new classical models of aggregate fluctua5ons · new classical models of aggregate fluctua5ons the...
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![Page 1: New Classical Models of Aggregate Fluctua5ons · New Classical Models of Aggregate Fluctua5ons The Stochas5c Growth Model and a New Classical Model without Capital. Prof George Alogoskoufis,](https://reader031.vdocuments.us/reader031/viewer/2022011809/5d5428ff88c99384648b85bb/html5/thumbnails/1.jpg)
Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
NewClassicalModelsofAggregateFluctua5ons
TheStochas5cGrowthModelandaNewClassicalModelwithoutCapital
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
AggregateFluctua5ons• Economiesarecharacterizedbyfluctua5onsinrela5ontotheirlong-termtrends.In
someperiodsoutput,consump5onandemploymentgrowathighrates,whileatother5mestheygrowatloworevennega5verates.Insomeperiodsunemploymentislowandinothersquitehigh.Infla5ondisplayssignificantfluctua5onsaswell.
• Understandingthedeterminantsofaggregatefluctua5onsisthesecondmainobjec5veofmacroeconomics.Inthis,andthelecturesthatfollow,wepresentthemaintheoriesregardingthenatureofaggregatefluctua5ons.
• Inthislecturewestartbyintroducingclassicalmodelsofaggregatefluctua5ons.“New”classicalmodelsareessen5allydynamicstochas3cgeneralequilibriummodels(DSGE),basedonop5mizinghouseholdsandfirms,flexiblewagesandpricesandfullycompe55vemarkets.Fluctua5onsinthesemodelsarecausedbyrealshockstoproduc5vity,householdpreferencesandgovernmentexpenditure,andtheeffectsoftheseshocksarepropagatedthroughendogenousdynamicprocesses,suchasconsump5onandinvestment.
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
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ΑΕΠ"κατά"κεφαλήν"ΗΠΑ"(σταθερές"τιμές)"
PerCapitaGDPintheUSA(logscale)
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
GrowthRateofPerCapitaGDPintheUSA
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
UnemploymentRateintheUSA
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
Infla5onRateintheUSA
!30.0%&
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
TheNatureandKeyCharacteris5csofAggregateFluctua5ons
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• Aggregatefluctua5onsarenotcharacterizedbysomesimplerepe55veregularityandseemtobecharacterizedbyrandomness.
• Theprevailingviewtoday,whichdatesbacktoFrisch(1930)andSlutsky(1937),isthateconomiesaresubjecttovariouskindsofrandomdisturbances,which,throughtheopera5onofeconomictransmissionmechanismssuchastheopera5onofmarkets,affectoutput,employment,realwages,realinterestrates,thepricelevelandinfla5on,andsetinmo5ondynamicstochas5cadjustmentprocesses.
• Thedynamicstochas5capproachtoaggregatefluctua5onsowesalottothecontribu5onsofLucas(1977)andKydlandandPresco[(1982).
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
Lucas(1977)ontheNatureofAggregateFluctua5ons
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“Technically,movementsabouttrendingrossna5onalproductinanycountrycanbewelldescribedbyastochas5callydisturbeddifferenceequa5onofveryloworder.Thesemovementsdonotexhibituniformityofeitherperiodoramplitude,whichistosay,theydonotresemblethedeterminis5cwavemo5onswhichsome5mesariseinthenaturalsciences.Thoseregulari5eswhichareobservedareintheco-movementsamongdifferentaggrega5ve5meseries…Oneisledbythefactstoconcludethat,withrespecttothequalita5vebehaviorofco-movementsamongseries,businesscyclesareallalike.Totheore5callyinclinedeconomists,thisconclusionshouldbea[rac5veandchallenging,foritsuggeststhepossibilityofaunifiedexplana5onofbusinesscycles,groundedinthegenerallawsgoverningmarketeconomies,ratherthaninpoli5calorins5tu5onalcharacteris5csspecifictopar5cularcountriesorperiods.”(p.9-10).
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
TheStochas5cGrowthModel:AnExtensionoftheRamseyModel
• Westartwiththesocalledstochas3cgrowthmodel,whichisanextendedstochas5cversionoftheRamseymodel.
• Thisisacompe55vedynamicstochas5cgeneralequilibriummodel,withoutexternali5es,asymmetricinforma5on,fric5onsandotherimperfec5onsofmarkets.Thereforeitisanaturalstar5ngpointfortheinves5ga5onofaggregatefluctua5ons.
• Thismodelisnothingbutageneraliza5onoftheRamseymodel.Itnotonlyexcludesanymarketimperfec5ons,butalsoallissuesrelatedtoheterogeneityofeconomicagents.TheRamseymodelisthereforethenaturalstar5ngpointforthestudyofaggregatefluctua5ons,likeitisthe“natural”star5ngpointforthestudyofthelongrungrowth.
• However,inordertostudyaggregatefluctua5ons,oneneedstoextendtheRamseymodel.
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
ExtensionsoftheRamseyModel
• First,oneshouldallowforrandomdisturbances,whichcancausefluctua5ons.Withoutrandomdisturbances,theRamseymodelconvergestoauniquesteadystate.ThedisturbancesusuallyintroducedintheRamseymodelaredisturbancesintotalfactorproduc5vity(technologyshocks),aswellasrealdemandshocks,suchasshockstothepreferencesofconsumersorrealgovernmentexpenditure.Sincebothkindsofshocksarereal-unlikemonetaryornominalshocks-thismodelturnsouttobearealbusinesscyclemodel.
• Second,inordertoallowthemodeltoexplainfluctua5onsnotonlyintotaloutput,butalsoemployment,employmentmustbecomeendogenous.Thisisachievedthroughtheintroduc5onofemploymentintheu5lityfunc5onofarepresenta5vehousehold,inordertoestablishanendogenouslaborsupplyfunc3on.
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
HouseholdsandFirmsintheStochas5cGrowthModel
• Thereareanumberofiden5calhouseholdsandfirms,sothisisacompe55verepresenta5vehouseholdmodel.
• Firmsuselaborandcapitalinorderproduceahomogeneousproduct.Theychooseinvestmentandemploymentinordertomaximizetheirprofits.
• Householdschooseconsump5onandlaborsupplyinordertomaximizetheirinter-temporalu5lity.
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
Firms,Produc5onandInvestment
Yt = Ktα (AtLt )
1−α
Yt = Ct +Gt + Kt+1 − Kt +δKt
Kt+1 = Kt +Yt −Ct −Gt −δKt
Produc5onFunc5on
Consump5on,InvestmentandGovernmentExpenditure
CapitalAccumula5onEqua5on
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
Maximiza5onofProfitsofFirms
wt = (1−α )Kt
AtLt
⎛⎝⎜
⎞⎠⎟
α
At
rt =αAtLtKt
⎛⎝⎜
⎞⎠⎟
1−α
−δ
TheDetermina5onofRealWages
TheDetermina5onoftheRealInterestRate
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
TheRepresenta5veHousehold
U = E01
1+ ρ⎛⎝⎜
⎞⎠⎟
t
u(ct ,1− lt )Nt
Ht=0
∞∑
ut = lnct + b ln(1− lt )
Therepresenta5vehouseholdmaximizesitsexpectedinter-temporalu5lityfunc5on,whichdependsonthepathofrealconsump5onofgoodsandservicesandleisure.Theu5lityfunc5onisdefinedby,
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
Popula5on,EfficiencyofLaborandGovernmentExpenditure
lnNt = N_+ nt
Popula5onincreasesexogenouslyataratenperperiod
lnAt = A_+ gt + vt
A vtA =ηAvt−1
A + ε tA
TheEfficiencyofLaborGrowsananexogenousrategandissubjectedtoAR(1)stochas5cdisturbances
GovernmentExpenditureGrowsatanexogenousrateg+nandissubjectedtoAR(1)stochas5cdisturbances
lnGt = G_+ (n + g)t + vt
G vtG =ηGvt−1
G + ε tG
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
LaborSupplyoftheRepresenta5veHousehold:TheOnePeriodCase
u = lnc + b ln(1− l)
ThefirstdifferenceofthismodelfromtheRamseymodelarisesfromtheintroduc5onofleisure5meintheu5lityfunc5onofthehousehold,whichmakeslaborsupplyendogenous.Toanalyzetheimportanceofthisaddi5on,letusfirstconsiderthesta5onaryproblemofahouseholdlivingforasingle5meperiodandhasnoassets.Theproblemofthathouseholdisdefinedasthemaximiza5onof,
undertheconstraint
c = wl
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
Fromthefirstordercondi5onsforanop5mum,
Itfollowsthat,
1c− λ = 0 − b
1− l+ λw = 0 c = wl
− b1− l
+ 1l= 0
Laborsupplyisindependentoftherealwage.Thisisbecauseoftheassump5onoflogarithmicpreferences,implyingthattheelas5cityofsubs5tu5onbetweenconsump5onandleisureisequaltounity.Thus,thesubs5tu5oneffectfromachangeintherealwageiscounteractedbytheincomeeffect.However,thisdoesnotmeanthattemporarychangesinrealwagesdonotaffectlaborsupply.Thiscanbeseenifwelookatthebehaviorofahouseholdlivingfortwoperiods.
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LaborSupplyoftheRepresenta5veHousehold:TheOnePeriodCase
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 18
lnc1 + b ln(1− l)1 +1
1+ ρlnc2 + b ln(1− l)2( )
c1 +11+ r
c2 = w1l1 +11+ r
w2l2
Weshallnowanalyzethebehaviorofahouseholdlivingfortwoperiods,hasnoini5alwealth,andnouncertaintyabouttherealinterestrateortherealwageofthesecondperiod.
LaborSupplyoftheRepresenta5veHousehold:TheTwoPeriodCase
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
b1− l1
= λw1
Fromthefirstordercondi5onsforlaborsupply:
b1− l2
= 1+ ρ1+ r
λw2
Rela5velaborsupplyinthetwoperiodsdependsposi5velyontherela5verealwageinthetwoperiods,aswellastherealinterestrate.
1− l11− l2
= 1+ ρ1+ r
w2w1
Inter-temporalSubs5tu5oninLaborSupply
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
Implica5onsofInter-temporalSubs5tu5onforLaborSupply
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• Thehighertherealwageofthefirstperiodinrela5ontotherealwageofthesecondperiod,thehigherthelaborsupplyofthefirstperiod,inrela5ontothatofthesecond.Thehouseholdsubs5tuteslaborbetweenperiods,dependingonrela5verealwagesbetweenperiods.Becauseoflogarithmicpreferences,theinter-temporalsubs5tu5onelas5cityisequaltoone.
• Moreover,thehighertherealinterestraterthegreaterthelaborsupplyofthefirstperiodcomparedtothesecondperiod.Theincreaseintheinterestrateincreasesthea[rac5venesstoworktodayandsave,comparedtoworkinginthefuture.Ithastheoppositeeffectofthepurerateof5mepreferencerateρ.
• Theseeffectsofrela5vewagesover5meandtherealinterestrateonlaborsupplyareknownasinter-temporalsubs3tu3oninlaborsupply.
• Consequently,fluctua5onsinrealwagesandtherealinterestratecancausefluctua5onsinemployment,althoughpermanentchangesinrealwagescannot.
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
UncertaintyandtheEulerEqua5onforConsump5on
1ct
= 11+ ρ
Et1ct+1
1+ rt+1( )⎡
⎣⎢
⎤
⎦⎥
21
• Thesecondelementthatdifferen5atesthestochas5cgrowthmodelfromtheRamseymodelisuncertaintyarisingfromthestochas5cdisturbances.Therefore,theexpecta3onsoftherepresenta5vehouseholdforfuturedevelopmentsplayasignificantrole.
• Itcanbeshownthat,forthegeneralcasewhenthehouseholdmaximizestheexpectedinter-temporalu5lityfunc5on,theEulerequa5onforconsump5ontakestheform,
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
UncertaintyandtheBehavioroftheRepresenta5veHousehold
1ct
= 11+ ρ
Et1ct+1
⎡
⎣⎢
⎤
⎦⎥
⎧⎨⎩⎪
Et 1+ rt+1( )+Cov 1ct+1, 1+ rt+1( )⎛
⎝⎜⎞⎠⎟⎫⎬⎭⎪
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Themathema5calexpecta5onoftheproductoftworandomvariablesisnotequaltotheproductofmathema5calexpecta5ons.Itisequaltotheproductofmathema5calexpecta5onsplusthecovarianceoftworandomvariables.
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
ct1− lt
= wt
b
Thisconditionlinkslaborsupply(leisure)andconsumptionwiththerealwage.Itincludesonlycurrentvariables,asthereisnouncertaintyinthecurrentperiod.
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Fromthefirst-ordercondi5onsforconsump5onandlaborsupply,thera5oofconsump5ontoleisureisposi5vefunc5onoftherealwageoftheform,
TheFirstOrderCondi5onsforConsump5onandLaborSupply
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
Kt+1 = Yt −Ct 1+ rt =αAtLtKt
⎛⎝⎜
⎞⎠⎟
1−α
24
• Thismodelisnoteasytosolveanaly5cally,asitcontainsfactorsthatarelinear,andfactorsthatarelog-linearinitsvariables.Theproper5esofthemodelcanbedescribedifwesimplifyitfurther,orifweusealog-linearapproxima5onaroundthebalancedgrowthpath,andsolveitnumericallyforspecificvaluesoftheparameters.
• InaspecialAnnex,wepresenttheCampbell(1994)log-linearapproxima5onofthefullmodel,arounditsbalancedgrowthpath.Thisallowsustodescribethefullproper5esofthemodel.
• Intheremainderweshallconcentrateontheproper5esofasimplifiedversionofthemodel,withoutgovernmentexpenditureandadeprecia5onrateof100%..
ASpecialCaseoftheModel
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
ct = (1− st )Yt / Nt 1+ rt+1 = aYt+1 /Kt+1
Kt+1 = stYt
s^= α (1+ n)1+ ρ
l^= 1−α
(1−α )+ b(1− s^)
Itfollowsthatthesavingsrateandlaborsupplyareconstantinthisspecialcase,becauseoflogarithmicpreferencesandtheCobbDouglasproductionfunction.
25
TheSpecialCaseoftheStochas5cGrowthModel
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 26
Laborsupplyisconstantbecausetheimpactoftheshocksintechnology(laborefficiency)ontherealwageandtherealinterestratecanceleachotherout,sothereisnointer-temporalsubs5tu5on.Thisisduetothespecificassump5onthatwemadeinordertosimplifythemodel,andasonecanseefromtheanalysisofthefullmodelintheAnnexisnotageneralfeatureofthemodel.
TheConstancyofLaborSupply
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 27
Fluctua5onsinRealOutput
lnYt =α lnKt + (1−α )(lnAt + lnLt )
Kt = s^Yt−1
Fromtheproductionfunction,
Giventhat, Lt = l^Nt
lnYt =α ln s^+α lnYt−1 + (1−α )(lnAt + ln l
^+ lnNt )
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
lnYt =α ln s^+α lnYt−1 + (1−α ) (A
_+ gt)+ vt
A + (ln l^+ N
_+ nt)⎡
⎣⎢⎤⎦⎥
SubstitutingforAandN,
Expressingrealoutputaslogarithmicdeviationsfromitslongruntrend,
Y~t = (α +ηA )Y
~t−1−αηAY
~t−2+ (1−α )ε t
A
Y~t =αY
~t−1+ (1−α )vt
A
whichcanbesolvedas,
28
Fluctua5onsinRealOutput
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015
Y~t =αY
~t−1+ (1−α )ε t
A
29
• Thepercentage(logarithmic)devia5onsoftotalrealoutputfromtrendfollowasecondorderautoregressiveprocess(AR(2)).
• Becauseαislow(about1/3),thedynamicbehavioroftotalrealoutputdependsprimarilyonthedegreeofpersistenceofproduc5vityshocks.Ifthepersistenceofproduc5vityshocksηAishigh,thenwehaveconsiderablepersistenceinthefluctua5onsofoutput.Otherwise,thepersistenceofoutputfluctua5onsaroundtrendislow.
• Ifrealshocksdisplaynopersistence(i.eifηA=0),then,
ConclusionsfromtheSpecialCaseoftheModel
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 30
EconometricEs5matesforthelogofUSGDP1890-2014
lnYt=0,270+1,211lnYt-1-0,319lnYt-2+0,0036t(0,083)(0,086)(0,087)(0,0012)
R2=0,999,DW=2.020,T=125
Fromthesees5matesitfollowsthatα=0,386(s.e.0.132)andηΑ=0,824(s.e.0.082).Italsofollowsthatg+n=0,033(s.e.0.001).
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 31
• Thesimplifiedformofthemodel,containsmanyofitsessen5alelements,andprovidesthebasic“newclassical”accountoffluctua5onsintotaloutput(GDP)aroundtrend,mainlyonthebasisofpersistentproduc5vityshocksandcapitalaccumula5on.
• However,manyotherfeaturesofaggregatefluctua5onsarenotadequatelydescribedbythissimplifiedversionofthestochas5cgrowthmodel.
• Theconstantsavingsra3o.Thismeansthatconsump5onwilldisplaythesamedegreeofvariabilityasoutputandinvestment,whichdoesnottendtohappeninreality.
• Theconstantemploymentrate.Inreality,theemploymentrateisnotconstantoverthebusinesscycle.Employmentispro-cyclical,movinginthesamedirec5onasoutput.
• RealWagesovertheBusinessCycle.Inthesimplifiedstochas5cgrowthmodelrealwagesarepro-cyclicalandequallyvola5leasGDPpercapita,whichisnotalwaysthecase.
WeaknessesoftheSimplifiedDynamicGrowthModel
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 32
• Whenoneexaminesthemoregeneralformofthemodel,assumingalowdeprecia5onrate,aswedointheAnnextothislecture,manyoftheseweaknessesarecorrected,assavings,investmentandemploymentalsodisplayfluctua5onsinresponsetoproduc5vityshocks.
• Forexample,inthefullstochas5cgrowthmodel,analyzedintheAnnex,thesavingsrateisnotconstant,andconsump5ontendstobelessvariablethaninvestmentandoutput.Inaddi5on,inthefullstochas5cgrowthmodel,theemploymentrateispro-cyclical,andmovesinthesamewayasoutput.Moreover,theintroduc5onofpublicexpenditureshocksorpreferenceshockscouldrelaxthestrictdependenceoffluctua5onsinrealwagesonfluctua5onsinaggregateproduc5vity.
TheMoreGeneralVersionoftheStochas5cGrowthModel
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 33
TheImpactofaProduc5vityShockintheFullStochas5cGrowthModel
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 34
• Weshallnextfocusonananaly5callysimplerversionofthe“newclassical”modelofaggregatefluctua5ons,inwhichtheonlyvariablefactorofproduc5onislabor.Weshallthusabstractfromcapitalaccumula5on.
• Inthisanaly5callysimplermodelweallowforamoregeneralapproachtothepreferencesoftherepresenta5vehousehold,andalsodis5nguishbetweennominalandrealvariables.
• Thisallowsustoconsiderthedetermina5onofthelevelofpricesandwages,infla5onandnominalinterestrates,andtheroleofmonetaryfactorsinclassicalmodels.
ANewClassicalModelwithoutCapital
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 35
Therepresenta5vehouseholdisassumedtomaximize,
TheRepresenta5veHousehold
Et1
1+ ρ⎛⎝⎜
⎞⎠⎟
s
u(Ct+s ,Lt+s )s=0
∞∑
Subjectto,
PtCt +11+ it
Bt ≤ Bt−1 +WtLt −Tt limT→∞
EtBT ≥ 0
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 36
Weassumethattheperperiodu5lityfunc5onisgivenby,
FirstOrderCondi5onsfortheRepresenta5veHousehold
U(Ct ,Lt ) =C
t
1−θ
1−θ− Lt
1+λ
1+ λ
Thefirstordercondi5onsinthiscasetaketheform,
Wt
Pt= Ct
θLtλ 1
1+ it= 11+ ρ
EtCt+1
Ct
⎛⎝⎜
⎞⎠⎟
−θPtPt+1
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 37
Therepresenta5vefirmiscompe55ve,andchoosesemploymentinordertomaximizeprofits,forgivennominalwagesandprices.Profitsaremaximizedsubjecttoaproduc5onfunc5onwithlaborastheonlyvariablefactorofproduc5on.
TheRepresenta5veFirm
PtYt −WtLt = Pt (AtLt1−α )−WtLt
Profitmaximiza5onimpliesthatemploymentwillbedeterminedsoastoequatethemarginalproductoflabortotherealwage.Thus,
Wt
Pt= (1−α )AtLt
−α
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 38
TheModelinLogLinearForm
wt − pt = θct + λlt ct = Et (ct+1)−1θit − Et (π t+1)− ρ( )
wt − pt = at −αlt + ln(1−α )
yt = at + (1−α )lt
FirstOrderCondi5onsfortheHousehold
FirstOrderCondi5onfortheFirm
Produc5onFunc5onandProductMarketEquilibrium
ct = yt
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 39
ImposingEquilibriumintheLaborandProductMarkets,itfollowsthatalltherealvariablesdependonthestochas5cprocessdrivingtotalfactorproduc5vity.
Solu5onoftheModel
lt =ηLAat + l_
yt = ct =ηYAat + y_
wt − pt =ηWAat +ω_
rt = it − Et (π t+1) = ρ +θηYAEt (Δat+1)
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 40
Theparametersdefiningtheeffectsoftotalfactorproduc5vityare,
ParametersoftheSolu5on
ηLA =1−θ
θ(1−α )+α + λ
ηYA = 1+ (1−α )ηLA =1+ λ
θ(1−α )+α + λ
ηWA = 1−αηLA =θ + λ
θ(1−α )+α + λ
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 41
• Output,consump5onandrealwagesareposi5vefunc5onsofproduc5vity,whileemploymentisaposi5vefunc5onofproduc5vityonlyifθ<1,i.e.iftheinter-temporalelas5cityofsubs5tu5onofconsump5onisgreaterthanone.
• Ifθ>1,employmentisanega5vefunc5onofproduc5vity,whileifθ=1,employmentisindependentofproduc5vity.Thisisbecauseifθ<1thesubs5tu5oneffectdominatesovertheincomeeffect,aperachangeinproduc5vityandrealwages,andemploymentrises.Ifθ>1theincomeeffectdominatesoverthesubs5tu5oneffect,whileinthecaseθ=1thetwoeffectscanceleachotherout,andemploymentisnotaffected.
• Onlyrealfactors,suchasrealproduc5vity,affectfluctua5onsinrealvariables.Asinthestochas5cgrowthmodel,monetaryfactorssuchasmoneysupplyandnominalinterestrateshavenoimpactontheevolu5onofrealvariables.
Proper5esoftheSolu5on
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 42
Inordertoexaminetheimpactofmonetaryfactorsinthe“new”classicalmodel,weshallassumetheexistenceofamoneydemandfunc5onbyhouseholdsandfirms,which,inlogarithms,takestheform,
MonetaryFactorsintheNewClassicalModel
mt − pt = yt −ηit
whereηisthesemi-elas5cityofmoneydemandwithrespecttothenominalinterestrate,whichisdefined,bytheFisherequa5on,as,
it = rt + Et (π t+1)
Realvariables,suchasyandraredeterminedwithoutreferencetomonetaryfactors,asfunc5onsofshockstototalfactorproduc5vity.
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 43
Ifthecentralbankdeterminesanexogenouspathforthemoneysupply,then,fromthemoneymarketequilibriumcondi5onandtheFisherequa5on,itfollowsthat,
AnExogenousPathfortheMoneySupply
pt =η1+η
Et (pt+1)+1
1+ηmt −
11+η
yt −ηrt( )
Undertheassump5onthatη>0,thesolu5onofthispriceequa5onis,
pt =1
1+ηη1+η
⎛⎝⎜
⎞⎠⎟j=0
∞∑j
Et mt+ j − yt+ j +ηrt+ j( )
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 44
Ifweassumethatthecentralbankfollowsanexogenouspathforthenominalinterestrate,fromtheFisherequa5on,iffollowsthat,
AnExogenousPathfortheNominalInterestRate
Et (π t+1) = it − rtWithanexogenouspathforthenominalandtherealinterestrate,thisdoesnotdetermineinfla5on,butexpectedinfla5on,andisconsistentwithanypricelevelthatsa5sfies,
pt+1 = pt + it − rt + ξt+1 Et ξt+1( ) = 0foranyξforwhich
Thissuggeststhattherearemul5pleequilibriaforthepricelevelandinfla5on,dependingonξ.Thus,inthiscasewehavepricelevelandinfla5onindeterminacy.
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 45
• Centralbankspredominantlyusethenominalinterestrateastheirpreferredmonetaryinstrument.
• Ifsuchpoliciescausedindeterminacyofthepricelevelandinfla5on,i.epricebubbles,thiswouldhavebeenextremelyworrying.
• However,CentralBanksdonotfollowexogenousnominalinterestratepath,butpoliciesaccordingtowhichthepathofnominalinterestratesdependsonpast,currentandexpectedeconomicdevelopments,mainlyinfla5on.Forexample,ifinfla5onrises,centralbanksusuallyraisenominalinterestratesinordertoreduceit,andviceversa.ThiswasaperalltheessenceoftheWicksellrule.Whataretheimplica5onsofsuchpoliciesinthe“newclassical”model?
CentralBanksandInterestRateRules
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 46
Letusassumethefollowingrulefordeterminingnominalinterestrates:
AnInfla5onBasedInterestRateRule
it = ρ +φπ t
whereφ>0isthereac5onofthecentralbanknominalinterestratetoinfla5on.FromthispolicyruleandtheFisherequa5onitfollowsthat,
π t =1φEt π t+1( )+ 1
φrt − ρ( )
Undertheassump5onthatφ>1(Taylorprinciple),thiscanbesolvedas,
π t =1φ
⎛⎝⎜
⎞⎠⎟s=0
∞∑s+1
Et rt+s − ρ( )
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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 47
• Ifthecentralbankdeterminesanexogenouspathforthemoneysupply,thepricelevelandinfla5onaredeterminedasfunc5onsoftheexogenouspathofthemoneysupply,andthepathsofrealoutputandtherealinterestrate,whichareindependentofmonetaryfactorsinthe“new”classicalmodel.
• Ifthecentralbankfollowsanexogenouspathforthenominalinterestrate,theresultisindeterminacyofthepricelevelandinfla5on.
• Ifthecentralbanknominalinterestratesreacttoinfla5on,andthereac5onissufficientlypronounced(φ>1),thenthereisnoindeterminacyproblemforinfla5on.Ifthereac5onofthenominalinterestratestoinfla5onisnotsufficientlypronounced(φ≦1),thentheproblemofindeterminacyofinfla5onremains.
• Inanycase,inthe“new”classicalmodelofaggregatefluctua5onsonlyrealfactorsaffectfluctua5onsinrealvariables.Monetaryfactorsandmonetarypolicyonlyaffecttherealmoneybalances,andnominalvariablessuchasthepricelevelandinfla5on,nominalinterestratesandthemoneystock.
MonetaryFactorsandMonetaryPolicyintheNewClassicalModel