Understanding the New Keynesian Macroeco

Olivier

Blanchard

∗

May

2005

14.452.Spring2005.Topic9.

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Thepurposeofthisnoteistoputtogetheramodelinwhichmoneymatters in the short run,in whichthe economy returnsto thenatural ratein the mediumrun,andinwhichwecanthinkabouttherelationbetweennominal variables,in particularinflation,andrealvariables,in particularoutput.

The model has: Consumers whodecide about how much to spend versus howmuchtosave,howmuchoftheirwealthtokeepin theform ofmoney versus bonds, howmuch laborto supply; firmswho set prices and decide howmuchtoproduce, andpricedecisions whicharestaggeredovertime.

Theresultingmodel—anditsassociatedloglinearapproximation—iscalled theNewKeynesianModel.Whileitfallsshortofwhatonewouldwant toincorporateinanumberofimportantdimensions,itiscurrentlyseenas the minimal model with which to think about fluctuations, the effects of shocks,andtheroleofpolicy.

The organization of the note is slightly different from previous notes. I startwithtwo“ad­hoc”modelsofpricestaggering,thefirstonefrom Tay-lor,thesecond from Calvo.Thesemodels show mostclearlythepotential implications ofprice staggering.Then, I introduce pricestaggeringin the modelofmonopolisticcompetitiondevelopedintopic8,andderivetheNew Keynesianmodel.Iendwitha discussionofresults.

Price

staggering

The model of price setting introduced in topic 8 assumed that all prices were set simultaneously for one period. This is not particularly realistic, andwe donotobserve suchcoordinationofpricedecisions.Inmostcases, staggeringseemsmore appropriate.

1.1 Assumptions

Consideraneconomycomposedofnproducers/pricesetters,eachsellinga differentiatedgood.Atany giventime,thedesiredpriceofpricesetteriis givenby:

pi =p+ay−u

where pi and p are the logs of the nominal price of producer i and of

the price level respectively, y is the log of aggregate output, and u is a shock. The desired relative price increases with marginal cost, which is itself an increasingfunction ofoutput y andof the shock u. Allvariables aremeasuredaslogdeviationsfromthenonstochasticstationarystate.

(This specification is consistent with the derivation of the relative price chosenbyproduceriintheyeoman-farmermodeloftopic7.Inthatmodel, theloglinearizationgave:

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pi=p+[

1+σ(β−1)][(β−1)y−βz]

So,underthatinterpretation,theparameteracorrespondsto(β−1)/(1+

σ(β−1)). a increases with β, and is positive ifβ is greaterthan one.In that interpretationalso,theshock ucorrespondsto-β(1+σ(β−1))zand isa technologicalshock.)

Underflexibleprices,theequilibriumlevelofoutput(thesecondbestlevel ofoutput,astheeconomystillsuffersfromthedistortioncomingfromthe markup, itselfduetothemonopolypowerofthefirms), isgivenby:

1

yˆ = u a

so wecanrewritethepricesettingequationas:

pi=p+a(y−yˆ)=p+ax

and equilibrium (second best) output. (Note that, given the output gap, technologicalshocksorenergypricechangesdonotappearanymoredirectly in theequation.)

1.2 The Taylor version

TheTaylorformalizationofpricesettingisbasedontwoassumptions:

Each period,1/n of the price setters set their pricefor the current

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andthenextn−1periods.

Theirpriceisfixedbetweenreadjustments.

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Thisleads tothefollowingtwoequations:

n−1

1

qt=

�

Ep∗

t+i

n

i=0

n−1

1

pt=

�

qt−i

n

i=0

The first equation states that qt, the (log) price chosen in period t for

periods tto t+n−1 isequalto theaverageoftheexpected desiredprice for periods t to t+n−1. Here and below, for any z, Ezt+i stands for

E[zt+i|Ωt].From above,thedesiredpriceforeachperiodisgivenby

p∗

=pt +axt t

Thesecondequationsimplystatesthatthepricelevelisaweightedaverage ofallthepricescurrentlyin force.

Forthecasewheren=2,itiseasytoderivethebehaviorofprices.Write:

qt =0.5(pt +Ept+1)+0.5a(xt +Ext+1)

and

ReplaceptandEpt+1 fromthesecondequationinthefirst,andreorganize

toget:

qt=0.5qt−1 +0.5Eqt+1 +a(xt +Ext+1) (1)

or

(qt−qt−1)=(Eqt+1−qt)+2a(xt +Ext+1) (2)

Thefirstequationgivesthe(new)priceasafunctionofitselflagged,itself expected, and thecurrentand expected outputgap. Thesecond equation rewritesthe relationas a relation between(newprice) inflation, expected (newprice)inflation,andthecurrentandexpectedoutputgap.

Beforediscussingtheimplicationsofthesetwoequationsfurther,letuslook attheCalvo versionofpricestaggering.

1.3 The Calvo version

Assume that insteadofbeing fixedfor afixed lengthof time,prices,once set,arereadjustedwithprobabilityδeachperiod(a“Poisson”assumption which,asifoftenthecasewithPoisson,leads toniceaggregation)

Inthis case,theequations forindividual prices andthepricelevel can be written as:

∞

qt=δ

�

(1−δ)iEp∗

t+i

0

∞

pt=δ

�

(1−δ)i qt−i

0

where, asbefore:

p∗

=pt +axt t

Thepriceqtchoseninperiodtdependsonallfutureexpecteddesired

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prices, with weights corresponding to theprobability that the price isstill inplaceateachfuture date.

The price level is in turn a weighted average of current and past

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Tosolve,rewritethetwo equationsinrecursiveform as:

qt=δ(pt+axt)+(1−δ)Eqt+1

pt=δqt +(1−δ)pt−1

Use the second equation to express qt as a function of pt and pt−1 and

Eqt+1 as a function of Ept+1 and pt, and replacein thefirst equation to

get:

pt−(1−δ)pt−1=δ2(pt +axt)+(1−δ)(Ept+1−(1−δ)pt)

Reorganizing:

1 1 1 δ2

axt (3)

pt=

2pt−1 +

2Ept+1 +2 1−δ

Or,intermsofinflation:

δ2

(pt−pt−1)=(Ept+1−pt)+

1−δaxt (4)

Thecoefficientontheoutput gapdependsnotonlyona(theslopeofthe marginalcostcurve),butalsoonδ,theparameterreflectingtheproportion ofprices beingadjustedeachperiod.

Note two differences with the Taylor formalization: Theequations are in terms of the price level pt rather than in terms of the new price qt; and

only thecurrentoutput gap appears,insteadofthe currentand expected outputgaps.ThesetwocharacteristicsmaketheCalvomodelslightlymore tractable and explain why it has become more widely adopted than the Tayloralternative.

1.4 Implications. Price versusinflation stickiness

Staggering leads to price stickiness. This is clear from equations (1) or (3), in which the pricelevel at time t depends on the price level at time

Toseethismoreclearly,takeequation(3)andclosethemodelbyassuming

ut=0so yˆt =0,andyt=mt−pt,so

xt=yt−yˆt=mt−pt

Inthiscase,thepricelevelisgivenby:

pt=bpt−1 +bEpt+1 +(1−2b)mt

whereb≡(1−δ)/(2(1−δ)+(1−δ)2a)

Solvingthisequation(say,byfactorization)gives:

pt=λpt−1 +(1−λ)2

�

λiEmt+i

whereλ≡(1−√1−b2_)/b≤1

Note thatthecloseraisto 0,thecloserbisto1/2,andthecloserλ

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isto 1:Theflatterthemarginalcost,themorepricestickiness. Note also that the closer δ is to 0, the closer b is to 1/2, and the

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closer λ is to 1: The less often prices are adjusted, the higher the pricestickiness.

Consider thecase where mt follows a random walkwith innovation

�t.Inthiscase,thepricelevelfollows:

pt=λpt−1 +(1−λ)mt

andso outputisgivenby:

yt=λyt−1 +�t

Thecloserλistoone,thelongerlastingtheeffectsofanunanticipated shock inmoneyonoutput.

thisisnotexactlytrue;Newpriceinflationisalsofullyforwardlooking,but inflationisa weightedaverageofcurrentandlagged(newprice)inflation.)

IntheCalvoformalization,wecansolveequation(4)forwardtoget:

δ2 k

πt = [ a] lim

�

Ext+i

1−δ k→∞ 0

Inflation depends oncurrent and expected output gaps, not (directly)on thepast.

This is a strong result,and a warning that price stickinessdoes not nec-essarilyimplyinflationstickiness(stickinessina leveldoesnotnecessarily implystickinessina derivative).

It has important implicationsfor monetary policy: Assume xt =mt−pt

and supposethat moneygrowsat rateg.Thenverifythat thesolutionto equation (4) is that pt = mt, so the inflation rate is equal to g and the

output gap is equal to zero. Suppose that, at time t, the rate of money growth from t on decreases from g to g�

< g. Then inflation from t on decreases from g to g�

, and theoutputgap remainsequal to zero. (Check that this is indeed the solution to equation (4). In words, disinflation is achievedwithoutanyoutputloss.

Itraiseshowever twoquestions:

Isitrobust,orisitinsteadtheresultofthespecialassumptionsunderlying CalvoorTaylorstaggering?Theansweristhatitisveryrobust.Staggering tendstogeneratelittleinflationinertia.

Isitconsistentwithreality?Theevidenceisitdoesnotappear tobe,that thereisinflationinertia,i.e.directdependenceofinflationonpastinflation. Weshallcomebacktothislater.

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The

“New

Keynesian”

model

Let’sgobackto themodeldeveloped intopic 8andextenditto allowfor staggeredpricesettinga laCalvo.

2.1 Assumptions

Assume theeconomy is composedof a continuum of households, indexed byi,whomaximize:

∞

Mit+k+1

maxE[�βk(U(Cit+k)+V( _¯ )−Q(Nit+k)) Ωt]

Pt+k |

0

subjectto:

σ/(σ−1) 1/(1−σ)

Cit≡[

� 1

Cijtσ

−_1/σ

dj] P¯t= [

� 1

Pjt1

−_σ

dj]

0 0

� 1

PjtCijt +Mit+1 +Bit+1=WtNit +(1+it)Bit +Mit +Πit +Xit

0

Consumersderiveutilityfromaconsumptionbasket,realmoneybalances, and leisure. The consumption basket is a CES function of different con-sumptiongoods.Thepriceindex associatedwiththeconsumptionbasket, thepricelevel,isnowdenotedbya bar.

Consumerssupplylaborinacompetitivelabormarketandsoreceivelabor incomeWtNit.(Thisisa slightly differentformalizationthanthatused in

topic8.Theyeoman­farmerapproachtakenintopic 8,togetherwithprice staggering,wouldleadtoheterogeneityofconsumers­workers.Here,witha competitivelabormarket,allconsumerworkersarethesame.) Consumers equallyownallthefirmsproducingtheindividualconsumptiongoods,and receiveprofitsΠit.Thebudgetconstraintisotherwisethesameas before.

Yjt=NjtZt

(Noteanimplicationofintroducingacompetitivelabormarket.Thefirms takethewageandthereforemarginalcostasconstant,whereasbeforethey tookinto accountthemarginal disutilityofleisure).

Firmsset prices alaCalvo,withprobability δofchangingthepriceevery period.

2.2 Derivation

Thefirstorderconditionsforhouseholdsarestraightforwardandfamiliar:

Cijt =(Pjt)

Firms maximizethe expected presentvalue ofprofits theywill get atthe chosenpricePjt.Profitisequaltorealrevenuesminusrealcosts.Profitat

time t+kis discountedat themarginalrate ofsubstitutionof consumers (wherewecandroptheindexiasallconsumersfacethesamemaximization problemand thushavethesameconsumption),timestheprobability that thepricechosenat timetisstillin effectattimet+k,(1−δ)k.

MaximizationyieldsthefollowingexpressionforPjt:

σ E[�kA(k)(Wt+k/Zt+k)|Ωt]

So the pricechosen by price settersis a weighted average of current and expectedfuturemarginalcostsWt+k/Zt+k.ThisisverymuchlikeinCalvo,

except for the fact that the weights are much more complex, involving expectationsofproductsofrandomvariables.Notethat,incontrasttothe Calvo formalization, future periods are also discounted bythe discounted factorβk_.

Thepricelevel isgiveninturnby:

Wt

LSstandsforlaborsupply.PS standsforpricesetting. PFforproduction function.

Not a simplesystem, and weshall need to log linearize to make progress withdynamics.Butitiseasytocharacterizethesecondbestlevelofoutput (thelevelofoutputwhichwouldprevailabsentnominalrigidities).

Fromthelaborsupplyequation:

Wt

¯ =Q

�

(Yt/Zt)/U�(Yt)

Pt

Fromtheprice­settingequation,withoutnominalrigidities:

¯ σ Wt

Puttingthetwotogethergives:

Q�

(Yt/Zt)/U�(Yt)=

σ−1

Zt

σ

Thisdeterminesimplicitlyoutputas afunctionofthetechnologicalshock. If we assume that U(.) is log (as is required under separability to get a balancedgrowth path),thentheequationbecomes:

Q�

(Yt/Zt)Yt/Zt=

σ−1

σ

Thisdeterminesa uniquevalueofYt/Zt, orequivalentlya uniquevalueof

Nt. Sosecondbestemployment isconstant,andsecondbestoutputvaries

one forone withZt.

2.4 Log linearization

Loglinearizationaroundthenon stochasticsteadystategives:

IS: yt=Eyt+1−art+1

LM: mt+1−p¯t=byt−cit+1

LS: wt−p¯t=γnt +zt

PS: pt=(1−δ)βEpt+1 +(1−β(1−δ))(wt−zt)

p¯t:=(1−δ)¯pt−1 +δpt

PF: nt=yt−zt

Thelastthreeequationscanbecombinedtogivea“Phillipscurverelation”.

Use the labor supply relation to eliminate the wage in the price setting equation:

pt=(1−δ)βEpt+1 +(1−β(1−δ))(p¯t +γnt)

Denotethesecondbestlogdeviationsofoutputandemploymentbyyˆtand

bestlog deviationofoutput yt−yˆt byxt. Fromabove yˆt=zt andnˆt=0

so

xt=yt−yˆt=nt−nˆt=nt

Replacingin thepreviousequationgives:

pt=(1−δ)βEpt+1 +(1−β(1−δ))(p¯t +γxt)

Combiningwiththeequationforthepricelevelandreorganizinggives:

δ(1−β(1−δ))

PC: πt=βEπt+1 + γxt

1−δ

where πt ≡ pt−pt−1. Inflation depends on itself expected, and on the

output gap. If output is above its second best (or “natural”) level, then, given expected inflation, inflationincreases.If output isbelow its natural level, then, given expected inflation, inflationdecreases. As in the adhoc Calvo model,inflationis forwardlooking,andthere isnoinflationinertia. Unlike the Calvo model,the coefficientin frontof expected inflation is β

ratherthanone.This impliesa (small,asβ isclosetoone)long runtrade offbetweeninflationandoutput.

2.5 The New Keynesian model.

Puttingtheessentialequationstogetherone lasttime:

IS: yt=Eyt+1−art+1; rt+1=it+1−Eπt+1

LM: mt+1−p¯t=byt−cit+1

PC: πt=βEπt+1 +dxt; xt=yt−yˆt =yt−zt

topic11.But,beforeturningtothose,notesomeoftheimplicationsofthis model (andcomparethemto thestylized factsdescribedintopic 1)

Expectationsmatterverymuch.

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Demandshocks,i.e.anyshockwhichincreasesthedemandforgoods

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inthefirstequation,haveaneffectonoutputintheshortrun.Unless theyalsoaffectthenaturallevelofoutput,theireffectgraduallygoes away overtime.

Demand shocks may be shocks to nominal money, which affect the

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nominal interest rateand in turn thereal interestrate. These may be shocks to the discount factor, a preference for the present rela-tive tothefuture.Thesemay beshockswhichaffect expectationsof future income.These may be shockscoming from fiscal policy,from governmentspendingorchangesin taxation.

Inthelongrun,theeconomyreturnstoitsnatural(equivalently

sec-•

ond best) level of output (equivalently, its natural level of employ-ment; equivalently, its natural rateof unemployment). This natural levelisnotconstanthowever.Itvarieswithproductivityshocks,and more generally,withthe relativepriceofthe inputsusedin produc-tion (for examplethe relative priceof oilif production usesoil and labor).

Productivity shocks may well have effects in the shortrun, through

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theireffectondemand.Higherexpectedproductivityleadstohigher expected incomeandhigherconsumptiondemand today.

An increase in output above the natural level tends to be

associ-•

ated with higher inflation; a decrease in output below the natural level tends to be associated with lower inflation. The relation be-tween inflation and the output gap is however quite different from thetraditionalPhillipscurverelation.

Should we be satisfied with the model? The answerto any such question mustbe no.Amongtheplusesandtheminuses,takingitssize asgiven:

Plus:Thederivationofthemodel fromfirstprinciples,so wecando

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Plus:Themodelseemstocaptureinasimplewaymanyofthestylized

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facts offluctuations.

Minus:Thelackofinflationinertia.Themodelappearsatoddswith

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theevidence,both fromdisinflations,andfrom normaltimes. Minus: The model has a striking implication: There is no trade off

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between achievinga zero outputgap (that is keeping outputat the naturallevel)andstableinflation,eveninthepresenceofproductivity or priceof oil shocks. This can be seen from equation (PC):If π is constant, thenx=0.

The implication is striking and implies for example that monetary policy should aim at keeping constant inflation even in the face of largeoilpriceincreases.Thisfeelswrong,andsuggeststhatsomething importantismissingfrom themodel.

(Ontheselast twoissues,seemypaperwithGali)

Thereisobviouslyalonglistofelementsonewouldliketoaddtothemodel (butwouldincreaseitssize, andcomplexity):

Investmentandcapitalaccumulation.

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Governmentspendingandtaxes.

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Opennessin goodsandfinancialmarkets

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A non competitive labor market, so we can discuss unemployment

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ratherthanjustemploymentoroutput.

Nominal wagerigiditiesin additionto nominalpricerigidities.

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Assumptions leading to non Ricardian equivalence, and thus richer

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implicationsoffiscal policy,oftaxversusdeficitfinance. Heterogeneity,andtheimplicationsofincompletemarkets.

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