美联储-贷款价值约束的非线性效应 Nonlinear Effects of Loan-to-Value Constraints

FinanceandEconomicsDiscussionSeries

FederalReserveBoard,Washington,D.C.

ISSN1936-2854(Print)

ISSN2767-3898(Online)

NonlinearEfectsofLoan-to-ValueConstraints

C.BoraDurduandSergioVillalvazo

2024-081

Pleasecitethispaperas:

Durdu,C.Bora,andSergioVillalvazo(2024).“NonlinearEfectsofLoan-to-ValueCon-straints,”FinanceandEconomicsDiscussionSeries2024-081.Washington:BoardofGov-ernorsoftheFederalReserveSystem,

/10.17016/FEDS.2024.081

.

NOTE:StafworkingpapersintheFinanceandEconomicsDiscussionSeries(FEDS)arepreliminarymaterialscirculatedtostimulatediscussionandcriticalcomment.TheanalysisandconclusionssetfortharethoseoftheauthorsanddonotindicateconcurrencebyothermembersoftheresearchstafortheBoardofGovernors.ReferencesinpublicationstotheFinanceandEconomicsDiscussionSeries(otherthanacknowledgement)shouldbeclearedwiththeauthor(s)toprotectthetentativecharacterofthesepapers.

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NonlinearEffectsofLoan-to-ValueConstraints*

C.BoraDurduSergioVillalvazo

FederalReserveBoard

September2024

Abstract

Thispaperinvestigatestheimpactofloan-to-value(LtV)borrowingconstraintsinmodelswithoccasionallybindingcreditconstraints.TheseconstraintsgiverisetoaFisheriandebt-deflationmechanism,whereexogenousshockscantriggercascadingeffectsresultinginsignificantdeclinesinconsumption,assetprices,andborrowingreversals—characteristicoffinancialcrises.However,recentliteraturechallengestra-ditionalviewbysuggestingthatcollateralconstraintsmaynotalwaysexacerbatefi-nancialdisturbancesbutcouldinsteadfosterdynamicsleadingtomultipleequilibria.Buildingonthisdiscussion,thepaperexploresequilibriumassetpricingmodelswithLtVcollateralconstraints,identifyingcriticalthresholdsthatgovernassetpricedy-namics,consumptionpatterns,andcurrentaccountbehaviors.OuranalysisuncoversthatwhentheLtVlimitisclosetozero,tighterconstraintsinducesmallerdropsinconsumptionduringcrises.Conversely,whentheLtVlimitisclosetoone,weob-servethattighterconstraintsinducelargerdropsinconsumptionduringcrises.ThenonlinearrelationshipbetweentheLtVratioandadverseeffectsonmacroeconomicoutcomesalignswithcross-countryevidenceregardingtherelationshipbetweentheleveloffinancialdevelopmentandtheseverityofconsumptiondeclinesduringcrises.

J.E.L.classificationcodes:E31,E37,E52,F41,G01.

Keywords:Financialcrises,Loan-to-valueconstraints,Debt-deflation.

*WethankLucaGuerrieri,GastonNavarro,AlexVardoulakisandSarahZoiforhelpfuldiscussionsandcomments,andJulianWangforexcellentresearchassistance.Wearealsogratefulforcommentsbyconfer-enceandseminarparticipantsatthe2024MEAandITAMAlumniMeetings.TheviewsexpressedinthispaperarethoseoftheauthorsandshouldnotbeattributedtotheBoardofGovernorsoftheFederalReserveSystemoritsstaff.

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1Introduction

Inthewakeofthe1990s’suddenstopsexperiencedbynumerousemergingeconomiesandtheeconomicturmoilofthe2007-2008GlobalFinancialCrisis,thequestformodelscapableofelucidatingthedynamicsofthesecriseshasintensified.PioneeredbytheworkofMendoza(2010),oneprominentavenuehasbeentheexplorationofmodelsfeaturingoccasionallybindingcreditconstraints,bothfornormativeandpositiveanalysis.Centraltothesemodelsistheincorporationofloan-to-value(LtV)typeborrowingconstraints,whichcapborrowingatafractionofthemarketvalueofassetholdings.

Thesemodels,enrichedbythepresenceofoccasionallybindingconstraints,unveilaFisheriandebt-deflationmechanism.AsshownbySchmitt-GrohéandUribe(2017),invul-nerableeconomicstates,exogenousshockscantriggerthetighteningoftheseconstraints,precipitatingacascadeofeffects.Asassetpricesplummetinresponse,theconstrainttight-ensfurther,exacerbatingdeflationarypressures.Thisfeedbackloopoftenmanifestsinsharpdeclinesinconsumption,assetprices,andborrowingreversals—hallmarksoffinan-cialcrises.

However,recentresearch,epitomizedbytheworkofSchmitt-GrohéandUribe(2021),challengesconventionalwisdomregardingtheimplicationsofcollateralconstraints.Con-trarytopriorintuition,theseauthorsarguethattheeffectsofcollateralconstraintsmaynotalwaysamplifyfinancialdisturbances.Rather,theypositthatsuchconstraintscouldengenderdynamicsseeminglyatoddswithearlierliterature,includingtheemergenceofmultipleequilibria.

Inparticular,Schmitt-GrohéandUribe(2021)highlightthenonlinearityofadjustmentineconomiessubjecttocollateralconstraints.Thesedynamicsgiverisetotwodistinctequilibria:onecharacterizedbyself-fulfillingfinancialcrisesdrivenbypessimisticviewsoncollateralvalues,andanothermarkedbyunderborrowingandexcessiveprecautionarysavingsasawaytoself-insure.

Thispapercontributestothisevolvingliteraturebyrigorouslycharacterizingtheana-lyticalsolutionofequilibriumassetpricesinsmallopeneconomiesfeaturingloan-to-valuecollateralconstraints.Specifically,weinvestigatetheramificationsofvaryingcollateralfractionsonkeymacroeconomicandfinancialvariables.OuranalysisuncoversthatwhentheLtVlimitisclosetozero,tighterconstraintsinducesmallerdropsinconsumptionduringcrises.Conversely,whentheLtVlimitisclosetoone,weobservethattightercon-straintsinducelargerdropsinconsumptionduringcrises.

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ThenonlinearimpactofLtVlimitsisinfluencedbyageneralequilibriumeffectcausedbyassetprices.WhentheLtVlimitrises,twoconflictingoutcomesoccur.Firstly,in-creasedindebtednessmakestheeconomymorefragile,leadingtoagreaterdeclineincon-sumption.Secondly,amorerelaxedcollateralconstraintlessenstheimpactonassetprices.IneconomieswithlowLtV,thefirsteffectismoresignificant,whereasineconomieswithhighLtV,thesecondeffectprevails.

Toassesstheempiricalvalidityofourtheoreticalfindings,weexaminethecross-countryevidenceonhowtheleveloffinancialdevelopment,whichproxiesthemodel’sLtVlimits,iscorrelatedwiththeseverityofconsumptiondeclinesduringafinancialcrisis.Todothis,weconstructapaneldatabaseofSuddenStopepisodesfromBianchiandMendoza(2020),afinancialdevelopmentindexdevelopedbySvirydzenka(2016)andmacroeco-nomicaggregatesfromTheWorldBank(2024).Predictivequadraticregressionsvalidateourtheoreticalfindings.Inparticular,wefindaU-shapedrelationshipbetweenconsump-tiongrowthandfinancialdevelopmentduringcrises.

Ourpaperisrelatedtoseveralstrandsoftheliterature.Theinvestigationoffinancialcrises,particularlywithinthecontextofemergingeconomies,hasbeensignificantlyad-vancedbymodelsincorporatingoccasionallybindingcreditconstraints.Thesemodels,pi-oneeredbyMendoza(2010),haveprovidedessentialinsightsintothemechanismsdrivingsuddenstopsandeconomicdownturns.Centraltothisapproachistheloan-to-value(LtV)constraint,whichlimitsborrowingbasedonthemarketvalueofassets.Schmitt-GrohéandUribe(2017)furtherdevelopedthesemodelsbyexaminingstockcollateralconstraints,revealinghowexogenousshockscantightentheseconstraints,leadingtoaFisheriandebt-deflationcyclethatexacerbatesfinancialinstability.

Theliteraturehasalsoexploredthepossibilityofmultipleequilibriaarisingfromtheseconstraints.Schmitt-GrohéandUribe(2021)analyzedflowcollateralconstraints,demon-stratingthatthesemodelscanresultinnonlineardynamicsandthepotentialformulti-pleequilibria.Similarly,JeanneandKorinek(2019)discussedtheheuristicimplicationsofstockcollateralconstraints,suggestingthatundercertainconditions,theseconstraintscouldleadtoself-fulfillingfinancialcrisesorexcessiveprecautionarysavingsbehavior.

Inadditiontothetheoreticalcontributions,variousstudieshaveemployedthesemodelstoexplorethebroaderimplicationsofcollateralconstraints.Forinstance,BianchiandMendoza(2018)andJeanneandKorinek(2019)appliedsimilarmodelstoinvestigatetheroleofpecuniaryexternalitiesinfinancialcrises,whileotherscholarssuchasLorenzoni(2008),Mendoza(2010),Bianchi(2011),andBenigno,Chen,Otrok,Rebucci,andYoung

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(2013,2016),examinedhowtheseconstraintsinfluenceoverborrowingandthedesignofoptimalmacroprudentialpolicies.Devereux,Young,andYu(2019)alsocontributedtothisdiscussionbyexploringthecapitalflowdynamicsunderdifferentregulatoryframeworks.

Moreover,therelationshipbetweenfinancialdevelopmentandeconomicoutcomeshasbeenextensivelystudied.Levine(1997),LoayzaandRanciere(2006),andBordoandMeissner(2015)exploredhowfinancialdevelopmentaffectseconomicgrowth,businesscycles,andtheseverityoffinancialcrises.Thesestudiesprovideempiricalsupportforthetheoreticalmodels,suggestingthattheleveloffinancialdevelopmentcansignificantlyinfluencetheimpactofcollateralconstraintsonmacroeconomicstability.

Theremainderofthepaperisorganizedasfollows.Section2describesthesuddenstopmodelwithperfectforesightandcharacterizesitsanalyticalsolution.InSection3weextendthemodeltoanenvironmentwithuncertainty.Wesolvethemodelnumericallyandshowthatourresultsaredrivenbyageneralequilibriumeffectthroughtheasset’sprice.Section4describesthemodelvalidationexerciseandshowsthatourtheorypredictionsareinlinewiththedata.Finally,Section5concludes.

2ModelwithPerfectForesight

ThemodelissimilartoBianchiandMendoza(2018)andJeanneandKorinek(2019).

Thesmallopeneconomyispopulatedbyacontinuumofhomogeneoushouseholds.TherepresentativehouseholdmaximizesaCRRAutilityfunction,u(c)andonlyderivesutilityfromconsumption,c.Thehouseholdbuysnextperiodinternationalbondholdings,b,,thatpayanexogenousinterestrateRandalsobuysnextperioddomesticassetholdings,a,,withanendogenouspriceq.Thedomesticassetnetsupplyisfixedandnormalizedto1.Totalendowmentoutputintheeconomyisgivenbyy,whoseshareoflaborincomeisgivenby(1-α)andsharetocapital(dividends)incomebyα.Finally,thehousehold’screditisconstrainedbyastandardloan-to-valuecollateralconstraintinwhichdebtcannotexceedaconstantfractionhofthemarketvalueofnextperiodassetholdings.1Theproblemofthehouseholdinrecursiveformisgivenby:

1Thetimingofassetsusedascollateralontheright-handsideoftheconstraintreliesonassumptionsaboutthecharacteristicsofcreditcontractsandtheirenforcement.Bothtimingapproacheshavebeenwidelyadopted.FormoredetailsseeBianchiandMendoza(2020)

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V(a,b)=maxu(c)+βV(aI,bI)s.t.(1)

{c,b,,a,}

c+R-1bI+qaI=(1-α)y+a(αy+q)+bR-1bI≥-κqaI.

Letλ>0andµ≥0bethemultipliersonthebudgetconstraintandcollateralconstraint,respectively,anddefine=µ/λ≥0.Then,assumingthemarketclearingcondition,a=aI=1,theequilibriumconditionsare:

uc(c)=Rβuc(cI)+µ(2)

(3)

0=µ(R-1bI+κq)(

4)

c=y+b-R-1bI.(

5)

Whereuc(c)correspondstothefirstderivativeoftheutilityfunctionwithrespecttocon-sumption.

Nowassumeb>-RκqsuchthattheeconomyisnotconstrainedandRβ=1,toanalyzeastationaryequilibriuminwhichconsumptionisconstant:c=cI.FromEq.2-5weobtain:

c=cI=y+(1-β)b(

6)

µ==0(

7)

(8)

TheequilibriumcharacterizedbyEq.6-8isawellknownresult.Thehouseholdwillkeepaconstantconsumptionequaltothetotalendowment,y,plusorminustheinterestpaidorearneddependingonthesignoftheinitialbondposition,b.Thecurrentaccountisconstantandzero,CAt=bI-b=0,thecollateralconstraintisnotbindingbyassumptionandtheequilibriumexhibitsthestandardforwardlookingassetprice.

Nowassumethatthehouseholdismarginallyconstraint,i.e.,thedebtisatitshighestlevelandthecollateralconstraintmultiplieriszero:b=-Rκqandµ=0.Thenthe

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equilibriumallocationsbecome:

b=b,=-(9)

c=y(1-hα).(10)

2.1AnalyticalResults

Theanalyticalresultsinthissectionprovideaclearandconcisederivationoftheequi-libriumallocationsandassetpricesinasimplifiedmodelsetupwithalogarithmicutilityfunction(thecoefficientofriskaversionisequalto1).Byassuminganunexpectedwealth-neutralshocktotheeconomyandasequentialframework(Eq.11),wecanexplorehowtheLtVlimitinfluencestheoutcomesofthemodelunderdifferentscenarios.Theclosed-formsolutionsderivedherehelptouncoverkeyinsightsintothedynamicsofassetprices,con-sumption,anddebt,particularlywhentheeconomyfacesabindingcollateralconstraint.

yfort≤-1

Thesolutionofthemodelstartsbyconsideringtwoscenariosbasedontherelationshipbetweentheequilibriumassetpriceinperiod0andperiod1.Inperiodt=0,theendow-mentlevelintheeconomyislowerthaninthepreviousperiodandfromt=1onward,theendowmentislargersuchthatthenetpresentvaluestaysconstant.2Itisstraightforwardtoseethat,duetotheshock,thehouseholdwouldliketosmoothherconsumptionbyin-creasingherdebtinperiod0.However,notethatinthepreviousperiod,thehouseholdwasholdingthemaximumpossibledebt.Therefore,therearetwopossiblescenarios.

Scenario1:Iftheequilibriumpriceinperiod0~islessthanorequaltotheuncon-strainedequilibriumpriceinperiod1(q0≤q1=-0issupportedinanunconstrainedperiod1and,hence,thehouseholdcanmaintainacon-

2Notethatintheabsenceofthecollateralconstraint,suchwealthneutralshockwouldnothaveanyeffectonconsumption.Theeconomywouldbeabletoperfectlysmoothconsumptionbyadjustingthecurrentaccount.Duringperiod0,theeconomywouldborrowfromabroadandfromperiod1onward,theeconomywouldpaytheinterestsofsuchborrowingwiththeadditionalendowment.

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stantconsumptionfromperiod1onward.Theequilibriumallocationsaresuchthatthehouseholdconsumesasmuchaspossibleinperiod0sincesheisatthedebtlimitand,fromperiod1onward,thehouseholdconsumesaconstantlevel.Notethatq-1=hencethiscaseallowsforsomeassetpriceinflationinperiod0.Wheneverthereisassetpriceinflation,thecurrentaccountwillbepro-cyclical.

Scenario2:Iftheequilibrium~priceinperiod0isgreaterthantheunconstrainedequi-libriumpriceinperiod1(q0>-0isnotsupportedinanunconstrainedperiod1andhencethehouseholdcannotmaintainaconstantconsump-tionfromperiod1onward.Theequilibriumallocationsaresuchthatthehouseholdgetsasmuchdebtaspossibleinperiod0aslongasc-1≥c0and,fromperiod1onward,thehouseholdstartsadeleveragingprocess.Inthisprocess,thehouseholdholdsthemaximumamountofdebtpossibleineachperiodandslowlyincreasesherconsumptionuntilshereachesthesteadystateinwhichtheconstraintismarginallybindingandholdsaconstantconsumptionlevel.Notethatinthiscase,theeffectsoftheshockarepersistent,thecurrentaccountispro-c~yclical,theassetpriceincreasesinperiod0andslowlydecreasesuntilitconvergesto-.

NowweturntocharacterizingforwhichLtVlimitsweareinthefirstscenario.Let

z(h)=∈(0,1].Notethatz(·)isanimplicitfunctionofhsincetheequilibrium

consumptionchoicesdependonitand,z(·)≤1sincec0≤c1,becauseifnotthenthehouseholdcouldsaveinperiod0tosmoothherconsumption.FromEq.3wegetthefollowingrelation:

q0≤q1,

,

z(h)(1—h)≤1—h.

Thelastinequalityissatisfiedonlyforall0≤h≤1.Hence,for0≤h≤1weareinscenario1andforh>1weareinscenario2.

Assumption1.LettheLtVlimitbesuchthat0≤h≤1.

UnderAssumption1,thefollowingequationssummarizetheequilibriumallocations

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andassetprices:

q-1=(12)

c-1=y(1-κα)(13)

b0=-(14)

b1=-Rκq0=bτforτ≥2(16)

c1=qτforτ≥2.

Notethattheonlyunknownisthecurrentpriceq0.CombiningEq.2-3andsubtituingEq.15-18,weobtainaquadraticpolynomial:

,

,

,

,

0=Aq+Bq0+C.

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Where:

Whichcanbesolvedusingthegeneralquadraticformulasolution:

and(19)

Equation19istheanalyticalsolutionfortheassetpriceontheperiodwhentheunex-pectedwealthneutralshockhitstheeconomy.Inthenextsubsectionweusethisanalyticalsolutiontocharacterizetheparameterregionswheretheequilibriumisunique,multipleornon-existent.

2.1.1CharacterizationoftheEquilibriumAssetPriceandCases

Inthissubsection,weobtainthegeneralcasesforuniqueness,multiplicityornon-existenceoftheequilibrium.

Assumption2.Thedropintheendowmentisnottoolarge:√>α.

Assumption2isplausiblesince√correspondsto1minusthesizeoftheshocksoitisexpectedtobecloseto1andαcorrespondstothecapitalincomesharewhichisexpected.Nowweobtainthefollowingcases:

Cases:

>>

>:q>0,q>0butonlyq>0isconsistentwithc0≤c-1.

Multiple(pair){A>0,C>0andB2-4AC>0thenq>0,q>0.

Non-existence{B2-4AC<0.

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Insummary,0≤h<min{1,(1-β)}guaranteesauniqueequilibrium.Intuitively,morevolatileeconomies(lower√)andeconomieswithhighercapitalreturns(higherα)requirealowerLtVlimittoguaranteeauniqueequilibrium.ThisresultisimportantsinceitsaysthatauniqueequilibriumisguaranteedtoexistwhentheLtVlimitisclosetozero.

Havingaclosedformsolutionfortheequilibriumassetpriceallowsustogetthefol-lowingresults.

Proposition1.Iftheeconomyisunderauniqueequilibrium(0≤h<(1-β)≤1),thentheDebt-to-GDPratioislessthantheshocksize:<√.

Toobtainthisresult,weusethebondexpressioninEq.9dividedbytheendowmentandnotethatunderauniqueequilibrium,thenα<(1-β).CombiningthetwoexpressionsdeliversProposition1.

Proposition2.Whendebtisnotpossible(h=0)then:

•==√.

•CrisisamplificationordampeningfromincreasesintheLtVfromzerodependonthecapitalsharefortheassetpriceandunambiguouscrisisamplificationfordropinconsumption:

Thefirstpartofthisresultisobtainedbysubstitutingh=0inEq.12-15and19.ThesecondpartisobtainedafterdifferentiatingEq.19withrespecttohandevaluatingitath=0.

Proposition3.WhentheLtVlimitisatthemaximumandtheequilibriumisunique(h=1andα=(1-β)√)then:

•>1.

•CrisisamplificationfromdecreasesintheLtVfromoneinboththeassetpriceandconsumption:

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TheproofofthisresultisanalogoustohowweobtainedProposition2evaluatingthederivativesath=1andα=(1-β)√.

Proposition4.Whenα=(1-β)√andhencetheequilibriumisuniqueoverallh∈ ,thenpricedeflationhappenswhenh<andinflationwhenhisabovesuchthreshold.

Thisresultisobtainedbysubstitutingα=(1-β)√andcomparingq-1withq0fromEq.19,thensinceq0isincreasinginh(Proposition2)wesolvefortheh=1toobtainthethreshold.

Thederivedpropositionsfurtherelucidatetheimplicationsofthemodel.Forinstance,thefindingthatthedebt-to-GDPratioisconstrainedbytheshocksizeunderauniqueequilibriumunderscorestheimportanceofmanagingleverageintheeconomy.Moreover,theanalysisshowsthattheeffectsofincreasingtheLtVlimitarenuanced:whileitcandampencrisesinsomecases,itmayamplifytheminothers,dependingonthecapitalshareandcurrentLtVlevel.Thisresultemphasizesthetrade-offsinherentinfinancialregulationandtheneedforcarefulcalibrationofLtVlimitstobalanceeconomicgrowthandfinancialstability.

Overall,thissectiondemonstratesthepowerofanalyticalmodelinginuncoveringthecomplexinterplaybetweencollateralconstraints,assetprices,andmacroeconomicout-comes.Theinsightsgainedfromtheclosed-formsolutionsprovideasolidfoundationforunderstandingthebroaderdynamicsexploredinthemorecomplex,stochasticversionofthemodel.

2.2NumericalExercise

Tofurtherdissectthemechanicsofourmodel,weanalyzenumericallyaversionofthemodelwithoutuncertainty,allowingustoisolatethecoredynamicsatplay.Thisapproachhelpsinunderstandingthefundamentalbehaviorofthemodelunderdifferentlevelsoftheloan-to-value(LtV)limit,h,particularlyfocusingontheexistenceandnatureofequilibria.

TheparametersusedforthenumericalexerciseareshowninTable1.Thesearestan-dardvaluesintheliterature.

Theresultspresentedinthefollowingfigureshighlightthreedistinctscenarios:caseswherethemodelyieldsauniqueequilibrium,multipleequilibria,ornoequilibriumatall.Thesescenariosarecriticalforunderstandingthepotentialstabilityorinstabilityofthe

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Table1:ParameterValues

Parameter

Value

β,(R=β-1)

discountfactor

0.94

y

totalendowment

1.0

γ

dropendowment

0.95

α

capitalshare

∈{(1-β)λ=0.057;0.15;0.20}

κ

LtVlimit

∈[0;1]

economyundervaryingfinancialconstraints.

UniqueEquilibriumCase

Figure1specificallyillustratesthecasewherethecapitalshareα,isequalto(1-β)γ,themaximumvaluethatguaranteesauniqueequilibriumacrossallLtVlimitsrangingfrom0to1.ThisparticularsettingallowsustoobservehowtheeconomyrespondswhentheLtVlimitisvariedinacontrolled,predictableenvironment,freefromthecomplexitiesintroducedbyuncertainty.

Panela)ofFigure1demonstratesamonotonicdampeninginthedeclineofassetpricesastheLtVlimitincreases.ThisbehavioralignswiththeintuitionthathigherLtVlimits,bylooseningborrowingconstraints,mitigatethedownwardpressureonassetprices.Asthecollateralconstraintbecomeslessbinding,themarketisabletoabsorbshocksmoreefficiently,preventingasharpdropinassetvalues.

Panelb)showsaU-shapedresponseintheconsumptiondropastheLtVlimitincreases.Thisnon-monotonicpatternisparticularlynoteworthyasitreflectsthedelicatebalancebe-tweenincreasedborrowingcapacityandheightenedvulnerability.AtlowerLtVlimits,theeconomyislessleveraged,andtheimpactofshocksonconsumptionisrelativelycon-tained.However,astheLtVlimitincreases,theincreasedleveragemakestheeconomymoresusceptibletoshocks,leadingtoamorepronounceddeclineinconsumption.Be-yondacertainpoint,though,thelooseningofconstraintsallowsforgreaterconsumptionsmoothing,hencetheU-shapedpattern.

Panelc)presentsthecorrespondingchangeinthecurrentaccounttoGDPratio,whichexhibitsamirrorimageoftheU-shapedconsumptionresponse.Thisinverserelationshiphighlightsthetrade-offsfacedbytheeconomyasitadjuststoshocksunderdifferentLtVregimes.WhentheLtVlimitislow,thelimitedborrowingcapacityleadstosmallerswingsinthecurrentaccount.AstheLtVincreases,theeconomyfinancesconsumptionbybor-

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rowingmoreandwhentheeconomybecomesoverlyleveraged,theneedtocorrectthisimbalanceinthefaceofadverseshocksresultsinasharpreversal,hencethemirroredU-shape.

Thesefindingsfromtheuniqueequilibriumcasehighlightthecriticalrolethatcollateralconstraintsplayinshapingtheeconomy’sresponsetoshocks.Theexistenceofauniqueequilibriuminthesettinghereindicatesapredictableandstableeconomicenvironmentwherepolicyinterventionscanbemoreeffectivelytailored.TheU-shapedconsumptionresponseandthemirroredcurrentaccountbehaviorfurtheremphasizethenonlineareffectsofvaryingLtVlimits,suggestingthatwhileincreasingtheLtVlimitcanoffershort-termbenefitsintermsofconsumptionsmoothing,italsointroduceslonger-termrisksassociatedwithhigherleverage.

1

0

-1

-2

-3

-4

-5

-6

0

-1

-2

-3

-4

-5

-6

-7

00.10.20.30.40.50.60.70.80.91

(a)AssetPrice

00.10.20.30.40.50.60.70.80.9

(b)Consumption

2

1.5

1

0.5

0

-0.5

00.10.20.30.40.50.60.70.80.9

(c)CurrentAccounttoGDPratio

Figure1:Percentchangesduringcrisisforlowcapitalshare(α)

Wenowexplorethebehaviorofthemodelwhenthecapitalshare(α)issettomediumandhighlevels,specifically0.15and0.20,asillustratedinFigures2and3,respectively.Thesescenariosallowustoexaminehowincreasingthecapitalshareinfluencesthestabil-ityoftheeconomy,particularlyintermsoftheexistenceandmultiplicityofequilibria.

MultipleEquilibria

InFigure2,whereαissetto0.15,themodelbeginstoexhibitmorecomplexdynam-icscomparedtotheuniqueequilibriumcase.Atthismediumcapitalsharelevel,certainvaluesoftheLtVlimitleadtomultipleequilibria.Thesolidbluelinescorrespondtotheequilibriumobtainedwiththe“high”assetpricewhilethedashedredlinescorrespondtotheequilibriumwiththe“low”assetpricefromEq.19.Notethatthedynamicsofthe“high”assetpricearesimilartothedynamicsunderauniqueequilibriumandaremorestablerelativetothe“low”assetprice,whichgeneratestotalassetpricecollapsesforsome

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LtVlimits.

Thisoutcomereflectstheincreasedroleofcapitalintheproductionprocess,whichamplifiesthefeedbackeffectsbetweenassetpricesandcollateralconstraints.Astheca