Economic Studies, Vol. 9, No. 2. (Summer, 1942), pp. 89-110. Stable URL: http://links.jstor.org/sici?sici=0034-6527%28194222%299%3A2%3C89%3AAROTTO%3E2.0.CO%3B2-3

8 A Reformulation of Certain Aspects of Welfare Economics Abram Bergson The Quarterly Journal of Economics, Vol. 52, No. 2. (Feb., 1938), pp. 310-334. Stable URL: http://links.jstor.org/sici?sici=0033-5533%28193802%2952%3A2%3C310%3AAROCAO%3E2.0.CO%3B2-%23

11 A Note on Welfare Economics Gerhard Tintner Econometrica, Vol. 14, No. 1. (Jan., 1946), pp. 69-78. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28194601%2914%3A1%3C69%3AANOWE%3E2.0.CO%3B2-1

12 A Reformulation of Certain Aspects of Welfare Economics Abram Bergson The Quarterly Journal of Economics, Vol. 52, No. 2. (Feb., 1938), pp. 310-334. Stable URL: http://links.jstor.org/sici?sici=0033-5533%28193802%2952%3A2%3C310%3AAROCAO%3E2.0.CO%3B2-%23

14 A Reformulation of Certain Aspects of Welfare Economics Abram Bergson The Quarterly Journal of Economics, Vol. 52, No. 2. (Feb., 1938), pp. 310-334. Stable URL: http://links.jstor.org/sici?sici=0033-5533%28193802%2952%3A2%3C310%3AAROCAO%3E2.0.CO%3B2-%23

17 A Reformulation of Certain Aspects of Welfare Economics Abram Bergson The Quarterly Journal of Economics, Vol. 52, No. 2. (Feb., 1938), pp. 310-334. Stable URL: http://links.jstor.org/sici?sici=0033-5533%28193802%2952%3A2%3C310%3AAROCAO%3E2.0.CO%3B2-%23

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18 The High Cost of Food--A Suggested Solution D. Gale Johnson The Journal of Political Economy, Vol. 56, No. 1. (Feb., 1948), pp. 54-57. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28194802%2956%3A1%3C54%3ATHCOFS%3E2.0.CO%3B2-X

NOTE: The reference numbering from the original has been maintained in this citation list.