IMF-兩主體經(jīng)濟(jì)中的突然停止與最優(yōu)政策（英）

SuddenStopsandOptimalPolicyinaTwo-agentEconomyiljanovskaandAlexandrosPVardoulakisWP/147IMFWorkingPapersdescriberesearchinprogressbytheauthor(s)andarepublishedtoelicitcommentsandtoencouragedebate.TheviewsexpressedinIMFWorkingPapersarethoseoftheauthor(s)anddonotnecessarilyrepresenttheviewsoftheIMF,itsExecutiveBoard,orIMFmanagement.2022?2022InternationalMonetaryFundWP/22/147IMFWorkingPaperResearchDepartmentSuddenStopsandOptimalPolicyinaTwo-agentEconomyPreparedbyNinaBiljanovskaandAlexandrosP.Vardoulakis*AuthorizedfordistributionbyMariaSoledadMartinezPeriaJuly2022IMFWorkingPapersdescriberesearchinprogressbytheauthor(s)andarepublishedtoelicitcommentsandtoencouragedebate.TheviewsexpressedinIMFWorkingPapersarethoseoftheauthor(s)anddonotnecessarilyrepresenttheviewsoftheIMF,itsExecutiveBoard,orIMFmanagement.ABSTRACT:WeintroduceheterogeneityintermsofworkersandentrepreneursinanotherwisestandardFisherianmodeltostudySuddenStopdynamicsandoptimalpolicy.Weshowthatthedistinctionbetweenworkersandentrepreneursintroducesadistributiveexternalitythatisabsentfromtherepresentative-agentsetup.Whileintranquiltimesredistributionisdrivenbytherelativemarginalutilitiesofconsumption,theplanneradditionallyfavorsentrepreneursduringSuddenStopstomitigateFisheriandeation.Althoughagent-heterogeneitydoesnotaddmuchinexplainingtheSuddenStopphenomena,itaddstotheunderstandingofhowpoliciescanbestbedesignedtoalleviatethenegativeeffectsofSuddenStops.JELClassificationNumbers:E2,E44,G1Keywords:SuddenStops;agentheterogeneity;macroprudentialpolicy;payrolltaxpolicyAuthor’sE-MailAddress:NBiljanovska@andalexandros.vardoulakis@*WethankMiguelFaria-e-Castro,GitaGopinath,VeronicaGuerrieri,MariaSoledadMartinezPeria,andseminarparticipantsattheIMFandFederalReserveBoardforusefulcomments.TheviewsexpressedinthispaperarethoseoftheauthorsanddonotnecessarilyrepresentthoseoftheIMF,theFederalReserveBoard,oranyoneintheFederalReserveSystem.11Introductionmarketeconomiesandareassociatedwithbigdropsinexternal?nancing,asmeasuredbyareversalinthecurrentaccount,aswellasacollapseinassetpricesandbigdropsinconsumptionandrealeconomicactivity.AvastandactiveliteraturehasbeenstudyingthequantitativeimplicationsofSuddenStopsbyintroducingFisheriande?ationdynamicsintocanonicalsmallopeneconomymacroeconomicmodels(seeMendoza,2010,andMendoza,2020forarecentsurvey).TheFisherianapproachfeaturesoccasionallybindingcollateralconstraintsthatrestricttheabilitytoborrowuptoafractionofpledgedcollateral,thevalueofwhichdependsonmarketprices.ThesefeaturescancaptureSuddenStopepisodesastheygeneratequantitativelybigampli?cationwhentheconstraintbinds,buttheprobabilityofsucheventsoccurringissmallasinthedata.Importantly,thepresenceofmarketdeterminedpricesinthecollateralconstraintintroducesapecuniaryexternalityasprivateagentsdonotinternalizehowtheiractionsmatterfortheincidenceofbindingconstraints,whichintroducesaroleformacroprudentialpolicy(seeBianchi,2011;BianchiandMendoza,2018;andtheliteraturereviewedbelow).OurpaperbuildsonthesameFisheriande?ationapproachtocaptureSuddenStops,butattemptstocontributetotheliteraturebyintroducingagentheterogeneity.1Inparticular,weintroduceheterogeneitybetweenentrepreneursandworkersintoanotherwisecanonicalmodelofBianchiandMendoza(2018),wherebytherepresentativeagentmakesallproduction,investment,andborrowingdecisionsaswellassupplieslabor.Inourpaper,wesplitthetwofunctionsbyhavingentrepreneursmakedecisionaboutproduction,investment,andborrowing,whileworkersarethesuppliersoflaborforwhichtheyreceiveawagefromentrepreneurs.ThemainnoveltyofourpaperistoshowthatthisdistinctionhasimportantimplicationsforoptimalpolicyasitintroducesamotiveforredistributionofresourcesbetweenthetwoagentsthatmeaningfullyinteractswithFisheriande?ationdynamicsduringSuddenStops.Weillustratethisresultanalyticallybysolvingforthesocialplanner’sproblem,identifyingtheexternalitiesthatjustifyapolicyintervention,andderivingtheoptimaltaxesthatdecentralizeit.Moreover,wesolvethemodelquantitativelyusingglobalsolutionmethodsandshowthee?ectofheterogeneitybycomparingtheoptimalpolicybetweenthetwo-agenteconomyanditsrepresentative-agentcounterpart.1Mostoftheliteratureassumesarepresentative-agenteconomy.MendozaandQuadrini(2010)andVillalvazo(2022)aretwoexceptionsthat,nevertheless,considerdi?erenttypesofheterogeneitythaninourpaperand,moreimportantly,donotstudytheimplicationsofagentheterogeneityforoptimalpolicy.Wediscusshowourpaperdi?ersinmoredetailfromtheirswhenwereviewtheliteraturebelow.3muchinexplainingtheaggregatemomentsunderlyingtheSuddenStopphenomenon;atleastforthetypeofheterogeneityweconsiderinthispaper.However,weshowthateventhisminimaldegreeofheterogeneityaddsnon-triviallytoourunderstandingofhowpoliciescanbestbede-signedtoalleviatethenegativee?ectsofSuddenStops.Ouranalyticalnormativeresultsshouldbegeneralenoughtoholdunderricherformsofheterogeneity,wherethereisaredistributivemotive.Asintherepresentative-agenteconomy,theplannerwouldlevyapositiveborrowingtaxingoodtimesinthetwo-agenteconomyinordertoleanagainstcreditgrowthandmitigatetheseverityofFisheriande?ationdynamicsduringSuddenStops.Similarly,theplannerusesthepayrolltaxinbotheconomiestoaddressthepecuniaryexternalityoperatingthoughwageswhentheconstraintbinds.But,importantly,theplannercanusethepayrolltaxinthetwo-agenteconomytotacklethedistributiveexternalitydescribedabove.Duringnormaltimes,thepayrolltaxisusedtoimplicitlytransferresourcestotheagentwithhighermarginalutilityfacilitatingconsumptionsmoothingacrossagents.DuringSuddenStops,thepayrolltaxisalsogearedtowardshelpingtoaddressFisheriande?ationbytransferringmoreresourcesto(lessresourcesawayfrom)entrepreneurs.Naturally,thesignofthepayrolltaxdependsonthesocialwelfareweightsassignedtothetwoagentsandthemodelcanquantitativelyyieldpositiveornegativepayrolltaxes.But,irrespectiveofthecalibration,thefactthatthepayrolltaxincreasesduringSuddenStopepisodestohelpentrepreneursisaverygeneralresultthathighlightstheusefulnessofthispolicytomitigatethenegativee?ectsofSuddenStopsexpost.Importantly,usingthepayrolltaxexposttohelpentrepreneursdoesnotremovetheneedfora(macroprudential)borrowingtaxexante,astheanticipatedredistributioncanweakentheprecautionarymotiveofentrepreneursandurgethemtoborrowmoreinnormaltimes.Hence,thecombinationofborrowingandpayrolltaxesresultsinsomewhatbiggerdecreaseintheseverityofSuddenStopscomparedtothedecentralizedrepresentative-agenteconomy.Ouranalysishasimplicationsforthedebateonprovidingbailoutsorsubsidiesto?nanciallyconstrainedagents(or?rms)duringcrises,echoedalsointheanalysisinBianchi(2016).Weshowthatthereisamotivetosupport?nanciallyconstrainedagents,whichhasbeenempiri-callydocumentedinLaevenandValencia(2013).Wealsoshowthatthismotiveisevenmorepronouncedwhentheconstrainedagentsaretheholdersofcollateralintheeconomy.Suchpol-icyaimstoavertadestructive?re-sales—aconclusioninlinewiththe?ndingsinShleiferandVishny(2011)andcentralinFisherianmodelsofSuddenStops.Itistruethatwemodeltheex-tremecase,whereworkersdonotpurchaseanycapital.However,weshowthatthedistributive4externalitywouldstillbepresent,yetattenuated,evenifweallowedworkerstoholdcapital,oralternatively,ifweintroducedadditionalheterogeneityamongentrepreneurssuchthatsomearemoreandsomearelessconstrained(as,forexample,inVillalvazo,2022).Tofurtherhighlightthee?ectofheterogeneityonoptimalpolicy,weextendthebaselinemodeltoincorporatean(anticipated)laborsupplyshock.Ourmotivationistostudyashockthatwouldoperatesimilarlytootheradverseshocksintherepresentative-agenteconomy;butcanhavedi?erentimplicationsinthetwo-agenteconomybecauseita?ectsworkersdirectlyandentrepreneursindirectlythroughproductionandgeneralequilibriumchannels.Assuch,wedonottrytoarguethatlaborsupplyshocksareimportantingredientstounderstandtheSuddenStopphenomenon.ButatthesametimeonecouldthinkofacasethataSuddenStopmaybeaccompaniedbyalossoflaborforceduetoimmigrationtoothercountries,orlowerjobcreationandhigherjobdestructionasshowninGallegoandTessada(2012).SuchshockswouldintuitivelymaketheSuddenStopmoresevereasweshowbysimulatingtheextendedmodel.Yet,weadditionallyshowthatthewaythepayrolltaxadjuststotacklethelaborsupplyshockisasymmetricwithrespecttothetightnessoftheborrowingconstraintinthetwo-agenteconomy.Theplannerimplementsalowerpayrolltaxwhenthelaborsupplyshockhitsandtheconstraintdoesnotbindinordertohelpworkersthataredirectlyhitbytheshock.Bycontrast,whentheconstraintbinds,theplannerimplementsahigherpayrolltaxwhenthelaborshockhitsinordertohelpentrepreneursand,thus,supportassetpricesbysupportingtheirconsumption.ThisresultfurtherspeakstotheimportanceoftheredistributionchannelforSuddenStopsthatwehighlightedabove.Literaturereview—Ourpaperrelatestotheliteraturestudyingoptimalpolicyineconomieswith?nancialfrictionsmorebroadly,andtotheliteraturestudyingoptimalpolicyinresponsetoSuddenStopsmorespeci?cally.OurpapermostcloselyrelatestoBianchiandMendoza(2018),whichderivesoptimalmacroprudentialregulationthatcanbeusedtoalleviatethenegativee?ectsofFisheriande?ation.Asmentionedearlier,weextendtheirmodelbyintroducinghet-erogeneityintermsofworkersandentrepreneurs.Thismodi?cationgivesrisetoadistributiveexternalitythatoperatesthroughwagesandisabsentfromtherepresentative-agentframe-work.Whilethemacroprudentialpolicyrecommendationsremainlargelyunalteredbetweentherepresentative-agentandthetwo-agenteconomy,theoptimalpayrolltaxhasanadditionalroleinthetwo-agenteconomyframeworkasitservestoredistributeresourcesexpostbetweentheAnothercloselyrelatedpaperisBianchi(2016).Themodelinthatpaper,similarlytoours,5featuresapecuniaryexternalityarisingfromthefactthat?rmsdonotinternalizehowtheirlabordemanda?ectsthetightnessofthecollateralconstraint.Thisexternalityisaddressedbyapayrolltax.But,thecrucialdi?erencefromourpaperisthatourmodeladditionallyfeaturesadistributiveexternality,whichinteractsinanimportantwaywiththeFisheriande?ationmechanism,absentinBianchi(2016).Twootherpapersthatconsideragentheterogeneity,albeitdi?erentfromours,areMendozaandQuadrini(2010)andVillalvazo(2022).MendozaandQuadrini(2010)featuressavers,whoown?nancialintermediariesthatlendtoproducerssubjecttoacollateralconstraint.Contrarytoourpaper,thereisnolaborsuppliedbysaversandproductionjustusescapital.Hence,thedistributiveexternalitywedescribeinourpaperinabsentintheanalysis.Villalvazo(2022)con-sidersaBewley-typemodelofSuddenStops,inwhichtheeconomyispopulatedbyacontinuumofhouseholdsthatdi?erintheirincomeendowment.Inhismodel,unconstrainedhouseholdscanbuyassetsfromconstrainedhouseholds,dampeningthee?ectofSuddenStopsonassetprices.Aswehintedabove,thisdampeninge?ectwouldattenuatethedistributivemotivetowardsentrepreneurs.Inourmodel,workersdonotholdcapitaland,thus,supportingentrepreneursconsumptionisalwaysdesirabletomitigatethedropinassetpricesduringSuddenStops.Im-portantly,neitherofthetwoaforementionedpapersderivestheoptimalpolicyinthepresenceofheterogeneityinaFisherianmodel,whichisthefocusofourpaper.Ourpaper,morebroadly,relatestotheliteraturestudyingthedynamicsofSuddenStopeventsandtheoptimalpolicyresponsesdesignedtomitigatethee?ectsofsuchboom-bustcycles.BianchiandMendoza(2020)o?ersanexhaustivesurveyoftheempiricalandtheoreti-calliteratureonthistopic.Manywell-knownempiricalcontributionsincludeEdwards(2004),RothenbergandWarnock(2006),ForbesandWarnock(2012),Calvo,IzquierdoandTalvi(2006),Eichengreen,GuptaandMasetti(2018),whichmainlyfocusonperformingeventanalysisinacross-countrypaneldatasetsusingoneormore?lterstoidentifyandanalyzeSuddenStopepisodes.Thisempiricalworkalsocomplementsarelatedempiricalliteraturedocumentingthedeeprecessionsandpricecorrectionsthatfollowthecollapseofcreditbooms(e.g.,MendozaandTerrones,2012;SchularickandTaylor,2009).Onthetheoreticalside,UribeandSchmitt-Groh`e(2017)providesausefultextbookpresentationofmodelswithFisherianapproachtoSuddenStops.Thesemodelstypicallyfeatureaborrowingconstraint(e.g.AiyagariandGertler,1999;KiyotakiandMoore,1997),whichcanbederivedfromalimitedenforcementproblemorcostlystateveri?cation(e.g.,BianchiandMendoza,2018;MendozaandQuadrini,2010).Di?erentspeci?cationsofborrowingconstraints—eitherintheformofaLoan-to-ValueorDebt-to-Income6ratio—havebeenextensivelyusedinFisherianmodelsofSuddenStops.2Fromthesepapers,thosethataremorecloselyrelatedtooursincludeBianchi(2011),Benigno,Chen,Otrok,RebucciandYoung(2013),BenguiandBianchi(2018),HernandezandMendoza(2017),andMendozaandRojas(2019)giventheirfocusonoptimalpolicyanalysis;butnoneofthesepapersstudiestheroleofpolicyintacklingadistributiveexternalityinnormaltimesandduringSuddenStops. Moregenerally,ourpaperrelatestothetheoreticalliteraturestudyingtheimplicationsofpe-cuniaryexternalitiesthatariseduetothepresenceof?nancialfrictions.SomeexamplesofthesepapersincludeLorenzoni(2008),Stein(2012),andJeanneandKorinek(2020).Thesepapershavemainlyfocusedoncreditpoliciestotackletheexternalities.Wedi?erfromthesepapersbecause,apartfromcreditpolicies,wealsofocusonlaborpoliciestoaddressthedistributiveexternalityweidentify.2ModelEconomyTheeconomycomprisestwotypesofdistinctagents:workersandentrepreneurs.En-trepreneursowna?rmthatutilizeslabor,intermediategoods,andcapitaltoproduceacon-sumptiongood.Theyalsohaveaccesstointernationalcreditmarketswheretheycanborrowpledgingtheircapitalascollateral.Workerssupplylabortothe?rmandonlyconsumeoutoftheirlaborincome.3Weproceedbyoutliningthemodelandde?ningtheoptimalityconditionsforeachagents.SomederivationsarerelegatedtotheAppendixattheendofthepaper,whilewehavealsoincludedadditionalanalyticalandquantitativeresultsinanOnlineAppendix.2.1WorkersTheeconomyispopulatedbyaunitmassofidenticalworkerswhosepreferencesarerepre-sentedbytheutilityfunction2Thislistofpapersincludesmodelswithwithcapitalaccumulationandworking-capital?nancingaddedtothecreditconstraint(Mendoza,2010),modelsofreserveaccumulation(Durduetal.,2009;Arceetal.,2019),macroprudentialpolicy(Bianchi,2011),real-exchange-ratestabilizationpolicies(Benignoetal.,2013),ex-postinterventionwithindustrialpolicy(HernandezandMendoza,2017),self-ful?llingcrises(Schmitt-Groh`eandUribe,2020),imperfectenforcementincapital-?owmanagementpolicies(BenguiandBianchi,2018),modelswithbanksintermediatingcapitalin?ows(MendozaandRojas,2019),andmodelsofexchange-ratepolicywithnominalrigiditiesandcreditfrictions(Ottonello,2021;Coulibaly,2018;FarhiandWerning,2016).3Weoptforhand-to-mouthworkersthatcannotpledgetheirlaborincometoborrowinter-temporallyinordertorepresentasegmentofthepopulationthatdoesnothaveaccesstocreditmarketsandthepossibilitytosmoothconsumptionovertime.Kaplan,ViolanteandWeidner(2014)documentthatone-thirdofallUShouseholdslivehand-to-mouth.Themodelcanalsobeextendedtoallowforworkers’accesstocreditmarketsatthecostofcomplicatingtheanalysisandmakingthederivationofoptimalpolicymoreconvoluted.WeelaboratelaterinSection3.3whattheimplicationsofsuchanextensionwouldbe.7/E0匕βtU(ct_G(ht)),(1)t=0whereE(.)denotestheexpectationsoperatorandβisthesubjectivediscountfactor.TheutilityfunctionU(.)isstandardconcave,twicecontinuouslydi?erentiable,andsatis?estheInadaconditions.Itdependsonconsumption,ct,andlaborsupply,ht,whicharecombinedinacompositecommodityct_G(ht),de?nedbyGreenwood,Hercowitz,andHu?man(1988).ThetermG(ht)isaconvex,strictlyincreasing,andcontinuouslydi?erentiablefunction,measuringthedisutilityoflabor.Thisformofautilityfunctionremovesthewealthe?ectonlaborsupply,whichpreventsacounterfactualincreaseinlaborsupplyduringcrisis.WeconsiderastandardCRRAutilityfunctionU(ct_G(ht))=[(ct_G(ht))(1—生)_1]/(1_σ)withriskaversioncoe?cientσ,whileG(ht)=ψh1+。/(1+φ)with1/φbeingtheFrischelasticityoflaborsupply.Asworkersarehand-to-mouthagents,theirbudgetconstraintisgivenbyctswtht,(2)wherewtdenotesthewagetheyreceiveforthelaborsupplied.Workersmaximizetheutilityfunction(1)subjecttotheirbudgetconstraint(2),whichyieldsthefollowingconsumptionandlaboroptimalityconditions,respectively,ct:uc]t=λ,(3)ht:Gh]t=wt,(4)whereλdenotestheLagrangemultiplierassociatedwithworkers’budgetconstraint(2).Fornotationconvenience,wehavescaledallLagrangemultipliersthroughoutthepaperbyβt.Forexample,themultiplierontheworker’sbudgetconstraintisλ=βtλintheLagrangean.WeadoptthesamenotationforallotherLagrangemultipliers.Xi]tdenotesderivativesofafunctionX(.)withrespecttoavariableiattimet.Thisnotationfordenotingderivativesoffunctionswillbepreservedthroughoutthepaper.2.2EntrepreneursInadditiontoworkers,theeconomyispopulatedbyaunitmassofidenticalentrepreneursgivenby8/E0匕βtU(xt),t=0(5)wherextisconsumption,βisthetimediscountfactor—equaltotheoneofworkers—andU(x)=[x(1—生)_1]/(1_σ)isastandardCRRAutilityfunctionwiththesameriskaversioncoe?cientσtoworkers.Entrepreneursproduceyt=F(zt,kt,lt,vt)eachperiod.F(.)isaCobb-Douglasproductionfunction,whichcombineslabor,lt,withthestockofcapitalpurchasedinthepreviousperiod,kt,andanintermediategood,vt;ztisanaggregateproductivityshock.Aggregatecapitalisinunit?xedsupply:K=1.Theintermediategoodistradedincompetitiveworldmarketsata?xedexogenousprice,pu.Thebudgetconstraintofentrepreneursisgivenbyxt+bt+puvt+wtlt+qtkt+1=yt++qtkt,(6)wherebtdenotesthebeginning-of-periodborrowingusingone-period,non-statecontingentbondsissuedlastperiod,qtisthepriceofcapital,andRistheworld-determinedgrossrealinterestratetakenasgiveninthesmallopeneconomy.Entrepreneurs’consumption,xt,equalsoutputnetoftheoutlaysforthefactorsofproduction,vtandlt,thenetcapitalexpenditure,qt(kt+1_kt),andthenetdebtissuance,bt+1/R_bt.Weassumethatentrepreneurscannotraiseequityandthattheirborrowingdecisionislimitedbyacollateralconstraint,endogenouslyderivedfromarenegotiationproblembetweenborrowersandlenders(seesection(A)intheAppendix).Entrepreneursobtaintwotypesofloans:Aninter-temporal,bt+1/R,andanintra-temporalloan.Theyneedthelatterto?nance,aheadofproduction,aportionθuoftheintermediategoodpurchasesandaportionθlofthelaborexpenses.Hence,thetotalliabilitiesatthebeginningoftheperiodcompriseofbt+1/R+θupuvt+θlwtlt.Whilebt+1isaninter-temporalloanandbearsaninterestpayment,θupuvt+θlwtltdoesnotasitisrepaidwithinthesameperiod(werelaxthisassumptionintheOnlineAppendix).Allborrowedfundscanbediverted,asituationthatisprecludedbyimposingthefollowingcollateralconstraintbt+1R+θupuvt+θlwtltsκtqtkt.(7)Constraint(7)limitsthesizeoftotalborrowingtoafractionκtofthebeginningofperiodholdingsofphysicalcapital.κtfollowsatwo-state,regime-switchingMarkovprocesswithκl<κh,whereκhandκlrepresentlooserandtighterlendingstandards,respectively(seeSection49fordetailsaboutthespeci?cationoftheprocess).Entrepreneursmaximize(5)subjectto(6)and(7).ThismaximizationproblemleadstothefollowingoptimalityconditionsU北]t=λ,(8)Fu]t=pu(1+θuμt),(9)Fl]t=wt(1+θlμt),(10)U北]t(1_μt)=βREtU北]t+1,(11)qtU北]t=βEt[U北]t+1(Fk]t+1+qt+1)+κt+1U北]t+1μt+1qt+1],(12)whereλdenotestheLagrangemultiplieronentrepreneurs’budgetconstraintandU北]tμtdenotestheLagrangemultiplieronthecollateralconstraintscaledbyentrepreneurs’marginalutility.Thepresenceofthecollateralconstraintdistortsboththeoptimalinter-andintra-temporalmarginaldecisionswhenbinding.Conditions(9)and(10),de?ningentrepreneurs’optimalchoiceoftheintermediategoodandlabor,embedanadditionalcost,i.e.thecostofcollateral?nancingequaltoθuμtpuandθlμtwt,respectively.Inaddition,bothEulerequations—withrespecttoborrowingandcapital—aredistorted.TheEulerequationforborrowing(11)impliesthatthemarginalbene?tfromincreasingborrowingtodayisequaltotheexpectedfuturemarginalcostfromrepayingthedebtplusthecostoftighteningthecollateralconstrainttoday,givenbytheshadowvalueμt.Similarly,theEulerequationwithrespecttocapital(12),equatingthemarginalcostofanextraunitofcapitaltodaywithitsfuturemarginalbene?t,embedsanadditionalbene?tobtainedbyrelaxingthefuturecollateralconstraints,valuedatκt+1U北]t+1μt+1qt+1.2.3CompetitiveEquilibriumThecompetitiveequilibriumoftheeconomycanbede?nedasfollows.De?nition1.Forgiveninitialvaluesoftheendogenousstatevariable,b0,andexogenouspro-cesseszttacompetitiveequilibriumforthetwo-agenteconomyisasequenceofallo-cations{ct,xt,vt,ht,lt},anassetpro?le{kt+1,bt+1},andapricesystem{qt,wt,pu},suchthat1.Giventhepricesystem{qt,wt,pu},theallocationsandtheassetpro?lesolveworkers’andentrepreneurs’optimizationproblemsasde?nedinsections2.1and2.2,respectively,2.Labor,asset,andgoodsmarketsclear,satisfyingconditionsht=ltAt,(13)kt=K=1At,(14)ct+xt+bt+puvt=yt+,At,(15)ct=wtlt,At.(16)3OptimalPolicyInthissection,we?rstde?netheoptimizationproblemofaconstrainedsocialplanner,andthendiscussthepropertiesoftheoptimalpoliciesthatimplementtheplanner’sallocations.3.1SocialPlanner’sEconomyWeformulatethesocialplanner’sprobleminasimilarmannertoBianchiandMendoza(2018),whofollowthe“primalapproach”tooptimalpolicyanalysis.Theplannerchoosesallo-cationsinordertomaximizeasocialwelfarefunctionsubjecttoagents’budget,implementability,andcollateralconstraints.Weassumethat,whenchoosingallocations,theplannerlackstheabilitytocommittofuturepolicies.Therefore,itchoosespolicyrulesatanygivenperiodwhiletakingasgiventhepolicyrulesthatrepresentfutureplanners’decisions.Formally,theoptimizationoftheconstrainedsocialplannercanbede?nedasfollows.De?nition2.Theplannermaximizesthein?niteweighted-sumofagents’futurediscountedutilities,!βt[ωU(ct_G(lt))+U(xt)],subjecttoallconditionscomprisingthecompetitiveborrowing,(11),andtheoptimaldecisionwithrespecttoentrepreneurs’labordemand,(10),chthattheseconditionsdonotrepresentbindingconstraintsfortheplanner.ωdenotes