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CIRJE-F-761

Non-Self-Averaging in Macroeconomic Models:A Criticism of Modern Micro-founded Macroeconomics

Masanao AokiUniversity of California, Los Angels

Hiroshi YoshikawaUniversity of Tokyo

September 2010

Non-Self-AveraglnglnMacroeconomicModels:ACriticismofModernMicro-founded

MacroeconomlCS

MasanaoAoki'

DepartmentofEconomlCS

UnlVerSltyOfCallfornla,LosAngeles

FaxNumberi-310-825-9528,aokl◎econuclaedu,

andHlrOShlYoshlkawa

FactlltyofEconomlCS,UnlVerSltyOfT〇kyo

FaxNumber81-3-5841-5521,yoshlkawa@e.u-tokyoacJp

September13,2010

Abstract

WllCllthccocthcllmLofvallatlOu;llalllelytIICStalldalddcvlatlO"ldaLIVCLomeanapproacheszeroasthenumberofecononllCagentsbecomeslarge,t･hesystemlSCalledself-averagingOtherwISe,1日snor)-sc】f-averagingMosteconomicmodelsL礼keltrOrgr(lnt･edl･hateconomicSySt･emlSSelf-averaglngHowever,theyalebasedonanextremelyunreallStlCassumptionthatallLheeconomlCagentsfacethesameprobabllltydlStrlbut)onOncethIsunrcallStlCassumptionlSdropped,non-self-aveLaglngnaturallyemerges-Usmgaslm-PlestochaStlCgrowthrnodelthisPaperdemonstratesthatthecoefR｡1entOfvarlatlOnOfaggregateoutputorGDPdoesnotgotozeroevenlfthenum-beroLsectorsoreconormCagentsgoest01nfinltyNon-self-averaglnglrnPlleSthatevenlfthenumberofeconomlCagentsISlarge,dtsperslOnCanremainslglllflCa】lt,alld,thcICforc,tllatW C CallllOtkgltllnatClyfocusoilthelllCallSofaggregatevanableslt,tnturn,meansthatthestandardmlCrOeCOnOrnlCfoundationsbasedonrepresentativeagentshasllttlevaluefortheyaremeanttoprovldeusw】thaccuratedyna,mlCSOfthemeansofaggregatevarlablesContrarytothemanstreamvleW,mlCrO-foundedmacroeconomlCSSuchasadynamicgeneralequlllbrlummodeldoesnotprovidesolidmicrofoundations

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TnthlSPaper,Weforma)12.eSolo､V'scriticISm､Vlththehelpoftheconceptofnon-scl/-averagmg.andarguethattheresearchprogrambasedonrep-resentatlVeagentsWhlChprevailslnmodernmacroeconomlCSISmlSguldedWhctllerlngrowthorbuslnCSSCyclemodels,thefundamentalcauseForoftencomplexopt)m】zatlOnexercisesISthattheyareexpectedtoleadustoourbetterunderstandmgdynamlCSOfthemeansofaggregatestochastlCVa.r】一ablesThestandardmodelthusbeglnSWlthoptlmlZatlOnOfrepresentaLIVeagent,andhornothetlCallytlanSlatesltIntotheanalystsOftheeconomyasawllOle

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4

canbewellcapturedbytheanalystsbasedonarepresentatlVeagentThesameasumpt10nlSCOmmOmlymadeinthestandardsearchmodelssuchas

MortensenandPISSarldes(1994),LucasandPrescott(1994)Though apparentlyheterogeneousagents,firmsorsectorsareIntroduced

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Tnthenextsect10n,We6rstexplaintheconceptofnon-set/-aveTlagm9,thecrucialconceptforourpurposelnthepresentpaper AfterdlSCuSSlngthefundamentalproblemofthestandardmicro-foundedmodelslnSecも10n

3,Section4demonstratesthatnon-self-averagingemergesVerynaturallylnaslmPlemodelofeconomicgrowth

Tnnon-self-averaglngmodels,evenlfthenumberofeconomicagentsISlarge,thebehaviorOfthemacl.OeCOnOmyCannotbegenerallyweHappL'OXI-matedbythemeansThelmPllCatlOrHSthatanalysesbasedonrepresentatlVeagentwhlChgeneratethemeansofstochastlCtimePathsofaggregatevarトables,havellttlevalue Puいtanotherway,thesta･ndardmlCO-foundatlOnS

alenottruemlCrO-foundat10nSThefinalsectlOnO斤el.SCOnCludmgremarks

2 Non-self-averaglng

lnthlSSection,WeeXPlalntheconceptor"non-selhveragJng" ItlSnotknownmeconomlCS,but)sthecruClalcollCePtfoI.OurPurpose.Theterm

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averagmgAsolf-FLVC.raglngbcllaV-0日.SOneIl一WhlCht,hcc恥cr,sorthelndlVldualeventsaddLIPtOProduceanalmostdetermlmStlC

outcome AnexampleofthisISthepressuleOnamaCrOSCOPICsurface ThisPressure)SProducedbyhugenumbersOflndlVld-ualcolllS10nS,Whichproduceamomentumtransferperunittime

whlChsccmscsscntlallywlf,holltnuぐt.llatlOnSThelar酢rt.hcnllm-berofcollislOnS,thelessistheuncertalntylnthepressure

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5

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mean,lS,therefore,､/文面/入N-1/､乃両 Thus,lnthePoISSOnmodel,whenthenumberofeconomlCagentsN becomeslarge(N-∞),thecoe用clentOfvarlat10napproachesze10ThlSPrOPertyknownasself-averagingProvidesuswithJuStl員CatlOnfo上OutCOrlCClltlatlllgOilthemcallSOfvallabieslnmaCIOrnodels Now,becausethemeandependsbasicallyonA,ltlSnaturaltoexplorehowAlSdeterminedlnmodels Indeed.1nStandardmodels.入lS

endogenouslydeterminedasallOutcomeOfeconomlCagentS'optlmlZratlOnandmarketequlllbrlum ThePoISSOnmodellSJustanexampleThenormal

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goestolnfinlty こくon-self-aNeraglnglSimportantbecausenon-self-averagingmodelsaresampledependent.andasaconsequence,somedegreeoflm-precIS10nOrd1SPerSIOnremalnSaboutthetlmetraJeCtOrleSevenWhenthe

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W1-llP

W2-(1-p)p

W,I-(1-p)pn-1

whereWnlStheslZeOfthepleCekeptatthen-thstep

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t10nlSIb)Thera･ndomvalLablepISdrawnandfixedforeachfragmentaLtlOn

7

historyWewlllconslderavarlableYdefinedby

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Tldegreeoffragmentation(0<Y<i)YclosetooneIndicatesthepresenceofalargepleCeWhereasYclosetozeromeansthatallthepleCeSaremlnuteYISarandomvariablebecausepISrandom ltcanbeeasilyshownthatY,andconsequently,thesizeOfpleCeW,I1SnOn-Self-averaglng(Sorne比e(2000,p370)

WenotethattheprocessoffragmentatlOnlSPath-dependent ThlSIS110thlngbuta一lCXarnplcAs Kadal10ff(2000)states,theLealernanyexam-pleslnnaturalsclenCeS Wewュllla･terobservetha･tnon-self-averagingVerynaturallyariseslneconomicsaSWellBeforedoingSO,WeWilldiscussthefundamentalproblemofthestandardrnlCrO-foundedmacroeconomlCS

3 StandardMicro-foundedMacroeconomics

AlmostalltheeconomicOPPOltumtleSSuchasJoboffer.discoveriesOfnewtechnology,market,andresoul･CeSareStOChastlC.Thestandardmicro-foundedmacroeconomlCS- Lucas'ratlOnalexpectat10nSmodel,realbuslneSSCycletheoly,laboISearchtheory,andendogenousgrowththeory- rightlytakesIntoaccountStOChastlCeventsHowever,lnthesemlCrO-foundedmodels,ltlStakellforgrantedtllataSthe-つumberofage-1tSgoestO]nfhlty,anymlCTOor"ldyosynclatlC"flucttlatlOnSVan-Sh,andthatwelldefineddetcnT"nlStlCrnacroeconornJCrelaLIOnSPrCVa]lThatlS,Self-averaginglStaCltlypresumed

SeLf-averagingemergesWheneveryagentWassumedLo/ace仏esameunchanged"uJelL-behaved"pTTObabtlltydtstrlbuLwnsuchasthenormalandPoISSOndlStrlbuL10nS This)SIndeedthest,andardassumptionlnmlCrO-foundedmacL･OeCOllOlnlCSLucas(1972,73)'sfamousmodelofbuslne;SCyclesISaPrlmar.yeXarnl)】e ltlSmStruCtlVetOtlaCehlSmodellndetailLucasbeglnStOmodelasLIPpller'sbehaviorlneachlndlVl(lualmarketasfollows,

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ofthesecomponents;supplylnmarketIIS

(1) yt(I)-y山+yet(I)

ThecycllCalcomponentvarleSWithpercelVed,reLatweprlCeSandwithItsOwnlaggedvalue

(3) yct(I)- JylPt(I)-E(PHIt(I))1+入yc､トl(I)

8

WherePt(I)lStheactualpricelnIatiandE(PH It(.Z))lSthemeancurrent,generalprlCelevel,condltlOnedonlnformatlOnavallablemIati,It(2=)･"

GIVenth】sframework,hegoesontothelnformatlOnStructureOftheeconomy

''ThelnformatlOnavailabletosupplierslnIatiCOmeSfromtwosourcesFlrSt,tradersenterper10diwlthknowledgeofthepastcourseofdemandshifts,ofnormalsupplyy,W,andofpastdevlatlOnSyc.ト 1,yc,ト2,･WhllethlSInformationdoesnotper-mltexactmferenceofthelogofthecurrentgeneralprlCelevel,PI,ltdoesdeterm】neaL"Pr10r"dlSt,rlbut10nOnPt,commontotraderslnallmal･ketsWeassumethatthlSdlStrlbut10nlSknwon

tobenormal⊃wlthrneandPt(dependinglna.knownwayontheabovehlStOry)andaconstantva,rlanCeO12 second,wesupposetha･ttheactualpricedeviatesformthe(geometric)economy-WideaveragebyanamountwhlChlSdlStrlbutedlndependentlyofPtSpec)Scally,lettllePercentagedevlatlOnOfthepricelnIfromtheaveragePlbedenotedbyI(SothatmarketsarelndexedbytheirprlCedeviatlOnSfromaverage)､vhereIISnormallydlSLrlbuted,lndependenLoFPL,WILhmeanrZeroandvarlanCeT2 Thentheobservedpr)celnI,PL(I)(lnlogs)-sthesumol-ndependent,rlOmlalvaLlateS

(4) PL(I)-PL+I

ThelnformatlOnIL(I)lelevanttorest)matlOnOItheunobserved

(bysuppllerSlnIatt),PLCOnSIStSt_henoftheobservedprlCePL(I)andtheh.storysumrnarはedlnPL(Lucas(1973,p-328))"

ⅠいsextremelyImportanttOnotethatthough zvarleSfrommarkettomarket,thed)str】butlOl-OfI】suniquelyglVenandsharedbyauthemarketsWealsonotethatmthlSmOdeい11arketsarelndexednotbylntrlnS】C】dentltysuchassteelIndustrybutbythelrPl】CedevlatlOnSfromaverage Eachmarketfindsareal】zatlOnOrIeachperiodglVenthesamedlStrlbutlOnOfITheassumptionOfrationalexpectatIonsthenpet.mltSSuPPlleL'SlnlndlVldualJLLalketstolllakeefhclerlいuferetlCeSOILthelelatlVCpllCeS TlllSleadstomlCrOSupplyfunctionsGivenmlCrOSupplyfunctions,theaggregatesupplyfunction,lStriviallyderlVed一

"ToutlllZethislnformatlOn,SupPllerSuse_(4)tocalculatethedlStrlbutlOnOfPt,condltlOnalonPL(I)andPtThlSdlStrlbutlOnlS(bystraightforwardcalculation)normalwlthmean

(5) E(PHlt(I))-E(PtlPt(2･),Pt)-(1-8)Pt(I)+Opt

9

where0-72/(q2+72),andvar)anceOcr2 comblmng(1),(3),and(5)yleldsthesupplyfunctionformarketI

(6) yt(I)- ynt+OrytPt(I)-Pt】+Aye.t_l(I)

Avera,glngOvermarkets(IntegratingwithrespecttothedlStr)bu-tl｡nOfI)givestheaggregatesupplyfunction

(7) yl-ynt十07(Pt-PL)+袖 t_1-yn.i_ll

(Lucas(1973,p328))"

TheaggregatesupplyfunctionlSthecoreofLucas'rat10nalexpecta.tlOnSmodelofbus)nesscyclesJnthismodel,thecruclalassumptlOnlSh)sequation(4)aboveMorespcClnCally,tIICrandomv'TLrlablcIISassumedtobenormallydlStrlbutlOnWithmeanzel･OandvarianceT2 As polntedoutearlier.eachsupplierfacesthesameprObabllltydlStrlbutlOnOfmlCrOShockalthougl1areal1ZatlOnOrSuchashocl(,Ofcourse,differsacrosssuppllerS

Thesameassumption)SrOutlnelymade,andtakenbymosteconomlStSaS

InnOCl10uSTtlSaSlfthcrcwc.rcnootherwayoFdc6nlng仇 emtCrOOrtdyosyn-craLICShocksflowever,thestar)dar° aLSSumptlOnmeansthatMIC10SOft,andsmallgrocerystoreonthestreetfacemicroShocksdrawnfromthesameunchangedprobabllltyd】StrlbutionlltpresumeshomogeneityWlthrespect

Lotheprobabilitydw tmbutwno/mlCrOShocksThoughltlSrOutlnelymade)nstandardmCrO-foundedmodels,LhlSassumptionlSextremelyumeallStlC,anclasSolow(2004)notes,almosta,bsurd

Lucas'modelemphasIZeSthel､01eofmCrOShockswhlChbydefimt10rLdlだeracrosssectorsoragelltSInthisSense,ltapparentlyrejectsrepreSen-LatlVeagentHowever,llkeothelmlC10-roundedmacromodels,1日sbulltonthecrucla)premisethateveryagentfacesthesameunchangedprobabLlltydlStrlbutlOnOfmlCrOShocksThlSaSSumptlOnenLallsself-a.veraglngSpec汁･lCally,1nhlSmodel,onecaneasilyobtalntheaggL.egateSupplyfunctionby"averaglng OVermarkets(IntegratingWlthrespecttothedlStr)buttonOfI)"NotethathlSagglegateOutputyL(h】sequatlOn(7))1SnOthlngbutthemeanc･fstochastlCaggregateOutput

T llClulldalTIClltalplOblcllllSILOtCOllt"Cd toLucas(1972,73)lat1011alexpectationsrnOdelbutalsoappliestOSearchtheory LucasandPrescott(1974)'smodelofequllllnlumSearchandunemploymentISaPrlmaryexam-ple.Themodel,lntheauthors'owndescrlptlOn,lSaSfollows

"WethlnkofaneconomylnWhlChproductionandsa一eofgoodsoccurinalargenumberofspatlallydlStlnCtmarketsProd-uctdemandlneachmarketshiftsstochastlCally,drlVenbyshockswhlChareIndependentovermarkets(SothataggregatedemandlSconstant)butautocorrelatedwlthmasinglemarketOutputtosatlSfycurrentperioddemandlSProducedlnthecurrentperiod,wlthlaborastheonlyInputEachproductmaJketlSCOmpetltlVe.

10

TherelSa.COnStantWOrkforcewhichatthebeginningOfape-rlOdlSdlStrlbutedlnsoれeWayovermarketsIneachmarket,la-borlSallocatedover伝rmscompctltlVelywlthactua.1moneywagesbelngmarketclearingEachworkermayeitherworkatthlSWagerate,lnWhlChcasehewlllremainlnthlSmarketIntothenext

penod,orleave Ifheleaves,heeamsnothingthisPeriodbutentersa"pool"ofunemployedworkerswhicharedlStrlbutedlnsomewayovermarketsforthenextperlOd lnthlSWay,anewworkforcedlStrlbutlOnlSdetermined,newdemandsare''drawn".andtheprocesscontinues

Inthisprocess,allagentぶareassumedtobehaveoptlmallylnlightoftheirob｣ectlVeSandthemrormatlOnaVallabletothemFol丘lmS,thlSrllea】lSSllllplythatla.bollSelllployedtotllePOllltatwhlChItsma･rglnalvalueproductequalsthewagerateForwork-ers,thedecISIOntOWOrkortosearchlStakensoastomaxlmlZetheexpected,dlSCOuntedpresentva.lueoftheearnlngSStream lncarrylngOutthisCalctllatlOn,WOrkersareasstlmedtobeawaleofthevalues0fthevarla･blesaffectmgthemarketwheretheycurrentlyare(1e,demarLdaridworkforce)andofthetruep10ba-bllltydlSLrlbutionsgovernlngthefutuleStateOfthlSmarketandthepresentaLndfuturestatesofallothersThatlS,eXPeCtat)OnS

aretakentoberalwnaL(Lucas andPrescott,1974,p-190)"

MarketsareallcompetltIVe,SOthatthemarginalvalueproductoflaboreqllalsthewagemeverymarketHowever,thestateofdemandrepresentedbyareal1-/,atlOnOfasLochastlCVariablesdl庁elSaCIOSSmarketswhlleatthesametlme,mObllltyOHaborlSnotinstantaneousAsaconsequence,themaLglna】valllぐr)rOdllCfsoflahol,Tuldwagesdl軌 ､r,TtCrOSSm,Vkct5;Incolltr礼Sttothestandardgeneralequll)br】ummodel,productlVltydlSPerSionexIStSlnequlllbrlumaSaLCtuauyObserved)ntheeconomyOnceagain,theproblemlSthenatureofstochastlCequll)brlumlnthelrmodel

ThestochastJCdisturbances-nLuca5al-dPrescott(1974)arethedemandshlrtSSTheyareassumedtobeIndependentacrossmarketsandthenumberofrr]arket,sISlarge

"Bylarge,wemeaneitheracontinuumOfmarketsoracount-ablemfhltyEconomlCally,then,theassumpt10nOflndeper)dentdemandshlftsmeansthata99re9atedemandJStakentobecon-stantthroughi)me (Lucas andPrescott,1974;Footnote8onp192)"

ThemlCrOdlSturbancesareassumedtocarlCeleachother Self-averagingensues lnthlSWay,LucaSandPrescottcandescrlbethedetermlnatlOnOfthestatlOnalydlStrlbutlOnOfemployment,workforce】andwagesormarglnalvalueproductslnarePreSenta如emarket0nthelrOwnassumption,theystateasfollows

ll

"ThedistnbutlOnOftheworkforceoverlocal)ons(lndexedby(S,y))wouldlnthisCasebethesameasthesea.t10narydlStrlbu-tlOnOf(S,y)lnanyOnemarket(ThlSfollowsfromourassump-t10nSthatthenumberofmarketsISlargeandthatdemandshlftsareIndependentacrossmarkets) (LucasandPrescott,1974,p202)"

Thesa.meassumpt10nallowsLucas andPleSCOtttOfocusonthemeans

characterlStlCSOfwhlCharedescrlbedlnarepresentativemarketSpecifically,worker'ssearchdependscruclallyontheexpectedpresentvalueofsearch,入 Themaxlm12ratlOnexercises(SectlOn30fthelrPaper)aredoneontheaLSSumPt10nthat入iscommontoallthemarkets,andthat'つthesearchprocesselimlnaterentsOnaverage"

ModernmlCrO-foundedmacroeconomlCSemphasIZlngtheroleofmlCrO/ld10SynCratlCShocksisbu】ltonPhelps(1970)island-paradlgm Wecanuse-fullyInterpretPhelPSislandmodellntermsOfultrlameれctrees(seechaLpter50fAoklandYosh)kawa(2007))UltrarnetrlCtreesmeasurethe"distance"be-tweenmarketsoragentsWhatthe"dlStanCe''measuL.eSdependsonmodel,ofcourseForexample,thedistancebetweentwostatesaaridbmaylndlCatetransltlOnratefromatoblr)MarkovmodelNow,lngeneral,thedlStanCebctwcellhvol11とukcts/agclltSdlf上oISdCpClldlllgOILapallhiFlgulC1(a),a一lexampleofthree-leveltreelSShown lnthlSexample,thedlStanCebetweenmarkets/agents1and2)smucllCloserthanthaい)etween1and3,Grocery(agentl)andlaundryonthesanleStreet(agent2)are-Lcloser"toeachotherthantoMIC10SOrL(agent3)OrGM(agent1)andFord(agent2)areclosertoeachotherいlantOgrocery(agent3).

lncontrast,lnLheone-leveltreeshownlnF)gore1(b),al一themar-kets/agentsarcsymmetric.OnemaythinkofNmaL･ketslnthestandardnllCrO-foundedmodelsdiscussedlnthepresentsect10naSleavesoraone-1eveltreewlthNbranchesfromLhcrootThlSOrgaJllZat10nlSaVeryspec)alcaseofmulLl-leveltrees lntheone-leveltreearl･angementOfN markets,eachbranchlSthesam e asanyoLherbrancllbecausemarketsareldentICalbyasslJmPtlOn Then,everyoneofthelVmalketscanserveasarepresen-tativemat-ketMIxlngthesemarketsrandomlybylntrOducmgaprobabllltydlStr-butlOll,aSLucasandPrescottdolnthellpaper(tobespecific,thelrprobablllLydlStrlbutlOn◎011P198)doesnothlngtOthemodelThernlX-turelSldentlCa-ItoanyoneofthebrancheslthatlS,themlXturelSagaJnarepresentatlVemarketThis)StheessenceorPhell)S(1970)lSla血d-paradigmonwhichmodernmlCrO-foundedmacroeconomlCSISbullt

Tosumup,staLndardmlCrO-foundedmodelsotherthanrealbuslneSSCy-clemodelsallIntroduceapparentheterogeneityOfagentsandmarkets,andemphasIZ,etheroleofmicroOrldlOSynCratlCShocks However,contrarytoもhclmplCSSIOntheyglVC,thetacltaSSumPtlOllOfself-avclagHlgCffcctlVClymakessuchframeworksmodelsofarepresentativea.gentO1-market.

12

4 Non-self-averaglnglnaGrowthModel

InthisSection,WePresentaSlmPle)nnovatlOndrivengrowthmodellnWhich

aggregateoutputorGDP】snon-self-averaglng ThemalllPurposeOfthisexerciselSnottOadvanceamodelofeconomicgrowth,butrathertodemon-

stratehownaturallynon-self-a.veraglngemergeslneconomicmodel

TheM odel

FollowingtheliteratureOnendogenousgrowth,WeassumethattheeconomygrowsbyInnovationsInnovationsareShochastlCeventsTherearetwokmdsofinnovationslnOurmodelNamely,anlnnOVatlOn,WhenltOccurs,either

raisesPrOductlVltyOfoneoftheexIStlngSectors,OrCreatesanewSector

Thus,thenumberofsectorsISnotglVen,butIncreasesovertime

BythetlmenthmnovatlOnoccurs.thetotalofK.1Sectorsareformed

lntheeconomylVherelnthet-thsectorhasexperlenCedTい nnOVatlOnS(8=

1,2, jr.I) Byde･Gnlt･10n,thefollowmgeqllalltyholds

nl+n2+ +77,た-n (1)

whenKn - k lfn-thユnnOVatlOnCreatesanewSector(secotoric).thenTIL-I

TheaggregateoutputorGOPwhennInnovationshaveoccuredlSdenoted

byYn YnlSSimplythesumofoutputslnallthesectors,yl

J{,I

yn -写 yl. (2)

OutpuいnsecotrigrowsthankstolnnOVat10mSWhlChstochastlCaHyoccurlnthatsectorSpeclGcally,weassml-e

yt-･rn nt (rl>0,7>1)

Forourpurpose,ltlSCOnVenlenttOreWllteequatlOn(1)asfollows

71n-Eja,(n)

Inequation(4),a,(n).sthenumberofsectorswhereJinnOVatlOnShave

occurred-Thevectora(n)consIStlngOfa,(n),lSCalledparttlwnvecLorlWlththlSpartlt10nVector,a(n),I(,ICanbeexpressedas

Kn-E a,(n)

1seechaptc120fAoklandYoshlkawa･(2007)forpartltlOnVector

13

(5)

tl

Sl叩OulJOSaSSでl〇oLq∂SaqlJOUOI,rtHleAJOlu∂lJWt)OJ叫IJOIOl八叫aqJttOldutJ(5花叫1PJL【LuLCXaLtla=VJVd-uLOつ恥でq▲J払恥OqSl10lSnOtaldclqlJOauONaTelLtlJhOtBrelluDuOdx〇3uで碍【tO〇〇叩1SI(pqs3rLOqltlSl〇POuIJOSSrePl〇PeOlqeOl冨uolaqA〇LIMnqSIStutOuO33031でtltLLlqSTaPOut

ql1012t叩uC･uOdxc･10u310JCua耶31lZSPpOulaSDqLSllC朋でla耶attLt爪3ul3℃JaLYe-JlaS一oustlaulJOlat4つaSn'E'〇aquOtSlaJalau柑陀d-att()aTPutOJj叩alaDtPL(latlelTtf'rtbal'F!Slap(1Ul

latLPutローuOSStOdl叩uelVd一心叫D耶叩耶u凡OtlSSq(900乙)tlOVSlaW03uoLLrei(966T)JOputtreL岬dPu℃(乙00乙■666t)uttLultd∫(9661)addoHPuでBuad(666t)uo叫で〇■(C661)uqeuL凱lt〉I∂aSueLultdJ〔qsuot叩q叫STPlalq〇uTO-uOSSIOdl叩ItreJでd･o叫aLIIOIPaPllal-xouaLlISett〇pOulJ叩ULLでI12d-つuoaqエuOL叩ueldxcIl〇LtTlrg101(qDOO石l℃ooo乙)IIOV〇asI((ZL6T)su〇Jhヨ)PpoLLISua13SVu仇OuqOStでSLt叩Otul叩PIJlq-uOSStOdJ叩ureJでd-cILtO〇LILSl〇1Snt〇1〇LlllSll)01ulS叩｡8℃sno〇u〇Bol叩LtJO凱10T叩TndodJOSLtOtmredtuop

-ttet〇ql'〇S叩0日10■1nqu･JSIP一叩t〇ltlO-uOSSIOdt叩LLLで1℃d-DtTO〇耶P叩[〇ulLL℃L【t9ut〉l乙

(o)cld'uot3n(1･JlStPCdユ叩ul.C1'L,d-au°'aql

olu爪OPSt10quOtlnqIJISLPaLllpuretOlaIatul!J℃(I∂揮ulSでSt∂laL110-DW爪

t>V>oput!くo<･t)+C

･SuOTITpuOつ

3utJAOttOJaqlJ勺STl12S0PUtZVSlalauI℃1℃do叫aLILHO13aSulPaJJn〇〇OJぐpで∂･tTB∂∧t'Ll叩t11SuOl叩Ouut30Jaqulnu叫l01llヲnbast叫Lt101〇aSJO"arZIS"all

(9)三宝-ld

lda叩JI(ltllq12qO-1dLllT/Atu,【aZIS"JOS101〇∂S3utqSTXaaqlJOau°ut

s･1n〇〇OuOtreOuutu℡可u++乙u十Iu〒upuが(qlLZ--甘)ttusaZtSJOslalSnl〇31aJ℃aJa耶u叫lLuol叩qulS-P(C●V)UdJ叩tuで1℃(トoLq叩uat〇

石uOt7nqu7叩rCTcl)抑/〇uもCT-uOSSWdJa?∂uLVJVd-o叫叩S爪OllOJuOl叩∧OuuZ

uvTat)Ou∂耶uLJn〇〇OI(llで叩Sq｡01SSuO【叩∧OtJuT∧＼OLIaqtlつS叩JuOu;地I

SuO!T甘AOuuIJOuO!lnq!叫S!GlattIつtJtG-uOSS!OdatIL

lnつつOSuOtlでAOuul

＼otluOSPuadaptL'叩dlnOa叩3aJBBでatllJOJOt℃tlaqatllEL(tsnol∧qOauO

st血1叫1∂unSSreaJu'J(1ttVJaU3330SSOt叩OtJlt爪91∂H0<(i)ut-99Jaq爪

(i)

(9)

(u)EDr二三d+u)I籍utL

uTqqOa/AI(9)pu℡(S)く(ウ)i(Z)i(I)suol叩nbaUTOJJsnqL

Tu(ん)utLL+LL-鳩

sB(C)uotTenbaalu爪aluや〇aLVL

叱(i)ut+IRj(Ltlllu)dx9-...ん

uo耶tLm(OJd加3utAtotloJatI13utsn

qt

TaPOulCTdIapu陀red-∂uo〇耶uTu?qlt昨outCTdl叩uJでred10叫attlulJC･qBtLtStS娼IGrulaJOlつaS八〇u叫で耶Lqlll叩qOtd叫H叫13人laSqOa八

･Qil<_旦土竺CDq+0

spzoLI付1℃nb〇uT凱‖AoHOl叩〇Sn12〇咽ウ(SVddv

901｡Ds◆乙00乙)■'tOVutP〇SSn〇STP｡t叫｡PotltSLT〇1ElBLtlI叩【〇S〇dJq爪〇ulOS〇Tm一叩｡lOLdc

叩0爪aJqaJa耶乞sTII3u伽-1∂Ne-JPSST(0-ィ,)1叩OuUd-叩u,･eユpL,dau°

a耶tLtlgutlSa-1叫uISuT3℃JaBJTaS-uOuStTaPOuIローcTJal∂ult,ユTZd-o叫aLtL

xtptIaddvaLtlu川atBsTJOOldatIL

BuT3℃Ja℃JtaS-uOu

sttt叩dlnOa叩BaJ富加印やpotLI叩TLP"tO-uOSStOd-TalaLu,L,ユ官d-o叫凱Ilul

tIO!VSOdoJd

uotltSOdoJd3ut爪OTtO3aqluTqqOa爪

TkI(TaUJでu'd〔lOJOIOINeLPCtattluTPa1SaLTalUta柑∂爪ltaPOLu叩uat三)

S7aPOuLPaPuTWJ

-oJ〇'LuLuJ叩OuLuヱaPVuL句∂u叩乃OJ叩OySOJ〇iuLuOuOi7duLnSSDuOuLuLOD∂耶

077gVJ7uODP∂qJVuLu之SiuOt7nqu7S一之P7∂7叩uEU-UOS-SWcT叩EPadsaJS叩uI

∂ultlJ∂AOSa芳u叩〇一usB句sTWuaeopu∂∂6uvy〇S叩ItTqtZqOJdaJ叩tJa0∂10N

s∂tno〇aqHO1〇aSu【SJn〇〇OUOl叩∧OuUIU℃叩tll巾tTtqreqOl(IaLrlJ叩∂J冒atTlst

もJOIDaSJa冨JpPtatlLgJ,n〇〇OuOi7mOuu…V7叩伽77qVqOJd∂7n077〇ads∂JWmsno∂u∂6ouLO7日0u∂1でS101〇∂ST叩OLuSTLllut叩ql∂10uOl叩叩OdLulSTll

･＼

stpa3u飢PLtnSulTZtu∂J

sJOl〇∂SJOJaqumUatll叩qlLrlll叩qOJdSut叩uOds∂110〇aql'pumtl叩03t11uO

(ot)蓋荒-a-(q-u,I'瑚T十-+u,I)Jd

vJ(qu∂At冨st

L31-uXUOT甘uO叩PuO〇I+u01uUIOIJ∂uOLqs∂sLe∂J〇ulSJOlDaSJOlaqtLmu

∂t11叩WJCTllTtqでqOld∂Lt1uol叩q叩Stp(etv)ltdl∂阿LIでJでd10叫叩uI

･6)崇芝芸i-I-a†

dc叩JLqTTl叩qCudW爪S∂宕J∂tualOl〇aS爪auで爪ONSJOl〇aS

男uTISTXaaqlJOauOulSln〇〇OuO耶∧OtTutu℃叩耶JqTT-q℃qOJd∂WSrtd

lnqulrlngWhythetwo-parameterPDmodellSnOn-SeLfaveraglngTheanswerlleSユn(10)a.nd(ll) Inth)smodel,lnnOVatlOnSOccurlnOneOfthetwodlffcrcnttypesofsectors,one,thelleWtypeandtheothcLrknownOrprC-exIStlngtypes Theprobab)lltythatan lnnOVatlOngeneratesanewSeCtOl.lS(e+Knα)/(n+8),andtheprobabllltythata･nlnnOVatlOnOccurslnOneoftheexIStlngSectorsIS(n-Knα)/(n+8),whereKTLISthenumberoftypesofsectorslnthemodelbythetlrnenmnOVat10nSoccurred TheseprobabllltleSandtheirratioVaryendogenously,dependingOnthehlStOrleSofhowInnovationsOCCured lnotherwords,themixOfoldandnewsectors

evolveendogenously,andlSPath-dependent-ThlSISthereasonwhynon-selfaveragingemergeslnthetwoparameterPDmodelInoneparameterPD

modellnWhichα-0,twoprobabllltleS(10)and(ll)becomeIndependentofK..,andthatthemodelbecomesself-averaglngThetwo-parameterPDmodellSfllndamc.ntallydlffcrcnt,fromst.'TLlldardmlCrO-foundedmodelswcdiscussedlnthepreviousSeCtlOn

5 Conclusion

Macroeconoml岱 lSmeanttoa.nalyzetherealeconomylくeyneslanmaCrOeCO-nomlCSWasIndeedbornarmdtheGreatDepresslOnlntheseriousendevourtoITleetthegraveChallellgeSfaclllgtheworldeconomy AftertheGnallClalcrlSISandglobalrecessionduring2008-10,modernmlCrO-foundedmacroe-conomlCShasbeenjustlfyablysubjecttOSerlOuSCrltlCISmS PaulKrugmanlnhlSL10neIRobblnSLectulleheldatLondonSchoolofEconorn1CSmJune,2009saidthat"mostmac【oeconomlCSOfthepast30yearswasspectacu-

larlyuselessatbest,andposltlVelyharmfulaいvoISt/'5stlglltZ(2010)alsocrlt.lCIZeSrn)CrO-foundedmacroeconomlCSaSfollows

worstrecesslOnSlnCetlleGreatDepresslOn- thefinancICrSWhodldsuchabadjobofmanagingrlSkortheregulatorswhofallcdtostopthem Buttheeconom】csplOfesslOnbearsmorethanallttlecLllpablllty

Iいshardfornon-economlStStOunderstandhowpeculiarthepredomlnantmaCrOeCOllOrnlCmodelswere Manyassumedde-mandhadtoequalsupply- andthatmeanttherecouldbenounemployment.(RlghtnowalotofpeopleareJustenJOy111ganextradoseofleisure.WhytheyareunhappylSamatterforpsych1-atry,noteconomlCS)Manyused"representatrveagentmodels"- alllnd】vldualswereassumedtobeldent)cat,andthismeant

therecou一dbe】10rnea11111gfulfinanclalrnarkets(whowouldbelendlngmoneytO､Vhom?)

Ssccalso′LModcmEconormcTheory-whcICltWCnいvl0ng,andhowthecllSISISchanglngltTheEconomist,JuLy18-242009

16

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Certalnly,changingparadlgmSISneCeSSalyTbachleVethegoal,ltlSIm-perativetOunderstandprecISelywhatlSthefundamentalproblemofmodern

mlCrO-foundedmacroeconomlCS AsSolow(2004)andStlgllt2r(2010)bothnghtlypointOut･theassumptlOnOfrcprcscntatlVCagentsISSerlOuSlyflawedSeeAoklandYoshlkawa(2007)forourowncrltlCISm

TheassumptlOnOfrepresentatlVeagentsISCrystalclearlnSuchmodelsasrealbusinessCycletheorywherellterallytherepresentatlVeCOnSumerand丘rlTLareIntroduced However,Lucas(197273)ratlOnalexpectatlOnSmodelandsearchtheorysuchasLucas/Prescott(1974)andMortensen/PISSarldes(1994)allassumeapparentlyheterogeneotlSeCOnOmlCagentsandemphasIZetheroleofmicro/IdyosyncratlCShocksInthispaper,Weshowedthatdespiteofapparentheterogeneity,thesemodelsarebasedlnessenceOnrepresentativeagentsPutltanotherway,theso-calledmicro/ldyosynctratlCShockslnthestandardmicro-foundedmodelsarenott.ruemicroShocks

Thecruclalconceptforourthesisisnon-self-averaglng.Self-averaglnglStacitlyassumednotonlymcrudercprescntat)veagentmodelssuchasrealbusmesscycletheorybutalso)nLucasrationalexpectationsmodelsandsearchtheoryltJustlGesustofocusonmeallSThestandardoptlmlZatlOllexercISeSbasedonrepresentatlVeagentsELreme且-nttOanalyzethebehav10rormeans

Self-averag】nglS,however,obtainedonlyontheextl.emelyunreallStlCassumptlOnthatal1theeconomicagent,SSha.rethesameprobab111tydlStrl-but10n As wedlSCuSSedltlndeatll,1nLucasratlOnalexpectatlC･nSm0Cle】sandsearchtheory,aunlquerePrTeSCnla伽epI.Obabll】tydlStLllbutlOnOrmトcro/ldyosyncratlCShockssharedbyalltheeconomica･gentSISassumed lnSection4,wedemonstratedthatoncethisunreallStlCaSSurnPtlOnlSdropped

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17

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18

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References

fl]Agh10n,P,a･ndPHowltt,(1992),"Amode一ofgrowththrough creatrve

destructユOnlEconometrwa,60,323-351

[2]Aokl,M (2006),"ThermodynamlCLlmltSOfMacroeconom lCOrFlnan-CialModels･One-andTwo-ParameterPoISSOn-DlrlCheltModels'′,forth-comlngJEconDyn Control

l3]- ,(2003),''ModelswithRandomExchangeableStructuresandCo-exIStenCeOfSeveralTypesofAgentslntheLong-Run Newlmplemen-ta.tLOnSOfSchumpeter'sDynamics"Tech Report,ResearchlmtlatlVealldDevelopment,ChuoUmVTbkyo-Dec2003

[4】- ,(2002),ModelmgAggregateBehavwrandFluctua如nsm Eco-nom乞CSSLochasttcVtewso/InLeractmgAgents,NewYork-CambridgeUnlVPress

l5】- ,(2000a),"OpenmodelsofsharemarketswlthtwodominanttypesofpartlCIPantS,:;,JEcon Behav.Or9491991216

【6】- ,(2000b),"Clusters】ZedlStrlbutlOnSOfeconomlCagentsOfmanytypeslnamarket',JMaLhAnalAppL.249,32-52

[7】- (1996),NewApprlOaChesLoMac710eCOnOmlCModeLmg Evolutwn-aryStochasttCDynamtcs,MuuゆLeEquiLlbma,andExicrylalittesasFwLd

EHecls,NewYorkCarnbrldgeUnlVPress

l8】- ,andR HawklnS(2010),'JNon-self-averaglngandthestatlStlCalmechamcsofendogenousmaCrOeCO110mlCfluctllatlOnS,EconomicMod-eLLm9

19】- ,andnYoshlkawa(2007),月econsLruc加 gMacroeconomもCSAPer-spec如elrvmSLatwわCalPhysICSandCombmalorwISLochastwProcesses

,NewYorkCambrldgeUmVerSltyPleSS

l10】Blngham,NHIC M Go】dle,andJLTeugels(1987),ReguLarvm a一

如n,Cambrldge,UK CambrldgeUnlVPress

flllBlumenfeld,R,and】〕B Mandelbrot.(1997),LevydtlStS,Mlttag-Le庁lerstatlStlCS,massrraCtallacunarlty,aJldpelCelVeddumenslOn,"PhyRevE56112-118

t12】Chafambldes,A C,EnumeratweComblnatOmCS,Chapma,nHall/CRC,2002,london

l13JCarlton,M A (1999),AppLtcatwnso/theTwo-ParameterPowson-DmchLetDは tribulwnPhD thesis,DeptMach UnlVCallfornla,LosAngeles

22

I147Devroye,L,(1993),"AtriptychofdlSCretedlStrlbutlOnSrelatedtothestablelaw,'つSialwIProbabLea18,349-351

[15】Eggenberger,FandG Polya,(1923),UberdieSlaltslikverketicterVorg

ange,ZeltSChrlftfur.AngewandteMathernatlkandMechanlk,Vo13

[16】ElddylA (cd)(1955)H29her77anscendenLaLFunciwns,vol3,NewYork McGrawIHlll

L17】Ewens,W ∫(1972)∫"ThesamplingtheoryofseLectlVelyneutralalleles,'′TheorPopBwl-,3787-112

t181Feller,W (1968),AnlntroductwnProbabiLtLyTheoryandItsAppLtca-

Lwns,VolI,3rdEdlt10n,NewYorkJohnWlley良Sons

(19】Feng.S,andFM Hoppe,(1998),"LaL-gedevlatlOnPrlnCIPlesforsomel'andomcomblnatOrlalstructureslnpopulationgeneticsandbrownlaLnmotion,r'AnnAppIProbab8,975-994.

【20lGrossman,G-M,andE Helpman,(1991),JnnovatwnandGrowtlHnLheGlobalEconomy,Cambr)dge,MA MITPress

l21】Janson,S(2006),LmtLtheorems/OrlmanguLarum sche7neS,Probab11-1tyTheoryandRelatedFields134.417-452

[221KadallO托一L,(20OO),SLalwttcalPhysics,SlugapOIC WorldScICllt･flCPul)llShlng

(23】Karl川,S,and∫ McGregor,(1967),"ThenumbeIOfmutantfromsmalntalnedlnapOpulatlOn:'Proc5thBerkeleySymponMachStabandProbabVo14

【24】Klngman,JFC (1978),"TherepresentL)onoflつalrtltlOnStructure".JLondonMaulSoc,18,374-380

[25]- ,(1993),PowsonProcesses,OxfordUK ClarendonPress

[26]Kydland,F andE PleSCOtt,(1982),"TimetOBulldandAggregat･eFlucutalOnS:'Economeimca,November

l27】Lucas,RE,(1972),"ExpectatlOnandtheNeutralltyOfMoney,'Jour-naLofEconomtCTheory,Vo14,Issue2.103-124

[28]- ,(1973),"SomeInternationalEvidenceonOtltPut-InflationnaLdeofis"TheAmer甘CanEconomlCRevww,Vo163,No3,3261334

[291- ,(1987),ModelsofBusm essCycles,NewYork･Blackwell

【30】- andE Prescott,(1974),"Equlllbrlum SearchandUnemploy-ment,;∫JoumalofEconomicTheory,77,721-754

23

(31lMantegna,RN andHE Stanley(2000),AnlniroductwntoEcono-

physもCS CorrelalwrLSandCompleェiLym Fmance, CambrldgeUKCambrldgeUnlVerSltyPress

[32】Mortel-Ben,DT andPISSarldes(1994),

[33】Phelps,ES,(1970),MwroeconomもCFoundatwnsofEmpLoymenlandln-j7aLwnTheor7J,NewYorkNorton

l34lPltrnan,J,(2002),LectureNoteso/thesummer･schoolonprlObab仙y,StFlour,France,forthcomingfromSprlngeトVerLag

l35]Plosser,C,(1989),'iUnderstandlngRealBuslneSSCycles,"Joumalo/EconomlCPerspectwes,Summer

(36】Puyhaubert,V (2003)一"AllalytlCUrnsofTriangulaIForm,}'AL90mlhms

Semmar,2002-2004,FChyzk(ed),INRIA,(2005);p61-64

(37]RDmer,PM,(1986),"IncreasingReturnsandLongRunGrowth,"Jour-naloIPo仙calEconomy

t38】Solo､V,R h'Ⅰ一(2004),"TheToblnApproachtoMonetaryEconornlCS',Joumalo/Money,Credt土andBankm9,36,659-663

[391Somctte,D,(2000),CmtwalPhenomenam Natu7TalSclenCeS,BerlinHeidelbergSp川1ger

t40lStlgl･tz,J,(2010),Needed ANewEconom-cParadlgm,'つFmancwITmes,Frlday,August20,2010

Ⅰ411Sutto",∫,(2OO2)"ThevallallCCOH11111glOWtlllatCS tllC "seal.】lgpuzzle"",PhystCaA,312,577-590

[42】Wlnter,SG,andYM KalllOVSkl,G Dosュ,(2000),"Modellnglndus-trlalDynamicsWlthhnovatlVeEntrants:-StructuralChangeandEco-nomtCDy77,amu:S.Volll,255-293

[43]Yamato,H,andM Slbuya,(2000),"MomentsofsomestatlStlCSOfPltmanSamPllngformula,",BullIn/orm,CybemeLICS,6.463-488

[44】- (2003),"SometopicsOnPltman'sprobablllStlCPaXtlt10n"Stai乞3-

ttCalMaih51,351-372(1nJapanese)

24

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