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September24,2018

LIQUIDITYDEFLATION:Supply-SideLiquidityTrap,DeflationBiasandFlatPhillipsCurve

GuillermoCalvoColumbiaUniversity

Abstract.ThepapershowsthatLiquidityDeflation,aliquidity-drainagemechanismassociatedwithSafeAssetShortage,helpstorationalizesupply-sideliquiditytrapandinvoluntaryunemploymentunderwage/priceflexibility,especiallyineconomiesthatsufferedmassiveliquiditydestruction,likeadvancedeconomiesinthewakeoftheLehmancrisis.Moreover,LiquidityDeflationgeneratesdeflationbiasandflatPhillipscurveinaNewKeynesianmodel.

I.INTRODUCTIONTheobjectiveofthisnoteistoshowthatwhatIamtemptedtocallthe"crucialmissingpiece"inKeynes'sGeneralTheory,GT,namely,LiquidityDeflation—aliquidity-drainagemechanism,discussedinCalvo2016and2018—canhelptoresolvetwocentralpuzzlesincurrentmacroeconomicdebate,namely,(1)persistenceofdeflationaryforcesdespitealargeincreaseinthesupplyofcentralbankliquidliabilities,deflationbias,and(2)flatteningofthePhillipscurve.Tosetthediscussiononfamiliargrounds,letusconsidertheIS-LMapparatusinwhichfiatmoneyMisassumedtohavenointrinsicvalue,andtobetheonlyliquidasset(or,moreprecisely,theonlyassetprovidingliquidityservices).IncontrastwiththestandardmodelIwillincludeLiquidityDeflationbyassumingthattheliquidityofMmaybeadecreasingfunctionoftheaveragemarketholdingsofrealmonetarybalances,𝑀 𝑃,wherePstandsforthepricelevel.SomemicrofoundationsarediscussedinCalvo(2018).However,giventherelativeunfamiliarityofLiquidityDeflation,itisworthtostartrollingtheballwithaninformalexample.Considerarepresentative-individualeconomyinwhichallindividualsareexantealike.Thus,realmonetarybalancesheldbyeachindividualwouldbe𝑀/𝑃𝑛,wherenisthenumberofindividuals.Therefore,ifnissufficiently'large',realmonetaryholdingsbyeachindividualwouldbe'small'.ThismayresultinindividualsfeelingthatMisSafe.However,individuals'perceptionmaychangeifM/Pbecomes'large'.Thiswouldberational,forexample,ifliquidationofmoneyholdingsissubjecttorandomsystemicshocks.Thus,therewouldnowbestatesofnatureinwhichmoneyissubjectto"runs",makingmoneylesssafeanddeterioratingitsliquidityservices,implyingthatthequalityofmoneyasaliquidassetworsensastheeconomy-widestockofmoneybecomes'large'inrealterms.Clearly,thiseffectis

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formallyequivalenttoanexternality.Individualmoneyholdingsmayhaveanegligibleeffectontheliquidityofmoneyand,yet,thelattermaysignificantlydeclineastheeconomy-widerealmonetarybalances,𝑀/𝑃,becomes'large'.1Tocapturethisexternalityinasimplemanner,IwillwritethemoneymarketLMequilibriumconditionasfollows:2

𝑚 + 𝑍 𝑚! = 𝐿 𝑖,𝑦 , 𝑍′ < 0, 𝐿! < 0, 𝐿! > 0, (1)

whereiandydenote,respectively,nominalinterestrateandoutput(asinthestandardIS-LMmodel);mstandsfortheatomisticindividual'sdemandforrealmonetarybalances𝑀 𝑃,while𝑚! standsfortheaveragemarketholdingofrealmonetarybalances—anegativeexternalityfromthepointofviewoftheatomisticindividual.FunctionZcorrespondstoLiquidityDeflation.Thus,expression(1)givesrisetoastandardLMcurve,exceptfortheLiquidityDeflationterm𝑍(𝑚!).Inequilibrium𝑚 = 𝑚! and,consequently,anincreaseinmmayfailtoincreasemarketliquidityif𝑍'islargeenough(inabsolutevalue).ThiswouldgiverisetowhatIwillcallSupply-SideLiquidityTrap.SectionII.1willapplyequation(1)toshowthepossibilityofSupply-SideLiquidityTrapandinvoluntaryunemployment,evenifpricesandwagesareperfectlyflexible.TheexamplesummarizesresultsinCalvo(2018)andfocusesonacoupleofimportantapplications.SectionII.2,inturn,couchesthediscussionintermsofasimpledynamicNewKeynesianmodelandshowsthatLiquidityDeflationmaygeneratepricedeflationbiasandaflatteningofthePhillipscurve.SectionIIIclosesanddrawsfurtherlinkswithcurrentmacroeconomicdebates.

II.SIMPLEMODELS1.Price/WageDeflationConsiderequation(1),andsupposeaclosedeconomywithperfectlyflexibleprices.Iwillassumethatatfull-employmentoutput,𝑦! ,and𝑖 = 0wehave:

𝑚 + 𝑍 𝑚 < 𝐿 0,𝑦! (2)

and

1+ 𝑍′(𝑚) ≤ 0. (3)

1ThisargumentcouldbecouchedintermsofDiamondandDybvig(1983).SeerelatedexamplesinGoldsteinandPauzner(2005),Holmström(2015),andGorton(2016).2Inwhatfollows,andtosimplifythenotation,Iwillassumeanatomisticeconomyinwhichthe'measure'oftotalpopulation𝑛 = 1,andremainsconstantovertime.

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ThisisasituationinwhichtheeconomyhasreachedtheZeroLowerBoundoninterestrates,ZLB,buttherestillisexcessdemandforliquiditythatpreventsachievingfullemploymentequilibriumandgeneratesasituationlabeledinvoluntaryunemploymentinKeynes(1936)GeneralTheory,GT.Increasingmoneysupplywillfailtorestorefullemploymentbecauseinequality(3)holds.Pumpingmoremoneyintotheeconomymaysucceedinincreasingrealmonetarybalances,m,butfailstoincreasethesupplyofliquidassets.ThisgivesrisetoaphenomenonthatcouldeasilybemistakenforLiquidityTrapinGT,althoughthesourceslieonthesupplyside,notontheexistenceofaninfinitelyelasticdemandformoneywithrespecttothenominalinterestrate(ademand-sidephenomenonemphasizedintheGT).Iwillcallthesituationdepictedbyinequalities(2)and(3)Supply-SideLiquidityTrap,SSLT.Asafirstapproximation,recenteventsfitwellintothiskindofscenario.DuringtheGreatRecession,centralbanksresortedtoincreasingcentralbankliquidity,QE,butresults,althoughpositive,werelargelydisappointing.Theexampleaboveshowsthatthissituationcouldholdeventhoughpricesareperfectlyflexible.Moreover,Calvo(2018)showsthatattemptstoincreasethenominalinterestratebyraisingtherateofgrowthofmoneysupplymaynotsucceed(assuggestedinKrugman1998),becausetheremayexistarational-expectationsequilibriuminwhichindividualsaccumulatetheentireadditionalamountofmoneyflow.Incontrast,ifmoneypaysinterest,𝑖!notsubjecttoZLB,thenlowering𝑖!maysucceedinpushingtheeconomybacktofullemployment.However,low𝑖!maygiverisetoCurrencySubstitution(sincelowering𝑖!isequivalenttotheimpositionofaninflationtax).Inextremecases,thismaysimplymakecentralbank'smonetarypolicyarsenaluselessforalleviatingLiquidityTrap.Themodelcanbeextendedtotakeexplicitaccountofthelabormarket.Lettheproductionfunctionbe𝑓(𝑠),wheresstandsforemployment,andfunctionfsatisfiesallstandardproperties.Let𝑦 < 𝑦! ≡ 𝑓(𝑠!),where𝑠! standsforperfectlyinelasticlaborsupply(=full-employmentlabor),andassume

max![𝑚 + 𝑍 𝑚 ] = 𝐿 0,𝑦 . (4)Moreover,let𝑠besuchthat𝑓 𝑠 = 𝑦.Thus,inwords,theaboveequationsimplythatthelargestamountoflaborthattheeconomycanemployatZLB,𝑠,islessthanfullemployment𝑠! .Ifpricesandwagesareperfectlyflexible,thiswillleadtochronicdeflation.Downwardwageflexibilitywillbeofnohelpbecausetheirdeclinewillquicklybefollowedbylowerprices.ThemodelhelpstorationalizeGTChapter19view,frequentlyquoted,butseldommodeled,accordingtowhichafallinwageswillnotsucceedinreestablishingfullemployment.For,inthepresentmodel,aslongasequation(4)holds,aggregatedemandwillnotbeabletomatchfull-employmentoutput.Afallinnominalwages

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maytemporarilyraiseemploymentandoutput,buttheliquidityconstraintimpliedin(4)preventssalesfromrising.Amorerealisticversionofthemodelmightincludeprice/wageadjustmentfrictions,buttheghostofdeflationisunlikelytoentirelydisappear:fullemploymentmayneverbereached,unlessmarketsareundergostructuralchangeand/orthereisahealthyinfusionofliquidassets.2.DeflationBiasandFlatPhillipsCurveTheabovemodelassumesthatLiquidityDeflationisinstantaneous.HereIwillextendthemodelbyassumingthatthenegativeliquidityexternalitytakestimeand,thus,moneyprintingiscapableofincreasingliquidityintheshortrun.Iwillcouchtheseassumptionsintermsofasimple,perfect-foresightstandardNewKeynesianmodelwithliquidity-in-advanceconstraint.Let𝑋!

! denotenominalliquidityheldbyindividualjattimet.Followingthepreviousdiscussion,Iassumethat

𝑋!! = 𝑀!

! + 𝑍! , (5)

where,𝑀!! isthestockofmoneyheldbyindividualj,and𝑍!standsfortheLiquidity

Deflationcomponent,which,forsimplicityishomogeneouslyandexogenouslydistributedacrossagents.Thus,Zcouldbethoughtasalump-sumliquidityendowmentthatcouldpossiblybenegative.Iassumethat𝑍!ispredeterminedattimet(forallt)and,therefore,unliketheprevioussubsection,𝑍!isinvarianttochangesintheeconomy-widedemandformoneyattimet.Therepresentativeindividualfacesaperfectcapitalmarketandisthusfreetochoosethestockofmoneyholdings,butthestockofliquidityisgivenbyequation(5).IassumethatZsatisfiesthefollowingdifferentialequation

𝑍! = 𝛾 ln 𝑥 − ln 𝑥! 𝑋! , 𝛾 > 0, 𝑥 > 0, (6)

where𝑋!iseconomy-wideliquidity,𝑥 = 𝑋/𝑃,and𝑥isanexogenousconstant.Aswillbecomeclearinamoment,equation(6)introducesaforcethatpusheseconomy-widerealliquiditytowards𝑥.Thus,timedifferentiatingequation(5)andaggregatingacross(identical)individuals,weget,inequilibrium,

𝑋! = 𝑀! + 𝛾 ln 𝑥 − ln 𝑥! 𝑋! . (7)Thus,nominalliquidityisfedbymoneysupplybutthereexistsanindependentforce,associatedwithLiquidityDeflation,whichgraduallypushesrealeconomy-

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wideliquiditytowards𝑥.3Iwillfirstexaminecasesinwhichfullemploymentrealliquiditywouldcallfor𝑥 > 𝑥,aphenomenonakintoSafeAssetShortage.Iwillfocusonmonetarypolicyaimedatcontrollingmonetaryaggregates(asQE).Iassumethatmoneysupplysatisfies:

𝑀! = 𝜇𝑋! , (8)

where𝜇isapositiveparameter.NoticethatintheIS-LMmodelinwhich𝑀 = 𝑋,𝜇wouldcorrespondtotherateofgrowthofmoneysupply.Hence,dividingbothsidesofequation(7)byXandtakingequation(8)intoaccount,wehave

𝑥! = 𝜇 − 𝜋! 𝑥! + 𝛾 ln 𝑥 − ln 𝑥! 𝑥! , (9)

where𝜋 = 𝑃 𝑃=instantaneousrateofinflation.Thus,denotingℎ = ln 𝑥,equation(9)canbeexpressedas

ℎ! = 𝜇 − 𝜋! + 𝛾(ln 𝑥 − ℎ!). (10)Thisisaclosedeconomywithhomogeneousoutput."Fullemployment"outputis,again,denotedby𝑦! .Equilibriumoutputisdemand-determinedasinNewKeynesianmacroeconomicmodels.Moreover,Iwillassumethatconsumption,c,equalsaggregatedemand(investmentandgovernmentexpenditurearezero)andconsumptionissubjecttothefollowingliquidity-in-advanceconstraint:

𝑐! = 𝑥! , (11a)

or,equivalently,

ln 𝑐! = ℎ! . (11b)

ThepricelevelisstickyandsatisfiesCalvo(1983)equation.Thus,recallingequation(11b),Iassume4

3Thelagmaybetheresultofsignal-extractionproblems(Lucas1972),someformofInattention(MankiwandReis2002,Sims2003),orfamiliarmoneyillusion(Shafir,DiamondandTversky1997).ThelatteristhemoreappropriateinterpretationforthepresentmodelbecauseIassumethatallindividualslagbehindcurrentrealizationsinlockstep.4IfthereaderwonderswhyIshifttologsinsteadofexpressingvariablesintheirnaturalunits,theansweristhatIwantthesystemofdifferentialequationstobelinear.InthiscasetheequilibriumPhillipscurveislinearandcaneasilybe

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𝜋! = 𝛽 ln𝑦! − ℎ! , 𝛽 > 0. (12)

Anyequilibriumpathmustsatisfyequations(10)and(12).5Moreover,Iwillfollowconventionandassumethatequilibriumpathsmustconvergetosteadystate.Theseassumptionsimplythat,foralltimet,(logof)realliquidityℎ!ispredetermined.Ontheotherhand,Calvo(1983)showsthattherateofinflationcorrespondingtoanequilibriumpathshouldbeexpectedtobecontinuousforallt.However,the"present"rateofinflationisfreetojump.Asiswellknown,thepossibilitythat𝜋!canjumpwhentheeconomyisattimet,carriesnonon-uniquenessconsequences,anduniquenessisensuredifsystem(10)and(12)displayssaddle-pathstability,whichinthisinstanceissatisfiedbecausethedeterminantofthecorrespondingsteady-stateJacobeanmatrixisnegative.Thephasediagramforequations(10)and(12)isdepictedinFigure1,wherethearrowedlinepointingtothesteadystateistheequilibriumpath.Thus,inequilibriumthereexistsapositiveassociationbetweenℎand𝜋,i.e.,recallingequation(11b),apositiveassociationbetweenoutputandinflation.Hence,themodel'sreduced-formPhillipscurvedisplaystheconventionalslope,despitetheadditionofthenewLiquidityDeflationingredients.

Figure1.EquilibriumDetermination

computed.Theinquisitivereadercouldeasilyverifythatsubstantiveimplicationsarenotaffectedbythesemathcontortions.5Theliquidity-in-advanceassumptionsimplifiestheanalysisenormouslycomparedtoCalvo(1983),forexample:Thereisnoneedtorefertoutilityfunctions!AsfarasIknow,Reinhart(1992)isthefirstpaperthatexploresthisassumption.However,forthesakeofcompleteness,AppendixA1showsthemissingequations.

π

lnyF

�̇� = 0

ℎ̇ = 0

equilibriumpath

h

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Atsteadystatewhereℎ = 𝜋 = 0,wehave,byequation(12),thatℎ = ln𝑦! ,and,byequation(10):6

𝜋 = 𝜇 + 𝛾(ln 𝑥 − ln𝑦!). (13)If𝑥 = 𝑦! ,therateofinflationequalstherateofgrowthofliquiditydrivenexclusivelybytheexpansionofmoneysupply(recallequation(8)).However,if𝑥 < 𝑦! ,inflationatsteadystate𝜋 < 𝜇,reflectinga"deflationbias."Thisisaninterestingcasebecauseitbringsusclosertothecurrentdeflationdebate.Inaddition,thelargeristheLiquidityDeflationparameter𝛾,thesmallerwillbetherateofinflationatsteadystate,andchronicdeflationcannotbediscounted,evenifthecentralbanktakesanaggressiveQEstance(i.e.,ahigh𝜇inthemodel)!Somuchfortheformalanalysis;nowitisagoodtimetoreflectonthetwonewparameters𝑥and𝛾.Byequation(10),theloweris𝑥,thelesseffectivewillbemoneysupplyinincreasingmarketliquidity.Ontheotherhand,if𝑥 < 𝑦! ,thehigheris𝛾,thestrongeristheLiquidityDeflationeffectonsteady-stateinflation.Theseresultscanhelptocastsomelightonrecentevents.Forinstance,severalanalysesoftheLehmancrisispointtothefactthatitinvolvedamassivedestructionofSafeAssets(e.g.,Gorton2010,Calvo2012,Caballeroetal2016and2017),whichinthepresentsetupcouldbeinterpretedasresultinginalowersteadystatecapacitytosupplyliquidityservices.Thiscouldbeinterpretedasafallin𝑥.Ontheotherhand,themassivedestructionofliquiditymayhavemadethemarketmoreawareoftherelevanceoftheliquidityofcollateralbehindpresumedSafeAssetsand,therefore,becomemoresensitivetodeparturesofxfrom𝑥,givingrisetoahigher𝛾.Thus,theLehmancrisismayhaveenhancedtherelevanceofLiquidityDeflationanditsconsequentdeflationbias.Iwillshowbelowthatanincreasein𝛾offersanewinsightfortheexistenceofaflatPhillipscurve.7Arelatedimplicationofthemodelisthatdeflationarybiasgoeshandinhandwithadeclineinthevelocityofcirculationofmoney,m,acrossstablesteadystates—anothersalientfactindevelopedeconomiesaftertheLehmancrisis.Byequation(8),(11aandb),and(12),wehavethatif𝑥and𝜋areatsteadystate8

!!!!= 𝜇 !!

!!− 𝜋. (14)

6Thisimpliesthatinthelongruntheeconomyconvergestofullemployment.However,thiscouldbemodifiedintroducingelementshighlightedinSectionII.1.7TheFlatPhillipscurveissueisattractingwideattentioninmacroeconomiccircles.See,forinstance,Borio2017,BlanchardandSummers2017.8Noticethattheequilibriumpathsof𝑥and𝜋convergetotheirrespectivesteadystateequilibriumvalues(recallFigure1),irrespectiveof𝑚.

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Therefore,byequation(14),anecessaryconditionexistenceandstabilityofasteadystateforrealmonetarybalances,m,is𝜇 > 0, and𝜋 > 0.Considerthecaseinwhich𝑚, 𝑥and𝜋aresimultaneouslyatsteadystate,i.e.,𝑚 = 𝑥 = 𝜋 = 0.Byequations(11a,b),(12),(13)and(14),itfollowsthat

velocityofcirculationofmoneyatsteadystate=

= !!

!= !

!= !!!(!"!!!"!!)

!. (15)

Therefore,acrossstablesteadystates,velocityfallsasdeflationaryconditionsgetexacerbated,i.e.if𝛾(ln 𝑥 − ln𝑦!)goesdown.Themodelisalsoconsistentwithsituationsinwhichmgrowswithoutbound(implyingthatvelocitysteadilydeclinestowards0),whichisanextremecaseofLiquidityTrapthatIhavediscussedintermsofasimplermodel(seeCalvo2018).Toillustrate,considerthecaseinwhich𝜇 > 0,andsteadystate 𝜋 < 0.Byequation(13),steadystatepricedeflationarisesifLiquidityDeflationisstrongenough(i.e.,if𝛾(ln 𝑥 − ln𝑦!)isnegativeenoughtooffset𝜇).Interestingly,theresultingpathdoesnotviolatetransversalityconditionsbecauserealmonetarybalancesarenota"fundamental".Itsrolehavebeenappropriatedbyrealliquidity,i.e.,x!Ontheotherhand,asinastandardNewKeynesianmodel,thenominalinterestrateonnon-liquidassets,i,satisfiesatsteadystate(seeAppendixA1fordetails):

𝑖 = 𝜌 + 𝜋 = 𝜌 + 𝜇 + 𝛾(ln 𝑥 − ln𝑦!), (16)where𝜌isthesubjectiverateofdiscount,andtheright-handequalityfollowsfromequation(13).Thus,strongerdeflationaryconditionsresultinlowernominalinterestrates,anotherphenomenonobservedinthewakeoftheLehmancrisis.IwillnowturntoexaminetheslopeofthePhillipscurve.System(10)and(12)islinear.Therefore,theequilibriumsaddlepathcaneasilybecomputed.LetBdenotetheslopeofthereduced-formPhillipscurve(i.e.,𝐵 = !"

!!alongthearrowed

equilibriumpathinFigure1).Itcanbeshown(seeAppendixA2)that

𝐵 = !!! !!!!!!

. (17)

Hence,byequation(17),proofthatthePhillipscurvebecomesflatterastheLiquidityDeflationparameterincreasescanbeinferredfromthefollowingexpressions:

!"!"< 0andlim!→! 𝐵 = 0. (18)

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Moreover,byAppendixA2,

ℎ! = −(𝐵 + 𝛾)(ℎ! − ln𝑦!) = − !! !!!!!!

(ℎ! − ln𝑦!). (19)

Hence,thespeedatwhichoutputconvergestofullemploymentisinvarianttotheliquiditydrag𝑥,andacceleratesasthedragbecomesstronger,i.e.,𝛾goesup.Therefore,LiquidityDeflationfailstoslowdownconvergencetofullemployment,andmayactuallyaccelerateit!Thissoundsimplausiblebuttheseresultsaretiedtotheassumptionthatfullemploymentoutput,𝑦! ,isinvarianttoliquidityfrictions.Relaxingthisassumption,however,neednoteliminateDeflationBiasandFlatPhillipscurves,aswouldbeillustratedbythecaseinwhich𝑦! suffersaonce-and-for-allnegativeshock.Asitstands,themodelimpliesthatastockchangeinmoneysupplycouldplacetheeconomysmackintofullemployment.Thiscontradictstheviewthatliquidityofmoneysupplydeclineswithitsstockinrealterms.However,thiscaneasilyberemediedbymodifyingequation(7)and,forinstance,replacing𝑀byaconcavefunctionof𝑀subjecttoanupperbound.

III.FINALWORDS

Thepaperexploresfinancialfrictionsthatgiverisetochronicdeflation,aflatPhillipscurve,andsituationsthatareobservationallyequivalenttoaKeynesianLiquidityTrap.Moreover,thisholdseveniftheinterestelasticityofthedemandformoneyisbounded.Theapproachisrelevantforsituationsinwhich,say,theeconomysufferedamassivedestructionofliquiditythatinterfereswithtradeflows,andrestorationofliquidityto"normal"levelscannotberapidlyachievedbypumpingcentralbankliquidity,e.g.,high-poweredmoney.Thereasonforthis,thepaperattributestothefactthattheliquidityqualityofliquidassetsmaydeteriorateastherealvalueofthoseassetsbecomeslarge.Thiseffectisaresultofanegativeexternalityrootedinthefactthatbyitsverynatureliquidityisopenedtoself-fulfillingrunsàlaDiamond-Dybvig(1983),especiallyifthesupplyofliquidassetsincreaseswithoutbeingaccompaniedbyanequivalentriseofrealcollateral.Aninstanceofthissortisthelargeincreaseinhigh-poweredmoneyinthewakeoftheLehmancrisis.Thus,theeconomymay,atleastintheshortrun,endupinasituationthatseveralauthorshavelabeledSafeAssetShortage(e.g.,Caballeroetal2016,2017,Gorton2016,Barroetal2017)inwhich,beyondacertainpoint,liquiditycannotbeincreasedbyQE.ItisaSupply-SideLiquidityTrapthatcanevenholdunderperfectlyflexiblepricesandwages,andmaynotbealleviatedbyexpansionaryfiscalormonetarypolicyunlessitresultsinanincreaseinmarketliquidity.Inclosing,itisworthpointingoutthattheeffectshighlightedinthispaperholdwhenliquidityisscarcerelativeto"normal"situations.Thisisimportantbecause,forinstance,theslopeofthePhillipscurvecouldsteepensharplyasthereisgreater

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confidenceinthestabilityofthefinancialsector.Unfortunately,returntonormalityissurroundedbyalargedegreeofuncertainty,andforecastsarelikelytobeapoorguideforpolicy.Butcentralbankscouldstillplayavaluableroleiftheysucceedinbetterunderstandingandmanagingnewliquidassetsthat,asshownbyrecentevents,couldcauseinefficientfrictionsormassiveliquiditycrunch.

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APPENDIXA1

HereIwillsketchoutthemicrofoundationsofthemodelconsideredinSectionII.2.ThemodelisclosetotheoneinCalvo(1983),exceptthatmoneyisremovedfromtheutilityfunctionandreplacedforacash-in-advancecondition,andtheexistenceofLiquidityDeflation.Therepresentativeconsumer'sutilityfunctionfromtheperspectiveoftime0satisfies:9

𝑢 𝑐!! 𝑒!!"𝑑𝑡,!

! (A1.1)

where,again,jreferstoindividualj,functionuisdefinedonthepositiverealline;itisincreasing,strictlyconcave,andtwice-continuouslydifferentiableeverywhere.Parameter𝜌 > 0standsforthesubjectiverateofdiscount.Thebudgetconstraintsatisfies(assuming0initialwealth),

[𝑦!! + 𝑠!

!! − 𝑐!

! − 𝑖!𝑚!!]𝑒! !!!"

!! 𝑑𝑡 ≥ 0, (A1.2)

wheresstandsforlump-sumsubsidyemployedtorebatecentralbankseignioragetothepublic,astandardassumptioninmonetarytheory;ristheinstantaneousrealrateofinterest,andthenominalinterestrate𝑖 = 𝑟 + 𝜋.Equations(5)and(11)implythat

𝑐!! = 𝑥!

! = 𝑚!! + 𝑧! , (A1.3)

implyingthat

𝑚!! = 𝑐!

! − 𝑧! . (A1.4)

Insertingequation(A1.4)inbudgetconstraint(A1.2)yields

[𝑦!! + 𝑠!

!! − 𝑐!

! − 𝑖!(𝑐!! − 𝑧!)]𝑒! !!!"

!! 𝑑𝑡 ≥ 0. (A1.5)

Variablezisexogenoustotherepresentativeindividual.Therefore,thefirst-orderconditionforconsumption,takingintoaccount(A1.1)and(A1.5),takesthefollowingfamiliarform:9Variablesdefinedinthemaintextwillnotbedefinedhere,unlessnecessaryforthesakeofclarity.

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𝑢′ 𝑐!

! 𝑒!!" = 𝜆(1+ 𝑖!)𝑒! !!!"!! , (A1.6)

where𝜆isthetime-invariantLagrangemultiplier.Themodelcannowbecomplementedbythezdynamicsdiscussedinthetext.

APPENDIXA2

Considerequations(10)and(12).Thesystemislinearand,hence,thereareparametersdenotedDandB,𝐵 > 0,suchthatalongthesaddlepathdepictedinFigure1,wehave

𝜋 = 𝐷 + 𝐵ℎ,forallh. (A2.1)

Droppingtimesubscripts,equations(10),(12)and(A2.1)imply:

𝜋 = 𝐵ℎ = 𝐵[𝜇 − 𝐷 − 𝐵ℎ + 𝛾 ln 𝑥 − ℎ = 𝛽(ln𝑦! − ℎ). (A2.2)

Therightmostequationmustholdforallh,whichimplies:

𝐵! + 𝐵𝛾 − 𝛽 = 0. (A2.3)

SolvingforBandrecallingthat,byFigure1,𝐵 > 0,yieldsequation(17)inthetext.TosolveforparameterDin(A2.2),setℎ = ln𝑦! .Itfollowsthat

𝐷 = 𝜇 − 𝐵 ln𝑦! + 𝛾 ln 𝑥 − ln𝑦! . (A2.4)

Byequations(A2.2)and(A2.4),onecanderiveequation(19)inthetext.

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REFERENCES

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