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

BDerivationoftheExtendedModel

B.1PatientHouseholds
LetΛ_t^pbethepatienthousehold’smarginalutility.Thefirst-orderconditionswith

respecttoconsumption,housingservices,deposits,andasset-backedsecuritiesare

0=Λ_t^p(1+τ^c)P_t−ζ_c,t/(C_t^p−b_c^pC_t^p₋₁)+b_c^pβ^pE_tζ_c,t+1/(C_t^p₊₁−b_c^pC_t^p),

0=1/H_t^p−Λ_t^pQ_t^h+β^pE_tΛ_t^p₊₁Q_t^h₊₁,
0=Λ_t^pP_t−β^pP_tE_tΛ_t^p₊₁R_t.
0=Λ_t^pP_t−β^pP_tE_tΛ_t^p₊₁R_t^s/ν_t.

B.2ImpatientHouseholds
Workers.—LetΛⁱ_tbetheimpatienthousehold’smarginalutility.Thefirst-ordercon-

ditionsforconsumptionandhousingservicesare

0=Λⁱ_t(1+τ^c)P_t−ζ_c,t/(C_tⁱ−bⁱ_cC_tⁱ₋₁)+bⁱ_cβⁱE_tζ_c,t+1/(C_tⁱ₊₁−bⁱ_cC_tⁱ),
0=1/H_tⁱ−Λⁱ_tP_tr_t^h.

Homeowners.—Ahomeownermaximizesthepresentdiscountedvalueofdividends

_Vt_i=maxi□_Δit_+βiEt_Λi/Λit_{max{0,Vi} ,}

H_¯i t+1 t+1

t^,Bt
subjecttoN_tⁱ=R_t^hωⁱQ_t^h₋₁H^¯_tⁱ₋₁−R_tⁱB_tⁱ₋₁,

Q_t^hH^¯_tⁱ+Δⁱ_t=N_tⁱ+P_tr_t^hH^¯_tⁱ+B_tⁱ,

andthebankparticipationconstraint.Substitutethetwoconstraintsinthevaluefunction anddefineη_tⁱ≡B_tⁱ/H^¯_tⁱandg_tⁱ≡H^¯_tⁱ/H^¯_tⁱ₋₁

_βi_Et i _0,Vi_}.

_¯i t +1

t−1^{+(Ptrt+ηtQt)gt]H¯}

i _Λ/Λi

₋⁺ t^max{

_H i ^{1 t t+1}

t^,B

t

ThevaluefunctionV_tⁱislinearlyhomogeneousinhousingH^¯_tⁱ₋₁.Therefore,allhome-

ownersselectthesameleverageanddefaultthresholdregardlessoftheirhousingnet worth.Wecanrewritetheproblemintheformofthescaledvaluefunctionv_tⁱ≡V_tⁱ/H^¯_tⁱ₋₁

vti=max(RthωiQth−1−Rtiηi−+(Ptrh+ηi−Qth)gi+giβiEtΛi/Λi_Z∞vi iωi).

_Vt_i=max□_[Rt_hωiQt_h−1_−Rt_{iηi h i−h i}

dF()

g_t,η_t^{i t1 t t t} _¯^{i t+1}

t t _t+1

i

ω_t+1

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Thefirst-orderconditionswithrespecttog_tⁱandη_tⁱare

0=Prh+ iQh+ iEΛiΛiZ_∞

i i i
t_t^η_t⁻ _t^β t _t+1^/ _t ^v_t+1^dF(ω_t

),



ω¯^{i +1}
t+1

1=βⁱE_tΛⁱ_t+1/Λⁱ_t[1−Fⁱ(ω¯_tⁱ₊₁)](R_tⁱ₊₁+η_tⁱ∂R_tⁱ₊₁/∂η_tⁱ).
Substitutetheoptimalconditionforg_tⁱintothevaluefunctionandmultiplybyH^¯_tⁱ₋₁

V_tⁱ={R_t^hωⁱQ_t^h₋₁H^¯_tⁱ₋₁−R_tⁱB_tⁱ₋₁}=N_tⁱ.
Adefaultthresholdω¯_tⁱissuchthatN_tⁱ<0,thatisassetsareworthlessthanliabilities

R_t^hω¯_tⁱQ_t^h₋₁H^¯_tⁱ₋₁=R_tⁱB_tⁱ₋₁.

_R t+1⁾ t+1_.

[1−Fⁱ(ω¯_tⁱ₊₁)]η_tⁱ

Computethepartialderivative∂R_tⁱ₊₁/∂η_tⁱandplugitintotheoptimalconditionforη_tⁱ

_1=βiEt_Λi/Λi□_{− i′ i i ˜h h h i i′ i i}□

¯

i ^t

t+1⁼
Now,rewritetheparticipationconstraint
R_tηⁱ−Fⁱ(ωⁱQ^˜h

t+1 ^tRtF(ω¯t+1^)Rt+1^Qt+1^/(Rt+1^Qt)+ω¯t+1^F(ω¯t+1^)Rt+1^.

RealEstateBroker.—Acompetitiverealestatebrokeractsasamiddlemanbypur- chasinghousinggoodsfromhousingproducersandsellingthemtothehomeowners. Intheprocessofacquiringvastamountofrealestate,thebrokerissubjecttohousing adjustmentcosts.Thesecostsareimportantbecausetheysmooththedynamicsof housingandhenceofhouseholdcredit,whichisanobservablevariable,andthushelp ourmodelfitthedata.Therealestatebrokermaximizesprofit

_EX^∞

_{0 β}i,t_Λi^{□ □ □}

h_¯

i₋

h

t ^(ζh,tH¯

h_H¯i

i i
t^/H¯_t−1^{) ,}

t^Q

t^H

t^Q

t^1+S

t=0

whereS^h(t)isanincreasingconvexfunctionandζ_h,tisahousingshock.Thefirst-order

conditionis

Hⁱ

= _{t h,t}H_¯iS(t)+βⁱE_tΛⁱ_t+1Q_t^h₊₁ζ_{h,t+1 ¯}^t+1S^h′(t+1).

_ζ t _h′

i

− ^H

t1 ^t
0ΛitQh"Sh(t)+H_¯i # _¯!₂

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B.3Entrepreneurs
Defineleverageasassetsoverequity,L_t^e≡Q_t^kK^¯_t/N_t^e,andletΓ^e(ω¯_t^e₊₁)betheexpected

grossshareofentrepreneurialreturnsgoingtocreditors

Γe(ω¯te+1)≡[1 Z_ω¯e

t

_{ωω ω , ω ω}e e_ωe _.

+1

−F^e(¯^{e e e e e e}
t+1^)]¯t+1^+G(¯t+1^)G(¯t+1^)≡t+1^dF(t+1⁾

0

Expectedpre-dividendnetworthis

EZ<sub>∞


t [R</sub>k_ωek

_t+1^Qt^K¯t−R

e_Be_ω ω_¯e

e₍e_)=Et[1−Γe₍

k e

L_tN_t^e.

t+1 ^t]dF

t+1^)]R

ω^{e +1}

t
¯_t+1
DefineΓ₂^e(ω¯_t^e₊₁)≡[1−F^e(ω¯_t^e₊₁)]ω¯_t^e₊₁+F^e(ω¯_t^e₊₁)andrewritetheparticipationconstraint

usingthedefinitionsofleverageanddefaultcutoff

Γ2e(ω¯e Q_˜k !

t+1 _eωe ^Lt^e−1Rt_.

t+1^{)+ −1F(¯}
Q_t^kR_t^k₊₁ ^t+1)=L_t^eR_t^k₊₁

t+1^{)regardlessoftheirnetworth.Maximizationyields}

Theproblemofanentrepreneurinperiodtistochooseapairofleverageanddefault cutoff(L_t^e,ω¯_t^e₊₁)tomaximizeexpectednetworthint+1subjecttothebankparticipation constraint.SincecurrentnetworthN_t^edoesnotappearintheconstraintandispresent intheobjectiveonlyasafactorofproportionality,allentrepreneursselectthesame(L_t^e, _ω¯e

_0=Et□_1−Γe(ω_¯e□R_t^k_+1E Γ^e′(ω¯_t^e₊₁) <sub>1



t+1</sub>) − .
R_t ^t_Γ2^e′_(ω¯t^e_+1)+[Q^˜_t^k_/Q^kk₋ ^e′_ω^eL^e
+1^{( tR}t+1^)1]F(¯t+1^)t

UtilizationRate.—Theentrepreneuralsodeterminestheutilizationrateofcapitalu_t. Sincethemarketforcapitalservicesiscompetitive,theusercostfunctionisequalto thereturnonrentingoutcapitalservices

P_tΥ^−ta(u_t)ω_t^eK^¯_t−1=P_tr˜_t^ku_tω_t^eK^¯_t−1.

Optimalutilizationimplies

a^′(u_t)=Υ^tr˜_t^k=r^kexp(σ_a[u_t−1]).

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B.4ProductiveSector

FinalGoodProducers.—Arepresentative,competitivefinalgoodfirmcombinesin- termediategoodsY_j,t,j∈[0,1],toproducefinaloutputY_tusingthetechnology

Y_j^λ_,t^p,tdj ,

0
Yt=□_{Z1 1} □_λp,t

dj=P_tY_t.

Optimizationleadstothefamiliardemandfunctionandaggregatepriceindex

0 ^j,t
_whereλp,t_{≥1isamarkup}shoc_{k.Thefirm’s}budge_tcons_traintisR_1Pj,t_Y

□ □₋^λp,t □

_P 1λ ^{Z1 1 □1−λp,t}



p,t

= ^jt
Y_j,t ^, Y_t; P_t=P_j¹_,⁻_t^λp,tdj .
P_{t 0}

IntermediateGoodProducers.—Eachintermediategoodjisproducedbyamonopolist accordingtotheproductionfunction

_Yj,t_=max□_εt_(ut_Kj,t−1_)α(zt_lj,t_{)1−α−θz}t_{∗;0 ,α∈(0,1),}

whereε_tisastationarytechnologyshockandθisafixedcost.Therearetwosources ofgrowthinthemodel,namelyagrowthtrendintechnologyz_tandaninvestment- specificshockµ_Υ,tthatchangestherateatwhichfinalgoodsareconvertedintoΥ^tµ_Υ,t investmentgoods,withΥ>1.Thefixedcostθisproportionaltoz_t^∗=z_tΥ^(1−αα)t,

whichcombinesthetwotrends.TheintermediategoodproducerfacesstandardCalvo frictions.Everyperiod,afraction1−ξ_pofintermediatefirmssetsitspriceP_j,toptimally. TheremainingfractionfollowsanindexationruleP_j,t=π^ιpπ_t¹₋⁻₁^ιpP_j,t−1,whereι_p∈(0,1) andπ_t≡P_t/P_t−1isinflation.Avariablewithoutthesubscripttdenotesitssteady-state

value.
Laborinputoffirmjisacombinationofpatientandimpatientlabor

,1−κ_{,κ∈ ,}

j,t ^(01].

Parameterκisdecisive:ifκ=1wearebacktoarepresentativeagentmodel.Profit writesP_j,tY_j,t−P_tr˜_t^ku_tK^¯_j,t−1−W_t^pl_j^p_,t−W_tⁱlⁱ_j,t,whereP_tr˜_t^kdenotesthenominalrentalrate

ofcapital.Costminimizationimplies

P_tr˜_t^k=MC_tαε_t(u_tK^¯_t−1)^α−1(z_tl_t^p,κl_t^i,1−κ)^1−α,

W_t^pl_t^p=MC_t(1−α)κε_t(u_tK^¯_t−1)^α(z_tl_t^p,κl_t^i,1−κ)^1−α,

W_tⁱl_tⁱ=MC_t(1−α)(1−κ)ε_t(u_tK^¯_t−1)^α(z_tl_t^p,κl_t^i,1−κ)^1−α,

t^l
l_j,t=l_j^p_,^,κi

29

whereMC_tisthemultiplierontheproductionfunction,iethemarginalcost.Wehave droppedthejsubscriptbecauseallfirmschoosethesameproportionofinputsand henceshareacommonmarginalcost.

Turningtoprices,theintermediategoodsproducerchoosesanoptimalpriceP_j,tto maximizethepresentvalueoffutureprofits

_maxEX^∞

,s_Λ

p^β t+ ⁻

s^{Yj,t+s(Pj,tΠt,t+sMCt+s)}

P_{j,t s=0
t ξ}sp ^p _{˜ ,}

k=1^˜t+kand

_Πt,t+s_≡Q_sπ

k=1 ^{t+k.Notethefirmusesthediscountfactorofthepatienthousehold,its}
owner.Thefirst-orderconditionis
_{subjecttothedemandfunction.Here,}Π_˜t,t+s_≡Q_sπ

1

π˜_t=π^ιpπ_t−⁻₁^ιp.Let

λ_p

_X∞ ^{□ □}

_˜˜ λ ^□

1− ^□

,t+s



E_t ξ_p^sβ^p,sΛ_t^p_+sY_t+s^{PtΠt,t+s 1}Π^˜_t,t+s−λ_p,t+s^MCt+s

p,t+s

0=
_s=0 P_tΠ_t,t+s 1−λ_p,t+sP^˜_t^.

TheoptimalpriceP^˜_t≡P_j,tdependsonlyonaggregatevariablesandistherefore commontoallproducers.Rearranging,weobtainP^˜_t=P_tK_p,t/F_p,twhere

pΛp_λp,t_MCt □_π□₁₋^λp,t+1

K_p,t≡P_{t t}Y_t +ξ_pβ^pE ^{˜t+1 p,t+1p} ,

λ


1−λ_p,tP_t ^tπ_t+1 ^Kp,t+1
FpPΛp □ □₁¹



1 π˜ ^−λp,t
_p,t≡ _{t t}Y_t +ξ_pβ^pE_t ^{t+1 +1}F^p .

1−λ_p,t π_t+1 ^p,t+1