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

LaborContractors.—Arepresentative,competitivelaborcontractoraggregatesspe- cializedlaborservicesl_k^o_,t,wherek∈[0,1]ando∈{p,i},intohomogeneouslaborl_t^o

usingthetechnology

lo=□_Z1 □_λ


o,^{1 w}
_t l_k,t^λwdk ,o∈{p,i},λ_w≥1.

0

0 ^tt
demandfunctionforintermediatelaborandtheaggregatewageindex
_Itsbudge_tcons_traintisR_1Wk_o,t_lk_o,t_{dk=Wolo,o∈{}p_{,i}.Optimization}lead_stothe

_Wo!₁₋^λw _□Z

λ_{w 1} ^□1−λ



1 ^w

l^o= ^{k,t o}

t ⁰
_k,tWo l_t^o;W_t^o=W_k,^,_t^1−λwdk ; o∈{p,i}.

MonopolyUnions.—Eachworkeroftypekisrepresentedbyamonopolyunionthat setsitsnominalwagerateW_k^o_,t,whereo∈{p,i}.AllunionsaresubjecttoCalvo frictionsinasimilarfashiontointermediatefirms.Afraction1−ξ_wofmonopoly

30

unionschoosesitswageoptimally.Theremainingfractionfollowsanindexationrule _Wo_=µ ι_w¹

k ^z∗

_,t ππ_t−⁻₁^ιwW_k^o_,t−1,whereo∈{p,i},ι_w∈(0,1),µ_z^∗≡z^∗/z₋^∗₁isthesteady-state growthrateoftheeconomy,andµ_z^∗_,tisashock.Anoptimizingunionmaximizes

X_∞ so,s"_Z1lk_o,1+σl #

,t+s
E_t ξ_wβ−ψ_l dk+Λ_t^o_+s(1−τ^l)W^o˜ ,

w

t,t+s^l

o _,},




1+σ ^{k,tΠ ∈{}

k,t+s ^opi

s=0 ^{0 l}

_Πt_w,t+s_≡Q_sπ ^w

k=1^{w,t+k.Theoptimalwageconditionis}
subjecttothedemandfunction.Here,Π^˜_t^w_,t+s≡^Qs_k=1µ_z^∗π˜_w,t+kandπ˜_,t=π^ιwπ_t¹₋⁻₁^ιw.Let

X_∞ W_˜ !_λ^ _˜ !^{λ }

w^σ

λ_w

ow 1−_w ^l


_ψλWo_Π˜w ^1−λw 

Π^˜

0=E_t ξ_w^sβ^o,sl_t^o_+sW^t_oΠ^t_w^,t+s _Λ_t^o_+s(1−τ^l)Π^˜_t^w_,t+s− ^{lw t t,t+s} l^o,σl_^




_{s=0 t t,t+s  W}˜^o_W^{o t+s .}

t ^t+s _^

TheoptimalwageW˜o≡  

t^W_ko_,t^{iscommontoallworkerunions.Thatis,thereisone}
optimalwageW^˜_t^pforpatientworkersandanotherW^˜_tⁱforimpatientworkers.Rearranging,


o

_{t l w,t}1−λ_w(1+σ_l)
W_t^oW_t^o/P_tF_W^o_,t ^{,whereo∈{p,i}and}

K_w^o_,t≡l^{o,1+ 1 (1+)}

o

= ^ψK

weobtain^W
_˜ h i ^1−λw

_σl λ

t ⁺

_ξwβo_Et(π_˜w,t+1π−_{µ 1λ}^wσ_lo _,

w,t+1^z∗)−wKw,t+1


_FW_o,t_{≡(1−τl)λ}w_−1lt_oPt_Λt_o+ξw_βoEt



_∗ λ_wππ
w,t+1^t+1FW,t+1.

CapitalProducers.—Arepresentative,competitivecapitalproducerbuildsrawcapital accordingtoastandardtechnology

K_¯t_=(1−δ)K_¯t−1₊□_1−Sk(ζi,t_It_/It−1₎□_It_,δ∈(0,1),

whereI_tisinvestment,S^k(t)isanincreasingfunctiondefinedbelow,andζ_i,tisashock

tothemarginalefficiencyofinvestment.Optimalinvestmentimplies

0=ΛtpQtk□1−Sk(t)−ζi,t_ItSk′(t)□−_Λt_pPt pEpQk □_It+1□₂k′( .



+β _tΛ ζ )

t
I_t−1 Υµ_Υ,t ^{t+1 t+1i,t+1}I_t^St+1

HousingProducers.—Housingisinfixedsupply.Thetotalhousingstockis
H^¯=H^¯_t^p+H^¯_tⁱ.
31

B.5Government

ThemonetaryauthorityfollowsastandardTaylorrule

_Rt_−R=ρp_(Rt−1_{−R)+(1−ρ}p₎□_απ_(Et_πt+1_−π)+αΔy_(gy,t_−µz∗₎□_+εt_p,ρp_∈(0,1),

whereα_π,α_Δy>0areweights,g_y,tisquarterlyGDPgrowthindeviationfromsteady state,andε_t^pisamonetarypolicyshock.Thefiscalauthoritycollectstaxestofinance publicexpendituresG_tandtransferlump-sumamountsT_ttohouseholds

G_t+T_t=τ^k([u_tr_t^k−a(u_t)]Υ^−tP_t−δQ_t^k₋₁)K_t−1+τ^l(W_tⁱl_tⁱ+W_t^pl_t^p)+τ^cP_tC_t.

GovernmentspendingisgivenbyG_t=z_t^∗g_t,whereg_tisanexogenous-spendingshock.12

Transfersaredistributedtobothtypesofhouseholdsaccordingtotheirrespectiveshare intotallaborincome,T_t=κT_t^p+(1−κ)T_tⁱ.

B.6AggregationandMarketClearing

Production.—Clearinginthegoodsmarketimposes

Y_t=G_t+C_t+Υ^−tµ⁻_Υ¹_,tI_t+a(u_t)Υ^−tK^¯_t−1,
whereC_t≡C_t^p+C_tⁱistotalconsumption.WedefineGDPasY_t^gdp≡G_t+C_t+Υ^−tµ⁻_Υ¹_,tI_t.

ImpatientHouseholds.—Asexplainedabove,allhomeownerschoosethesamelever- ageanddefaultthreshold.Perfectinsurancewithinthehouseholdensurestheybegin thenextperiodwiththesamelevelofnetworth,whichinaggregateisgivenby

N_tⁱ=[1−Γⁱ(ω¯_tⁱ)]R_t^hQ_t^h₋₁H^¯_tⁱ₋₁,Γⁱ(ω¯_tⁱ)≡[1−Fⁱ(ω¯_tⁱ)]ω¯_tⁱ+Gⁱ(ω¯_tⁱ).

equa_ltheq_uantitypurchase_db_yentrepreneurs_,K_¯t₌R₁K_¯

Entrepreneurs.—Thequantityofphysicalcapitalproducedbycapitalproducersmust

dj.Theaggregatesupply

ofcapitalservicesprovidedbyentrepreneursmustequalthedemandfromintermediate

0 ^j,t

0

Kt=Z₁Z_∞
firms,Kt₌R_1Kj,t_dj.Sinceωeha_sunitmean_,tha_tsupp_lyis

u_t+1ω^eK^¯_j,tdF^e(ω^e)dj=u_t+1K^¯_t.

00

Asexplainedabove,allentrepreneurschoosethesameleverage,defaultcutoffand utilization.Toprevententrepreneursfromaccumulatingnetworthtothepointwhere

12Thisshockcapturesbothchangesingovernmentexpendituresandchangesinnetexports.
32

theyarecompletelyself-financed,werequirethattheypayafixeddividendδ^eeach periodtopatienthouseholds.Wealsoincludeanequityshockγ_t^ethatshiftstheir

aggregatenetworth.Aggregatenetworthafterdividendpaymentsis

N_t^e=γ_t^e[1−Γ^e(ω¯_t^e)]R_t^kQ_t^k₋₁K^¯_t−1−δ^eN_t^e.

Banks.—Theaggregatebalancesheetofthebankingsectoris

B_t≡B_tⁱ+B_t^e=D_t.

B.7SummaryofEquilibriumConditions

Westationarizeourmodelbydefiningthefollowingscaledvariables
b_t^e=B_t^e/(z_t^∗P_t), h_t=H^¯_t/z_t^∗, nⁱ_t=N_tⁱ/(z_t^∗P_t), w_tⁱ=W_tⁱ/(z_t^∗P_t),

bⁱ_t=B_tⁱ/(z_t^∗P_t), h_tⁱ=H^¯_tⁱ/z_t^∗, q_t^h=Q_t^h/P_t, w_t^p=W_t^p/(z_t^∗P_t),
c_t=C_t/z_t^∗, h_t^p=H^¯_t^p/z_t^∗, q˜_t^h=Q^˜_t^h/P_t, y_z,t=Y_t/z_t^∗,
c_tⁱ=C_tⁱ/z_t^∗, i_t=I_t/(z_t^∗Υ^t), q_t^k=Q_t^kΥ^t/P_t, y_t=Y_t^gdp/z_t^∗,
c_t^p=C_t^p/z_t^∗, k_t=K^¯_t/(z_t^∗Υ^t), q˜_t^k=Q^˜_t^kΥ^t/P_t, µ_z^∗_,t=z_t^∗/z_t^∗₋₁,
d_t=D_t/z_t^∗, λ_tⁱ=Λⁱ_tP_tz_t^∗, r_t^k=Υ^tr˜_t^k, _zt^∗_=ztΥ⁽1−^αα^)t_.

F_wⁱ_,t=F_Wⁱ_,tz_t^∗, λ_t^p=Λ_t^pP_tz_t^∗, s_t=S_t/(z_t^∗P_t),
F_w^p_,t=F_W^p_,tz_t^∗, mc_t=MC_t/P_t, t_t=T_t/(z_t^∗P_t),

e

g_t=G/_t, ⁿ_t^=N_t^/(z_t^P_t^{), t}_t^=T_t^i/(z_t^∗P_t^),
e ∗ i

t^z∗

33

Pricesandwages

1

F_p^p_,t=λ_t^py_z,t+ξ_pβ^pE_t(π˜_t+1π_t^{−11−λp,t+1p}

+1^{) F}_p,t+1^{. (1)}
λ_p,t+1

K_p^p_,t=λ_t^py_z,tλ_p,tmc_t+ξ_pβ^pE_t(π˜_t+1π_t^{−11−λ+1p}

+1^{) p,tK}_p,t+1^.(2)

_−ξ− p_.

p,t ^{p t}t^{) (1 p) F}p,t ⁽³⁾

1 ^λ

F^p=(1

w 1

_−τl_)λ−1_λ^p_l^p_+ξβp_µ1−λ_wEµ−1_πλ_w−11−λ_w−1^p

w,t w tt ^w _z^{∗ t}z^∗,t+1_w,t+1π^˜_w,t+1^πt+1^F_w,t+1^{. (4)}
K^p=l^p,1+σ+ ^pE(˜ ¹ )
K_p=□h1−ξ(π˜π₋₁1−λ₁p,ti ₁i_1−λp,t

λ_w

l ^p
,t t ^{ξwβ tπw,t+1π}

−_µ ^(1+σ_l⁾
w w,t+1^z∗1−λwK_w,t+1^{. (5)}

_−ξ pp

w^(π˜w,t^π_w,t^µz^{∗)1−λw(1−ξ}w^{) w}_t^F_w,t^{. (6)}

1 ^λ 1

Fⁱ=(1−τ^l)λ^{−1 i 1}

w

_w,_{t wλt}i_lt+ξwβi_µ1−λ_wEµ−_πλ_w−11−

λ_wπ−1 i _.

z z,t+1w,t+1w,t+1t+1^Fw,t+1 ⁽⁷⁾
Kⁱ=lⁱ¹⁺+ ⁱE(˜ ¹ )

∗ t ∗ ^π˜
Kw_p,t=ψl₋₁h□1 _{−1 1}□ ₋₁i<sub>1−λw(1+σl)

,σl</sub> λ<sub>w

w,t t ξwβ tπw,t+1π</sub>⁻_t+1µz∗1−λ_w^(1+σl)_Kⁱ _.

t+1 ⁽⁸⁾

Kw_i,t=ψ₋₁h□1−ξw(π˜w,tπw₋₁^w, □ i^w₁^,_−λ(1+σ)

1 w

1−λ_w(1ξw)⁻¹

l _i
− w_tF_wⁱ_,t. (9)

_l ,t^µz∗)

Production,resourceconstraints,andgovernment

p_.

t ^tt z^∗,t^tt−1 t^/l

t ⁽¹²⁾
r_t^k=r^kexp(σ_a[u_t−1]). (10) r_t^k=αε_t(Υµ_z^∗_,tl_t)^1−α(u_tk_t−1)^α−1mc_t. (11) _wp_{=(1−α)κmcεΥ−α(µ−1uk )αl1−α}

_wi

_tεt−α_(µ−1 α1 i

−α
z ^t−1)lt^/lt^{. (13)}

t^{=(1−α)(1−κ)mcΥ}

∗_,t^u_t^k

k_t=(1−δ)Υ⁻¹µ^{−1 k}

_z∗_,t^k_t−1^+[1−SΥµ_z^∗_,t^ζ_i,tⁱ_t^/i_t−1^]i_t^{. (14)}

_Rt_k=□_(1−τk)[ut_rt_k−a(ut_)]+(1−δ)qt_k□_Υ−1πt_qt<sub>k,−1 k.



Υt z</sub>∗_,t^k_t−1^{. (18)}
c_t=c_t^p+c_tⁱ. (19) _lt=l^p,κ_li,1−κ_.
−1^+τδ(15)
R_t^h=π_tq_t^h/q_t^h₋₁.(16) y_z,t=ε_t(Υ⁻¹µ⁻_z∗¹_,tu_tk_t−1)^αl_t^1−α−θ.(17) y_z,t=g_t+c_t+µ⁻¹_,i_t+a(u_t)Υ⁻¹µ⁻¹
t t ⁽²⁰⁾ h=h_t^p+h_tⁱ. (21) y_t=g_t+c_t+µ⁻_Υ¹_,ti_t.(22)

gt=_k□[ur_ka(u)]Υ_{−1 −1}q_k□₋₁k+ _l w

_{τ t−} ii ^pp c

t ^{t −π}t^δ_t−1^µ∗,t^t−1τwt^lt⁺ t^lt^{)+τct−tt. (24)}

z ⁽
_Rt_=R+ρp_(Rt−1_{−R)+(1−ρ}p₎□_απ_(Et_πt+1_−π)+αΔy_(gy,t_−µz∗₎□_+εt_{p, (23)}

_0=λp_qk^□ _k

_{i,titit−1−Υµz∗,tζi,tit/it−1S}′_{Υµz∗,tζi,tit/}

t ^{it−1 (25)}

t^{1−SΥµz∗,tζ/}


₋−1^p p −1^p_qk

2

,^+βE(Υ
µ_Υtλ_{t t}µ_z^∗_,t+1)λ_t+1t+1ζ_i,t+1Υµ_z^∗_,t+1i_t+1/i_tS^′Υµ_z^∗_,t+1ζ_i,t+1i_t+1/i_t.

34

Patienthouseholds
0=(1+τ^c)λ_t^p−µ_z^∗_,tζ_c,t/(µ_z^∗_,tc_t^p−b_c^pc_t^p₋₁)+b_c^pβ^pE_tζ_c,t+1/(µ_z^∗_,t+1c_t^p₊₁−b_c^pc_t^p).(26)

0=1/h_t^p−λ_t^pq_t^h+β^pE_tµ⁻_z∗¹_,t+1λ_t^p₊₁q_t^h₊₁. (27)
0=λ_t^p−β^pE_t(π_t+1µ_z^∗_,t+1)⁻¹λ_t^p₊₁R_t. (28)
0=λ_t^p−β^pE_t(π_t+1µ_z^∗_,t+1)⁻¹λ_t^p₊₁R_t^s/ν_t. (29)

Impatienthouseholds

0=(1+τ^c)λ_tⁱ−µ_z^∗_,tζ_c,t/(µ_z^∗_,tc_tⁱ−bⁱ_cc_tⁱ₋₁)+bⁱ_cβⁱE_tζ_c,t+1/(µ_z^∗_,t+1c_tⁱ₊₁−bⁱ_cc_tⁱ). (30)
0=1+λ_tⁱbⁱ_t−λ_tⁱq_t^hh_tⁱ[1+S^h(µ_z^∗_,tζ_h,th_tⁱ/h_tⁱ₋₁)+µ_z^∗_,tζ_h,th_tⁱh_t^i,₋⁻₁¹S^h′(µ_z^∗_,tζ_h,th_tⁱ/h_tⁱ₋₁)]

+βⁱE_tµ⁻_z∗¹_,t+1λ_tⁱ₊₁q_t^h₊₁h_tⁱζ_h,t+1(µ_z^∗_,t+1h_tⁱ₊₁/h_tⁱ)²S^h′(µ_z^∗_,t+1ζ_h,th_tⁱ₊₁/h_tⁱ) (31)
+βⁱE_t(π_t+1µ_z^∗_,t+1)⁻¹λ_tⁱ₊₁[1−Γⁱ(ω¯_tⁱ₊₁)]R_t^h₊₁q_t^hh_tⁱ.

0=λ_tⁱ−βⁱE_t(π_t+1µ_z^∗_,t+1)⁻¹λ_tⁱ₊₁[R_t+F^i′(ω¯_tⁱ₊₁)R_tⁱ₊₁(ω¯_tⁱ₊₁−π_t+1Q^˜_t^h₊₁/[R_t^h₊₁Q_t^h])].

(32)
0=(1−τ^l)w_tⁱl_tⁱ+(π_tµ_z^∗_,t)⁻¹[1−Γⁱ(ω¯_tⁱ)]R_t^hq_t^h₋₁h_tⁱ₋₁+bⁱ_t+t_tⁱ−(1+τ^c)c_tⁱ−q_t^hh_tⁱ.(33) 0=R_t−1bⁱ_t−1−[1−Fⁱ(ω¯_tⁱ)]R_tⁱbⁱ_t−1−Fⁱ(ω¯_tⁱ)π_tq˜_t^hh_tⁱ₋₁.(34)

ω¯_tⁱ=R_tⁱbⁱ_t−1/(R_t^hq_t^h₋₁h_tⁱ₋₁). (35)
nⁱ_t=(π_tµ_z^∗_,t)⁻¹[1−Γⁱ(ω¯_tⁱ)]R_t^hq_t^h₋₁h_tⁱ₋₁. (36)

Entrepreneurs

E_tΓ₁^′(ω¯_t^e₊₁) ₁
t^[1−Γ₁^(ω¯_t+1^)]− _′ ^.(37)

R_tΓ^e_(ω¯t^e_)+[πt+1q˜t^k_/(ΥRt^k_q^k

L^e

+1 +1 ^t

0=R_t−1b_t^e₋₁−[1−F^e(ω¯_t^e)]R_t^eb_t^e₋₁−F^e(ω¯_t^e)Υ⁻¹π_tq˜_t^kk_t−1. (38)

ω¯_t^e=R_t^eb_t^e₋₁/(R_t^kq_t^k₋₁k_t−1). (39)
n_t^e=γ_t^e(π_tµ_z^∗_,t)⁻¹[1−Γ^e(ω¯_t^e)]R_t^kq_t^k₋₁k_t−1−δ^en_t^e. (40)
L_t^e=q_t^kk_t/n_t^e. (41)
b_t^e=q_t^kk_t−n_t^e. (42)

Financialsector

s_t=Fⁱ(ω¯_tⁱ)µ⁻_z∗¹_,tq˜_t^hh_tⁱ₋₁+F^e(ω¯_t^e)Υ⁻¹µ⁻_z∗¹_,tq˜_t^kk_t−1. (43)

0=(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−π_tµ_z^∗_,tR_t^ss_t. (44)

q^{h i}_ωⁱ q^{k e}_ω^e

0=(1−µ^h)R_t^{h t−1G(¯t)}−(1−µ^k)ΥR^{k t−1G(¯t)}. (45)

q˜_t^hFⁱ(ω¯_tⁱ) ^tq˜_t^kFe(ω¯_t^e)

e

2 t+1⁾

+1 ^t)−1]Fe^′(ω¯
R^{k e}
0=E ^{e e t+1}

35