Macro Lect 15 New Keynesian Model Wage Contracts · Prof George Alogoskoufis, Dynamic Macroeconomic...

ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015

Labor Market “Insiders”, Nominal Wage Contracts and

Unemployment

A New Keynesian Model with Periodic Wage Contracts

ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 2

• Sincethe1970s,themacroeconomicsofaggregatefluctua?onshasbeenemphasizingthemicroeconomicfounda?onsofallbehavioralrela?ons,andinpar?culartheconsump?onandinvestmentfunc?onsandtheshort-termdetermina?onofwages,pricesandtheequilibriumunemploymentrate.

• Inaddi?on,the“ra?onalexpecta?ons”hypothesis,whichrequiresthathouseholdsandfirmsformtheirexpecta?onsaboutfuturevariables,takingintoaccounttheactualprocessdeterminingtheevolu?onofthesevariables,hasbecomethedominantexpecta?onshypothesis.Thehypothesisofadap?veexpecta?ons,wasgraduallyabandoned.

• Thus,macroeconomicmodelsofaggregatefluctua?onsgraduallyevolvedintodynamicstochas3cgeneralequilibriummodelsbasedonra?onalexpecta?ons.

DynamicStochas?cGeneralEquilibriumModels


• The“newclassical”modelisadynamicstochas?cgeneralequilibriummodel,inwhichwagesandpricesareperfectlyflexibleandequilibrateboththeproductandlabormarkets.

• In“newclassical”models,onlyrealshocks,suchasshockstoproduc?vity,canaffectthefluctua?onsofoutput,employmentandotherrealvariables.

• Monetaryshocksonlyaffectnominalvariables,suchasthepricelevelandinfla?onbutnorealvariables.

• Inaddi?on,employmentfluctua?onsarebasedonlabormarketequilibriumandintertemporalsubs?tu?oninlaborsupply,and,thus,thereisnoinvoluntaryunemploymentinthe“newclassical”model.

Weaknessesofthe“New”ClassicalModel


• Theshortrunneutralityofmoneyimpliedby“newclassical”modelswasini?allytroublesomefortheirproponents,asthesemodelswerenotcompa?blewiththeexistenceofaposi?veshortrunrela?onbetweeninfla?onandemployment,assuggestedbytheexpecta?onsaugmentedPhillipscurve.

• Lucas(1972,1973),developeda“newclassicalmodel”whichwasconsistentwithaposi?veshortrunrela?onbetweeninfla?onandemployment.Thismodelwasbasedontheassump?onthatfirmsdidnothavefullinforma?onaboutthepricelevelatthe?metheymadetheirproduc?ondecisions,andtheya[ributedpartofanychangeinthepriceleveltoachangeintherela?vepriceoftheirproduct.Thus,wheninfla?onwasunexpectedlyhigh,allproducersthoughttherela?vepriceoftheiroutputhadgoneup,andthusincreasedproduc?onandemployment.Theoppositehappenedwheninfla?onwasunexpectedlylow.

• However,this“newclassical”explana?onoftheshortrunrela?onbetweeninfla?onandoutputandemploymentwass?llincompa?blewithinvoluntaryunemployment,andcouldonlyaccountfortemporarydevia?onsofoutputandemploymentfromtheir“naturallevels”duetointer-temporalsubs?tu?oninlaborsupplyandunan?cipatedinfla?on.

RealEffectsofMoneyinthe“New”ClassicalModel


• Analterna?veapproach,duetoGray(1976),Fischer(1977)andTaylor(1979),emphasizedperiodicnominalwagecontracts.

• ThisapproachdescendeddirectlyfromtheGeneralTheory,trea?ngnominalwagesastemporarilyfixed.

• IntheGray-Fischermodel,nominalwagecontractsareassumedtobenego?atedatthebeginningofeveryperiod,oratthebeginningofalternateperiods.Inaddi?on,nominalwagesareassumedtoremainfixedforthedura?onofthecontract.Thus,nominalwagesdependonpriorexpecta?onsabouttheevolu?onofthepricelevel,produc?vityandallothershocks.

• Ifinfla?onturnsouttobehigherthanexpected,thenrealwagesfall,firmsdemandmorelabor,andemploymentrises.Theoppositehappenswheninfla?onturnsouttobelowerthanexpected.Thus,thesemodelshavekeynesianfeatures,andhaveformedthebasisofthesocallednewkeynesianapproachtoaggregatefluctua?ons.

The“New”KeynesianApproach:PeriodicNominalWageContracts


• The“new”keynesianapproachalsoemphasizedlabormarketdistor?onsthatmadethe“natural”rateofunemploymentinefficientlyhigh.

• Onesuchdistor?onthatweshallemphasizeisthedis?nc?onbetween“insiders”and“outsiders”inthelabormarket.

• “Outsiders”aredisenfranchisedfromthelabormarket,and,asaresult,wagecontractsdonotaimtomaintainfullemployment.Evenwithoutshocks,thereisan“inefficiently”high“natural”rateofunemployment.

• Sincefirmsdetermineemployment,giventhewagecontracts,forthosewhoturnoutunemployed,the“outsiders”,unemploymentisinvoluntary.

The“New”KeynesianApproach:AnInefficientlyHigh“Natural”Rate


• Inthislectureweanalyzea“newKeynesian”modelbasedonsuchperiodicnominalwagecontracts,whichiscomparabletothe“newclassical”modelwithoutcapital.

• Itnotonlyallowsfornominalshocksandmonetarypolicytoaffectthefluctua?onsofrealvariables,butitalsoallowsfortheexistenceof“involuntary”unemployment.

• ThemodelbuildsononeofthekeyinsightsoftheGeneralTheory,namelytheshortrunrigidityofnominalwages,asenvisagedbyGrayandFischercontracts.

• Inallotherrespectsitisbasedoninter-temporalop?miza?ononthepartofbothhouseholdsandfirms.

• Themodelisinessenceadynamicstochas?cgeneralequilibriummodelthatincorporatesmanyofthefeaturesoftheAS-ADversionoftheKeynesianmodel.

A“New”KeynesianModelwithaPosi?veNaturalRateofUnemploymentandPeriodicNominal

WageContracts


Weconsideraneconomyconsis?ngofcompe??vefirms,indexedbyi,wherei∈[0,1].Laboristheonlyvariablefactorofproduc?on,andfirmsdetermineemploymentbyequa?ngthemarginalproductoflabortotherealwage.Theproduc?onfunc?onoffirmiisgivenby,

Output,EmploymentandLaborDemand

Y (i)t = AtL(i)t1−α

Firmsdetermineemploymentbyequa?ngthemarginalproductoflabortotherealwage.Labordemandbyfirmiisgivenby,

(1−α )AtL(i)t−α = W (i)t

Pt


Inlog-linearform,theproduc?onfunc?onandlabordemandcanbewri[enas,

Output,EmploymentandLaborDemandinLog-linearForm

y(i)t = at + (1−α )l(i)t l(i)t = l_− 1α(w(i)t − pt − at )

Aggrega?ngacrossfirms,aggregateoutputandemploymentaredeterminedby,

yt = at + (1−α )lt lt = l_− 1α(wt − pt − at )


Nominalwagesaresetby“insiders”ineachfirm,atthebeginningofeachperiod,beforevariables,suchascurrentproduc?vityandthecurrentpricelevelareknown.Thus,nominalwagesaresetonthebasisofthera?onalexpecta?onsof“insiders”abouttheseshocks.Nominalwagesremainconstantforoneperiod,andtheyareresetatthebeginningofthefollowingperiod.

Objec?veofNominalWageContracts

minEt−1 β ss=0

∞∑ 12l(i)t+s − n

_(i)t+s

⎛⎝

⎞⎠

2⎡

⎣⎢

⎤

⎦⎥

Et−1l(i)t = n_(i)t


Weassumethatthetotalnumberof“insiders”intheeconomyisalwaysstrictlysmallerthanthelaborforce.Wethusassumethat,

“Insiders”andthe“NaturalRate”ofUnemployment

n_(i)t dii=0

1

∫ = n_t < nt

Sincewagecontractsmakeexpectedemploymentequaltothenumberof“insiders”,itfollowsthat,

Et−1lt = n_t < nt

The“natural”rateofunemployment,whichisinvoluntary,becauseoutsiderswouldbepreparedtoworkattheprevailingrealwage,isdefinedas,

u_t ! nt − n

_t > 0


Thewageissetsoastomakeexpectedemploymentequaltothenumberofinsiders,andisbasedononeperiodaheadexpecta?onsaboutthepricelevelandproduc?vity.Itthussa?sfies,

NominalWageContractsandtheDetermina?onofEmployment

wt = Et−1pt + Et−1at −α (n_t− l

_)

Oncethewagecontracthasbeensigned,firmsdetermineemployment,afertheyhaveobservedthecurrentpricelevelandcurrentproduc?vity.Thus,employmentisdeterminedby,

lt = n_t+1α

pt − Et−1pt + at − Et−1at( ) = n_t+1α

π t − Et−1π t + at − Et−1at( )


Employmentdeviatesfromits“natural”leveltotheextentthatthereareunan?cipatedshockstoinfla?onandproduc?vity.

Unan?cipatedincreasesininfla?oncauseareduc?oninrealwagesandincreaselabourdemandandemployment.

Unan?cipatedincreasesinproduc?vityincreaseproduc?vityrela?vetorealwages,andthusalsoincreaselabourdemandandemployment.

Thisemploymentfunc?onisthebasisforan“expecta?onsaugmentedPhillipscurve”inthismodel.

Themodelis“Keynesian”,butitalsoincorporatesthe“naturalrate”hypothesisofFriedman.

Infla?onandEmploymentwithNominalWageContracts


Thecurrentunemploymentrateisdefinedby,

NominalWageContractsandtheExpecta?onsAugmented“PhillipsCurve”

ut ! nt − lt

Usingtheemploymentfunc?ontosubs?tuteforcurrentemployment,

ut = u_t−1α


Theunemploymentratedeviatesfromits“natural”rateasaresultofunexpectedshockstoinfla?onandproduc?vity,becausebothreducerealwagesrela?vetoproduc?vity,comparedwiththepriorexpecta?onsofwagese[ers.


Unexpectedshockstoinfla?onandproduc?vitycauseoutputtobehigherthanits“natural”level,astheycauseemploymenttobehigherthanitsown“natural”level.Thisequa?oncanbeseenastheoutputversionofthe“expecta?onsaugmentedPhillipscurve”,orasashortrun“outputsupplyfunc?on”.

NominalWageContractsandFluctua?onsofOutput

Usingtheemploymentfunc?ontosubs?tuteforcurrentemploymentintheproduc?onfunc?on,

yt = y_

t+1−αα


where,y_

t = (1−α )n_t+ at


Itisworthdis?nguishingbetweenthe“natural”levelofoutputandthe“fullemployment”leveloutput.Fullemploymentoutputisgivenby,

The“NaturalRate”ofUnemploymentandthe“NaturalLevel”ofOutput

ytf = (1−α )nt + at

Fullemploymentoutputisalwayshigherthanthe“natural”levelofoutputinthismodel.Thereasonisthatequilibriumemploymentislowerthanfullemployment,sincethepoolof“insiders”,whoaretheoneswhodetermineequilibriumemploymentthroughtheirwagesegngbehavior,issmallerthanthelaborforce.Thus,becauseofthisrealdistor?oninthelabormarket,the“natural”levelofoutputisinefficientlylow,andthe“naturalrate”ofunemploymentisinefficientlyhigh.Theirrela?onisgivenby,

ytf − y

_

t = (1−α )(nt − n_t ) = (1−α )u

_t


• Weassumethattheeconomyconsistsofalargenumberofiden?calhouseholdsj,wherej∈[0,1].

• Eachhouseholdmembersuppliesoneunitoflabor,andunemploymentimpactsallhouseholdsinthesamemanner.

• Thus,ifHisthenumberofhouseholdsandNistheaggregatelaborforce,eachhouseholdhasN/Hmembers.Ofthose,someare“insiders”inthelabormarket,andtherestare“outsiders”.

• Thepropor?onofinsidersisthesameforallhouseholds.Inaddi?on,thepropor?onoftheunemployedisalsoassumedtobethesameforallhouseholds.

Households,EmploymentandUnemployment


Therepresenta?vehouseholdchooses(aggregate)consump?onandrealmoneybalancestomaximize,

Households,Consump?onandMoneyDemand

Et1

1+ ρ⎛⎝⎜

⎞⎠⎟s=0

∞∑s

11−θ

Vt+sC Ct+s

1−θ +Vt+sM M

P⎛⎝⎜

⎞⎠⎟ t+s

1−θ⎛

⎝⎜⎞

⎠⎟⎛

⎝⎜

⎞

⎠⎟

subjecttothesequenceofexpectedbudgetconstraints,

Et Ft+s+1 − (1+ it+s ) Ft+s −it+s1+ it+s

Mt+s + Pt+s Yt+s −Ct+s −Tt+s( )⎛⎝⎜

⎞⎠⎟

⎛

⎝⎜⎞

⎠⎟= 0

Ft = Bt +Mt


Fromthefirstordercondi?onsfortherepresenta?vehousehold,

FirstOrderCondi?onsoftheRepresenta?veHousehold

VtCCt

−θ = λt (1+ it )Pt VtM M

P⎛⎝⎜

⎞⎠⎟ t

−θ

= λtitPt Etλt+1 = Et1+ ρ1+ it+1

⎛⎝⎜

⎞⎠⎟λt

Attheop?mumthehouseholdequatesthemarginalu?lityofconsump?ontothevalueofsavings.Italsoequatesthemarginalu?lityofrealmoneybalancestotheopportunitycostofholdingmoney.Finally,therealinterestrate,adjustedfortheexpectedincreaseinthemarginalu?lityofconsump?on,isequaltothepurerateof?mepreference.


Elimina?ngλfromthethreefirstordercondi?ons,wederivethemoneydemandfunc?onandtheEulerequa?onforconsump?on.

FromtheFirstOrderCondi?onstotheEulerEqua?onforConsump?onandtheMoney

DemandFunc?on

MP

⎛⎝⎜

⎞⎠⎟ t

= CtVt

C

VtM

it1+ it

⎛⎝⎜

⎞⎠⎟

−1θ

EtVt+1

C Ct+1( )−θPt+1

⎛

⎝⎜

⎞

⎠⎟ =

1+ ρ1+ it

⎛⎝⎜

⎞⎠⎟Vt

C Ct( )−θPt

⎛

⎝⎜

⎞

⎠⎟


Log-linearizingtheEulerEqua?onforConsump?onandtheMoneyDemandFunc?on,andimposingtheproductmarketequilibriumcondi?onCt=Yt,

ALog-linearVersionoftheEulerEqua?onforConsump?onandMoneyDemand,andProduct

MarketEquilibrium

mt − pt = yt −1θln it1+ it

⎛⎝⎜

⎞⎠⎟+ 1θvtM − vt

C( )

yt = Etyt+1 −1θit − Etπ t+1 − ρ( )+ 1

θ(vt

C − Etvt+1C )

Thosetwoequa?onsareofenreferredtoasthe“newkeynesian”IScurve,andthe“newkeynesian”LMcurve,describingaggregatedemandforoutputandequilibriuminthemoneymarket.


• Sinceoutputdemanddependsondevia?onsoftherealinterestfromthepurerateof?mepreference,therealinterestrateistherela?vepricethatadjuststoequilibrateoutputdemandwithoutputsupply.

• Nootherrela?vepricecanplaythisrole,astherealwageisdeterminedinordertomakeexpectedlabordemandequaltothenumberof“insiders”inthelabormarket.

AdjustmentoftheRealInterestRatetoEquilibratetheProductMarket

TherealinterestrateisdefinedbytheFisherequa?on,

rt = it − Etπ t+1


The“natural”realinterestrateisdeterminedbytheproductmarketequilibriumcondi?on,whenoutputisatits“natural”level.Fromthe“newkeynesian”IScurveandthedefini?onofthe“natural”levelofoutput,

The“Natural”RealInterestRate

r_t = ρ −θ (1−α ) n

_t− Et n

_t+1

⎛⎝

⎞⎠ + at − Etat+1( )⎛

⎝⎜⎞⎠⎟ + vt

C − Etvt+1C( )

The“natural”realinterestrateisequaltothepurerateof?mepreference,butalsodependsondevia?onsofcurrentrealshocksfroman?cipatedfutureshocks.Realshocksthatcauseatemporaryincreaseinthe“natural”levelofoutputreducethe“natural”realrateofinterest,inordertobringaboutancorrespondingreduc?oninconsump?onandmaintainproductmarketequilibrium.Ontheotherhand,realshocksthatcauseatemporaryincreaseinconsump?on,requireanincreaseinthe“natural”realrateofinterest,inordertoinducelowerconsump?on,andmaintainproductmarketequilibrium.


Becauseofthenominalrigidityofwagesforoneperiod,thecurrentequilibriumrealinterestdeviatesfromits“natural”rate.Thecurrentrealinterestrateisdeterminedbytheequa?onofthe“newkeynesian”IScurvewiththeshortrunoutputsupplyfunc?on.Itisthusdeterminedas,

TheCurrentRealInterestRate

rt = r_t−

θ 1−α( )α


Unan?cipatedshockstoinfla?onorproduc?vity,whichcauseatemporaryriseincurrentoutputrela?vetoits“natural”level,alsoreducethecurrentrealinterestraterela?vetoits“natural”rate.Thisisthe“Wicksellian”mechanisminthismodel.


Inwhatfollows,weshallassumethatthelogarithmsoftheexogenousshockstopreferencesandproduc?vityfollowsta?onaryAR(1)processes.

Assump?onsaboutExogenousShocks

vtC =ηCvt−1

C + ε tC vt

M =ηMvt−1M + ε t

M at =ηAat−1 + ε tA

Weshallfurtherassumethatthe(logofthe)laborforceisfixedatn,andthattheexogenousnumberof“insiders”alsofollowsasta?onaryAR(1)process,oftheform,

n_t = (1−ηN )n

_+ηN n

_t−1+ ε t

N


Sincethenumberof“insiders”fluctuates,the“natural”rateofunemploymentalsofluctuates,accordingto,

Fluctua?onsinthe“Natural”RatesofUnemploymentandOutput

u_t = (1−ηN )u

_+ηN u

_t−1− ε t

N

Sincethenumberof“insiders”andtotalfactorproduc?vityfluctuate,the“natural”levelofoutputalsofluctuates,accordingto,

y_

t = (1−α )n_t+ at


Fluctua?onsinEmployment,UnemploymentandOutput

lt = n_t+1α

π t − Et−1π t + ε tA( )

ut = u_t−1α

π t − Et−1π t + ε tA( )

yt = y_

t+1−αα

π t − Et−1π t + ε tA( )


Fluctua?onsintheRealWageandtheRealInterestRate

wt − pt = (w − p_)t − π t − Et−1π t( )

rt = r_t−

θ 1−α( )α

π t − Et−1π t + ε tA( )

where,

(w − p_)t =ηAat−1 −α (n

_t− l

_)

r_t = ρ −θ (1−α )(1−ηN )(n

_t− n

_)+ (1−ηA )at

⎛⎝

⎞⎠ + (1−ηC )vt

C


• The“natural”rates(orlevels)ofrealvariablesevolveasfunc?onsoftheexogenousrealshocks.

• Intheabsenceofthenominalrigidityduetotheassump?onthatnominalwagesaresetinadvanceandremainfixedforoneperiod,theevolu?onofrealvariableswouldbeequaltotheir“natural”levels.Themodelwouldinallrespectsbesimilartoa“newclassical”model.

• However,unan?cipatedinfla?on,andinnova?onsinproduc?vity,byreducingrealwagesrela?vetoproduc?vity,causeatemporaryincreaseinemploymentandoutputabovetheir“natural”level,andatemporaryreduc?oninunemploymentandtherealinterestratebelowtheir“natural”rates.

• Sinceinfla?onisalsoaffectedbynominalshocks,unan?cipatednominalshockshaverealeffectsinthismodel.

Proper?esoftheModel


Weshallini?allyassumethatthemoneysupplyfollowsarandomwalkwithdrif,oftheform,

AnExogenousMoneySupplyRule

mt = µ +mt−1 + ε tS

Withthisassump?on,thesteadystaterateofgrowthofthemoneysupplyisequaltoμ,andsincegrowthisequaltozerointhismodel,steadystateinfla?onisalsoequaltoμ,andthesteadystatenominalinterestrateisequaltoρ+μ.

Τhemoneydemandfunc?oninlogsisgivenby,

mt − pt = yt −1θln it1+ it

⎛⎝⎜

⎞⎠⎟+ 1θ(vt

M − vtC )


TheMoneyDemandFunc?oninLog-linearForm

Τhemoneydemandfunc?oncanbeapproximatedaroundthesteadystatenominalinterestrateρ+μas,

mt − pt ! m0 + yt −ζ (rt + Etπ t+1)+1θ(vt

M − vtC )

where,

m0 = − 1θln ρ + µ1+ ρ + µ

⎛⎝⎜

⎞⎠⎟− 11+ ρ + µ

⎛⎝⎜

⎞⎠⎟

ζ = 1θ(ρ + µ)(1+ ρ + µ)

> 0


Subs?tu?ngforrealoutputandtherealinterestrateinthemoneydemandfunc?on,andsolvingforthepricelevel,wegetthat,

TheDetermina?onofthePriceLevelunderaMoneySupplyRule

pt 1+ζ + (1+ζθ )(1−α )α

⎛⎝⎜

⎞⎠⎟ − Et−1pt

(1+ζθ )(1−α )α

⎛⎝⎜

⎞⎠⎟ −ζEt pt+1 = zt

where,

zt = mt − y_

t+ζ r_t−

(1+ζθ )(1−α )α

⎛⎝⎜

⎞⎠⎟ ε t

A − 1θ(vt

M − vtC )−m0


where,

TheSolu?onforthePriceLevelunderaMoneySupplyRule

pt = p_+mt−1 + χAat−1 + χCvt−1

C + χMvt−1M + χN (n

_t− n

_)+ψ Aε t

A +ψ Cε tC +ψ Sε t

S +ψ Mε tM

p_= µ −m0 − (1−α )n

_+ζρ

χA = −1+ζθ(1−ηA )1+ζ (1−ηA )

ηA χC = 1+ζθ(1−ηC )1+ζ (1−ηC )

ηC

θχM = − 1

1+ζ (1−ηM )ηM

θ χN = −1+ζθ(1−ηN )1+ζ (1−ηN )

(1−α )

ψ A = − (1+ζθ )(1−α )α

+α 1+ζθ(1−ηA )( )

α (1+ζ (1−ηA ))+ (1+ζθ )(1−α )⎛⎝⎜

⎞⎠⎟

ψ C =α 1+ζθ(1−ηC )( )

α (1+ζ (1−ηC ))+ (1+ζθ )(1−α )1θ

ψ S =α

α + (1−α )(1+ζθ )ψ M = − α

θ α 1+ζ (1−ηM )( )+ (1−α )(1+ζθ )( )


Alltherelevantshocks,realandnominal,affectunan?cipatedinfla?on.Thus,alltherelevantshocksaffectoutputandunemploymentfluctua?onsaswell.Fluctua?onsofoutputandunemploymentaroundtheir“natural”ratesaregivenby,

Unan?cipatedInfla?onandFluctua?onsinOutputandUnemploymentunderaMoney

SupplyRule

π t − Et−1π t =ψ Aε tA +ψ Cε t

C +ψ Sε tS +ψ Mε t

M

yt = y_

t+1−αα

(1+ψ A )ε tA +ψ Cε t

C +ψ Sε tS +ψ Mε t

M( )

ut = u_t−1α(1+ψ A )ε t

A +ψ Cε tC +ψ Sε t

S +ψ Mε tM( )


Inreality,moderncentralbanksdonotallowthemoneysupplytofollowanexogenousprocess.Monetarypolicyusuallyreactstodevia?onsofinfla?onfromtargetanddevia?onsofoutputandunemploymentfromtarget.

Inaddi?on,centralbanksusuallyconductmonetarypolicybycontrollingthenominalinterestrateratherthanthemoneysupply.Thisisbecauseofthedifficul?esincontrollingthemoneysupply,andbecausethemoneydemandfunc?onissubjecttoshocksduetofinancialinnova?ons.

Inwhatfollowsweshallthusexaminethebehaviorofthemodelundertheassump?onthatthecentralbankfollowsafeedbackruleforthenominalinterestrate.Inpar?cular,weshallassumethatthecentralbankfollowsafeedbackruleoftheTaylor(1979)form,

AFeedbackNominalInterestRateRule

it = r_t+ µ +φ1(π t − µ)+φ2 (yt − y

_

t )+ ε ti

φ1,φ2 > 0


• TheTaylor(1993)ruleisageneraliza?onoftheWicksellrulethatweexaminedinthecaseofthe“newclassical”model.

• Accordingtothisrule,thecentralbankaimsforanominalinterestratewhichisequaltothe“natural”realrateofinterest,plusatargetinfla?onrateequaltoμ.

• Ifactualinfla?onishigherthanthetargetμ,thenthecentralbankraisesinterestratesinordertoreduceinfla?on.

• Inaddi?on,ifoutputishigherthanits“natural”levelandunemploymentlowerthanits“naturalrate”,thenthecentralbankalsoraisesinterestrates,inordertobringoutputbacktoits“natural”levelandunemploymentbacktoits“naturalrate”.

Proper?esoftheTaylorRule


Subs?tu?ngtheTaylorRuleintheFisherequa?on,andusingtherealinterestrateequa?onandtheoutputsupplyfunc?onofthemodel,wegetthefollowingprocessforinfla?on.

TheInfla?onProcessunderaTaylorRule

π t = γ 1Et−1π t + γ 2Etπ t+1 + (φ1 −1)γ 2µ − γ 1ε tA − γ 2ε t

i

γ 1 =(φ2 +θ )(1−α )

φ1α + (φ2 +θ )(1−α )γ 2 =

αφ1α + (φ2 +θ )(1−α )


• Theinfla?onaryprocessdependsonthepolicyparametersoftheTaylorruleandtheotherstructuralparametersofthemodel,suchasαandθ.

• Itisdrivenbytwoshocks.Shockstoproduc?vity,astheseshockscausedevia?onsofoutputfromits“natural”level,duetothefactthatnominalwagesweredeterminedbeforetherealiza?onoftheseshocks,andalsoshockstothepolicyrule(monetaryshocks).

• Noothershocksaffecttheinfla?onaryprocessunderthisrule,asthenominalinterestrateadjuststoreflectchangesinthe“natural”rateofinterest,whichisaffectedbytheothernominalandrealshocks.

• Theinfla?onprocessinstableifγ1+γ2<1.

TheInfla?onProcessunderaTaylorRule


Theinfla?onprocessunderaTaylorruleisstableifγ1+γ2<1.Anecessaryasufficientcondi?onforthisisthatφ>1.

TheCondi?onforStabilityoftheInfla?onProcessunderaTaylorRule

γ 1 + γ 2 =α + (φ2 +θ )(1−α )φ1α + (φ2 +θ )(1−α )

<1⇒φ1α >α ⇒φ1 >1

Thiscondi?onisusuallyreferredtoastheTaylorprinciple,andrequiresthatthenominalinterestratereactsmorethanonetoonetodevia?onsofcurrentinfla?onfromitstargetμ.Weshallassumethatitisalwayssa?sfiedbythecentralbank.


AssumingthattheTaylorprincipleissa?sfied,thenthera?onalexpecta?onssolu?onoftheinfla?onprocesstakestheform,

SolvingfortheInfla?onProcessundertheTaylorRule

π t = µ − γ 1ε tA − γ 2ε t

i

Infla?ondeviatesfromthecentralbanktargetμ,onlyinresponsetocurrentshockstoproduc?vityandshockstothenominalinterestrate(monetaryshocks).Thus,unan?cipatedinfla?onisgivenby,

π t − Et−1π t = −γ 1ε tA − γ 2ε t

i


Subs?tu?ngforunan?cipatedinfla?oninthe“Phillipscurve”andtheoutputsupplyfunc?on,

Fluctua?onsofOutputandUnemploymentunderaTaylorRule

ut − u_t =

1α

−(1− γ 1)ε tA + γ 2ε t

i( ) yt − y_

t =1−αα

(1− γ 1)ε tA − γ 2ε t

i( )

UnderaTaylorrule,onlyproduc?vityandmonetarypolicyshocksaffectfluctua?onsinrealvariables,suchasoutputandunemployment,aroundtheir“natural”level.Thisisincontrasttotheexogenousruleformonetarygrowth,whichresultsinallshocksaffec?ngdevia?onsofoutputfromits“natural”rate,andthusahigherpoten?alvarianceofunemploymentandoutput.Furthermore,theimpactoftheseshocksdependsontheparametersoftheTaylorrule.Thus,inthismodelthereisscopeformonetarypolicytoaffecttheshortrunfluctua?onsofrealvariablesbyappropriatechoiceofthepolicyparameters.


• ThemodelwehavepresentedbuildsononeofthekeyinsightsoftheGeneralTheory,theshortrunrigidityofnominalwages,butinallotherrespectsitisbasedoninter-temporalop?miza?ononthepartofbothhouseholdsandfirms.

• Themodelischaracterizedbyanexpecta?onsaugmented“Phillipscurve”,inwhichdevia?onsofoutputandunemploymentfromtheir“natural”leveldependonunan?cipatedcurrentinfla?on,whichreducesrealwagesrela?vetoproduc?vity,andunan?cipatedproduc?vityshocks,whichalsoaffecttherela?onbetweenrealwagesandproduc?vity.

• Nominalshocksand,byextension,monetarypolicyareabletoaffectfluctua?onsinbothinfla?onandrealvariablessuchasoutput,employment,unemployment,realwagesandtherealinterestrate.

NominalandRealShocksandAggregateFluctua?ons


• Weanalyzedaggregatefluctua?onsinthismodelundertwoalterna?vemonetaryrules.Thefirstisanexogenousprocessfortherateofgrowthofthemoneysupplyandthesecondisafeedbackinterestraterule,accordingtowhichthenominalinterestraterespondstodevia?onsofinfla?onfromthetargetofthecentralbank,anddevia?onsofoutputfromits“natural”level.

• Thevarianceofsuchdevia?onsdependsonthemonetaryrule.Underanexogenousprocessfortherateofgrowthofthemoneysupply,allshocksaffectaggregatefluctua?ons.UnderaTaylorfeedbackinterestraterule,onlyproduc?vityshocksandshockstomonetarypolicyaffectaggregatefluctua?ons,andtheireffectdependsontheparametersoftheTaylorrule.

• Wehavethusdemonstratedthedependenceofaggregatefluctua?onsnotonlyonexogenousshocks,butalsoontheformofthemonetarypolicyrulefollowedbythecentralbank.

MonetaryPolicyandAggregateFluctua?ons


• Oneweaknessofthemodelaswehaveanalyzeditsofaristhatincannotaccountforpersistentdevia?onsofinfla?onfromtarget,andoutputandunemploymentfromtheir“natural”rates.

• Fluctua?onsofinfla?onfromtarget,andoutputandunemploymentaroundtheir“natural”levelsandarethesumoftwowhitenoiseprocesses,i.ewhitenoiseprocessesthemselves.Alldevia?onslastforoneperiodandthereisnopersistence.

• Thislackofpersistenceisaseriousweaknessofthemodel,aspersistenceofaggregatefluctua?onsisoneofthemaincharacteris?csofbusinesscycles.

• However,themodelcanbeextendedtoaccountforpersistence.

ExplainingPersistenceinInfla?onandAggregateFluctua?ons


• FollowingBlanchardandSummers(1986),weassumethattheemploymentobjec?vewhichdeterminesthenominalwageinthecontractdependsonboththeexogenousnumberof“coreinsiders”ineachfirm,butalsothosewhowereemployedinperiodt-1.Theexpecta?onsonthebasisofwhichwagesaresetdependoninforma?onavailableun?ltheendofperiodt-1,butnotoninforma?onaboutpricesandproduc?vityinperiodt.

• Onthebasisoftheabove,weassumethattheobjec?veof“insiders”istomakeexpectedemploymentsa?sfyapaththatminimizesthefollowingquadra?cinter-temporallossfunc?on,subjecttothesequenceofexpectedlabordemandcurvesoffirms.ωistherela?veweightofrecentemployeesinthewagesegngprocess.

AGeneralizedBlanchardandSummersModelofUnemploymentPersistence

minEt−1 β ss=0

∞∑ 12l(i)t+s − n

_(i)⎛

⎝⎞⎠

2

+ ω2l(i)t+s − l(i)t+s−1( )2⎡

⎣⎢

⎤

⎦⎥


Fromthefirstordercondi?onsforaminimum,wagesaresetsothatexpectedemploymentforeachfirmsa?sfies,

FirstOrderCondi?onsforaMinimumandtheDetermina?onofExpectedEmployment

1+ω (1+ β )( )Et−1l(i)t − βωEt−1l(i)t+1 −ωl(i)t−1 = n_(i)

Integra?ngoverthenumberoffirmsi,expectedaggregateemploymentmustthensa?sfy,

1+ω (1+ β )( )Et−1lt − βωEt−1lt+1 −ωlt−1 = n_

Weshallassumethatthetotalnumberofcore“insiders”intheeconomyisalwaysstrictlysmallerthanthelaborforce.Weshallthusassumethat,

n_(i)di

i=0

1

∫ = n_< n


Expectedemploymentandexpectedunemploymentarethusdeterminedby,

ExpectedEmploymentandExpectedUnemployment

Et−1lt =1

1+ω (1+ β )n_+ ω1+ω (1+ β )

lt−1 +βω

1+ω (1+ β )Et−1lt+1

Et−1ut =1

1+ω (1+ β )u_+ ω1+ω (1+ β )

ut−1 +βω

1+ω (1+ β )Et−1ut+1

wherewehaveusedthedefini?onsthat, ut ! n − lt u_! n − n

_

Wecanusethose2ndorderdifferenceequa?onstosolveforexpectedemploymentandunemployment.Thesehavetworootsthatlieoneithersideofunity.


Solvingforexpectedemploymentandexpectedunemployment,weget,

SolvingforExpectedEmploymentandExpectedUnemployment

Et−1lt = λ1lt−1 +λ1

ω (1− βλ1)n_= λ1lt−1 + (1− λ1)n

_

Et−1ut = λ1ut−1 +λ1

ω (1− βλ1)u_= λ1ut−1 + (1− λ1)u

_

λ1isthesmallerroot,andthetworootssa?sfythecondi?ons,

λ1 + λ2 =1+ω (1+ β )

βωλ1λ2 =

1β


Itisstraighnorwardtoshowthatanincreaseinω,therela?veweightofrecentemployeesinthewagesegngprocess,resultsinanincreaseinλ1,thecoefficientthatdeterminesthepersistenceofexpectedunemployment.Fromthecondi?onswhichdefinethetworoots,itfollowsthat,

ThePersistenceofEmploymentandtheRela?veWeightofRecentEmployeesintheWageSegng

Process

∂λ1∂ω

= λ1ω

⎛⎝⎜

⎞⎠⎟2

> 0

Thus,thehighertheweightofrecentemployeesrela?vetocore“insiders”inthewagesegngprocess,thehigherthepersistenceofunemployment.

Forexample,assumingβ=0.99,withω=1,λ1=0.38.Withω=2,λ1=0.50,withω=10,λ1=0.73andwithω=100,λ1=0.91.


Weshallagainassumethatwagese[ersdeterminethenominalwageisordertomaketheexpectedrealwageconsistentwithexpectedemployment.Employmentisdeterminedonthelabordemandcurve.

WageSegngandEmployment

lt = l_− 1α(wt − pt − at )

Thus,thewageissetat,

wt = Et−1pt + Et−1at −α Et−1lt − l_⎛

⎝⎞⎠

Subs?tu?ngforthewageintheemploymentequa?on,employmentfollows,

lt = λ1lt−1 + (1− λ1)n_+ 1α(pt − Et−1pt + at − Et−1at )


Subtrac?ngthelogofemploymentfromthelogofthelaborforcen,andrearranging,theunemploymentrateevolvesaccordingto,

ADynamicModelofthePhillipsCurve

ut − u_= λ1(ut−1 − u

_)− 1

α(π t − Et−1π t + at − Et−1at )

Devia?onsofunemploymentfromits“natural”leveldependnega?velyonunan?cipatedshockstoinfla?onandproduc?vity,asthesecauseadiscrepancybetweenrealwagesandproduc?vity,duetothefactthatnominalwagesarepredeterminedfortheperiod.

Followinganunan?cipatedshocktoinfla?onorproduc?vity,unemploymentwillconvergegraduallybacktoits“natural”rate,withthespeedofadjustmentbeingequalto1-λ1perperiod.Thus,followingshockstoinfla?onorproduc?vity,devia?onsofunemploymentfromits“natural”ratewilldisplaypersistence.


Thepersistenceofemploymentandunemployment,willalsobetranslatedintopersistentoutputfluctua?ons.Subs?tu?ngforemploymentintheproduc?onfunc?on,devia?onsofoutputfromits“natural”levelwillevolveaccordingto,

TheRela?onbetweenthePersistenceofUnemploymentandOutput

yt − y_

t = λ1(yt−1 − y_

t−1)+1−αα

(π t − Et−1π t + at − Et−1at )

Thisisadynamicoutputsupplyfunc3on.Devia?onsofoutputfromits“natural”leveldependposi?velyonunan?cipatedshockstoinfla?onandproduc?vity,asthesecauseadiscrepancybetweenrealwagesandproduc?vity,duetothefactthatnominalwagesarepredetermined.Thisdiscrepancyaffectsemploymentandtranslatesintooutput,throughtheproduc?onfunc?on.


Fluctua?onsinEmployment,UnemploymentandOutput,whenthereisEmploymentPersistence

lt = (1− λ1)n_+ λ1lt−1 +

1α

π t − Et−1π t + ε tA( )

ut = (1− λ1)u_+ λ1ut−1 −

1α

π t − Et−1π t + ε tA( )

yt = y_

t+ λ1(yt−1 − y_

t−1)+1−αα

π t − Et−1π t + ε tA( )


Fluctua?onsintheRealWageandtheRealInterestRate

wt − pt = (w − p_)t +αλ1(ut−1 − u

_)− π t − Et−1π t( )

rt = r_t+θ 1−α( )(1− λ1)(ut − u

_)

where,

(w − p_)t = at −α (n

_− l_)

r_t = ρ −θ(1−ηA )at + (1−ηC )vt

C


WeshallassumethatthecentralbankfollowsaTaylorruleoftheform,

TheTaylorRulefortheNominalInterestRate

it = r_t+ µ +φ1 π t −π *( )−φ2 (ut − u

_t )+ ε t

i

WehavenowexpressedtheTaylorruleintermsofdevia?onsofunemploymentandnotoutputfromits“natural”rate.Thisdoesnotaffectthenatureoftheresults,asinthismodeldevia?onsofunemploymentfromits“naturalrate”arealinearfunc?onofdevia?onsofoutputfromits“naturallevel”.


AssumingthatthecentralbankfollowsaTaylorrule,subs?tu?ngforthenominalinterestrateintheFisherequa?on,andusingthedynamicPhillipscurvewehavederived,theinfla?onaryprocessisdeterminedby,

TheInfla?onaryProcessunderaTaylorRule

π t = γ 1Etπ t+1 + γ 2Et−1π t + γ 3π t−1 + γ 4µ + γ 5ε tA + γ 6ε t

i + γ 7ε t−1i

where,

γ 1 =α

φ1α +φ2 +θ(1− λ1)(1−α )+ λ1αγ 2 =

φ2 +θ(1− λ1)(1−α )φ1α +φ2 +θ(1− λ1)(1−α )+ λ1α

γ 3 =λ1φ1α

φ1α +φ2 +θ(1− λ1)(1−α )+ λ1αγ 4 =

(φ1 −1)(1− λ1)αφ1α +φ2 +θ(1− λ1)(1−α )+ λ1α

γ 5 = −γ 2 γ 6 = −γ 1 γ 7 = λ1γ 1


• Becauseofthepersistenceofunemployment,theinfla?onaryprocessalsodisplayspersistence.

• Italsodependsoncurrentexpecta?onsaboutfutureinfla?on,throughthedefini?onoftherealinterestrate.

• ItalsodependsonbothparametersoftheTaylorrule,asunan?cipatedinfla?oncausestheunemploymentrateandtherealinterestratetodeviatefromtheir“naturalrates”.

• Finally,becauseofthepersistenceinunemploymentbothcurrentandpastnominalinterestrateshocksaffecttheinfla?onaryprocess.

• Theeffectsofproduc?vityandnominalinterestrateshocksoninfla?onalsodependontheparametersoftheTaylorrule.

TheInfla?onaryProcessunderaTaylorRule


Solvingforinfla?onunderra?onalexpecta?ons,onecanshowthat,iftheTaylorprincipleφ1>1issa?sfied,infla?onfollowsastableprocessoftheform,

SolvingforInfla?onunderRa?onalExpecta?ons

π t = (1− λ1)µ + λ1π t−1 −ψ 1ε tA −ψ 2ε t

i +ψ 3ε t−1i

where,ψ 1 =

φ2 +θ(1− λ1)(1−α )φ1α +φ2 +θ(1− λ1)(1−α )

<1

ψ 2 =φ1 − λ1φ1

αφ1α +φ2 +θ(1− λ1)(1−α )

> 0

ψ 3 =λ1φ1

> 0


• Fluctua?onsofinfla?onaroundthetargetofthemonetaryauthori?esμarepersistent,anddependonthecurrentinnova?oninproduc?vityandcurrentandpastnominalinterestrateshocks.

• Furthermore,thepersistenceofinfla?onisequaltothepersistenceofdevia?onsofunemploymentandotherrealvariables,suchasoutput,fromtheir“natural”level.

• Thereasonisthatboththeequilibriumrealinterestrateanddevia?onsofunemploymentfromits“natural”ratedisplaypersistence.Thus,underaTaylorrule,thenominalinterestrateandinfla?onwillalsodisplaypersistence.

ThePersistenceofInfla?onunderaTaylorRule


Unan?cipatedinfla?onunderaTaylorrulewillbegivenby,

Unan?cipatedInfla?onandFluctua?onsofDevia?onsofUnemploymentandOutputfrom

their“Natural”Rates

π t − Et−1π t = −ψ 1ε tA −ψ 2ε t

i

Fluctua?onsofdevia?onsofunemploymentandoutputfromtheir“natural”rateswillbegivenby,

(ut − u_) = λ1(ut−1 − u

_)− 1

α(1−ψ 1)ε t

A −ψ 2ε ti( )

(yt − y_

t ) = λ1(yt−1 − y_

t−1)+1−αα

(1−ψ 1)ε tA −ψ 2ε t

i( )


• UndertheTaylorrule,onlyinnova?onsinproduc?vityandnominalinterestrateshocksinducefluctua?onsofdevia?onsofunemploymentandoutputfromtheir“natural”rates.

• Otherdemandshocks,suchasshockstoconsump?onpreferences,arefullyneutralizedbymonetarypolicy,sincethenominalinterestrateisassumedtofullyaccommodatechangesinthe“natural”rateofinterest.

• However,becauseofthepersistenceindevia?onsofunemploymentandoutputfromtheir“natural”levels,theeffectsoftheseshocksarenolongershortlived,buttheydisplaypersistence.Thehigherthepersistenceofdevia?onsofunemploymentfromits“natural”rate,thehigherthepersistenceoftheeffectsoftemporarynominalandrealshocks.

AggregateFluctua?onsunderaTaylorRule


• Inordertodemonstratetheimpulseresponsefunc?onsofthemodeltonominalandrealshocks,wepresenttheresultsofadynamicsimula?onofthemodel,followinganunan?cipatedtemporary1%shocktothenominalinterestrate,andanunan?cipated1%shocktoproduc?vityrespec?vely.

• Inthesimula?onswehaveassumedthefollowingvaluesoftheparameters:α=0.333,ρ=0.02,θ=1,ω=2,implyingavalueofλ1=0.5,φ1=1.5,φ2=0.5andηΑ=0.75.Wehavealsoassumeda“naturalrate”ofunemploymentequalto5%andatargetinfla?onrateof2%.

ImpulseResponseFunc?onsoftheModel


A1%Unan?cipatedTemporaryShocktotheNominalInterestRate


A1%Unan?cipatedShocktoProduc?vity


• Inthislecturewehaveintroducedadynamicstochas?c“newKeynesian”model,whichnotonlyallowsfortheexistenceofinvoluntaryunemployment,butalsofornominalshocksandmonetarypolicytoaffectthefluctua?onsofallrealvariables.

• ThemodelbuildsononeofthekeyinsightsoftheGeneralTheory,theshortrunrigidityofnominalwages,butinallotherrespectsitisbasedoninter-temporalop?miza?ononthepartofbothhouseholdsandfirms.

• Themodelischaracterizedbyanexpecta?onsaugmented“Phillipscurve”,inwhichdevia?onsofoutputandemploymentfromtheir“natural”leveldependonunan?cipatedcurrentinfla?on,whichreducesrealwagesrela?vetoproduc?vity,andunan?cipatedproduc?vityshocks,whichalsoaffecttherela?onbetweenrealwagesandproduc?vity.

• Nominalshocksand,byextension,monetarypolicyareabletoaffectfluctua?onsinbothinfla?onandrealvariablessuchasoutput,employment,unemployment,realwagesandtherealinterestrate.

Characteris?csof“NewKeynesian”ModelsofAggregateFluctua?ons


• Weanalyzedaggregatefluctua?onsinthismodelundertwoalterna?vemonetaryrules.

• Thefirstisanexogenousprocessfortherateofgrowthofthemoneysupplyandthesecondisafeedbackinterestraterule,accordingtowhichthenominalinterestraterespondstodevia?onsofinfla?onfromthetargetofthecentralbank,anddevia?onsofoutputfromits“natural”level.

• Contrarytothe“newclassical”model,monetaryshocksaffectrealvariablesinthismodel,causingtemporarydevia?onsofoutput,employment,unemployment,realwagesandtherealinterestratefromtheir“natural”levels.

• Theexactvarianceofsuchdevia?onsdependsonthemonetaryrule.Underanexogenousprocessfortherateofgrowthofthemoneysupply,allshocksaffectaggregatefluctua?ons.UnderaTaylorfeedbackinterestraterule,onlyproduc?vityshocksandshockstomonetarypolicyaffectaggregatefluctua?ons.

• Wehavethusdemonstratedthedependenceofaggregatefluctua?onsnotonlyonexogenousshocks,butontheformofthemonetarypolicyrulefollowedbythecentralbank.

AggregateFluctua?onsunderAlterna?veMonetaryPolicyRules


• Wehavealsoextendedthemodeltoaccountforpersistenceindevia?onsofunemploymentandoutputfromtheir“natural”levels.

• Theextensionisbasedonadynamicmodelofthe“PhillipsCurve”,inwhichunan?cipatedshockstoinfla?onandproduc?vityhavepersistenteffectsonunemployment,andthesepersistenteffectsarecompa?blewithfullinter-temporalop?miza?ononthepartoflabormarket“insiders”.

• Thepropaga?onmechanismthatcausesunan?cipatednominalandrealshockstoproducepersistentdevia?onsofunemploymentandoutputfromtheir“natural”rateisthepar?aladjustmentoflabormarketinsiderstoemploymentshocks.

• WedemonstratedthatunderaTaylorrule,theonlyshocksthatcannotbecompletelyneutralizedbymonetarypolicyareproduc?vityshocksand,ofcourse,monetarypolicyshocks.Fluctua?onsofdevia?onsofunemploymentandoutputfromtheir“natural”ratesdisplaypersistenceandaredrivenbythesetwotypesofshocks.Becauseoftheendogenouspersistenceofdevia?onsofunemploymentfromits“natural”rate,theequilibriuminfla?onratealsodisplayspersistencearoundtheinfla?ontargetofthecentralbank.

ThePersistenceofAggregateFluctua?ons

Macro Lect 15 New Keynesian Model Wage Contracts · Prof George Alogoskoufis, Dynamic Macroeconomic...

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