Keynesian Models and the Phillips Curve - … George Alogoskoufis, Dynamic Macroeconomic Theory,...

ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015

The Original Keynesian Models of Aggregate Fluctuations

Keynesian Models and the Phillips Curve

ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 2

• Aswehavealreadymen=oned,followingtheGreatDepressionofthe1930s,theanalysisofaggregatefluctua=onsdevelopedintomacroeconomics,onthebasisoftheso-calledKeynesianmodels.

• Thesemodels,werederivedfromtheGeneralTheoryofKeynes(1936),whoarguedagainstthethenprevailing“classical”equilibriumtheoryofemploymentandaggregatefluctua=ons.

• Keynesproceededtoarguethatageneraltheory,asopposedtotheclassicaltheory,oughttobeabletoexplaininvoluntaryunemployment,whichheproceededtodulydefineasasitua=oninwhichtherealwageishigherthanthemarginaldisu=lityoflabor.

• TheKeynesianapproachwasdevelopedthroughasequenceofmodelsintheGeneralTheory.

TheKeynesianApproach


• TheKeynesiancrossfocusedonthecondi=onsforshortrunequilibriuminthemarketforgoodsandservices,whenpricesarefixed.

• Asecondmodel,againbasedontheassump=onoffixedprices,focusedonthecondi=onsforsimultaneousequilibriuminthemarketforgoodsandservicesandthemarketformoney.Thismodel,wascodifiedastheIS-LMmodelbyHicks(1937),andwasformanyyearsthedominantKeynesianparadigm,bothamongacademiceconomistsandpolicymakers.

• Athirdmodel,combinedtheIS-LMmodelwithtwoaddi=onalassump=ons.First,thatthenominalwageisfixedintheshortrun,and,second,thatemployersdetermineemploymentbyequa=ngtherealwagewiththemarginalproductoflabor.ThismodelwascodifiedastheAD-ASmodel,orneoclassicalsynthesis,andwasusedtodeterminebothunemploymentandthepricelevel.

TheThreeModelsoftheGeneralTheory


• IntheKeynesianmodels,theimmediateadjustmentofpricesandwagesasequilibra=ngmechanismswasreplacedbytheassump=onthat,intheshort-term,pricesand/ornominalwagesarefixedoradjustonlypar=ally.

• Thus,thelevelofincomeandemploymentbecomesanaddi=onaladjustmentmechanismforthemarketforgoodsandservicesandthemarketforlabor.

• InthislecturewepresentthestructureofthebasicKeynesianmodels,assumingini=allyaconstantlevelofpricesand/ornominalwages,andthengradualadjustmentofpricesandnominalwages,basedonthePhillipscurve,whichwasasubsequentaddi=ontothebasicKeynesianmodels.

TheDis=nguishingDifferenceoftheKeynesianApproach


• TheKeynesianmodelstartsbyconsideringthedetermina=onofaggregatedemand.Theassump=onisthattotalrealoutputandincomeY,isequaltothesumofrealconsump=onC,realgrossinvestmentΙandrealgovernmentexpenditureG.

• Inthesimplestversion,thatoftheKeynesiancross,investmentandgovernmentexpenditureareconsideredexogenous,andconsump=onisassumedtobeaposi=vefunc=onofcurrentrealdisposableincome.

TheKeynesianCross


TheModeloftheKeynesianCross

Yt = Ct + It +Gt

Ct = C Yt −Tt( )

0 <CY =∂C

∂(Y −T )<1


ShortRunEquilibriumintheModeloftheKeynesianCross


TheEquilibriumCondi=onintheProductMarketandtheMul=pliers

Yt = C Yt −Tt( )+ It +Gt

dYtdIt

= dYtdGt

= 11−CY

>1

dYtdGt dGt=dTt

= 1


• Realoutputisdeterminedbytheequalityofaggregatesupplyandaggregatedemand,whichisthesumofrealconsump=on,investmentandgovernmentexpenditure.Pricesareassumedfixedorsluggishtoadjust,soachangeinaggregatedemandbringsaboutachangeinaggregatesupplythroughcorrespondingchangesinemployment.

• Anexogenouschangeinaggregatedemand,eitherthroughinvestmentorthroughgovernmentexpenditure,resultsinachangeinrealoutputwhichisamul=pleoftheoriginalchangewhichishigherthanone,sincethemarginalpropensitytoconsumeislessthanone.

• Whenaggregatedemandincreases,realoutputandincomeincreasesinordertomaintainequilibriumbetweenrealoutputandexpenditure.

TheEquilibriumCondi=onintheProductMarketandtheMul=pliers


• Anincreaseininvestment,orgovernmentexpenditure,causesacorrespondingonetooneincreaseinaggregatedemand,andaggregaterealoutput.

• Theincreaseinrealoutputinducesanincreaseinprivateconsump=onthroughtheconsump=onfunc=on,whichbringsaboutafurtherincreaseinaggregatedemandandrealoutputandincome.

• Consequently,agivenexogenousriseinaggregatedemandhasamul=pleeffectonaggregaterealincome,duetothesecondroundeffectsthroughaggregateconsump=on,whichinturnproducefurtherincreasesinrealdemandandrealoutputandemployment.

WhyistheMul=plierHigherThanOne


• Anincreaseinpublicexpenditure,fundedbyanequalincreaseintaxes,increasestotalaggregateexpenditureandrealoutputandincomebythesameamount.

• ThisissomethingthatwasprovenbyHaavelmo(1945)whoderivedthebalancedbudgetmulAplier.

• Governmentexpenditureincreasesaggregatedemandonetoone,butthetaxincreasereducesprivateconsump=onbylessthanonetoone,becausethemarginalpropensitytoconsumeislessthanone.Consequently,theini=alimpactonoverallexpenditureisequalto1-CY=mesthechangeingovernmentexpenditure,andtheoveralleffectofachangeingovernmentexpenditurefinancedbyincreasedtaxesisequaltounity.

WhyistheBalancedBudgetMul=plierEqualtoUnity


• Inanimportantpaper,Samuelson(1939)combinedthemodeloftheKeynesiancrosswithaninvestmentfunc=onbasedontheprincipleofaccelera=on,toderiveendogenouscycles.

• IntheSamuelsonmodel,theinvestmentfunc=onisbasedonthesocalledacceleraAonprinciple,whichimpliesthatwhenthereisaposi=vechangeinconsump=on,firmswillinvestmore,inordertoproducethehigherquan=tyofconsumergoodsdemanded.Theacceleratormeasuresthesensi=vityofinvestmenttochangesinaggregateconsump=on.

• Theconsump=onfunc=onislinear,accordingtowhichconsump=onisafunc=onoflaggedandnotcurrentincome,andgovernmentexpenditureisassumedexogenousandconstant.

• TheSamuelsonmodelresultsinasecondorderlineardifferenceequa=on,which,dependingonthevalueoftheaccelerator,resultsineithermonotonicconvergenceorendogenouscycles.

TheSamuelsonModeloftheInterac=onoftheMul=plierwiththeAccelerator


• ThebasicmodeloftheKeynesiancross,assumesthatinvestmentisexogenous.

• AmoregeneralformoftheKeynesianmodelconsidersthataggregateinvestmentdependsontherealinterestrate,andintroducestheequilibriumcondi=oninthemarketformoney,inordertoanalyzethesimultaneousdetermina=onofrealoutputandtheinterestrate.ThisformofthemodelwaspresentedinChapters10-18oftheGeneralTheory,andwascodifiedastheIS-LMmodel,intheimportantcontribu=onofHicks(1937).ThiswasthemostwidelyusedversionoftheKeynesianmodel.

• ThemaindifferencefromthepreviousmodeloftheKeynesiancrossisthatinvestmentceasestobetreatedasexogenous,andisconsideredtodependnega=velyontherealinterestrate.Assumingthatinfla=onaryexpecta=onsaregiven,investmentwilldependnega=velyonthecurrentnominalinterestrate.

TheIS-LMModel


• Thiscondi=ondescribesthecombina=onsofrealoutputandthenominalinterestratethatensureequilibriuminthemarketforgoodsandservices.Itisdepicteddiagramma=callyasadownwardslopingIS(Investment-Savings)curve.

EquilibriumCondi=onintheMarketforGoodsandServices

Yt = C Yt −Tt( )+ It (it )+Gt

∂I∂i

< 0


• Thiscondi=ondescribesthecombina=onsofrealoutputandthenominalinterestratethatensureequilibriuminthemoneymarket,inthesensethatmoneydemandisequaltotheexogenousmoneysupply.Itisdepicteddiagramma=callyasanupwardslopingLM(Liquidity-Money)curve.

EquilibriumCondi=onintheMoneyMarket

Mt

Pt= m Yt ,it( )

∂m∂Y

> 0, ∂m∂i

< 0


TheIS-LMModelofSimultaneousEquilibriumintheOutputandMoneyMarkets


• Realoutputandthenominalinterestratearedeterminedatthepointwhereboththemarketforgoodsandservicesandthemarketformoneyareinequilibrium.

• Atthepointofintersec=onoftheIScurve,whichdescribesequilibriuminthemarketforgoodsandservices,andtheLMcurve,whichdescribesequilibriuminthemoneymarket,theeconomyisthusinshortrunequilibrium.

• Sincethepricelevelisassumedtobefixed,thisequilibriumcouldbeatlessthanfullemployment.Infact,thisisassump=onusuallymadeinKeynesianmodels.

ShortRunEquilibriumintheIS-LMModel


• Anincreaseingovernmentexpenditure,orataxcut,shifstheIScurvetotheright,andcausesanincreaseinrealoutputandthenominalinterestrate.

• AnincreaseinthemoneysupplyshifstheLMcurvetotheright,andcausesanincreaseinrealincomeandareduc=oninthenominalinterestrate.

• Bothfiscalandmonetarypoliciescanthusleadtoanincreaseinaggregatedemand,andanincreaseinrealoutputandemployment.

• Thisisthera=onaleforaggregatedemandpoliciesintheKeynesianmodel.

TheEffectsofMonetaryandFiscalPolicyintheIS-LMModel


• Thelastandmoresophis=catedformofthebasicKeynesianmodelisanalyzedaferChapter19oftheGeneralTheory.

• Inthisformofthemodel,thepricelevelceasestobeexogenous,andisallowedtochangeinordertoequilibrateaggregatedemandforgoodsandserviceswithalessthanperfectlyelas=caggregatesupply.However,nominalwagesares=llconsideredtobefixedintheshortrun.

• TheaggregatedemandfuncAon(AD)isderivedfromthesimultaneoussa=sfac=onoftheequilibriumcondi=oninthemarketforgoodsandservices(IS)andtheequilibriumcondi=oninthemoneymarket(LM).

• TheaggregatesupplyfuncAon(AS),isderivedfromthemaximizingbehaviorofarepresenta=vecompe==vefirm,selec=ngthelevelofemploymentandoutput,takingnominalwagesandpricesasgiven.

TheAD-ASModel


Ahigherpricelevel,giventhemoneysupply,meanslowerrealmoneybalances,highernominalinterestratesandlowerinvestmentdemand.Thus,giventhemoneysupply,anincreaseinthepricelevelreducesaggregatedemand,andafallinthepricelevel,increasesaggregatedemand.TheADscheduleisthusdownwardsloping.

TheAggregateDemandSchedule(AD)

Yt = DMt

Pt,Gt ,Tt

⎛⎝⎜

⎞⎠⎟

∂D∂Pt

< 0


Sincemaximizingbehaviorbytherepresenta=vefirmimpliesadownwardslopinglabordemandcurvewithrespecttotherealwage,employmentandoutputareanega=vefunc=onoftherealwage.Ifthenominalwageisexogenouslyfixed,asassumedinthisversionoftheKeynesianmodel,thenaggregatesupplyisaposi=vefunc=onofthepricelevel,asahigherpricelevelisassociatedwithalowerrealwage.

TheAggregateSupplySchedule,AS

Yt = SWt

Pt

⎛⎝⎜

⎞⎠⎟

∂S∂Pt

> 0


TheAD-ASModel


ShortRunEquilibriumwithLowEmploymentintheAD-ASModel


• Assumeanoriginalequilibriumisat“high”employmentoutput.Anega=vedisturbanceinaggregatedemand,shifstheaggregatedemandcurveADtothelef.Inthenewshort-runequilibrium,incomeandemploymentdeclines,andthepricelevelfalls.Becauseoftherigidityofnominalwages,theeconomymovestoanequilibriumwithlowerrealoutputandemploymentandhigherunemployment.

• Ifnominalwageswerenotfixedintheshortrun,andadjustedtoachievehighemployment,theaggregatesupplyfunc=onwouldbeperpendiculartothelevelofhighemployment,andchangesinaggregatedemandwouldonlyresultinchangesinthepricelevel.Consequently,withfullflexibilityofwagesandpricesandwages,thismodelhasthesameproper=esasa“classical”model.

• Incontrasttothe“classical”modeloffulladjustmentofwagesandprices,intheKeynesianmodelwithnominalwagerigidity,evenmonetarydisturbancescanshifaggregatedemandandcausefluctua=onsinrealoutput,employmentandotherrealvariablessuchasrealwagesandinterestrates.

AggregateDemandDisturbancesandAggregateFluctua=onsintheAD-ASModel


• Howcantheimpactofshockstoaggregatedemandandsupplybeaddressed?AccordingtotheKeynesianapproach,anappropriatesolu=oncancomefrommacroeconomicpolicy.Anincreaseofgovernmentexpenditure,areduc=onintaxesoranincreaseinthemoneysupplycanmovetheaggregatedemandcurvetotheright,andcounteracttheconsequencesofanini=aldemandorsupplyshockonrealoutputandunemployment.Inthecaseofdemandshocks,thepricelevelreturnstoitsoriginalequilibrium.Inthecaseofanega=vesupplyshocktherearefurthereffectsonthepricelevel.

• Thisisthejus=fica=onandtheeffectsofaggregatedemandpoliciesintheKeynesianmodel.Inprinciple,aggregatedemandpolicies,suchasmonetarypolicy,couldbeusedtocounteracttheeffectsofbothdemandandsupplyshocksonaggregateoutputandunemployment.Thiswouldimplythataggregatedemandpoliciescouldinprinciplebeusedtostabilizeeconomiesaroundlevelsofhighemploymentandhelpavoidtherepe==onofphenomenasuchastheGreatDepression.

TheJus=fica=onforAggregateDemandPoliciesintheKeynesianModel


RecessionsandRealOutputintheUSA,1854-2014

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RecessionsandUnemploymentintheUSA,1889-2014

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RecessionsandInfla=onintheUSA,1854-2014

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Recession% Infla8on%(based%on%CPI)%


• Theassump=onofabsolutenominalrigidityineitherpricesorwagesisobviouslynotrealis=c.Economiesarecharacterizedbythesimultaneousexistenceofinfla=onandunemployment,aphenomenonwhichimpliesthatbothpricesandwagesadjustgradually.

• Sincethelate1950s,acentralpointofreferencefortheKeynesianmodelwasthePhillipscurve.Thiswasanega=verela=onshipbetweenunemploymentandwageinfla=on,documentedbyPhillips(1958).

• ThePhillipscurve,wascombinedwiththeIS-LMmodelofaggregatedemand,tosimultaneouslydeterminebothinfla=onandunemployment.

TheKeynesianApproachandthePhillipsCurve


• ThePhillipscurvewasanempiricalrela=onship,es=matedeconometricallybyA.W.Phillipsinthelate1950s:

ThePhillipsCurve

• whereπdenotesinfla=on,utheunemploymentrateandu0theequilibriumunemploymentrate,i.etheonecorrespondingtozeroinfla=on.Thefunc=onalrela=onshipφbetweeninfla=onandunemploymentes=matedbyPhillipswasnonlinear.

π =ϕ(u), ϕ(u0 ) = 0, ′ϕ (u) < 0


• ΗκαμπύλητηνοποίαεκτίμησεοικονομετρικάοPhillipsείχετηνεξήςμορφή:

TheOriginalPhillipsCurve,1861-1913


AGeneralRepresenta=onofthePhillipsCurve


• InthecontextofthebasicKeynesianmodel,combiningtheIS-LMmodelofaggregateddemandwiththePhillipscurve,onecoulddeducethatanincreaseinaggregatedemandwouldleadtohigherrealincomeandemployment,lowerunemployment,andariseininfla=onalongthePhillipscurve.

• Conversely,adeclineinaggregatedemandwouldleadtoalowerlevelofrealincomeandemployment,higherunemploymentandareduc=onininfla=onalongthePhillipscurve.

• Usingthiscombinedapproach,SamuelsonandSolow(1960),arguedthattheshort-termproblemofmacroeconomicpolicycouldbeseenasthedetermina=onofthelevelofaggregatedemand,inordertoselectthesociallydesiredcombina=onofinfla=onandunemployment.

• In=mesofrecession,anincreaseinaggregatedemandwillleadtoareduc=oninunemployment,butatthecostofhigherinfla=on.In=mesofeconomicboomandhighinfla=on,infla=oncouldbereducedthroughareduc=oninaggregatedemand,butthiswouldalsoresultinhigherunemployment.

MacroeconomicPolicyintheKeynesianModelCombinedwiththePhillipsCurve


• Sincethemid-1960s,thenega=verela=onshipbetweeninfla=onandunemploymentbegantoshif.Higherinfla=onledtoareduc=onofunemploymentonlytemporarily,asunemploymentroseaferawhiletoreturntoitsoriginallevelwithoutareduc=onininfla=on.

• Thiswasa�ributedtotheimpactofinfla=onaryexpecta=ons.AsarguedbyPhelps(1967)andFriedman(1968),asustainedincreaseininfla=onwouldleadtoexpecta=onsofhigherfutureinfla=ononpartofhouseholdsandfirms.Theresultofthiswouldbethatinfla=onwouldhavetoincreaseevenfurtherinordertoachieveareduc=oninunemployment.

TheInstabilityofthePhillipsCurveandInfla=onaryExpecta=ons


• Essen=ally,PhelpsandFriedmanarguedthatthePhillipscurvetakestheform,

AnExpecta=onsAugmentedPhillipsCurve

• whereπisinfla=on,πeexpectedinfla=on,utheunemploymentrate,andu0theequilibriumunemploymentrate,correspondingtotheequalitybetweenactualandexpectedinfla=on.

π = π e +ϕ(u), ϕ(u0 ) = 0, ′ϕ (u) < 0


Infla=onaryExpecta=onsandtheShifingPhillipsCurve


• AccordingtoFriedman(1968),eacheconomytendstowardsanequilibriumunemploymentrateu0,whichcanbelabelledthe“natural”rateofunemployment.

• The“natural”rateofunemploymentdependsonlyonrealfactors,includinglabormarketfric=ons,distor=onsandinefficiencies.

• Tryingtoreducetheunemploymentratebelowthis“natural”rate,byincreasingaggregatedemandandinfla=on,wouldonlymeetwithtemporarysuccess.

• Asinfla=onaryexpecta=onsadjusttohigherinfla=on,unemploymenttendstoreturntothe“natural”rate.Onewouldneedcon=nuousincreasesininfla=oninordertokeepunemploymentbelowits“natural”rate.

The“Natural”RateofUnemployment


InordertoanalyzetheFriedman-Phelpsargument,assumethatthePhillipscurveislinearandgivenby,

The“Natural”RateHypothesisandthePhillipsCurve

π t = π te + a − but

whereaandbareposi=veparameters.Theeconomyisatits“natural”ratewheninfla=onaryexpecta=onsareequaltoactualinfla=on.Consequently,the“natural”rateofunemploymentinthismodelisgivenby,

u0 =ab


Formanyyears,thedominantapproachtotheforma=onofexpecta=onsinmacroeconomicswastheadap=veexpecta=onshypothesisofCagan(1956).

Accordingtothishypothesis,expecta=onsineachperiodareadjustedbyapercentageofthedevia=onoftheactualfromtheexpectedvalueofavariableinthepreviousperiod.Consequently,theadjustmentofinfla=onaryexpecta=onsinlinewiththishypothesiswouldtaketheform,

Infla=onaryExpecta=onsandtheAdap=veExpecta=onsHypothesis

π te −π t−1

e = (1− λ)(π t−1 −π t−1e )

Infla=onaryexpecta=onsareadjustedbyapercentage1-λofthedivergencebetweenactualandexpectedinfla=oninthepreviousperiod.1-λisthespeedofadjustmentofexpecta=ons.λ=0impliesfulladjustmentwithinaperiod,andλ=1impliesnoadjustment(staAcexpectaAons).


Whatwouldhappeninthismodelifthegovernmenthadafixedtargetforunemployment,whichwaslowerthanthe“natural”rateofunemployment?Inthiscase,thegovernmentandthemonetaryauthori=eswouldseekapathforinfla=oninordertomaintainunemploymentbelowits“natural”rateu0atuA,whereuA<u0.

Infla=onineachperiodwouldbedeterminedby,

ThePathofInfla=onandUnemploymentunderAdap=veExpecta=ons

π t = π te + a − buA = π t

e + b(u0 − uA )

Infla=onaryexpecta=onsineachperiodwouldbedeterminedby,

π te = (1− λ)π t−1 + λπ t−1

e = 1− λ1− λL

π t−1 = (1− λ) λ iπ t−1−ii=0

∞∑


Solvingforinfla=on,subs=tu=ngouttheadap=veexpecta=onshypothesis,infla=onineachperiodwouldbedeterminedby,

ThePathofInfla=onandUnemploymentunderAdap=veExpecta=ons

π t = π t−1 + b(1− λ)(u0 − uA )

Infla=onhasaunitroot,andthusnotonlydoesitnotconverge,butitincreasesineachperiodbyapercentagewhichdependsonthedifferencebetweenthe“natural”rateofunemploymentrateu0fromthegovernment'stargetunemploymentrateuA.Asthegovernmenta�emptstokeepunemploymentbelowits“natural”rate,itmustberelyingonconstantlyincreasinginfla=on,sothatinfla=onisalwayshigherthaninfla=onaryexpecta=ons.


TheShifingPhillipsCurveandthe“Natural”RateofUnemployment


TheinstabilityofthePhillipscurvederivesdirectlyfromthea�emptbythegovernmenttokeepunemploymentbelowits“natural”rate.Otherwise,infla=onwouldconverge.

Assumethatthegovernmenthadafixedinfla=ontargetπ0,andkeptinfla=onconstantatthistarget.

Theninfla=onaryexpecta=onswouldfollow,

TheInstabilityofthePhillipsCurveandGovernmentPolicy

π te = (1− λ)π 0 + λπ t−1

e

which,forλ<1,convergestoπ0.


• TheinstabilityoftheoriginalPhillipscurveobservedinthelate1960s,andtheinterpreta=ongiventoitbyPhelpsandFriedman,sparkedarevolu=onintheanalysisofaggregatefluctua=onsandmacroeconomicpolicy.

• Theliteraturehassincemovedintwodirec=ons.

• First,greatemphasiswasplacedonthemicroeconomicfounda=onsofthedetermina=onofwages,prices,thesupplyanddemandforgoodsandservicesandthedetermina=onoftheequilibriumunemploymentrate.Theresultisthattodayboth“newclassical”andthe“newKeynesian”modelsofaggregatefluctua=onsarebasedondynamicstochas=cgeneralequilibriummodelswithdetailedandconsistentmicroeconomicfounda=ons.

• Secondly,thisdevelopmentledtotheadop=onofthehypothesisofra=onalexpecta=onsinsteadofadap=veexpecta=ons.

TheInstabilityofthePhillipsCurveandMacroeconomicTheoryandPolicy


InthislecturewehaveintroducedthebasicKeynesianmodels,whichwereassociatedwiththedevelopmentofmacroeconomicsfromthemid-1930stotheendofthe1960s.

WehaveshownthatwhatseparatesclassicalfromKeynesianmodelsistheassump=onofnominalrigidityand/orgradualadjustmentinnominalwagesandthepricelevel.ThisiswhatallowsKeynesianmodelstoexplaintherealeffectsofmonetaryandaggregatedemandshocks,andprovideara=onaleaggregatedemandpolicies.

Muchofthetheore=calresearchofthelastfivedecadeshasfocusedonthemicroeconomicfounda=onsofKeynesianmodels,andinpar=cularthePhillipscurve.Thisisanimportantpartoftheso-callednewKeynesianapproach.

Inthenextlectureweanalyzeasimpledynamicmodelthatbelongstothiscategoryandlookatthepredic=onsofthe“new”Keynesianapproachforaggregatefluctua=onsandtherela=onshipbetweeninfla=onandunemployment.

Evalua=ngtheTradi=onalKeynesianModels


Inanimportantpaper,Samuelson(1939)combinedthemodeloftheKeynesiancrosswithaninvestmentfunc=onbasedontheprincipleofaccelera=on,toderiveendogenouscycles.SamuelsonconsideredthefollowingversionoftheKeynesianmodel.

AnnexontheSamuelsonMul=plierAcceleratorModel

Yt = Ct + It +Gt

It = a(Ct −Ct−1)+ b

Ct = cYt−1 + d

Gt = e


• a,b,c,dandeareconstantparameters.aistheaccelerator,whichdetermineshowachangeinconsump=onaffectscurrentinvestment,andbisautonomousinvestment.c<1isthemarginalpropensitytoconsume,anddisautonomousconsump=on.Finallyeisgovernmentexpenditure,assumedconstant.

• Theinvestmentfunc=onisbasedonthesocalledacceleraAonprinciple,whichimpliesthatwhenthereisaposi=vechangeinconsump=on,firmswillinvestmore,inordertoproducethehigherquan=tyofconsumergoodsdemanded.Whenthereisanega=vechangeinconsump=on,firmswillreduceinvestment,astheyneedlesscapitaltomeetthelowerdemandforconsumergoods.Theacceleratorameasuresthesensi=vityofinvestmenttochangesinaggregateconsump=on.

• Theconsump=onfunc=onislinear,accordingtowhichconsump=onisafunc=onoflaggedandnotcurrentincome.

• Governmentexpenditureisassumedexogenousandconstantate.

TheParametersoftheSamuelsonModel


Subs=tu=ngtheconsump=onfunc=onintotheinvestmentfunc=on,

TheDynamicsoftheSamuelsonModel

It = ac(Yt−1 −Yt−2 )+ b

Subs=tu=ngthisinvestmentfunc=onandtheconsump=onfunc=onintotheequilibriumcondi=onintheoutputmarket,

Yt = c(1+ a)Yt−1 − acYt−2 + b + d + e

Ifthisdifferenceequa=onconverges,equilibriumisgivenby,

Y*= b + d + e1− c


Equilibriumoutputdependsonlyonautonomousexpenditureandthemul=plier1/(1-c),assuggestedbythemodeloftheKeynesiancross.However,thepathofoutputisdeterminedbyasecondorderdifferenceequa=on,whichdependsonboththemul=plierandtheaccelerator.

Forthedifferenceequa=ontobestable,andfortheeconomytoconvergetoequilibriumoutput,acmustbelessthanone,ortheacceleratormustsa=sfya<1/c.Ifthiscondi=onisnotsa=sfied,outputwilldiverge.

Thus,forstabilitytheacceleratormustbesmallerthantheinverseofthemarginalpropensitytoconsume.

Proper=esoftheSamuelsonModel


Fortherootstoberealandforincometoconvergemonotonicallytoitsequilibriumvalue,thecondi=onis,

ConvergenceintheSamuelsonModel

4a(1+ a)2

≤ c < 1a

Ifthecondi=ononthelefhandsideisnotsa=sfied,butthecondi=onontherighthandsideis,convergencetakesplacenon-monotonicallyanddisplaysoscilla=onsastherootsarenotreal.


ConvergenceintheSamuelsonModelforc=0.75andDifferentValuesofa

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

a=0.10 a=0.333 a=1.10

Keynesian Models and the Phillips Curve - … George Alogoskoufis, Dynamic Macroeconomic Theory,...

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