Sen, Contributions

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Amartya K. Sen's Contributions to the Study of Social WelfareAuthor(s): Kenneth J. ArrowSource: The Scandinavian Journal of Economics, Vol. 101, No. 2 (Jun., 1999), pp. 163-172Published by: Blackwell Publishing on behalf of The Scandinavian Journal of EconomicsStable URL: http://www.jstor.org/stable/3440690Accessed: 16/10/2009 12:25

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Scand. J of Economics 101(2), 163-172, 1999

AmartyaK. Sen's Contributions o theStudyof Social WelfareKennethJ ArrowStanfordUniversity, tanford,CA94305-6072,USA

I. IntroductionThe oeuvreof AmartyaK. Sen is veryconsiderablen magnitude,earlytwentybooks(withtwo forthcoming)nd about250 articles,mostly nlearnedournalsbut also in the higher evel of the popular ress,andlectures. t has nevertheless very considerableegreeof unity.It ismotivatedhroughouty concern or the welfare f the individualsn aneconomy,withspecial nterestn the lowerpartof the income pectrum;starvation,overty, nd economicdevelopmentavebeenthe objectsoffruitful,mportant,ndsocially aluabletudy ndhavemotivatedisbasicresearchnto heconceptsf individualnd ocialwelfare.

Allof thiswork s ofveryhighqualityndmakesmportantontributionsto ourunderstandingndto theformationf policy.But I will isolate orspecial onsiderationhat arge ubsetwhich evolves bout heconceptualquestion:what s meantby, better ff socially ?A majorportion f hiswork allsdirectlyntothiscategory. hevalueof Sen'sanalysis f socialwelfare nd heconceptualuestionselevanto it havenowbeenrecognizedbytheawardftheNobelMemorial rizenEconomic cience.Unlike omeotherworkwhichhasbeenawardedhisPrize, speciallyneconomicheory,we cannot ojusticeo Sen'swork nsocialwelfare n thebasisofoneortwoseminal apers, lthoughhere avebeen everaluch,asIwillpointout.Rathertis thework s a wholeand heway hevarious artsinterplayhatmustbe understoodo seetheimportancef Sen'scontribu-tion.Hisexplorationf thenotionsof socialwelfare akesplaceat everylevel of analysis,ormal-mathematical,onceptual,ndempirical.t is byfarthemostcomprehensivetudyof itskind,drawingnprofoundnder-standingf botheconomics ndmoral hilosophy.t isusefulonote hatheis regarded y philosopherss one of theirleading ights,and he haspublishedxtensivelynphilosophicalournals.Inview of thecomprehensivendfar-flungature f hiswork,t is notsurprisinghat his role has beensomewhat ifferentn various pecificaspects.nsome, ndeed, e hasopened nddefined ewfieldswithstrongresults.nothers, ehasperfectedndenrichedarlieresults rcontributedalongwithothers.n stillothers,he has raisednewinquirieswhosechiefvaluehasbeen hestimulationo others.What s strikings theconsistency? The editorsof the Scandinavianournalof Economics1999.Published y BlackwellPublishers, 08 CowleyRoad,OxfordOX4 1JF,UK and350 MainStreet,Malden,MA 02148, USA.


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164 K.J Arrowof his thought, o thateventhemost formalwork s motivated yphilosophi-cal and ethical considerations,while his most interpretiveand broadlyrangingpapers emaincompatiblewithrigorousechnicalanalysis.Becauseof thewiderangeof his work on socialwelfare,I will discussitin moredetail under ive headings: 1) formal heoriesof socialchoice;(2)the formaltheoryof individualpreferenceand choice; (3) the conceptualmeaningsof choice, welfare,and utility; (4) the measurement f socialwelfare as reflected n inequality,poverty,and real social income;and(5)empiricalanalysesof famineand nutrition elated o Sen'swork on povertyandits moralimplications. n SectionVII, I brieflyreviewa sampleof hiswork n fieldsother han heanalysisof socialwelfare.Onegeneralcharacteristicf Sen'sworkmustbe stressed.He is especiallyconcernedwith the distribution f welfare however hatconceptis under-stood.Averages or him concealinformation elevant o the formulation fsoundethical udgments.He hastherefore mphasizedhe measurementndmeaningof inequalityand in particular overtyas a morallyand economic-allyspecialcategory.

II. Formal Theory of Social ChoiceA majorpartof Sen'sworkhas dealtwith the formal heoryof social choice.Thebasicparadigmn this field has been that of Arrow 1951).1It is in thetraditionof welfare economics,which has sought to providea rationaljustificationor choice amongalternative ossibleeconomicpoliciesbasedon the preferencesof individualmembersof society.Assume that socialchoice, like individualchoice, is expressedby a preferenceorderingover(social) alternatives,o thatthe alternative hosenfromany given feasibleset of alternativess the most preferredone. Hence, one formulationofsocialchoice is thatof defininga socialordering overning ocialchoice foreach profile of orderings,one for each individual. n otherwords,socialchoice is definedby a mapping romprofilesto social orderings.One canthen state some desirablepropertiesof this mappingand ask if any suchmapping social welfareunctionin Arrow'sand Sen'sterminology) xists;if social welfarefunctionsdo exist,one canattempto characterizeheminsomeusefulway.Arrow'sconditions can be stated roughlyas follows: (U) the socialwelfarefunction is definedfor all profiles;(P) if the profileis such thateveryoneprefersalternative to alternative , thenthesocialorderingetsxabove y (the Paretoprinciple);(I) the choice between two alternativesdependsonly on the individualpreferencesbetweenthose two alternatives'Dates in parenthesesrefer to references listed at the end of this article.? The editors of the Scandinavian Journal of Economics 1999.


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AmartyaK.Sens contributionsothestudyofsocialwelfare 165(independencef irrelevant lternatives);nd(ND) the socialorderingdoesnotalwaysagreewiththepreferences f anysingleindividualnon-dictator-ship).Arrow hen showed hatthere is no socialwelfarefunctionsatisfyingall of these conditions.This resulthas givenriseto a largeliterature,o which Sen has beentheoutstandingontributor;hebestsurvey s Sen'sbookCollectiveChoiceandSocial Welfare1970].2Thereare severaldirections o go for changing henegativeresultto a positiveone;they all obviouslyrequire omechange nthe conditionsmposed.One is to enrich he informationalase, so thatthesocialorderingdependsnotonlyon individual rderings utalso on cardinalvaluesand,moresignificantly,n some kind of interpersonal omparisons.A second is to restrict he rangeof profilesfor which the social welfarefunction is defined,on the basis of some a priori assumptionsas to thepossible kinds of orderings ndividualsmay have. Indeed,even beforeArrow'sTheorem,Black (1948) had in effect shownthatmakingpairwisechoicesbymajority otingdetermines socialorderingobviously atisfying(P), (I), and(ND)) if the social alternatives anbe thoughtof as arrayedalongone dimensionandif each individual's referenceorderingover thisarrays single-peaked.Sen has developedboth of these researchdirections,but, more impor-tantly,he has introduced new statementof the problemwhichbrilliantlycombinessimplicityand depthin his discussionof the impossibilityof aParetian iberal [I, 1970a]. As will be discussedin Section IV, Sen hasobjected o theview that socialchoice shouldbe completelydetermined yindividualwelfarecomparisons.n particular,his broaderperspectivehasdrawnhim intostudyingheconceptionof rights.Each ndividuals taken ohave the right o makecertainchoicesamongsocial states.Totakethe leastcontroversialxamples, he choice between wo social statesthatdifferonlyin what one individual eadsor in the color of thepaint n theinteriorof anindividual's ouseshouldbe madeby that ndividual.Senthenshowedverysimplythat,even withoutrequiring ndependence I), the conditions U),(P) andthe minimaliberalism equirement,hatfor each individualhereshouldbe at leasttwo socialstatesbetweenwhichthat ndividual's hoice isdecisive, lead to a contradiction.This is very surprising; oth the Paretojudgmentand the idea that each individualhas some privatedomainofchoice, even if otherswouldmakedifferentchoices over thatdomain,arehard to deny; and independence,which on the whole is centralto mostvariations f the ImpossibilityTheorem,s not assumedhere.Theparadoxarises because nosy preferencesof others aboutchoices that are in an2Dates n brackets efer opublications y Senlisted n theBibliography. hoseprecededbyaRomannumeral efer o thesubjectheadingsof articleswrittenby Sen; hosewithouta Romannumeral efer o his books.

? The editors of the Scandinavian Journal of Economics 1999.


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166 K. J Arrowindividual'sdomainof privatechoice enter into the Pareto udgment.Theresult s not only surprising nalyticallybut also addressesprofound thicalquestionson the relationbetweeneventhevestigialremnant f utilitarianismcontainedn the Paretoprincipleandthe existenceof individual rights,scope(however mall)overwhich the individual ascompletecontrol.Sen'swork has sparkedboth a technical literature,e.g., Gibbard(1974) andSuzumura1978),and contributedo philosophical mphasison rightsfromverydiversecriticalviewpoints, .g.,Nozick(1974)andDworkin1978).A very different line of analysis pursuedby Sen has concerned thepossibilityof resolving he socialchoiceproblemby assuming hat herangeof individualpreferenceswhichneed to be aggregateds narrowedromtheuniversalrangepostulated n condition(U). In particular,he has statedconditions on profilesunder which pairwise majorityvoting leads to anordering. nada 1964) had found two conditionsother than Black'ssingle-peakednessof preferences, nd Ward 1965)hadgeneralizedBlack'scondi-tion. Sen [I, 1966]introduced condition extremalvaluerestriction)whichincluded all previousones: in any triple of alternatives, here are onealternative nd one value(best,middle,orworst)suchthatno oneranks hatalternativeat that value. Later, Sen and P. Pattanaik[I, 1969) foundnecessaryand sufficientconditionswherebymajorityvoting yields a well-definedfirst choice. Sen thusbothmade a majorearlycontributiono thisparticularubfieldandcollaboratednestablishinghe definitive esults.Morerepresentativef Sen'sgeneralposition s his systematizationf theinformationconstraintson social choice. As he observes in his articleInterpersonal ggregationand PartialComparability I, 1970b],Arrow'sassumption f interpersonallyncomparablerdinalisms an extremecase.Interpersonallyncomparable ardinalismdoes not get us any further navoiding mpossibilityresults[1970, Theorem8*2], a resultgreatly deep-ened by Kalai and Schmeidler(1977). In effect, the assumptionthatindividualudgments,ordinalor cardinal, annotbe comparedmpliesthatsocial judgmentsmust be invariantunder a wide range of independenttransformationsf individualpreferences all monotone ransformationsnthe ordinalcase, all affine transformationsn the cardinalcase). For anydegree of interpersonal omparabilitye.g., that the ratio of unit utilitychangesfor two individuals s judgedto lie between wo prescribedimits),the invariance equirementsn the social welfare functionarecorrespond-ingly lightened,andthe possibilityof findingan acceptablesocial choiceprocedure ncreased.For a survey of this point of view in subsequentresearch, ee Sen[I, 1977b].The research temming romSen'sargument as indeedbeenrich. This isone of the areaswhereSen'scontribution as been the formulation f thequestionand the inspirationof others.It is, of course,well knownthatcompleteinterpersonal omparability f unit utility differencespermits-? The editors of the Scandinavian Journal of Economics 1999.


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AmartyaK.Sens contributionsothestudyofsocialwelfare 167andmakesmeaningful the sum-of-utilitiesriterion s satisfying onditions(P), (ND), and (I) (the last suitablyreinterpretedo admit the additionalinformation).FollowingSen's stimulus,Hammond 1976) and Strasnick(1976)formulated different nformationalasis.Theyadmittednterperso-nal ordinalcomparisons, .g., that individual in state x is betteroff thanindividual in statey. Thisassumptionmplies hatsocialwelfareudgmentsbe invariantunder the same monotonetransformation f all individualpreferences.Hammondand Strasnickndependentlyhowedthat underthis informa-tional constraint, he leximin and leximax principlessatisfy all thesocial choice conditionsabove.Theyarein fact theonlypossibleprinciplessatisfying hese conditionsand some mild additional nes.By leximin smeantthe following:for each of the social alternatives, andy, ranktheindividualsn increasing rderof utility(thisoperations meaningfulunderthe informationalonstraint).f the worst-offunderx is betteroff than theworst-offundery, choose x (orvice versa).If the two worst-off ndividualsareequallywell off, thencomparehe two second-worst-offndividuals ndso forth.The leximin principle s reminiscentof Rawls's differenceprinciple(1971), but it must be acknowledged hat Rawls gives it a differentinterpretation.oRawls, hecomparisons in termsnot of well-beingbut ofprimary oods, those thatgive rise to the possibilityof well-being.Senhasdevelopedhis themefurthersee SectionIVbelow).Sen has contributedo the technicaldevelopment, s well as settingtheterms of the argument.He notes that the leximin principleis not verypersuasive y itself for a largesociety.It is, however, eryreasonablef thereareonly two individuals.But, in conjunctionwith conditions U) and(I),leximinfor two-memberocieties entails eximinforanysociety;see Sen[I,1977b,Section6].A last exampleof Sen's seminal role in the area of formaltheoriesofsocialchoiceis his formulation f the WeakEquityAxiom in his lectures[1973]: an individualgetting less utility out of any given income thananothershould get a higher income (incentiveeffects aside). That is,disabilities houldbe compensatedor. Thisaxiom s in general nconsistentwith utilitarianismand has revived interest in equality as a criterionindependentf diminishingmarginal tility.III. Formal Theories of Individual Choice and RationalityReflectionon socialchoiceprovideda new stimulus orstudying ndividualchoiceand the meaningof rationality.Gossen,Jevons,Walras,Menger,andEdgeworthad ntroducedhehypothesishatconsumer hoiceof commoditybundleswas determined y maximization f a utilityfunctionsubject o a

? The editorsof the Scandinavian ournalof Economics1999.


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168 K. J.Arrowbudgetconstraint.rvingFisherandVilfredoPareto bserved hat hecardinalnatureof the utilityfunctionwas irrelevant;nly an ordering f commoditybundles need be postulated.Samuelson(1938) introduceda change inorientationn his conceptof revealedpreference.The choicesmade,not anunderlying rderingor utilityfunction,wereprimary.Conditionsof ration-ality were imposedon the choices. As shown laterby Ville (1946) andHouthakker1950), suitablystrong conditionson the demandfunctionsimplied he existenceof anunderlying rderingwhichrationalized hoice.Arrow(1959) observed hatif choice fromfinitesets (ratherhanbudgetsets) was assumed o be defined, he equivalencebetween choice functionsdefinedby maximizinga given orderingand choices satisfyingcertainconsistencyconditionsook a somewhatdifferentorm.This line of analysisinvites taking apartthe conditions on the choice function(the functiondefining he choices or choices madefromany given opportunityet) andshowing heequivalence f each to someform of rationalityweaker hananordering.Thisstudyhas beenpursued y severalwriters,nparticularUzawa(1956), Herzberger1973), and Sen. Sen'sarticle, ChoiceFunctionsandRevealedPreferences IV, 1971], is the epitomeof thiswork,summingupandextendingallprevious tudies.Therearemanyresults; hefollowing s atypical example.Startwith a choice function,C(S), mappingeach finiteopportunity et S into the subset of chosen elements.Define, revealedpreference, R,to mean:(xRy) f andonlyif, for someS, x belongsto C(S) andy belongsto S.State he WeakCongruenceAxiom(WCA):If xRy, henforanyS suchthaty belongsto C(S) andx belongsto S, x mustalsobelongto C(S).Also statepropertiesa) and(/3):(a): If S is a subsetof T andx belongsto both S andC(T), then x belongsto C(S).(/3): If S is a subsetof T andx andy anytwo elementsof C(S), theneitherbothbelongto C(T) or neitherdoes.Then hefollowing heoremholds:Theorem. A choice function satisfies WCAif and only if it satisfies bothproperty (a) andproperty (/3).Note: Senalso showsthatWCA is equivalento the statementshatR is anordering and that C*(S) = C(S), where C*(S) is the set of elements of Swhicharemaximalwithrespect o R.? The editors of the Scandinavian Journal of Economics 1999.


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AmartyaK. Sen s contributions o thestudy of social welfare 169IV. Clarificationof the Aims of Social PolicySocialchoicetheoryandutilitarianismeekto base socialjudgmentson thewelfaresof the individualmembers f society; n different ariants, ifferingassumptionsremadeabout heproperties f welfaremeasurementsordinal,cardinal,nterpersonallyomparableardinal, ndso forth). nsomearticles,e.g., [II, 1979],Sen hasdevelopeda fundamentalritiqueof this doctrineofwelfarism, s he calls it. He hasargued ordistinguishingggregation fjudgmentsof differentpeopleaboutsocialpolicyfromaggregation f inter-ests. Forthe former,only information bout ndividualpreferencess avail-able,andtheImpossibilityTheorems relevant.Foraggregation f interests,on the otherhand,there is additionalnformation eyond any measureofindividualwelfares,in particular,measurement f inequality n objectiveterms.Also, the forms of consumptionhatgive rise to pleasuresmightbemorallyrelevant orgiven utilitylevels (e.g., pleasuresarising rom sadismandmasochism).Even the Paretoprinciplecan be questionedalongtheselines,and theparadoxof the Pareto iberal see SectionII) showsthattherecanbeprincipleswhichweregard soverridingheParetoprinciple.This lineof argument as beendeveloped urther y Sen[1985, 1987,and1992].A given set of commoditiesmaybe utilizedin differentwaysby aconsumer,hough he rangeof possibleutilizationmodesmaybe restrictedby an individual'spersonal imitations.Utility in the usual sense may beidentifiedwithhappinessor fulfillment f desires.Butneither hevaluationto be placedon thechosenutilizationnorthechoice madeby theindividualneed be related o theutilitynorindeed owell-beingnanysense.Further,udgmentof a given stateof affairsmay rationallydepend,notmerelyon the alternative tilizationchosen but on the rangeof alternativesavailable o theindividual,ncludinghose notchosen.There s a preferencefor capability or freedomof choice);povertyand disabilityare infringe-mentsoncapability.

V. Measurement of Inequalityand PovertySen'sgeneralconcernsaboutthe meaningof social policy and individualvariationsnthecapacity orfunctioning ave foundapplicationsn develop-ing appropriatemeasures of income inequality.His paper with ParthaDasguptaandDavidStarrettIII,1973b;see also III, 1978]introduced ewcriteria for measures of income inequality,essentially formalizingthetransfer rinciple of H. Dalton,thata transfer rom a richindividual o apooronemust be regarded s a reductionn inequality.Mathematically,hisleads to the implicationhatthe functionexpressing nequalityn termsofindividualncomesmustbe S-concave,a weakercondition hanconcavityoreven quasi-concavity. romthis, they show that in orderfor one income



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170 K.J.Arrowdistributiono be better hananother orany S-concave nequalitymeasure,it is necessaryand sufficient hattheLorenzcurvefor thesecond ie entirelybelow that of the first(or,equivalently,hat the firstbe obtainableromthesecondby a sequenceof transfers romtherichto thepoor).Theimportanceof this paper lies not merely in this characterization f the strongeststatements hatcan be made n theabsenceof moredefinitecriteria,but alsoin its improvedestsforinequalitymeasures.Thispapergeneralizeshe earlierworkof Atkinson 1970).However,Senhas objectedthat Atkinson'smeasures,which are additivelyseparable nindividual ncomes, tend not to describeinequalityvery well in certaincircumstances.Thus,the Gini coefficient,which is not separablen indivi-dual ncomes,nevertheless atisfiesa number f interestingonditions.Closelyrelated o measures f income nequality re measuresof poverty,which have played such a large role in evaluationsof public economicpolicies,nationalandinternational. gain,Sen [III, 1976b]has contributedimportantly hroughan axiomatic characterization f poverty measureswhich take into account nequalityamong hepooras well as theproportionbelowa givenlevel. As anapproximationor a largepopulation,he derivesthepovertyndex,

P= H[I + (1 -I)G],whereH is the proportionn poverty, the ratio of total income shortfallsfrom the povertylevel to total income, and G is the Gini coefficientofincomedistributionmong hepoor.He has pursuedthis analysis into the correction of national incomecomparisons or distributional easons [III, 1976a]. Again an axiomaticcharacterizationeadsto the use of the Gini coefficient o modifytheusualmeasures f per capitanationalncome,specificallyo measuredistribution-ally-adjustedper capita national income as e (1 - G), valid for largeeconomies,wheree is mean income and G is the Gini coefficient.This isactuallya specialcaseof a muchbroaderet of results.VI. Empirical Studies of Distributionand Its ConsequencesThenormativemeasuresof distributionnd Sen'semphasison the function-ing of individualsas the criterion or social policy have led to importantstudiesof the interactionbetweenincome distribution nd severe incapa-cities,includingdeath.This is moststrikingly roughtoutin his remarkablestudiesof famine[1981 and 1989].His carefulempiricalanalysesof fourfaminesbringoutclearly hatneithervariationn foodsupplydue to naturalcausesnorphysicalobstacles o fooddistributionlayedsignificant olesinthese famines.Rather,heyweredueto shifts n thereal incomedistribution,? The editors of the Scandinavian Journal of Economics 1999.


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AmartyaK. Sen scontributionso thestudyofsocialwelfare 171particularly loweringof incomesof thevery poor; t wasinability o affordthepurchase f food thatcausedhungerandfamineonmassivescales.

He has also studied hedistributionf foodwithinthefamily,particularlyunderconditionsof stress,andhas shownthat thereis strongevidence ofdiscriminationgainstfemalefamilymembers n some partsof the world[VIII passim]. In particular,demographicanalysis indicates the highermortality f womenatyoungeragelevels.VII. Other Areas of StudySen haswrittenonmanyother opicsthansocialwelfare,broadas thattopicis. Tojudge by the references, his workhas hadvery important ffects. Ihave followedonly partof this work;in this section, I make very briefreferenceso thequestionshatI havefollowed.One is the meaningof rationalityn individual hoice.Perhapsno singlepaperof Sen'shas been more cited thanhis RationalFools [IV, 1977],inwhichhe showshow limited heordinaryonceptof rationalitys incoveringtherangeof motivesin individual hoice;in particular,heroleof commit-ment s completelygnored.Morerecently,n Maximization ndtheAct ofChoice [IV, 1997], he has shown thatconceptsof maximizationare notnecessarily onnectedwiththeimplicationf anordering f alternatives.Anothercontinuing nteresthas been the achievementof efficiencyineconomicdevelopment.While the generaltheoretical hrusthas been lessnovel thanin his workon socialwelfare,his contributions ave been of avery high order,especiallysince they frequently lew in the face of thencurrentpolitical dogmas accepted by many economists. This work com-mencedwith his earlybook on choosingamongtechniquesof production[1960a] and continuedwith many paperson the evaluationof projectsculminatingn the handbookon projectevaluation or the UnitedNationsInternationalDevelopmentOrganization,written ointly with ParthaDas-guptaandStephenA. Marglin1972].Twopaperson the rate of discountappropriateo governmentnvestment[XI, 1967 and 1982]have stressed he role of externalities n the formofpublicconcern or the future.Closelyrelatedhasbeenhiscontributiono thetheoryof optimal avings IX,1961a, 1967,and1975].

Ina verydifferent ein, Senhasaddressedheideathatthevaluesystemsof Indiaand otherAsiancountries re so different romthatof theWest hatthe polity and economy cannotbe organizedalong Western ines, thatdemocracy,orexample, s inappropriaten SouthandEastAsia. Hisarticles[XIV,1993a, 1993b,and 1996]haveshownthat a deeperknowledgeof theIndian and Asiatic pasts would reveal far less uniformityand far morestrands f rationalanalysisanddemocratichinkinghan s assertedby self-interested articipantsn the debate.? The editors of the Scandinavian Journal of Economics 1999.


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172 K. J ArrowVIII. SummaryOne couldpointto still otheraccomplishments,specially n severaldiffer-ent fields of philosophy,but the area whereSen'scontributions ave beentruly uniqueis his extraordinaryynthesisof economicandphilosophicalreasoningon the bases for social policy. No one has combineddifferentapproaches,ormalanalysis,conceptual larification,heoryof measurementandempiricalworkas hasSen.ReferencesArrow,K. J.(1951), Social ChoiceandIndividualValues,Wiley,New York.Arrow,K. J. (1959), Rational Choice Functions and Orderings,Economica, N.S. 26,121-127.Atkinson,A.B. (1970), On the Measurementof Inequality,Journal of Economic Theory2, 244-263.Black, D. (1948), On the Rationale of Group Decision-making, Journal of PoliticalEconomy56, 23-34.Dworkin,R. (1978), TakingRightsSeriously,Duckworth,London.Gibbard,A. (1974), A Pareto-consistentLibertarianClaim, Journalof Economic Theory7, 338-410.Hammond, P. (1976), Equity, Arrow's Conditions, and Rawls' Difference Principle,Econometrica44, 793-804.Herzberger, . (1973), OrdinalPreferences nd RationalChoice,Econometrica1,187-238.Houthakker,H. S. (1950), Revealed Preferenceand the Utility Function,Econometrica17, 159-174.Inada,K.-I. (1964), A Note on the Simple MajorityDecision Rule, Econometrica32,490-506.Kalai, E. and Schmeidler,D. (1977), AggregationProcedure or CardinalPreferences:AFormulationand Proof of Samuelson'sImpossibility Conjecture,Econometrica44,

793-804.Nozick, R. (1974), Anarchy,State,and Utopia,Blackwell,Oxford.Rawls,J.(1971), A Theoryof Justice,HarvardUniversityPress,Cambridge,MA.Samuelson,P. A. (1938), A Note on the PureTheoryof Consumer'sBehavior,Econom-ica, N.S. 5, 61-71.Strasnick, S. (1976), Social Choice Theory and the Derivation of Rawls' DifferencePrinciple,Journalof Philosophy73, 85-99.Suzumura,K. (1978), On the Consistencyof LibertarianClaims, Review of EconomicStudies45, 329-342.Uzawa, H. (1956), A Note on PreferencesandAxioms of Choice, Annalsof the Instituteof StatisticalMathematics8, 35-40.Ville, J. (1946), Surles conditionsd'existence d'une ophelimit6totale et d'un indice duniveau des prix, Annalesde l'Universitede Lyon9, 32-39.Ward, B. (1965), Majority Voting and Alternative Forms of Public Enterprise, inJ. Margolis (ed.), Public Economy of Urban Communities,Johns Hopkins Press,Baltimore,112-126.


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