Transcription of Part I General Equilibrium - Harvard University
1 HartNotesMatthewBasilicoApril,2013 PartIGeneralEquilibriumChapter15- General EquilibriumTheory:Examples PureexchangeeconomywithEdgeworthBox Pro ductionwithOne-Firm,One-Consumer [SmallOp enEconomy] :TheEdgeworthBoxDe nitionsandSetUpPureExchangeEconomy Aneconomyinwhichtherearenopro ductionopp ossessendowments,eco-nomicactivityconsis tsoftradingandconsumptionEdgeworthBoxEco nomy Preliminaries Assumeconsumersactaspricetakers Twoconsumersi= 1,2;Twocommo ditiesl= 1,2 ConsumptionandEndowment Consumeri sconsumptionvectorisxi= (x1i,x2i) Consumeri sconsumptionsetisR2+ Consumerihaspreferencerelation%iovercons umptionvectorsinthisset Consumeri sendowmentvectoris i= ( 1i, 2i) Totalendowmentofgo o dlis l= l1+ l2 Allo cation1 Anal locationx R4+isanassignmentofanonnegativeconsumpti onvectortoeachcon-sumer x= (x1,x2) = ((x11,x21),(x12,x22)) Afeasibleal locationis xl1+xl2 lforl= 1,2 Anonwastefulal locationis xl1+xl2= lforl= 1,2 Thesecanb edepictedinEdgeworthBox WealthandBudgetSets Wealth.
2 Notgivenexogenously, p i=p1 1i+p2 2i BudgetSet:Givenendowment,budgetsetisafun ctionofprices Bi(p) ={xi R2+:p xi p i} Graphically,drawbudgetline[slop e= (p1p2)].Consumer1'sbudgetsetconsistsofal lnonnegativevectorsb elowandtotheleft;consumer2'sisab oveandtotheright Graph: Axes:horizontalisgo o d1,vertialisgo o d2 Origins:consumer1inSWcorner(asusual),con sumer2inNEcorner(uniquetoEdgeworthb oxes) O erCurve=Demand(asafunctionofp) (Adepictionofpreferences%iofeachconsumer ) Asumestrictlyconvex,continuous,stronglym onotone2 Aspvaries,budgetlinepivotsaround .Giventhisline,consumerdemandsmostprefer redp ointinB(p).
3 Thissetofdemandedconsumptionsmakesupo ercurve Justlikedemandfunction:x1(p,p 1) Curvepassesthroughtheendowmentp ,sinceateachp ointendowmentisa ordable,everyp ointono ercurvemustb eatleastasgo o das i. WalrasianEquilibrium( etitiveEquilibrium),foranEdgeworthBox: De nition:Pricevectorp andanallo cationx = (x 1,x 2)intheEdgeworthb oxsuchthatfori= 1,2 x i%ixiforallx i Bi(p ) Atequilibrium,theo ercurvesofthetwoconsumersintersect Anyintersectionoftheo ercurvesoutsideof corresp ondstoaWE Onlyrelativeprices(p 1p2 )aredeterminedinequilibrium,sinceeachcon sumer'sdemandishomogenousofdegreezero p = (p 1,p 2)isaWE= p = ( p 1, p 2)isaWE :Maxutility,usingbudgetconstrainttoreduc enumb :SolveforDemand(OC)asafunctionofpricexli (p).
4 Settotaldemandforago o dequaltoendowment:xl1+xl2= l(a)AKAsolveforexcessdemandfunctionzl(p) =xl1+xl2 l, (p 1p 2)(a)Todeterminep ,onlyneedto SeePS1Q63 MultpleWE CancertainlyhavemultipleWE,asyouwillhave aWEanytimetheO erCurvesintersectmorethanonce ecttodi erentnumeraires(do ctoredabit,MWG521) Non-existanceofWE (a)Endowmentontheb oundary(b)Consumer2onlydesiresgo o d1andhasallgo o d1.(c)Priceofgo o o d1(butwon't).Butalsopreferstoreceivemore ofgo o d2atprice0.(d)Nop atwhichdemandsarecompatiblei.[SinceConsu mer1'sdemandforgo o d2isin niteatp2= 0] :thereisanx 1 1x 1, x 1 B(p ), (a)Inexample,consumer1'so ercurveisdisconnected,soitdo esnotintersectconsumer2's4 ParetoOptimum De nition:Anallo cationxintheEWBisParetoOptimalifthereisn ootherallo cationx intheEdgeworthb oxwithx i%ixiandi= 1,2andx i ixiforsomei.
5 AneconomicoutcomeisParetooptimaliftherei snoalternativefeasibleoutcomeatwhichever yindividualintheeconomyisatleastaswello ,andsomeindividualisstrictlyb ettero . Interiorsolution:consumers'indi erencecurvesthroughxmustb etangent Cornersolution:tangencyneednothold ParetoSet SetofallParetoOptimalallo cations ContractCurve ThepartoftheParetoSetwhereb othconsumersdoatleastaswellasattheirinit ialendowments 1stWelfareTheorem AnyWalrasianequilibirumallo cationx necessarilyb elongstotheParetoSet 2ndWelfareTheorem Verbally:Underconvexityassumptions(notre quiredfor1stWT),aplannercanachieveanydes iredParetooptimalallo cationbyappropriatelyredistributingwealt hinalump-sumfashionandthen lettingthemarketwork De nition.
6 Anallo cationx intheEWBissupp ortableasanequilibriumwithtransfersifthe reisapricesystemp andwealthtransfersT1andT2satisfyingT1+T2 = 0suchthatforeachconsumeriwehave:x i%x i x i R2+suchthatp x i p i+Ti Notethatp do esn'tchange;itisessentiallytheendowment thatismovedbythetransferstogettothenewx Failures Nonconvexities:Mayfailwhenpreferencesare n'tconvex(givenbudgetline,consumerwithno nconvexpreferencemaypreferdi erentp ointtointendedx ) Monotonicity:(Figure10(a)ab ove). isPO,butcannotb esupp ortedasanequilib-riumwithtransfers515C:T heOne-Consumer,One-Pro ducerEconomy Setup Singlepro ducer,single rm;b otharepricetaking.
7 X1=leisure(andlabor=z= L x1);x2=go o dpro ducedby rm Endowment: Lofleisure,0ofx2(althoughconsumerowns rmandgetspro ts,whicharep ositive) Continuous,convexandstronglymonotone%de nedoverx1(leisure)andx2 Pro ductionmaxz 0pf(z) wz Givenprices(p,w),[pispriceofx2,wiswage] rmhasoptimallab ordemandz(p,w),outputq(p,w)and (p,w) Consumptionmax(x1,x2) R2+u(x1,x2) w( L x1)+ (p,w) Budgetconstraintisfromwageearningandpro tfrom rm WalrasianEquilibrium Pricevector(p ,w )atwhichconsumptionandlab ormarketsclear:x2(p ,w ) =q (p ,w ) L x1(p ,w ) =z(p ,w )6 (b)isnotanequilibrium.
8 Aparticularconsumption-leisurecombinatio nisanWE itmaximizestheconsumer'sutilitysub jecttotheeconomy'stechnologicalandendowm entconstraints Graph Of(pro ducer'sorigin)inSEcorner;takesx axisas z Oc(consumer'sorigin)inSWcornerasusual Budgetconstraintlinehasslop e wpandintercept (p,w) tlineofthe rm'spro tmaximizationproblem:( z,q) :pq wz= (p,w)16:EquilibriumandItsBasicWelfarePro p ertiesBasicMo delDe nitions Iconsumers,Jconsumers,Lgo o ds ConsumericonsumptionsetXi RL Preferences%ide nedonXi,rational(complete&transitive) Endowmentvector = ( 1,.., L) RL Firmpro ductionset(or technology )Yj (nonempty), ,technologiesandresourcesforthiseconomya resummarizedby({(Xi,%i)}Ii=1,{Yj}Jj=1, )FeasibleAllo cation(x,y) De nition:Afeasibleallo cationisavector(x1.)
9 ,xI,y1,..,yI) Xi Yj j3. ixi= + jyjThesetoffeasibleallo cationsisdenotedbyA={(x,y) X1 XI Y1 YI: ixi= + jyj} RL(I+J)ParetoOptimum De nition:Afeasibleallo cation(x,y) AisParetoOptimalif@(x ,y ) i%ixi iandx i Endowment:Eachconsumerihasinitialendowme nt i RL Shareholding:Eachconsumerihasshareholdin g ijin rmj ij [0,1] i,j i ij= 1 jDataonpreferences,technology,resourcesa ndownershiparesummarizedby({(Xi,%i)}Ii=1 ,{Yj}Jj=1,{ i, i1,.. iJ}Ii=1)WalrasianEquilibriumGivenaprivat eownershipeconomysp eci edby({(Xi,%i)}Ii=1,{Yj}Jj=1,{( i, i1,.., iJ)Ii=1}),Anallo cation(x ,y )andapricevectorp= (p1.
10 PL)constituteaWalrasian(orcomp etitive) yj p y j yj Yj(a)[ max:Thatis,foreveryj, y jmaximizespro tsinyj] ,x iismaximalfor%iinthebudgetset{xi Xi:p xi p i+ j ijp y j}(a)[Hencexi ix i= p xi> p i+ j ijp y j](b)[ Consumersmaximizeutilitysub jecttobudetconstraint,andanythingstrictl yb etterisunattainable ]3. ix i= + jy j(a)[Demand=supply(Lequations,oneforeach go o d)]Notethatifp = 0,thenallaresatis edexcept(2),sincecanb eanotherbundlewithmoreofago o dthatisa calNonsatiation De nition:Thepreferencerelation%iontheconsu mptionsetXiisLNSifforeveryxi xiandevery >0, x i x i xi andx i ixiNotethatthisassumesdivisibilityofgo o ds,soitcan'thapp : Supp ose%iisLNS i Let(x ,y ,p )b eaWE Then(x ,y )isPOPro of: Supp osethat(x ,y ,p)isapriceequilibriumwithtransfersandth attheasso ciatedwealthlevelsare(w1.
