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LIQUIDITYDEFLATIONANDLIQUIDITYTRAPUNDERFLEXIBLEPRICESSomeMicrofoundationsandImplications

GuillermoCalvo1

September24,2018Abstract.ThepaperdiscussessomesimplemicrofoundationsforLiquidityDeflation,andshowsthatitcouldgiverisetoliquiditytrapconditionsunderperfectlyflexibleprices.UnlikeinKeynes(1936),thisisaSupplySideLiquidityTrap,SSLT,whichchallengesthePigoueffect,e.g.,itmaynotbeeliminatedbyamassivehelicopter-typeincreaseinliquidgovernmentliabilitiesorlargefallinpricesandwages.However,asinstandardmodels,lowpolicyinterestratescouldhelpreachingfullcapacityutilization.Moreinteresting,however,isthatheterodoxpolicylikedirectpricecontrols,forinstance,couldwork.Moreover,themodelshowsthatunderLiquidityDeflationtheOptimalQuantityofMoney(Friedman1969,Chapter1)doesnotcallforachievingliquiditysatiation,anditissituateddangerouslyclosetoaSSLT.

I.INTRODUCTIONSeveralrecentpaperssuggestthatshortageofveryliquid(alsocalledsafe)assetscouldbeacentralexplanatoryfactorforthedeepandlong-lastingGreatRecessionthatfollowedtheLehmancrisis.ThisistracedbacktothemassivedestructionofsafeassetsassociatedwiththeLehmancrisis(seeCalvo2012,Caballero,FarhiandGourinchas2016and2017),largedemandforandinsufficientcreationofsafeassetsduetoconstraintsliketheZeroLowerBoundonpolicyinterestrates,ZLB,andsterilizedintervention.TheZLBishardtobypassbecauseitinvolvesthornyoperationalproblemsbut,inprinciple,unsterilizedintervention,likehelicoptermoney,shouldbelessproblematic.Therefore,partofthedifficultycouldbefoundincentralbanks'reluctancetoutilizeunsterilizedintervention,particularlyinasituationinwhichlong-termTreasurybondsappeartobehighlysubstitutablebyTreasurybillsandbankreserves—and,therefore,sterilizedintervention,likeOperationTwistintheUS,mightbeineffectiveinrelievingsafe-assetshortage.Buttheeffectivenessofhelicoptermoneyisnotfreefromcritics.Keynes(1936),forinstance,raisedsomedoubtsabouttheeffectivenessofunsterilizedinterventionbyconjecturingthatthereexistsapositivenominalinterestrateatwhichthedemandformoneybecomesinfinitelyelastic—asituationlabeledLiquidityTrap.However,Keynes'sconjectureishardtojustifyinconventionalmodelssince,giventhepricelevel,helicoptermoneycouldincreaserealwealthwithoutboundand,ifthereisnoconsumptionsatiation,utilitymaximizationshouldleadtoariseinaggregatedemandthatmatchesfullcapacityoutput(asimpliedbythePigoueffect).ThiscriticismtoKeynes'sconjecture,though,isduetoanassumptionthatistakenfor

1IamthankfultoRicardoCaballero,SaraCalvoandMartínUribeforusefulcommentsonapreviousversion.Errorsandomissionsareexclusivelymine.

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grantedinconventionalmacromodels,namely,thattheliquidityservicesprovidedbymoneyareproportionaltothestockofrealmonetarybalances.Thisassumptionsoundsreasonableinnormalcircumstances,awayfromLiquidity-Traptypeepisodes,butitmaybequestionablewhen,asintheGreatRecession,thevelocityofcirculationofmonetaryaggregates(e.g.,M1andM2)intheUS,hasfallentolevelsnotseeninmorethanhalfacentury.2InrecentworkIhaveexploredthepossibilityofadisconnectbetweenrealsafeassets(i.e.,thenominalsupplyofsafeassetsdividedbythepricelevel),andtheirliquidityortransactionservices—andarguedthatbeyondacertainpointanincreaseinrealsafeassetsmayfailtoincreasetotalliquidityinthesameproportion.IlabeledthisphenomenonLiquidityDeflation.AsshowninCalvo(2016aandb)anddiscussedbelow,LiquidityDeflationhelpstovalidateKeynes'sconjectureeventhoughexpansionofmonetaryaggregatesistriggeredbyhelicoptermoney,andtheinterestelasticityofthedemandformoneyisbounded.TheideasbehindLiquidityDeflationarestraightforward,butIhaveencounteredsomeintellectualresistanceinconsideringitsrelevance.Partofthereasonforthisisthatinmacroeconomicstheconceptof“money”isassociatedwithanobjectwhichliquidityistarnishedbynothingexcept“inflation”.Inparticular,increasingthesupplyof“reserveassets,"(i.e.,assetsemployedbycentralbanksasinternationalreserves,likeUSTreasuryobligations)israrelyassociatedinmodelswithalossoftheir“liquidity”ortransactionservices.Thus,IbelievethattimeishighfordiscussingthemicrofoundationsofLiquidityDeflationand,onthatbasis,getnewinsightsonmacropolicyaroundLiquidityTrapepisodes.Thesearethecentralissuestackledinthepresentpaper.Tomotivatethediscussion,IwillfirstbrieflypresentinplainEnglishtworelatedLiquidityDeflationscenarios.ThenIwillspelloutaformalmodel,whichwillallowustodiscussthemechanicsofLiquidityDeflationingreaterdetailandshow,underconventionalassumptions,thatLiquidityDeflationcouldcompletelycrowdouttheliquidity-enhancingimpactofanincreaseinmoneysupply.Moreover,themodelshowsthatoperatingnearthecomplete-crowd-outequilibriumisalsoproblematic.Underthoseconditions,evenaslow-pacedreturntonormalitymaybringaboutasharpandunwelcomeincreaseinthepricelevel.Asabonus,andtoinsertLiquidityDeflationatthecoreofreceivedmonetarytheory,IexaminetheOptimalQuantityofMoney,OQM,issueoriginallyraisedinFriedman(1969,Chapter1).Ishowthat,unlikeFriedman'sOQM,LiquidityDeflationdoesnotcallforreachingliquiditysatiationandthatthenewOQMislocateddangerouslyclosetothenewLiquidityTrapconceptdevelopedinthispaper.2InthispaperIwillindistinguishablyspeakofsafe,highlyliquidassetsandmoney.Thefocusisonassetsthatareessentialfortradeandfinancialtransactions,includingassetsthatareemployedforcreditcollateral.Theircompositionisnotcentralforthepresentdiscussion,andwillbeignored.

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IntheclosingremarksIpresentashortandtentativeextensionoftheseideastoEmergingMarketeconomies,EMs,inwhichdomesticassetscouldhardlybeclassifiedassafe.Arguably,however,thesearchforyieldtriggeredbylow-interest-rateofsafeassetsmayhaveturnedsomeEMassetssafer(perhapsduetohigherturnover),helpingtoexplainthenoticeablerecentdownwardtrendinEMinflation.

II.LIQUIDITYDEFLATION

a.IntuitiveScenarios

1) Consideranatomisticeconomyinwhichcarryingcashtothemallsavesshoppingtime(seeVégh1989).However,timesaveddeclinesascashheldbytheothermallcustomersgoesup:acongestioneffect.Therefore,fromthepointofviewofeachatomisticagent,thetime-savingimpactofcashholdingsincreaseslessthaninproportiontocashheld:LiquidityDeflation.

2) Alternatively,andmoreinlinewithpopularnarrativesoftheGreat

Recession(seeGorton2010),considerthecaseinwhichahighlyliquidasset(e.g.,USTreasurybond)isusedascreditcollateral.ThecollateralvalueofthosebondsdependsontheamountofgoodsandservicesthattheUSgovernmentcouldseizeby,say,raisingemergencytaxes.Therefore,ifthelatterhasanupperbound,thevalueofTreasurybondsascollateralmayincreaselessthaninproportiontotheincreaseinthe(real)marketvalueofTreasurybonds:anotherexampleofLiquidityDeflation.

RemarkI.Theaboveexamplesassumethatliquidgovernmentliabilitiesaresafe.However,thisassumptionishighlydebatableforeconomies,likeJapanandtheUS,thatexhibitlargefiscaldeficitsanddebt-to-GDPratios(especially,ifunfundedsocialsecuritybenefitsaretakenintoaccount).Keynes(1936)offersanalternativeexplanation,whichIlabeledthePriceTheoryofMoney,PTM,inCalvo(2016b).3Theconjectureisbasedontheobservationthatstickypricesprovideanunintendedoutputbackingtomoney,evenifthepublicsectoroffersnone.IfindthePTMmoreappealingthanexplanationsthatrelyontheabilityorpredispositionofthepublicsectortoprovideabackstoptomoneysupplyintermsofgoodsandservices.Furthermore,ifrealmonetarybalancesare‘small’relativetothevalueoftransactionssubjecttostickyprices,thePTMmayensurethatmoneyofferssafeliquidity,e.g.,freefrom'runs.'4However,safetymaydeteriorateasthestockofreal

3Toquotethemaster(emphasismine):"[...]thefactthatcontractsarefixed,andwagesareusuallysomewhatstableintermsofmoney,unquestionablyplaysalargepartinattractingtomoneysohighaliquidity-premium"Keynes1936,Chapter17.4Thishelpstoexplainwhycurrenciesthatareemployedforinvoicingandunitsofaccountacrossworldeconomy,liketheUSdollar,becomestrongerintermsofoutputduringaLiquidityCrunchepisode.

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monetarybalancesbecomeslarge,forthesimplereasonthat,realistically,theflowofrealtransactionssubjecttostickypriceshasanupperbound.b.ParsimoniousMicrofoundedFlexiblePricesModelSupposeastandardinfinite-horizonopen-economymodelinwhichinstantutilityindexsatisfies:

𝑢 𝑐 − 𝑙, (1)

wherecstandsforconsumption,uistwice-continuouslydifferentiableoverthepositiverealline,𝑢′ > 0,𝑢" < 0,andlislaborrequiredtoconsumec,e.g.,shoppingtime(seeVégh1989).Iassumethatshoppingtimeincreaseswithconsumptionanddeclineswiththeholdingsofrealmonetarybalances.Thelatterprovidetransactionsservicesandthussaveonshoppingtime.However,theeffectivenessofmoneytoprovidethoseservicesdeclinesasmarketholdingsofrealmonetarybalancesincrease,inlinewiththeaboveexamples.InexampleII.a.1theeffectcanbeinterpretedas"congestion,"whileinexampleII.a.2itcanbeinterpretedasadeclineinmoney'scollateralvalue.5Iassumethat

𝑙 = 𝑐 − 𝑉 𝑚 + 𝑍 𝑚! , 𝑉! > 0,𝑉′′ < 0,𝑍′ < 0,𝑍′′ ≤ 0, (2)

whereVandZare,respectively,thetimesavingandLiquidityDeflationfunctions;botharetwice-continuouslydifferentiableoverthepositiverealline,andmand𝑚! are,respectively,therepresentativeindividual'sholdingofrealmonetarybalancesandmarketequilibriumrealmonetarybalances(individualsareatomisticandtotalpopulationisnormalizedto1).Thus,inequilibrium,𝑚 = 𝑚! .Individualscansingle-handedlydeterminem,but,sincetheyareatomistic,areconstrainedtotake𝑚! asgiven.Forsimplicity,Iwillassumeanopeneconomy,rationalexpectations(=perfectforesightbecausethereisnouncertainty),notradebarriersandperfectcapitalmobility.Therepresentativeindividualhasaconstantendowment,y,of(perishable)exportablesthatarenotconsumedathome.Ontheotherhand,consumptioncisentirelycomposedof(perishable)importables.Therelativeinternationalpricebetweenimportablesandexportablesisconstantandequalto

5Itisworthnotingthat,strictlyspeaking,thePriceTheoryofMoneydoesnotapplyifpricesareperfectlyflexible.However,Iassumepriceflexibilityheretoshowthat,grantedLiquidityDeflation,pricestickinessisnotindispensableforexistenceofLiquidityTrap.Extensionstostickypricesarestraightforward,but,atthisjuncture,wouldcloudtheanalysiswithsuperfluousdetails.Foranattemptinthatdirection,seeCalvo(2018).

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unity.Moreover,theinternationalrealinterestrateispositive,equalstherepresentativeindividual'ssubjectiverateofdiscount,𝜌,and(importantlyforourdiscussionhere)exchangerateandpricesareperfectlyflexible.Thus,thebudgetconstraintoftherepresentativeindividualintermsoftradablegoodssatisfies:

[𝑦 + 𝑠! − 𝑐! − 𝑖!𝑚!]𝑒!!"𝑑𝑡 ≥ 0,!! (3)

where,withoutlossofgenerality,initialwealthissetequaltozero,andiandsstand,respectively,forthemarketinstantaneousnominalinterestrateandgovernment’slump-sumsubsidiestorebateseignioragefrommoneycreation.Weabstractfromothergovernmentactivitiesand,therefore,assumethatothertaxesandexpendituresaresetequaltozero.Thegovernmentsetsnominalmoneysupplyateachmomentoftime.6Pluggingequation(2)intoequation(1),maximizingutility(= [𝑢 𝑐! − 𝑙!]𝑒!!"𝑑𝑡])

!!

withrespecttocandmsubjecttobudgetconstraint(3),andfocusingoninteriorsolutions,i.e.,𝑐 > 0, 𝑙 > 0,weget:

𝑢′ 𝑐! − 1 = 𝜆, (4)

and

𝑉′ 𝑚! + 𝑍 𝑚!! = 𝜆𝑖! , (5)

where𝜆standsfortheLagrangemultiplier,whichisconstantovertimeanddeterminedbyequalitybetweenpresentdiscountedvalueofendowmentandconsumption.Alloftheseassumptionsarestandardinmonetarymodelsaimedathighlightingfundamentalmonetaryphenomena—inthepresentcaseLiquidityDeflation.Toruleoutconsumptionsatiation,Iwillassume𝑢! 𝑦 > 1.Thus,by(4)and(5),wegetequilibriumconsumption𝑐 = 𝑦,and(droppingtimesub-indexes):

!! !!! !!

!! ! !!= 𝑖, (6)

Thus,byequation(6),andrecallingthat𝑉′ > 0,thereexistsafunction𝐿(𝑖,𝑦),suchthat

6Asiswellknown(Olivera1970,SargentandWallace1975),whenpricesareperfectlyflexibleandfiscalconstraintsarenotbinding,asinthepresentmodel,settingidoesnotanchorthepricelevel.However,belowIwillextendthemodeltothecaseofinterest-bearing"money,"wherethelattercouldbeidentifiedwiththepolicyinterestrate,asinCalvoandVégh(1995).

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𝑚 = 𝐿 𝑖,𝑦 − 𝑍 𝑚! , 𝐿! < 0, 𝐿! > 0, (7)

Equation(7)isafamiliarexpressionforequilibriuminthemoneymarket,exceptfortheterm𝑍(𝑚!).Clearly,by(7),recallingthat𝑍′ < 0,thedemandformoneyincreaseswiththeequilibriumrealstockofmoney,aconventional"network"effect(see,e.g.,Uribe1987).Thus,althoughtheZfunctionishereassociatedwithanovelformofexternality(i.e.,liquiditydeflation),theeffectswouldbesimilarifZisinterpretedasamorefamiliarnetworkexternality.Atequilibriuminthemoneymarketwehave𝑚 = 𝑚! ,implying,by(7),

𝑚 + 𝑍 𝑚 = 𝐿 𝑖,𝑦 , 𝐿! < 0, 𝐿! > 0. (8)Therefore,recallingexpression(2),transactionsservices,i.e.,𝑚 + 𝑍 𝑚 ,mayincreasewithrealmonetarybalancesdespitetheLiquidityDeflationeffect,butthemodeldoesnotprecludethepossibilitythat,beyondacertainpoint,transactionsservicesdecline.Condition(8)isdepictedinFigure1,where𝑚∗isassumedtomaximize𝑚 + 𝑍(𝑚)withrespectto𝑚(hence𝑍′ 𝑚∗ = −1).Theslopeoftheequilibriumtransaction-servicesequilibriumconditionisdownwardslopingwithrespecttothenominalinterestratetotheleftof𝑚∗,asinconventionalmodelsthatignoretheZcomponent.However,theslopebecomespositivetotherightof𝑚∗.Noticethat𝑚∗isassociatedwith𝑖∗inFigure1.Moreover,if𝑖 > 𝑖∗,theremaybetwovaluesofrealmonetarybalancesthatareconsistentwithfullequilibrium.ThisisillustratedinFigure1by𝑖 = 𝑖',atwhichpointtheequilibriumrealmonetarybalancescansettleat𝑚1or𝑚2.Thus,givennominalmoneysupply,thepricelevelmaybeundetermined.Moreover,IwillshowinAppendixAthatthereexistsacontinuumofequilibriumpathsconvergingtom2,eveniftherateofexpansionofmoneysupplyisconstrainedtobeconstantovertime.7ThisshowsthatLiquidityDeflationcanbringaboutequilibriumindeterminacy,amajorcomplicationforthedesignofmonetarypolicy.Despitethis,however,fullcapacityutilizationisstillreachable.However,Iwillshownextthatfullcapacityutilizationmaynotbereachableif𝑖 < 𝑖∗,duetotheexistenceofanewtypeofLiquidityTrap.Considerthecaseinwhichthenominalinterestrateequals𝑖" < 𝑖∗(seeFigure1).Byequation(8)andFigure1,𝑚∗ + 𝑍 𝑚∗ < 𝐿(𝑖",𝑦).Hence,𝑖"isincompatiblewithgeneralequilibrium.Iffullcapacityutilizationholds,i.e.,𝑐 = 𝑦,forinstance,themoneymarketwouldexhibitexcessdemand,asituationthatisnotresolvedbyafallinthepricelevelorincreaseinmoneysupply(given𝑖 = 𝑖"),becauseLiquidityDeflationpreventstherelevantliquidityconceptfromrising.Thus,ifforsomereasoniisstuckat𝑖",theeconomywoulddisplaycharacteristicsakintoaKeynesianLiquidityTrap,exceptthatthelatterwouldbegeneratedbyliquiditysupply

7ThemathinvolvedhereisverysimilartothatintheOptimumSeigniorageproblem.

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phenomena.IwillcorrespondinglycallitSupplySideLiquidityTrap,SSLT.8NextIwilldiscusssituationsinwhichitmaybedifficultfortheeconomytogetridofSSLT,andevencasesinwhichequilibriumdisplayssomekindofcapacityunderutilization.PriortotheGreatRecession,adominantviewwasthatdeflationandLiquidityTrapproblemscouldbecuredbyincreasingmoneysupplyandraisinginflationexpectationsinacrediblemanner(see,e.g.,Krugman1998).Inthepresentmodelthiswouldcorrespondtobeingabletoraisingthenominalinterestratesuchthat𝑖 ≥ 𝑖∗.True,LiquidityDeflationcanstillcauseequilibriummultiplicitybut,asnoted,fullcapacityutilizationisnotbeyondreach.Howtogetthere?Byequation(6),insteadystateequilibriumwehave𝑖 = 𝜌 + 𝜇.Hence,allittakestomovetheeconomyoutofSSLTistoset𝜇suchthat𝜌 + 𝜇 ≥ 𝑖∗.Inwords,allittakesishighenoughinflationexpectations,whichcouldbeachievedbyjackinguptherateofexpansionofmoneysupplyinacrediblemanner(asproposedinKrugman1998).Unfortunately,asIarguenext,LiquidityDeflationcouldmaketheKrugmanproposalhardtoaccomplish.Tosimplifythediscussion,considerthecaseinwhich𝑚 + 𝐿 𝑚 = 𝑚∗ + 𝑍 𝑚∗ ,forall𝑚 ≥ 𝑚∗.9ThissituationisdepictedinFigure2,wherethesolidcurveisnowflatforall𝑚 ≥ 𝑚∗,andm2-typeequilibriumisruledout(seegraphicalproofinAppendixB).NoticethatthesolidcurvelookslikethetypicaltextbookKeynesianLiquidityTrap,which,afteracriticalpoint(𝑚∗inthepresentcase),becomesinfinitelyelasticwithrespecttotheinterestrate.Supposethattheincreasein𝜇failstochangeinflationexpectations.Hence,givenrationality,atsteadystatetherepresentativeindividualshouldexpectinflationtobeconstantovertime,andsuchthat𝜋 = 𝑖"− 𝜌.Thus,if𝜇 > 𝑖"− 𝜌,thenmwillincreasewithoutbound,butitwouldfailtochangetherelevantliquidityconcept(whichwillremainconstantat𝑚∗ + 𝑍 𝑚∗ ).Excessdemandformoneywouldcontinuetoprevail,asituationthatmayplausiblyleadagentstoexpectthat,ifanything,inflationwillfall,makingmoneymoreattractive—asituationthatwouldlookasifaLiquidityDeflation"blackhole"isswallowingtheentireexpansionofmoneysupply!Iwillshownextthatbymodifyingtheequilibriumconceptinasimplebutnotimplausiblemanner,themodelcouldexhibitcapacityunderutilizationorinefficientallocationofresources,withoutviolatingtransversalityconditionsand,thus,qualifyingasarationalexpectationsequilibrium.IwillassumethatunderSSLTandexcessmoneydemand,individualsdiverttheirattentiontofinding/developingalternativeliquidassets(e.g.,crypto-currencies,searchingforyield,etc.)indetrimentoftheirendowments,𝑦.Moreover,thesenegativeeffectsonoutputdonotstopuntileffectiveendowmentshrinksandrestoresequilibriuminthemoney

8SeeFigure1,wherethesetofnon-negativeinterestratesassociatedwithSSLTareindicated.9Thisholdsif𝑍′ 𝑚 = −1,forall𝑚 ≥ 𝑚∗.

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market.10Thus,iftheeconomyisstuckat𝑖 = 𝑖", theequilibriumeffectiveendowment,denotedbyy',mustsatisfy:

𝑚 + 𝑍 𝑚 = 𝐿 𝑖",𝑦′ . (9)Clearly,y'<yand𝑚 ≤ 𝑚∗.Thehigheris𝑚,thehigherwillbe𝑦′.Considerthecase,depictedinFigure2,inwhich𝑚 = 𝑚∗and,therefore,𝑦′attainsitsmaximumvaluesubjectto𝑖 = 𝑖" but still 𝑦! < 𝑦 . 11 Thisexcesscapacityequilibriumisnotaffectedbychangesin𝜇.Tobesure,if𝜇 > 𝜋,mwillgrowwithoutboundbut,aspointedoutbefore,thisdoesnotviolateanytransversalityconditionbecausethestockofrelevantliquiditywillremainconstantat𝑚∗ + 𝑍(𝑚∗).Sinceoutputstaysconstantandmincreaseswithoutbound,velocityfallsovertime.12ThisisinlinewithwhathappenedduringtheGreatRecession,asthevelocityofcirculationexhibitedasteepdeclineinreserve-currencyeconomies.Admittedly,thepresentmodelisstillquiteunsophisticatedandshouldgreatlybenefitfromplausibledynamicextensions,butIwouldbesurprisedifitscentralresultscouldbeeasilyoverturned.13RemarkII.Itisworthkeepinginmindthatiftherateofexpansionofmoneysupply,𝜇,ishighenough,thereexistsafullcapacityutilizationequilibrium.Thus,recallingFigure1,itwouldbecorrecttosayinthiscontextthatfullcapacityutilizationcouldbeachievedbycrediblymanipulatingmoneysupplysothat(insteadystate)𝑖 ≥ 𝑖∗.However,thediscussionalsorevealedthatif,forsomereason,theeconomyisstuckat𝑖 < 𝑖∗,itmaynotbepossibletoachievefullcapacityutilizationbylargeinfusionsofmoneysupply,iftherepresentativeindividualexpectslowinflationconsistentwith𝑖 < 𝑖∗.Inthatsituation,therewillbeexcessdemandformoneycombinedwithexcesssupplyoffullcapacityutilizationoutput,𝑦—whichthemodelassumeswillleadagentstodiverttheirattentiontounproductiveactivitiesaimedatincreasingthesupplyofliquidassets.Thisisaveryinterestingimplicationofthemodel,becauseitoffersanewrationaleforwhyitmightbehardtohaveagentsbelievethatinflationwillbehighenoughandgeneratefullcapacityutilization.Krugman(1998)madeimportantstridesinthatdirectionbyclaimingthattheprobleminJapanwasthat,forsomereason,thegovernment

10Thisisakintodemand-determinedoutputassumptioninKeynesianmodels,althoughsupplyfactorsareatwork!11InthestandardIS/LMapparatus,theadjustmentmechanismisparalleltotheoneoutlinedabove,butrelyingonstickypricesanddemand-determinedoutput.SeeCalvo(2018)forananalysisofLiquidityDeflationintermsofaNewKeynesianmodelwiththosecharacteristics.12ItisworthnotingthatthesametypeofSSLTequilibriumwouldholdifthecurveinFigure1becomesflatatsome𝑚 > 𝑚∗or,inotherwords,ifthereexistssome𝑚 = 𝑚∗∗ > 𝑚∗,suchthattheliquidityofm,i.e.,𝑚 + 𝑍(𝑚),becomesconstantforall𝑚 > 𝑚∗∗.13ForaNewKeynesianMacroeconomicsmodelinwhichLiquidityDeflationbringsaboutafallinthevelocityofcirculation,seeCalvo(2018).

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wasperceivedbyagentsasbeingstronglyinflationaverse,whichconspiredagainstcredibilityandhenceeffectivenessofaninflationarystrategy.Thepresentmodeloffersanotherreason:ifagents'inflationexpectationsare"toolow,"thecentralbankwillbeunabletojackupinflationbecausetheeconomywillbestuckataSSLTwheremonetarypolicyisineffective.Itisnotthegovernment'sinflationaversionthatisatstakehere,itistheexcessivelylowinflationexpectationsoftherepresentativeindividual—expectationsthatturnouttoberational—givingrisetoexcesssupplyoffullcapacityoutputandaninsatiabledemandforliquidity!However,recallingFigure1,themodelimpliesthatfullcapacityutilizationcouldbereachedifthegovernmentcouplesmonetarypolicywithregulationsandarrangementsthatensurethatinflationissuchthat𝑖 = 𝜌 + 𝜋 > 𝑖∗or,equivalently,that𝜋 > 𝑖∗ − 𝜌.Thisinvolvesheterodoxproceduresthatmanypolicymakerstry,quiteunderstandably,toavoid.Butwhenalltheorthodoxprocedureshavefailed,priceguidancelooksincreasinglyattractive.Besides,controllingjustahandfulofprices,e.g.,thoseofpublicsectorservicesandsomelargeprivateenterprises,maybeenough.Thishasbeencommonpracticeineconomiesfacingtheoppositeproblem:stoppingchronicinflation.14!RemarkIII.SSLTshowsthepossibilityofpersistentexcessdemandforliquidityatfullcapacityutilization,whichinthepresentmodelinducesunproductivesearchforalternativeformsofliquidity.Isuspectthathererestsawealthofnewinsightsthatcouldbefoundby,ononeend,studyingempiricallyplausibleendogenousmechanismsofprivate-sectorliquiditycreation—and,ontheotherend,modelingrealisticpolicymakers'responsefunctionsinLiquidityTrapcrisisepisodes.Astothelatter,itisnotunusualforpolicymakersto"fightthelastwar,"andtakemeasures(i.e.,reactionfunctions)thatarecounterproductiveintheshortrun.TheGreatRecession,forinstance,hasinducedtighterbankingandfinancialregulationsthatsomeobserversseeashavingexacerbatedliquidityshortage.Moreover,theveryexistenceofrationalexpectationsSSLTequilibriummayleadthecentralbanktoprematurelystoppursuingQEanddebilitatingtheeffectivenessofQEifresumed.15!IguessthatKeyneswouldhavebeenpleasedbytheseresults.HespentmuchofChapter19ofhisGeneralTheorydiscussingseveralinstancesinwhichevendownwardflexiblewageswouldfailtoachievefullcapacityutilization.However,heutilizesargumentsthatareabsentintheprevioussectionsofthebook.Thisisunfortunatebecausethelackoffoundationsleavesthisimportantconjecturehangingintheair.Itisprobablythereasonwhypost-Keynesianmodelsrelyexclusivelyonpricestickinessassumptions.Incontrast,LiquidityDeflationisa14SeeBrunoetal1991.15Recently,theECBandtheBOJappeartohavegivenupfightingdeflationbymeansaggressiveQEandswitchedinsteadtolowinterestrates.OpinioninthefinancialpresswasunanimousinconcludingthatQEhadrunoutofsteam,anadditionalfactormilitatingagainsttheeffectivenessofQE.

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naturalcorollarytothePriceTheoryofMoneyputforwardinChapter17(recallfootnote3,andRemarkI).Although,aspointedoutinfootnote6,interesttargetingresultsinpricelevelindetermination,themodelcouldbeextendedtothecaseinwhichthecentralbankpaysinterestonm,andmisidentifiedwithanaggregateofhighlyliquidassets.Denotetheinterestrateonmbyim.16Itiseasytoshowthatthemoney-marketequilibriumcondition(7)wouldbecome:

𝑚 + 𝑍 𝑚 = 𝐿 𝑖 − 𝑖!,𝑦 , 𝐿!!!! < 0, 𝐿! > 0. (10)Clearly,lowering𝑖!reducesthedemandforliquidityservicesandpushestheeconomytowardsfullcapacityutilization.However,thispolicymaynotbefeasibleorgiverisetoproblemsofitsown.Forinstance,𝑖! < 0mayleadtodevisingcrypto-currencieswithoutaneffectiveLenderofLastResortthateventuallygeneratesevereliquiditycrises.17TheOptimalQuantityofMoney:ADigression.Friedman(1969)showsthatabstractingfromtaxdistortions,andassumingthatthemarginalnominalcostofissuingnon-interest-bearingmoneyisnil,theoptimumquantityofmoney—i.e.,realmonetarybalancesthatmaximizesocialwelfare—mustbesuchthattheopportunitycostofholdingmoney,i.e.,themarketnominalinterestrate,isequaltozero,i.e.,𝑖 = 0intheabovenotation.IwillshowthatthisdoesnotnecessarilyholdsunderLiquidityDeflation(ornetworkexternality).Iwillfocusonsteadystate.Byequations(1)and(2),thesocialplannermaximizes:

𝑢 𝑐 − 𝑐 + 𝑉(𝑚 + 𝑍 𝑚 ) (11)

withrespectto𝑐 ≤ 𝑦,and𝑚 ≥ 0.Since𝑢! 𝑦 > 1,optimalconsumption𝑐 = 𝑦.Thefirst-orderconditionwithrespecttomsatisfies:

𝑉! 𝑚 + 𝑍 𝑚 1+ 𝑍′(𝑚)] = 0. (12)

InabsenceofLiquidityDeflation,theOQMrequiresmoneysatiation,i.e.,𝑉! = 0,whichholds,recallingequation(6),ifi=0.ThisisFriedman’sresult.18Noticethat

16Forarelatedmodel,seeCalvoandVégh(1995).Variable𝑖!couldalsobeinterpretedascentralbank'sinterestonbankreserves.17Itisworthnotingthatanegative𝑖!isequivalenttoaninflationtax.ThelatterhasfosteredCurrencySubstitution,particularlyineconomiessufferingfromChronicInflation.18ItisworthnotingthatFriedman'sresultisnotwelldefinedundertheassumptioninequation(2)thatrulesoutliquiditysatiation.

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sinceatsteadystate𝑖 = 𝜌 + 𝜋,inthisstaticmodeltheOQMcallsfordeflation.Moreprecisely,𝜋 = −𝜌.ConsidernowthecaseinwhichLiquidityDeflationholds.Thus,sincebyexpression(2),𝑉! > 0everywhere,itfollowsthattheOQMcallsforsetting1+ 𝑍! 𝑚 = 0,which,recallingFigure1,holdsat𝑚 = 𝑚∗.Thus,byequation(6),decentralizingtheOQMcallsforsettingthenominalinterestratesuchthat(recallingFigure1)𝑖 = 𝑖∗,where

!! !∗!! !∗

!! ! !!= 𝑖∗ = 𝜌 + 𝜋 > 0. (13)

Hence,if𝑖∗ ≥ 𝜌,theOQMwouldcallforinflation,notdeflationasinFriedman'sOQM.Moreover,ifmoneyyieldsarateofreturn𝑖!,equation(13)becomes:

!! !∗!! !∗

!! ! !!= 𝑖∗ = 𝑖 − 𝑖! = 𝜌 + 𝜋 − 𝑖!, (14)

implyingthattheOQMisachievedif

𝜋 = 𝑖∗ − 𝜌 + 𝑖!. (15)Thus,theinflationrateisfreetotakeanyvalueaslongasitsatisfiesequation(15).However,aspointedoutabove,startingfromaSSLTmaymakeithardforthecentralbanktoimplementtheOQM.Besides,theOQM𝑚 = 𝑚∗seatsattheborderlinetotheSSLTregion(seeFig.1),toocloseforcomfort!!Thusfar,ourdiscussionhasfocusedonthelimitcaseinwhichLiquidityDeflationcompletelyemasculatestheimpactofQE(QuantitativeEasing).However,LiquidityDeflationproblemsmaystillberelevantnearthelimit,whereQEisstillcapableofincreasingliquiditysupply.Forthesakeofconcreteness,letusgobacktothecaseofnon-interest-bearingmandrewriteequation(7)as

𝑚 + 𝛽𝑍 𝑚 = 𝐿(𝑖,𝑦), 𝛽 ≥ 0, (16)andconsidertheeffectofaslightdropinparameter𝛽.If𝑍 𝑚 < 0,thelatteramountstoloweringthedragimpliedbyLiquidityDeflation,andcanbeinterpretedasashifttowards"normality."Since,byassumption,QEstillworks,beingnearthepointwhereQEwouldbeineffective,impliesthatthederivativeof𝑚 + 𝛽𝑍 𝑚 withrespecttomispositivebut≈ 0.Hence,recallingthat𝑍 𝑚 < 0,onecaneasilyshowthatafallinparameter𝛽impliesa"large"contractioninequilibriumrealmonetarybalancesm.Thus,forinstance,ifnominalmoneysupplywereexogenous,afallinparameter𝛽wouldbringabouta"large"increaseinthepricelevel,given𝑖 − 𝑖!andoutput,whichmaybringaboutasuddenunwelcomesurgeofinflation.

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III.CLOSINGREMARKSHavingreachedthispoint,thereadermayfeelthat,ifoneisreadytoassumeLiquidityDeflation,theaboveresultsaretrivial(inthepejorativesenseoftheword).Ibegtodisagree.Resultsareindeedtrivial—asithappensinanylogicallyconsistenttheory—butimportant.Firstly,becauseLiquidityDeflationunveilsanewtypeofLiquidityTrap(SSLT)fromwhichitismuchhardertoescapecomparedtothetextbookcase,andhelpstorationalizesomeofthekeyproblemsfacedbypolicymakersintherecoveryfromtheGreatRecession.Secondly,becauseitbringsuptothesurfaceresultsthatclashwiththepre-crisisconventionalwisdom.Forinstance,ifyouaskyour"representative"economist(especiallypriortheGreatRecession):"Whathappensifmoneysupplydisplaysalargeincreaseinashortperiodoftime?"theanswerwilllikelybesomethinglike"priceswilltakeabigjump."Insymbols,abigincreaseinMwillresultinabigincreaseinP.Incontrast,theabovediscussionfocusesonthepossibilitythatabigincreaseinMwillprovokeabigfallinthe"quality"ofM—hereidentifiedasabigfallinthe"liquidityofM."Thus,inasituationlikethat,amarginalincreaseinMmayhavenoeffectonPorinthe"realvalueofM/P,adjustedforliquidityservices."Thirdly,becauseLiquidityDeflation—asimpleadd-ontostandardmodels—mighthelptodevelopa"NewSynthesis"fromwhichmorerealisticandcomplexmodelswillgrow(seeCalvo2018).Inturn,theoppositestrategy,i.e.,developingmodelswithavarietyofidiosyncraticshocksandassumptions,mayactuallyleavemacrotheoryinlimbo,unabletoreachprofessionalconsensus,andthusmakingitoflittleuseforpolicymaking.TheSSLTphenomenonisespeciallyrelevantforreservecurrenciesineconomiesthathaveundergoneasevereboutofLiquidityCrunch.Until2008,thiswasnotthecaseforEMsbecauseLiquidityCrunchofEMassets,droveagentsawayfromdomesticallydenominatedassets—akindofAnti-LiquidityTrap—causingasharpspikeinexchangeratesandtriggeringhighinflation(seeCalvo2018b).ThereasonforthisisthatadeteriorationofEMassets'liquiditydidnothaveanegativeimpactontheliquidityofglobalsafeassets.ThisincentivizedportfolioshiftagainstEMassets.However,thisstartedtochangewiththeLehmancrisisandensuinglongperiodoflowinterestratesinAdvancedMarketeconomies,AMs.A'searchforyield'followed,increasingturnoverofEMassets,conceivablytransformingtheminclosersubstitutesforAMsafeassets.Asaresult,monetaryconditionsinsomeEMsstartedtolooklikethoseinAMs.Israelisaninterestingcaseinpoint.Inthe1980sIsraelwasstrugglingtogetridofchronicinflation,occasionallyreachingstaggeringlevels(e.g.,over350percentin1985).Atpresent,however,Israelisfacingtheoppositeproblem:deflation!Inflationisbelowthecentralbanks'inflationtargetbyawidemargin,andisperilouslyteeteringaroundzero.Thiscould,ofcourse,beduetoprudentfiscalandmonetarypolicy.Butevenso,theexternalconditionsmayhavecontributedtoincreasingtheliquidityofIsraelliabilities,puttingdownwardpressureoninflationrates,possiblythrowingtheeconomyintoaSSLT.

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Inclosing,itisworthreiteratingthattheSSLTphenomenonisassociatedwithsituationsinwhichitishardforpolicymakerstoincreasethestockofliquidassetstolevelscompatiblewithfullcapacityutilization.ThephenomenonislikelytoloseitsrelevanceaseconomiesrecoverandfindnewSafeAssets.Incapitalisteconomiesthisislikelytooccurbythehandoftheprivatesector.Once'normality'isrecovered,theoldghostofinflationislikelytorevive,andtakeourattentionawayfromLiquidityTrap,asithappenedafterthe1930s.Thiscouldbeaseriousmistakeifthegenieisoutofthebottleandthefinancialsectorcontinuesgeneratingnewformsofgloballiquidassets,notprotectedbyaLenderofLastResort.Thus,tokeeptheseissuesinfocus,myhumbleproposalisthattextbooksshouldaddtheLiquidityDeflationphenomenonontheirfavoritemodels.19

19NoticethatFigureB1inAppendixBmakestheSSLTeasytoincorporateinanIS-LMmodel.

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15

AppendixA

Theanalysisinthemaintextfocusedonsteady-stateequilibria.Iwillsketchoutanextensiontothegeneralcase,butrulingoutexplosiveequilibriumsolutions.Iwillconstraintheattentiontonon-interest-bearingmoney.Droppingtimesubindexes,andrecallingthatnominalmoneystockisassumedtogrowataconstantrate𝜇,wehave

!!= 𝜇 − 𝜋. (A1)

Moreover,sincethisisaneconomysubjecttoperfectcapitalmobility,notradebarriers,andinternationalrelativepricebetweenimportablesandexportablesisconstantovertime,itfollowsthatthenominalinterestratesatisfiesFisher'sequation,i.e.,

𝑖 = 𝜌 + 𝜋. (A2)

Hence,byequations(6)and(A2),atequilibriumwehave

−𝜋 = 𝜌 − !! !!! !!

!! ! !!. (A3)

Thisisarepresentative-individualeconomyinwhich𝑚 = 𝑚! .Thus,plugging(A3)inequation(A1),wehave

!!= 𝜇 + 𝜌 − !! !!! !

!! ! !!. (A4)

Therefore,atsteadystate(where𝑚 = 0),wehave

!!!"

= −𝑚 !"(!!!(!))[!!!!(!)] !! ! !!

. (A5)

Letusexaminethesteadystatesm1andm2inFigure1.Sincethecurveisdownwardslopingatm1,itfollowsthat𝑉"(𝑚 + 𝑍(𝑚))[1+ 𝑍′(𝑚)] < 0.Therefore,𝑑𝑚 𝑑𝑚 > 0,andm1isunstable.Butthesameprocedureleadstotheconclusionthatm2isstableandgivesrisetoacontinuumofequilibriumpaths.LiquidityDeflationisbehindthisindeterminacy,eventhoughtheseequilibriumpathsdonotdisplayLiquidityTrap.Uniquenesswouldbeensured,however,if,forinstance,𝑖 > 𝑖∗,and𝑉 𝑚 + 𝑍 𝑚 = 𝑉(𝑚∗ + 𝑍 𝑚∗ ),forall𝑚 ≥ 𝑚∗,asassumedinFigure2.

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AppendixB

FigureB1.GraphicalDerivationofFigure2

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