Financial Regulation in a Quantitative Model of the Modern ......Nov 07, 2018 · This Paper I Model I comm. banks and shadow banks provide liquidity services I both have limited

Financial Regulation in a Quantitative Model of theModern Banking System

Juliane Begenau1 Tim Landvoigt2

1Stanford & NBER

2Wharton & NBER

Indiana UniversityNovember 7, 2018

Begenau & Landvoigt Financial Regulation 1 / 33

Motivation

I Financial System:regulated (commercial) banks & unregulated (shadow) banks

I provide access to “intermediated” assets, e.g. long term creditI balance sheet: risky & illiquid assets funded with money-like liabilities

I Effects of financial regulation on a subset of banks?I Does tighter regulation cause shift to shadow banks?I Does it make financial system more risky?

I Requires quantitative general equilibrium analysis

I Study effect of capital requirements


This PaperI Model

I comm. banks and shadow banks provide liquidity servicesI both have limited liability & costly bankruptciesI comm. banks: deposit insurance, subject to capital regulationI shadow banks: risky debt, no regulationI focus on (1) risk taking through bank leverage

(2) liquidity provision by banks

I Calibration matchesI aggregate liquidity premium of safe debtI size of shadow banking sectorI default risk of both types of banksI greater fragility of shadow banks (runs)

I Tighter capital requirementI causes shift to shadow sectorI only small increase in risk taking (leverage) by shadow banksI trade-off between financial fragility and liquidity provision


Model Overview

Y Asset

Own Funds

Equity

Equity

Commercial Banks

Shadow banks

Intermediaries

Capital

Capital

Deposits

Debt

Capital

produced by

banks

C. Equity

C. Deposits

S. Equity

S. Debt

Households

Y Asset

(not intermediated)

Deposit Insurance

𝜚 deposits

withdrawn

early &

bailout

probability


Overview of Talk

I Static ModelI What pins down size and leverage of shadow banks?I Effect of tighter capital requirementI Efficient allocation vs. equilibrium

I Dynamic quantitative modelI Differences to simple modelI Calibration highlightsI Quantitative results


Setup

I Dates t = 0 and t = 1I Unit mass of households endowed with 1 unit of capital at t = 0I Unit mass of C-banks and S-bank purchase capital

and issue equity and debt to households

I Capital produces 1 unit of consumption at t = 1 if held by banksI Capital much less productive if held by households

I Household preferences: bank deposits provide liquidity services

U = C0 + β(C1 + ψH(AS ,AC ))

with Aj , j = S ,C , are deposits of banks held by households


Setup


and issue equity and debt to householdsI Capital produces 1 unit of consumption at t = 1 if held by banksI Capital much less productive if held by households


U = C0 + β(C1 + ψH(AS ,AC ))



Setup


and issue equity and debt to householdsI Capital produces 1 unit of consumption at t = 1 if held by banksI Capital much less productive if held by households


U = C0 + β(C1 + ψH(AS ,AC ))



S-banks

I Each bank solves

maxKS≥0,BS≥0

qS(BS ,KS)BS − pKS︸︷︷︸equity raised at t = 0

+βmax ρSKS − BS , 0︸︷︷︸dividend paid at t = 1

I Bank issues risky debt at price qS(BS ,KS)I Creditors price default riskI Bank internalizes effect of choice (BS ,KS) on debt price

I Limited liability with costly bankruptcy: if default, equity is wiped outand all assets lost (no recovery for creditors)

I Bank-idiosyncratic payoff shock ρS ∼ Uniform[0, 1]


S-banks

I Each bank solves

maxKS≥0,BS≥0







S-banks

I Each bank solves

maxKS≥0,BS≥0







S-banks

I Each bank solves

maxKS≥0,BS≥0







S-Bank Problem

I Write time-1 dividend as

max ρSKS − BS , 0 = KS (1− FS(LS)) (E(ρS |ρS > LS)− LS)

with bank leverage LS = BS/KS and FS() is c.d.f. of ρS

I Using ρS ∼ Uniform[0, 1], FS(LS) = LS is default probability, andcontinuation value further simplifies

1

2KS (1− LS)2

I Using scale independence

vS = maxLS∈[0,1]

qS(LS)LS − p + β1

2(1− LS)2

maxKS≥0

KSvS


S-Bank Problem





1

2KS (1− LS)2


vS = maxLS∈[0,1]


2(1− LS)2

maxKS≥0

KSvS


S-Bank Problem





1

2KS (1− LS)2


vS = maxLS∈[0,1]


2(1− LS)2

maxKS≥0

KSvS


C-banks

I Each bank solves

maxKC≥0,BC≥0

qCBC − pKC︸︷︷︸equity raised at t = 0

+βmax ρCKC − BC , 0︸︷︷︸dividend paid at t = 1

subject toBC ≤ (1− θ)E(ρC )KC

I Differences to S-bank problemI Government-insured debt is riskfree to creditorsI Regulatory capital requirement θ


C-banks

I Each bank solves

maxKC≥0,BC≥0

qCBC − pKC︸︷︷︸equity raised at t = 0

+βmax ρCKC − BC , 0︸︷︷︸dividend paid at t = 1

subject toBC ≤ (1− θ)E(ρC )KC

I Differences to S-bank problemI Government-insured debt is riskfree to creditorsI Regulatory capital requirement θ


Households and Government

I HH choose purchases of debt and equity of each bank to max utility

maxAj ,Sjj=S,C

C0 + β(C1 + ψH(AS ,AC ))

s.t. C0 = p︸︷︷︸sell cap.

−qSAS − qCAC − pSSS − pCSC︸︷︷︸buy securities

C1 = (1− LS)AS + AC − T

+ SS1

2KS (1− LS)2︸︷︷︸

div. from S-bank

+SC1

2KC (1− LC )2︸︷︷︸

div. from C-bank

whereT = LCBC

are lump-sum taxes required to bail out liabilities of failing C-banks


Equilibrium

I Market clearing

SS = 1

SC = 1

AC = BC

AS = BS

KS + KC = 1.

I Resource constraints: C0 = 0 and

C1 =1

2

(1− KCL

2C − KSL

2S

)I Time-1 consumption clarifies fundamental trade-off

I Bank leverage causes bankruptcies and deadweight lossesI But some leverage necessary to produce liquidity services


Planner Problem (1/2)

I Planner solves

maxKS ,LS ,LC

1

2

(1− (1− KS)L2

C − KSL2S

)+ ψH

LSKS︸︷︷︸=AS

, LC (1− KS)︸︷︷︸=AC

I Liquidity preference function

H(AS ,AC ) = (αAεS + (1− α)AεC )1/ε

I Marginal liquidity benefit (= “convenience yield”)

∂H(AS ,AC )

∂Aj= Hj(RS), where RS = AS/AC

I Solution isKS

KC=

(α

1− α

) 11−ε

, and LS = LC

with LS = ψHS(RS) and LC = ψHC (RS)



I Planner solves

maxKS ,LS ,LC

1

2

(1− (1− KS)L2

C − KSL2S

)+ ψH

LSKS︸︷︷︸=AS

, LC (1− KS)︸︷︷︸=AC




∂H(AS ,AC )


I Solution isKS

KC=

(α

1− α

) 11−ε

, and LS = LC




I Planner solves

maxKS ,LS ,LC

1

2

(1− (1− KS)L2

C − KSL2S

)+ ψH

LSKS︸︷︷︸=AS

, LC (1− KS)︸︷︷︸=AC




∂H(AS ,AC )


I Solution isKS

KC=

(α

1− α

) 11−ε

, and LS = LC




I How does planner set leverage of each bank?

Lj = ψHj(AS/AC )

I RHS is marginal benefit to household from liquidityI LHS is marginal cost from failing banks (consumption losses)I LS = LC means that same marginal benefit from both types

I How does planner allocate capital?

KS

KC=

(α

1− α

) 11−ε

I Both banks have same production technology for consumption goodI So capital allocation only based on liquidity production technologyI Greater elasticity 1/(1− ε) tilts allocation towards bank with greater

weight in CES aggregator



I How does planner set leverage of each bank?

Lj = ψHj(AS/AC )

I RHS is marginal benefit to household from liquidityI LHS is marginal cost from failing banks (consumption losses)I LS = LC means that same marginal benefit from both types

I How does planner allocate capital?

KS

KC=

(α

1− α

) 11−ε

I Both banks have same production technology for consumption goodI So capital allocation only based on liquidity production technologyI Greater elasticity 1/(1− ε) tilts allocation towards bank with greater

weight in CES aggregator


Equilibrium Characterization: S-Bank Problem

I S-bank FOC for leverage

q′S(LS)LS + q(LS) = β(1− LS)

I Household FOC for S-bank debt

q(LS) = β(1− LS︸︷︷︸payoff

+ ψHS(RS)︸︷︷︸liq. premium

)

I Combining givesLS = ψHS(RS)

⇒ S-bank leverage choice is same as planner solution!

I Constant returns also imply zero expected profits

vS = 0⇔ p − qS(LS)LS = β1

2(1− LS)2


Equilibrium Characterization: C-Bank ProblemI Simplify problem as for S-bank

vC = maxLC∈[0,1]

qCLC − p + β1

2(1− LC )2

subject to

LC ≤1

2(1− θ) ,

and maxKC≥0

KCvC

I Household FOC for C-bank debt

qC = β( 1︸︷︷︸payoff

+ ψHC (RS)︸︷︷︸liq. premium

)

I C-bank constraint always binds if ψ > 0: LC = 12 (1− θ)

⇒ Moral hazard due to limited liability and deposit insurance


vC = 0⇔ p − qCLC = β1

2(1− LC )2



vC = maxLC∈[0,1]

qCLC − p + β1

2(1− LC )2

subject to

LC ≤1

2(1− θ) ,

and maxKC≥0

KCvC




)




vC = 0⇔ p − qCLC = β1

2(1− LC )2



vC = maxLC∈[0,1]

qCLC − p + β1

2(1− LC )2

subject to

LC ≤1

2(1− θ) ,

and maxKC≥0

KCvC




)




vC = 0⇔ p − qCLC = β1

2(1− LC )2


Equilibrium Characterization: Relative Size of SectorsI Rewrite zero-profit conditions

S-bank: p = β1

2(1 + L2

S)

C-bank: p = β1

2(1 + L2

C ) + βLCψHC (RS)

I Combining both conditions to get indifference condition

LS = (L2C + LCψHC (RS))1/2 (L1)

I Condition implies that S-bank leverage is always higherthan C-bank leverage

I C-bank has key competitive advantage: deposit insuranceI To deliver same profit to equity owners, S-banks need to have higher

leverage (holding constant S-bank default risk)

I Also recall optimal S-bank leverage

LS = ψHS(RS) (L2)



S-bank: p = β1

2(1 + L2

S)

C-bank: p = β1

2(1 + L2

C ) + βLCψHC (RS)


LS = (L2C + LCψHC (RS))1/2 (L1)





LS = ψHS(RS) (L2)



S-bank: p = β1

2(1 + L2

S)

C-bank: p = β1

2(1 + L2

C ) + βLCψHC (RS)


LS = (L2C + LCψHC (RS))1/2 (L1)





LS = ψHS(RS) (L2)


Equilibrium Characterization: Relative Size of Sectors

1

𝐿𝑆

𝑅𝑆

𝐿𝑆0

𝑅𝑆0

I Red line: indifference conditionLS = (1/4(1− θ)2 + 1/2(1− θ)ψHC (RS))1/2

I Blue line: leverage condition LS = ψHS(RS)

I Key property: decreasing returns, i.e. H ′S(RS) < 0 and H ′C (RS) > 0


Effect of Higher θ

1

𝐿𝑆

𝑅𝑆𝑅𝑆0

𝐿𝑆0

I Indifference condition LS = (1/4(1− θ)2 + 1/2(1− θ)ψHC (RS))1/2 ↓I Higher θ makes C-banks less profitable and S-banks relatively more

profitable ⇒ S-bank sector expands: RS ↑I But decreasing returns ⇒ lower S-bank liquidity premium ⇒ LS ↓


How Should Regulator Set θ?

Proposition

Index competitive equilibria by the factor m > −1, such that

LC = (1 + m)ψHC (RS),

and the function θ = f (m) that determines the value of θ implementingequilibrium m.

(i) There is no θ ∈ [0, 1] that implements the planner allocation.

(ii) In any equilibrium with m ≥ 0, an increase in the capital requirementθ is welfare-improving.

I Planner wants LC = ψHC (RS), so could choose θ = f (0)

I But always have LS > LC in equilibrium due to deposit insurance andcompetition ⇒ need additional policy tool to regulate S-banks

I Still welfare-improving to raise θ in world with m > 0


Effect With More General Preferences

1

𝐿𝑆

𝑅𝑆𝑅𝑆0

𝐿𝑆0

𝐻 𝐴𝑆, 𝐴𝐶 =𝛼𝐴𝑆

𝜀 + (1 − 𝛼)𝐴𝐶𝜀 1/𝜀 1−𝛾𝐻

1 − 𝛾𝐻

I Ambiguous net effect of higher θ

1. makes C-banks less profitable and shadow bank equity more attractive

2. shadow bank share expands, liqu. premium on S-bank debt declinesrelative to C-bank debt

3. with γH > 0, marginal benefit of total liquidity goes up as H(AS ,AC )falls and leverage curve shifts up


Overview of Talk

I Static ModelI What pins down size and leverage of shadow banks?I Effect of tighter capital requirementI Efficient allocation vs. equilibrium

I Dynamic Quantitative ModelI Differences to simple modelI Calibration highlightsI Quantitative results


Dynamic Model: Key Differences

1. Infinite horizon model with bank-independent sector (endowment)and bank-dependent sector (production)

I Banks have access to standard investment technologyI Convex capital adjustment costs

2. Riskier S-banks: runs and implicit bail-out guaranteesI S-banks subject to stochastic deposit redemption shocks %t More Details

I Introduces additional losses through fire-saleI Government bails out S-bank liabilities with probability πB

3. Risk averse households with preferences

U(Ct ,H

(ASt ,A

Ct

))=

C 1−γt

1− γ+ ψ

([α(AS

t )ε + (1− α)(ACt )ε] 1ε

)1−γH

1− γH

I Portfolio choice of equity and debt of both types of banksI Inelastic labor supply


State Variables and Solution Method

I Exogenous states

log(Yt+1) = (1− ρY )log(µY ) + ρY log(Yt) + εYt+1

Zt = φZYt exp(εZt )

and %t follows a two-state Markov-process

I Endogenous states

1. Capital stock2., 3. C-bank and S-bank debt

4. S-bank capital share

I Solve using non-linear projection methodsI Probability of default bounded in [0, 1]I Nonlinear dynamics because of bankruptcy option

I Report results for simulated model


Calibration: Consolidated View of Shadow Banks

Assets

Comm.

Paper

Equity

MMMF

Shares

Comm.

Paper

AssetsDebt

Equity

Shadow banks

Finance Company Money Market Mutual Funds

Consolidated


Key Parameters: Quarterly data 1999− 2015Values Target Data Model

Bank leverage and defaultδS 0.300 Quarterly corp. bond default rate 0.36% 0.31%δC 0.175 Quarterly net loan charge-offs 0.25% 0.26%ξC 0.515 Recovery rate Moody’s 63% 62%ξS 0.415 Recovery rate Moody’s 63% 63%πB 0.905 Shadow bank leverage 93% 93%

Liquidity preferencesβ 0.993 C-bank debt rate 0.39% 0.39%α 0.330 Shadow banking share (Gallin 2013) 35% 34%ψ 0.0103 Liquidity premium C-banks; KV2012 0.18% 0.17%γH 1.700 Corr(GDP, C-bank liquid. premium) -0.28 -0.39ε 0.420 S-bank liquidity elasticity 0.17% 0.16%

RunsδK 4 ×δK Max. haircut (GM 2009) 20% 19%Z 26% × Z Forecl. discount (Campbell et al 2011)% [0, 0.3] Fraction run

Prob%

[0.97 0.030.33 0.67

]Uncond. run prob. (Covitz et al 2013) 3% 3%


Liquidity Preference Parameters (1/2)

I How are key liquidity preference parameters disciplined by data?

I ψ: level of liquidity premiumI Krishnamurthy & Vissing-Jorgenson 2012 estimate annual premium of

75 bpsI ψ directly scales marginal liquidity benefit in model

I α: market share of S-banksI Higher α raises S-bank relative to C-bank premiumI Lowers funding cost, increases demand for capital of S-banks

I γH : comovement of premium with GDPI Countercyclical in data: liquidity is abundant in good times,

so premium is lowI Model matches countercyclical premium with γH = 1.7 ⇒

downward-sloping demand curve for liquidity





















qCt − qSt = β

(MRSC

qC−

MRSS

qS

)γCt+1 +

βF S

qSF St+1

+ β

((1− ε− γH)

(MRSC

qC−

MRSS

qS

)α

(AS

H

)ε+ (1− ε)

MRSS

qS

)ASt+1

+ β

((1− ε− γH)

(MRSC

qC−

MRSS

qS

)(1− α)

(AC

H

)ε− (1− ε)

MRSC

qC

)ACt+1

I ε: elasticity of substitution between S- and C-bank debtI Log-linear approximation of spread 1/qS − 1/qC (shadow rate −

deposit rate)I If ε = 1 (perfect substitutes) and γH = 0 (CRS in liquidity), quantities

of debt (AS ,AC ) do not matter for spreadI If ε < 1, would expect negative sign on AC and positive sign on AS

I Regression of CP − Tbill spread on Tbill supply, shadow debt supply(and controls) gives elasticity of 17 bp

I Matched in model with ε = 0.42 (net substitutes)


Increasing Capital Requirement

Larger shadow banking share, C-banks “exit”, S-bank “enter”

Benchmark 13% 17% 20% 30%mean mean mean mean

Capital and Debt1. Capital 4.005 +0.2% +0.6% +0.9% +2.2%2. Debt share S 0.349 +3.4% +6.2% +8.1% +16.0%3. Capital share S 0.342 +1.1% +0.6% -0.1% -1.7%4. Leverage S 0.933 +0.1% +0.4% +0.6% +1.0%5. Leverage C 0.900 -3.3% -7.8% -11.1% -22.4%6. Early Liquidation (runs) 0.004 +0.3% +0.8% +1.2% +2.3%

Prices7. Deposit rate S 0.49% -0.7% -2.0% -3.0% -6.5%8. Deposit rate C 0.39% -4.1% -9.1% -13.0% -28.6%9. Convenience Yield S 0.23% +2.0% +5.3% +8.0% +17.8%10. Convenience Yield C 0.31% +5.2% +11.5% +16.4% +36.2%



C-banks become safer, but S-banks riskier






Interest rates fall as liquidity premia rise ⇒ more investment






DWL from C-banks decline, from S-banks rise

BM 13% 17% 20% 30%mean mean mean mean

Welfare11. DWL S 0.001 +5.2% +11.7% +16.7% +31.8%12. DWL C 0.003 -68.6% -94.6% -98.8% -100.0%13. Default Rate S 0.31% +3.7% +9.9% +15.1% +29.7%14. Default Rate C 0.26% -67.2% -94.1% -98.7% -100.0%

15. GDP 1.365 +0.0% +0.1% +0.1% +0.2%16. Liquidity Services 1.969 -2.3% -5.2% -7.3% -14.4%17. Consumption 1.261 +0.13% +0.19% +0.20% +0.24%18. Vol(Liquidity Services) 0.068 -2.5% -6.1% -8.9% -18.7%19. Vol(Consumption) 0.005 +0.4% +0.6% +0.7% +10.5%

20. HH Welfare -114.596 +0.106% +0.129% +0.116% +0.048%



More consumption and lower liquidity provision

BM 13% 17% 20% 30%mean mean mean mean

Welfare11. DWL S 0.001 +5.2% +11.7% +16.7% +31.8%12. DWL C 0.003 -68.6% -94.6% -98.8% -100.0%13. Default Rate S 0.31% +3.7% +9.9% +15.1% +29.7%14. Default Rate C 0.26% -67.2% -94.1% -98.7% -100.0%

15. GDP 1.365 +0.0% +0.1% +0.1% +0.2%16. Liquidity Services 1.969 -2.3% -5.2% -7.3% -14.4%17. Consumption 1.261 +0.13% +0.19% +0.20% +0.24%18. Vol(Liquidity Services) 0.068 -2.5% -6.1% -8.9% -18.7%19. Vol(Consumption) 0.005 +0.4% +0.6% +0.7% +10.5%

20. HH Welfare -114.596 +0.106% +0.129% +0.116% +0.048%


Welfare

10 15 20 25 30

%

0

0.05

0.1

0.15

Wel

fare

: con

sum

ptio

n eq

uiv.

uni

ts %


Transition Dynamics

0 4 8 12 16 200

0.05

0.1

0.15Consumption

0 4 8 12 16 20-6

-5

-4

-3

-2

-1

0Liquidity

0 4 8 12 16 20-0.1

0

0.1

0.2

0.3

0.4Capital

0 4 8 12 16 200

0.1

0.2

0.3

0.4

0.5S Cap Share

0 4 8 12 16 200

0.1

0.2

0.3

0.4

0.5

0.6S Leverage

0 4 8 12 16 200

2

4

6

8S Debt Share


Other Policies

I Set time-varying insurance fee such that fund breaks evenI Fair κ does not reduce C-bank leverage, but shifts activity to S-bankI ⇒ Less liquidity provision and higher deadweight losses

θ=17% fair κ Corr(θt ,Yt) Corr(θt ,Yt) Minn. plan< 0 > 0

Debt share S +6.18% +9.04% +5.95% +6.12% -63.88%Leverage S +0.39% -0.22% +0.42% +0.37% +0.02%Leverage C -7.78% -0.00% -7.77% -7.79% -14.45%

DWL S +11.70% +2.57% +11.83% +11.99% -70.70%DWL C -94.63% -5.49% -80.47% -83.73% -99.72%Liquidity Services -5.16% -1.43% -5.08% -5.16% -14.02%Consumption +0.19% -0.00% +0.16% +0.17% +0.28%Welfare +0.129% -0.011% +0.115% +0.118% +0.144%


Other Policies

I Set cyclical capital req’s with mean 17%I similar effects as with static optimal θ





Other PoliciesI “Minneapolis plan”: θ =23% and tax on S-bank debt of 30 bps

I Shrinks S-banks while making C-bank saferI Large drop in liquidity production, but greatest overall welfare gainI Consistent with welfare analysis in simple model: both banks have too

high leverage in status quo





Conclusion

I Tractable quantitative GE model with two types of banks

I Increasing capital requirement on commercial banks

I makes C-banks less, S-banks more profitableI leads to larger and riskier S-bank sectorI less liquidity provisionI no negative effects on production and investment in total

I Welfare trade-off: greater consumption (fewer bank failures)versus reduced liquidity provision

I Key Model Lessons

I GE effects (e.g. deposit rate response to higher cap reg)I Nature of competition between S-bank & C-bankI Slight increase in S-bank risk does not undermine intended benefits of

tighter capital regulation


Fraction of Liquid Wealth in MMA at Household Level

0 20 40 60 80 100

Percentile of Wealth Distribution

0

0.1

0.2

0.3

0.4

0.5

0.6

Fra

ctio

n M

MA

+ M

MM

F

Source: 2013 SCF Back


S-bank problem Back

vS(Zt) = maxbSt+1≥0,kS

t+1≥0kSt+1

(qS(bSt+1)bSt+1 − pt

)− φK

2

(kSt+1 − 1

)2

+ kSt+1Et

[Mt,t+1 ΠS

t+1ΩS(LSt+1)],

with

ΩS(LSt ) = (1− F Sρ,t)

(ρS,+t

(1− `St

(1− xSt

))− LSt + (1− `St )

vS(Zt)

ΠSt

)− F S

ρ,tδS

I Endogenous liquidation (fraction of assets)

`St =%St B

St

KSt ΠH

t

I Probability of default F Sρ,t = F S

ρ (ρSt ) with threshold

ρSt =LSt − (1− `St ) vS (Zt)

ΠSt− δS

1− `St(1− xSt

)increasing in leverage, liquidation fraction, and fire sale discount


Bankruptcy & Deposit InsuranceI C-bank default

I Government bails out liabilities of failing C-banksI Recovers

rC (LCt ) = (1− ξC )ρC ,−t

LCtper bond issued by C-banks

I S-banks defaultI Benchmark: government does not bailout failing S-banks bails out

liabilities of failing S-bank with probability πBI Recovery value per bond

rS(LSt ) = (1− ξS)(1− `St (1− xt))ρS,−t

LSt

I Required taxes in addition to deposit insurance revenue

Tt = FCρ,t(1− rC (LCt ))BC

t − κBCt+1︸︷︷︸

Net payment to defaulting C-bank depositors

+ πBFSρ,t(1− rS(LSt ))BS

t︸︷︷︸Bailout for S-bank




rC (LCt ) = (1− ξC )ρC ,−t




rS(LSt ) = (1− ξS)(1− `St (1− xt))ρS,−t

LSt



t − κBCt+1︸︷︷︸







rC (LCt ) = (1− ξC )ρC ,−t




rS(LSt ) = (1− ξS)(1− `St (1− xt))ρS,−t

LSt



t − κBCt+1︸︷︷︸





Financial Regulation in a Quantitative Model of the Modern ......Nov 07, 2018 · This Paper I Model I comm. banks and shadow banks provide liquidity services I both have limited

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