The frontier of indeterminacy in a neo-Keynesian model with staggered prices and wages

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The frontier of indeterminacy in a neo-Keynesian model with staggered prices and wages

Alexis Blasselle, Aurélien Poissonnier

To cite this version:

Alexis Blasselle, Aurélien Poissonnier. The frontier of indeterminacy in a neo-Keynesian model with staggered prices and wages. 2013. �hal-00915913�

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THE FRONTIER OF INDETERMINACY IN A NEO- KEYNESIAN MODEL WITH STAGGERED PRICES AND

WAGES

Alexis BLASSELLE Aurélien POISSONNIER

December 2013 First version July 2011

Cahier n°

2013-28

ECOLE POLYTECHNIQUE

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE

DEPARTEMENT D'ECONOMIE

Route de Saclay 91128 PALAISEAU CEDEX

(33) 1 69333033

http://www.economie.polytechnique.edu/

mailto:chantal.poujouly@polytechnique.edu

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pries and wages

∗∗

Alexis Blasselle

∗

Aurélien Poissonnier

†

This version Deember 2013,First version July 2011

Abstrat

We onsidera neo-Keynesian modelwithstaggered priesand wages. Whenboth ontratsexhibit sluggish

adjustmenttomarketonditions,thepoliymakerfaesatrade-obetweenstabilizingthreewelfarerelevantvari-

ables: output,prie inationand wageination. We onsider amonetary poliy ruledesigned aordingly: the

Central Bankeran reattobothinations andthe outputgap. Wegeneralize theTaylor prinipleinthis ase:

it embeds the frontier ofdeterminay derivedwith staggeredpries only, it isalso symmetriinprie and wage

inations. It follows that whenstaggered labourontrats are onsidered,wageination is alsoan illegibleand

eienttargetfortheCentralBanker.

Keywords: DynamiStohastiGeneralEquilibriummodel,MonetaryPoliyRule,SunSpotEquilibria,Taylor

Priniple

JEL:C62,C68,E12,E58,E61

∗

whenthispaperwaswritten,LaboratoireJaques-LouisLions,UniversitéPierreetMarieCurie,Paris

†

Crest-LMA-aurelien.poissonnierensae.fr

∗∗

Weare grateful toJordi Galí forraisingthisproblem duringthe 2009 BarelonaMaroeonomi SummerShool;to YvonMaday

(LJLL)andothermathematiiansatLJLLforproofreadingandomments;toOlivierLoiselforsuggestions.

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In(Taylor,1993), JohnTayloradvoatestheuse of monetarypoliy ruleswhere theCentral Bankerreatstoboth

prieinationandoutputasabenhmarktobeusedjudgementally. HisdesignofWiksellianrulehasbeenextensively

studiedsinethenin theontextofneo-Keynesianmodels. Insuh models,twonormativequestionsarise:

1

Whatkindofpoliyrulean ahieveasoialwelfareoptimum?

Howanoneruleoutsun-spot utuations(asdesribedby(Woodford,1987))?

In both respets, it has been shown that the Taylor rule has appealing properties (Woodford, 2001): in the sim-

plest neo-Keynesian model, the Taylor rule an be proved optimal in terms of welfare under some assumptions

(RotembergandWoodford, 1999). It isalso keyin enforingsolution determinay: the Taylorpriniplestates that

theCentralBanker'sreationtoinationmustbelargeenoughtoensuretheuniquenessofthesolutionunderrational

expetations.

2

Theseresultsholdunderstaggeredpriesand exiblewages. Whenonsideringbothstaggeredpries

andwages,someoftheappealingpropertiesofthestandardneo-Keynesianmodelareweakened. (BlanhardandGali,

2007)showthat allowingforbothrigiditiesgeneratesatrade-obetweenstabilizinginationandoutputeveninthe

abseneofost-pushshoks:

3

thesoialoptimumouldbeahievedwhenonlystaggeredprieswereonsidered,itis

nolongertheasewith bothstaggeredontrats. (Ereget al.,2000)study thewelfare impliationsof theaddition

ofstaggered wages. They show that is notpossiblefor the monetary poliy to fully stabilize morethan oneof the

threeobjetives: prie ination, wageination oroutput, but the varianeof eahis detrimental to welfare. Using

numerialsimulations,theyalso showthat soleprieorwageinationtargetingis suboptimalinthis ontext, buta

poliyrulesuhassuggestedbyTaylororwithreationstobothprieandwageinationsperformsnearlyaswellas

theoptimalrule.

Inthispaper,weonsiderthesamemodelas(Galí,2008,hapter6) or(Ereget al.,2000)butaremainlyonerned

withtheproblemofsunspotutuationsinsteadofwelfareoptimization. Weonsideramonetarypoliyrulein line

withEregetal.'sresults: theCentralBankeranreattobothinationsandtheoutputgap. With straightforward

notations,themonetarypoliyruletakesthefollowingform:

it= Φpπ^p_t+ Φwπ_t^w+ Φyyt

Wendthat theneessaryandsuientonditiontoruleoutsun-spotequilibriaissymmetriininations:

Φp+ Φw+1−β

˜

κ Φy>1

withβ households'disountfatorandκ˜^a^oeient^dependingsymmetriallyonbothslopesofthepriesandwages Phillipsurves.

ThefrontieroftheTaylorpriniplewithstaggeredpriesonlyisΦp+¹⁻_κ^βΦy>1^withκ^the^slope^of^the^Phillips^urve

onpries(Woodford,2001). Ourresultsthusgeneralizesthefrontierderivedisthissimplerase. Thoughthemodel's

symmetrymaynotappearstraightforward,similarsymmetryariseswhenstudyingtheoptimalmonetarypoliy (see

thefuntionalformofthewelfareriterionderivedbothbyGalíandEregetal.). Theintuitionforthissymmetryis

givenbyBlanhardandGali'sommenton(Eregetal., 2000). Inthesimplemodelwithstaggeredpriesonly,the

Phillipsurveimpliesthat stabilizingprieinationis equivalenttostabilizingtheoutputgap,aresulttheypresent

asadivineoinidene beauseitallowstheCentral Bankerto enforethesoialoptimum. But,asaforementioned,

theyshowthatwith theadditionofstaggeredwages,thisresultnolongerholds. InEregetal.'smodel, theynotea

weakerformofthisoinidene: ombiningthetwoPhillipsurvesyieldsthatstabilizingtheoutputgapisequivalent

tostabilizingaweightedaverageofprieandwageination(withtheweightoneahinationbeingtheslopeofthe

othersPhillips urve).

Intheremainderofthis paper,the rstsetionreallsthemodel. Weexpose somegeneralmathematial properties

ofthis model in setion2when theCentral Bankeranonlyreatto priesandwagesination(Φy = 0^). ^We ^then

1

Thesequestionsareindependentofoneanother:optimalrulesdonotneessarilyavoidsun-spotutuations(Claridaetal.,1999)

2

(BullardandMitra,2002)showsthatthepropertiesofthispriniplearealsokeyinamodelwithadaptivelearning

3

Inpreseneofost-pushshoksthereisashortruntrade-obetweenthetwoobjetives(Claridaetal.,1999)

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studythe uniqueness of itssolution in this ase(Φy = 0⁾^(setions ^3, ⁴ând ^5). ^We ^rstônsider ^the ^limit^subase Φp+ Φw = 1^(setion ^3). În ^setion^4, ^we^study ^the ^deviations ^from ^this ^subase ⁽Φp+ Φw ≷1^). În ^setion⁵ ^we

derivethefrontierof theTaylorpriniplewhenΦy = 0^. ^Finally^we^expand^this^result^to ^the^ase^where^the^Central

Bankeranalso reattothe outputgap (Φy 6= 0⁾ⁱⁿ ^setion ^6. ^Readers ^not^familiar ^with^this ^literature^an ^ndⁱⁿ

appendixsomegeneralelementsonneo-Keynesianmodelsformonetarypoliy solvedunder rationalexpetationsin

whihweexposethegeneralset-upofthisproblem.

1 A monetary model with stiky wages and pries

Westudythe model exposed in(Galí,2008, hap 6)and (Ereget al.,2000). This model extends thestandardneo-

Keynesianmodel formonetary poliy analysis whih onsist ofan IS urverelatingtheoutput gapto the expeted

real interestrate, aPhillips urverelating ination, expeted ination and output gap and a monetary poliy rule

desribing how the interest rate is set by the Central Banker. The present extension of the model onsiders wage

rigidities under the form of Calvo ontrats. It follows from this rigidity that real wages may deviate from their

exibleequivalentduetoexogenousdisturbanes.

Themodeltakesthefollowinglinearform:

4

π^p_t =βE(π^p_t+1|t) +κ_py_t+λ_pω_t ⁽¹⁾

π_t^w=βE(π^w_t+1|t) +κwyt−λwωt ⁽²⁾

ω_t−1=ω_t−π_t^w+π^p_t+ ∆ω_tⁿ ⁽³⁾

y_t=E(yt+1|t)− 1

σ(it−E(π^p_t+1|t)−r_tⁿ) ⁽⁴⁾ it= Φpπ_t^p+ Φwπ_t^w+ Φyyt+vt ⁽⁵⁾

Inthissystem,ateahdatet^,â^setôf^variables⁽π^p, π^w, ω, y, i⁾âre^determined^by^theirûrrentând^past^valueând

theirexpetedvalueatthefollowingdate(E(.|t)^, îs^the^rationalexpetationsoperatoratdatet^,î.e. ^theêxpetation

onditionalonthevaluesofeveryvariablesupto datetand themodel itself). Equations(1)and(2)arethePhillips

urvesonprieination(π^p⁾^and^wage^ination⁽π^w^). ^They^desribe^theprogressiveadjustmentofpriesandwagesto marketonditions. Priesmayinreasewithexpetedinationorthemarginalostofprodution. Thisostdepends

positivelyontheoutputgap(y_t^,^denedâs^the^deviationôfôutput^fromîts^fullyêxibleequivalent)andtherealwage gap(ω_t^, ^dened âs^the^deviation ôf ^real^wage^from îts^fully êxible equivalent). Wages may inreasewith expeted wageinationordereasewiththewagemark-up(takenin deviationfromtheexibleontratsase). Thismark-up

dependspositivelyontherealwagegapandnegativelyontheoutputgap.Equation(3)desribesthefatthatbeause

ofnominalrigidities,realwagesdepartfromtheirfullyexibleounterpart. Exogenousshokstotheeonomyaeting

thereal wage(∆ωⁿ⁾^are^notinstantaneouslytransmitted totheatualreal wagebutonlyto itsexibleounterpart, henedrivingawedgebetweeninationsandthedynamioftherealwagegap.Equation(4)desribestheevolution

oftheoutput gap (y⁾âs â^funtion ôf înterest^rate⁽i⁾ ândêxpetedînation. ^Theîmpliit âssumption^hereîs ^that

output is driven, in the short run, by private demand. rⁿ_t îs ^the ^natural ^rate ôf înterest, ^that îs ^the ^real înterest

ratewhihwouldprevailunderfullyexibleontrats. Equation(5)desribestheinterestratedeisionoftheCentral

Banker.ItisaTaylorrulemodiedtoaountforthefatthattheCentralBankermayreattowagesinationaswell

aspriesination. Thehigherinationsoroutputare, thehighertheCentralBankerwillset theinterestratein or-

dertotempertheeonomigrowth. Moreover,theCentralBankermaydepartfromthisruleforexogenousreasons(v^).

Theparametersofthismodelare:

0< β <1^,^is^the^disount^fator^ofhouseholds.

σ≥0^, ^is^the^inverseintertemporalelastiityofsubstitutionofonsumption.

Φp>0^,îs^the^Central^Banker's^reation^to^prieînation.(Tâylor,¹⁹⁹³⁾ônsidersΦp = 1.5

4

Theompletederivationofthemodelisexposedinfulldetailsin(Galí,2008,hap6)withthesamenotations

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Φw≥0îs^the^Central ^Banker's^reation^to^wageînation. În^the^standardâses^the^Central^Bankerônly^reat

toprieination(Φw= 0⁾

Φy≥0îs^the^Central^Banker's^reation^to^theôutput^gap. ^(Taylor,¹⁹⁹³⁾ônsidersΦy= 0.5^.

λp =⁽¹⁻^θ^p⁾⁽¹_θ ⁻^βθ^p⁾

p

1−α 1−α+αεp

,where

0< θ_p <1^, îs^the^Calvo^parameterôn^pries, ⁱⁿ ôther^words^the ^stikinessôf ^pries^(if ^0,^priesâre^fully

exible)

0< α <1^,^with1−α^theêlastiityôfôutput^with ^respet^to^labour

0< ε_p≤1^,îs ^theêlastiityôfsubstitution amonggoods

⇒ 0< λp

λw= ⁽¹⁻_θ^θ^w⁾⁽¹⁻^βθ^w⁾

w(1+ϕεw) ^,^where

0< θ_w <1^,îs ^the^Calvo^parameterôn^wages,ⁱⁿôther ^words^the^stikinessôf ^wages^(if ^0,^wagesâre^fully

exible)

0< ϕ^,îs^the^F^rishêlastiity,ⁱⁿôther^words^theônvexityôf^theôstôf^labourⁱⁿ^termsôf^welfare.

0< ε_w≤1^,îs^theêlastiityôfsubstitutionamong labourtypes

⇒ 0< λw

κp =₁^αλ₋_α^p^,^we^will^also^denote^laterλpnp=κp ^withnp>0

κw=λw(σ+₁₋^ϕ_α)^whihîmpliesκw ≥ λwσ^. ^We^willâlso^denote^laterλwnw=κw^withnw>0ôrκw=λw(σ+ν)

withν >0^.

Denotingx_t= [yt, π_t^p, π^w_t, ω_t−1]^T^,^theêndogenous^variables,ândz_t= [r_tⁿ−v_t, ∆ω_tⁿ]^T^,^theêxogenous^variables,^the

equations(1)to (5)anbewritten intheform:

x_t=A⁻¹(E(xt+1|t) +B z_t) ⁽⁶⁾

Intheequation(6),thematrixof interestA^is:

A=





 1 + κp

σβ +Φy

σ

βΦp−1−λp

σβ

βΦw+λp

σβ

λp

σβ

−κ_p β

1 +λ_p β

−λ_p β

−κ_w β

−λ_w β

1 +λ_w β

λ_w β

0 −1 1 1







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Therearethreeforwardlookingvariablesin thismodels: ([yt, π^p_t, π_t^w]^).

Lemma1 Aording to(Blanhard andKahn,1980), the system (6)has aunique solutionif andonly if the matrix

Adenedby (7)has 3eigenvaluesstritlylargerthanoneinmodulusandoneeigenvaluestritlysmallerthanonein

modulus.

Inthis ase, there is numerial evidene that the sumΦp+ Φw ^should ^be^larger ^than1 ^when Φy = 0 ^to ^meet ^this

ondition. WhenΦy 6= 0^,^theônditionôn Φp+ Φwîs ^dereasing^with Φy ^(Galí,^2008). Nevertheless,aformalproof tothesepropertieshasnotbeengivenyet,itisthemain objetiveofthis paper.

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Φp+ Φw+ Φy

(1−β) (nw+n_p)

1 λ_p + 1

λ_w

> 1 ⁽⁸⁾

rulesoutsunspotequilibria.

Theadmissibilityofapoliyrulesymmetriallydependsonwageinationandpriesination: whentheentralbank

doesnotrespondto hanges inoutput, theonditionformonetary poliy omes downto Φp+ Φw > 1ⁱⁿ ^line ^with

Galí'snumerialinvestigations.

Alsoinline withGalí'snumerialinvestigations,whentheentralbankreatstohangesin output,doingsorelaxes

theonstraintabove,proportionallyto Φy ^with^a^fator (1−β) (nw+np)

1 λp +_λ¹

w

. Thisoeientruiallyandsymmetri-

allydepends onthePhillips urvesofpriesandwages: moreimpatientagents(smallerβ⁾ôrâtter^Phillipsûrves

(smallerλôrn^),^failitate^the^taskôf^the^Central^Banker^to^prevent^sun^spotûtuations.

Inthis model, a permanentshift in prie ination (˜π⁾ împliesân îdential ^permanent ^shift ⁱⁿ ^wageînation ^(equa-

tion(3)). ThePhillipsurves(equations(1)and(2))implyaproportionalshiftinoutputgapy˜=_(n⁽¹⁻^β)

w+np)

1 λp +_λ¹

w

π˜^.

Inturn, the Taylorrule (5)implies that the reationof theCentral Bankeris to raise thenominal interest rateby

˜i = h

Φp+ Φw+ Φy (1−β) (nw+np)

1 λp+_λ¹

w

i˜π^. ^Thus, ^as ⁱⁿ ^the ^standard neo-Keynesian model without wage rigidities (Woodford, 2011, hapter 4), our frontierof indeterminayanalso beinterpreted in termsof theTaylorpriniple:

theCentral Bankerreating morethanone forone topermanenthanges in ination.

UsingDynare (Adjemianet al., 2011), it is possibleto verify numeriallyfrontier(8).

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(Galí, 2008, hapter 6) and

(Eregetal., 2000) show that wage ination targeting ompares with prie ination targeting in terms of welfare.

UsingDynareitispossibletoonrmtheirresultofsymmetrybyomputingtheoptimaloeientsforthemonetary

poliy ruleonsideredhere.

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WhenonsideringaCentralBankerreatingto bothinationsand theoutputgap,we

ndΦp = 47.1,Φw = 67.8,Φy = 231.9^. ^This ôptimal ^rule împlies â ^very ^sensitive înterest^rate ^whih îs ^standard

whenthebenetsofasmoothedmonetarypoliyarenotonsidered. Moreinterestingly,thereationsoftheoptimal

interestratetobothinationsareomparable.

Wageinationand prieinationplaysimilarrolesforthedesignoftheoptimalmonetarypoliy,weshowthat they

alsoplaysymmetriroles foreliminating sun-spot utuations. This extended onlusionremains"at odds with the

pratieof mostentral banks, whih seemtoattahlittle weighttowageination asatargetvariable"(Galí,2008).

Outlineoftheproof Deningthefrontierofindeterminayisbasedonthestudyoftherootsoftheharateristi

polynomialofmatrixA^,^a^fourth^degreepolynomial. Thoughitisnotomplexmathematis,itisratherumbersome.

Wearepartiularlygrateful toYvonMadayandothermathematiians atLaboratoireJaques-LouisLionsforproof-

readingandomments.

In setions 2 to 5 we develop the proof in the ase Φy = 0^. ^In ^setion ² ^we ^study ^the ^general ^properties ^of ^this

polynomial and its oeients. We use the intuition that in this asethe frontier of determinay is Φp+ Φw = 1

and deompose the polynomial as afourth degree polynomialorresponding to this ase plus deviations from this

ase in both diretions (Φp,Φw)^. ^In ^setion ³ ^we ^study ^the ^polynomial ⁱⁿ ^the ^ase Φp + Φw = 1 ^to ^show ^that:

1 îs â ^root ôf ^this polynomial; its real roots are non-negative; its omplex roots have a modulus stritly greater thanone; and at most onereal root is in ]0,1[^. În ^setion ⁴^we^study ^the ^deviations ^from Φp+ Φw = 1^; ^we^show

thatthese deviations areseond degreepolynomialswithpositivereal roots, onestritly greaterthan onetheother

stritly smaller than one. In setion 5 we study how the deviations from Φp+ Φw = 1 ^modies ^the ^roots ^of ^the

harateristi polynomial. The omplexroots annot enter theunit irle. The real roots stritly greater orlower

5

Codeavailableuponrequest

6

ThewelfareriteriontobeoptimizedisderivedinGalí,weusehisbenhmarkalibrationandinlinewithhismethodologyonsider

tehnologyshoksonly.

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thanoneare keptaway from 1^. ^The^root 1 ^movesⁱⁿ ^the ^diretion ênsuring ^theûniqueness ôf ^the ^model's ^solution

(dependingontheexisteneofanotherrootsmallerthanone)ifandonlyifthedeviationfromΦp+ Φw= 1^is^positive.

Insetion6weshowthattheaseΦy 6= 0ân ^be^treatedîdentially^to^theâseΦy = 0^. ^Weônsider^the^frontierôf

indeterminayundertheformΦp+Φw= 1−θ^and^show^that^settingθ= Φy (1−β) (nw+np)

1 λp+_λ¹

w

allowsadeomposition

oftheharateristipolynomialwhihhasthesamepropertiesasintheaseΦy= 0^. ^Weânônlude^thatêquation

(8)generalizesthefrontierofindeterminay.