P aper No. 907 Banks, shadow banks, and business c

Proposition1.ForagiveninterestrateR^b,andprovidedthatγQ^˜>ω_minQ,thebalance sheetofbanksislargerwithsecuritization,B^s>B^ns.

2.2ShadowBanks

˜

AmeasureoneofshadowbanksbuymanyclaimsKfrombanksatpriceQandbundle themintoasset-backedsecuritiesS.Thesecuritiesareissuedtoinvestorsinperiodt=1 andpromisetorepayR^sinperiodt=2.Poolingtogetheralargenumberofclaims eliminatesthebank-levelrisk.Letω_meanbetheaveragevalueofωoveralldefaulting

borrowers.Thebalancesheetoftheshadowbankinperiodt=1is

ω_meanQ^˜K(B)=S.

Inperiodt=2,themarketvalueoftheseclaimsisω_meanQK(B).Profitisthen ω_meanQK(B)−R^sω_meanQ^˜K(B).Profitmaximizationyields

Q^{˜. (3)}

5Expectedprofitwhenbehavingoptimallyunderthetwoscenarioswrites

R^s=^Q

□_{R−(1−p)Rb}□_γ¹₋₁ □_{R−(1−p)Rb}□_γ^γ
−1
E[Π^ns]=[(1−p)R^b−R] +pE[ω]Q,

pω_minQpω_minQ

pγQ^˜ pγQ^{˜ .}
E[Πs]=[(1−p)Rb−R]□_{R−(1−p)Rb}□_γ¹₋₁+pQ˜□_{R−(1−p)Rb}□_γ^γ₋₁

6

2.3Investors

AmeasureoneofinvestorsreceiveaperishableendowmentWinperiod1andwishto consumeinbothperiods.Theirutilityis

E[U(C₁)+U(C₂)],

whereCisconsumptionandUisanincreasingandconcavefunction.Thebudget constraintsare

C₁=W−D−S; E[C₂]=RD+E[R^s]S.
OptimizationyieldsE[M]R=E[MR^s],whereM≡U^′(C₂)/U^′(C₁)isthestochastic discountfactor.Definethecovariancetermν≡−Cov[M,R^s]/E[M]andrewritethe

first-orderconditionas

E[R^s]=R+ν,(4)

Weassumethatνisexogenousandrefertoitasthemarketsentimentshock.

2.4Equilibrium

Sofarwehavesaidnothingabouttheeconomy’sultimateborrowers,exceptthatthey defaultwithprobabilityp.Inthenextsection’sextendedmodel,theactionstaken bytheseagentswillresultinendogenousdefaultprobability,loaninterestrate,and collateralvalue.Fornow,wesimplyassumethatthedemandforloansBisadecreasing functionoftheloaninterestrateR^b.

Combiningthefirst-orderconditionsfrombanks(2),shadowbanks(3),andinvestors (4),weget

pγQB^γ−1
R+ν= . (5)

R−(1−p)R^b
Allelsebeingequal,anincreaseinνeitherleadstofallinBoranincreaseinR^b.

Equation(5)bringsustooursecond,andkey,result.

Proposition2.Inequilibrium,adeteriorationinmarketsentiment,thatisariseinν, reducestheamountofsecuritiesSandloansBandincreasestheloaninterestrateR^b.

Theintuitionisasfollows.Lowersentimentmakesinvestorsmoreriskaverse,prompt- ingthemtodemandahigherreturnonriskysecurities.Shadowbankscutbackon securitization,drivingdownthepriceofsecuritizedassetsQ^˜.Tocompensatethisdrop

inincome,banksmusteitherreducetheirbalancesheetbylendinglessorincreasetheir spreadR^b−R,whichinturnreducesthedemandforloans.

7

3ExtensionandEstimation

Havingillustratedhowinvestorsentimentaffectscreditspreadsinaparsimoniousen- vironment,weturntoaquantitativedynamicgeneralequilibriummodeldesignedtofit theEuropeandata.Therearetwotypesofhouseholds,patientandimpatient.Patient householdsaretheeconomy’sultimatesaversandcorrespondtotheinvestorsofthe previoussection.Impatienthouseholdsarenetdebtors:theyobtainloansfrombanksto purchasehousingandconsume.Inthebusinesssector,entrepreneursalsoobtainloans frombankstopurchasephysicalcapital.Thesetwotypesofborrowersaresubjectto agencyproblems,similartoBernanke,Gertler,andGilchrist(1999,hereafterBGG), andafractionofthemdefaultseachperiod.Financialinstitutionsactasthemiddlemen, asbefore.

Weembedthisframeworkintoastandardmodelofbusinesscyclessuchastheone estimatedbySmetsandWouters(2007).Althoughsomefeatureslikepriceindexation andadjustmentcostshavebeencriticizedforlackingsupportingmicroevidence,they helpthemodelmatchthepersistenceofmacroaggregates.

3.1TheExtendedModel

Wedescribethenonstandardelementsandrelegatethewell-knownpartstotheOnline Appendix.

Households.—Denotepatientandimpatienthouseholdsbysuperscriptspandi,re- spectively.Eachhouseholdcontainsalargenumberofworkersindexedbyk∈[0,1] andenjoyslifetimeutility

X_∞o,t( ln(o oo)+lno _Z1lk_o,1+σl)

,t

E₀ β ζ_c,tC_t−b_cC_t−1H_t−ψ_l dk ,o∈{p,i},
_t=0 0^1+σl

whereC_t^odenotesconsumption,H_t^odenoteshousingservices,l_k^o_,tisspecializedlabor, andζ_c,taispreferenceshock.Theparameterb^o_cdetermineshabit,ψ_lisaweight coefficient,andσ_listheinverseelasticityoflaborsupply.Toensurethatimpatient householdsarenetborrowers,weimposeβⁱ<β^p.Thepatienthousehold’sbudget

constraintis

(1+τc)PCp+QthH¯p Z₁

p

k,t^l

p

k,t^{dk+Rt−1Pt−1Dt−1}

_{t −τ}l

t ^)W

t^{+PtDt+PtSt≤(1}

0

+ν_t⁻₋¹₁R_t^s₋₁P_t−1S_t−1+Q_t^hH^¯_t^p₋₁+Δ_t^p+T_t^p,

whereP_tisthepriceoffinalgoods,H^¯_t^pisahousinggoodthatprovidesH_t^punitsof

housingservices,Q_t^histhepriceofhousing,D_tisdeposits,S_tissecurities,W_k^p_,tisthe

8

nominalwageofworkerk,R_tisthenominalrisk-freerate,R_t^sistheinterestrateon securities,Δ_t^pbundlesdividendsfromfirmsandshadowbanks,T_t^pisatransferfrom thegovernment,andτ^candτ^lareconsumptionandlabortaxrates.Asintheprevious section,ν_tisamarketsentimentshock,thedecisiveexogenousvariableinouranalysis.

ImpatientHouseholds.—Thebudgetconstraintofimpatientworkersis

(1+τc)PtCti+PtrthHti≤(1−τl)Z₁

W_kⁱ_,tl_kⁱ_,tdk+Δⁱ_t+T_tⁱ,

0

wherer_t^histherentalrateofhousinganddividendsΔⁱ_taredescribedbelow.

Besidesworkers,theimpatienthouseholdscomprisesalargenumberofhomeown- ers.6Attimet,ahomeownercombineshernetworthN_tⁱandaloanB_tⁱfromabankto

acquirehousingfromhousinggoodproducers.Shereceivesastandarddebtcontractand promisestorepayR_tⁱ₊₁B_tⁱinthenextperiod.Atthestartofperiodt+1,eachhomeowner ishitbyanidiosyncraticshockωⁱdrawnfromaunit-meanlognormaldistributionwith cumulativedistributionFⁱ.Atthispoint,networthisgivenby

N_tⁱ₊₁=R_t^h₊₁ωⁱQ_t^hH^¯_tⁱ−R_tⁱ₊₁B_tⁱ,
whereR_t^h₊₁≡Q_t^h₊₁/Q_t^histhereturnonhousing.Thebudgetconstraintis

Q_t^h₊₁H^¯_tⁱ₊₁+Δⁱ_t+1=N_tⁱ₊₁+P_t+1r_t^h₊₁H^¯_tⁱ₊₁+B_tⁱ₊₁.

ThegoalofthehomeowneristomaximizethedividendΔⁱ_t+1shepaystoherfamily

subjecttothebudgetconstraintandabankparticipationconstraint,givenbelow.A defaultthresholdω¯_tⁱ₊₁separateshomeownerswhoareabletopayofftheirdebtfrom

thosewhoarenot

R_t^h₊₁ω¯_tⁱ₊₁Q_t^hH^¯_tⁱ=R_tⁱ₊₁B_tⁱ.

Finally,weassumehousingadjustmentcostssimilartoinvestmentadjustmentcosts.

Entrepreneurs.—Thereisalargenumberofentrepreneurs(superscripte).Theseare analogoustohomeownerssofarastheirrelationshipwiththebankisconcerned.Each combineshernetworthN_t^eandaloanB_t^etoacquirecapitalfromcapitalproducers

Q_t^kK^¯_t=N_t^e+B_t^e,

6Wesplittheimpatienthouseholdintoworkersandhomeownerstoensurethattheproblemofthe borrowingagent—thehomeowner—islinearinnetworth,whichfacilitatesaggregation(Ferrante,2019).

9

whereQ_t^kisthemarketpriceofcapital.Afterbeinghitbyanidiosyncraticshockω^e drawnfromdistributionF^e,theentrepreneurearnsrevenuesbyrentingoutcapitalser- vicesω^eK_ttoproductivefirmsandsellingdepreciatedcapitalbacktocapitalproducers afterproduction.FollowingChristiano,Motto,andRostagno(2014),weletσ_t^edenote thestandarddeviationoflogω^eandrefertoitasthefirmriskshock.Thereturnper

unitofcapitalis

_Rt_k=□_(1−τk)[ut_rt_k−a(ut_)]Υ−tPt_+(1−δ)Qt_k+τkδQt_k−1□_/Qt_k−1_,

whereu_tiscapitalutilization,aisautilizationcost,andτ^kistaxoncapital.Theobject

Υ>1accountsforinvestment-specifictechnicalchange,iefinalgoodsconvertinto Υ^tµ_Υ,tinvestmentgoods,whereµ_Υ,tisashock.

Similarlytohomeowners,entrepreneursdefaultifthecostofservicingdebtexceeds thevalueofcollateral,R_t^k₊₁ω¯_t^e₊₁Q_t^kK^¯_t=R_t^e₊₁B_t^e.Thegoalofanentrepreneurinperiodt

istomaximizeexpectednetworth

EZ_∞

_{t [R}k_ωek

t^Qt¯

e e e_ωe_,

+1 ^Kt−R

t+1^{Bt]dF( )}

e
ω¯_t+1

subjecttoaparticipationconstraintsetbythebank.

Banks.—Arepresentative,competitivebanktransformsdepositsfrompatienthouse- holdsintomortgageloansB_tⁱtoimpatienthouseholdsandbusinessloansB_t^etoen- trepreneurs.Loansarebackedbycollateral—housingQ_t^hH^¯_tⁱformortgagesandcapital Q_t^kK^¯_tforbusinessloans.Asinthetwo-periodmodel,thebankinsuresitselfagainstany

lossbytransferringtheriskypartofitsloanportfoliotoashadowbank.Thus,Equation (1)oftheprevioussection,togetherwiththezero-profitcondition,duplicatesintoapair ofparticipationconstraintswhichthebankimposesonitsborrowers,oneforeachtype

[1−Fⁱ(ω¯_tⁱ₊₁)]R_tⁱ₊₁B_tⁱ+Fⁱ(ω¯_tⁱ₊₁)Q^˜_t^h₊₁H^¯_tⁱ≥R_tB_tⁱ, (6)
[1−F^e(ω¯_t^e₊₁)]R_t^e₊₁B_t^e+F^e(ω¯_t^e₊₁)Q^˜_t^k₊₁K^¯_t≥R_tB_t^e. (7)

Theleft-handsidecorrespondstothebank’srevenues.Itssecondtermmakesclear thatpartoftheserevenueshingeonthepricesofasset-backedsecurities,Q^˜_t^h₊₁and Q^˜_t^k₊₁,whichthemselvesdependonthedemandforABScomingfromshadowbanks.

Thisconnectsthetraditionalandshadowbankingsectorsandiswhatdifferentiatesthe constraintsfromtheiroriginalformulationinBGG.

ShadowBanks.—TherepresentativeshadowbankemploysfundingS_tfrompatient householdstoacquiremortgageandbusinessloanportfoliosfrommainstreambanks.
10

Itsbudgetconstraintinperiodtis

S_t=Fⁱ(ω¯_tⁱ)Q^˜_t^hH^¯_tⁱ₋₁+F^e(ω¯_t^e)Q^˜_t^kK^¯_t−1.

assets_,name_lyhousingGi(ω¯_i)≡R_ω¯ti_{ωidFi(ωi)RhQ}th−1H_{¯i eωe≡}

Astheclaimowner,theshadowbankisentitledtothereturnontheunderlying

R_ω¯te ^{0 t t−1,andcapitalG(¯)}

_ωe_dFe_(ωi_)Rk

0 ^t
Q_t^k₋₁K^¯_t−1.Weassumetheshadowbankenjoyssomedegreeofmarket
powerinthetwomarketsforasset-backedsecurities,intheformofmarkups,µ^hand _µk_{.Theseparametersreplacethemonitoringcostµinthestandardfinancialaccelerator}

mechanismofBGG.InOnlineAppendixSectionA,wediscussthedifferencesbetween ourmodifiedframeworkandtheoriginalcostlystateverificationmodel.

Shadowbankprofitistransferredtopatienthouseholdsasdividends.Thus,the shadowbankmaximizesafter-dividendprofit

(1−µ^h)Gⁱ(ω¯_tⁱ)R_t^hQ_t^h₋₁H^¯_tⁱ₋₁+(1−µ^k)G^e(ω¯_t^e)R_t^kQ_t^k₋₁K^¯_t−1−R_t^sS_t,

subjecttoitsbudgetconstraint.Thefirst-orderconditionequalizesthemarginalbenefit ofthetwoassets

(1 ^{t−1 t}=(1−µ^k) ^{t t−1 (¯t)}

Q^˜_t^hFⁱ(ω¯_tⁱ) Q^˜_t^kFe(ω¯_t^e).

Thisequationbindstogetherthepricesofmortgage-backedandcapital-backedsecu- rities.Thesecuritizeddebtmarketthusactsasacentralizingforcethatsynchronizes assetprices.

ω
R^hQ^h_Gⁱ_(¯ⁱ₎ R^kQ^k_G^e_ω^e

_−µh₎ ^t
OtherAgents.—TherestofthemodelispresentedandderivedinOnlineAppendix SectionB,wherewealsolistalltheequilibriumconditions.

Shocks.—Weconsider12shocks:themarketsentimentshock,permanentandtransi- torytechnology,permanentandtransitoryinvestment-specifictechnology,preference, housing,markup,firmequity,firmrisk,governmentspending,andmonetarypolicy. Allhavethesamestructureandfollowtheprocessln(x_t/x)=ρ_xln(x_t−1/x)+ε_t^x,with _εx_∼N(0,σx).

11

Figure2:TheRoleoftheMarketSentimentShock
3.2Estimation

Weestimatethemodelonquarterlyaggregatedatafor12countriesoftheeuroarea.7 Thesampleperiodis1999Q1–2019Q4.Weuse11series:GDP,consumption,invest- ment,workhours,inflation,thenominalinterestrate,householdcredit,businesscredit, householdspread,businessspread,andhouseprices.8Anumberofparametersarecal- ibratedbasedonourdatasetandothertargets.Weestimatetheremainingparameters withBayesiantechniques.TheOnlineAppendixprovidesadetaileddescriptionofthe dataanditstreatment(SectionC),thecalibrationandestimationofparameters,and measuresofmodelfit(SectionD).