GeneralEquilibriumpricetaking1(伦敦经济学院高级

MicroeconomicsGeneralEquilibrium:pricetakingExcessDemandDecentralis-ationandTradeAmany-personeconomyRobinsonCrusoe'sProblemExcessDemandDecentralis-ationandTradeRobinsonCrusoe'sProblemGeneralEquilibriumWhyprice-taking?Amany-personeconomyOverview...Sofarwe'vefocusedoncompetitiveequilibriumanalysis.Butwhy?Whyconcentrateonequilibrium?Whyassumecompetitivebehaviour?Let'sstartbyhavinganotherlookattheexchangeeconomy.We'llredrawtheEdgeworthboxSomepuzzlesincompetitiveequilibriumanalysisGeneralequilibriumRobinsonCrusoe’sproblemDecentrali-sationandTradeAmany-personEconomyWhyprice-taking?ExcessdemandThesolutionconceptPricesandtheCoreThesolutionconceptAnexchangeeconomyPricesandtheCoreWhyprice-taking?DEOax2ax1aDEx1ax2ax1bObOaDEx2bx1bObWhatwouldIdooutofequilibrium?Andwhatwouldhedo?Let'slookattheinnerworkings...BuildinganEdgeworthboxx1aR1aR2ax2aOaThesituationfromAlf'spointofviewHowAlfrespondsasp1/p2changes...x1aR1aR2ax2aOaAlf'soffercurveWhataboutBill?x1bR1bR2bx2bObThesituationfromhispointofview...asanAustralianx1bR1bR2bx2bObHowBillrespondsasp1/p2changes...x1bR1bR2bx2bObBill'soffercurveThetwooffercurves1x2bObx1ax2aOax1bR1bR1aR2aR2bOffersareonlyconsistentwherecurvesintersectBut,byconstruction...ThisisthecompetitiveequilibriumSatisfiesmaterialsbalanceEDequilibriumhereendowmentherePrice-takingutility-maximisingBillendowmenthereequilibriumherePrice-takingutility-maximisingAlf•Butwhycompetitivebehaviour?•WhyshouldAlfandBillbehavethatway?•Wheredothepricescomefrom?Givencompetitivebehaviour,theCEistheonlyconsistentallocationsubjecttohhSpixiMi=1U(x)maxnhhSpixiMSowheredothepricescomefrom?WeneedamoregeneralsolutionconceptThenexaminehowprice-takingequilibriumrelatestothis.Let'sdeveloptheapproachfortheexchangeeconomy,thoughitcouldapplytomoreinterestingeconomies.AfreshapproachGeneralequilibriumRobinsonCrusoe’sproblemDecentrali-sationandTradeAmany-personEconomyWhyprice-taking?ExcessdemandThesolutionconceptPricesandtheCoreAnexchangeeconomyPricesandtheCoreAnexchangeeconomyWhyprice-taking?Viewthepopulation...Q2Q1...asnseparateindividualsQ0Acoalition...Thecoalitionconceptgivesusanimportantclue....isformedbyanysubgroupwe'vegotourendowmentswithusOnedayyoutakeyourbundlealongtotheswapshop......andsomebossypersonistherewhoproposes(insistson?)aparticularfeasibleallocation.Youandsomeothersdon'tlikethexbundleyouwouldgetunderthisallocation......andyoufindthatbyusingjusttheresourcesyouallownyoucouldallgetautilitylevelatleastashighasthatofferedtoyouundertheproposedallocation.Youguysthereforerefusetoacceptit.Theideaofblocking:alittlestoryAnallocationisblocked...Q2^xb^xc^xd^xa^xe^xf^xg^xl^xk...whichcandobetterforitselfifwecanfindacoalition...AproposedallocationforthewholeeconomyhhU(x)^hhU(x)...andthatarefeasibleQ{1,2,...,nh}^h[x]isblockedbyQhQ,forsomehhQhSxhSRhQAcoalitionBundlesthatarepreferredbymembersofthecoalition...^h[x]AformalapproachTheCoreisthesetofunblocked,feasibleallocationsUsetheideaofblockingtointroduceabasicsolutionconcept{Alf&Bill}{Alf}{Bill}Ina2-personworldthereareveryfewcoalitionspossibleUsetheEdgeworthbox...x2bObx1ax2aOax1b...togetthecontractcurve{Alf}blockstheseallocations...D{Bill}blockstheseallocations...{Alf,Bill}(together)blockthese...EndowmentNo-onecanblockanallocationhereThe2-personcoreGeneralequilibriumRobinsonCrusoe’sproblemDecentrali-sationandTradeAmany-personEconomyWhyprice-taking?ExcessdemandThesolutionconceptPricesandtheCoreAnexchangeeconomyThesolutionconceptAnexchangeeconomyWhyprice-taking?x2bObx1ax2aOax1bDEThecompetitiveequilibriumagainx2bObx1ax2aOax1bDEAnycompetitiveequilibriummustlieinthecoreEveryCEallocationmustbelongtothecoreOfcourseitispossiblethatnoCEexists(seenextlecture)ButwhatofothercoreallocationswhicharenotCE?Let'stakeaclose-uplookAsimpleresultDBAlfBillAlfBillATheextremesofthe2-personcoreCoreCEIntroducingAlfandBill'slong-losttwinbrothers...Rememberwearedealingwitha2-personmodelButwilltherealwaysbenon-CEpointsinthecore?Abasicrelation...AlfandhistwinbrotherArthurhavethesamepreferencesandendowmentsLikewisethetwinsBillandBenNowofcoursetherearemorepossibilitiesofformingcoalitionsSolet'sclonetheeconomy{Arthur&Ben}{Arthur}{Ben}{Alf&Arthur}{etc,etc}{Bill&Ben}{Alf&Bill}{Alf}{Bill}{Alf,Arthur&Bill}...plussomenewonesAlltheoldcoalitionsarepossible...Willthesestillbecoreallocationsinthe4-personeconomy?BAAlf'n'ArthurBill'n'BenAlf'n'ArthurBill'n'BenDAlf,Arthur,BenBillG...leavingtheBentwinoutsidethecoalitionThisallocationisnotasolution...ButitshowsthatthecorehasbecomesmallerNo.{Alf-Arthur-Bill}canblockA...Alf½[xa+Ra](G)Arthur½[xa+Ra](G)Billxb(A)BenRb(D)2Ra+RbThecoalitionFeasibleHowtheblockingcoalitionworksABDCanthisallocationbeblocked?AnA-personwouldbebetteroffsomewherehereBingo!Whatiftherewere500ofeachtype...?500oftheAtribe450oftheBtribe500oftheAtribe400oftheBtribe500oftheAtribe360oftheBtribe500oftheAtribe310oftheBtribe500oftheAtribe250oftheBtribeIfNisbiggertheremaybemoreblockingcoalitionsABDETheonlyallocationleftinthe2N-personcore(eventually)...Asyouclonethe