倫敦經濟學院高微一般均衡倫敦經濟學院高微一般均衡倫敦經濟學院高微一般均衡倫敦經濟學院高微一般均衡倫敦經濟學院高微一般均衡倫敦經濟學院1ExcessDemandDecentralis-ationandTradeAmany-personeconomyRobinsonCrusoe'sProblemExcessDemandDecentralis-ationandTradeRobinsonCrusoe'sProblemGeneralEquilibriumWhyprice-taking?Whyprice-taking?Overview...ExcessDemandDecentralis-ation2There'sonlyoneeconomicactorThereforetherecanbenointeractionPricesareeitherexogenous(fromthemainland?theworld?Mars?)orhypotheticalWhyisRobinsonCrusoesoboring?There'sonlyoneeconomicacto3Manyactorsintheeconomy......andthepossibilityoftheirinteraction.Theendogenisationofpricesintheeconomy.Butwhatdowemeanby"aneconomy"...?weneedthistogivemeaningto"equilibrium"OnwardfromCrusoe...Manyactorsintheeconomy...w4Theseencapsulatealmostallthestepsinthecoursesofarresourcestockstechnologypeople'stastesThreemajorcomponentsTheseencapsulatealmostallt5Resources

nhoftheseHouseholds(preferences)U1,U2,...Firms(technologies)F1,F2,...nfofthesenoftheseR1,R2,...Whatisaneconomy?Resources

nhoftheseHousehold6GeneralequilibriumRobinsonCrusoe’sproblemDecentrali-sationandTradeAmany-personEconomyWhyprice-taking?ExcessdemandIncomesEquilibriumIncomesAllocationsEquilibriumAmany-personeconomyGeneralequilibriumRobinsonDec7ofconsumptionbasketsforeachhousehold...Thinkofthisastwocollections...andofnetoutputsforeachfirm.…likethis.[x1,x2,x3,...][y1,y2,y3,...]Thestateofaneconomyisdescribedbyanallocation...ofconsumptionbasketsThinkof8Formally,anallocationis...Acollectionofbundles(oneforeachhousehold)Acollectionofnet-outputvectors(oneforeachfirm)Asetofprices(usedby

householdsandfirms)p[xh][yf]profit-maximisingutility-maximising..andacompetitiveallocationis...

Formally,anallocationis...9p

yf(p)(profitmaximisation){yf(p):f=1,2,...,nf}Eachfirm'sbehaviouralresponsemapspricesintonetoutputsp,Mhxh(p)(utilitymaximisation){xh(p):h=1,2,...,nh}Eachhousehold'sbehaviouralresponsemapspricesandincomesintodemandsTheallocationajustaminute!wheredotheseincomescomefrom???Howdoesacompetitiveallocationwork?pyf(p)(profit{yf(p):f=1,2,..10GeneralequilibriumRobinsonCrusoe’sproblemDecentrali-sationandTradeAmany-personEconomyWhyprice-taking?ExcessdemandIncomesEquilibriumAllocationsAllocationsEquilibriumAmany-personeconomyGeneralequilibriumRobinsonDec11ownershiprightsofeverythingontheislandWheredidR.Crusoe's"income"comefrom...?ownershipWheredidR.Crusoe's12ResourcesR1h,R2h,...V1h,V2h,...Sharesin

firms'profitsSowhatdoeshouseholdhpossess?0Vfh1

Rih0ResourcesR1h,R2h,...V1h,V2h13ResourcesSharesinfirms

NetoutputsPricesProfitsRentsThecomponentsofincomelet'slookat

theroleof

pricesIncomesResourcesProfitsRentsThecompo14Netoutputsdependonprices...yif=yif(p)(supplyofnetoutputs)P

f(p):=S

piyif(p)ni=1directlyindirectlyAndsoprofitsdependonprices...Netoutputsdependonprices..15...andIncomes=resourcerents+profits

Mh

=S

piRih

+

S

VfhP

f(p)

nnfi=1f=1P

f(p):=S

piyif(p)ni=1Mh

=Mh(p)...soincomesdependonpricesProfitsareafunctionofprices......andIncomes=resourcere16{yf(p):f=1,2,...,nf}{xh(p):h=1,2,...,nh}So,taketheallocationasacollectionofresponses...pp,MhTheallocationa...andsubstitutetheprice-incomerelationintohouseholdresponses...

p...whichimplies:Mh

=Mh(p){yf(p):f=1,2,...,nf}{xh(p):h=117[yf(p)][xh(p)]TheallocationapLet'slookatthewholeprocess.Writtenmoresimply...[yf(p)][xh(p)]Theallocation18dR1a,R2a,...R1b,R2b,...V1a,V2a,...V1b,V2b,...Propertydistribution

resourcedistributionshareownershipCompetitivesystemtakesasgiven...dR1a,R2a,...R1b,R2b,...V1a19...andthroughthepricemechanismdresourcedistributionshareownershippriceshousehold

incomesaconsumptionbundlesnetoutputvectors...determinesanallocation...andthroughthepricemecha20dapricesasummaryoftheprocessdapricesasummaryoftheproce21GeneralequilibriumRobinsonCrusoe’sproblemDecentrali-sationandTradeAmany-personEconomyWhyprice-taking?ExcessdemandIncomesEquilibriumAllocationsAllocationsIncomesAmany-personeconomyGeneralequilibriumRobinsonDec22Crusoeagain...Mustbeutility-maximising(consumption)......profit-maximising(production).......andsatisfymaterialsbalance(thefactsoflife)OK,let'sdothisforthemany-personcase.Whatkindofallocationisanequilibrium?Crusoeagain...Mustbeutility23foreachhouseholdsubjecttoUh(xh)maxS

pixih

￡Mhni=1subjecttoFf(yf)￡

0maxS

piyifni=1foreachfirmxi

￡yi+Riandforeachgood...Butwhatdeterminestheseaggregates?ThebasicsofcompetitiveequilibriumforeachhouseholdsubjecttoUh24Canweaddup

acrosshouseholdslikethis...?S

xih

=xinhh=1Yesifiisa

rivalgoodCounterexamples:

goodslikeTV,

nationaldefence...Aggregateconsumptionof

goodi...??Canweaddup

acrosshousehold25S

yif

=:yinff=1definitionifalltheyfarefeasiblewillybefeasible?YesiftherearenoproductionexternalitiesCounterexample:

productionwithcongestion...AggregatenetoutputSyif=:yinfdefinitionifall26Assumeincomesaredeterminedprivately

allgoodsare"rival"commodities

TherearenoexternalitiesTomakelifesimple:Assumeincomesaredetermined27(a)Itmustbeacompetitiveallocation

x￡y+RThematerialsbalanceconditionmustholdCompetitiveequilibrium:asummary(a)Itmustbeacompetitive28Exchangeeconomy(noproduction)2traders(Alf,Bill)2GoodsAnexampleExchangeeconomy(noproductio29resourceendowmentconsumptionutilityUa(x1a,x2a)(x1a,x2a)(R1a,R2a)Alfsellssomeofhisendowmentof2forgood1bytradingwith...AdescriptionofAlf...resourceconsumptionutilityUa(x30Bill,ofcourse,sellsgood1inexchangefor2Ub(x1b,x2b)(x1b,x2b)(R1b,R2b)resourceendowmentconsumptionutility...BillBill,ofcourse,sellsgood131x1aR1aR2ax2aOaAlf'soptimisationproblemIncreasingpreferenceAlf'sbudgetconstraint

endowmentpoint

x1aR1aR2ax2aOaAlf'soptimisati32Bill'sbudgetconstraintx1bR1bR2bx2bOb

IncreasingpreferenceBill'soptimisationproblem

endowmentpointBill'sbudgetconstraintx1bR1b33Dx2aOax1a

E

equilibriumendowmentTheequilibriumofAlfDx2aOax1aEequilibriumendowm34x1bR1bR2bx2bOb

E

equilibriumendowmentDTheequilibriumofBillx1bR1bR2bx2bObEequilibriume35x1bR1bR2bx2bOb

E

DFlipped...x1bR1bR2bx2bObEDFlipped...36R1bR2bObx2bx1b

E

DFlipped...R1bR2bObx2bx1bEDFlipped...37...togivetheEdgeworthBoxDEx2bObx1bx2aOax1aR1a+R1bR2a+R2bThesizeoftheEdgeworthBox...togivetheEdgeworthBoxDE38DEx1ax2ax2bx1bObOaendowmentcompetitiontakesAlf&Billfromhere...

DEx1ax2ax2bx1bObOaendowmentcom39EDx1ax2ax2bx1bObOaequilibrium...tohere.

EDx1ax2ax2bx1bObOaequilibrium.40EDx1ax2ax2bx1bObOaIncomesinthedistributionmatchexpendituresintheallocationTheroleofpricesiscentral...EDx1ax2ax2bx1bObOaIncomesint41TheCrusoeequilibriumstorytranslatestoamany-personeconomy.Roleofpricesinallocationsandequilibriumiscrucial.Equilibriumdependsondistributionofendowments.MainfeaturesareinthemodelofAlfandBill.But,why,why,whydotheseguysjustacceptthegoingprices...?…that’sfornexttime.AlfandBillasamicrocosm.TheCrusoeequilibriumstoryt42謝謝觀賞！2020/11/543謝謝觀賞！2020/11/543倫敦經濟學院高微一般均衡倫敦經濟學院高微一般均衡倫敦經濟學院高微一般均衡倫敦經濟學院高微一般均衡倫敦經濟學院高微一般均衡倫敦經濟學院44ExcessDemandDecentralis-ationandTradeAmany-personeconomyRobinsonCrusoe'sProblemExcessDemandDecentralis-ationandTradeRobinsonCrusoe'sProblemGeneralEquilibriumWhyprice-taking?Whyprice-taking?Overview...ExcessDemandDecentralis-ation45There'sonlyoneeconomicactorThereforetherecanbenointeractionPricesareeitherexogenous(fromthemainland?theworld?Mars?)orhypotheticalWhyisRobinsonCrusoesoboring?There'sonlyoneeconomicacto46Manyactorsintheeconomy......andthepossibilityoftheirinteraction.Theendogenisationofpricesintheeconomy.Butwhatdowemeanby"aneconomy"...?weneedthistogivemeaningto"equilibrium"OnwardfromCrusoe...Manyactorsintheeconomy...w47Theseencapsulatealmostallthestepsinthecoursesofarresourcestockstechnologypeople'stastesThreemajorcomponentsTheseencapsulatealmostallt48Resources

nhoftheseHouseholds(preferences)U1,U2,...Firms(technologies)F1,F2,...nfofthesenoftheseR1,R2,...Whatisaneconomy?Resources

nhoftheseHousehold49GeneralequilibriumRobinsonCrusoe’sproblemDecentrali-sationandTradeAmany-personEconomyWhyprice-taking?ExcessdemandIncomesEquilibriumIncomesAllocationsEquilibriumAmany-personeconomyGeneralequilibriumRobinsonDec50ofconsumptionbasketsforeachhousehold...Thinkofthisastwocollections...andofnetoutputsforeachfirm.…likethis.[x1,x2,x3,...][y1,y2,y3,...]Thestateofaneconomyisdescribedbyanallocation...ofconsumptionbasketsThinkof51Formally,anallocationis...Acollectionofbundles(oneforeachhousehold)Acollectionofnet-outputvectors(oneforeachfirm)Asetofprices(usedby

householdsandfirms)p[xh][yf]profit-maximisingutility-maximising..andacompetitiveallocationis...

Formally,anallocationis...52p

yf(p)(profitmaximisation){yf(p):f=1,2,...,nf}Eachfirm'sbehaviouralresponsemapspricesintonetoutputsp,Mhxh(p)(utilitymaximisation){xh(p):h=1,2,...,nh}Eachhousehold'sbehaviouralresponsemapspricesandincomesintodemandsTheallocationajustaminute!wheredotheseincomescomefrom???Howdoesacompetitiveallocationwork?pyf(p)(profit{yf(p):f=1,2,..53GeneralequilibriumRobinsonCrusoe’sproblemDecentrali-sationandTradeAmany-personEconomyWhyprice-taking?ExcessdemandIncomesEquilibriumAllocationsAllocationsEquilibriumAmany-personeconomyGeneralequilibriumRobinsonDec54ownershiprightsofeverythingontheislandWheredidR.Crusoe's"income"comefrom...?ownershipWheredidR.Crusoe's55ResourcesR1h,R2h,...V1h,V2h,...Sharesin

firms'profitsSowhatdoeshouseholdhpossess?0Vfh1

Rih0ResourcesR1h,R2h,...V1h,V2h56ResourcesSharesinfirms

NetoutputsPricesProfitsRentsThecomponentsofincomelet'slookat

theroleof

pricesIncomesResourcesProfitsRentsThecompo57Netoutputsdependonprices...yif=yif(p)(supplyofnetoutputs)P

f(p):=S

piyif(p)ni=1directlyindirectlyAndsoprofitsdependonprices...Netoutputsdependonprices..58...andIncomes=resourcerents+profits

Mh

=S

piRih

+

S

VfhP

f(p)

nnfi=1f=1P

f(p):=S

piyif(p)ni=1Mh

=Mh(p)...soincomesdependonpricesProfitsareafunctionofprices......andIncomes=resourcere59{yf(p):f=1,2,...,nf}{xh(p):h=1,2,...,nh}So,taketheallocationasacollectionofresponses...pp,MhTheallocationa...andsubstitutetheprice-incomerelationintohouseholdresponses...

p...whichimplies:Mh

=Mh(p){yf(p):f=1,2,...,nf}{xh(p):h=160[yf(p)][xh(p)]TheallocationapLet'slookatthewholeprocess.Writtenmoresimply...[yf(p)][xh(p)]Theallocation61dR1a,R2a,...R1b,R2b,...V1a,V2a,...V1b,V2b,...Propertydistribution

resourcedistributionshareownershipCompetitivesystemtakesasgiven...dR1a,R2a,...R1b,R2b,...V1a62...andthroughthepricemechanismdresourcedistributionshareownershippriceshousehold

incomesaconsumptionbundlesnetoutputvectors...determinesanallocation...andthroughthepricemecha63dapricesasummaryoftheprocessdapricesasummaryoftheproce64GeneralequilibriumRobinsonCrusoe’sproblemDecentrali-sationandTradeAmany-personEconomyWhyprice-taking?ExcessdemandIncomesEquilibriumAllocationsAllocationsIncomesAmany-personeconomyGeneralequilibriumRobinsonDec65Crusoeagain...Mustbeutility-maximising(consumption)......profit-maximising(production).......andsatisfymaterialsbalance(thefactsoflife)OK,let'sdothisforthemany-personcase.Whatkindofallocationisanequilibrium?Crusoeagain...Mustbeutility66foreachhouseholdsubjecttoUh(xh)maxS

pixih

￡Mhni=1subjecttoFf(yf)￡

0maxS

piyifni=1foreachfirmxi

￡yi+Riandforeachgood...Butwhatdeterminestheseaggregates?ThebasicsofcompetitiveequilibriumforeachhouseholdsubjecttoUh67Canweaddup

acrosshouseholdslikethis...?S

xih

=xinhh=1Yesifiisa

rivalgoodCounterexamples:

goodslikeTV,

nationaldefence...Aggregateconsumptionof

goodi...??Canweaddup

acrosshousehold68S

yif

=:yinff=1definitionifalltheyfarefeasiblewillybefeasible?YesiftherearenoproductionexternalitiesCounterexample:

productionwithcongestion...AggregatenetoutputSyif=:yinfdefinitionifall69Assumeincomesaredeterminedprivately

allgoodsare"rival"commodities

TherearenoexternalitiesTomakelifesimple:Assumeincomesaredetermined70(a)Itmustbeacompetitiveallocation

x￡y+RThematerialsbalanceconditionmustholdCompetitiveequilibrium:asummary(a)Itmustbeacompetitive71Exchangeeconomy(noproduction)2traders(Alf,Bill)2GoodsAnexampleExchangeeconomy(noproductio72resourceendowmentconsumptionutilityUa(x1a,x2a)(x1a,x2a)(R1a,R2a)Alfsellssomeofhisendowmentof2forgood1bytradingwith...AdescriptionofAlf...resourceconsumptionutilityUa(x73Bill,ofcourse,sellsgood1inexchangefor2Ub(x1b,x2b)(x1b,x2b)(R1b,R2b)resourceendowmentconsumptionutility...BillBill,ofcourse