TheFirstWelfareTheorem
•Inthelastlecturetwoconceptswereintroduced:Paretoefficiencyandgeneralequilibrium.Howdotheyrelate?•Theorem:ThefirstwelfaretheoremstatesthateverygeneralequilibriuminvolvesaParetoefficientallocation.
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x2B
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A
•Theproofworksbycontradiction.Supposetherewasanon-Paretoefficientallocationwhichwasalsoageneralequilibrium.Thispointcannotlieonthecontractcurve(thedashedline)sinceitisnotefficient.
•SupposeitwasapointlikexA.Ifitisageneralequilibriumaswell,consumerAmustbemaximisinggivenprices(illustratedbythebudgetline).ButconsumerBmustbemaximisingalso—sayatpointxB.•xAcannotbeequaltoxBasitisnotonthecontractcurve.ThereforexBisadifferentpoint.
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•Butnowthemarketforgood2doesnotclear:x2A+xB>ωA+ωB—acontradiction(thisisnotanequilibrium).
Market—Welfare2
MarketFailure
•Assumptionsneedtobemadeforthistheoremtowork.Therearethreecrucialones.
1.NoExternalities:Eachagent’sconsumptiondecisiondoesnotaffecttheutilityofanyotheragent.2.PriceTakingBehaviour:Eachagentintheeconomybehavesasapricetaker.
3.PricesareKnown:Allthepricesforeachofthegoodsmustbeknowntoeachoftheagents.Importantly,the
consumersdonothavedifferent(asymmetric)informationconcerningthegoods.•Thefirstassumptioniscritical.Thenextlecturedealswiththecaseofexternalitiesinmoredepth.
•Thelastistheminimalinformationrequirement.Agentsneedonlyknowprice.Theyneednotknowthedemandoroutputdecisionsofothers,orhowmuchofagoodisavailable.Theybehave“selfishly”giventheknownprices.•Marketfailureariseswhenanyoftheseassumptionsisnotmet.Externalities,marketpowerandasymmetricinformationareallexamplesofmarketfailure.
Market—Welfare3
TheInvisibleHand
•Whataretheimplicationsofthistheorem?
•Inageneralequilibriumeveryonemaximisesutility“selfishly”givenprices.Firms“selfishly”maximiseprofits.•However,asaresultofthisselfishbehaviour(giventheassumptionsdiscussedearlier)asociallydesirableoutcomearises.Anallocationisachievedwhereno-onecanbemadebetteroffwithoutmakingsomeoneelseworseoff.•ThisisaformalisationoftheinvisiblehandargumentofAdamSmith.
•Whatroleisthereforgovernmentinterventioninsuchanefficientworld?Therearetwopossibilities:1.Marketfailure:Ifoneoftheassumptionsfail,theallocationmaynolongerbeefficient.2.DistributiveGoals:Paretoefficiencysaysnothingaboutdistributionalfairness.
Market—Welfare4
TheSecondWelfareTheorem
•Anequilibriumisefficient,areefficientallocationsalwayspartofanequilibrium?
•Theorem:ThesecondwelfaretheoremstatesthateveryParetoefficientallocationcanbesupportedbyageneralequilibriumsetofpricesgivenasuitablereallocationoftheendowment.
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EquilibriumPrices
•Thedemonstrationproceedsconstructively.WhichsetofpricescansupporttheaboveParetoefficientallocation(x)asageneralequilibrium?Theansweristhegivenbythebudgetlinethatseparatesthetwoindifferencecurves.•Inordertosupportxtheendowmentwouldneedtoreallocatedfromωtoω.
Market—Welfare5
ConvexityandtheTheorem
•Thecrucialassumptionhereisthatofconvexity.Preferencesneedtobewell-behavedforthetheoremtowork.•Whatwouldhappeniftheywerenot?Considerthefollowingcase.Ahaswell-behavedpreferences.Bdoesnot.
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•Canapointlikexbesupportedasageneralequilibrium?Thebudgetlineisasetofprices,separatingthetwo
indifferencecurves.Aismaximising.Bisnot—theycoulddobetter,bychoosingxB.
•xBmaximisesutilitygiventhesepricesforconsumerB,xmaximisesutilityforconsumerA.Thisisnotageneralequilibriumsincethemarketforgood2isnotclearing.
Market—Welfare6
ImplicationsoftheTheorem
•Noticethatconvexityisonlyrequiredforthesecondtheorem.Thefirsttheoremholdsforanypreferences.•Whataretheimplicationsofthesecondtheorem?
•Distributionalissuescanbeseparatedfromefficiencyissues.Agovernment,operatinginsuchaworldcan“simply”transferendowmentstoachieveanydistributionalgoalstheymighthaveandleavethemarkettoattainefficiency.•Pricesplaytworolesinthemarket,(i)allocative—relativescarcityofthetwogoodsand(ii)distributive—howmuchagentscanafford.Thesecanbeseparated.Donotusepricestoattaindistributivegoals,useendowments.•Ofcourse,suchatransferofendowmentsisnotsosimple—itmaybeimpossible.
•Note:Althoughonlypureexchangeeconomieshavebeenconsideredsofar,everythinggoesthroughinananalogouswayforaproductioneconomy.Theassumptionsneedtoapplybothtoconsumersandfirms.
Market—Welfare7
EfficiencyandWelfare
•ThecontractcurveisthesetofallocationsthatareParetoefficient.
•Byapplyingthefirstandsecondwelfaretheoremsageneralequilibriumwilllieonthiscurveandmoreoveragovernmentcouldtransferendowmentstoachieveanyofthesepoints.Thequestionis:Whichone?•Paretoefficiencysaysnothingabout“fairness”or“justice”.Indeed,theallocationwhereconsumerAgetseverythingandconsumerBgetsnothingisParetoefficient.Itisprobablynotfairhowever.
•Supposethegovernmentcouldrankthevariousallocationsavailable,simplyattachinganumbertoeachoutcome.Theonewiththehighestnumberwouldthenbethebestallocationfromsocietiespointofview.
•Howshouldthegovernmentconstructtheirrankingoverdifferentallocations?Inotherwords,howshouldthegovernmentturnindividualpreferencesintosocialwelfare?Howcanpreferencesbeaggregated?
Market—Welfare8
AggregatingPreferences
•Supposetherearethreeagents(1,2and3)intheeconomyandthreepossibleallocations:a,bandc.•Supposethefollowingtableillustratesthepreferencesthethreeindividualshaveoverthevariousallocations.
Agent1abc
Agent2
bca
Agent3
cab
•Howshouldtheirpreferencesbeaggregated?Supposeamajorityvotingmechanismisproposed.Givenachoicebetweenaandb,awouldwin(agents1and3wouldvotefora).Givenachoicebetweenbandc,bwouldwin(agents1and2wouldvoteforb).Givenachoicebetweenaandc,cwouldwin(agents2and3wouldvoteforc).•Sothissocialorderingisnottransitiveeventhoughtheutilityfunctionofeachagentis.
•Isthereabetterwaytorankthethreealternatives?Whatpropertiesshouldsuchasocialwelfarefunctionhave?
Market—Welfare9
Arrow’sImpossibilityTheorem
•Supposetheaggregationmethod(orsocialwelfarefunction)hadthreeproperties:
1.Givenasetofcomplete,transitiveandreflexivepreferences,aggregationshouldresultinsocialpreferencesthatarecomplete,transitiveandreflexive(unlikemajorityvoting).2.Ifeveryonepreferssomeallocationatoanother,b,thenashouldbesociallypreferredtob.
3.Socialpreferencesbetweenaandbshouldonlydependonthewaythatagentsrankaandbandnotontherelativerankofanyother“irrelevant”allocationc.•Theorem:Arrow’simpossibilitytheoremstatesthatanymechanismfortheaggregationofpreferences(socialwelfarefunction)whichsatisfiestheabovethreepropertiesisadictatorship.
•Inotherwords,thesocialrankingscorrespondexactlywithoneindividual’srankings—thesocialwelfarefunctionissimplytheutilityfunctionofaparticularagent.
•Thereisno“perfect”waytorankallocations.Howcanthegovernmentchoosebetweenallocations?
Market—Welfare10
SocialWelfare
•Obviously,theyusealessthan“perfect”socialwelfarefunction.Eitheroneofthepropertiesisnotsatisfiedortheyuseadictatorshipwelfarefunction.Surelythisisbad?Perhapsnot.
•Formally,asocialwelfarefunction(SWF)mapsindividualutilityfunctionstoanumber—socialwelfare.•ConsidertheRawlsianandUtilitariansocialwelfarefunctionswhicharegivenrespectivelyby:
WR(u1,...,un)=minui
i
andWU(u1,...,un)=
ni=1
ui
•ConsidertheRawlsianSWF.Itsatisfiesproperty1—sinceitissimplyautilityfunctionandhencecomplete,
transitiveandreflexive.Itsatisfiesproperty2,sinceifeveryoneprefersatobthentheminimumutilityindividualprefersatobandhencetheSWFranksaaboveb.
•Itisindependentofirrelevantalternatives,c,sinceallindividuals(includingtheonewiththesmallestutility)rankaandbindependentlyofc.Hence,theremustbeadictator.Thereis.Theleastwelloffagent.Isthissobad?•TheUtilitarianSWFalsoeitherbreaksoneofthethreeconditionsorisadictatorship.Exercise:Which?
Market—Welfare11
WelfareMaximisation
•ThegovernmentwishestochoosethebestallocationaccordingtotheSWF—itwishestomaximisewelfare.•Notallallocationsarepossible.Theutilitypossibilitiessetisthesetoffeasibleallocations.
u2u2
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•ThefeasibleallocationsliewithinthesetsUinthegraphabove.TheboundaryofthissetiscalledthePareto
frontier.Theisowelfarelines—linesofequalsocialwelfareareillustratedfortwoexamples.
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•ThefirstexampleisforaRawlsianSWF—itresultsinequality,u∗1=u2.ThesecondexampleisforaUtilitarian
∗
SWF—“thegreatestgoodtothegreatestnumber”—itdoesnotresultinequality,u∗1>u2.
•NoticethateveryParetoefficientpoint(apointontheParetofrontier)isthemaximumofsomeSWF.
Market—Welfare12
Fairness,EnvyandEquity
•Anotherapproachmightbetoproposeaparticulartypeofallocation—afaironeforexample.
•Whatisafairallocation?Anenvy-freeorequitableallocationisonewhereconsumerAdoesnotpreferthebundleconsumerBgetsandvice-versa.IntheEdgeworthboxtheotherconsumer’sbundleisthe“mirrorimage”bundle.
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•xisafairallocation,(if)itisequitableandefficient.Neitherconsumerwouldprefertobeaty—thatis,neither
consumerwantstoswapbundleswiththeother.Noticethatωliesonthisbudgetlinealso—equalendowments.•Infactifbothconsumersstartwithanequalamountofthetwogoodseachtheywilltradetoafairallocation.Acompetitiveequilibriumfromequaldivisionmustbeafairallocation.
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