ESSAYS ON FINANCIAL INTERMEDIATION AND MACROECONOMIC

POLICY

A Dissertationsubmitted to the Faculty of the

Graduate School of Arts and Sciencesof Georgetown University

in partial fulllment of the requirements for thedegree of

Doctor of Philosophyin Economics

By

Arsenii Olegovich Mishin, M.A.

Washington, DCApril 13, 2020

Copyright c© 2020 by Arsenii Olegovich MishinAll Rights Reserved

ii

ESSAYS ON FINANCIAL INTERMEDIATION AND

MACROECONOMIC POLICY

Arsenii Olegovich Mishin, M.A.

Dissertation Advisor: Behzad Diba, Ph.D.

Abstract

This dissertation studies the role of capital requirements in combating excessive

risk-taking incentives of banks in two settings.

In the rst chapter, we build a quantitatively relevant DSGE model with

endogenous risk-taking in which deposit insurance and limited liability can lead banks

to make socially inecient risky loans. This excessive risk-taking can be triggered

by aggregate or sectoral shocks that reduce the return on safer loans. Excessive

risk-taking can be avoided by raising bank capital requirements, but unnecessarily

tight requirements lower welfare by limiting liquidity producing bank deposits.

Consequently, optimal capital requirements are dynamic (or state contingent). We

provide examples in which a Ramsey planner would raise capital requirements: (1)

during a downturn caused by a TFP shock; (2) during an expansion caused by an

investment specic shock; and (3) during an increase in volatility that has little

eect on the business cycle. In practice, the economy is driven by a constellation of

shocks, and the Ramsey policy is probably beyond the policymaker's ken; so, we also

consider implementable policy rules. Some rules can mimic the optimal policy rather

well but are not robust to all the calibrations we consider. Basel III guidance calls

for increasing capital requirements when the credit to GDP ratio rises, and relaxing

them when it falls; this rule does not perform well. In fact, slightly elevated static

capital requirements generally do about as well as any implementable rule.

iii

In the second chapter, we incorporate shadow banks into a quantitative general

equilibrium model to study the impact of non-bank nancial intermediation on capital

regulation policies. Shadow banks do not have deposit insurance and have a relatively

lower price of taking risk compared to regulated banks. We nd that when the

returns to safer projects are depressed, migration of credit from shadow banks toward

traditional banks boosts excessive risk-taking incentives of regulated banks. Shadow

banks magnify the impact of business cycle shocks on capital requirements that

prevent excessive risk-taking. Moreover, the failure to account for the interaction

between regulated banks and shadow banks does not only blunt policy prescriptions,

but may also lead to calling for increases in capital requirements when a decrease

would be warranted. We show that capital requirements should also react to shocks

that originate in the shadow banking sector without having a direct inuence on

regulated banks. The Ramsey planner, equipped with capital requirements and a tax

on shadow bank debt, implements the welfare-maximizing optimal policy by moving

both instruments signicantly to stabilize the eect of a TFP shock on the loans of

regulated banks. Our results stress the importance of considering the interactions of

banks and shadow banks for designing macroeconomic policies.

Index words: Capital Requirement, Risk-taking, Liquidity Provision, DepositInsurance, Policy Rules, Shadow Banks, Basel III, MinneapolisPlan.

iv

Dedication

To my parents and my brother.

v

Acknowledgments

I would like to thank my advisor and committee members, Behzad Diba, Matthew

Canzoneri, and Luca Guerrieri, for their help and support. The rst chapter of this

dissertation is based upon my joint work with them. I beneted a lot from insightful

discussions with them. Behzad and Matthew were always very generous with their

time. Luca, being an unlimited source of ideas, was also a great example for me of how

to do quantitative work in economics research. I am also grateful to the participants

of the macro advising group organized by Toshihiko Mukoyama for useful comments

and suggestions.

I am also thankful to the organizers of Netherlands Carillon live Concerts in

summer 2019. I really enjoyed guest carillonneurs playing two-hour long concerts of

dierent pieces of music on the carillon's 50 bells on summer Saturdays. This gave

me the strength to nish my thesis. The music of this dissertation is Gnossienne No.

1 by Erik Satie. Finally, I would like to thank my mother who, despite being 10,000

kilometers away from me, mentally accompanied me throughout the long journey of

the Phd program.

vi

Table of Contents

Chapter

1 Optimal Dynamic Capital Requirements and Implementable CapitalBuer Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Calibration and Steady-State Capital Requirements . . . . . . . 171.4 Numerical Methods and Ramsey Policy . . . . . . . . . . . . . . 191.5 Optimal Dynamic Capital Requirements . . . . . . . . . . . . . 221.6 Implementable Buer Rules . . . . . . . . . . . . . . . . . . . . 281.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.8 Tables and Figures . . . . . . . . . . . . . . . . . . . . . . . . . 36

2 Dynamic Bank Capital Regulation and Optimal MacroprudentialPolicies in the Presence of Shadow Banks . . . . . . . . . . . . . . . . . 472.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.3 Calibration and Experiments . . . . . . . . . . . . . . . . . . . . 812.4 Quantitative Results I. Responses to Shocks . . . . . . . . . . . 852.5 Quantitative Results II. Comparison with the Model of Commercial

Banks Only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 902.6 Quantitative Results III. Additional Instrument and Permanent

Increase in Capital Requirements . . . . . . . . . . . . . . . . . 942.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002.8 Tables and Figures . . . . . . . . . . . . . . . . . . . . . . . . . 103

Appendix

A Appendix for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 118A.1 The Bank's Problem . . . . . . . . . . . . . . . . . . . . . . . . 118A.2 The Non-Financial Firm's Problem . . . . . . . . . . . . . . . . 128A.3 The Government . . . . . . . . . . . . . . . . . . . . . . . . . . 134A.4 Choice of Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136A.5 Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . . . . 141A.6 Calibration of τ . . . . . . . . . . . . . . . . . . . . . . . . . . . 143A.7 Robustness Checks . . . . . . . . . . . . . . . . . . . . . . . . . 144

vii

B Appendix for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 145B.1 Expression of Net Cash Flow . . . . . . . . . . . . . . . . . . . 145B.2 Share of Non-Defaulted Deposits . . . . . . . . . . . . . . . . . . 147B.3 Choice of Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147B.4 The Shadow Bank's Problem . . . . . . . . . . . . . . . . . . . . 153B.5 The Commercial Bank's Problem . . . . . . . . . . . . . . . . . 156B.6 The Firm's Problem . . . . . . . . . . . . . . . . . . . . . . . . 160B.7 The Government . . . . . . . . . . . . . . . . . . . . . . . . . . 165B.8 Household . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165B.9 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

viii

List of Figures

1.1 Negative TFP Shock. . . . . . . . . . . . . . . . . . . . . . . . . . . . 421.2 Positive Investment Shock. . . . . . . . . . . . . . . . . . . . . . . . . 431.3 Volatility Shock. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441.4 Sensitivity Analysis, TFP Shock. . . . . . . . . . . . . . . . . . . . . 451.5 Robustness Checks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.1 Timeline for Two Types of Bankers. . . . . . . . . . . . . . . . . . . . 1072.2 Impulse Responses to a TFP Shock. . . . . . . . . . . . . . . . . . . 1082.3 Impulse Responses to a Capital Quality Shock. . . . . . . . . . . . . 1092.4 Impulse Responses to a Shadow Bank Shock. . . . . . . . . . . . . . 1102.5 Endogenous Responses to a TFP Shock Depending on the Presence of

Shadow Banks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1112.6 Sensitivity Analysis: Relative Responses of Capital Requirements to a

TFP Shock. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1122.7 Endogenous Responses to a Capital Quality Shock Depending on the

Presence of Shadow Banks. . . . . . . . . . . . . . . . . . . . . . . . 1132.8 Endogenous Responses to a Commercial Bank Shock Depending on the

Presence of Shadow Banks. . . . . . . . . . . . . . . . . . . . . . . . 1142.9 Comparison of Policy Reactions to a TFP Shock. . . . . . . . . . . . 1152.10 Policy Responses to a TFP Shock Depending on Instruments. . . . . 1162.11 A Capital Requirement Permanent Shock: 2 % Rise in γt. . . . . . . 117

ix

List of Tables

1.1 Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.2 Calibration 1, Shock Processes. . . . . . . . . . . . . . . . . . . . . . 371.3 Calibration 1, Variance Decompositions. . . . . . . . . . . . . . . . . 371.4 Calibration 1, Matching Moments. . . . . . . . . . . . . . . . . . . . . 371.5 Calibration 2, Shock Processes. . . . . . . . . . . . . . . . . . . . . . 381.6 Calibration 2, Variance Decompositions. . . . . . . . . . . . . . . . . 381.7 Calibration 2, Matching Moments. . . . . . . . . . . . . . . . . . . . . 381.8 Simple Rules with Calibration 1. . . . . . . . . . . . . . . . . . . . . 391.9 Simple Rules with Calibration 2. . . . . . . . . . . . . . . . . . . . . 401.10 The Eciency of Static Buers. . . . . . . . . . . . . . . . . . . . . . 412.1 Comparison of Two Types of Banks. . . . . . . . . . . . . . . . . . . 1032.2 Illustrating the Eects of Higher Risk on Dividends. . . . . . . . . . . 1042.3 Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052.4 Steady-State Values. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

x

Chapter 1

Optimal Dynamic Capital Requirements and Implementable Capital

Buffer Rules

1.1 Introduction

A protracted period of low returns on safe assets followed in the wake of the global

nancial crisis, and this trend is expected to continue for the foreseeable future. These

low returns have raised concerns that nancial intermediaries will be tempted to reach

for higher yields by taking excessive (or socially inecient) risks. We formalize these

concerns by developing a dynamic macroeconomic model in which limited liability

and deposit insurance provide incentives for excessive risk-taking: a sudden fall in the

returns on safe assets or more precisely, a wider spread between expected returns

on safe and risky assets can trigger an extended period of excessive risk-taking,

with consequences for consumption and business investment. Prudential policy can

curb these incentives by raising bank capital requirements; indeed, dynamic (or state

contingent) capital requirements can eliminate the incentives entirely. But this may

come at the expense of reducing bank deposits, which provide liquidity services to

households. We will explore this tradeo, both theoretically and quantitatively.

The theoretical literature on the dynamic nature of optimal capital requirements

in a full macroeconomic framework is rather thin. The paucity of results may be due

to the technical complications that arise when we combine endogenous risk-taking,

aggregate economic uctuations, and Ramsey policies. Our DSGEmodel combines key

1

elements of the literature on nancial frictions and macroeconomic stability. Following

Van den Heuvel (2008), banks can lend to safe rms or risky rms. The return

on a risky loan is the same as the return on a safe loan except that the risky loan is

also exposed to an idiosyncratic shock with negative expected value; risky rms are

therefore socially inecient. The only reason a prot maximizing bank would fund

a risky rm is that limited liability shields it from downside risk; if the expected

return on safe loans are expected to fall, the bank may take a ier on a risky loan.

Banks fund their lending by issuing deposits and equity to households. Deposits are

the cheaper source of funding since they provide liquidity services and government

deposit insurance makes them a safe asset. Capital requirements increase funding costs

and make banks keep more skin in the game. This eect reduces their temptation to

take excessive risks. Van den Heuvel's model is static; it does not allow for aggregate

economic uctuations or increases in market volatility. In our model, macroeconomic

shocks lead to business cycles, and they can trigger excessive risk-taking by decreasing

the expected return on safe loans. We provide examples in which a Ramsey Planner

would raise capital requirements during both cyclical booms and busts, and raise

capital requirements in response to an increase in volatility that has little consequence

for the business cycle.

In practice of course, the economy is driven by a multiplicity of shocks. Policy

makers have to respond to the full stochastic structure of the economy, which may

prove daunting. So, we also consider simple policy rules that try to mimic the optimal

policy by responding to just one or two endogenous variables. To this end, we use the

simulated method of moments to calibrate our model's dynamic structure, which in

turn allows us to calculate optimal dynamic capital requirements when the model

economy is driven by a full constellation of shocks. We generate model data in

that stochastic environment, and we regress the optimal capital requirements on

2

candidate sets of endogenous variables. Some simple rules capture the optimal capital

requirements rather well; that is, they have an R-square statistic close to 1, at least

for some calibrations.

The Basel III accords advocated cyclical capital buer: during credit booms (of

increases in the credit to GDP ratio), capital requirements would be tightened; during

contractions they could be loosened. These prescriptions which we will call the

Basel rule sound sensible, and they should be implementable in practice. But in

out model, the Basel rule does not come close to mimicking the optimal response

of the Ramsey Planner; other simple rules, or even static capital requirements, do

better. In fact, slightly elevated static capital requirements generally do about as well

as any implementable policy rule.

A number of contributions to the literature address the pro-cyclical bias of Basel II

guidelines and the counter-cyclical buers of Basel III.1 Referring to earlier arguments,

Kashyap and Stein (2004) argue that capital requirements should be lower when

bank capital is scarce, and they suggest this is more likely to be the case during

recessions; thus, the pro-cyclical bias of Basel II guidelines would seem undesirable.

The normative models of the banking literature, however, highlight frictions in bank

funding and in lending relationships that can aect the optimal cyclical behavior of

capital requirements. Repullo and Suarez (2013) develop a model in which optimal

policy can imply cyclical variations very similar to those of Basel II. In their model,

the tightening of capital requirements during recessions does have the drawbacks

noted in earlier commentary on Basel II, but it can be nonetheless optimal because

it reduces the frequency of bank failures during recessions.

1The procyclical bias of Basel II guidelines is attributed to risk-based capitalrequirements, which eectively tighten during recessions as the default risk on bank assetsincreases.

3

By contrast, Gersbach and Rochet (2017) present a model in which bank funding

frictions lead to higher optimal capital requirements during economic expansions,

and lower requirements during recessions. In their model, funding frictions make

bank lending too low (compared to the ecient benchmark) during expansions, and

even more so during contractions. Optimal policy raises the capital requirement

to curb lending during expansions because this improves the funding capacity of

banks during contractions (under complete markets). Malherbe (2020) develops a

quantitative model with the same policy implication. In his model, a pecuniary

externality makes bank lending too high compared to the ecient benchmark. A

positive TFP shock increases bank capital (proportionally) more than it increases the

optimal level of lending. The capital requirement, which is binding in equilibrium,

must rise to curb bank lending during TFP-driven booms.

The models of optimal cyclical variation above focus on the eects of capital

requirements on the volume of bank credit. By contrast, our model focuses on the

composition of bank credit we have in mind risk-taking decisions like the choice

between prime and subprime mortgages before the 2007-2009 nancial crisis, or

participation in syndicated loans to highly leveraged rms more recently. Our focus,

of course, is not intended to negate the importance of risks associated with bank

leverage (and the volume of bank credit). High leverage can, for example, increase

the risk of bank runs in environments like the models of Angeloni and Faia (2013) or

Gertler and Kiyotaki (2015). Our focus, we think, is complementary to the emphasis

on leverage and the volume of credit in much of the literature. It makes, for example,

a case for cutting capital requirements in a TFP-driven boom; policymakers may

have to weigh this consideration against, say, the risk of a run on the liabilities of

shadow banks. Gomes, Grotteria, and Wachter (2018) share our emphasis on risks

that arise from endogenous changes in the composition of bank credit, but not our

4

focus on what this implies about optimal capital requirements. They construct a

model that deliberately decouples their macro economy from the nancial side and

banking-sector activities. In their model, output (consumption) follows an exogenous

stochastic process with an exogenous and time-varying risk of a large drop in output.

Banks can make risky loans to rms or hold less risky government bonds, but their

portfolio decisions have no macroeconomic consequences in the (partial equilibrium)

model. They show that although credit expansions have no causal eect, they predict

output declines in the model. The connection arises from the optimal response of

leverage and the composition of bank credit to anticipated macroeconomic risks.2 An

econometrician working with data generated by this model would observe periods of

rapid credit expansion followed by periods of higher default rates on bank loans, and

declines in output. More specically, Gomes, Grotteria, and Wachter (2018) show that

their model can replicate the empirical evidence presented by Schularick and Taylor

(2012), Jordà, Schularick, and Taylor (2016), and Mian, Su, and Verner (2017). This

type of evidence is often cited in support of Basel-III style counter-cyclical regulation.

The main point of Gomes, Grotteria, and Wachter (2018) is to question the causal

interpretation of this evidence.

The papers by Martinez-Miera et al. (2012), Collard et al. (2017) and Begenau

(2019) also examine capital requirements from a perspective similar to ours, but

they do not share our focus on cyclical variation in optimal capital requirements.

Martinez-Miera et al. (2012) develop a model with systemic risk abstracting from

aggregate shocks. Collard et al. (2017) focus on interactions of optimal monetary

and prudential policies, in a setting that keeps bank failures o the equilibrium path.

2In the model, a higher risk of a future output decline increases the risk premium andalso erodes the franchise value of banks. The optimal response of bankers has an element ofgambling for resurrection, they increase leverage and tilt their asset portfolios towards riskyloans.

5

Begenau (2019) develops a quantitative business-cycle model to determine the optimal

level of a constant capital requirement.

Our work is also related to analyses that evaluate simple rules for capital

requirements, but which may not call for capital requirements in the long run;

we shall see that limited liability implies an ongoing need for capital requirements.

The rest of the paper proceeds as follows. Section 1.2 describes the model. Section

1.3 discusses the model calibration, including the choice of steady-state capital

requirements. Section 1.4 describes our numerical methods for the model solutions.

Section 1.5 presents the responses to dierent shocks and discusses the Ramsey-

optimal capital requirements. Section 1.6 considers some simple implementable rules.

Section 1.7 concludes.

1.2 The Model

Here we augment a standard RBC model to include banks that enjoy limited

liability and government deposit insurance. These are the main features that allow

for excessive, or socially inecient, risk-taking, and of course the RBC framework

allows for macroeconomic shocks that cause business cycles. Our model consists

of households, banks, non-nancial rms, and a government whose sole purpose is

to provide bank deposit insurance. Banks are at the heart of our model, but the

exposition is smoother if we begin with the less exciting rms and households.

But rst, a note on notation: There are a measure one continua of households,

banks and non-nancial rms. In what follows, small letters denote individual

households, banks or rms; capital letters represent aggregate values. Safe rms

(dened below) carry a superscript s; risky rms carry a superscript r.

6

1.2.1 Non-Financial Firms

Non-nancial rms are competive, and they earn zero prots in the end. There

are goods producing rms and capital producing rms. We begin with the former.

Goods Producing Firms:

Firms live for just two periods. A rm born in period t, obtains a bank loan, lt, to

buy the capital, kt+1, that it will use for production in period t + 1; so, lt = Qtkt+1,

where Qt is the price of capital. The ex-post return on the loan is Rt+1lt = Rt+1Qtkt+1,

where we shall soon see that Rt+1 is the rate of return on capital ownership. So, these

bank loans might be better described as equity positions.

There is a continuum of rms of measure 1. But the rms come in two types:

safe rms face only aggregate shocks, while risky rms face both aggregate shocks

and idiosyncratic shocks.

In period t+1, a safe rm hires labor, hst+1, to produce yst+1 = At+1(kst+1)α(hst+1)1−α,

where At+1 is an aggregate TFP shock. When a safe rm takes the loan in period t, it

knows that the rm will hire the optimal hst+1 next period. So, the safe rm chooses

lst and kst+1in period t, and then hst+1 in period t+ 1, to

maxlst ,k

st+1

Et

ψt,t+1

[maxhst+1

(yst+1 + (1− δ)Qt+1k

st+1 −Wt+1h

st+1 −Rs

t+1lst

)], (1.1)

where δ is the depreciation rate of capital, Wt+1 is the real wage rate, and ψt,t+i =

β λct+iλct

is the stochastic discount factor of the households, which are described further.

The rst order conditions for this maximization problem imply

Etψt,t+1R

st+1

= αEt

ψt,t+1

[At+1

Qt

(hst+1

kst+1

)1−α

+ (1− δ)Qt+1

Qt

], (1.2)

where the rst term within the brackets is the rental rate on a unit of capital, and

the second term is the capital gain on a non-depreciated unit of capital.

7

A risky rm employs the technology yrt+1 = At+1

(krt+1

)α(hrt+1

)1−α+ εt+1k

rt+1,

where εt+1 is an idiosyncratic shock that follows a Normal distribution G with a

negative mean, − ξ, and standard deviation τ :3

PDF of εt+1 : g(εt+1) =1√

2πτ 2e−

(εt+1+ξ)2

2τ2 ,

CDF of εt+1 : G(εt+1) =1

2

[1 + erf

(εt+1 + ξ

τ√

2

)].

(1.3)

The risky rm chooses lrt and krt+1, and then hrt+1, to

maxlrt ,k

rt+1

Et

ψt,t+1

[maxhrt+1

(yrt+1 + (1− δ)Qt+1k

rt+1 −Wt+1h

rt+1 −Rr

t+1lrt

)]. (1.4)

The rst order conditions for this maximization, and equation (1.5) below, imply that

EtRrt+1 = EtR

st+1−

ξQt

where we use Etεt+1 = −ξ. The idiosyncratic shock makes the

risky rm a bad bet (absent limited liability). Moreover, the zero prot conditions

actually imply that Rrt+1 = Rs

t+1 + εt+1

Qt. So the idiosyncratic shock lowers the expected

value, and increases the variance, of the return on a loan to a risky rm. These loans

are socially inecient, or in our language excessively risky.

Note nally that the marginal product of labor for safe and risky rms is (1 −

α)A(kit+1/hit+1)α where i denotes the type of rm (i ∈ s, r). Labor is mobile across

rms, and both types of rms face the same wage rate. So, the rst order conditions

for labor in period t+ 1 imply the capital labor ratios equalize across sectors.

krt+1/hrt+1 = kst+1/h

st+1. (1.5)

Appendix A.2 provides details on aggregation across rms. We show that there is a

representative safe rm that produces

Y st+1 = At+1(Ks

t+1)α(Hst+1)1−α.

There is also a representative risky rm that produces

Y rt+1 = At+1

(Krt+1

)α(Hrt+1

)1−α − ξKrt+1.

3erf(x) = 1√π

x−x e

−v2dv = 2√

π

x0 e−v

2dv.

8

Capital Producing Firms

At the end of period t, goods producing rms sell their capital to (competitive)

capital producing rms. Letting Igt denote gross investment, the evolution of capital

depends on

It = ηt

[1− φ

2

(IgtIgt−1

− 1

)2]Igt , (1.6)

where ηt is an investment specic technology shock, and φ is a measure of the severity

of investment adjustment costs. The aggregate capital stock evolves according to

Kst+1 +Kr

t+1 = It + (1− δ) (Kst +Kr

t ) . (1.7)

The capital producing rms are owned by households, and solve the problem

maxIgt+i

Et

∞∑i=0

ψt,t+i

Qt+iηt+i

[1− φ

2

(Igt+iIgt+i−1

− 1

)2]Igt+i − I

gt+i

. (1.8)

1.2.2 Households

The household's problem is to

maxCt,Dt,Es,t,Er,t

E∞∑t=0

βt

[(Ct − κCt−1)1−ςc − 1

1− ςc+ ς0

D1−ςdt − 1

1− ςd

], (1.9)

subject to

Ct +Dt + Est + Er

t = Wt +Rdt−1Dt−1 +Re,s

t Est−1 +Re,r

t Ert−1 − Tt, (1.10)

Est ≥ 0,

Ert ≥ 0.

Households value consumption, Ct, and they value bank deposits, Dt, which

provide liquidity services. We put deposits in the utility function in lieu of modeling

a particular transactions technology. And for simplicity, we assume that households

supply labor inelastically; Wt is the real wage, and we have normalized the supply of

9

labor to be one.4 Household assets include deposits,Dt, which pay a gross real rate Rdt ,

and two types of bank equity: Es,t is equity in a safe bank, which lends to a safe rm

and pays Re,st+1 next period; E

rt is equity in a risky bank, which lends to a risky rm

and pays Re,rt+1. The returns on equity are of course not known when the household

invests. By contrast, the return on deposits is known, and deposits are protected by

deposit insurance; deposits are the safe asset in our model. Finally, households pay

lump sum taxes, Tt, to fund the government's deposit insurance program.

The household's rst order conditions include:

C : (Ct − κCt−1)−ςc − βκEt (Ct+1 − κCt)−ςc − λct = 0, (1.11)

D : ς0D−ςdt − λct + Etβλct+1R

dt = 0, (1.12)

Es : −λct + Etβλct+1Re,st+1 + ζst = 0, (1.13)

Er : −λct + Etβλct+1Re,rt+1 + ζrt = 0, (1.14)

where λct, ζst and ζrt are the Lagrangian multipliers for the budget constraint and the

two non-negativity constraints.

If households did not value deposits for their liquidity services (ς0 = 0), (1.12)

would be the standard RBC Euler equation, and Rdt would be the standard CAPM

rate. But households do value deposits in our model (ς0 > 0), and Rdt is below the

CAPM rate. Equity is not a safe asset, and it does not provide liquidity services.

So, deposits will be the cheaper source of funding for banks.5 This fact will play an

important role in what follows.

4While the total supply of labor is xed, its distribution across safe and risky rms ismarket determined.

5More precisely, the rst order conditions imply Etβλct+1

(Re,jt+1 −Rdt

)= ς0D

−ςdt −ζjt > 0

for j = s, r.

10

1.2.3 Banks

Banks are at the heart of our model, and we set the stage by describing their

incentives to take excessive risk. Then, we discuss the banking sector in some detail.

Incentives to Take Excessive Risk and Capital Requirements

We saw from the section on rms that EtRrt+1 < EtR

st+1. So, why would a prot

maximizing bank ever invest in a risky rm? Limited liability and government deposit

insurance are the culprits here. Limited liability shields the bank from downside risk.

Moreover, and perhaps counter intuitively, deposit insurance actually subsidizes risk-

taking; it makes bank deposits the safe asset, lowering the cost of issuing deposits,

and allowing the bank to expand its portfolio of safe or risky loans. In what follows,

we shall see that if the expected return on investment in a safe rm falls, due say to

a negative TFP shock, the bank may be tempted to take a ier on the risky rm.

As we shall see, capital requirements are a potential remedy for excessive risk-

taking. In what follows, we will consider a requirement that says equity nance cannot

fall below a fraction γt of the bank's loans. A high γt requires the bank and its equity

holders to keep more skin in the game, and it shrinks the bank's portfolio since equity

nance is more expensive than deposit nance.

The Banking Sector

Now we can specify the banking sector in more detail. A measure one continuum

of perfectly competitive banks are born each period, and they live for two periods.

In the rst period, a bank issues equity and deposits to households, and uses the

proceeds to make loans to rms; in the second period, the bank receives the return

on its investments and liquidates its assets and liabilities.

11

More specically, in period t the bank creates a loan portfolio by directing a

fraction σt of its loans to a risky rm; the remainder of its loans go to a safe rm.6

Since Rrt+1 = Rs

t+1 + εt+1

Qt,the ex-post return on the portfolio will be Rs

t+1 +σtεt+1

Qt. Note

that nwt+1 ≡(Rst+1 + σt

εt+1

Qt

)lt−Rd

t dt is the bank's net worth in period t+ 1. If it is

positive, the bank pays its depositors and distributes the rest to its equity holders. If

it is negative, the bank declares bankruptcy; its depositors are protected by deposit

insurance, but its equity holders get nothing. So, the bank's dividends payment is

divt+1 = max [nwt+1, 0] . (1.15)

The bank seeks to maximize the expected return of its equity holders, whose

stochastic discount factor is β λct+1

λct. Let ε∗t+1 be the realization of the idiosyncratic

shock below which the bank's net worth is negative; i.e.(Rst+1 + σt

ε∗t+1

Qt

)lt−Rd

t dt= 0.

Since the distributions of aggregate and idiosyncratic shocks are independent of each

other, we can nest expectations with respect to the idiosyncratic shock within the

expectation of the aggregate and idiosyncratic shocks, and the representative bank's

maximization problem can be written as:

maxlt,dt,et,σt

Et

βλct+1

λct

∞

ε∗t+1

nwt+1 dG(εt+1)

− et, (1.16)

subject to

lt = et + dt,

et ≥ γtlt, (1.17)

lt ≥ 0,

σ ≤ σt ≤ σ,

6Our assumption that a bank only deals with one safe and one risky rm comes at noloss of generality because all the safe rms are identical, and diversication among the riskyrms does not take full advantage of the bank's limited liability. See Collard et al (2017) fora more formal exposition of this result.

12

where et is equity issued to households. The rst constraint is the bank's balance

sheet, and the second is the bank's capital requirement. The fourth imposes limits on

the fraction of a bank's portfolio that can go to safe or risky loans. In our calibrations,

σ is set equal to 0.99 and σ is set equal to 0.01; so, banks can get very close to totally

safe or totally risky portfolios if they so choose.7

The bank's rst-order conditions can be found in Appendix A.1. In the next

subsection, we discuss the bank's basic tradeo when it decides how risky to make its

portfolio of loans.

The Bank's Dividends, and Its Choice of σt.

In the Appendix, we derive the bank's expected (discounted) dividend function,

Ω(σt; lt, dt, et) = Et

[βλct+1

λctdivt+1

], (1.18)

where

ω1 ≡(Rst+1 −Rd

t (1− γt)−ξσtQt

)(1−G(ε∗t+1)

),

ω2 ≡(σtQt

)τ√2πe−(ε∗t+1+ξ

τ√

2

)2

.

Remember that ε∗t+1 is the realization of the bank's idiosyncratic shock below which

the bank's net worth is negative; more precisely, ε∗t+1 = − (Qt/σt)[Rst+1 −Rd

t (1− γt)].

1−G(ε∗t+1) is the probability that the bank will not default.

The rst component, ω1, is the return on a loan portfolio with a fraction σt going

to a risky rm; −ξ is the (negative) expected value of the idiosyncratic shock. The

second component, ω2, is a bonus attributable to the bank's limited liability; the

higher is the standard deviation of the idiosyncratic shock, τ, the higher is the upside

potential, while the downside risk is protected by limited liability.

7These limits on σt are necessary for the numerical methods that follow.

13

Increasing σt makes the portfolio more risky. More risk decreases the ex-post return

on the bank's portfolio, but it increases the bonus from limited liability. This is the

tradeo that a bank faces.

1.2.4 The Government

The government provides deposit insurance, and collects taxes to pay for it. Given

the Ricardian nature of the model, a lump sum tax, Tt, can balance the budget each

period without distorting private decision making. In the Appendix, we show the tax

necessary to support the insurance scheme is

Tt = σt−1Lt−1

Qt−1

τ√2πe−(Rdt−1(1−γt−1)Qt−1−R

stQt−1+ξσt−1

σt−1√

2τ

)2

− (1.19)

12

(RstLt−1 − σt−1ξ

Qt−1Lt−1 −Rd

t−1Dt−1

) [1 + erf

(Rdt−1(1−γt−1)Qt−1−RstQt−1+ξσt−1

σt−1

√2τ

)],

where Lt is the aggregate amount of loans provided by the banking sector. As might

be expected, more risk-taking (σt−1) and/or a higher variance (τ) of the idiosyncratic

shock increases the tax required to protect deposits.

1.2.5 Analytical Characterization of Equilibrium

We are able to derive some analytical results that enhance our understanding

of the model's equilibrium, and how to calculate it. Deriving the Ramsey planner's

dynamic capital control policies will however require numerical methods.

Two Propositions and a Corollary

As discussed in the section on households, deposits are a cheaper source of bank

funding than equity. So, a bank will fund as much of its loans by issuing deposits

as is allowed by the capital requirements. We formalize this argument and prove the

following proposition in Appendix A.1.2.

14

Proposition 1. In equilibrium, capital requirements always bind; that is, et = γtlt.

The next proposition, and its corollary, show that we need only consider two values

of the bank's portfolio risk parameter, σt, when we derive the model's equilibrium.

The proposition is established in Appendix A.4.

Proposition 2. The expected dividends function of banks, Ω(σt; lt, dt, et) is convex

in σt. This result holds for arbitrary (and not necessarily continuous) distributions of

the idiosyncratic shock.

Corollary. There are no equilibria with σ < σt < σ.

The intuition for this proposition and its corollary is as follows: If σt is high enough,

the bank will be bankrupt for low values of εt anyway, so it might as well take on

as much risk as possible to maximize the portfolio's upside potential (due to limited

liability) for high values of εt. If σt is low enough, the bank will not be bankrupt even

for low values of εt, and the value of limited liability is negated; the bank might as

well take on the minimum risk to raise the expected value of the portfolio.

Equilibrium and Aggregation

We consider a competitive equilibrium in which each bank takes aggregate prices

as given. The Appendix lists all the equilibrium conditions of our model. In this

subsection, we only present the equilibrium conditions that are not already included in

the preceding subsections. We let µt denote the fraction of banks with risky portfolios

(banks that choose σt =σ) at date t; the remaining fraction 1− µt of banks are safe

banks (σt = σ).

The fraction µt is endogenously determined by the bank-capitalization decision

(equity positions) of households: we have µt =Ert

Ert+Est. At any point in time, the

15

economy may be in a safe equilibrium (with µt = 0), a risky equilibrium (with µt = 1),

or a mixed equilibrium (with 0 < µt < 1).

Each bank within a group (safe or risky) is alike and solves the same maximization

problem in which it chooses lit, dit, e

it according to its type i ∈ s, r. The aggregate

loans to the (representative) safe rm come from two sources: 1) from all safe banks

(of measure 1−µt) that allocate 1−σ share of their loan portfolio to safe projects and

2) from all risky banks (of measure µt) that allocate 1− σ share of their loan portfolio

to safe projects. Therefore, the equilibrium restrictions linking our bank-level and

rm-level variables representing loans are

QtKst+1 = (1− σ) (1− µt) lst + (1− σ)µtl

rt .

Similarly,

QtKrt+1 = σ (1− µt) lst + σµtl

rt .

The aggregate bank loans are linked to the individual bank loans by: Lrt = µtlrt and

Lst = (1−µt)lst . Therefore, we can describe the latter two equations by using aggregate

loans

QtKst+1 = (1− σ)Lst + (1− σ)Lrt ,

QtKrt+1 = σLst + σLrt .

The equity positions taken by households, in turn, determine the equity positions

of individual banks: Ert = µte

rt and Es

t = (1 − µt)est . The returns on the equity

positions taken by households at date t are linked to the dividends paid by banks at

date t+ 1. We have:

ErtR

e,rt+1 =µtEε,t divt+1 = (ωr1 + ωr2)Lrt ,

EstR

e,st+1 =(1− µt)Eε,t divt+1 = (ωs1 + ωr2)Lst ,

16

where Eε,t is the expectation taken with respect to the idiosyncratic uncertainty. We

use that Eε,t divt+1 is linear in loans and ω1 and ω2 are dened in equation (1.18).

The superscript index of the variables corresponds to the choice of risk in equation

(1.18): σr = σ and σs = σ, respectively. Deposits held by households are issued by

(safe and risky) banks: Dt = Dst +Dr

t where Dst = Lst − Es

t and Drt = Lrt − Er

t .

The equilibrium restrictions linking our aggregate and individual rm-specic

variables are straightforward but cumbersome in terms of notation. We state the

restrictions in Appendix A.2.3. The market-clearing conditions for labor, capital, and

goods are

Hst +Hr

t = 1,

Kst +Kr

t = Kt,

and

Y st + Y r

t = Ct + Igt .

1.3 Calibration and Steady-State Capital Requirements

Our calibrated parameters are reported in Table 1.1. We use standard values for

the discount factor β, the capital share α, the intertemporal elasticity of substitution

%c, and the depreciation rate δ. We choose ςd = 1.1 that makes our deposit utility

function close to the log case. In the literature, its values range from 0 (linear utility

as in Stein (2012)) to 1.4 in Begenau (2019)8 and 1.7 in the model with shadow banks

as in Begenau and Landvoigt (2018). So our choice of 1.1 is consistent with the range

of values considered by the literature.

We consider loans to be risky if they are made by banks with a debt-to-EBITDA

ratio above 6 in the leveraged loan market.9 We choose τ, the standard deviation

8Note that Begenau (2019) has a slightly dierent specication of the utility function.9EBITDA is earnings before interest, taxes, depreciation, and amortization.

17

of the risky rm's idiosyncratic shock, to match the conditional variance of returns

on a risky project to the conditional variance of returns from lending to a rm with

a debt-to-EBITDA ratio of 6 around the non-stochastic steady state. Appendix B.9

provides the details of our procedure. Given τ , we x the value of ξ, the average

penalty from nancing risky projects, so that a 10% steady state capital requirement

prevents lending to risky rms. Why do we not try to calculate an optimal steady

state capital requirement? We show in Appendix A.7 that alternative choices of τ

and ξ would support a wide range of steady state capital requirements. This suggests

that a model like ours is not suitable for any attempt to pin down the optimal steady

state value. We note that our choice of 10% is consistent with the static values of

capital requirements proposed by Basel III; it also lies within a span of values usually

considered in the literature on optimal capital regulation

To match the data on interest rate spreads, we introduce costs of banking in our

quantitative model. Costs of banking include operating expenses; we assume they

are linked to the provision of loans. In particular, each period the bank incurs an

additional cost, flt, that is paid out of its current prots. And when a bank defaults,

the household has to pay a higher tax to the deposit insurance fund to cover this cost

of banking. Both Appendix A.3.2 and Appendix A.1.6 provide further details on the

implications of this cost for the lump sum tax, Tt, and on the rst order conditions for

the optimization problem of banks. We choose f to make the average spread between

the safe loan rate and the deposit rate equal to 2.26 percent per annum, a value taken

from Collard et al. (2017). The parameter ς0 measures the utility of deposits in the

steady state. We set the value of ς0 to make the interest rate on bank deposits equal

0.86% per quarter, a value we borrow from an estimate in Begenau (2019). Finally,

our setup for investment adjustment costs mimics the one analyzed in Altig et al.

18

(2011). We pick the value of φ consistent with the broad range from their analysis

and related literature.

1.4 Numerical Methods and Ramsey Policy

Since our model involves occasionally binding nonnegativity constraints on bank

loans, we need to rely on nonlinear solution methods. We apply the Occbin toolkit

developed in Guerrieri and Iacoviello (2015). This solution algorithm modies a rst-

order perturbation method and employs a guess-and-verify approach to obtain a

piecewise linear solution.10 The solution reects the endogenous transition between

regimes, depending on the size of a shock and the state vector, and thus it is highly

nonlinear. The algorithm has advantages over nonlinear projection methods because

it is computationally fast and can be applied to nonlinear models with a large number

of state variables, such as ours.

We need to impose constraints in our algorithm to rule out short-selling of assets

(or negative loans). To see why, suppose banks are in the safe equilibrium; in this case,

risky loans are overpriced compared to safe loans (because expected returns to risky

loans are relatively lower in the safe equilibrium); absent short-selling restrictions,

each bank would want to short risky loans (i.e., acquire a negative amount of risky

loans). Similar reasoning applies to the risky equilibrium, in which the banks in our

model would short safe loans if we did not impose a short-selling restriction. In either

of these cases, arbitrageurs would force the expected returns on safe and risky loans

to equality. And this would result in the mixed equilibrium described in 1.2.5

To compute optimal capital requirements, we focus on the Ramsey problem,

conditional on the restrictions of the decentralized equilibrium. The Ramsey program

10See Guerrieri and Iacoviello (2015) for a discussion of the accuracy of this type of solutionmethod.

19

selects the path of capital requirements that maximizes the conditional expectation

of households' utility as of time zero. More precisely, following a "dual" approach, the

Ramsey planner chooses the sequence of capital requirements γ∗t ∞t=0 to maximize

the household utility function, (1.9), subject to the equilibrium conditions implied

by the optimality conditions of households, rms and banks, and the market clearing

conditions. The non-negativity and short-selling restrictions that we noted above

complicate this Ramsey problem. We proceed by proposing a natural candidate for

the solution and then verifying that the proposed solution does indeed maximize the

objective function, (1.9).

Our proposed solution is to consider the sequence of capital requirements γ∗t ∞t=0

that is set at the lowest level necessary to prevent risk-taking given the realizations

of the shocks at any date t. This sequence dominates any alternative pathγAt∞t=0

in

which γAt = γ∗t for t 6= tk and γAt = γ∗t + ∆ for t = tk and some ∆ 6= 0. When ∆ > 0,γAt∞t=0

is welfare dominated by γ∗t ∞t=0 because a higher capital requirement in

period tk leads to welfare losses from the reduced amount of liquidity services without

altering risk-taking incentives. This holds for any tk and does not depend on the size

of ∆ > 0. When ∆ < 0, banks switch to funding socially inecient risky projects in

period tk underγAt∞t=0

. The decrease in the capital requirement involves an output

loss of ξK from making risky loans, but it may increase the liquidity services that enter

into household utility. The trade-o between these two considerations determines the

impact on welfare. For a small decrease in capital requirements (i.e. negative values

of ∆ close to zero), the former consideration is more important. Since banks jump to

the risky equilibrium, the lower capital requirement entails a discrete drop in welfare,

arising from the drop in output. By contrast, the welfare gain (or loss) associated

with liquidity provision is a second-order change.

20

Our reasoning above establishes that the Ramsey planner's objective function

has a local maximum along the path γ∗t ∞t=0. To show that this is indeed a global

maximum, it remains to consider the welfare eect of a larger decrease in capital

requirements, since in this case liquidity considerations will not be of second order. To

see how liquidity considerations compare to the welfare loss associated with inecient

risk-taking, we compare (numerically) the welfare measure under our candidate for

optimal policy to welfare under an alternative policy that maximizes the benet of

liquidity provision under the risk-taking regime. All the equilibria under the risk-

taking regime have he same level of expected output; so, we only need to consider

the policy that maximizes liquidity provision. The gains from liquidity services are

maximized when γAtk = 0. Therefore, we need to compare conditional welfare under

γ∗t ∞t=0 to the alternatives that let the capital requirement go down to zero, in some

periods.

To check quantitatively if setting capital requirements to zero becomes optimal in

response to shocks, we use a variant of the OccBin algorithm. We consider a horizon

K and construct all possible combinations of periods from 1 to K in which capital

requirements are hardwired to go to zero whenever the switch to the risk-taking

regime is made and are set at γ∗t ∞t=0 otherwise. Then, for each combination, we

calculate the conditional welfare and compare it against the conditional welfare of

keeping capital requirements at γ∗t ∞t=0. We verify that the proposed path of γ∗t

∞t=0

that makes capital requirements just large enough to prevent excessive risk-taking

incentives is, in fact, optimal in our setup.

21

1.5 Optimal Dynamic Capital Requirements

To see what triggers a shift to excessive risk-taking (σt → σ), and how dynamic

capital requirements can oset it, we compare the expected dividends for safe and

risky rms, Ωs ≡ Ω(σ; lt, dt, et) and Ωr ≡ Ω(σ; lt, dt, et) respectively. In turn, as

established in equation (1.18), the term Ω(σt; lt, dt, et) is

Ω(σt; lt, dt, et) = Et

[βλct+1

λctlt (ω1 + ω2)

], (1.20)

where

ω1 ≡(Rst+1 −Rd

t (1− γt)−ξσtQt

)(1−G(ε∗t+1)

),

ω2 ≡(σtQt

)τ√2πe−(ε∗t+1+ξ

τ√

2

)2

,

(In the discussion that follows, we will assume that 1− G(ε∗t+1) < 1). It will also be

recalled that the return from investing in a safe rm is

Rst+1 = α

At+1

Qt

(Hst+1

Kst+1

)1−α

+ (1− δ)Qt+1

Qt

. (1.21)

Finally, recall that ε∗t+1 which also enters the term ω2 is given by

ε∗t+1 = − (Qt/σt)[Rst+1 −Rd

t (1− γt)]. (1.22)

The term ε∗t+1 is the realization of the bank's idiosyncratic shock below which its

net worth is negative. The value of the shield of limited liability is reected in the

presence of the term ε∗t+1 instead of −∞ in the bank's expected prots; see the bank's

maximization problem, (1.16). Accordingly, any force that pushes up the cut-o point

ε∗t+1 will increase the value of the shield of limited liability. All else equal, a decrease

in Rst+1 or an increase in Rd

t will lower the expectation of Ωst+1 relative to Ωr

t+1; so will

an increase in τ or a decrease in ξ; and so will a decrease in Qt. Any of these events

22

could trigger a switch to excessive risk-taking if γt does not counteract these change

and lets Ωrt+1 becomes larger than Ωs

t+1. We elaborate on this description for the case

of three shocks in the examples below.

This is the mechanism at play for the rst period of our TFP (with adjustment

costs) and of our ISP shock (with or without adjustment costs). Moreover, uctuations

inQt can also aect the value of the shield of limited liability. From(Rst+1 + σt

ε∗t+1

Qt

)lt =

Rdt dt it is clear that uctuations in Qt can change the distribution of returns from

risky projects. Decreases in Qt, in particular, open up the distribution of idiosyncratic

returns. The cuto point ε∗t+1 and the value of the shield of limited liability will

increase as long as Rdt dtlt− Rs

t+1 < 0. This is the mechanism at play for ISP shocks

beyond the rst period.

Suppose for the moment that γt is held constant at its steady state value, γ = 10%.

Our calibration makes Ωs > Ωr in the steady state. So, a shock in period t that lowers

expectations of Ωst+1 relative to Ωr

t+1 can, if the spread St+1 ≡ Re,st+1−R

e,rt+1 falls enough,

lead to excessive risk-taking. On the other hand, a state contingent increase in γt can

eliminate excessive risk-taking. Note that capital requirements will have to be state-

contingent. The Planner will not simply set steady state requirements high enough

to avoid any excessive risk-taking no matter what the shock; overly stringent capital

controls are inecient.

1.5.1 A Contractionary TFP Shock

To make contact with the vast literature on real business cycles, we rst consider

the eects of a TFP shock. Figure 1.1 illustrates the eects of a contractionary

TFP shock of one standard deviation (calibrated at 1.5 percent). We use a standard

calibration of the auto-regressive shock process by setting the persistence parameter

to 0.95. In each panel, the dashed line shows what would happen if γt were to be

23

held constant at its steady state value, 10%. St+1 rises enough that banks shift to

risky loans. Also notice that the solid line in each panel shows the response under the

Ramsey-optimal response for capital requirements, an increase, in this case.

The share of risky investments is predetermined, which implies that the response

of output is identical in the rst period for the two cases shown. However, starting

in the second period, the fraction of risky projects nanced shoots up when capital

requirements are kept constant. Some of the risky project nanced will fail, further

depressing output and the returns to equity for households relative to the case with

optimal regulation.

All else equal, a contractionary TFP shock in period t leads to expectations

of lower TFP in period t + 1, which, in the absence of adjustment costs would

depresses the returns for safe projects, Rst+1. However, with investment adjustment

costs this drop in safe returns needs not last more than one period. With investment

adjustment costs (as in our calibration) the price of investment drops, which can

boost At+1

Qt

(Hst+1

Kst+1

)1−α, the rst component of Rs

t+1. Furthermore, as can be seen

from equation (1.21), returns from investing in a safe rm also include capital gains

(1 − δ)Qt+1

Qt. As shown in the gure, the price of investment, Qt, rises monotonically

with optimal capital requirements, further boosting the returns from investing in safe

rms.

Given that safe returns go up, why do capital requirements have to increase then

to avoid excessive risk-taking? Because, as shown in the gure, the expected returns

from investing in risky banks (and therefore risky dividends Ωr) rise even more. As

can be seen from equation (1.22), the drop in Qt pushes up ε∗t+1, which boosts the

value of limited liability. That term is relatively more important for risky project

than for safe projects, which explains why risky projects can become more attractive.

Intuitively, a lower price of investment lowers the costs of gambling on risky projects.

24

From the periods in which risky projects are nanced, lower returns on risky loans

and the increase in taxes to pay for the deposit insurance, will cause a sizable gap

to open up between consumption, investment and output in the case with constant

capital requirements relative to the case with optimal requirements. However after a

few quarters, banks shift back to safe loans.

Why? Recall that the capital labor ratio equalizes across sectors. The falling

capital stock and the xed supply of labor boost the marginal returns for safe

investments. Moreover, while still lower than its steady state, the price of investment

rises, compressing the value of the shield of limited liability. At constant capital

requirements, as the nancing of risky projects abates, output rebounds, leading

to a jump in consumption. Adjustment costs make the jump less prominent for

investment.

Turning more fully to consider the actions of the Ramsey planner, bank capital

requirements jump just enough to keep safe loans attractive. As the falling capital

stock and the xed supply of labor raise the expected return on safe loans and as

the rising investment prices lower the attractiveness of risky projects, the capital

requirements can be lowered. There are no risky loans; so there are no defaults. These

IRF's look familiar from the RBC literature. Notice that the gap between the paths

for output with and without the optimal capital requirements during the period in

which risky loans are nanced is prominently inuenced by the size of the expected

loss on risky loans, ξ in the model.

Takeaways: A one standard deviation shock to TFP causes a 1.5% decrease in

output. However, the optimal capital requirement needs only a modest adjustment;

an increase from 10% to 10.15%. Note also that the Planner increases capital

requirements as the economy goes into recession. Basel III's cyclical buers envisioned

a decrease.

25

1.5.2 An Expansionary Investment Technology Shock.

Here we study a positive ηt shock in equation (1.6) for net investment; the shock

follows an AR(1) process with an auto regressive coecient of 0.8. We calibrate the

size of the shock to increase output by 1% at its peak; so, it is on the same order of

magnitude as the TFP shock above.

Figure 1.2 illustrates the eects of this shock. Once again, the dashed lines show

what would happen if γt were to be held constant at 10%. The shock to investment in

period t increases the supply of capital, Kt+1, and lowers its price, Qt+1. The return

from investing in safe projects, Rst , falls on impact, but then rises. Consequently the

expected return from holding equity in safe banks rises. Nonetheless, as seen for the

TFP shock above, decreases in the price of investment also boost the value of the

shield of limited liability. The expected returns from holding equity in risky banks

rise by more than those from holding equity in safe banks and, without an increase

in capital requirements, there is a switch to nancing risky projects.

Once again, some of these risky loans fail; households lose their equity and their

taxes go up. But here, a wealth eect increases consumption (after a few periods).

The same wealth eect and a strong substitution eect boost investment. After a

few quarters, investment wanes as the eects of the shock abate and as households

increase their consumption. The capital stock falls, increasing the return on capital,

and eventually safe loans become attractive again; at this point the process reverses

itself. Consumption jumps and deposits fall.

The solid lines illustrate what would happen if the Ramsey Planner set the path

γt. The Planner raises the capital requirement just enough to oset the switch to

excessive risk-taking. Consumption and investment rise more in this case since there

are no bankruptcies and equity losses to lower household income.

26

Takeaways: In this example, the Planner raises capital requirements as the

economy goes into a boom period, which is in line with Basel III's cyclical buers.

The required adjustment in the capital requirement is again small; γt rises from 10%

to a little over 10.2%. Here, the dierence in consumption and output with or without

optimal capital requirements is substantial. Perhaps the eort is worthwhile.

1.5.3 A Volatility Shock

In the steady state, the standard deviation of the idiosyncratic shock aecting

risky rms is 5%. This volatility shock increases the standard deviation to 5.5%,

after which it follows an AR(1) process back to 5%. An increase in volatility raises

Ωrt+1 since it enhances the upside potential of risky loans while the downside risk is

protected by limited liability.

Figure 1.3 illustrates the macroeconomic consequences of this volatility shock. As

before, the dashed lines show what would happen if γt were to be held constant. This

rather small shock is big enough to entice banks to switch to risky loans, some of

which will fail, increasing taxes and destroying bank equity. The story that follows

is by now familiar. Consumption and investment fall. Eventually the falling capital

stock raises Rs enough to make safe loans attractive again. As the solid lines illustrate,

the Ramsey Planner would increase capital requirements just enough to eliminate the

excessive risk-taking.

Takeaways: As with the sectoral productivity shock, the eect of this shock

on consumption and output is small, while the adjustment in the optimal capital

requirement is relatively large, from 10% to a little over 11%

27

1.5.4 Sensitivity Analysis

The size of the optimal adjustments in capital requirements is strongly inuenced

by two parameters: the standard deviation of the idiosyncratic shocks, τ , and the

average penalty for taking a ier on a risky loan, ξ. In Figure 1.4, we focus on the

TFP shock. The circles in these diagrams represent the baseline calibrations. The

maximum adjustment in the optimal capital requirements is especially sensitive to

increases in τ . At the outer range of the values of τ that we consider we can boost

the change in capital requirements to a more substantive 0.75 percent in response to

a TFP shock that, at its peak, still contract output by 1.5 percent, just as in Figure

1.1.

1.6 Implementable Buffer Rules

The three Ramsey policies derived in Section 1.5 were in response to three

dierent shocks, each of which was considered in isolation. They may, or may not, be

individually implementable. But in practice policymakers face a much more dicult

challenge: the economy is actually driven by a multiplicity of shocks, all occurring at

the same time; policymakers have to respond to the full stochastic structure of the

economy. In our model, we can derive the Ramsey policy when the economy is hit by

a full constellation of shocks, but it is implausible to think that policymakers would

be able to implement it. So, in this section, we consider simple policy rules in which

the capital requirement responds to one or two observable endogenous variables, and

we ask which, if any, of these rules can closely mimic the actual Ramsey policy. Of

particular interest will be Basel III's capital buer rule in which capital requirements

respond positively to the credit to GDP ratio.

28

This exercise is neither easy nor straightforward. The rst step is to decide which

shocks drive the macroeconomy, and we will see that this decision is not innocent: a

dierent choice of shocks can alter the results. Here, we will consider two calibrations

which t the U.S. data rather well. In Calibration 1, we use the two macroeconomic

shocks TFP and ISP (investment specic) that were considered in the last section;

in Calibration 2, we expand the set of shocks to include a volatility shock.

The statistics we choose to match in the date are selected moments of chained

real GDP, chained real private investment, and the implicit price deator (divided by

price deator for consumption) for chained investment. In the second calibration, we

add selected moments for the DOW Total Stock Market returns.

The next step is to calibrate the shocks to make model moments match moments

in the U.S. data. We allow each shock to follow auto-regressive processes of order

1, and we need to size the persistence parameters and the standard deviations of

the innovations. We also want to size the investment adjustment cost parameter, φ,

and the habits perimeter, κ. To do this, we use a simulated method of moments

(SMM) procedure. For these calibrations, we are focusing on variances, covariances,

and auto-covariances of all the observed variables, with the estimation sample starting

in 1980. We experiment with the SMM optimal weighting matrix. We match observed

moments from bandpass-ltered data (selecting standard business cycle frequencies)

against analogous moments simulated from a sample of 2,000 model observations (also

bandpass ltered).

Finally, it should be noted that we are also calculating and imposing the Ramsey

policy for capital requirements in our model simulations. So, the model output gives

us data for the optimal dynamic capital requirements, and model data are generated

under the assumption that the optimal capital requirements are in place.

29

1.6.1 Moments, Shock Processes and Variance Decompositions

Tables 1.2 and 1.5 show that the calibrations are virtually indistinguishable, as is

also indicated by the values of the distance functions reported at the bottom of each

table. So, both calibrations are worth considering.

The other tables show the calibrated shock processes and variance decompositions

associated with them. In Calibration 1, both shocks are very persistent. But in the

variance decompositions, the TFP shock does all of the work for GDP and investment;

the ISP shock only matters for the investment price. Note also that the ISP shock

explains all the variation in the Ramsey policy setting, γt. In Calibration 2, the TFP

shock once again explains all of the variations in GDP and investment, while now the

volatility shock explains the variation in the Ramsey policy settings.

1.6.2 Implementable Capital Buffer Rules

The Ramsey policy requires full knowledge of all the shocks. It makes its

implementation virtually impossible in practice. We focus next on simple rules

that may be able to mimic the optimal policy; these rules are based on one or

two observable variables, and they are clearly implementable. The Basel III cyclical

buer, which runs o of the credit to GDP ratio, will be of particular interest. We

will also compare these simple rules to more complex rules that are probably not

implementable; the more complex rules may not be much better than the simple

rules.

To derive the policy rules, we use data generated by our simulations. That is, we

regress the Ramsey policy settings on one or more of the endogenous variables (and

a constant). Then, we use a variety of measures to rank the alternative rules. The

rst, and perhaps the most obvious, measure is the R-square of the regression; the

30

higher the R-square, the more closely the rule captures the optimal Ramsey settings.

But there are other measures performance measures that focus on what the rule

actually achieves. A good rule should minimize the frequency excessive risk episodes;

the Ramsey policy eliminates them altogether. But recall that there is a tradeo here.

The frequency of episodes can be minimized, or even eliminated, by simply setting a

static capital requirement at a very high level. This cannot be the only performance

measure that we consider since a very high capital requirement forces banks to limit

the deposits they issue, and deposits are valued for their transactions services. So, a

third measure of a good rule is the average level of deposits that it achieves.

Simple Rules Under Calibration 1

Table 1.8 shows our results for various policy rules under Calibration 1. The rst

column lists the variables in the rule; the second column shows the R-square for the

regression; the third and fourth columns show the regression coecients; the fth

and sixth columns show the average number of risk-taking quarters per 100 years

and the average level of deposits when the steady state capital requirement is 10.1

percent; the seventh and eighth columns show the same statistics when the steady

state capital requirement is raised to 10.3 percent. The last column shows the average

level of deposits achieved by the Ramsey policy; there are no episodes of risk-taking

under this policy. The Ramsey policy sets the standard for the implementable rules

to aspire.

The best implementable rule has capital requirements responding to the investment

price. The R-square in 0.96, so it tracks the Ramsey policy quite well. And in fact, this

simple rule does virtually as well as the Ramsey policy no risk-taking episodes, and

an average level of deposits of approaching 16.25. It is easy to see why this rule does

so well. Figures 1.1 and 1.2 show that for both of the shocks that drive the economy,

31

the investment price falls while the Ramsey capital requirement rises. Moreover, in

Table 1.3, the ISP shock explains all the variation in the Ramsey requirement and

most of the variation in the investment price. So the investment price is a good signal

for what should be done with the capital requirement.

By contrast, the Basel III rule does very poorly. The Basel III prescription is to

tighten capital requirements when the debt to GDP ratio is rising and relax them when

the ratio is falling. In Table 1.8, the R-square is only 0.25 for the regression of Ramsey

policy settings on the credit to GDP ratio. Moreover, the number of risk-taking

quarters per 100 years is very high when the steady state capital requirement is 10.1

percent, and the average level of deposits is very low. Note also the sign of regression

coecient is wrong, at least from the perspective of the Basel III recomendations.

In the next row, we impose a positive coecient, and the results are even worse, as

might have been expected.

With the Basel III rule, raising the steady state capital requirement to 10.3%

brings a huge improvement. The higher steady state capital requirement is doing all

of the work here: the number of risk-taking quarters falls dramatically, and the level of

deposits rises dramatically. The latter result may seem counter intuitive; higher capital

requirements force banks to decrease the proportion of loans that are funded by bank

deposits. The answer to this puzzle is that the level of output and loans is lower

during risk-taking episodes. Limiting the number of risk-taking episodes increases the

average amount of extended credit, and this can raise the level of deposits even when

they account for a lower fraction of the bank's funding.

The next rule in the list has capital requirement responding to the spread

between the expected safe return and the deposit rate. As might be expected from

our discussion at the beginning of Section 1.5, this rule works relatively well, but it

is not clear that it would be implementable in practice. The R-square is 0.83, but

32

the performance measures (where we can calculate them) are not as good as the rule

that reacts to the investment price.

The next rule implausibly assumes that the policymaker can also observe the

individual shocks. Armed with all this information, the R-square is 1.0. However,

this rule does not do any better that the simple investment price rule. The nal rule

conveys the same message.

Simple Rules Under Calibration 2

Calibration 2 adds a volatility shock. Table 1.9 shows that this changes our results

dramatically. Neither the investment price rule nor the Basel III rule works well. The

R-squares are close to zero, and the steady state capital requirements have to be raised

to 11% to obtain good results; here again, the work is being done by the steady state

capital results, and not the rules themselves. The reason for the poor performance of

these rules can be seen in the variance decompositions of Table 1.6. Once again, the

ISP shock explains most of the variance of the investment price, but the volatility

shock explains virtually all of the variance of the capital requirement.

The rule based upon the spread between the expected return on safe loans and

the deposit rate does a better job of tracking the Ramsey settings with an R-square

of 0.82, but once again the steady state capital requirement has to be increased to

11% before the rule does well by the performance measures. The same is true with

the rules based upon much more information. Most of the work seems to be done by

the higher steady state reserve requirements.

The Efficiency of Static Capital Requirements

The results reported above seem to indicate that the static capital requirement is

an important instrument in the regulator's tool kit. Table 1.10 bears that out. Here,

33

there are no rules, just static (or steady-state) capital requirements. The last row

gives the performance measures achieved by the Ramsey Planner. The rst row with

numbers reports the performance measures if the static capital requirement is raised

from the benchmark value of 10 percent to 10.1 percent; they are not good. However,

if the requirement is raised to 10.4 percent for Calibration 1, or 11.5 percent for

Calibration 2, the results are almost as good as those achieved by the Ramsey Planner.

This suggests that the regulator need not bother with dynamic capital controls. If

the static capital requirement is raised to 11.5 percent, the performance measures for

both calibrations are very close to the optimal ones.

Takeaways: Calibrations 1 and 2 show that changing the shock structure that

drives the economy can radically alter the ability of simple rules to perform well.

Simple rules, like the Basel rule, do not perform well across the two calibrations.

However, eschewing policy rules and increasing the static capital requirement by as

little as 1 percent nearly achieves the performance standards set by the Ramsey policy.

1.7 Conclusions

We have built an RBC model in which bank risk-taking is endogenous, and a

temptation to take excessive (or socially inecient) risk is enabled by limited liability

and government deposit insurance. Both macroeconomic shocks and market volatility

shocks can trigger bouts of excessive risk-taking by lowering the expected return on

safer investments. Capital requirements can eliminate that temptation by making

banks keep more skin in the game, but this comes at the cost of limiting liquidity-

producing deposits.

We provide examples in which a Ramsey Planner would raise capital requirements

in response to shocks that cause both cyclical booms and busts, and raise capital

34

requirements in response to increases in volatility that have little consequence for

the business cycle. In practice, the policymaker's problem is more dicult than

responding to a single well-identied shock. The policymaker has to respond to the full

constellation of shocks that drive the macroeconomy. We can calculate the Ramsey

policy in this environment, but it is not plausible to believe that a policymaker

could implement it in practice. So, we employed a SMM procedure to calibrate the

shock process that drives our model economy, calculate the Ramsey policy in that

environment, and evaluate implementable policy rules against it.

We found that a slightly elevated static capital requirement outperformed most

simple rules, and indeed most complicated (or informationally demanding) rules. Some

nely tuned policy rules such as the Basel III prescriptions sound like they make

sense, but they do more harm than good in our model.

35

1.8 Tables and Figures

Table 1.1: Parameters.

Value Description

Conventional

β 0.99 Discount rate

α 0.3 Capital share in production

%c 1.1 Elasticity of substitution for consumption

δ 0.025 Depreciation rate

ςd 1.1 Interest rate elasticity of supply of deposits

Specic Target/Explanation

τ 0.05521 Standard deviation of idiosyncratic shock DebtEBITDA

= 6

ξ 0.00076 Minus mean of idiosyncratic shock Cap. requirement= 10%

ς0 0.015 Relative weight on liquidity in the utility function Quarterly rate on bank debt= 0.86%

f 0.0055 Linear Cost of Banking Rs −Rd = 2.26%

φ 100 Investment adjustment costs VAR evidence

σ 0.01 Minimum risk that banks can take needed for numerical solution method

σ 0.99 Maximum risk that banks can take needed for numerical solution method

Note: See Section 1.3 for the calibration strategy.

36

Table 1.2: Calibration 1, Shock Processes.

AR(1) param. Innov. St. Dev.TFP 0.79 0.0093ISP 0.95 0.0052

Distance Function 0.0020334

Note: This table shows the parameter values of the shock processes in Calibration 1.

Table 1.3: Calibration 1, Variance Decompositions.

var(GDP) var(invest.) var(invest. p.) var(gamma)TFP 100 99 8 0ISP 0 1 92 100

Note: This table shows the variance decompositions of output, gross investment,investment price, and capital requirement under Calibration 1.

Table 1.4: Calibration 1, Matching Moments.

Data ModelVar(GDP) 0.92 0.97Corr(GDP,Investment) 0.96 1.00Corr(GDP,Investment Price) 0.08 0.08Var(Investment) 27.68 27.68Corr(Investment,Investment Price) 0.02 0.06Var(Investment Price) 0.40 0.37Autocorr(GDP) 0.93 0.88Autocorr(Investment) 0.93 0.88Autocorr(Investment Price) 0.87 0.88

Note: This table compares the selected moments in the data and in our model underCalibration 1.

37

Table 1.5: Calibration 2, Shock Processes.

AR(1) param. Innov. St. Dev.TFP 0.79 0.0093ISP 0.95 0.0052Volatility∗ 0.80 0.0015

Distance Function 0.0020332

Note: This table shows the parameter values of the shock processes under Calibration2.

Table 1.6: Calibration 2, Variance Decompositions.

var(GDP) var(invest.) var(invest. p.) var(gamma) shareTFP 100 99 8 0ISP 0 0 92 2Volatility 0 1 0 98

Note: This table shows the variance decompositions of output, gross investment,investment price, and capital requirement under Calibration 2.

Table 1.7: Calibration 2, Matching Moments.

Data ModelVar(GDP) 0.92 0.97Corr(GDP,Investment) 0.96 1.00Corr(GDP,Investment Price) 0.08 0.08Var(Investment) 27.68 27.68Corr(Investment,Investment Price) 0.02 0.06Var(Investment Price) 0.40 0.37Autocorr(GDP) 0.93 0.88Autocorr(Investment) 0.93 0.88Autocorr(Investment Price) 0.87 0.88

Note: This table compares the selected moments in the data and in our model underCalibration 2.

38

Table 1.8: Simple Rules with Calibration 1.

Simple rule R square First variable

Second variable

Number quarters with excessive risk-

taking (per 100 years)

Average deposit under simple rule

Number quarters with excessive risk-

taking (per 100 years)

Average deposit under simple rule

Invest. p. (best state variable) 0.960 -0.087 0 16.23 0 16.20Expected banking spread 0.881 0.842 148.8 10.26 10.4 15.80GDP 0.002 -0.001 149.6 10.21 10.4 15.79Credit/GDP 0.250 -0.005 149.2 10.18 4.4 16.02Credit/GDP wih positive coef 0.005 158.8 9.87 38 14.68Expected safe return and deposit rate

0.826 594.284 -594.312 20.4 15.83

All shock processes, innovations, expected safe return and deposit rate

1.000 0 16.23 0 16.20

All shock processes, innovations, and lagged capital requirement

1.000 21.2 15.35 0 16.17Too many to show

Non convergence problems

Static buffer = 10 basis points Static buffer = 30 basis pointsRegression coefficients

Too many to show

Note: This table reports the performance of various policy rules under Calibration 1. The rst column lists the variablesin the rule; the second column shows the R-square for the regression; the third and fourth columns show the regressioncoecients; the fth and sixth columns show the average number of risk-taking quarters per 100 years and the averagelevel of deposits when the steady state capital requirement is 10.1 percent; the seventh and eighth columns show thesame statistics when the steady state capital requirement is raised to 10.3 percent.

39

Table 1.9: Simple Rules with Calibration 2.

R Square First variable

Second variable

Number quarters with excessive risk-

taking (per 100 years)

Average deposit under

simple rule

Number quarters with excessive risk-

taking (per 100 years)

Average deposit under

simple rule

Number quarters with excessive risk-taking (per

100 years)

Average deposit under

simple rule

Invest. p. (best state variable) 0.002 -0.031 205.6 7.892 74.8 13.092 6.0 15.833Expected banking spread 0.847 0.908 214.0 7.547 78.4 12.965 6.8 15.799GDP 0.035 -0.022 204.0 7.909 83.2 12.765 8 15.756Credit/GDP 0.002 -0.003 210.0 7.679 77.2 13.019 7.2 15.787Credit/GDP wih positive coef 0.003 80.0 12.892 7.2 15.788expected safe return and deposit rate

0.826 607.668 -607.648 7.6 15.858

All shock processes, innovations, expected safe return and deposit rate

1.000 145.6 10.271 0.0 16.158 0 16.068

All shock processes, innovations, and lagged capital requirement

1.000 147.2 10.297 3.2 16.025 0 16.066

Static buffer = 100 basis pointsRegression coeffiecients

Too many to show

Too many to show

Non convergence problems

Non convergence problems Non convergence problems

Static buffer = 10 basis points Static buffer = 50 basis points

Note: This table reports the performance of various policy rules under Calibration 2. The rst column lists the variablesin the rule; the second column shows the R-square for the regression; the third and fourth columns show the regressioncoecients; the fth and sixth columns show the average number of risk-taking quarters per 100 years and the averagelevel of deposits when the steady state capital requirement is 10.1 percent; the seventh and eighth and the ninth andtenth columns show the same statistics when the steady state capital requirement is raised to 10.5 percent and 11 percent,respectively.

40

Table 1.10: The Eciency of Static Buers.

Static Buffer

Number of quarters with excessive risk-

taking (per 100 years)

Average deposit

Number of quarters with excessive risk-

taking (per 100 years)

Average deposit

10 bp 149.2 10.269 211.2 7.67820 bp 66.8 13.526 172.0 9.21630 bp 10.4 15.800 140.8 10.47940 bp 0 16.189 108.8 11.78450 bp 0 16.171 79.2 12.920100 bp 0 16.081 6.8 15.805150 bp 0 15.991 0 15.991Optimal Rule 0 16.251 0 16.241

Calibration 1 (excludes volatility shocks)

Calibration 2 (includes volatility shocks)

Note: This table reports the performance of static buers for two types of Calibration.The rst column lists the proposed static buers; the second and third columns showthe average number of risk-taking quarters per 100 years and the average level ofdeposits when the steady state capital requirement is 10 percent plus the buer underCalibration 1; the fourth and fth columns show the average number of risk-takingquarters per 100 years and the average level of deposits when the steady state capitalrequirement is 10 percent plus the buer under Calibration 2.

41

10 20 30 40 50 60

-2

-1.5

-1

-0.5

Per

cent

1. Total output

Endog. Capital RequirementsFixed Capital Requirements

10 20 30 40 50 60

0

0.05

0.1

0.15

Per

c. P

oint

2. Bank capital requirement

10 20 30 40 50 60-2.5

-2

-1.5

-1

-0.5

Per

cent

3. Consumption

10 20 30 40 50 60

-0.3

-0.2

-0.1

0

0.1

0.2

Per

c. P

oint

4. Credit/GDP ratio

10 20 30 40 50 60

-0.02

0

0.02

Per

c. P

oint

5. Expected Equity Return Spread (risky-safe)

10 20 30 40 50 60

-0.5

-0.4

-0.3

-0.2

-0.1

Per

cent

6. Total capital

10 20 30 40 50 60

Quarters

-0.5

0

0.5

Per

c. P

oint

7. Expected Safe Equity Return

10 20 30 40 50 60

Quarters

-2

-1.5

-1

-0.5

0

Per

cent

8. Investment Price

Figure 1.1. Negative TFP Shock.

Note: This gure plots the responses to a 1.5 percent fall in At. The shock followsAR(1) process with persistence 0.95. The dashed line shows what would happen if thecapital requirement were to be held constant at its steady state value. The solid lineshows the responses under the Ramsey-optimal response for capital requirements.

42

10 20 30 40 50 60

0

0.5

1

Per

cent

1. Total output

Endog. Capital RequirementsFixed Capital Requirements

10 20 30 40 500

0.05

0.1

0.15

0.2

Per

c. P

oint

2. Bank capital requirement

10 20 30 40 50 60

-0.5

0

0.5

1

Per

cent

3. Consumption

10 20 30 40 50 60

-3

-2

-1

0

1

2

Per

c. P

oint

4. Credit/GDP ratio

10 20 30 40 50 60

-0.02

0

0.02

0.04

Per

c. P

oint

5. Expected Equity Return Spread (risky-safe)

10 20 30 40 50 60

1

1.5

2

2.5

3

3.5

Per

cent

6. Total capital

10 20 30 40 50 60

Quarters

-0.5

0

0.5

Per

c. P

oint

7. Expected Safe Equity Return

10 20 30 40 50 60

Quarters

-2.5

-2

-1.5

-1

-0.5

Per

cent

8. Investment Price

Figure 1.2. Positive Investment Shock.

Note: This plots the response to a positive ηt shock. The shock follows AR(1) processwith persistence 0.8. The shock is sized to lead to an increase in output by 1% at itspeak. The dashed line shows what would happen if the capital requirement were tobe held constant at its steady state value. The solid line shows the responses underthe Ramsey-optimal response for capital requirements.

43

10 20 30 40 50 60-0.6

-0.4

-0.2

0

Per

cent

1. Total output

Endog. Capital RequirementsFixed Capital Requirements

10 20 30 40 50 600

0.1

0.2

0.3

Per

c. P

oint

2. Bank capital requirement

10 20 30 40 50 60

-0.6

-0.4

-0.2

0

Per

cent

3. Consumption

10 20 30 40 50 60

-0.2

0

0.2

0.4

Per

c. P

oint

4. Credit/GDP ratio

10 20 30 40 50 60

0

0.05

0.1

Per

c. P

oint

5. Expected Equity Return Spread (risky-safe)

10 20 30 40 50 60

-0.03

-0.02

-0.01

0

Per

cent

6. Total capital

10 20 30 40 50 60

Quarters

-0.5

0

0.5

Per

c. P

oint

7. Expected Safe Equity Return

10 20 30 40 50 60

Quarters

-0.6

-0.4

-0.2

0

0.2

Per

cent

8. Investment Price

Figure 1.3. Volatility Shock.

Note: This gure plots the response to a 50 basis points increase in the standarddeviation of the idiosyncratic shock. The shock follows AR(1) process with persistence0.8. The dashed line shows what would happen if the capital requirement were to beheld constant at its steady state value. The solid line shows the responses under theRamsey-optimal response for capital requirements.

44

0.05 0.1 0.15 0.2Standard deviation of idiosyncratic returns for risky projects

0

0.2

0.4

0.6

0.8

1P

erc.

Pt.,

dev

. fro

m s

tead

y st

ate

Max. in capital requirements, sensitivity

0.05 0.1 0.15 0.2Standard deviation of idiosyncratic returns for risky projects

0

0.5

1

1.5

2

% d

ev. f

rom

ste

ady

stat

e

Max. in output, sensitivity

0 0.2 0.4 0.6 0.8 1Average penalty on returns to risky projects, perc. pt.

0

0.2

0.4

0.6

0.8

1

Per

c. P

t., d

ev. f

rom

ste

ady

stat

e

Max. in capital requirements, sensitivity

0 0.2 0.4 0.6 0.8 1Standard deviation of idiosyncratic returns for risky projects

0

0.5

1

1.5

2

% d

ev. f

rom

ste

ady

stat

e

Max. in output, sensitivity

Figure 1.4. Sensitivity Analysis, TFP Shock.

Note: The gure plots the maximum adjustment in the Ramsey-optimal capitalrequirements and the maximum response of output to a 1.5 percent fall in Atdepending on the various choices of two parameters: the standard deviation of theidiosyncratic shocks, τ , and the average penalty for taking a ier on a risky loan,ξ. We keep all other parameters unchanged. The shock follows AR(1) process withpersistence 0.95. The circles in the diagrams represent the baseline calibration.

45

0 0.05 0.1 0.15 0.2Standard deviation of idiosyncratic returns for risky projects

0

5

10

15

20

25

30

35

40

45

50

55S

tead

y st

ate

capi

tal r

equi

rem

ent,

%

0 0.2 0.4 0.6 0.8 1Average penalty on returns for risky projects, PPt.

0

5

10

15

20

25

30

35

40

45

50

55

Ste

ady

stat

e ca

pita

l req

uire

men

t, %

Figure 1.5. Robustness Checks.

Note: This gure plots the steady-state capital requirement depending on the variouschoices of two parameters: the standard deviation of the idiosyncratic shocks, τ , andthe average penalty for taking a ier on a risky loan, ξ. We keep all other parametersunchanged. The circles in the diagrams represent the baseline calibration.

46

Chapter 2

Dynamic Bank Capital Regulation and Optimal Macroprudential

Policies in the Presence of Shadow Banks

2.1 Introduction

How should macroprudential policy be set when policymakers also take into

account that as a result of their actions some of the banking activities might shift

towards lighter or non-regulated nancial institutions? The objective of this paper

is to quantify the relevance of the shadow banking sector1 lenders that are not

regulated like banks for conducting capital regulation policies. We provide examples

which emphasize the importance of considering the interactions of regulated banks

and shadow banks when designing macroprudential policies. We show that abstracting

from shadow banks does not only underestimate the role of capital requirements for

attenuating excessive risk but can also mislead policymakers about the direction of

their response to macroeconomic shocks.

We dierentiate between two types of questions which we ask in this paper. First,

we look at dynamic capital requirements to provide guidance on imposing capital

ratios over the business cycle. How should policy respond to macroeconomic shocks?

Should capital requirements react to the shocks that arise in the shadow banking

1It might sound like shadow banks carry pejorative connotation. The equivalent terms arenon-bank nancial intermediation and market-based nance. In fact, the Financial StabilityBoard, which oversees risks in the global nancial system, had previously used word shadowbanks in their reports, but it has switched to nonbanks in the recent publications. We usethese terms interchangeably throughout the paper.

47

sector? How should capital requirements together with some kind of regulation of

shadow banks2 be set to maximize the benets of macroprudential policy? From

the practical side, this direction of research is motivated by the proposals of the

Basel III committee on the countercyclical capital buer (CCyB) framework. When

vulnerabilities are accumulating, CCyBs call for a larger buer to absorb potential

losses. Our contribution is to evaluate the macroeconomic implications of CCyBs

when shadow banks are also taken into account. Second, we consider unintended

consequences of tighter regulation of commercial banks for risk-taking incentives of

shadow banks. Does tighter regulation of traditional banks necessarily imply less

fragile nancial system as a whole? This line of research is driven by rising concerns

of policymakers about migration of activities and risk from disproportionately more

regulated conventional banks toward lightly regulated shadow banks. We examine

all these questions through the lenses of a quantitative general equilibrium model

augmented with a nancial sector.

Before giving an overview of our model, we rst clarify the terminology around

shadow banks and recent developments in the sector. The denition of shadow banking

has been evolving over the years and depends on the context. Before the crisis, the

term was meant to capture mainly those companies such as broker-dealers, mortgage

nance rms, asset-backed commercial paper (ABCP) conduits, and money market

mutual funds (MMMF) that participate in wholesale-funded, securitization-based

lending process. According to Gorton (2010), shadow banks were one of the main

culprits of the 2007-09 nancial crisis. Their blithe investments into subprime related

products helped trigger a run on assets at the heart of the downturn.

2For example, apart from tightening the capital requirement on traditional banks, theso-called Minneapolis plan proposes a tax on shadow bank debt.

48

The post-crisis measures were intended to increase the resilience of the nancial

system to macroeconomic shocks. The Dodd-Frank Wall Street Reform, the Consumer

Protection Act and creation of the Financial Stability Oversight Council (FSOC)

introduced tighter standards on the amount and quality of capital that nancial

intermediaries hold. However, the measures directed at regulating shadow banks

were less powerful and less eective due to mainly intense lobbying activities by the

nancial industry.3 Nowadays, many companies that used to be very dierent from

banks have started getting involved in activities earlier associated with conventional

banks. These include ntech lenders, insurance companies, private equity funds, hedge

funds, and many others, all of which provide a signicant source of credit to the

economy. The common denominator of these practices is that they function outside

the protective umbrella of public guarantees by the Federal Reserve and the FDIC in

the event of liquidity and funding problems.

The lightly regulated shadow banking system has been expanding since the crisis,

and policymakers have become more concerned about the risks that shadow banks

spread over to the nancial system. According to the recent data from the Financial

Stability Board, in the years since the global nancial crisis, assets of the narrow

measure of shadow banking, which is estimated to pose the greatest nancial stability

risks, have increased by 75% globally. The U.S. constitutes the biggest share of the

sector, amounting to 29% or $15 trillion, the number comparable to the size of assets

of commercial banks. Moreover, the market of loans extended to non-investment

grade rms and co-funded by group of banks and shadow banks, also known as the

leveraged loan market, has trebled in size since 2012, making a $1.3 trillion market.4

3For an overview of regulatory changes linked to shadow banking, consult Adrian andAshcraft (2012).

4This measure comprises the loans in the S&P leveraged loan index. Broader estimatesinclude institutional loans not in the S&P index and amortizing term loans, increasing the

49

Increased securitization activity through collateralised loan obligations (CLOs) and

strong demand from investment funds were the main drivers of the growth in leveraged

lending. In particular, more than half of the leveraged loans outstanding around

the world have been bundled into CLOs. Deterioration of underwriting standards of

leveraged lending has also increased risk-taking opportunities of nancial institutions

in the environment of low interest rates. Shadow banks stepped in, holding two-thirds

of global CLOs.

In our model, commercial banks and shadow banks borrow from households,

issue equity and nance rms that undertake projects of dierent risk. Each type

of bank lends to two types of rms. Safe rms have a production technology that

is only exposed to aggregate shocks; while risky rms have a production technology

exposed to aggregate and idiosyncratic shocks. Idiosyncratic shocks have no inuence

on the average expected output of a risky rm but increase its variance. We make

exposure to the idiosyncratic risk dierent across two types of nancial intermediaries.

Commercial banks must incur a cost to hide risky projects from regulators, so they

eectively receive a smaller average return on lending to risky rms. In contrast,

shadow banks are not regulated as much and do not incur this penalty on nancing

risky rms. Therefore, shadow banks have a relatively lower price of taking risk.

Another way of justifying this modeling feature is to dierentiate between the

expertise on the asset side of two types of banks, such that shadow banks have

a better screening system for selecting more creditworthy risky rms. In contrast,

commercial banks, protected by deposit insurance, have less incentives to develop a

screening tool for this purpose. It is consistent with the recent empirical evidence

size of global leveraged loans outstanding in 2018 to $2.2 trillion. Its scale and growthare comparable to the ones observed in the U.S. market of subprime mortgages before theGlobal Financial Crisis. For more details, see Box 4 in the Financial Stability Report of theBank of England, available at https://www.bankofengland.co.uk/-/media/boe/files/

financial-stability-report/2018/november-2018.pdf.

50

by Buchak et al. (2018) that shows that shadow banks (represented by ntech

lenders) serve more creditworthy borrowers. It is also related to the historical fact

that regulated megabanks, which were formed after the period of mergers in the

1990s, started oering a range of standardized loan products that required very

little evaluation of prospective borrowers.5 Moreover, securitization allows shadow

banks to bundle loans together, eectively decreasing the risk of an individual loan.

To account for this fact in our model, we consider two dierent variances for the

idiosyncratic process depending on the type of nancial intermediary.

Deposits of commercial banks are insured by the government, so no depositor

experiences a loss if the corresponding bank fails. Deposit insurance acts as a subsidy

for taking excessive risk: commercial banks, also protected by the shield of limited

liability, do not internalize the probability of their default on the cost of borrowing.

This increases their incentives to nance risky rms that undertake socially inecient

projects. The regulator imposes time-varying capital requirements on commercial

banks. In contrast, unregulated shadow banks have no access to public source of

liquidity and face endogenously determined balance sheet constraints.

Households have preferences for liquidity oered by deposits of commercial and

shadow banks. Deposits generate convenience yield, i.e. the non-pecuniary premium

of holding safe and liquid assets that can perform money-like functions, for which

there is strong empirical evidence documented in recent papers by Gorton, Lewellen,

and Metrick (2012), Krishnamurthy and Vissing-Jorgensen (2012), and Nagel (2016).6

5To mention more work that considers emergence of shadow banks based on dierences inproduction technology of nancial services, we can refer to Gertler, Kiyotaki, and Prestipino(2016), Ordoñez (2018), and Martinez-Miera et al. (2019), among others.

6In reality, shadow banks issue REPOs, commercial paper, MMMFs, and other short-term instruments in the money market to fund their assets. The recent empirical papersby Pozsar et al. (2013), Chernenko and Sunderam (2014), and Sunderam (2015) nd theimportance of shadow banks for liquidity creation. To capture those benets in the model,

51

This assumption makes debt a relatively cheaper source of nancing than equity. It

is also consistent with the stylized facts that shadow banks rely on short-term credit.

Excessive risk-taking incentives arise when banks have low net worth. In a

downturn, when the returns to safer projects fall, limited liability, which bounds

the payo to zero in the worst case scenario, makes risk more attractive by pushing

up the upside potential. Higher capital requirements attenuate excessive risk by

forcing commercial banks to keep more skin in the game. Thus, our model calls for

increases in capital requirements during an economic downturn. There is also another

source of risk-taking associated with greater volatility of risky projects that increases

protability of commercial bank equity. Since we can think of equity as a call option

on the underlying assets of the bank, this source of risk-taking is akin to the positive

eect of volatility on the price of a call option. Higher capital requirements impose

less taxation on households to protect banks against default. However, they reduce

liquid and safe assets which households value. We explore this trade-o quantitatively.

Apart from the familiar nancial accelerator mechanism that amplies the eect

of shocks on balance sheet conditions of banks, an additional channel of risk-taking,

which is activated in a general-equilibrium setting, comes from inclusion of shadow

banks into our framework. It consists of the reintermediation of credit when assets

migrate from shadow banks to commercial banks in response to a contractionary

shock. As documented in He, Khang, and Krishnamurthy (2010), this spillover eect

accompanied the recent nancial crisis. In our model, since shadow banks are more

leveraged, their net worth is more negatively aected than the net worth of commercial

banks after the shock. With constant prices, the equity of commercial banks becomes

relatively more protable, inducing households to invest in commercial banks, and

we put shadow bank deposits in the utility function. It corresponds to the approach takenin Begenau and Landvoigt (2018).

52

thus commercial banks issue more debt. In a general equilibrium, as long as deposits of

the two types of banks are imperfect substitutes, households require a higher deposit

rate to substitute away from shadow bank deposits to commercial bank deposits,

increasing the relative costs of commercial banks and thus pushing down the net

worth. This magnies excessive risk-taking incentives beyond those which are present

in the model without shadow banks.

Although our baseline framework abstracts from capital regulation of shadow

banks, it still embeds the features that leave room for a non-negligible role in

regulating shadow banks. Since both aggregate and idiosyncratic uncertainty are

resolved simultaneously, an individual shadow bank borrows at the same pre-

determined interest rate applied to all shadow banks, and so its cost of borrowing

does not fully reect the default risk. Making the cost of borrowing contingent on

the realization of the idiosyncratic uncertainty would decrease attractiveness of risk

and thus lower the spillover eect of a contractionary shock from shadow banks to

commercial banks. Therefore, a policy that provides incentives for shadow banks to

hold more equity and have smaller leverage is benecial because it makes commercial

banks less aected by a contractionary shock.

To evaluate proposals regarding regulation of shadow banks, we consider a tax

on shadow bank debt as a supplementary policy tool to combat the risk and provide

recommendations on using it jointly with capital requirements to respond to a

contractionary TFP shock within the Ramsey framework. Being the only instrument,

an infusion of cash into more leveraged shadow banks (i.e., a negative shadow

bank tax) counteracts excessive risk-taking incentives by inuencing protability

of commercial banks relatively less negatively. The reintermediation channel turns

its direction, and commercial banks decrease the demand for deposits, pushing

down their costs of nancing loans and, thus, excessive risk. At the same time, the

53

Ramsey planner would choose to minimize the eect of the shock on the amount of

commercial bank loans by increasing both capital requirements and a tax on shadow

bank deposits. This would require large magnitudes of changes in both instruments

due to repercussion eects of one tool on the other to equalize the returns on equity

in equilibrium.

Finally, we assess rising concerns about migration of risk and loans from

commercial banks to shadow banks because of tighter regulation. Specically, we

start from the state in which commercial banks take excessive risk and then consider

the eects of a permanent increase in capital requirements. We nd that although

more credit is intermediated through shadow banks, this shift does not feed back

into excessive risk-taking incentives of commercial banks because a fall in total

liquidity provision leads to a drop in capital stock that raises the marginal product

of capital and pushes up the attractiveness of safe projects. Consumption increases,

and the eect on output is inuenced by the parameter that governs relative losses

on nancing risky projects.

2.1.1 Related Literature

This paper draws from dierent strands of a growing literature that lies at

the intersection of macroeconomics and banking. The modeling of the risk-taking

incentives of the banking sector is related to work by Van den Heuvel (2008) that

shows how to exploit the shield of limited liability and deposit insurance to consider

the nancing choice, on the part of banks, of risky and safe projects. Van den

Heuvel (2008) focuses on a setup that excludes aggregate risk. Here we embark on

the nontrivial extension of considering aggregate risk, which enables us to study

how risk-taking incentives vary over the business cycle. We model endogenously

54

determined balance sheet constraints of shadow banks in the spirit of Gertler and

Karadi (2011), to which we add default.

Although many models of nancial frictions address macroprudential policies, they

do not necessarily provide theoretical guidance for capital regulation of banks in a

general equilibrium setting. To model the benets of capital requirements, we refer

to the papers that make equity nance more expensive than debt nance due to

either a tax distortion of prots or liquidity benet of deposits and then propose a

moral hazard problem created by deposit insurance and limited liability. For example,

Collard et al. (2017) examine jointly optimal prudential and monetary policies, which,

unlike our setup, make excessive risk-taking an o-equilibrium outcome. Davydiuk

(2018) studies time-varying capital requirements within the Ramsey framework where

banks risk-shift in terms of quantity of lending. We also consider that banks risk-shift

in terms of quality of projects. At the same time, these papers leave no room for

shadow banks in the analysis. Begenau (2019) introduces the government subsidy of

a particular functional form. One of the key elements that distinguish our work is that

we explicitly derive, from rst principles, the government subsidy associated with the

provision of deposit insurance to banks and a share of non-defaulted shadow bank

deposits that we use to calculate the net worth.

Only few attempts have been made to dierentiate between traditional and

shadow banks in a general-equilibrium framework. The approach that the literature

usually takes is to introduce the run-like behavior on shadow banks in the crisis time

when banks are forced to liquidate assets at resale prices. This amplies the shocks

and captures the highly nonlinear nature of collapse. The papers that share these

features include Gertler, Kiyotaki, and Prestipino (2016), Begenau and Landvoigt

(2018), Ferrante (2018), and Gertler, Kiyotaki, and Prestipino (2019). Begenau and

Landvoigt (2018) nd that a general equilibrium mechanism reduces the funding

55

costs of banks following tighter capital requirements, and thus shadow banks expand

their scale without becoming more risky as long as households care more about the

overall liquidity than its composition. Unlike our work, they focus on static capital

requirements and make no distinction between the technologies possessed by two

types of banks. Gertler, Kiyotaki, and Prestipino (2016) introduce a wholesale market

where shadow banks borrow from retail banks and then characterize runs as self-

fullling rollover crises in an innite horizon endowment economy. Gertler, Kiyotaki,

and Prestipino (2019) extend that framework to a conventional macroeconomic

model, in which the banking sector is aggregated to include both investment banks

and some large commercial banks. A pecuniary externality related to bank's partial

internalization of the eect of its leverage on the run probability of the whole banking

system leads to a non-negligible role for capital regulation. The authors address the

role of macroprudential policy for future research. Ferrante (2018) constructs a model

based on Gertler, Kiyotaki, and Prestipino (2016) and introduces an informational

friction that allows him to capture securitization and endogenous loan quality. He

also considers the reintermediation channel, but there is no direct link between his

proposed policies and capital requirements.

Although we do not explore bank runs in our framework, we emphasize the role

of shadow bank liquidity for capital requirements considerations. Our model also

captures nonlinearities associated with the inequality constraints on the amount of

commercial bank equity and loans that bind occasionally. Overall, our work attempts

to bridge the gap in the literature on the interconnectedness between commercial

and shadow banking sectors for studying time-varying capital requirements in a

quantitative general equilibrium model.

56

2.2 Setup

2.2.1 Households

There is a continuum of identical innitely lived households of mass one. Each

household consumes, saves and supplies labor. Households save in the form of deposits

and equity supplied to nancial intermediaries.

Each household consists of two types of members: workers and bankers. At any

moment in time, workers compose a fraction f of the household members and bankers

make the remaining 1− f fraction of each household. Each banker manages either a

commercial bank or a shadow bank. The share of bankers who manage shadow banks

is equal to fS, and the share of bankers who manage commercial banks adds up to

1− fS.

Workers and shadow bankers can switch occupations each period. In particular,

a shadow banker continues operating in the shadow banking sector next period

with probability θ. This modeling approach guarantees that shadow banks do not

accumulate enough capital, becoming nancially unconstrained in the following

periods. Thus, the fraction (1− f) fS (1− θ) of shadow bankers exit and switch to

workers every period. The corresponding equal measure of workers randomly become

shadow bankers, keeping the composition of household members in each group

constant and time-invariant. This ensures that shadow bankers and workers stay

equally relevant across periods. Upon exiting, shadow bankers return any earnings

back to the household. There is perfect consumption insurance within the family.

Households solve the following problem:

maxCt,DCt ,D

St+1,E

Cs,t,E

Cr,t,E

St

E∞∑t=0

βt[C1−σct − 1

1− σc+ σ0Ψ

(DCt , D

St+1

)],

57

subject to

Ct +DCt +DS

t+1 + ECs,t + EC

r,t + ESH,t =

WtH +RdCt−1D

Ct−1 +RdS

t D?St + ΠS

t +ReCs,tE

Cs,t−1 +ReC

r,tECr,t−1 +ReS

t ESH,t−1 − Tt,

D?St = x?tD

St ,

ECs,t ≥ 0,

ECr,t ≥ 0.

The representative household values consumption, Ct, and the mix of two types

of deposits which enter the function Ψ(DCt , D

St+1

). The parameter σ0 > 0 captures

the utility weight on bank debt relative to consumption. Preferences for consumption

are described by iso-elastic preferences with σc > 0 governing the inverse of the

intertemporal elasticity of substitution for consumption. We consider the following

specication of derived utility from deposits:

Ψ(DCt , D

St+1

)=

(α1

(DCt

)σd + (1− α1)(DSt+1

)σd) 1−ζσd

1− ζ,

where α1 ∈ [0, 1] is the weight on commercial bank liquidity, 11−σd

> 0 is the elasticity

of substitution between commercial and shadow bank debt, and ζ > 0 is the measure

of the elasticity of households supply of total deposits with respect to changes in the

interest rate.

Workers supply labor, H, to rms inelastically for a wage, Wt. The labor supply

is normalized to two. Households save by supplying deposits, DCt and DS

t+1, and

acquiring three types of bank equities, ECs,t, E

Cr,t, and ES

H,t. The superscript stands

for the type of bank. Banks dier by the riskiness of their investments.7 Households

7Banks can choose any level of risk (volatility) between a minimum of σ and a maximumof σ. We will show below that commercial banks select one or the other corner, and shadowbanks always opt for the maximum risk corner. Accordingly, three types of equity span allpossible equilibrium risk choices for banks.

58

invest into shadow bank equity and deposits through two funds, which collect and

distribute all the returns to the household family members. In particular, the equity

fund supplies equal amount of equity to all existing shadow bankers. This ensures

that both entering (new) and existing (old) shadow bankers start with the same

(shared) amount of equity, which makes them all ex-ante identical.8 The deposit fund

holds deposits in the shadow banks that it does not own.9 In the next period, the

deposit fund receives returns from non-defaulted deposits and shares them equally to

depositors. Commercial bank deposits are risk-free due to complete deposit insurance.

In period t, they pay a non-contingent gross return RdCt−1 on the full amount of deposits

DCt−1; shadow bank deposits pay a gross return RdS

t on non-defaulted deposits D?St .

Households make losses on the deposits of those shadow banks that default. Let

x?t be a non-defaulted share of shadow bank deposits and ΠSt be the value that

households receive after liquidating the assets of defaulted shadow banks. We derive

those expressions from the optimization problem of shadow banks in Appendix B.8.

x?t =1

2

[1 + erf

((Rlt −RdS

t

)Qt−1

σS√

2τS+

RdSt Qt−1

σSφSt−1

√2τS

)],

ΠSt = Rl

tlSt−1 (1− x?t )− lSt−1σ

S τS

Qt−1

√2πe−(

(Rlt−RdSt )Qt−1

σS√

2τS+

RdSt Qt−1

φSt−1σS√

2τS

)2

,

where erf denotes the error function described by erf(x) = 2√π

x0e−v

2dv. Neither the

share of non-defaulted deposits nor the payment that the deposit fund receives after

liquidating the assets of defaulted banks depend on any household choice variables

in the current period, so they are taken as given in the optimization problem. The

terms ReSt , ReC

s,t and ReCr,t , are the gross returns to equity for shadow banks, safe

8In the similar vein to perfect consumption insurance, it can be viewed as full equityinsurance. Without this assumption, the framework calls for a model of heterogeneous agentsin which distribution of equity holdings matters for equilibrium. It goes beyond the scopeof this paper.

9In other words, these shadow banks are owned by a dierent family of households. Weuse this assumption for modeling a nancial friction in the shadow banking sector.

59

and risky commercial banks, respectively. The choices of the household are subject to

the budget constraint and to non-negativity constraints for commercial bank equity.

Households pay lump-sum taxes, Tt, collected by the government to provide deposit

insurance. The rst-order conditions are:

∂

∂DCt

= σ0

(α1

(DCt

)σd + (1− α1)(DSt+1

)σd) 1−ζσd−1α1

(DCt

)σd−1−

λct + βEtλct+1RdCt = 0,

(2.1)

∂

∂DSt+1

= σ0

(α1

(DCt

)σd + (1− α1)(DSt+1

)σd) 1−ζσd−1

(1− α1)(DSt+1

)σd−1−

λct + βEtλct+1x

?t+1

RdSt+1 = 0,

(2.2)

∂

∂Ct= C−σct − λct = 0, (2.3)

∂

∂ECs,t

= −λct + βEtλct+1R

eCs,t+1

+ µCs,t = 0, (2.4)

∂

∂ECr,t

= −λct + βEtλct+1R

eCr,t+1

+ µCr,t = 0, (2.5)

∂

∂ESH,t

= −λct + βEtλct+1R

eSt+1

= 0, (2.6)

µCs,tEs,t = 0, (2.7)

µCr,tEr,t = 0, (2.8)

Equations (2.1) and (2.2) are the intertemporal Euler equations for the consumption-

deposit choice which equates the marginal costs and benets of increasing deposits of

each type of bank in terms of consumption numeraire. There is an additional benet

for holding deposits represented by the term∂Ψ(DCt ,DSt+1)

∂Dwhich can be interpreted

as a liquidity premium. Equation (2.3) expresses the shadow price of relaxing the

budget constraint, λct, as the marginal utility of consumption. Equations (2.4), (2.5)

and (2.6) determine the state-contingent required rates of return on three types of

equity. They play an important role for allocation of capital across three technologies

60

in equilibrium. Equations (2.7) and (2.8) are two complementary slackness conditions

associated with no-short selling constraint on commercial bank equity.

2.2.2 Banking Sector

There is a continuum of measure one of each type of bank, i = C, S, indexed

by j ∈ [0, 1] . Banks have limited liability and make loans to production rms.10

They nance these loans by raising deposits and equity from households. Deposits of

commercial banks (C-banks) are fully insured by the government and, therefore, safe

to households, while deposits of shadow banks (S-banks) are risky.

Banks lend to a mix of two types of production rms. One type is subject to

aggregate shocks only (safe rms, for short). Safe loans earn Rlt+1. Another type is

subject to both aggregate and idiosyncratic shocks (risky rms, for short). Risky loans

yield Rlt+1 +

εij,t+1

Qtwhere Qt is the price of capital. The term εij,t+1 is a shock process

that is related to the choice of production technology and varies across each type

of bank. Both commercial and shadow banks lend to risky rms that are exposed

to the idiosyncratic shocks which have no inuence on average expected output.

However, commercial banks have a penalty on the returns to hide risky projects

from regulators. The penalty is associated with the average loss ξC of nancing risky

projects. We directly put this loss into the idiosyncratic shock process of C-banks.

Moreover, to account for the ability of shadow banks to diversify idiosyncratic risk of

a loan because of securitization, we dierentiate between two values of the variance

of the idiosyncratic process depending on the type of bank. A banker j creates a loan

10We call the nancial intermediaries banks and their contracts with the rms loans,but in truth the intermediaries issue equity contracts to the rms, just like in Gertler andKaradi (2011).

61

portfolio11 with riskiness σij,t and earns total returns Rlt+1 + σij,t

εij,t+1

Qtby directing a

fraction σij,t of loans to a risky rm that is chosen from a continuum of risky rms

of measure νr,it and 1 − σij,t of loans to a safe rm that is chosen from a continuum

of safe rms of measure νs,it .12 Table 2.1 shows the main dierences in the modeling

approach of each type of bank.

Shadow Banks

In period t, a shadow bank j receives eSj,t units of equity from the equity fund,

demands dSj,t+1 units of deposits from the deposit fund and lends lSj,t to rms. The

amount of net worth at the end of period t is described by the following balance sheet

identity:

lSj,t = dSj,t+1 + eSj,t.

At time t + 1, after realization of the shocks, the bank receives the net cash ow

ωSj,t+1 from the loan portfolio with riskiness σSt :

ωSj,t+1 = max

[(Rlt+1 + σSj,t

εSj,t+1

Qt

)lSj,t −RdS

t+1dSj,t+1, 0

].

The net cash ow comprises earnings on assets, Rlt+1 + σSj,t

εSj,t+1

Qt, minus payments

on deposits. Deposits pay the non-state contingent gross return RdSt+1. The term dSj,t+1

is thought to be debt of the shadow banker. Non-defaulted banks pay the return in

full amount. Defaulted banks liquidate their assets and partially reimburse depositors.

Households account for the possibility of bank's default and therefore require a higher

11The rm section describes the production functions of each type of rm, from whichwe derive the returns to loans and show how they compose the portfolio returns postulatedhere.

12The statement that a bank only deals with one safe and one risky rm comes at no lossof generality since diversication is useless given constant returns to scale technology of saferms and detrimental for loans to risky rms. You can consult Collard et al. (2017) for amore formal exposition of this result.

62

deposit rate. The household's problem captures internalization of the default risk. If

the net cash ow is positive, the bank pays it out to households in dividends. If the

net cash ow is negative, limited liability protects the bank from making any losses,

so the bank gets zero. The timeline for shadow bankers is plotted in Figure 2.1.

The net cash ow can be re-written to capture excess returns on assets above the

pre-determined cost the shadow bank pays on deposits:

ωSj,t+1 = max

[(Rlt+1 + σSj,t

εSj,t+1

Qt

−RdSt+1

)lSj,t +RdS

t+1eSj,t, 0

].

Since households have preferences for the bank debt, shadow banks have incentives

to nance their assets by issuing deposits only. To motivate shadow banks to

accumulate equity, we consider a costly enforcement problem in the spirit of Gertler

and Karadi (2011).13 In particular, at the start of each period, the shadow banker

can choose to transfer a share λ of loans back to the household. If the banker opts

for the transfer, the deposit fund that represents the interests of the depositors of

another household can force the banker to liquidate the remaining fraction 1 − λ of

the assets. In eect, depositors provide funds to the shadow bank if the following

incentive-compatibility constraint holds:

V Sj,t ≥ λlSj,t,

where V Sj,t is the value of the bank measured by the expected terminal wealth. We

make the following assumptions:

Assumptions (Shadow Banks):

1. At least a positive fraction σS of the value of total loans will go to risky rms

and at least a positive fraction 1− σS of the value of total loans will go to safe

rms.13The moral hazard problem also allows us to study allocation of capital across production

rms in a meaningful way.

63

2. εSj,t+1 follows a Normal distribution with mean zero and variance τ 2S.

3. The amount of equity that the shadow banker receives from the equity fund is

the same across all shadow banks, i.e. eSj,t = ESt for all j ∈ [0, 1] .

4. Shadow banker exits with i.i.d. probability 1− θ next period.

The rst assumption takes the form of the minimum scale condition for the

allocation of capital across two production functions. It ensures that the returns to

both safe and risky loans are always dened. The second assumption allows us to nest

expectations taken over the idiosyncratic shock and derive explicit expressions that

involve expectations over the aggregate shock only. Moreover, unbounded support of a

Normal distribution is necessary for having no restrictions on parameter values when

measuring the impact of aggregate shocks in the quantitative analysis. The third

assumption allows us to model the shadow banking sector within a representative

agent framework. The nal assumption is used to limit the ability of shadow banks

to get rid of the nancial friction.

Let Et denote expectation taken over the joint distribution of εCj,t+1 and Rlt+1,

subject to information known at time t. The objective of the bank is to maximize its

value. Let Λt,t+i = βi λct+iλct

be the stochastic discount factor the bank at t applies to

earnings at t+ i. The maximization problem of the bank can be written as follows:

V Sj,t = max

lSj,t+i,σSj,t+i

Et

∞∑i=0

(1− θ) θiΛt,t+1+iωSj,t+1+i

subject to

V Sj,t ≥ λlSj,t,

ωSj,t+1+i = max

[(Rlt+1+i + σSj,t+i

εSj,t+1+i

Qt+i

−RdSt+1+i

)lSj,t+i +RdS

t+1+ieSt+i, 0

],

σS ≤ σj,t ≤ σS.

64

Since the distributions of aggregate and idiosyncratic shocks are independent of

each other, we can represent the expectation operator Et as an expectation taken

over the aggregate shock, nesting expectations taken with respect to the idiosyncratic

shock. To deal with the nonlinear nature of the max operator, notice that for each

realization of the aggregate shock(s), there is a realization of the idiosyncratic shock

below which the max operator guarantees that bank prots are zero. Accordingly,

EtωSj,t+1

= Et

∞

ε∗Sj,t+1

((Rlt+1 + σSj,t

εSj,t+1

Qt

−RdSt+1

)lSj,t +RdS

t+1eSt

)dG(εSj,t+1)

,

where ε∗Sj,t+1 is a cuto level of the idiosyncratic shock below which the bank defaults

on its deposits obligation, i.e.(Rlt+1 + σSj,t

ε∗Sj,t+1

Qt

−RdSt+1

)lSj,t +RdS

t+1eSt = 0.

Commercial Banks

A commercial banker j enters period t + 1 with lCj,t lending units, nanced by

issuing dCj,t units of deposits and eCj,t units of equity. Therefore,

lCj,t = eCj,t + dCj,t,

that is the balance sheet condition. The timeline for commercial bankers is plotted in

Figure 2.1.

Deposits, dCj,t, pay the full non-state-contingent gross return RdCt at time t + 1.

The net worth, eCj,t, available to banks at the end of period t (going into period t+ 1)

evolves according to:

eCj,t+1 = max

[(Rlt+1 + σCj,t

εCj,t+1

Qt

−RdCt

)lCj,t +RdC

t eCj,t, 0

]− zj,t+1

where zj,t+1 is the net payout to the bank's shareholders in t+ 1 after the realization

of the idiosyncratic shock. We make the following assumptions:

65

Assumptions (Commercial Banks):

1. At least a positive fraction σC of the value of total loans will go to risky rms.

2. The bank supervisory authority will prevent risky loans in excess of nancing a

share σC of total loans where σC < σC < 1.

3. εCj,t+1 follows a Normal distribution with mean −ξC (where ξC > 0) and variance

τ 2C .

The rst assumption is needed for the same reasons as discussed in the shadow

banking section. The second assumption states that the regulator observes excessive

risk-taking imperfectly. The threshold for the share of risky loans from which the

authority starts detecting excessive risk is described by σC . The interpretation is

that the regulator has power to penalize banks signicantly enough to make them

never nd it optimal to nance the fraction of risky rms greater than σC . The third

assumption formalizes inecient risk-taking by making the expected return to a loan

portfolio decrease in risk.

The banker's objective is to maximize the expected discounted sum of equity

payouts:

V Cj,t = max

lCj,t+i,eCj,t+i,σ

Cj,t+i

Et

zj,t +

∞∑i=0

Λt,t+1+izj,t+1+i

,

subject to

eCj,t+i ≥ γt+ilCj,t+i,

zj,t+1+i = max

[(Rlt+1+i + σCj,t+i

εCj,t+1+i

Qt+i

−RdCt+i

)lCj,t+i +RdC

t+ieCj,t+i, 0

]− eCj,t+i,

lCj,t+i ≥ 0,

σC ≤ σCj,t+i ≤ σC .

66

The capital requirement stipulates that equity needs be at least a fraction γt of

total loans for the bank to operate in each period. The non-negativity constraint on

the amount of loans excludes the possibilities of short-selling.14

2.2.3 Production Firms

Competitive production rms are owned by households and produce goods using

capital and labor as inputs. There are two classes of production rms, i = C, S, each

having measure one. Firms borrow from banks to purchase capital. Commercial banks

nance C-rms; shadow banks lend to S-rms. Next period, after realization of shocks,

rms collect income from production activity and from the sale of undepreciated

capital. They distribute the resulting payo to workers and the banks they serve.

Within each class, there are two types of rms. Firm j in class i, with j ∈ [0, νit ], can

only access a risky technology subject to both aggregate and idiosyncratic shocks;

rm j in class i, with j ∈ [νit , 1] has access to a safe production technology subject to

aggregate shocks only. The bank perfectly observes the choice of technology and the

value of realized idiosyncratic shock, εij,t+1, so it can enforce the payment of the full

value of rm's payo.

Let πj,t+1 denote the revenue of rm j in period t+ 1 net of expenses:15

πj,t+1 = Yj,t+1 + (1− δ)QtKj,t+1 −Wt+1Hj,t+1 −Rlj,t+1lj,t.

The term Yj,t+1 is output in period t + 1, Qt is the price of capital in terms of the

nal good, δ is the depreciation rate, Hj,t+1 is the labor input in production, Wt+1

is compensation for labor, and Rlj,t+1 is the rate at which the rm of type j borrows

14The reasons for no short-selling constraint come from the fact that bank's objectivefunction is discontinuous in risk. This becomes clearer when we characterize the problem ofthe bank.

15Both C-rms and S-rms face the same maximization problems, so we will omit theindex of the class in the equations.

67

from the bank. Firms maximize expected prots, knowing that they will be able to

choose the optimal quantity of labor Hj,t+1 next period:

maxlt+i,Kj,t+i+1

Et

∞∑i=0

Λt,t+i maxHj,t+i+1

πj,t+i+1

In period t, rms choose the amount of loans and the capital input. In period

t+ 1, rms get to observe the aggregate shock and choose the labor input. Purchases

of capital are entirely nanced with loans from banks, so

Qt+iKj,t+i+1 = lj,t+i

Since there are no sector-specic frictions in physical markets for capital, Qt+i is the

same across two types of rms. The production function of risky rms is

Y rj,t = AtK

αj,tH

1−αj,t + εj,tKj,t,

where εj,t is an idiosyncratic shock specic to rm j, as already introduced when

discussing the optimization problem of banks. The term At is an aggregate technology

shock. The production function of safe rms is

Y sj,t = AtK

αj,tH

1−αj,t .

Appendix B.6 derives the rst-order conditions and shows that individual rms

can be aggregated into three representative rms. It also nds that the following

zero-prot conditions:

Rlt ≡ Rl

j,t =αAtQt−1

(Kj,t

Hj,t

)α−1

+ (1− δ) Qt

Qt−1

, j ∈ [0, νt] ,

Rlj,t = Rl

t +εj,tQt−1

, j ∈ [νt, 1] .

imply the optimality conditions. To interpret it, notice that 1Qt−1

is the capital

obtained by giving up one unit of nal good (consumption). That quantity of capital

68

obtains a rental rate αAt(Kj,tHj,t

)α−1

. After production takes place, the underpreciated

portion and can be resold at price Qt, so the same quantity of capital 1Qt−1

yields

additionally capital gains equal to QtQt−1

. The rate on risky loans includes the extra

return/loss from the realization of the idiosyncratic shock normalized by the price of

capital.

Therefore, at the time the bank j is making a loan to the safe rm, the bank

expects to receive the total returns on safe loans equal EtRlt+1l

sj,t and the total returns

on risky loans equal EtRlj,t+1l

rj,t = Et

(Rlt+1 +

εj,t+1

Qt

)lrj,t. Summing them up yields

EtRlt+1l

sj,t + EtR

lj,t+1l

rj,t =

EtRlt+1 (1− σj,t) lj,t + Et

(Rlt+1 +

εj,t+1

Qt

)σj,tlj,t =

(Rlt+1 + σj,t

εj,t+1

Qt

)lj,t,

where lj,t = lsj,t + lrj,t and lsj,t = (1− σj,t) lj,t. So, the returns in the bank's

maximization problem are consistent with the rm's problem.

2.2.4 Capital producing firms

At the beginning of period t, after realization of the shocks, competitive capital

producing rms buy capital from production rms and then repair depreciated capital

and build new capital. They sell both the new and re-furbished capital at the end of

period t. The cost of replacing worn out capital is unity. There is a common market

for capital for safe and risky rms.16 Thus, the value of a unit of new capital is Qt.

Let Igt denote aggregate gross investment expenditures. There are quadratic

adjustment costs measured in units of investment. Aggregate investment expenditures

16When the idiosyncratic shock becomes known, it only aects the quantity of capital,so capital is homogeneous across two technologies of each class of rms. In this framework,the quality of capital is best represented by the risk parameter σ, which is a share of loansdirected to risky rms and which is chosen by banks before the realization of the idiosyncraticshock.

69

of size Igt yield net investment of size Int which is related to Igt as follows:

Int =

[1− φ

2

(IgtIgt−1

− 1

)2]Igt ,

where φ is a parameter that governs adjustment costs. The functional form is based

on Christiano, Eichenbaum, and Evans (2005). The aggregate capital stock evolves

according to:

Kt+1 = Int + (1− δ)Kt,

where Kt+1 is the total capital allocated to two representative C-rms and two

representative S-rms, i.e.

Kt+1 = KsafeC,t+1 +Krisky

C,t+1 +KsafeS,t+1 +Krisky

S,t+1.

Capital producing rms solve:

maxIgt+i

Et

∞∑i=0

Λt,t+i

[Qt+i

[1− φ

2

(Igt+iIgt+i−1

− 1

)2]Igt+i − I

gt+i

],

where Qt is given and Λt,t+i = βi λct+iλct

is the stochastic discount factor of households

who own the capital producing rms. The rst order condition of the capital producing

rms gives the following relation for the gross investment:

Qt = 1 + φ(

IgtIgt−1− 1) [

IgtIgt−1

+ 12

(IgtIgt−1− 1)]Qt − Λt,t+1Qt+1φ

(Igt+1

Igt− 1)

Igt+1

(Igt )2 I

gt+1.

2.2.5 The Government

Deposit insurance requires the government to raise taxes. Given the Ricardian

nature of the model, positing the availability of lump sum taxes Tt implies that

the government budget can be balanced period by period without loss of generality.

Appendix B.7 shows the equilibrium tax necessary to support the insurance scheme.

70

2.2.6 Equilibrium Characterization

We calculate the integrals in Appendix B.1. Appendix B.3 shows that the function

that enters the values of each type of bank is convex in the risk parameter. Moreover, it

states that if ξ = 0, then the function is also increasing in risk. Both results generalize

to arbitrary (not necessarily continuous) distribution of the idiosyncratic shock.

Shadow Banks

In Appendix B.4.1 we derive the recursive expression for the shadow banker's

objective function and separate it into a component that depends on loans, υj,t, and

a component that depends on net worth, ηj,t:

V Sj,t = υj,tl

Sj,t + ηj,te

Sj,t.

Appendix B.4.2 shows that the objective function is increasing in σSt , so each

shadow banker opts for nancing the maximum share of risky projects. Therefore,

V Sj,t = max

lSj,t+i

Et

∞∑i=0

(1− θ) θiΛt,t+1+iωSj,t+1+i

subject to

υj,tlSj,t + ηj,te

St ≥ λlSj,t,

ωSj,t+1+i = max

[(Rlt+1+i + σS

εSj,t+1+i

Qt+i

−RdSt+1+i

)lSj,t+i +RdS

t+1+ieSt+i, 0

].

Note that the value function depends on one variable only. Thus each banker faces

the same problem and chooses the same amount of loans, i.e. lSj,t = lSt .17

17The sucient condition for generating this result is that starting amount of equity is thesame across all bankers. Otherwise, depending on the realization of the idiosyncratic shock,each banker will be subject to the nancial friction of dierent strength. In fact, thosebankers who receive a relatively large favorable idiosyncratic shock and continue operatingin the shadow banking sector have more skin in the game and thus can borrow morecheaply. The equity fund that shares equity across all bankers is key to justify our analysiswithin the representative agent framework.

71

Moreover, dSj,t+1 = lSj,t−eSj,t = lSt −eSt = dSt+1. It implies that there is a representative

shadow banker.

Note that υt is the expected discounted value of expanding loans by one unit

holding equity constant; ηt is the expected discounted gain of having an additional

unit of equity holding loans constant. As long as υt ≥ λ, shadow banks are nancially

unconstrained and would like to increase loans as much as possible. To make the

nancial friction relevant, consider the range of υt where 0 < υt < λ.18 Then

υtlSt + ηte

St = λlSt ,

which states that the bank equates the benet of diverting funds with its cost. Let

φSt =lSteSt

be the leverage of the banker. Dene:

νt = Et

(1− θ) Λt,t+1

σS τS

Qt

√2πe−(

(Rlt+1−RdSt+1)Qt

σS√

2τS+

RdSt+1Qt

φSt σS√

2τS

)2

+

((Rlt+1 −RdS

t+1

))2

[1 + erf

((Rlt+1 −RdS

t+1

)Qt

σS√

2τS+

RdSt+1Qt

φSt σS√

2τS

)]]+ θΛt,t+1

lSt+1

lStνt+1

and

ηt = Et

(1− θ) Λt,t+1

[(RdSt+1

)2

[1 + erf

((Rlt+1 −RdS

t+1

)Qt

σS√

2τS+

RdSt+1Qt

φSt σS√

2τS

)]+

θΛt,t+1

eSt+1

eStηt+1

].

Therefore, the incentive constraint can be written as

lSt =ηt

λ− υteSt = φSt e

St .

The nancial friction endogenously restricts lending by the amount of equity that

the bank accumulates and receives from the equity fund. Even though shadow banks

18Note that υt < 0 implies that no bank makes any loans.

72

do not choose equity on their own, they still have incentives to accumulate equity in

order to limit the eect of the nancial friction. These incentives are shared with

the equity fund that represents their interests perfectly. In fact, the equity fund

redistributes existing equity and gets additional equity which is optimally supplied by

the household. The household optimally supplies equity to ensure that shadow banks

can lend to rms by borrowing from the depositors of another household. Moreover,

the household solve the portfolio problem to determine the allocation of equity across

commercial and shadow banks.

The evolution of the banker's net worth is given by

eSt+1 =

σSφSt τS

Qt

√2πe−(

(Rlt+1−RdSt+1)Qt

σS√

2τS+

RdSt+1Qt

φSt σS√

2τS

)2

+

(Rlt+1φ

St −RdS

t+1φSt +RdS

t+1

)2

[1 + erf

((Rlt+1 −RdS

t+1

)Qt

σS√

2τS+

RdSt+1Qt

φSt σS√

2τS

)]]eSt .

Therefore, the terms described in νt and ηt evolve according to:

zt+1 ≡eSt+1

eSt=

σSφSt τS

Qt

√2πe−(

(Rlt+1−RdSt+1)Qt

σS√

2τS+

RdSt+1Qt

φSt σS√

2τS

)2

+

(Rlt+1φ

St −RdS

t+1φSt +RdS

t+1

)2

[1 + erf

((Rlt+1 −RdS

t+1

)Qt

σS√

2τS+

RdSt+1Qt

φSt σS√

2τS

)]],

xt+1 ≡lSt+1

lSt=φSt+1e

St+1

φSt eSt

.

Note that the dividends are paid by the banks that exit. The dividends compose a

constant share 1 − θ of today's period net worth of the equity fund after realization

73

of the shocks. Therefore, the gross return on equity is given by:

ReSt =

(1− θ) eSt+1

eSt= (1− θ)

σSφSt τS

Qt

√2πe−(

(Rlt+1−RdSt+1)Qt

σS√

2τS+

RdSt+1Qt

φSt σS√

2τS

)2

+

(Rlt+1φ

St −RdS

t+1φSt +RdS

t+1

)2

[1 + erf

((Rlt+1 −RdS

t+1

)Qt

σS√

2τS+

RdSt+1Qt

φSt σS√

2τS

)]].

The total shadow bank equity, ESt , comprises equity supplied by households, ES

H,t,

and retained earnings of the shadow banking sector, i.e.

ESt = ES

H,t + θztESt−1.

Commercial Banks

We derive the rst-order conditions and characterize the bank's problem conditional

on the choice of risk in Appendix B.5.1. The next proposition establishes that each

bank considers only two values of risk (σCj,t = σC and σCj,t = σC) to maximize the

objective function.

Proposition 3. The are no equilibria with σC < σCj,t < σC.

We prove Proposition 3 in Appendix B.5.2. Note that it does not imply that each

bank chooses the same amount of risk. Let µt be a share of the commercial banks

that nance a fraction σ of risky projects. Three types of equilibria are possible:

1. Safe equilibrium: µt = 0.

2. Risky equilibrium: µt = 1.

3. Mixed equilibrium: 0 < µt < 1.

If short-selling were allowed, and banks were in the safe equilibrium, risky loans

would be overpriced over safe loans because expected returns to risky loans are

74

relatively smaller in the safe equilibrium. Hence, each bank would want to short

risky loans (which means that it would acquire a negative amount of risky loans),

leading to an arbitrage opportunity. Similar reasoning applies to risky equilibrium,

in which shorting safe loans would result in arbitrage prots. To exclude arbitrage

opportunities, we need to impose the condition that loans cannot be negative. The

mixed equilibrium arises when the expected dividends of risky and safe banks are

equal, so a positive measure of commercial banks (call them risky banks) choose to

nance the maximum share, σC , of risky projects and a positive measure of commercial

bankers (call them safe banks) opt for nancing the minimum share, σC , of risky

projects.

Note that µt also corresponds to the share of household's equity invested into

risky banks out of the total amount of equity allocated to commercial banks.19 Thus,

following the notation from the households section:

µt =ECr,t

ECr,t + EC

s,t

.

Since each bank within a group (safe or risky) is alike and receives an aliquot share

of nancing, the bank-specic terms and aggregate terms are related as follows:

ECr,t =

µt

0

eCj,tdj,

and

ECs,t =

1

µt

eCj,tdj.

Since households have preferences for bank debt, they require a smaller return on

deposits than on equity due to a greater convenience yield. Therefore, viewing debt

19Note that to justify the duality of denition of µt, we consider equity invested into riskybanks beyond the amount required by the minimum scale in the safe/risky equilibrium. Thus,in the safe/risky equilibrium, the share of household's equity invested into risky banks is0/1.

75

as a cheaper source of funding, the bank prefers taking as much debt as it is allowed

by regulation. We formalize this argument and prove the following proposition in

Appendix B.5.3

Proposition 4. In equilibrium, capital requirements always bind, i.e. eCj,t = γtlCj,t.

The combined rst-order condition for the choices of loans, deposits and equity is

written as (taken σCj,t as given):

γt − χ2j,t = Et

βλct+1

λct

σCj,t τC

Qt

√2πe−(RdCt (1−γt)Qt−R

lt+1Qt+ξCσ

Cj,t

σCj,t

√2τC

)2

+

1

2

(Rlt+1 − σCj,tξC − (1− γt)RdC

t

) [1− erf

(RdCt (1− γt)Qt −Rl

t+1Qt + ξCσCj,t

σCj,t√

2τC

)]],

χ2j,tlCj,t = 0,

where χ2j,t ≥ 0 is the Lagrange multiplier that is appended to the constraint lCj,t ≥ 0.

The right-hand side of the above equation is the expected discounted benet from

making an additional unit of loans net of expenses on 1 − γt units of deposits. It

equates to the expected discounted cost of raising γt units of equity in equilibrium.

Note that the expected dividend that enters the value function of the bank,

Ω (µt, σj,t; lj,t) = Et Λt,t+1zj,t+1 =

− Et

Λt,t+1 (1− γt) lCj,t

+ Et

βλct+1

λctlCj,t

σCj,t τC

Qt

√2πe−(RdCt (1−γt)Qt−R

lt+1Qt+ξCσ

Cj,t

σCj,t

√2τC

)2

+

1

2

(Rlt+1 − σCj,tξC − (1− γt)RdC

t

) [1− erf

(RdCt (1− γt)Qt −Rl

t+1Qt + ξCσCj,t

σCj,t√

2τC

)]],

is linear in loans.20 It implies that the value function is also linear in loans. Therefore,

we can aggregate across the banks as follows: there is a representative bank of type

20Bear in mind that µt is determined endogenously and enters implicitly into the equationby aecting the aggregate returns.

76

i = s, r that nances a fraction σC of risky projects when i = s and nances σC share

of risky projects when i = r. Aggregate safe and risky loans are equal to, respectively,

lC,st =

1

µt

lCj,tdj = (1− µt) lCs,t

and

lC,rt =

µt

0

lCj,tdj = µtlCr,t,

where we use the result that each risky individual bank chooses lCr,t and each safe

individual bank chooses lCs,t.

2.2.7 Discussion of the Excessive Risk-Taking Mechanism

Following the result derived in Appendix B.2, we can express the erf function in

terms of the share of non-defaulted deposits of the representative commercial bank21

and then decompose the expected dividend into two components (omitting the index

of the commercial bank):

Ω (µt, σt; lt) = Et Λt,t+1lt [ω1 + ω2 − (1− γt)] , (2.9)

where

[ω1 + ω2] =

(Rlt+1 −Rd

t (1− γt)− ξσt) (

1−G(ε∗t+1))︸ ︷︷ ︸

non-defaulted︸ ︷︷ ︸ω1 ≡ returns from a loan

portfolio with riskiness σt

+ σtτ√2πe−(ε∗t+1+ξ

τ√

2

)2

︸ ︷︷ ︸ω2 ≡ bonus from

projects volatility

,

21Deposits of commercial banks are repaid fully. By non-defaulted deposits we mean thosedeposits that are not provided by the deposit insurance fund.

77

and the cuto point ε∗t+1 is dened by Rdt (1− γt)Qt −Rl

t+1Qt = σtε∗t+1.

The rst component, ω1, distinguishes loan returns of riskiness σt controlling for

the variance of idiosyncratic shock (when τ is taken as given). The bank trades o

the benets from limited liability and deposit insurance with a smaller protability

of riskier projects. The term ξσt reects, in expectation, the reduction of loan returns

for the bank holding σt share of risky projects. The bank receives net income on

loans, Rlt+1−Rd

t (1− γt)− ξσt, if it does not default on deposits which happens with

probability 1−G(ε∗t+1). If the bank defaults, it gets zero, i.e. 0 ·G(ε∗t+1) which is not

shown in the expression explicitly.

The second counterpart of the above decomposition, ω2, comprises the extra eect

of σt on expected dividends that comes from more dispersed returns from projects. In

fact, ω2 is strictly increasing in τ : the bank views projects as a call option the value

of which rises with volatility associated with higher upside. Limited liability bounds

the payo to zero in the worst case scenario.

Risk-taking incentives depend on the dierence between returns on safe loans

and returns on deposits. Table 2.2 illustrates the eects of greater risk-taking on

two components of dividends for each realization of the aggregate returns. We map

aggregate returns into states of nature and consider two cases depending on the sign

of ε∗t+1. The aggregate returns inuence the value of the shield of limited liability.

Risk amplies the eect of the idiosyncratic shock. So, in every state of nature, the

bank's choice of risk is determined by the expected eect of the idiosyncratic shock

on the value of the shield of limited liability and returns on loans. The up-turn

arrow, ⇑, indicates that greater risk-taking increases the corresponding component of

bank's dividends. The down-turn arrow, ⇓, means that the corresponding component

of bank's dividends decreases with greater risk-taking. Two arrows turned in the

78

opposite directions, ⇑⇓, signify that the eect of greater risk-taking is undetermined

and depends the parameterization.

First, ε∗t+1 > 0 indicates that the bank makes losses on safe loans. It happens in

those states of nature where the net income from the zero-risk portfolio is negative,

so the bank is behind the shield of limited liability. By accepting more risk, the

bank is more likely to get a positive net return under a favorable realization of the

idiosyncratic shock as risk acts like a leverage on the size of the shock. Therefore,

1 − G(ε∗t+1) rises. This balances with smaller returns on a portfolio with more risky

loans, i.e. Rlt+1 −Rd

t (1− γt)− ξσt goes down. Similarly, gambling on more dispersed

returns allows the bank to move away from a zero return that comes from the limited

liability to some positive return that is accompanied by less frequent defaults. So, the

eect of σt on expected dividends from ω2 is positive.

Second, ε∗t+1 < 0 shows that the bank makes positive prots on safe loans. The

bank is more likely to default when it takes on more risk because any negative

idiosyncratic shock would be amplied by risk. The bank internalizes that riskier

projects are less protable. Therefore, the overall eect of greater risk on ω1 is negative

when ε∗t+1 < 0.

Then consider the bonus from projects volatility. If −ξ < ε∗t+1 < 0, there are two

contrasting forces. On the one hand, the bank always benets from limited liability

that makes the variance of projects returns attractive. On the other hand, the bank

is more concerned about (and more vulnerable to) the variability of returns in the

situation when taking on more risk would result in zero payo instead of some positive

payo achieved by smaller risk. It occurs when −ξ < ε∗t+1 < 0. In these states of

nature, the bank requires greater than average realization of the idiosyncratic shock

in order to get a positive return. Call this type of shock a good idiosyncratic shock.

This shock happens with probability smaller than 0.5. Dene a bad idiosyncratic

79

shock as a complement to a good idiosyncratic shock. An increase in risk increases

the prots under a good shock. It captures the benets from greater upside. At the

same time, an increase in risk makes it more likely to get a bad shock. The bank trades

o marginal prots coming from a good shock with marginal losses coming from the

reduction of prots due to more defaults. Since the probability of the latter is greater

than the probability of the former, the losses from defaults can dominate the benets

from greater volatility. This force goes in the opposite direction when ε∗t+1 6 −ξ.

The dierence is that here the bank is more likely to get a good shock than a bad

shock. Therefore, the bank puts more weight on the benets from risk-taking than on

its costs. It is veried mathematically that the eects of σt on ω2 is unambiguously

positive when ε∗t+1 6 −ξ.

In sum, we nd that net returns on safe loans, Rlt+1 − Rd

t (1− γt), is the main

driver for the bank's choice of risk. In the partial-equilibrium setting, we dierentiate

between three cases that characterize incentives for risk-taking.

First, Rlt+1 < Rd

t (1− γt) applies to the states of nature where a relatively large

negative aggregate shock is realized. Two forces against the one that seems to be

of lesser relevance make the bank benet most from taking risk. Second, −ξ <

Rdt (1− γt)−Rl

t+1 < 0 applies to the states of nature where intermediate values (not

too large and not too small) of either negative or positive aggregate shock are realized.

There are more forces that lower incentives for risk. Third, Rdt (1− γt) − Rl

t+1 < −ξ

applies to the states of nature where a positive aggregate shock of a larger size

is realized. Interestingly, there is a force associated with the bonus from projects

volatility that makes it possible for the bank to increase risk. The choice of risk

depends on the strength of that force, ω2, relative to the negative exposure of returns

from a loan portfolio to risk, ω1. It still remains a quantitative question to nd out

how risk-taking is determined in the general equilibrium set-up.

80

Capital requirements aect risk-taking through a change in ε∗t+1. When γt

increases, ε∗t+1 falls. It means that the bank will be more likely to nd itself in

the states of nature where ε∗t+1 is negative. It forces the bank to keep more skin in

the game, make the shield of limited liability less attractive and prevent the switch

into nancing risky projects.

2.3 Calibration and Experiments

2.3.1 Calibration

To best match the data and to account for the missing elements in the baseline

model, we introduce costs of banking that are linear in the amount of loans. These

costs include operating expenses which are associated with the provision of loans. In

particular, each period the bank of type i = C, S repays F it = f ilit from current prots

to households. Accordingly,

ωSt+1 = max

[(Rlt+1 + σSt

εSt+1

Qt

− fS)lSt −RdS

t+1dSt+1, 0

],

ej,Ct+1 = max

[(Rlt+1 + σj,Ct

εCt+1

Qt

− fC)lj,Ct +RdC

t dj,Ct , 0

]− zjt+1,

where j = s, r stands for the safe and risky representative commercial bank,

respectively.

Table 2.3 shows the parameters used for the baseline model. We choose conventional

values for the discount factor β, the capital share α, the inter-temporal elasticity of

substitution for consumption σc, and the depreciation rate δ. We use estimates from

Begenau and Landvoigt (2018) to pick up the value of the elasticity of substitution

between commercial and shadow bank liquidity σd. We choose the interest rate

elasticity of household's demand for total liquidity ζ to be close to the lower bound of

81

the range [1, 2] of values used by the literature in the models with commercial banks

only. We perform sensitivity analysis to nd out how σd and ζ aect the results.

The remaining parameters are specic to our framework. We calibrate the model

to U.S. data from 2009 Q1 to 2015 Q4. This period, characterized by low interest

rates, reects the concern that nancial intermediaries may reach for yields by taking

on excessive risk. We choose the parameters to match the steady-state equilibrium

conditions with the selected data moments. The variables are real in our model, so we

deduct the average level of ination of 2% over that period from the corresponding

nominal data counterparts.

We choose the steady-state capital requirement to support the safe equilibrium

in which all the commercial banks optimally take the minimum risk and nance

a zero amount of risky loans in excess of the amount required by the minimum

scale. This approach nds the static capital requirement that is pseudo-optimal in

a sense that a small decrease in the capital requirement makes banks nance socially

sub-optimal projects,22 while a small increase in the capital requirement results in

liquidity losses without changing risk-taking incentives. Two parameters, which enter

the idiosyncratic process of the risky technology, are crucial in determining the value

of the steady-state level of the capital requirement. We relate the standard deviation

of the idiosyncratic shock τC to leveraged lending that is associated with nancing

corporations with high leverage - dened as those with a debt-to-EBITDA ratio grater

than 6. We choose the other parameter ξ, which is interpreted as an average penalty

on the returns to risky projects, to hit a 10% level of the capital requirement in the

steady state. It reects the levels proposed by Basel III regulation with which U.S.

banks are compliant. It also lies within a span of values usually considered in the

22Ignoring any general equilibrium eects on prices.

82

literature on optimal capital regulation. Appendix B.9 contains the description of the

choice of τC .

We calibrate σ0, fC , and fS to hit the return to safe loans and the spread of

the return to capital over the deposit rate for the commercial and shadow banking

sectors. We use the bank prime loan rate as a measure of the return to safe loans.

The data counterparts for the commercial bank deposit rate and the shadow bank

money market borrowing rate are the national rate on non-jumbo deposits and the

3-month nancial commercial paper interest rate, respectively. We set τS to match

the default rate on corporate bonds the value of which we borrow from Begenau and

Landvoigt (2018). We map the utility weight on shadow bank deposits in the liquidity

function α1 to the share of shadow bank assets. We use the estimates of the size of

the U.S. shadow banking sector from the Financial Stability Board and divide it by

its sum with the commercial bank assets, the value of which come from the Board

of Governors of the Federal Reserve System. We set θ = 0.9 that implies a shadow

bank dividend payout of 10% that is also considered in Ferrante (2018). We make the

value of the investment adjustment cost parameter φ much greater than it is in the

conventional literature. We nd that relatively higher values of φ imply a better match

between the model's impulse responses of the price of investment to the technology

shock and their estimated VAR-based counterparts.23 The corresponding responses

of output, investment, and consumption lie within 90% condence bounds for our

choice of φ.We calibrate the fraction of capital that can be diverted by shadow banks

λ to hit the shadow bank leverage of 25 which is in line with the leveraged ratios of

broker-dealers reported in the literature. Our steady-state capital requirement implies

the commercial bank leverage of 10, so these two values correspond to the evidence

23The VAR identication of technology shocks is similar to the scheme used in Guerrieri,Henderson, and Kim (2019).

83

the leverage of intermediaries belonging to the shadow banking system is about three

times as large as the one of depository institutions obtained from the Flow of Funds by

Ferrante (2018). Finally, we consider the share of risky projects within a wide range

of [0.01, 0.99] for both types of banks. This makes the minimum scale assumption

technical, and thus it does not contaminate the model's main mechanisms.

2.3.2 Experiments

We run several experiments that illustrate how our model responds to shocks.

We show that both aggregate and sectoral shocks that reduce the net worth of banks

move the economy into nancing risky projects. We compute the time-varying capital

requirements that avoid excessive risk-taking. We also evaluate how a permanent

increase in capital requirements aects reintermediation of credit and the amount

of excessive risk in the system. Then we compare the impulse response functions

of the baseline model with the responses of the economy in which commercial banks

intermediate all the assets. This comparison allows us to study how addition of shadow

banks may change policy prescriptions of tighter capital requirements to counteract

excessive risk. In the sensitivity analysis, we nd the parameters that are key to the

spillover of risk from shadow banks to commercial banks. We also include sectoral

shocks that increase the relative attractiveness of risky projects. To nd out how

introduction of an alternative policy tool aects the results, we consider a tax/subsidy

on shadow bank debt. We evaluate proposals to use both capital requirements and

tax on shadow bank activities as policy instruments within the Ramsey framework.

From the computational side, nding numerical solutions requires nonlinear

methods because the model includes the inequality constraints on the amount of

commercial bank equity and loans that bind occasionally. For the experiments that

do not involve policies, we solve the model by applying the OccBin toolkit developed

84

in Guerrieri and Iacoviello (2015). OccBin modies a rst-order perturbation method

and employs a guess-and-verify approach to obtain a piecewise linear solution under

perfect foresight.24 The solution reects the endogenous transition between the

regimes, depending on the state vector, and thus it is highly nonlinear. The algorithm

has advantages over nonlinear projection methods because it is computationally fast

and applies to nonlinear models with a large number of state variables.25

To adapt our framework to OccBin, we break it into three dierent models. The

constraints that capture the signs of the Lagrange multipliers on the loan and equity

constraints of commercial banks control the switching from one model to another.

Accordingly,

1. The Safe model describes the reference regime with µt = 0 in which χs2,t = 0,

χr2,t > 0, µCs,t = 0, and µCr,t > 0.

2. The Risky model describes the alternative regime with µt = 1 in which χs2,t > 0,

χr2,t = 0, µCs,t > 0, and µCr,t = 0.

3. The Mixed model describes the alternative regime with 0 < µt < 1 in which

χs2,t > 0, χr2,t > 0, µCs,t > 0, and µCr,t > 0.

2.4 Quantitative Results I. Responses to Shocks

TFP Shock: We consider a negative 1.5 percent innovation in the AR(1) TFP

process with a persistence of 0.95. Figure 2.2 shows the response of the model

economy to the TFP shock for two experiments. In the rst experiment, the capital

requirement is xed at the steady-state level (dashed line). In the second experiment,

24The reader should also be aware that OccBin solution does not capture precautionarybehavior linked to the possibility of moving away from the reference regime in the future. SeeGuerrieri and Iacoviello (2015) for discussion of the accuracy of the resulting approximations.

25Our baseline model includes 10 state variables.

85

the government sets the capital requirement endogenously and dynamically at the

levels needed to prevent excessive risk-taking in every period following the shock

(solid line). In the language of our model, this endogenous capital requirement is

the minimum capital requirement that supports the safe equilibrium. Any capital

requirement above this level would result in liquidity losses, while any level below

this level would be suboptimal due to nancing socially inecient risky projects.

A fall in At decreases the net worth of banks and launches the familiar nancial

accelerator mechanism described by conventional models of nancial frictions: a drop

in the net worth of banks increases agency costs, forcing banks to sell their assets, thus

depressing asset prices and further worsening their balance sheet conditions. However,

there are several elements that make our framework stand out in the literature.

First, there is reintermediation of credit from shadow banks to commercial banks.

The TFP shock reduces the returns on loan portfolios. Since shadow banks are more

leveraged, their net worth is more negatively aected than the net worth of commercial

banks. With constant prices, the equity of commercial banks becomes relatively more

attractive for households. Consequently, commercial banks start demanding more

loans, and shadow banks transfer some of their assets to commercial banks.

Second, the reintermediation is not full. Prices adjust to restore equilibrium in

the markets. In particular, since two types of deposits are imperfect substitutes,

households require a higher deposit rate to substitute away from shadow bank deposits

to commercial bank deposits. This increases the costs of banks, and the total capital

and loans fall. The return to safe projects goes down.

Third, the expected interest rate spread of commercial banks EtRlt+1 − RdC

t

decreases, making risky projects more attractive and leading to excessive risk-taking.

The government has to increase the lump-sum tax to provide deposit insurance. Over

time, lower capital stock raises the marginal product of capital, pushing up the return

86

to safe projects. Moreover, the deposit rate is gradually restoring to initial values as

the wealth eect of the shock is dissipating, and households switch back to shadow

bank deposits. These two forces activate the mechanism that makes commercial

banks nance safe rms only. Note that the kinks observed for certain variables are

related to the discontinuity of the risk in the periods when the switch occurs.

From the policy side, an increase in the capital requirement subdues risk-taking

incentives and prevents a part of the fall in output, consumption, and investment.

The parameter on the relative penalty of risky projects ξC controls the distance

between the two lines in Figure 2.2. Note that there is a relatively smaller amount of

reintermediation of credit across two sectors because the higher capital requirement

pushes down protability of commercial bank equity.

Capital Quality Shock: Our model calls for a 25 basis points increase in the capital

requirement, on impact, to tackle excessive risk after a contractionary 1.5% TFP

shock. Here we consider an alternative business cycle shock which is commonly used

in the literature of nancial frictions.

Let ιt denote the quality of capital. At the beginning of the period, each unit of

capital transforms into ιt units of eective capital used in production. Then the safe

and risky production technologies are now given by:

Y sj,t = At (ιtKj,t)

αH1−αj,t ,

Y rj,t = At (ιtKj,t)

αH1−αj,t + εj,tιtKj,t.

The quality of capital now provides additional variation in the returns on safe and

risky loans:

Rlt ≡ Rl

i,t =αAtι

αt

Qt−1

(Kt

Ht

)α−1

+ (1− δ)ιtQt

Qt−1

=αY s

i,t

Ki,tQt−1

+ (1− δ)ιtQt

Qt−1

,

Rlj,t = Rl

t +ιtεj,tQt−1

.

87

We consider an AR(1) process for the quality of capital and x the size of the shock so

that it leads to the same percentage drop of output, on impact, as for the TFP shock.

We pick up the autoregressive coecient of 0.66 from Gertler and Karadi (2011).

The results are reported in Figure 2.3. The qualitative mechanisms of the capital

quality shock are similar to the ones described for the TFP shock. However, the

magnitudes now reect a greater drop in the price of capital that signicantly pushes

down the return on safe loans. Risk becomes relatively more attractive for commercial

banks. Moreover, the net worth of shadow banks decreases by more, making the

reintermediation of assets from one sector to another sector even more pronounced.

This all leads to a greater fall in the expected interest rate spread of commercial

banks and calls for a greater rise in the capital requirement needed to avoid excessive

risk-taking. Note that since the shock is relatively less persistent, without any policy,

the economy switches into excessive risk-taking for relatively less number of periods.

Shadow Bank Shock: Having considered business cycle shocks, we now turn into

understanding how developments relevant for the shadow bank sector may aect

risk-taking incentives of commercial banks. In particular, we consider a shock to the

idiosyncratic variance of risky returns of shadow banks τSt , the parameter that only

shows up in the problem of shadow banks. The shock follows AR(1) process with the

autoregressive parameter 0.9. We size the shock, so that it increases the quarterly

default rate of shadow bank loans by 1% on impact. We can interpret this experiment

as the one that reects the rise in subprime defaults that set the recent nancial

crisis in motion or the one that captures the expected increase in the riskiness of loan

portfolios held by shadow banks due to widespread adoption of leveraged lending

practices.

Figure 2.4 shows the results of this experiment. The shock decreases protability

of shadow bank equity. As before, this leads to the reintermediation of capital from

88

shadow banks to commercial banks. The size of this reintermediation happens to be

large because there is a three-fold dierence in the leverage across the banks, and

the shock per se has no eect on commercial banks. The prices need to adjust in

order to make households supply more commercial bank deposits. Thus, the deposit

rate of commercial banks goes up. This increases the costs of making loans, and the

demand for capital falls. With relatively large adjustment costs, the price of capital

drops by more, and this decreases the return to safe projects. This all pushes down

the expected interest rate spread, making risky projects more attractive. Commercial

banks switch into nancing risky projects for about 20 periods. Note that the shock

has limited inuence on aggregate output, consumption, and investment.

The policy that fully prevents excessive risk-taking is to increase the capital

requirement by around 20 basis points on impact and then follow a smooth path

depicted in panel 2 of Figure 2.4. A higher capital requirement decreases the demand

for deposits, and households prefer to consume a little bit more than to save. Output

drops by less. The fall in the expected interest rate spread does not lead to excessive

risk because commercial banks internalize a penalty on the returns to risky projects,

and the rst-order eect of the shadow bank shock on commercial banks is zero. The

rise in the capital requirement does just enough to tame the risk and maximize the

available supply of liquidity.

The results show that excessive risk-taking incentives may arise from the shocks

that trigger a general-equilibrium eect of prices in response to the reintermediation

of assets from shadow banks to commercial banks. This even applies to the shocks that

have moderate eects on aggregate variables. The policy implication is that capital

regulation of commercial banks is still a useful tool that can be used to react to the

sectoral shocks that occur in the shadow banking sector.

89

2.5 Quantitative Results II. Comparison with the Model of Commercial

Banks Only

We quantify the relevance of the reintermediation channel by comparing capital

requirements that prevent excessive risk in two models:

1. The baseline model with shadow banks.

2. The model with only commercial banks.

Except for having no shadow banks, the framework with only commercial banks

follows the presentation of the baseline model. The commercial banking sector

intermediates all the assets, so the the weight on commercial bank liquidity α1 is one.

Substituting this value into the derived utility function from deposits results in the

following liquidity preferences:

Ψ2

(DCt

)=

((DCt

)σd) 1−ζσd

1− ζ=

(DCt

)1−ζ

1− ζ.

The parameter ζ controls the interest rate elasticity of deposit supply, which equals

1ζ. We make it identical for both models. Note that this comparison strategy leaves

the steady-state capital requirement and the interest rates the same across the two

models. The only dierence is the absence of the reintermediation channel in the

model with only commercial banks.

We consider the same TFP and capital quality shocks as before and add one more

shock related to the commercial banking sector. Then we compare the reactions of

capital requirements that avoid excessive risk-taking in every period after each of the

shocks. The solid line matches the responses of the baseline model with shadow banks.

The dashed line represents the reactions of the variables in the model of commercial

banks only. The following exercises show the dierences.

90

TFP Shock: As Figure 2.5 illustrates, in the model without shadow banks, the

TFP shock produces a smaller rise in the capital requirement. Unless there are

shadow banks, commercial banks decrease the demand for loans and deposits as

capital becomes less productive. This contrasts with the increased amount of loans

nanced by commercial banks in the baseline model. This substitution mechanism is

supported by a greater rise in the deposit rate that pushes down the expected interest

rate spread.

Intuitively, due to higher leverage of shadow banks, commercial banks bear a

part of the brunt of the crisis. Their protability decreases by more, making risk

more attractive. The implication of this result for policy analysis is that models that

exclude shadow banks may underestimate the role of capital requirements.

We perform sensitivity analysis on the liquidity preference parameters that

determine the strength of the reintermediation channel. Figure 2.6 shows the

maximum cross-period distance between the response of the capital requirement

to the TFP shock in the baseline model and the response of the capital requirement

to the same shock in the model with only commercial banks. The horizontal axis

denotes the range of values of σd , for each of which there is a value of ζ, each marked

by a dierent color. For example, we can infer from the red line, which xes the value

of ζ to 1.3, that avoiding excessive-risk taking requires about a 10 basis points greater

increase in the capital requirement for the model with shadow banks relative to the

model with only commercial banks when σd = −2, while it calls for around 10 basis

points smaller rise in the capital requirement in the baseline framework compared

with the framework without shadow banks when σd = 0.9.

In Figure 2.6 each line slopes downward, implying that the more substitutable the

deposits of commercial and shadow banks are, the less cross-model rise in the capital

requirement is needed to combat the excessive risk. Intuitively, a greater elasticity

91

suggests that households incur smaller utility costs of switching from shadow bank

deposits into commercial bank deposits, so they require a smaller increase in the

deposit rate. This decreases the expected interest rate spread by less. Note that a

relative rise can turn into a fall for large enough elasticities. We calculate that for the

chosen value of ζ in the calibration, the cuto elasticity of substitution that makes

two models indistinguishable in terms of the reaction of the capital requirement is

equal to 11−0.87

= 7.69.

As can be deduced from the relative location of the lines in Figure 2.6, the eects

of parameter ζ are also predominantly monotone across the range of values of σd. The

more elastic supply of deposits, the more capital regulation of commercial banks is

required. With a higher elasticity, the same drop in the demand for deposits leads

to a smaller fall in the deposit rate. There is also the wealth eect associated with a

corresponding greater decrease in the total liquidity about which households care. To

support liquidity, households allow for a larger drop in consumption that activates the

wealth eect that decreases the supply of deposits, pushing up the deposit rate. Both

eects result in a relatively larger increase in the deposit rate when the preferences

for total liquidity are more elastic. At the same time, the strength of the wealth

eect depends on the change in liquidity when the demand for deposits falls. This

is inuenced by ζ. Moreover, a higher σd makes the wealth eect relatively less

important. This may lead to a larger drop in the relative capital requirement for

smaller values of ζ when σd is large enough.

Capital Quality Shock: We consider the capital quality shock of the same size but

of the dierent direction, so the symmetrically opposite forces attenuate excessive

risk-taking incentives of commercial banks in the model with shadow banks. This

leaves some room for the economy to benet from greater liquidity if the capital

requirement falls. However, as Figure 2.7 shows, in the model without shadow banks,

92

the expansionary capital quality shock increases risk-taking incentives of commercial

banks. In the absence of the reintermediation channel, a rise in the demand for

loans cannot be subdued by the outow of assets from commercial banks. Total

capital has to increase. Investment adjustment costs weaken the initial response of

capital but push up capital in expectation. Firms would decrease investments to

smooth the impact of the shock on capital. Moreover, with adjustment costs, the

price of capital does not rise as much as before, having a relatively smaller positive

inuence on the loan rate. Consumption in expectation goes up, so it makes the wealth

eect on the supply of deposits strong enough to push up the deposit rate. Both

mechanisms increase the expected interest rate spread. Thus, the general equilibrium

eects inherently rise risk-taking incentive in the model of only commercial banks.

Therefore, the policy calls for higher capital requirements.

The implication of this nding is that a failure to account for shadow banks may

lead to incorrect recommendations for the direction of a change in capital requirements

in response to macroeconomic shocks.

Commercial Bank Shock: There is no counterpart for the shadow bank shock in

the model with only commercial banks, so there is no direct way to compare the

responses of two models to the sectoral shock chosen before. Instead, we examine

a shock to the idiosyncratic variance of risky returns of commercial banks τCt . We

can regard this shock as the one that increases the exposure of commercial banks to

leveraged lending risk. Greater risk-taking raises protability of commercial bank at

the expense of higher costs of providing deposit insurance by the government. The

size of the shock to τCt and its process correspond to the characteristics considered

for the shock to τSt .

As Figure 2.8 demonstrates, the two models produce very similar responses to the

shock. First, the size of the reintermediation mechanism is small. The shock has no

93

rst-order eect on shadow banks. However, in contrast to a shock to τSt , a possible

repercussion from greater attractiveness of the equity of commercial banks is limited

because higher capital requirements attain both objectives: they fully tame higher

risk-taking incentives and decrease protability of commercial banks. Second, the

capital requirement policies can fully stabilize the impact of the shock on the aggregate

variables. The sectoral nature of the shock increases the comparative advantages of

nancing risky rms, calling for a large rise in the capital requirement of nearly 1.3

percent.

A corollary of this experiment is that not all the shocks that redress the relative

attractiveness of the two types of projects activate the reintermediation channel that

leads to a greater response of capital requirements in the model with shadow banks.

2.6 Quantitative Results III. Additional Instrument and Permanent

Increase in Capital Requirements

Suppose that the macroprudential policy toolbox also includes an instrument to

regulate shadow banks. Here we analyze how it can be used separately from capital

requirements to attenuate excessive risk. Let τSd,t denote the tax on shadow bank

deposits. The shadow bank pays τSd,t to the government for each unit of deposits

raised from households. The government transfers the tax back to households in a

lump-sum fashion. The balance sheet transforms into:

lSt = dSt(1− τSd,t

)+ eSt .

Expressing dSt and substituting it into the expression of the net cash ow:

ωSt+1 = max

[(Rlt+1 + σSt

εSt+1

Qt

−RdSt+1

1− τSd,t

)lSt +

RdSt+1

1− τSd,teSt , 0

].

94

Therefore, taking into account the tax on shadow banks, the net worth evolves

according to:

eSt+1 = eSt

σSφSt τS

Qt

√2πe−(

((1−τSd,t)φSt Rlt+1−RdSt+1(φSt −1))Qt

(1−τSd,t)φSt σS

√2τS

)2

+

((1− τSd,t

)Rlt+1φ

St −RdS

t+1φSt +RdS

t+1

)2(1− τSd,t

) [1 + erf

(((1− τSd,t

)φSt R

lt+1 −RdS

t+1

(φSt − 1

))Qt(

1− τSd,t)φSt σ

S√

2τS

)]].

In the steady state, the tax on shadow bank debt is zero and the capital

requirement is 10%. Figure 2.9 compares the responses of three policies to the

same contractionary TFP shock as before. The black solid line and the red dashed

line repeat the results from the policies considered so far: one with time-varying

endogenous capital requirements that prevent excessive risk and another with xed

capital requirements, respectively. The blue dash-dotted line shows the response of

the tax on shadow bank deposits which is needed to force commercial banks to

nance the minimum share of risky projects.

Following the shock, a subsidy to shadow banks increases the relative attractiveness

of their equity. This leads to a change in the direction of the reintermediation channel:

loans, deposits, and capital now migrate from commercial banks to shadow banks.

Commercial banks decrease the demand for deposits, pushing down the deposit rate.

As before, the wealth eect on the supply of deposits pushes up the deposit rate. In

aggregate, these two forces balance each other, and the deposit rate of commercial

banks is barely aected. The price of capital is almost non-responsive because the

subsidy lessens the impact of the shock on the demand for investment due to a smaller

negative eect of a higher leverage of shadow banks on the returns. Thus, as shown in

panel 10 of Figure 2.9, a positive expected interest rate spread of commercial banks

decreases their risk-taking incentives. Intuitively, the subsidy to deposits of shadow

95

banks drags down protability of commercial banks relatively less, making the shield

of limited liability less attractive to use to nance risky projects.

We measure welfare, W0, as the conditional expectation of households utility as

of time zero:

W0 = E0

∞∑t=0

βtU(Ct, D

Ct , D

St+1

)=C1−σc

0 − 1

1− σc+ σ0Ψ

(DC

0 , DS1

)+ βE0W1.

Since the three policies begin from the same initial point being the non-stochastic

steady state, we can evaluate them by comparing their welfare responses to the shock.

This strategy avoids spurious welfare shifts associated with transitional dynamics

leading to the steady state. As panel 3 of Figure 2.9 illustrates, the xed capital

requirement is dominated by each of the other two policies. At the same time, there

is no ubiquitous welfare ranking of the endogenous policies. The endogenous capital

requirement achieves a smaller welfare loss in the rst 10 periods but a higher welfare

loss in the following periods compared with the endogenous tax on shadow bank

deposits. This motivates the experiment in which both instruments are introduced

simultaneously to improve welfare. We describe it in the next section.

To summarize, the model suggests that subsidizing more leveraged banks can

serve as another policy tool to decrease risk-taking incentives of commercial banks in

response to a contractionary shock.

2.6.1 Optimal Policy

Having added an additional policy instrument, we evaluate proposals on a

joint use of capital requirements, γt, and a tax on shadow banks, τSd,t, within the

Ramsey framework. We consider cooperative policies which maximize utility of the

representative household. We derive the Ramsey policy with full commitment from

96

the maximization problem written in general terms

maxxt,γt,τSd,t

∞t=0

E0

∞∑t=0

βtU (xt−1, xt, ςt) ,

subject to

Etg (xt−1, xt, xt+1, ςt) = 0,

where xt is a vector of endogenous variables and ςt is a vector of exogenous variables.

The N × 1 vector xt is partitioned as xt =(x′t, γt, τ

Sd,t

)′. Given the sequence of

the policy instrumentsγt, τ

Sd,t

∞t=0

, the remaining N − 2 endogenous variables need

to satisfy the N − 2 structural equilibrium conditions which are captured by the

vector g in the baseline (safe) model.26 We employ a dual approach to solve for

the optimal Ramsey policy. We adopt the concept of optimality from a timeless

perspective to make equilibrium functions time-invariant. We attach the N − 2

Lagrange multipliers to the vector g, solve for the steady state numerically, and

use perturbation methods to nd the optimal policy. For that purpose, we apply a

toolbox developed in Bodenstein, Guerrieri, and LaBriola (2019) that automates the

analytical derivation of the conditions for an equilibrium under the cooperative and

open-loop Nash policies.

Table 2.4 reports the implied steady-state values for the baseline economy together

with those for the Ramsey economy. The structural parameters of the two models

are identical. We use the steady state of the original model as the initial guess for

the steady state of the Ramsey problem. Then we utilize the toolbox that employs

quasi-Newton methods to solve for the steady state. In theory, multiple solutions are

26Remember that, in the original problem, capital requirements are xed at 10%, and atax on shadow bank deposits equals 0. In the previous sections, we consider the policies thatprevent excessive risk-taking incentives by making one instrument endogenous and anothertool xed. For the Ramsey policy, there are two degrees of freedom associated with theunconstrained choice of both capital requirements and a tax on shadow bank debt.

97

possible.27 The diculty of steady-state calculation comes from the presence of the

idiosyncratic risk in our problem. We cannot exclude multiplicity of steady states.

Our experimentation with dierent initial guesses either supported the same steady

state or led to the failure of the chosen numerical algorithms to solve the problem.

We would expect that the Ramsey planner would like to maximize the benets

from liquidity services by choosing small capital requirements and a small level of

a tax on shadow bank debt in the steady state. However, the Ramsey problem is

based on the model which does not capture the possibility of a switch from a safe

equilibrium in the steady state. In fact, in the steady state that we capture, the

Ramsey planner imposes a relatively high taxes on shadow bank activity and favors

low capital requirements on commercial banks. Notice that the Ramsey steady state

features the same value of the leverage of shadow banks as in the baseline model. This

can explain a relatively high tax on shadow bank deposits to make shadow banks

hold enough equity for φS = 25. Moreover, our choice of the steady state makes the

incentives of commercial banks to switch into nancing risky projects comparable to

the ones in the baseline model.

Figure 2.10 shows the responses of economic variables to the same contractionary

TFP shock under three policies. The black solid line represents the reaction of the

Ramsey planner, while the red dashed line and the blue dash-dotted line repeat the

results of the one-instrument endogenous policies (capital requirements and a tax on

shadow bank deposits, respectively) which prevent excessive risk-taking.

Absent intervention from the policymaker, commercial banks are undercapitalized

after the shock. The Ramsey policy calls for large simultaneous increases in capital

requirements and in a tax on shadow bank deposits to stabilize the eect of the shock

27It is on a research agenda of numerical literature to evaluate how a particular steadystate aects the Ramsey policy. It goes beyond the scope of the current paper to provideguidance on that.

98

on commercial bank loans. A higher capital requirement spills over into the shadow

banking sector, making its equity more attractive. A higher tax on shadow banks

prevents the migration of assets out of commercial banks. Otherwise, this would lead

to the reallocation of liquidity services, activating general equilibrium mechanisms

which the planner considers more harmful to the overall welfare. A higher shadow bank

tax, in turn, amplies the eect of the contractionary shock on aggregate variables

and spills over into commercial banks, so a larger increase in the capital requirement is

needed to achieve the stabilization goal. This spiral of repercussion eects can explain

large magnitudes of the reactions in the policy instruments.

Although shadow bank loans drop signicantly in percentage terms, they fall

relatively little in absolute terms. Thus, it has relatively mild negative consequences

for the total liquidity services. Notice that the Ramsey policy is also prudential in

a sense that it combats excessive risk because the Lagrange multiplier, χ2, which

controls risk-taking incentives of commercial banks, is always positive throughout the

transition path. Overall, the Ramsey policy acts through stabilization of loans oered

by commercial banks.

2.6.2 Effect of a Permanent Change in Capital Requirements on Risk

To evaluate the eects of tighter regulation on excessive risk-taking incentives, we

begin with the steady state in which commercial banks are relatively undercapitalized

at 6% capital-to-assets ratio and choose to nance the maximum share of risky

rms. Then the regulator imposes a 2% higher capital requirement permanently. This

experiment intends to capture the post-crisis measures on strengthening the balance

sheets of commercial banks.

Figure 2.11 shows the results of the experiment for two dierent parameter values

of ξ: baseline ξ = 0.0003 and two-times larger ξ = 0.0006. First, tighter regulation

99

makes commercial banks less protable and causes permanent expansion in shadow

banking.28 Second, the increase in capital requirements is sucient to move the

economy away from nancing socially inecient risky projects. Liquidity becomes

relatively more expensive, so households prefer to substitute away from liquidity

services to consumption. Therefore, in contrast to the reintermediation mechanism

described so far that pushes up the deposit rates, here migration of credit does not

result in higher borrowing costs of commercial banks. Moreover, since intermediation

becomes more costly, banks lend less, so both capital and investment drop. The fall in

capital stock raises the marginal product of capital, pushing up the attractiveness of

safe projects. From panel 9 of Figure 2.11 you can see that the expected interest rate

spread of commercial banks goes up, completely attenuating excessive risk. Third,

the gains of permanently higher capital requirements depend on the parameter ξ that

governs the dierence in expected returns between safe and risky projects. A larger

value of ξ makes the initial excessive risk more expensive for society, and thus, there

is a greater positive eect of higher capital requirements on output and consumption.

2.7 Conclusions

In this paper, we develop a macroeconomic model with a nancial sector that

includes both traditional and shadow banks. We quantify the roles of capital

regulation and macroprudential policy for managing excessive risk-taking incentives.

A contractionary shock decreases the net worth of nancial intermediaries, making the

shield of limited liability more attractive and, thus, increasing risk-taking motives of

banks. We nd that the reintermediation of credit from shadow banks to commercial28Note that initial drop in shadow bank loans is explained by the risk being pre-

determinant for the eects of the shock in the rst period. Starting from period 2, a greaterrelative protability of shadow bank equity becomes the main force that explains the shiftof credit into shadow banks.

100

banks, operating through prices in general equilibrium, additionally stimulates banks

to nance socially inecient projects. Tighter capital regulation makes the benets

from limited liability smaller, counteracting excessive risk-taking incentives.

We consider three types of dynamic policies: the endogenous capital requirements

that prevent excessive risk, the endogenous tax on shadow bank deposits that

combats excessive risk, and the welfare-maximizing Ramsey policy that combines

both instruments. All the policies are welfare-improving relative to xed capital

requirements. In the steady state, the Ramsey planner makes the capital requirement

close to zero and establishes a relatively high tax on shadow bank deposits. To respond

to a 1 s.d. negative TFP shock, the Ramsey policy (implemented by signicantly

increasing both capital requirements and the tax because of the spillover eects of

one instrument on the other) stabilizes the amount of commercial bank loans.

Our results stress the importance of embedding shadow banks in general

equilibrium. Non-inclusion of shadow banks underestimates the role of capital

requirements for attenuating excessive risk and can even mislead policymakers

about the direction of their response to macroeconomic shocks. The model provides

evidence that capital requirements should also counter the disturbances arising in

the shadow banking sector which may only have moderate eects on aggregate

variables. At the same time, although purely sectoral shocks that make risk relatively

more attractive call for a greater change in capital requirements, not all such shocks

necessarily trigger the reintermediation mechanism.

One issue we leave for further work is extension of the model to include bank

runs. Although we make liquidity services largely relevant in our framework, we

expect that a more detailed modeling of bank runs would increase the strength of

the reintermediation channel and, thus, sharpen our conclusions about a greater

response of capital requirements. Another promising avenue for future research lies in

101

consideration of monetary policy. Because risk-taking incentives depend on the spread

between the loan rate and the deposit rate, monetary policy, setting the interest rate

and aecting the spread, can have a non-negligible role in our framework. This feature

would make it possible to answer the questions, which have recently attracted a lot

of attention both from academics and policymakers, about how to combine monetary

and capital regulation policies.

102

2.8 Tables and Figures

Table 2.1: Comparison of Two Types of Banks.

Bank

FeaturesDeposit Insurance Risk-Taking

Commercial Full Endogenous: σC ≤ σt ≤ σC

Shadow No Endogenous: σS ≤ σt ≤ σS

Bank

FeaturesRegulation Life Span

Commercial Capital Requirement Innitely-lived

Shadow No Finite horizon: continues with probability θ

Bank

FeaturesModeling Frictions

Commercial Moral hazard associated with deposit insurance and limited liability

Shadow Moral hazard associated with limited liability and costly enforcement

Bank

FeaturesLoan portfolio

CommercialReturns to Safe projects:

(1− σCt

)Rlt+1

Returns to Risky projects: σCt

(Rlt+1 +

εCt+1

Qt

)where εCt+1 ∼ N

(−ξC , τ2

C

)for ξC > 0

ShadowReturns to Safe projects:

(1− σSt

)Rlt+1

Returns to Risky projects: σSt

(Rlt+1 +

εSt+1

Qt

)where εSt+1 ∼ N

(0, τ2

S

)Note: This table shows the main dierences in the modeling approach of two types ofbanks.

103

Table 2.2: Illustrating the Eects of Higher Risk on Dividends.

States of nature whereEects on ω1

Eects on ω2Rlt+1 −Rdt (1− γt)− ξσt 1−G(ε∗t+1)

Rlt+1 < Rdt (1− γt) ⇔ ε∗t+1 > 0 ⇓ ⇑ ⇑

Rlt+1 > Rdt (1− γt) ⇔ ε∗t+1 < 0 ⇓ ⇓ if ε∗t+1 > −ξ, then ⇑⇓if ε∗t+1 6 −ξ, then ⇑

Note: This table reports the eects of greater risk-taking on two components ofdividends for each realization of the aggregate returns. The up-turn arrow, ⇑,indicates that greater risk-taking increases the corresponding component of bank'sdividends. The down-turn arrow, ⇓, means that the corresponding component ofbank's dividends decreases with greater risk-taking. Two arrows turned in the oppositedirections, ⇑⇓, signify that the eect of greater risk-taking is undetermined anddepends the parameterization.

104

Table 2.3: Parameters.

Value Description

Conventional

β 0.99 Discount rate

α 0.3 Capital share in production

σc 1.1 Elasticity of substitution for consumption

δ 0.025 Depreciation rate

σd 0.4 Substitution elasticity b/w C- and S-bank liquidity Begenau and Landvoigt (2018)

ζ 1.1 Interest rate elasticity of supply of total liquidity Within [1, 2] in the literature

Specic Target

τC 0.0481 Standard deviation of C-bank idiosyncratic shock DebtEBITDA

= 7

ξC 0.0003 Minus mean of idiosyncratic shock for C-banks Cap. requirement= 10%

σ0 0.28 Relative weight on liquidity in the utility function Real prime rate Rl = 1.01

fC 0.006 Linear Costs of Commercial Banking Rl −RdC = 3%

fS 0.003 Linear Costs of Shadow Banking RdS −RdC = 0.15%

τS 0.0187 Standard deviation of S-bank idiosyncratic shock Corp. bond default= 1.44%

α1 0.4729 Weight on S-bank deposits in the liquidity function Share of SB assets= 45%

θ 0.9 S-banker's survival probability 10% dividend payout of SB

φ 100 Investment adjustment costs VAR evidence

λ 0.229 Fraction of capital that can be diverted by S-banks Shadow bank leverage= 25

σC 0.01 Minimum risk that C-banks can take numerical solution method

σC 0.99 Maximum risk that C-banks can take numerical solution method

σS 0.01 Minimum risk that S-banks can take numerical solution method

σS 0.99 Maximum risk that S-banks can take numerical solution method

Note: See Section 2.3.1 for the calibration strategy.

105

Table 2.4: Steady-State Values.

Baseline Ramsey Remarks

γ 10% 0.145% Capital Requirement

τSd 0 16.8% Tax on Shadow Bank Deposits

W0 150.6 151.92 Welfare

Rl 1% 0.95% Annualized Real Return

Rl −RdS 3% 2.41% AnnualizedRdS

1−τSd−RdC 0.15% -0.468% Annualized

KCB

Ktot 55% 98.7% Share of Capital Intermediated by CBKtot

Y tot 2.735 2.741 Capital-Output Ratio (output is annualized)CY tot 72.65% 72.59% Consumption-Output Ratio

φS 25 25 Leverage of SB

x? 98.56% 1 Share of non-defaulted SB Deposits

Note: This table reports the steady-state values of the baseline economy and thesteady-state values of the Ramsey economy. The structural parameters are identicalin the economies.

106

Shadow Bankers

Period tshocks

are realized

t

Receive equityfrom the

equity fund

Choose riskinessof a loanportfolio

Borrow fromhouseholds andlend to rms t+ 1

Period t+ 1shocks

are realized

Share earnings/losseswith the equity and

deposit funds

A bankerexits with

probability 1− θ

The equity fund paysdividends from net worth

of exiting bankers

Commercial Bankers

Period tshocks

are realized

t

Issueequity anddeposits

Choose riskinessof a loanportfolio

Lend toproduction

rms t+ 1

Period t+ 1shocks

are realized

Receive thereturnto loans

Pay o depositfrom period tor default

Distributedividends tohouseholds

Figure 2.1. Timeline for Two Types of Bankers.

Note: This gure plots the timelines for two types of banks.

107

20 40 60 80 100

-1.5

-1

-0.5

Per

cent

1. Total output

Endog. Capital RequirementsFixed Capital Requirements

20 40 60 80 1000

0.1

0.2

Per

c. P

oint

2. Capital requirement

20 40 60 80 100

-2

-1.5

-1

-0.5

Per

cent

3. Consumption

20 40 60 80 1000

0.1

0.2

Per

c. P

oint

4. Tax (percent of total output)

20 40 60 80 100

-1.5

-1

-0.5

Per

cent

5. Investment

20 40 60 80 100

0.20.40.60.8

1

Leve

l

6. Share of CB risky projects

20 40 60 80 100

-0.6

-0.4

-0.2

Per

cent

7. Total Capital

20 40 60 80 100

5

10

15

Per

cent

8. Total Loans of CB

20 40 60 80 100-25-20-15-10-5

Per

cent

9. Total Loans of SB

20 40 60 80 100Quarters

-2

-1

0

Per

cent

10. Price of Capital

20 40 60 80 100Quarters

-1.5

-1

-0.5

Bas

is P

oint

11. CB Interest Rate Spread

20 40 60 80 100Quarters

949698

100102104

Leve

l

12. Share of non-defaulted SB

Figure 2.2. Impulse Responses to a TFP Shock.

Note: This gure plots the responses to a 1.5% fall in At. The shock follows AR(1)process with persistence 0.95. The dashed line shows what would happen if the capitalrequirement were to be held constant at its steady state value. The solid line showsthe responses under endogenous capital requirement that is set at the minimum levelto prevent excessive risk-taking in every period following the shock.

108

20 40 60 80 100-1.5

-1

-0.5

Per

cent

1. Total output

Endog. Capital RequirementsFixed Capital Requirements

20 40 60 80 1000

0.2

0.4

0.6

Per

c. P

oint

2. Capital requirement

20 40 60 80 100-2

-1

0

Per

cent

3. Consumption

20 40 60 80 1000

0.1

0.2

Per

c. P

oint

4. Tax (percent of total output)

20 40 60 80 100

-0.6

-0.4

-0.2

0P

erce

nt

5. Investment

20 40 60 80 100

0.20.40.60.8

1

Leve

l

6. Share of CB risky projects

20 40 60 80 100

-0.15

-0.1

-0.05

Per

cent

7. Total Capital

20 40 60 80 1000

10

20

Per

cent

8. Total Loans of CB

20 40 60 80 100

-40

-20

0

Per

cent

9. Total Loans of SB

20 40 60 80 100Quarters

-10

-5

0

Per

cent

10. Price of Capital

20 40 60 80 100Quarters

-2

-1

0

Bas

is P

oint

11. CB Interest Rate Spread

20 40 60 80 100Quarters

90

95

100

Leve

l

12. Share of non-defaulted SB

Figure 2.3. Impulse Responses to a Capital Quality Shock.

Note: This gure plots the responses to a 5% fall in ιt. The shock follows AR(1)process with the autoregressive coecient 0.66 as in Gertler and Karadi (2011). Theshock is sized to lead to the same percentage drop of output, on impact, as in thecase of the TFP shock. The dashed line shows what would happen if the capitalrequirement were to be held constant at its steady state value. The solid line showsthe responses under endogenous capital requirement that is set at the minimum levelto prevent excessive risk-taking in every period following the shock.

109

20 40 60 80 100

-0.2

-0.1

0

Per

cent

1. Total output

Endog. Capital RequirementsFixed Capital Requirements

20 40 60 80 1000

0.05

0.1

0.15

Per

c. P

oint

2. Capital requirement

20 40 60 80 100

-0.2

-0.1

0

0.1

Per

cent

3. Consumption

20 40 60 80 1000

0.1

0.2

Per

c. P

oint

4. Tax (percent of total output)

20 40 60 80 100

-0.6

-0.4

-0.2

0P

erce

nt

5. Investment

20 40 60 80 100

0.20.40.60.8

1

Leve

l

6. Share of CB risky projects

20 40 60 80 100

-0.25-0.2

-0.15-0.1

-0.05

Per

cent

7. Total Capital

20 40 60 80 100

10

20

30

Per

cent

8. Total Loans of CB

20 40 60 80 100-50-40-30-20-10

Per

cent

9. Total Loans of SB

20 40 60 80 100Quarters

-2

-1

0

Per

cent

10. Price of Capital

20 40 60 80 100Quarters

-3

-2

-1

Bas

is P

oint

11. CB Interest Rate Spread

20 40 60 80 100Quarters

949698

100102104

Leve

l

12. Share of non-defaulted SB

Figure 2.4. Impulse Responses to a Shadow Bank Shock.

Note: This gure plots the responses to a 55 basis point rise in τSt . The shock followsAR(1) process with the autoregressive coecient 0.9. The shock is sized to increasethe quarterly default rate of shadow bank loans by 1% on impact. The dashed lineshows what would happen if the capital requirement were to be held constant atits steady state value. The solid line shows the responses under endogenous capitalrequirement that is set at the minimum level to prevent excessive risk-taking in everyperiod following the shock.

110

20 40 60 80 100-1.5

-1

-0.5

Per

cent

1. Total output

Endogenous Capital Requirements (With Shadow Banks)Endogenous Capital Requirements (No Shadow Banks)

20 40 60 80 1000

0.1

0.2

Per

c. P

oint

2. Capital requirement

20 40 60 80 100-2

-1.5

-1

-0.5

Per

cent

3. Consumption

20 40 60 80 100

-2

-1

0

Per

cent

4. Price of Capital

20 40 60 80 100

-1

-0.5

0P

erce

nt

5. Investment

20 40 60 80 1000.0095

0.01

0.0105

Leve

l

6. Share of CB risky projects

20 40 60 80 100

-0.6

-0.4

-0.2

Per

cent

7. Total Capital

20 40 60 80 100

0

5

10

15

Per

cent

8. Total Loans of CB

20 40 60 80 100-25-20-15-10-5

Per

cent

9. Total Loans of SB

20 40 60 80 100Quarters

-0.8-0.6-0.4-0.2

00.2

Bas

is P

oint

10. CB Interest Rate Spread

20 40 60 80 100Quarters

0.1

0.2

0.3

Per

c. P

oint

11. CB Deposit Rate

20 40 60 80 100Quarters

-2

-1

0

Per

c. P

oint

12. Return to safe loans

Figure 2.5. Endogenous Responses to a TFP Shock Depending on thePresence of Shadow Banks.

Note: This gure plots the responses to a 1.5% fall in At under endogenous capitalrequirement (that is set at the minimum level to prevent excessive risk-taking inevery period following the shock) for two models. The shock follows AR(1) processwith the autoregressive coecient 0.95. The dashed line shows what would happen inthe model without shadow banks (commercial banks only). The solid line shows theresponses of our baseline model with shadow banks.

111

-20

-15

-10

-5

0

5

10

15

-2.00 -1.50 -1.00 -0.50 0.00 0.20 0.40 0.50 0.70 0.87 0.90

1.10

1.30

1.50

1.75

1.90

2.00

Max D

iff.

in

R

esp

on

se o

f C

ap

. R

equ

irem

ent

(basi

s p

oin

ts)

𝝈𝒅 , (elasticity of substitution between CB and SB liquidity is given by

𝟏

𝟏−𝝈𝒅)

𝜻,

(interest rate

elasticity of

deposits is

given by 𝟏

𝜻 )

Figure 2.6. Sensitivity Analysis: Relative Responses of CapitalRequirements to a TFP Shock.

Note: This gure plots the maximum cross-period distance between the response ofendogenous capital requirement to a 1.5% fall in At in the baseline model and theresponse of endogenous capital requirement to the same shock in the model with onlycommercial banks. The shock follows AR(1) process with the autoregressive coecient0.95. The horizontal axis denotes the range of values of σd , for each of which thereis a value of ζ, each marked by a dierent color.

112

20 40 60 80 100

1

2

3

Per

cent

1. Total output

Endogenous Capital Requirements (With Shadow Banks)Endogenous Capital Requirements (No Shadow Banks)

20 40 60 80 100

-0.6

-0.4

-0.2

0

0.2

Per

c. P

oint

2. Capital requirement

20 40 60 80 1000

2

4

Per

cent

3. Consumption

20 40 60 80 100

0

5

10

Per

cent

4. Price of Capital

20 40 60 80 100

-2

-1

0P

erce

nt

5. Investment

20 40 60 80 1000.0095

0.01

0.0105

Leve

l

6. Share of CB risky projects

20 40 60 80 100

2468

1012

Per

cent

7. Total Capital

20 40 60 80 100

-20

-10

0

10

Per

cent

8. Total Loans of CB

20 40 60 80 1000

20

40

Per

cent

9. Total Loans of SB

20 40 60 80 100Quarters

-0.4

-0.2

0

Bas

is P

oint

10. CB Interest Rate Spread

20 40 60 80 100Quarters

-0.50

0.51

1.5

Per

c. P

oint

11. CB Deposit Rate

20 40 60 80 100Quarters

0

5

10

15

Per

c. P

oint

12. Return to safe loans

Figure 2.7. Endogenous Responses to a Capital Quality Shock Dependingon the Presence of Shadow Banks.

Note: This gure plots the responses to a 5% rise in ιt under endogenous capitalrequirement (that is set at the minimum level to prevent excessive risk-taking inevery period following the shock) for two models. The shock follows AR(1) processwith the autoregressive coecient 0.66 as in Gertler and Karadi (2011). The shockis sized to lead to the same percentage change of output, on impact, as in the caseof the TFP shock. The dashed line shows what would happen in the model withoutshadow banks (commercial banks only). The solid line shows the responses of ourbaseline model with shadow banks.

113

20 40 60 80 100

1

2

3

4

Per

cent

10-4 1. Total output

Endogenous Capital Requirements (With Shadow Banks)Endogenous Capital Requirements (No Shadow Banks)

20 40 60 80 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Per

c. P

oint

2. Capital requirement

20 40 60 80 100

-10

-5

0

Per

cent

10-4 3. Consumption

20 40 60 80 100

0

5

10

Per

cent

10-3 4. Price of Capital

20 40 60 80 100

0

1

2

3

Per

cent

10-3 5. Investment

20 40 60 80 1000.0095

0.01

0.0105

Leve

l

6. Share of CB risky projects

20 40 60 80 100

5

10

15

Per

cent

10-4 7. Total Capital

20 40 60 80 100

-0.4

-0.3

-0.2

-0.1

0

Per

cent

8. Total Loans of CB

20 40 60 80 100

0.1

0.2

0.3

0.4

0.5

Per

cent

9. Total Loans of SB

20 40 60 80 100

Quarters

0.5

1

1.5

2

Bas

is P

oint

10. CB Interest Rate Spread

20 40 60 80 100

Quarters

-0.025

-0.02

-0.015

-0.01

-0.005

Per

c. P

oint

11. CB Deposit Rate

20 40 60 80 100

Quarters

0

5

10

Per

c. P

oint

10-3 12. Return to safe loans

Figure 2.8. Endogenous Responses to a Commercial Bank ShockDepending on the Presence of Shadow Banks.

Note: This gure plots the responses to a 55 basis point rise in τCt under endogenouscapital requirement (that is set at the minimum level to prevent excessive risk-takingin every period following the shock) for two models. The shock follows AR(1) processwith the autoregressive coecient 0.9. The dashed line shows what would happen inthe model without shadow banks (commercial banks only). The solid line shows theresponses of our baseline model with shadow banks.

114

20 40 60 80 1000

0.1

0.2

Per

c. P

oint

1. Capital requirement

Endog. Capital RequirementsFixed Capital RequirementsEndog. Tax on SB Debt

20 40 60 80 100

-0.8-0.6-0.4-0.2

0

Per

c. P

oint

2. Tax on SB Deposits

20 40 60 80 100-0.2

-0.1

0

Per

cent

3. Welfare

20 40 60 80 100

-1.5

-1

-0.5

Per

cent

4. Total output

20 40 60 80 100

-2-1.5

-1-0.5

Per

cent

5. Consumption

20 40 60 80 100

0.20.40.60.8

1

Leve

l

6. Share of CB risky projects

20 40 60 80 100

-20-10

010

Per

cent

7. Total Loans of CB

20 40 60 80 100

-20

0

20

Per

cent

8. Total Loans of SB

20 40 60 80 100Quarters

-2

-1

0

Per

cent

9. Price of Capital

20 40 60 80 100Quarters

-1

0

1

Bas

is P

oint

10. CB Interest Rate Spread

20 40 60 80 100Quarters

949698

100102104

Leve

l

11. Share of non-defaulted SB

Figure 2.9. Comparison of Policy Reactions to a TFP Shock.

Note: This gure plots the responses to a 1.5% fall in At for three policies. The shockfollows AR(1) process with the autoregressive coecient 0.95. The black solid lineand the red dashed line repeat the results from the policies considered in Figure2.2. The blue dash-dotted line shows the responses under endogenous tax on shadowbank deposits that is set at the level to prevent excessive risk-taking in every periodfollowing the shock.

115

20 40 60 80 10002468

Per

c. P

oint

1. Capital requirement

Ramsey Capital RequirementsEndog. Capital RequirementsEndog. Tax on SB Debt

20 40 60 80 100

02468

Per

c. P

oint

2. Tax on SB Deposits

20 40 60 80 100

-0.15-0.1

-0.050

Per

cent

3. Welfare

20 40 60 80 100-1.5

-1

-0.5

Per

cent

4. Total output

20 40 60 80 100-2

-1.5-1

-0.5

Per

cent

5. Consumption

20 40 60 80 1000

2

4

Per

c. P

oint

6. Lagrange Mult. on loans, 2

20 40 60 80 100

-20-10

010

Per

cent

7. Total Loans of CB

20 40 60 80 100-80-60-40-20

020

Per

cent

8. Total Loans of SB

20 40 60 80 100Quarters

-2

-1

0

Per

cent

9. Price of Capital

20 40 60 80 100Quarters

0

5

10

Bas

is P

oint

10. CB Interest Rate Spread

20 40 60 80 100Quarters

95

100

105

Leve

l

11. Share of non-defaulted SB

Figure 2.10. Policy Responses to a TFP Shock Depending on Instruments.

Note: This gure plots the responses to a 1.5% fall in At for three endogenous policies.The shock follows AR(1) process with the autoregressive coecient 0.95. The blacksolid line shows the reaction of the cooperative Ramsey policy that uses both capitalrequirements and a tax on shadow bank deposits. The red dashed line and the bluedash-dotted line repeat the results of one-instrument endogenous policies (capitalrequirements and a tax on shadow bank deposits, respectively) that prevent excessiverisk-taking.

116

50 100 150 200

0

0.1

0.2

0.3

Per

cent

1. Total output

Baseline penalty on risky projects, xiHigher penalty on risky projects, xi

50 100 150 2001.9

1.95

2

2.05

2.1

Per

c. P

oint

2. Change in Capital requirement

50 100 150 200

0.1

0.2

0.3

0.4

0.5

Per

cent

3. Consumption

50 100 150 200

-0.8

-0.6

-0.4

-0.2

Per

cent

4. Investment

50 100 150 200

0.0096

0.0098

0.01

0.0102

0.0104

Leve

l

5. Share of CB risky projects

50 100 150 200

-0.8

-0.6

-0.4

-0.2

Per

cent

6. Total Capital

50 100 150 200

Quarters

-4

-2

0

2

4

6

Per

cent

7. Total Loans of CB

50 100 150 200

Quarters

-8

-6

-4

-2

0

2

4

Per

cent

8. Total Loans of SB

50 100 150 200

Quarters

6

7

8

Bas

is P

oint

9. CB Interest Rate Spread

Figure 2.11. A Capital Requirement Permanent Shock: 2 % Rise in γt.

Note: This gure plots the responses of a 2% permanent increase in capitalrequirement from its steady-state level of 8%. There is excessive risk-taking in thesteady state. The solid line shows the responses when we consider the baseline penaltyof nancing risky projects, i.e. ξ = 0.0003. The red dashed line shows the responseswhen we consider a two-times larger penalty of nancing risky projects, i.e. ξ = 0.0006.

117

Appendix A

Appendix for Chapter 1

A.1 The Bank's Problem

A.1.1 Baseline: First-Order Conditions

Substituting dt = lt − et into equation (1.16) and writing dG(εt+1) explicitly turn

the objective into:

maxlt,et,σt

Et

ψt,t+1

∞

ε∗t+1

((Rst+1 + σt

εt+1

Qt

)lt −Rd

t (lt − et))

1√2πτ 2

e−(εt+1+ξ)2

2τ2 dεt+1

− et ,

subject to

et ≥ γtlt,

lt ≥ 0,

σ ≤ σt ≤ σ,

where ψt,t+1 = β λct+1

λctand ε∗t+1 =

(Rdt−Rst+1

σt− Rdt et

σtlt

)Qt that we expressed from(

Rst+1 + σt

ε∗t+1

Qt

)lt −Rd

t (lt − et) = 0 to get the lower limit of the integral.

Append the Lagrangian multiplier χ1t to the constraint et ≥ γlt and χ2t to the

constraint lt ≥ 0. Conditional on the optimal choice of σt, the rst-order conditions

118

are:

∂L∂lt

= Et

ψt,t+1

=0︷ ︸︸ ︷((Rst+1 + σt

(Rdt (lt − et)σtlt

−Rst+1

σt

))lt −Rd

t (lt − et))·∂ε∗t+1

∂lt

+ χ2t+

Et

∞

ε∗t+1

ψt,t+1∂

∂lt

((Rst+1 + σt

εt+1

Qt

)lt −Rd

t (lt − et))

1√2πτ 2

e−(εt+1+ξ)2

2τ2 dεt+1

− γχ1t = 0,

∂L∂et

= −Et

ψt,t+1

=0︷ ︸︸ ︷((Rst+1 + σt

(Rdt (lt − et)σtlt

−Rst+1

σt

))lt −Rd

t (lt − et))·∂ε∗t+1

∂et

+ χ1t+

Et

∞

ε∗t+1

ψt,t+1∂

∂et

((Rst+1 + σt

εt+1

Qt

)lt −Rd

t (lt − et))

1√2πτ 2

e−(εt+1+ξ)2

2τ2 dεt+1

− 1 = 0,

χ1t (et − γtlt) = 0,

χ2tlt = 0,

et − γtlt ≥ 0,

lt ≥ 0,

χ1t ≥ 0,

χ2t ≥ 0,

We are using the Leibniz integral rule above to nd the partial derivatives of the

prot function. Note that the rst term is zero in the dierentiation because the upper

limit of the integral does not depend on any of the choice variables.

Next, express the integrals in the rst-order conditions above using the erf

function, wherever possible. Note that in order to make the next expressions more

neat we omit the stochastic discount factor and the expectation operator from

consideration. We include them in the nal exposition.

119

Work on ∂∂lt:

∞(Rdt−R

st+1

σt−R

dt etσtlt

)Qt

∂

∂lt

((Rst+1 + σt

εt+1

Qt

)lt −Rd

t (lt − et))

1√2πτ 2

e−(εt+1+ξ)2

2τ2 dεt+1 =

∞(Rdt−R

st+1

σt−R

dt etσtlt

)Qt

(Rst+1 + σt

εt+1

Qt

−Rdt

)1√

2πτ 2e−

(εt+1+ξ)2

2τ2 dεt+1 =

σtQt

∞(Rdt−R

st+1

σt−R

dt etσtlt

)Qt

εt+11√

2πτ 2e−

(εt+1+ξ)2

2τ2 dεt+1+

(Rst+1 −Rd

t

) ∞(Rdt−R

st+1

σt−R

dt etσtlt

)Qt

1√2πτ 2

e−(εt+1+ξ)2

2τ2 dεt+1.

Break the calculation of the integral into two parts.

∞(Rdt−R

st+1

σt−R

dt etσtlt

)Qt

εt+11√

2πτ 2e−

(εt+1+ξ)2

2τ2 dεt+1 =

Introduce a change in variables to recast the integral in terms of the Standard Normal

distribution. Use v = εt+1+ξ√2τ

, or equivalently εt+1 = v√

2τ − ξ, and remember that

for the change x = ϕ(t), the integral ϕ(b)

ϕ(a)f(x)dx becomes

baf(ϕ(t))ϕ′(t)dt. Here we

use that dv = dεt+1√2τ, so we need to multiply dv by

√2τ to express dεt+1 in terms of

dv. Moreover, we need to transform the lower limit using v. So we need to add ξ to

the lower limit of the integral and divide the result by√

2τ .

∞

(Rdt (lt−et)−Rst+1lt)Qt+ξσtltσtlt√

2τ

(v√

2τ − ξ) √2τ√

2πτ 2e−v

2

dv =

120

√2τ√π

∞

(Rdt (lt−et)−Rst+1lt)Qt+ξσtltσtlt√

2τ

ve−v2

dv − ξ√π

∞

(Rdt (lt−et)−Rst+1lt)Qt+ξσtltσtlt√

2τ

e−v2

dv =

−√

2τ

2√πe−v

2

∣∣∣∣∣∞

(Rdt (lt−et)−Rst+1lt)Qt+ξσtltσtlt√

2τ

− ξ√π

∞

0

e−v2

dv −

(Rdt (lt−et)−Rst+1lt)Qt+ξσtlt

σtlt√

2τ

0

e−v2

dv

=

0 + ltτ√2πe−(

(Rdt (lt−et)−Rst+1lt)Qt+ξσtlt

σtlt√

2τ

)2

−

ξ√π

[√π

2erf(∞)−

√π

2erf

((Rdt (lt − et)−Rs

t+1lt)Qt + ξσtlt

σtlt√

2τ

)]=

τ√2πe−(

(Rdt (lt−et)−Rst+1lt)Qt+ξσtlt

σtlt√

2τ

)2

− ξ

2

[1− erf

((Rdt (lt − et)−Rs

t+1lt)Qt + ξσtlt

σtlt√

2τ

)],

where we used that erf(x) = 2√π

x0e−v

2.

Let's express∞(

Rdt−Rst+1

σt−R

dt etσtlt

)Qt

(1√

2πτ2e−

(εt+1+ξ)2

2τ2

)dεt+1 in terms of the error

function. Again, use the transformation v = εt+1+ξ√2τ

or εt+1 = v√

2τ − ξ

∞

(Rdt (lt−et)−Rst+1lt)Qt+ξσtltσtlt√

2τ

√2τ√

2πτ 2e−v

2

dv =1√π

∞

(Rdt (lt−et)−Rst+1lt)Qt+ξσtltσtlt√

2τ

e−v2

dv =

1

2

(1− erf

((Rdt (lt − et)−Rs

t+1lt)Qt + ξσtlt

σtlt√

2τ

)).

Therefore,

Et

∞

(Rdt−R

st+1

σt−R

dt etσtlt

)Qt

∂

∂lt

((Rst+1 + σt

εt+1

Qt

)lt −Rd

t (lt − et))

1√2πτ 2

e−(εt+1+ξ)2

2τ2 dεt+1

=

121

Et

σtQt

τ√2πe−(

(Rdt (lt−et)−Rst+1lt)Qt+ξσtlt

σtlt√

2τ

)2

− σtξ2Qt

[1− erf

((Rdt (lt−et)−Rst+1lt)Qt+ξσtlt

σtlt√

2τ

)]+

Et

[(Rst+1 −Rd

t

)12

(1− erf

((Rdt (lt−et)−Rst+1lt)Qt+ξσtlt

σtlt√

2τ

))]=

Et

σtQt

τ√2πe−(

(Rdt (lt−et)−Rst+1lt)Qt+ξσtlt

σtlt√

2τ

)2

+

(Rst+1−

σtξQt−Rdt

2

)[1− erf

((Rdt (lt−et)−Rst+1lt)Qt+ξσtlt

σtlt√

2τ

)]].

Similarly, work on ∂∂et

∞(Rdt−R

st+1

σt−Rdt+1et

σtlt

)Qt

∂

∂et

((Rst+1 + σt

εt+1

Qt

)lt −Rd

t (lt − et))

1√2πτ 2

e−(εt+1+ξ)2

2τ2 dεt+1 =

∞(Rdt−R

st+1

σt−Rdt+1et

σtlt

)Qt

Rdt

1√2πτ 2

e−(εt+1+ξ)2

2τ2 dεt+1 = Rdt

1

2

(1− erf

(Rdt (lt − et)−Rl

t+1lt + ξσtlt

σtlt√

2τ

)).

In sum, the FOCs can be written as follows:

Et

β λct+1

λct

σtQt

τ√2πe−(

(Rdt (1− etlt

)−Rst+1)Qt+ξσtσt√

2τ

)2

+

(Rst+1−

σtξQt−Rdt

2

)[1− erf

((Rdt

(1− et

lt

)−Rst+1

)Qt+ξσt

σt√

2τ

)]]+ χ2t = γχ1t,

Et

β λct+1

λct

[Rdt

12

(1− erf

((Rdt

(1− et

lt

)−Rst+1

)Qt+ξσt

σt√

2τ

))]− 1 + χ1t = 0.

There are complementary slackness conditions which can be described by:

(et − γlt)χ1t = 0,

ltχ2t = 0.

122

A.1.2 Proof of Proposition 1

Equations (1.13) and (1.14) can be expressed as

βEtλct+1

λctRe,it+1 = 1− ζ it

λct,

where i ∈ s, r denotes the type of equity. Using the expression, substitute for 1 in

the bank's FOC with respect to et. Therefore,

Et

β λct+1

λct

[Rdt

12

(1− erf

((Rdt

(1− e

itlit

)−Rst+1

)Qt+ξσit

σit√

2τ

))]−Re,i

t+1

− ζit

λct+ χi1t = 0.

Since the range of the erf function is between −1 and 1, i.e.−1 ≤ erf(x) ≤ 1, we

know that the following expression is between Ψ∗1 and Ψ∗2:

Ψ∗1 ≤ Et

β λct+1

λct

[Rdt

12

(1− erf

((Rdt

(1− e

itlit

)−Rst+1

)Qt+ξσit

σit√

2τ

))−Re,i

t+1

]≤ Ψ∗2,

where

Ψ∗1 = Et

βλct+1

λct

[0−Re,i

t+1

],

Ψ∗2 = Et

βλct+1

λct

[Rdt −R

e,it+1

].

∂

∂Dt

= ς0D−ςdt − λct + Etβλct+1R

dt = 0,

Use Etβλct+1Re,it+1 + ζ it = λct (that comes from the household's FOCs with respect

to eit for each i ∈ s, r) to substitute for λct in equation (1.12) . We get:

Etβλct+1

[Rdt −R

e,it+1

]= −ς0D−ςdt + ζ it .

Note that ς0D−ςdt > 0 under the usual (and mild) assumptions on the preferences for

liquidity. Moreover, the Lagrangian multiplier on the households budget constraint,

λct, is positive. It reects the fact that the budget constraint always binds given the

123

standard assumptions on the preferences (Inada conditions). The latest expression is

transformed into the following after dividing it by λct:

Et

βλct+1

λct

[Rdt −R

e,it+1

]︸ ︷︷ ︸

=Ψ∗2

− ζ itλct

= −ς0D−ςdt

λct< 0.

Thus, Ψ∗2 <ζitλct. Therefore,

Et

βλct+1

λct

Rdt

1

2

1− erf

(Rdt

(1− eit

lit

)−Rs

t+1

)Qt + ξσit

σit√

2τ

−Re,it+1

− ζ itλct

+ χi1t =

0 < Ψ∗2 −ζ itλct

+ χ1t <ζ itλct− ζ itλct

+ χi1t = χi1t.

Hence, χi1t > 0.

A.1.3 Combined First-Order Conditions

Et

β λct+1

λct

σtQt

τ√2πe−(

(Rdt (1− etlt

)−Rst+1)Qt+ξσtσt√

2τ

)2

+

(Rst+1−

σtξQt−Rdt

2

)[1− erf

((Rdt

(1− et

lt

)−Rst+1

)Qt+ξσt

σt√

2τ

)]]+ χ2t = γχ1t,

Et

β λct+1

λct

[Rdt

12

(1− erf

((Rdt

(1− et

lt

)−Rst+1

)Qt+ξσt

σt√

2τ

))]− 1 + χ1t = 0.

Since χ1t > 0, multiply the second equation by γt and add it to the rst equation

using etlt

= γt. Therefore, the FOCs can be combined into:

Et

β λct+1

λct

σtQt

τ√2πe−(

(Rdt (1−γt)−Rst+1)Qt+ξσt

σt√

2τ

)2

+

12

(Rst+1 −

σtξQt−Rd

t

)[1− erf

((Rdt (1−γt)−Rst+1)Qt+ξσt

σt√

2τ

)]]= γt − χ2t,

χ2tlt = 0.

124

A.1.4 Zero-Profit Condition

Consider the zero-prot condition under all states of nature. Since there is no

agency problem between banks and households, this condition captures the fact that

all the prots (or losses) are distributed to equity holders after realization of shocks

at the beginning of each period. In each aggregate state, banks whose investments in

risky rms pan out will have returns that satisfy on average (over the realizations of

the idiosyncratic shock)[(Rst+1 + σt

Qt

)lt −Rd

t (lt − et)]−

Ret+1,b(b) · et = 0, where

the bounds of the integral are chosen such that we integrate over banks for which

the prot is non-negative, while banks whose risky investments earn low (negative)

returns will have Ret+1,b = 0. Therefore,

Ret+1 =

∞(Rdt (1−γt)−Rst+1

σt

)Qt

((Rst+1 + σt

εt+1

Qt

)lt −Rd

t dt

)1√

2πτ2e−

(εt+1+ξ)2

2τ2 dεt+1

et+

(Rdt (1−γt)−R

st+1

σt

)Qt

−∞

0 · 1√2πτ 2

e−(εt+1+ξ)2

2τ2 dεt+1 =

1

et

∞(Rdt (1−γt)−Rst+1

σt

)Qt

(Rst+1lt −Rd

t dt) 1√

2πτ 2e−

(εt+1+ξ)2

2τ2 dεt+1 +

1

et

∞(Rdt (1−γt)−Rst+1

σt

)Qt

σtεt+1

Qt

lt1√

2πτ 2e−

(εt+1+ξ)2

2τ2 dεt+1 =

1et

[(Rst+1lt −Rd

t dt)

12

(1− erf

((Rdt (1−γt)−Rst+1)Qt+ξσt

σt√

2τ

))+

σtltQt

τ√2πe−(

(Rdt (1−γt)−Rst+1)Qt+ξσt

σt√

2τ

)2

− ξ2

[1− erf

((Rdt (1−γt)−Rst+1)Qt+ξσt

σt√

2τ

)] =

125

ltet

σtQt

τ√2πe−(

(Rdt (1−γt)−Rst+1)Qt+ξσt

σt√

2τ

)2

+

12

(Rst+1 −

σtξQt−Rd

t (1− γt))[

1− erf

((Rdt (1−γt)−Rst+1)Qt+ξσt

σt√

2τ

)].

Since ltet

= 1γt, we can rewrite the latter condition as (using that it holds for each

i ∈ s, r):

Re,it+1 =

σitQt

τ√2πe

−

(Rdt (1−γt)−Rst+1)Qt+ξσit

σit

√2τ

2

+ 12

(Rst+1−

σitξ

Qt−Rdt (1−γt)

)[1−erf

((Rdt (1−γt)−R

st+1)Qt+ξσit

σit

√2τ

)]γt

.

Note that the combined FOC from Appendix A.1.3 can be expressed as:

Et

β λct+1

λct

σitQt

τ√2πe−(

(Rdt (1−γt)−Rst+1)Qt+ξσit

σit

√2τ

)2

+

12

(Rst+1 −

σitξ

Qt−Rd

t

)[1− erf

((Rdt (1−γt)−Rst+1)Qt+ξσit

σit√

2τ

)]]=

γt − χi2t = γt

(Etβ

λct+1

λctRe,it+1 +

ζitλct

)− χi2t,

where we substitute for 1 from Household's FOC with respect to two types of equity:

βEtλct+1

λctRe,it+1 = 1− ζit

λct.

Notice that lit > 0 implies both χi2t = 0 and ζ it = 0 which say that the zero-prot

condition implies the FOC.

A.1.5 Expression of Expected Dividends

Expected dividends (valued on date t) are dened as

Ω (µt, σt; lt, dt, et) =

Et

βλct+1

λct

∞(Rdt (lt−et)

σtlt−Rlt+1σt

)Qt

((Rlt+1 + σt

εt+1

Qt

)lt −Rd

t (lt − et))

1√2πτ 2

e−(εt+1+ξ)2

2τ2 dεt+1

=

126

We have already calculated all the necessary integrals in the Appendix A.1.1.

Therefore,

Et

βλct+1

λct

σtltQt

τ√2πe−(

(Rdt (lt−et)−Rst+1lt)Qt+ξσtlt

σtlt√

2τ

)2

+

(Rst+1lt −Rd

t (lt − et)− σtξQtlt

)2

[1− erf

((Rdt (lt − et)−Rs

t+1lt)Qt + ξσtlt

σtlt√

2τ

)] .

A.1.6 Linear Cost of Banking: FOCs of Banks

We use(Rst+1 + σt

ε∗t+1

Qt

)lt−Rd

t dt− flt = 0 to get ε∗t+1 =(flt+Rdt (lt−et)

σtlt− Rlt+1

σt

)Qt.

Conditional on the optimal choice of σt, the rst-order conditions are:

Et

∞

ε∗t+1

ψt,t+1∂

∂lt

((Rst+1 + σt

εt+1

Qt

)lt −Rd

t (lt − et)− flt)

1√2πτ 2

e−(εt+1+ξ)2

2τ2 dεt+1

+

χ2t − γχ1t = 0,

Et

∞

ε∗t+1

ψt,t+1∂

∂et

((Rst+1 + σt

εt+1

Qt

)lt −Rd

t (lt − et)− flt)

1√2πτ 2

e−(εt+1+ξ)2

2τ2 dεt+1

−1 + χ1t = 0.

The derivations are similar to the ones described in Appendix A.1.1. The only

dierence is that the lower bound of the integral now contains the additional term

flt. Hence, adding ξ to the lower limit of the integral and dividing the result by√

2τ

make the terms in the nal expressions. Moreover, note that we should carry f in

the expressions of the FOC with respect to lt. In sum, the FOCs can be written as

127

follows:

Et

β λct+1

λct

σtQt

τ√2πe−(

(f+Rdt (1− etlt

)−Rst+1)Qt+ξσtσt√

2τ

)2

+

(Rst+1−

σtξQt−Rdt−f

2

)[1− erf

((f+Rdt

(1− et

lt

)−Rst+1

)Qt+ξσt

σt√

2τ

)]]+ χ2t = γχ1t,

Et

β λct+1

λct

[Rdt

12

(1− erf

((f+Rdt

(1− et

lt

)−Rst+1

)Qt+ξσt

σt√

2τ

))]− 1 + χ1t = 0.

A.2 The Non-Financial Firm's Problem

A.2.1 Safe firms

Let πst+1 denote the revenue of a safe rm in period t+ 1 net of expenses:

πst+1 = yst+1 + (1− δ)Qtkst+1 −Wt+1h

st+1 −Rs

t+1lst .

In this notation, the problem of the safe rm is to

maxlst ,k

st+1

Et

βλct+1

λctmaxhst+1

πst+1

.

The rst-order condition for maxhst+1πst+1 is

∂πst+1

∂hst+1= 0. It implies that

Wt+1 =∂yst+1

∂hst+1

= (1− α)yst+1

hst+1

= (1− α)At+1

(kst+1

hst+1

)α, (A.2.1)

hst+1 = (1− α)yst+1

Wt+1

= (1− α)At+1

(kst+1

)α (hst+1

)1−α

Wt+1

. (A.2.2)

Accordingly, the safe rm's Lagrangian is:

Lsafe =Et

βλct+1

λct

[At+1

(kst+1

)α (hst+1

)1−α+ (1− δ)Qt+1k

st+1 −Wt+1h

st+1 −Rs

t+1lst

]+

λshtEt

βλct+1

λct

[(1− α)

At+1

(kst+1

)α (hst+1

)1−α

Wt+1

− hst+1

]+ λslt

(lst −Qtk

st+1

).

Notice that there is no expectation operator on the Lagrangian multipliers because

those constraints hold under every state of nature. The problem implies the following

128

rst-order conditions

∂Lsafe

∂lst= −Et

βλct+1

λctRst+1

+ λslt = 0,

∂Lsafe

∂kst+1

= Et

βλct+1

λct

[αyst+1

kst+1

+ (1− δ)Qt+1

]+

λsht (1− α)αEt

βλct+1

λct

At+1

Wt+1

(kst+1

hst+1

)α−1− λsltQt = 0,

∂Lsafe

∂hst+1

= (1− α)At+1

(kst+1

)α (hst+1

)1−α

Wt+1

−Wt+1 + λsht

[(1− α)2 At+1

Wt+1

(kst+1

hst+1

)α− 1

]= 0.

Combining ∂Lsafe∂hst+1

= 0 with equation (A.2.2) yields λsht = 0. Then, plugging ∂Lsafe∂lst

=

0 into ∂Lsafe∂kst+1

for λslt, we get

Et

βλct+1

λctRst+1

Qt = Et

βλct+1

λct

[αyst+1

kst+1

+ (1− δ)Qt+1

].

Consider the zero-prot condition of the safe rm under all states of nature. Since

output function has constant returns to scale,

yst+1 =∂yst+1

∂kst+1

kst+1 +∂yst+1

∂hst+1

hst+1 = αAt+1

(kst+1

hst+1

)α−1

kst+1 +Wt+1hst+1,

where we use equation (A.2.2) to substitute for Wt+1 in the last equality. Plugging

the expression of yst+1 into πst+1 = 0 and using Qtk

st+1 = lst , we nd that:

αAt+1

(kst+1

hst+1

)α−1

kst+1 + (1− δ)Qt+1kst+1 −Rs

t+1Qtkst+1 = 0.

Since kst+1 > 0, we can divide by kst+1 to get

Rst+1Qt = αAt+1

(kst+1

hst+1

)α−1

+ (1− δ)Qt+1 (A.2.3)

under all states of nature. This condition implies the rst-order condition

Et

βλct+1

λctRst+1

Qt = Et

βλct+1

λct

[αAt+1

(kst+1

hst+1

)α−1

+ (1− δ)Qt+1

].

129

A.2.2 Risky Firms

Let πrt+1 denote the revenue of a risky rm in period t+ 1 net of expenses:

πrt+1 = yrt+1 + (1− δ)Qtkrt+1 −Wt+1h

rt+1 −Rr

t+1lrt .

In this notation, the problem of the risky rm is to

maxlrt ,k

rt+1

Et

βλct+1

λctmaxhrt+1

πrt+1

.

The rst-order condition for maxhrt+1πrt+1 is

∂πrt+1

∂hrt+1= 0. It implies that

Wt+1 =∂yrt+1

∂hrt+1

= (1− α)At+1

(krt+1

hrt+1

)α, (A.2.4)

hrt+1 = (1− α)At+1

(krt+1

)α (hrt+1

)1−α

Wt+1

. (A.2.5)

Accordingly, the risky rm's Lagrangian is:

Lrisky =Et

βλct+1

λct·[

At+1

(krt+1

)α (hrt+1

)1−α+ εt+1k

rt+1 + (1− δ)Qt+1k

rt+1 −Wt+1h

rt+1 −Rr

t+1lrt

]+

λrhtEt

βλct+1

λct

[(1− α)

At+1

(krt+1

)α (hrt+1

)1−α

Wt+1

− hrt+1

]+ λrlt

(lrt −Qtk

rt+1

).

Notice that there is no expectation operator on the Lagrangian multipliers because

those constraints hold under every state of nature. The problem implies the following

rst-order conditions

∂Lrisky

∂lrt= −Et

βλct+1

λctRrt+1

+ λrlt = 0,

∂Lrisky

∂krt+1

= Et

βλct+1

λct

[αAt+1

(krt+1

hrt+1

)α−1

+ εt+1 + (1− δ)Qt+1

]+

λrhtEt

βλct+1

λctα (1− α)

At+1

Wt+1

(krt+1

hrt+1

)α−1− λrltQt = 0,

∂Lrisky

∂hrt+1

= (1− α)At+1

(krt+1

hrt+1

)α−Wt+1 + λrht

[(1− α)2 At+1

Wt+1

(krt+1

hrt+1

)α− 1

]= 0.

130

Equation (A.2.4) together with ∂Lrisky∂hrt+1

= 0 yield λrht = 0. Plugging ∂Lrisky∂lrt

= 0 into

∂Lrisky∂krt+1

for λrlt, we get

Et

βλct+1

λctRrt+1

Qt = Et

βλct+1

λct

[αAt+1

(krt+1

hrt+1

)α−1

+ (1− δ)Qt+1 + εt+1

].

Combining equation (A.2.1) with equation (A.2.4):

kst+1

hst+1

=krt+1

hrt+1

(A.2.6)

under all states of nature. But remember that the rst-order condition of the safe

rm implies

Et

βλct+1

λctRst+1

Qt = Et

βλct+1

λct

[αAt+1

(kst+1

hst+1

)α−1

+ (1− δ)Qt+1

].

Therefore

Et

βλct+1

λctRrt+1

Qt = Et

βλct+1

λct

[Rst+1Qt + εt+1

].

Consider the zero-prot condition of the risky rm under all states of nature.

πrt+1 = yrt+1 + (1− δ)Qtkrt+1 −Wt+1h

rt+1 −Rr

t+1lrt =

yrt+1 + (1− δ)Qtkrt+1 − (1− α)At+1

(krt+1

)α (hrt+1

)1−α −Rrt+1l

rt =

αAt+1

(krt+1

)α (hrt+1

)1−α+ εt+1k

rt+1 + (1− δ)Qtk

rt+1 −Rr

t+1lrt =

αAt+1

(krt+1

hrt+1

)α−1

krt+1 + εt+1krt+1 + (1− δ)Qtk

rt+1 −Rr

t+1lrt = 0,

where we use equation (A.2.5) to substitute for Wt+1hrt+1. Using equation (A.2.3)

together with equation (A.2.6), we can express

αAt+1

(krt+1

hrt+1

)α−1

= Rst+1Qt − (1− δ)Qt+1,

that holds under all states of nature. Plugging it into the zero-prot condition and

using Qtkrt+1 = lrt , we nd that:

Rst+1Qtk

rt+1 − (1− δ)Qt+1k

rt+1 + εt+1k

rt+1 + (1− δ)Qtk

rt+1 −Rr

t+1Qtkrt+1 = 0.

131

Since krt+1 > 0, we can divide by krt+1 to get

Rrt+1Qt = Rs

t+1Qt + εt+1

under all states of nature. This condition implies

Et

βλct+1

λctRrt+1

Qt = Et

βλct+1

λct

[Rst+1Qt + εt+1

].

A.2.3 Aggregating across firms

Here we show that we can aggregate individual rms into two representative rms.

Let denote kij,t the capital chosen by rm i that is nanced by borrowing from bank j.

Both i and j lie within the continuum of measure 1 of banks and rms, respectively.

In this notation, the equation (A.2.6) is written as

kij,t+1

hij,t+1

=kt+1

ht+1

, (A.2.7)

for all j ∈ [0, 1] and i ∈ [0, 1]. Each rm chooses the same capital-to-labor ratio

independently of the type of bank it borrows from.

Notice is that σt is the fraction of risky rms at date t; the remaining fraction 1−σt

of rms are safe rms. Let's index rms as follows: rm j1, with j1 ∈ [0, σt], can only

access a risky technology subject to both aggregate and idiosyncratic shocks; rm

j2, with j2 ∈ [σt, 1] has access to a safe production technology subject to aggregate

shocks only. Since there are no equilibria with σ < σt < σ, the fraction of risky rms

is linked to the fraction of banks with risky portfolios as follows:

σt = (1− µt)σ + µtσ.

Dene the following objects: Let Kss,t+1 =

1

σt

1

µtkij,t+1djdi be the total capital

allocated to the safe technology and nanced by borrowing from the banks that

choose a fraction σ of risky projects. Let Ksr,t+1 =

1

σt

µt0kij,t+1djdi be the total capital

132

allocated to the safe technology and nanced by borrowing from the banks that choose

a fraction σ of risky projects. We let Kst+1 denote the total capital allocated to the

safe technology. Thus,

Kst+1 =

1

σt

1

0

kij,t+1djdi = Kss,t+1 +Ks

r,t+1,

Let Krs,t+1 =

σt0

1

µtkij,t+1djdi be the total capital allocated to the risky technology

and nanced by borrowing from the banks that choose a fraction σ of risky projects.

Let Krr,t+1 =

σt0

µt0kij,t+1djdi be the total capital allocated to the safe technology

and nanced by borrowing from the banks that choose a fraction σ of risky projects.

We let Krt+1 denote the total capital allocated to the risky technology. Thus,

Krt+1 =

σt

0

1

0

kij,t+1djdi = Krs,t+1 +Kr

r,t+1,

The same upper and lower case notation applies to labor, i.e.Hss,t+1 =

1

σt

1

µthij,t+1djdi;

Hsr,t+1 =

1

σt

µt0hij,t+1djdi; H

rs,t+1 =

σt0

1

µthij,t+1djdi; H

rr,t+1 =

σt0

µt0hij,t+1djdi.

Safe representative rm produces:

Y st =

1

σt−1

1

0

At(kij,t)α (

hij,t)1−α

djdi =

1

σt−1

1

0

F(kij,t, h

ij,t

)djdi =

Using that the technology has Constant Returns to Scale:

=

1

σt−1

1

0

[Fkij,t

(kij,t, h

ij,t

)kij,t + Fhij,t

(kij,t, h

ij,t

)hij,t

]djdi =

where Fkij,t(kij,t, h

ij,t

)and Fhij,t

(kij,t, h

ij,t

)denote the partial derivative of F

(kij,t, h

ij,t

)with respect to kij,t and h

ij,t, respectively. Since these partial derivatives are homogeneous

of degree zero, we can express them in term of capital-labor ratio, i.e.

=

1

σt−1

1

0

[fkij,t

(kij,thij,t

)kij,t + fhij,t

(kij,thij,t

)hij,t

]djdi =

133

Plugging equation (A.2.7)

=

1

σt−1

1

0

[fkt

(ktht

)kij,t + fht

(ktht

)hij,t

]djdi =

fkt

(ktht

) 1

σt

1

0

kij,tdjdi

+fht

(ktht

) 1

σt

1

0

hij,tdjdi

= fkt

(ktht

)Kst +fht

(ktht

)Hst =

SinceKss,t

Hss,t

=Ksr,t

Hsr,t

= ktht,then Ks

t

Hst

htkt

=(Kss,t+K

sr,t

Hss,t+H

sr,t

)Hsr,t

Ksr,t

= 1. Therefore Kst

Hst

= ktht.

= fKst

(Kst

Hst

)Kst + fHs

t

(Kst

Hst

)Hst = At (Ks

t )α (Hs

t )1−α .

Risky representative rm:

Y rt =

σt−1

0

1

0

[At(kij,t)α (

hij,t)1−α

+ εij,tkij,t

]djdi =

σt−1

0

1

0

F(kij,t, h

ij,t

)djdi+

σt−1

0

1

0

εij,tkij,tdjdi =

Note that the similar steps described above apply to the rst term in the summation,

so that σt−1

0

1

0F(kij,t, h

ij,t

)djdi = At (Kr

t )α (Hr

t )1−α. To express the second term,

notice that σt−1

0

1

0εij,tk

ij,tdjdi = −ξ. Moreover since each risky rm solves the same

maximization problem, it chooses the same amount of capital independently of the

type of bank it borrows from. Therefore, σt−1

0

1

0εij,tk

ij,tdjdi = −ξKr

t . Hence,

Y rt = At (Kr

t )α (Hr

t )1−α − ξKrt .

A.3 The Government

The government levies the tax to fully compensate for the loss to the deposit

insurance fund due to rescue of defaulted banks.

134

A.3.1 Baseline: No linear cost of banking

Tt = − (Rdt−1Dt−1

σt−1Lt−1− Rstσt−1

)Qt−1

−∞

((Rlt + σt−1εt

Qt−1

)Lt−1 −Rd

t−1Dt−1

)dG(εt) =

−[∞−∞

((Rlt + σt−1εt

Qt−1

)Lt−1 −Rd

t−1Dt−1

)dG(εt)−

∞(Rdt−1Dt−1

σt−1Lt−1− Rstσt−1

)Qt−1

((Rst + σt−1εt

Qt−1

)Lt−1 −Rd

t−1Dt−1

)dG(εt)

]=

Note that the rst term equals(Rst −

σt−1ξQt−1

)Lt−1 + Rd

t−1Dt−1 in the square bracket.

We have already calculated the second term. Therefore,

= σt−1Lt−1

Qt−1

τ√2πe−(Rdt−1(1−γt−1)Qt−1−R

stQt−1+ξσt−1

σt−1√

2τ

)2

−(Rst −

σt−1ξQt−1

)Lt−1 +Rd

t−1Dt−1 +

12Lt−1

(Rst −

σt−1ξQt−1

− (1− γt−1)Rdt−1

) [1− erf

(Rdt−1(1−γt−1)Qt−1−RstQt−1+ξσt−1

σt−1

√2τ

)]=

σt−1Lt−1

Qt−1

τ√2πe−(Rdt−1(1−γt−1)Qt−1−R

stQt−1+ξσt−1

σt−1√

2τ

)2

−

12

(RstLt−1 − σt−1ξ

Qt−1Lt−1 −Rd

t−1Dt−1

) [1 + erf

(Rdt−1(1−γt−1)Qt−1−RstQt−1+ξσt−1

σt−1

√2τ

)].

A.3.2 Linear Cost of Banking: Tax

The tax that accounts for the cost of banking is described as follows:

Tt = −

(Rdt−1dt−1σt−1lt−1

−Rst−fσt−1

)Qt−1

−∞

((Rst +

σt−1εtQt−1

− f)lt−1 −Rd

t−1dt−1

)dG(εt) =

σt−1lt−1

Qt−1

τ√2πe−(

(f+Rdt−1(1−γt−1)−Rst)Qt−1+ξσt−1

σt−1√

2τ

)2

−(Rlt −

σt−1ξ

Qt−1

− f)lt−1 +Rd

t−1dt−1+

1

2lt−1

(Rst −

σt−1ξ

Qt−1

− (1− γt−1)Rdt−1 − f

)[1− erf

((f +Rd

t−1 (1− γt−1)−Rst

)Qt−1 + ξσt−1

σt−1

√2τ

)]=

σt−1lt−1

Qt−1

τ√2πe−(

(f+Rdt−1(1−γt−1)−Rst)Qt−1+ξσt−1

σt−1√

2τ

)2

−

1

2

(Rst lt−1 −

σt−1ξ

Qt−1

lt−1 −Rdt−1dt−1 − flt−1

)[1 + erf

((f +Rd

t−1 (1− γt−1)−Rst

)Qt−1 + ξσt−1

σt−1

√2τ

)].

135

A.4 Choice of Risk

This appendix shows a proof that the expected dividends function of banks is

convex in the risk parameter σt. This result guarantees that banks choose either

the maximum risk, σ, or the minimum risk, σ, to maximize their prots, so all the

intermediate values of σt, which may result from the rst-order conditions with respect

to σt, are not optimal.

We generalize the proof taken from Van den Heuvel (2008) to the case with

aggregate uncertainty. The proof applies to an arbitrary distribution of the idiosyncratic

shock, εt+1, with non-positive mean, so our example of a Normal distribution

considered in the analysis is not a special case which can drive our results. It is

used for expositional reasons and quantitative work.

Assumption. ε has a cumulative distribution function Gε with support [ε, ε], with

ε < 0 < ε. The mean of ε is equal to −ξ (ξ > 0). ε is independent of the aggregate

shock. The aggregate shock does not depend on the choice of σt.

Note that we do not restrict the analysis to the bounded support1, so ε and ε can

take −∞ and +∞, respectively. Note that Gε need not be continuous.

Let ε(σt, Rst+1) ≡

(Rdt dtσtlt− Rlt+1

σt

)Qt =

Rdt (1−γt)−Rst+1

σtQt, where the latter equation

uses the result that the capital requirement constraint always binds. Therefore,(Rst+1 + σt

ε(σt)Qt

)lt −Rd

t dt = 0. Let π(σt, Rst+1) = Eε

[((Rst+1 + σtε

Qt

)lt −Rd

t dt

)+]be

a function of expected dividends (taken over the idiosyncratic shock only) under some

realization of Rst+1 which is considered to be xed in this function. To account for

the aggregate uncertainty, Rst+1 needs to be a random variable. Therefore, expected

1Unbounded support is more relevant if we consider aggregate risk

136

dividends taken into account both idiosyncratic and aggregate uncertainty are

Π(σt) =

Ω

π(σt, R

st+1(ω)

)P (dω) = Et

ε

ε(σt, Rst+1)

((Rst+1 +

σtε

Qt

)lt −Rd

t dt

)dGε

=

Et

ε

ε

((Rst+1 +

σtε

Qt

)lt −Rd

t dt

)dGε

− Et ε(σt, Rst+1)

ε

((Rst+1 +

σtε

Qt

)lt −Rd

t dt

)dGε

=

EtRst+1lt −Rd

t dt −σtξ

Qt

lt −σtltQt

Et

ε(σt, Rst+1)

ε

(ε− ε(σt, Rs

t+1))dGε

=

EtRst+1lt −Rd

t dt +ltQt

σtEt ε(σt, Rst+1)

ε

(ε(σt, R

st+1)− ε

)dGε

− σtξ .

Note that in the derivations above we express(Rst+1 + σtε

Qt

)lt − Rd

t dt in terms of

ε(σt, Rst+1) and ε using the denition of ε(σt, Rs

t+1).

The proof below shows that Π(σt) is convex in σt. Since the expression of Π(σt)

involves the term which is linear in σt and ltQt≥ 0, the sucient condition for Π(σt)

to be convex in σt is that

H(σt) ≡ Et

[ ε(σt)

ε(ε(σt)− ε) dGε

]σt

is convex in σt.

Claim. H(σt) ≡ ltEt

[ ε(σt)ε

(ε(σt, R

st+1)− ε

)dGε

]σt is convex in σt:

Proof. Steps of the proof:

1. Dene h(σt, Rst+1) ≡ σt

[ ε(σt, Rst+1)

ε(ε(σt, R

st+1)− ε

)dGε

]in which the aggregate

uncertainty is taken o. Consider 3 cases:

(a) Realization of Rst+1 is such that ε(σt, Rs

t+1) =Rdt (1−γt)−Rst+1

σt> 0, so Rs

t+1 <

Rdt (1− γt) ,

137

(b) Realization of Rst+1 is such that ε(σt, Rl

t+1) =Rdt (1−γt)−Rst+1

σt< 0, so Rs

t+1 >

Rdt (1− γt) ,

(c) Realization of Rst+1 is such that ε(σt, Rl

t+1) =Rdt (1−γt)−Rst+1

σt= 0, so Rs

t+1 =

Rdt (1− γt) ,

Show that h(σt, Rst+1) is convex in σt in cases 1a and 1b and h(σt, R

st+1) is linear

in σt in case 1c.

2. Employ the argument that convexity is preserved under non-negative scaling

and addition (guaranteed by the expectation operator over the aggregate

uncertainty) to nd that H(σt) is convex.

Let's show each step of the proof formally

1. Let σ1t < σ2t and, for λ ∈ (0, 1), dene σλt = λσ1t + (1 − λ)σ2t. Let εi =

ε(σit, Rst+1) ≡ Rdt (1−γt)−Rst+1

σitQt, for i = 1, 2, λ.

(a) Rst+1 < Rd

t (1− γt): it implies that ε2 < ελ < ε1,

h(σλt) = (λσ1t + (1− λ)σ2t)

ε(σλt)

ε(ε(σλt)− ε) dGε

=

λσ1t

ε1

ε(ελ − ε) dGε −

ε1

ελ

(ελ − ε) dGε

+

(1− λ)σ2t

ε2

ε(ελ − ε) dGε +

ελ

ε2

(ελ − ε) dGε

=

λσ1t

ε1

ε(ε1 − ε) dGε + (ελ − ε1)Gε(ε1) +

ε1

ελ

(ε− ελ) dGε

+

(1− λ)σ2t

ε2

ε(ε2 − ε) dGε + (ελ − ε2)Gε(ε2) +

ελ

ε2

(ελ − ε) dGε

≤

λσ1t

ε1

ε(ε1 − ε) dGε + (ελ − ε1)Gε(ε1) +

ε1

ελ

(ε1 − ελ) dGε

+

(1− λ)σ2t

ε2

ε(ε2 − ε) dGε + (ελ − ε2)Gε(ε2) +

ελ

ε2

(ελ − ε2) dGε

,

138

where the inequality sign comes from ε1ελ

(ε− ελ) dGε ≤ ε1ελ

(ε1 − ελ) dGε

and ελε2

(ελ − ε) dGε ≤ ελε2

(ελ − ε2) dGε. Substituting for the denitions

of h(σ1t) = σ1t

ε1ε (ε1 − ε) dGε and h(σ2t) = σ2t

ε2ε (ε2 − ε) dGε, we get:

h(σλt) ≤ λh(σ1t) + (1− λ)h(σ2t) + λσ1t (ελ − ε1)Gε(ελ)+

(1− λ)σ2t (ελ − ε2)Gε(ελ) = λh(σ1t) + (1− λ)h(σ2t)+

Gε(ελ) (λσ1t (ελ − ε1) + (1− λ)σ2t (ελ − ε2)) = λh(σ1t) + (1− λ)h(σ2t),

where we use that σ1t = lt(Rdt (1− γt)−Rs

t+1

)= σ2tε2 = σλtελ in the last

equality. So,

λσ1t (ελ − ε1) + (1− λ)σ2t (ελ − ε2) =

ελ (λσ1t + (1− λ)σ2t)−(Rdt (1− γt)−Rs

t+1

)(λ+ (1− λ)) =

σλtελ −(Rdt (1− γt)−Rs

t+1

)=(Rdt (1− γt)−Rs

t+1

)−(Rdt (1− γt)−Rs

t+1

)= 0.

Therefore, h(σt) is convex in σt for Rst+1 < Rd

t (1− γt).

(b) Rst+1 > Rd

t (1− γt): it implies that ε1 < ελ < ε2

h(σλt) = (λσ1t + (1− λ)σ2t)

ε(σλt)

ε(ε(σλt)− ε) dGε

=

λσ1t

ε1

ε(ελ − ε) dGε +

ελ

ε1

(ελ − ε) dGε

+

(1− λ)σ2t

ε2

ε(ελ − ε) dGε −

ε2

ελ

(ελ − ε) dGε

=

λσ1t

ε1

ε(ε2 − ε) dGε + (ελ − ε1)Gε(ε1) +

ελ

ε1

(ελ − ε) dGε

+

(1− λ)σ2t

ε2

ε(ε2 − ε) dGε + (ελ − ε2)Gε(ε2) +

ε2

ελ

(ε− ελ) dGε

≤

λσ1t

ε1

ε(ε1 − ε) dGε + (ελ − ε1)Gε(ε1) +

ελ

ε1

(ελ − ε1) dGε

+

(1− λ)σ2t

ε2

ε(ε2 − ε) dGε + (ελ − ε2)Gε(ε2) +

ε2

ελ

(ε2 − ελ) dGε

,

139

where the inequality sign follows ελε1

(ελ − ε) dGε ≤ ελε1

(ελ − ε1) dGε and ε2ελ

(ε− ελ) dGε ≤ ε2ελ

(ε2 − ελ) dGε. Substituting for the denitions of

h(σ1t) = σ1t

ε1ε (ε1 − ε) dGε and h(σ2t) = σ2t

ε2ε (ε2 − ε) dGε, we get:

h(σλt) ≤ λh(σ1t) + (1− λ)h(σ2t) + λσ1t (ελ − ε1)Gε(ελ)+

(1− λ)σ2t (ελ − ε2)Gε(ελ) = λh(σ1t) + (1− λ)h(σ2t)+

Gε(ελ) (λσ1t (ελ − ε1) + (1− λ)σ2t (ελ − ε2)) = λh(σ1t) + (1− λ)h(σ2t),

where the last equality follows from the same reasoning employed in the

previous case. Therefore, h(σt) is convex in σt for Rst+1 > Rd

t (1− γt).

(c) Rst+1 = Rd

t (1− γt). Hence, ε(σt) = 0 and

h(σt) = σt

[ 0

ε(0− ε) dGε

],

which is linear in σt

2. We found in 1 that h(σt, Rst+1) is convex in σt for each Rs

t+1 ∈ R. Consider

P (ω) ≥ 0 for each Rlt+1(ω) ∈ R. Then the following function2:

Ω

h(σt, R

st+1(ω)

)P (dω) = Eth(σt, R

st+1) ≡ H(σt)

is convex in σt. It follows directly from the linearity of the expectation operator

which puts a non-negative weight on every realization of Rst+1 and the fact that

the sum of convex functions is a convex function. Therefore, Π(σt) is convex in

σt.

2Linearity in σt for one particular value of Rst+1 can be considered as a weakly convex

function, so it does not change the nature of the argument

140

A.5 Equilibrium Conditions

For ∀i ∈ [s, r]:

(Ct − κCt−1)−ςc − βκEt (Ct+1 − κCt)−ςc − λct = 0 (A.5.1)

ς0D−ςdt − λct + Etβλct+1R

dt = 0, (A.5.2)

−λct + Etβλct+1Re,st+1 + ζst = 0, (A.5.3)

−λct + Etβλct+1Re,rt+1 + ζrt = 0, (A.5.4)

ζstEst = 0, (A.5.5)

ζrtErt = 0 (A.5.6)

γt − χi2t = Et

βλct+1

λct

σitQt

τ√2πe−(

(Rdt (1−γt)−Rst+1)Qt+ξσit

σit

√2τ

)2

+

1

2

(Rst+1 −

σitξ

Qt

−Rdt

)[1− erf

((Rdt (1− γt)−Rs

t+1

)Qt + ξσit

σit√

2τ

)]],

(A.5.7)

Re,it+1 =

1

γt

σitQt

τ√2πe−(

(Rdt (1−γt)−Rst+1)Qt+ξσit

σit

√2τ

)2

+

1

2

(Rst+1 −

σitξ

Qt

−Rdt

)[1− erf

((Rdt (1− γt)−Rs

t+1

)Qt + ξσit

σit√

2τ

)],

(A.5.8)

χi2tlit = 0, (A.5.9)

σs = σ, (A.5.10)

σr = σ, (A.5.11)

lit = dit + eit, (A.5.12)

eit = γtlit, (A.5.13)

Ω(σit; lit, d

it, e

it) = Et

[β λct+1

λctRe,it+1e

it

], (A.5.14)

µt =Ert

Est+Ert, (A.5.15)

141

Lst = (1− µt) lst , (A.5.16)

Lrt = µtlrt , (A.5.17)

Eit = γtL

it, (A.5.18)

Lit = Dit + Ei

t , (A.5.19)

Dt = Dst +Dr

t , (A.5.20)

Y st = At (Ks

t )α (Hs

t )1−α , (A.5.21)

Y rt = At (Kr

t )α (Hr

t )1−α − ξKrt , (A.5.22)

QtKst+1 = (1− σ)Lst + (1− σ)Lrt , (A.5.23)

QtKrt+1 = σLst + σLrt , (A.5.24)

Wt = (1− α)Y stHst, (A.5.25)

Rst = αAt

Qt

(Kst

Hst

)α−1

+ (1− δ) Qt+1

Qt, (A.5.26)

Rrt = Rs

t + εtQt−1

, (A.5.27)

Kst

Hst

=Krt

Hrt, (A.5.28)

Hst +Hr

t = 1, (A.5.29)

Kt = Kst +Kr

t , (A.5.30)

Kt+1 = It + (1− δ)Kt, (A.5.31)

It = ηt

[1− φ

2

(IgtIgt−1− 1)2]Igt , (A.5.32)

Y st + Y r

t = Ct + Igt , (A.5.33)

Tt = Lt−1

σt−1

Qt−1

τ√2πe−(

(Rdt−1(1−γt−1)−Rst)Qt−1+ξσt−1

σt−1√

2τ

)2

−

1

2

(Rst −Rd

t−1 (1− γt−1)− ξσt−1

Qt−1

)[1 + erf

((Rdt−1 (1− γt−1)−Rs

t

)Qt−1 + ξσt−1

σt−1

√2τ

)],

(A.5.34)

ηtQt

[1− φ

2

(IgtIgt−1

− 1

)2]− ηtQtφ

(IgtIgt−1

− 1

)IgtIgt−1

− 1 + ηt+1ψt,t+1Qt+1φ

(Igt+1

Igt− 1

)Igt+1

(Igt )2 Igt+1 = 0. (A.5.35)

142

A.6 Calibration of τ

To calibrate the variance of the idiosyncratic shock τ , we link the production

function of the risky rm to the production function of the safe rm that has a

preexisting debt. Remember that the next period returns to safe and risky loans are

given by

Rst+1 =

αAt+1

Qt

(Kt+1

Ht+1

)α−1

+ (1− δ)Qt+1

Qt

,

Rrt+1 = Rs

t+1 + σRFεt+1

Qt

,

respectively. The parameter σRF is needed to distill the exposure of banks (versus

other nancial intermediaries) to the risk arising in the leveraged loan market. It

captures the fact that a certain fraction of leveraged loans is held by the nonbank

sector which we do not model here. The risky bank that nances the maximum share

of risky projects earns

Ωriskyt+1 = Rr

t+1QtKrt+1.

It comprises EBITDA and what the bank makes or loses by selling capital to capital

producers. The safe bank with preexisting debt earns

Ωsafet+1 = Rs

t+1Qt (Kt+1 +Bt)−QtBtRBt =

(Rst+1

(1 +

Bt

Kt+1

)− Bt

Kt+1

RBt

)QtKt+1,

where Bt is a predetermined debt, measured in units of capital, and RBt is a

predetermined interest rate. We equate the conditional variances of the returns

to loans

V art(Rrt+1

)= V art

(Rst+1

(1 +

Bt

Kt+1

)− Bt

Kt+1

RBt

)to nd the variance of the idiosyncratic shock that matches Debt

EBITDA= 6. Note that

V art(Rrt+1

)= V art

(Rst+1

)+

(σRFQt

)2

τ 2,

V art

(Rst+1

(1 +

Bt

Kt+1

)− Bt

Kt+1

RBt

)=

(1 +

Bt

Kt+1

)2

V art(Rst+1

),

143

where Kt+1 is the steady-state level of capital of the safe rms that are nanced by

commercial banks and Qt = 1 in the steady state.

The conditional variance of the returns on safe loans is given by

V art(Rst+1

)= α2

(Kt+1

Ht+1

)2α−2

V art (At+1) + (1− δ)2V art (Qt+1) +

2α

(Kt+1

Ht+1

)α−1

(1− δ)Covt (At+1, Qt+1) .

We can calculate the conditional variance of Qt+1 by picking up its process from

the optimization problem of capital producers. However, our approach is meant to be

suggestive, and we equate the conditional variances of Qt+1 and the aggregate shock.

The covariance term is expected to be positive, but we drop it in our calculation

because the terms that multiply the covariance are small. The model's counterpart

for EBITDA is a total output net of compensation for labor. Thus

DebtEBITDA

=Bt

Y safet −WtH

safet

=Bt

αY safet

.

The data analog of σRF is the share of leveraged loans held by banks (where the

remaining fraction is held by nonbanks). We choose σRF = 45% from the Shared

National Credit Report issued by the Fed, OCC, and FDIC.

A.7 Robustness Checks

See Figure 1.5.

144

Appendix B

Appendix for Chapter 2

We omit the index and the type of bank in the expressions when it is evident from

the context which bank we refer to.

B.1 Expression of Net Cash Flow

Let us calculate the integral from the expression of net cash ows (suppressing

the index i):

Et

∞

ε∗t+1

((Rlt+1 + σt

εt+1

Qt

−Rdt+1

)lt +Rd

t+1et

)dG(εt+1)

,

where(Rlt+1 + σt

ε∗t+1

Qt−Rd

t+1

)lt +Rd

t+1et = 0.

Break calculation of the integral into two parts.

1.∞ε∗t+1

εt+1 dG(εt+1),

2.∞ε∗t+1

dG(εt+1).

Working on the rst part:

∞

ε∗t+1

εt+1 dG(εt+1) =

∞(Rdt+1−R

lt+1

σt−Rdt+1et

σtlt

)Qt

εt+11√

2πτ 2e−

(εt+1+ξ)2

2τ2 dεt+1 =

Introduce a change in variables to recast the integral in terms of the Standard Normal

distribution. Use v = εt+1+ξ√2τ

, or equivalently εt+1 = v√

2τ − ξ, and remember that

145

for the change x = ϕ(t), the integral ϕ(b)

ϕ(a)f(x)dx becomes

baf(ϕ(t))ϕ′(t)dt. Since

dv = dεt+1√2τ, multiply dv by

√2τ to express dεt+1 in terms of dv. Moreover, to transform

the lower limit of the integral, add ξ to the current expression of the lower bound of

the integral and divide the result by√

2τ .

∞

(Rdt+1−Rlt+1)ltQt−Rdt+1etQt+ξσtlt

σtlt√

2τ

(v√

2τ − ξ) √2τ√

2πτ 2e−v

2

dv =

∞

(Rdt+1−Rlt+1)ltQt−Rdt+1etQt+ξσtlt

σtlt√

2τ

(v√

2τ − ξ) 1√

πe−v

2

dv =

√2τ√π

∞

(Rdt+1−Rlt+1)ltQt−Rdt+1etQt+ξσtlt

σtlt√

2τ

ve−v2

dv − ξ√π

∞

(Rdt+1−Rlt+1)ltQt−Rdt+1etQt+ξσtlt

σtlt√

2τ

e−v2

dv =

−√

2τ

2√πe−v

2

∣∣∣∣∣∞

(Rdt+1−Rlt+1)ltQt−Rdt+1etQt+ξσtlt

σtlt√

2τ

− ξ√π

∞

0

e−v2

dv −

(Rdt+1−Rlt+1)ltQt−Rdt+1etQt+ξσtlt

σtlt√

2τ

0

e−v2

dv

=

0 + ltτ√2πe−(

(Rlt+1−Rdt+1)ltQt+Rdt+1etQt−ξσtlt

σtlt√

2τ

)2

−

ξ√π

[√π

2erf(∞)−

√π

2erf

((Rlt+1 −Rd

t+1

)ltQt +Rd

t+1etQt − ξσtltσtlt√

2τ

)]=

τ√2πe−(

(Rlt+1−Rdt+1)ltQt+Rdt+1etQt−ξσtlt

σtlt√

2τ

)2

− ξ

2

[1 + erf

((Rlt+1 −Rd

t+1

)ltQt +Rd

t+1etQt − ξσtltσtlt√

2τ

)],

where we use that erf(∞) = 1 and erf(−x) = −erf(x).

Working on the second part: Again, use the transformation v = εt+1+ξ√2τ

, so εt+1 =

v√

2τ − ξ

∞

ε∗t+1

dG(εt+1) =

∞

ε∗t+1

(1√

2πτ 2e−

(εt+1+ξ)2

2τ2

)=

∞

(Rdt+1−Rlt+1)ltQt−Rdt+1etQt+ξσtlt

σtlt√

2τ

√2τ√

2πτ 2e−v

2

dv =1√π

∞

(Rdt+1−Rlt+1)ltQt−Rdt+1etQt+ξσtlt

σtlt√

2τ

e−v2

dv =

1

2

[1 + erf

((Rlt+1 −Rd

t+1

)ltQt +Rd

t+1etQt − ξσtltσtlt√

2τ

)].

146

Combining both parts, nd

Et

∞

ε∗t+1

((Rlt+1 + σtεt+1 −Rd

t+1

)lt +Rd

t+1et)

dG(εt+1)

=

Et

σtlt τ

Qt

√2πe−(

(Rlt+1−Rdt+1)ltQt+Rdt+1etQt−ξσtlt

σtlt√

2τ

)2

+

((Rlt+1 −Rd

t+1

)lt +Rd

t+1et −ξQt

)2

[1 + erf

((Rlt+1 −Rd

t+1

)ltQt +Rd

t+1etQt − ξσtltσtlt√

2τ

)] .

B.2 Share of Non-Defaulted Deposits

First, the bank defaults on its deposit obligation whenever the idiosyncratic shock,

εt+1, is below a cuto level ε∗t+1, dened by:

ε∗t = −(Rlt −Rd

t

)lt−1Qt−1 +Rd

t et−1Qt−1

σt−1lt−1

.

Second, let's express 1 + erf

((Rlt−Rdt )lt−1Qt−1+Rdt et−1Qt−1−ξσt−1lt−1

σt−1lt−1

√2τ

)= 1− erf

(ε∗t+ξ

τ√

2

)in

terms of the CDF of the normal distribution. By denition,

G(εt) =1

2

[1 + erf

(εt + ξ

τ√

2

)]=⇒ erf

(εt + ξ

τ√

2

)= 2G(εt)− 1.

Therefore,

1 + erf

((Rlt −Rd

t

)lt−1Qt−1 +Rd

t et−1Qt−1 − ξσt−1lt−1

σt−1lt−1

√2τ

)= 1− (2G(ε∗t )− 1) = 2 (1−G(ε∗t )) .

Thus, the share of defaulted loans is given by

1−G(ε∗t ) =1

2

(1 + erf

((Rlt −Rd

t

)lt−1Qt−1 +Rd

t et−1Qt−1 − ξσt−1lt−1

σt−1lt−1

√2τ

))

B.3 Choice of Risk

Theorem. The expected dividends function of banks, Etωt+1, is convex in the risk

parameter σt. Moreover, if ξ = 0, then it is also increasing in σt.

147

Proof. We generalize the proof taken from Van den Heuvel (2008) to the case

with aggregate uncertainty. The proof applies to an arbitrary distribution of the

idiosyncratic shock, εt+1, so a Normal distribution considered in the analysis is not a

special case chosen to drive the results.

Assumption. ε has a cumulative distribution function Gε with support [ε, ε], with

ε < 0 < ε. The mean of ε is equal to −ξ. ε is independent of the aggregate shock. The

aggregate shock does not depend on the choice of σt.

Note that we do not restrict the analysis to the bounded support1, so ε and ε can

take −∞ and +∞, respectively. Note that Gε need not be continuous.

Let ε(σt, Rlt+1) ≡ Rdt dtQt

σtlt− Rlt+1Qt

σt, so

(Rlt+1 + σt

ε(σt)Qt

)lt −Rd

t dt = 0.

Let π(σt, Rlt+1) = Eε

[((Rlt+1 + σt

εQt

)lt −Rd

t dt

)+]be a function of expected

dividends (taken over the idiosyncratic shock only) under some realization of Rlt+1

which is considered to be xed in this function. To account for the aggregate

uncertainty, Rlt+1 needs to be a random variable. Therefore, the expected dividends

are given by (taking into account both idiosyncratic and aggregate) uncertainty:

Π(σt) =

Ω

π(σt, R

lt+1(ω)

)P (dω) = Et

[ ε

ε(σt, Rlt+1)

((Rlt+1 + σt

ε

Qt

)lt −Rd

t dt

)dGε

]=

Et

ε

ε

((Rlt+1 + σt

ε

Qt

)lt −Rd

t dt

)dGε

− Et [ ε(σt, Rlt+1)

ε

((Rlt+1 + σt

ε

Qt

)lt −Rd

t dt

)dGε

]=

EtRlt+1lt −Rd

t dt − σtξ

Qt

lt −σtltQt

Et

[ ε(σt, Rlt+1)

ε

(ε− ε(σt, Rl

t+1))dGε

]=

EtRlt+1lt −Rd

t dt +ltQt

(σtEt

[ ε(σt, Rlt+1)

ε

(ε(σt, R

lt+1)− ε

)dGε

]− σtξ

).

Note that(Rlt+1 + σt

εQt

)lt−Rd

t dt is expressed in terms of ε(σt, Rlt+1) and ε using

the denition of ε(σt, Rlt+1).

1Unbounded support is more relevant for aggregate shocks

148

The proof below shows that Π(σt) is convex in σt. Since the expression of Π(σt)

involves the term which is linear in σt and ltQt≥ 0, the sucient condition for Π(σt)

to be convex in σt is that

H(σt) ≡ Et

[ ε(σt)

ε(ε(σt)− ε) dGε

]σt

is convex in σt,.

Claim. H(σt) ≡ Et

[ ε(σt)ε

(ε(σt, R

lt+1)− ε

)dGε

]σt is convex. It is increasing in σt

when ξ = 0.

Proof. Steps of the proof:

1. Dene h(σt, Rlt+1) ≡ σt

[ ε(σt, Rlt+1)

ε(ε(σt, R

lt+1)− ε

)dGε

]and consider 3 cases:

(a) Realization of Rlt+1 is such that ε

(σt, R

lt+1

)=

Rdt dtQtσtlt

− Rlt+1Qt

σt> 0,

(b) Realization of Rlt+1 is such that ε

(σt, R

lt+1

)=

Rdt dtQtσtlt

− Rlt+1Qt

σt< 0,

(c) Realization of Rlt+1 is such that ε

(σt, R

lt+1

)=

Rdt dtQtσtlt

− Rlt+1Qt

σt= 0,

Show that h(σt, Rlt+1) is convex and increasing in σt in cases 1a and 1b and

h(σt, Rlt+1) is linear and increasing in σt in case 1c.

2. Employ the argument that convexity and monotonicity are preserved under non-

negative scaling and addition (guaranteed by the expectation operator over the

aggregate uncertainty) to nd that H(σt) is convex and increasing.

Here we show each step of the proof formally

1. Let σ1t < σ2t and, for λ ∈ (0, 1), dene σλt = λσ1t + (1 − λ)σ2t. Let εi =

ε(σit, Rlt+1) ≡ Rdt dtQt

σtlt− Rlt+1Qt

σt, for i = 1, 2, λ.

(a) ε(σt, R

lt+1

)> 0 implies that 0 < ε2 < ελ < ε1,

149

Claim. h(σt) is convex in σt.

h(σλt) = (λσ1t + (1− λ)σ2t)

ε(σλt)

ε(ε(σλt)− ε) dGε

=

λσ1t

ε1

ε(ελ − ε) dGε −

ε1

ελ

(ελ − ε) dGε

+

(1− λ)σ2t

ε2

ε(ελ − ε) dGε +

ελ

ε2

(ελ − ε) dGε

=

λσ1t

ε1

ε(ε1 − ε) dGε + (ελ − ε1)Gε(ε1) +

ε1

ελ

(ε− ελ) dGε

+

(1− λ)σ2t

ε2

ε(ε2 − ε) dGε + (ελ − ε2)Gε(ε2) +

ελ

ε2

(ελ − ε) dGε

≤

λσ1t

ε1

ε(ε1 − ε) dGε + (ελ − ε1)Gε(ε1) +

ε1

ελ

(ε1 − ελ) dGε

+

(1− λ)σ2t

ε2

ε(ε2 − ε) dGε + (ελ − ε2)Gε(ε2) +

ελ

ε2

(ελ − ε2) dGε

,

where the inequality sign comes from ε1ελ

(ε− ελ) dGε ≤ ε1ελ

(ε1 − ελ) dGε and ελε2

(ελ − ε) dGε ≤ ελε2

(ελ − ε2) dGε. Substituting for the denitions of h(σ1t) =

σ1t

ε1ε (ε1 − ε) dGε and h(σ2t) = σ2t

ε2ε (ε2 − ε) dGε, one can get:

h(σλt) ≤ λh(σ1t) + (1− λ)h(σ2t) + λσ1t (ελ − ε1)Gε(ελ)+ (1− λ)σ2t (ελ − ε2)Gε(ελ) =

λh(σ1t) + (1− λ)h(σ2t) +Gε(ελ) (λσ1t (ελ − ε1) + (1− λ)σ2t (ελ − ε2)) =

λh(σ1t) + (1− λ)h(σ2t),

where the last equality follows from the fact that σ1tε1 =Rdt dtQt

lt− Rl

t+1Qt =

σ2tε2 = σλtελ. So

λσ1t (ελ − ε1) + (1− λ)σ2t (ελ − ε2) =

ελ (λσ1t + (1− λ)σ2t)−(Rdt dtQt

lt−Rl

t+1Qt

)(λ+ (1− λ)) =

σλtελ −(Rdt dtQt

lt−Rl

t+1Qt

)= 0.

150

Claim. If ξ = 0, then h(σt) is increasing in σt.

h(σ2t)− h(σ1t) = σ2t

ε2

ε(ε2 − ε) dGε − σ1t

ε1

ε(ε1 − ε) dGε =

σ2t

ε2

ε(ε2 − ε) dGε − σ1t

ε2

ε(ε1 − ε) dGε − σ1t

ε1

ε2

(ε1 − ε) dGε =

ε2

ε(ε2σ2t − εσ2t − ε1σ1t + εσ1t) dGε − σ1t

ε1

ε2

(ε1 − ε) dGε =

− σ2t

ε2

εεdGε − σ1t

( ε1

ε2

(ε1 − ε) dGε − ε2

εεdGε

)=

− σ2t

ε2

εεdGε − σ1t

( ε1

ε2

ε1dGε − ε1

εεdGε

)=

− σ2t

(0−

ε

ε2

εdGε

)− σ1t

ε1

ε2

ε1dGε + σ1t

(0−

ε

ε1

εdGε

)=

σ2t

ε1

ε2

εdGε + σ2t

ε

ε1

εdGε − σ1t

ε1

ε2

ε1dGε − σ1t

ε

ε1

εdGε >

(σ2t − σ1t)

ε

ε1

εdGε+σ2t

ε1

ε2

ε2dGε−σ1t

ε1

ε2

ε1dGε = (σ2t − σ1t)

ε

ε1

εdGε > 0

(b) ε(σt, R

lt+1

)< 0 implies that ε1 < ελ < ε2 < 0

Claim. h(σt) is convex in σt.

h(σλt) = (λσ1t + (1− λ)σ2t)

ε(σλt)

ε(ε(σλt)− ε) dGε

=

λσ1t

ε1

ε(ελ − ε) dGε +

ελ

ε1

(ελ − ε) dGε

+

(1− λ)σ2t

ε2

ε(ελ − ε) dGε −

ε2

ελ

(ελ − ε) dGε

=

λσ1t

ε1

ε(ε2 − ε) dGε + (ελ − ε1)Gε(ε1) +

ελ

ε1

(ελ − ε) dGε

+

(1− λ)σ2t

ε2

ε(ε2 − ε) dGε + (ελ − ε2)Gε(ε2) +

ε2

ελ

(ε− ελ) dGε

≤

λσ1t

ε1

ε(ε1 − ε) dGε + (ελ − ε1)Gε(ε1) +

ελ

ε1

(ελ − ε1) dGε

+

(1− λ)σ2t

ε2

ε(ε2 − ε) dGε + (ελ − ε2)Gε(ε2) +

ε2

ελ

(ε2 − ελ) dGε

,

151

where the inequality sign follows from ελε1

(ελ − ε) dGε ≤ ελε1

(ελ − ε1) dGε and ε2ελ

(ε− ελ) dGε ≤ ε2ελ

(ε2 − ελ) dGε. Substituting for the denitions of h(σ1t) =

σ1t

ε1ε (ε1 − ε) dGε and h(σ2t) = σ2t

ε2ε (ε2 − ε) dGε, one can get:

h(σλt) ≤ λh(σ1t) + (1− λ)h(σ2t) + λσ1t (ελ − ε1)Gε(ελ)+ (1− λ)σ2t (ελ − ε2)Gε(ελ) =

λh(σ1t) + (1− λ)h(σ2t) +Gε(ελ) (λσ1t (ελ − ε1) + (1− λ)σ2t (ελ − ε2)) =

λh(σ1t) + (1− λ)h(σ2t),

where the last equality applies the similar reasoning used for the previous case.

Therefore, h(σt) is convex in σt for Rlt+1 > Rd

t (1− γ).

Claim. h(σt) is increasing in σt.

h(σ2t)− h(σ1t) = σ2t

ε2

ε(ε2 − ε) dGε − σ1t

ε1

ε(ε1 − ε) dGε =

σ2t

ε1

ε(ε2 − ε) dGε + σ2t

ε2

ε1

(ε2 − ε) dGε − σ1t

ε1

ε(ε1 − ε) dGε =

ε1

ε(ε2σ2t − εσ2t − ε1σ1t + εσ1t) dGε + σ2t

ε2

ε1

(ε2 − ε) dGε =

σ1t

ε1

εεdGε + σ2t

( ε2

ε1

(ε2 − ε) dGε − ε1

εεdGε

)=

σ1t

ε1

εεdGε + σ2t

( ε2

ε1

ε2dGε − ε2

εεdGε

)=

(σ1t − σ2t)

ε1

εεdGε + σ2t

( ε2

ε1

ε2dGε − ε2

ε1

εdGε

)=

(σ1t − σ2t)

ε1

εεdGε + σ2t

( ε2

ε1

(ε2 − ε) dGε

)> 0

because σ1t − σ2t < 0, ε1ε εdGε < 0 and

ε2ε1

(ε2 − ε) dGε > 0

(c) ε(σt, R

lt+1

)= 0 implies that h(σt) = σt

[ 0

ε (0− ε) dGε

]is linear and

increasing in σt.

152

2. Consider P (ω) ≥ 0 for each Rlt+1(ω) ∈ R. Then the following function2:

Ω

h(σt, R

lt+1(ω)

)P (dω) = Eth(σt, R

lt+1) ≡ H(σt)

is convex in σt and increasing in σt when ξ = 0. It follows directly from the

linearity of the expectation operator which puts a non-negative weight on every

realization of Rlt+1 and the fact that the sum of convex functions is a convex

function. Therefore, Π(σt) is convex in σt. Moreover, when ξ = 0,

Π(σt) = Rlt+1lt −Rd

t dt +ltQt

H(σt)

is also increasing in σt as ltQt> 0.

B.4 The Shadow Bank's Problem

B.4.1 Value function

Substituting ξ = 0 into the expression of net cash ow to write the value of the

bank:

V Sj,t = Et

∞∑i=0

(1− θ) θiΛt,t+1+i

σSj,t+ilSj,t+i τ

Qt+i

√2πe−(

(Rlt+1+i−RdSt+1+i)Qt+i

σSj,t+i

√2τ

+RdSt+1+ie

Sj,t+iQt+i

σSj,t+i

lSj,t+i

√2τ

)2

+

((Rlt+1+i −RdS

t+1+i

)lSj,t+i +RdS

t+1+ieSj,t+i

)2

[1 + erf

((Rlt+1+i −RdS

t+1+i

)Qt+i

σSj,t+i√

2τ+RdSt+1+ie

Sj,t+iQt+i

σSj,t+ilSj,t+i

√2τ

)]].

Dene

νj,t = Et

∞∑i=0

(1− θ) θiΛt,t+1+i

σSj,t+i τ

Qt+i

√2πe−(

(Rlt+1+i−RdSt+1+i)Qt+i

σSj,t+i

√2τ

+RdSt+1+ie

Sj,t+iQt+i

σSj,t+i

lSj,t+i

√2τ

)2

+

(Rlt+1+i −RdS

t+1+i

)2

[1 + erf

((Rlt+1+i −RdS

t+1+i

)Qt+i

σSj,t+i√

2τ+RdSt+1+ie

Sj,t+iQt+i

σSj,t+ilSj,t+i

√2τ

)])lSj,t+ilSj,t

],

2Linearity in σt for one particular value of Rlt+1 can be considered as a weakly convex

function, so it does not change the nature of the argument

153

ηj,t = Et

∞∑i=0

(1− θ) θiΛt,t+1+i

[ (RdSt+1+i

)2

[1 + erf

((Rlt+1+i −RdS

t+1+i

)Qt+i

σSj,t+i√

2τ+RdSt+1+ie

Sj,t+iQt+i

σSj,t+ilSj,t+i

√2τ

)]eSj,t+ieSj,t

]].

So V Sj,t = υj,tl

Sj,t + ηj,te

Sj,t. Write it recursively. Pulling out the rst term in each

summation:

νj,t = Et

(1− θ) Λt,t+1

σSj,t τ

Qt

√2πe−(

(Rlt+1−RdSt+1)Qt

σSj,t

√2τ

+RdSt+1e

Sj,tQt

σSj,tlSj,t

√2τ

)2

+

(Rlt+1 −RdS

t+1

)2

[1 + erf

((Rlt+1 −RdS

t+1

)Qt

σSj,t√

2τ+RdSt+1e

Sj,tQt

σSj,tlSj,t

√2τ

)])lSj,tlSj,t

]+

Et

∞∑i=1

(1− θ) θiΛt,t+1+i

σSj,t+i τ

Qt+i

√2πe−(

(Rlt+1+i−RdSt+1+i)Qt+i

σSj,t+i

√2τ

+RdSt+1+ie

Sj,t+iQt+i

σSj,t+i

lSj,t+i

√2τ

)2

+

(Rlt+1+i −RdS

t+1+i

)2

[1 + erf

((Rlt+1+i −RdS

t+1+i

)Qt+i

σSj,t+i√

2τ+RdSt+1+ie

Sj,t+iQt+i

σSj,t+ilSj,t+i

√2τ

)])lSj,t+ilSj,t

].

Transform the summations, so that they start from zero

νj,t = Et

(1− θ) Λt,t+1

σSj,t τ

Qt

√2πe−(

(Rlt+1−RdSt+1)Qt

σSj,t

√2τ

+RdSt+1e

Sj,tQt

σSj,tlSj,t

√2τ

)2

+

(Rlt+1 −RdS

t+1

)2

[1 + erf

((Rlt+1 −RdS

t+1

)Qt

σSj,t√

2τ+RdSt+1e

Sj,tQt

σSj,tlSj,t

√2τ

)])]+

Et

θ∞∑i=0

(1− θ) θiΛt,t+2+i

σSj,t+1+i

τ

Qt+1+i

√2πe−(

(Rlt+2+i−RdSt+2+i)Qt+1+i

σSj,t+1+i

√2τ

+RdSt+2+ie

Sj,t+1+iQt+1+i

σSj,t+1+i

lSj,t+1+i

√2τ

)2

+

((Rlt+2+i −RdS

t+2+i

))2

[1 + erf

((Rlt+2+i −RdS

t+2+i

)Qt+1+i

σSj,t+1+i

√2τ

+RdSt+2+ie

Sj,t+1+iQt+1+i

σSj,t+1+ilSj,t+1+i

√2τ

)])lSj,t+1+i

lSj,t

].

Remember that Λt,t+2+i = Λt,t+1Λt+1,t+2+i. Plugging this into the last expressions

νj,t = Et

(1− θ) Λt,t+1

σSj,t τ

Qt

√2πe−(

(Rlt+1−RdSt+1)Qt

σSj,t

√2τ

+RdSt+1e

Sj,tQt

σSj,tlSj,t

√2τ

)2

+

(Rlt+1 −RdS

t+1

)2

[1 + erf

((Rlt+1 −RdS

t+1

)Qt

σSj,t√

2τ+RdSt+1e

Sj,tQt

σSj,tlSj,t

√2τ

)])]+

154

Et

θΛt,t+1

lSj,t+1

lSj,t

∞∑i=0

(1− θ) θiΛt+1,t+2+i

σSj,t+1+i

Qt+1+i

τ√2πe−(

(Rlt+2+i−RdSt+2+i)Qt+1+i

σSj,t+1+i

√2τ

+RdSt+2+ie

Sj,t+1+iQt+1+i

σSj,t+1+i

lSj,t+1+i

√2τ

)2

+

((Rlt+2+i −RdS

t+2+i

))2

[1 + erf

((Rlt+2+i −RdS

t+2+i

)Qt+1+i

σSj,t+1+i

√2τ

+RdSt+2+ie

Sj,t+1+iQt+1+i

σSj,t+1+ilSj,t+1+i

√2τ

)])lSj,t+1+i

lSj,t+1

].

Therefore,

νj,t =Et

(1− θ) Λt,t+1

σSj,t τ

Qt

√2πe−(

(Rlt+1−RdSt+1)Qt

σSj,t

√2τ

+RdSt+1e

Sj,tQt

σSj,tlSj,t

√2τ

)2

+

(Rlt+1 −RdS

t+1

)2

[1 + erf

((Rlt+1 −RdS

t+1

)Qt

σSj,t√

2τ+RdSt+1e

Sj,tQt

σSj,tlSj,t

√2τ

)]]+ θΛt,t+1

lSj,t+1

lSj,tνj,t+1

,

ηj,t =Et

(1− θ) Λt,t+1

[ (RdSt+1

)2

[1 + erf

((Rlt+1 −RdS

t+1

)Qt

σSj,t√

2τ+RdSt+1e

Sj,tQt

σSj,tlSj,t

√2τ

)]+ θΛt,t+1

eSj,t+1

eSj,tηj,t+1

].

B.4.2 Risk Choice

Denoting ΓSj,t as expected dividends:

ΓSj,t = Et

(1− θ) Λt,t+1

σSj,tlSj,t τ

Qt

√2πe−(

(Rlt+1−RdSt+1)Qt

σSj,t

√2τ

+RdSt+1e

Sj,tQt

σSj,tlSj,t

√2τ

)2

+

(Rlt+1l

Sj,t −RdS

t+1lSj,t +RdS

t+1eSj,t

)2

[1 + erf

((Rlt+1 −RdS

t+1

)Qt

σSj,t√

2τ+RdSt+1e

Sj,tQt

σSj,tlSj,t

√2τ

)])].

The value function of the shadow bank can be written as:

V Sj,t = ΓSj,t + θEt

Λt,t+1V

Sj,t+1

= Et

∞∑i=0

θiΛt,t+iΓSj,t+i.

In Appendix B.3 we show that ΓSj,t is increasing in σt. Accordingly, each term in

the summation above is increasing in σt. Then, after applying the law of iterative

expectations, it is easy to see that V Sj,t is also increasing in σt.

155

B.5 The Commercial Bank's Problem

B.5.1 First-Order Conditions

The objective function can be written recursively

V Cj,t = max

lCj,t,eCj,t,σ

Cj,t

zj,t + Et

Λt,t+1V

Cj,t+1

,

subject to

eCj,t ≥ γCt lCj,t,

zj,t = max

[(Rlt + σCj,t−1

εCj,tQt−1

−RdCt−1

)lCj,t−1 +RdC

t−1eCj,t−1, 0

]− ej,t,

lCj,t ≥ 0,

σC ≤ σCj,t ≤ σC .

Dene

J (St−1) = maxlCj,t,d

Cj,t,e

Cj,t,σ

Cj,t

−ej,t + Et

[Λt,t+1V

Cj,t+1 (lj,t, ej,t, σj,t, St)

],

where St is the aggregate state of the economy. Then

V Cj,t (lj,t−1, ej,t−1, σj,t−1, St−1) = max

[(Rlt + σCj,t−1

εCj,tQt−1

−RdCt−1

)lCj,t−1 +RdC

t−1eCj,t−1, 0

]+ J (St) .

Since the rst term in the summation on the right hand side is taken as given and

εj,t is i.i.d. sequence of random variables, each banker faces the same maximization

problem

J (St−1) = maxlt,et,σt

−et + Et

Λt,t+1

∞

ε∗t+1

((Rlt+1 + σt

εt+1

Qt

−Rdt

)lt +Rd

t et

)dG (εt+1) + J (St)

,

et ≥ γtlt,(Rlt+1 + σt

ε∗t+1

Qt

−Rdt

)lt +Rd

t et = 0,

lt ≥ 0,

σ ≤ σt ≤ σ.

156

Append the Lagrange multiplier χ1t to the constraint et ≥ γlt and χ2t to the

constraint lt ≥ 0. Conditional on the optimal choice of σt, the rst-order conditions

are:

Et

Λt,t+1

=0︷ ︸︸ ︷((Rlt+1 + σt

ε∗t+1

Qt

)lt −Rd

t (lt − et))· ∂∂lt

(Rdt (lt − et)Qt

σtlt−Rlt+1Qt

σt

)− γχ1t + χ2t+

Et

∞

Rdt (lt−et)Qtσtlt

−Rlt+1Qt

σt

Λt,t+1∂

∂lt

((Rlt+1 + σt

εt+1

Qt

)lt −Rd

t (lt − et))

1√2πτ 2

e−(εt+1+ξ)2

2τ2 dεt+1

= 0,

Et

Λt,t+1

=0︷ ︸︸ ︷((Rlt+1 + σt

ε∗t+1

Qt

)lt −Rd

t (lt − et))· ∂∂et

(Rdt (lt − et)Qt

σtlt−Rlt+1Qt

σt

)− 1 + χ1t+

Et

∞

Rdt (lt−et)Qtσtlt

−Rlt+1Qt

σt

Λt,t+1∂

∂et

((Rlt+1 + σt

εt+1

Qt

)lt −Rd

t (lt − et))

1√2πτ 2

e−(εt+1+ξ)2

2τ2 dεt+1

= 0,

respectively. We use the Leibniz integral rule above to nd the partial derivatives.

Note that the rst term is zero in the dierentiation because the upper limit of the

integral does not depend on any of the choice variables. Complementary slackness:

χ1t (et − γtlt) = 0,

χ2tlt = 0,

et − γtlt ≥ 0,

lt ≥ 0,

χ1t ≥ 0,

χ2t ≥ 0,

157

Envelope theorem:

J ′l (lt−1, et−1, σt−1) = 0,

J ′e (lt−1, et−1, σt−1) = 0.

Using the expressions of the integrals from Appendix B.1, the FOCs can be described

by

χ2t + Et

βλct+1

λct

σtQt

τ√2πe−(Rdt (1− et

lt)Qt−Rlt+1Qt+ξσt

σt√

2τ

)2

+

+

(Rlt+1 − σtξ −Rd

t

2

)1− erf

Rdt

(1− et

lt

)Qt −Rl

t+1Qt + ξσt

σt√

2τ

= γχ1t,

Et

βλct+1

λct

[Rdt

1

2

(1− erf

(Rdt (lt − et)Qt −Rl

t+1ltQt + ξσtlt

σtlt√

2τ

))]− 1 + χ1t = 0.

B.5.2 Risk Choice

Denoting ΓCt as expected dividends:

ΓCt = −et + Et

Λt,t+1

σtQt

τ√2πe−(Rdt (1− et

lt)Qt−Rlt+1Qt+ξσt

σt√

2τ

)2

+

(Rlt+1lt −Rd

t lt +Rdt et)

2

1 + erf

Rdt

(1− et

lt

)Qt −Rl

t+1Qt + ξσt

σt√

2τ

.

The objective function of the commercial bank can be written as:

J (St−1) = ΓCt + Et Λt,t+1J (St) = Et

∞∑i=0

Λt,t+iΓSj,t+i.

In Appendix A.4 we show that ΓCt is convex in σt. Accordingly, each term in

the summation above is convex in σt. Then, after applying the law of iterative

expectations, it is easy to see that J (St−1) is also convex in σt.

This result guarantees that all the intermediate values σ < σt < σ, which may

result from the rst-order conditions with respect to σt, are not optimal.

158

B.5.3 Proof of Proposition 4

Equations (2.4) and (2.5) can be expressed as

βEtλct+1

λctRei,t+1 = 1− µi,t

λct,

where i ∈ s, r denotes the type of equity. Using this expression, substitute for 1 in

the bank's FOC with respect to et. Therefore,

Et

βλct+1

λct

[Rdt

1

2

(1− erf

(Rdt (lt − et)−Rl

t+1lt + ξσtlt

σtlt√

2τ

))]−Re

i,t+1

−µi,tλct

+χ1t = 0.

Since the range of the erf function is between −1 and 1, i.e.−1 ≤ erf(x) ≤ 1, we

know that the following expression is between Ψ∗1 and Ψ∗2:

Ψ∗1 ≤ Et

βλct+1

λct

[Rdt

1

2

(1− erf

(Rdt (lt − et)−Rl

t+1lt + ξσtlt

σtlt√

2τ

))−Re

t+1

]≤ Ψ∗2,

where

Ψ∗1 = Et

βλct+1

λct

[0−Re

i,t+1

],

Ψ∗2 = Et

βλct+1

λct

[Rdt −Re

i,t+1

].

Summing up equation (2.1) with Etβλct+1Rei,t+1 + µi,t = λct that comes from the

household's FOCs with respect to ei,t for each i ∈ s, r, one gets:

Etβλct+1

[Rdt −Re

i,t+1

]= −σ0d

−σdt + µi,t.

The Lagrange multiplier on the households budget constraint, λct, is positive.

It reects the fact that the budget constraint always binds given the standard

assumptions on the preferences (Inada conditions). Furthermore, σ0d−σdt > 0 under

the usual (and mild) assumptions on the preferences for liquidity. Dividing the latest

expression by λct it can be transformed into:

Et

βλct+1

λct

[Rdt −Re

i,t+1

]︸ ︷︷ ︸

=Ψ∗2

−µi,tλct

= −σ0d−σdt

λct< 0.

159

Thus, Ψ∗2 <µi,tλct

. Therefore,

Et

βλct+1

λct

[Rdt

1

2

(1− erf

(Rdt (lt − et)−Rl

t+1lt + ξσtlt

σtlt√

2τ

))]−Re

i,t+1

−µi,tλct

+χ1t =

0 < Ψ∗2 −µi,tλct

+ χ1t <µi,tλct− µi,tλct

+ χ1t = χ1t.

Hence, χ1t > 0.

B.6 The Firm's Problem

B.6.1 Safe firms

For i ∈ [νt, 1]:

The optimality condition on the choice of labor by the safe rm is:

Hi,t+1 = (1− α)Yi,t+1

Wt+1

= (1− α)At+1K

αi,t+1H

1−αi,t+1

Wt+1

. (B.6.1)

Accordingly, the safe rm's Lagrangian is:

Lsafe =Et

βλct+1

λct

[At+1K

αi,t+1H

1−αi,t+1 + (1− δ)Qt+1Ki,t+1 −Wt+1Hi,t+1 −Rl

t+1li,t]

+ λsHtEt

βλct+1

λct

[(1− α)

At+1Kαi,t+1H

1−αi,t+1

Wt+1

−Hi,t+1

]+ λslt (li,t −QtKi,t+1) .

Notice that there is no expectation operator on the Lagrange multipliers because

those constraints hold under every state of nature. The problem implies the following

rst-order conditions

∂Lsafe

∂li,t= −Et

βλct+1

λctRli,t+1

+ λslt = 0,

∂Lsafe

∂Ki,t+1

= Et

βλct+1

λct

[αYi,t+1

Ki,t+1

+ (1− δ)Qt+1

]+

λsHt (1− α)αEt

βλct+1

λct

Yi,t+1

Wt+1Ki,t+1

− λsltQt = 0,

∂Lsafe

∂Hi,t+1

= (1− α)At+1K

αi,t+1H

1−αi,t+1

Wt+1

−Wt+1 + λsHt

[(1− α)2 Yi,t+1

Hi,t+1Wt+1

− 1

]= 0.

160

Combining ∂Lsafe∂Hi,t+1

= 0 with equation (B.6.1) yields λsHt = 0. Plugging ∂Lsafe∂li,t

= 0 into

∂Lsafe∂Ki,t+1

for λslt, we get

Et

βλct+1

λctRli,t+1

Qt = Et

βλct+1

λct

[αYi,t+1

Ki,t+1

+ (1− δ)Qt+1

].

Consider the zero-prot condition of the safe rm under all states of nature. Due

to equation (B.6.1) we have:

At+1Kαi,t+1H

1−αi,t+1 = αYi,t+1 +Wt+1Hi,t+1.

Substituting this result for Yi,t+1 into πi,t+1 = 0 and using QtKi,t+1 = li,t yield:

αEtAt+1

(Ki,t+1

Hi,t+1

)α−1

Ki,t+1 + (1− δ)Qt+1Ki,t+1 −Rli,t+1QtKi,t+1 = 0.

Thus, Rli,t+1Qt = αEtAt+1

(Ki,t+1

Hi,t+1

)α−1

+ (1 − δ)Qt+1 under all states of nature. This

condition implies

Et

βλct+1

λctRli,t+1

Qt = Et

βλct+1

λct

[αYi,t+1

Ki,t+1

+ (1− δ)Qt+1

].

B.6.2 Risky Firms

For j ∈ [0, νt]:

The optimality condition on the choice of labor by the risky rm is:

Hj,t+1 =

((1− α)At+1

Wt+1

)1/α

Kj,t+1. (B.6.2)

Accordingly, the risky rm's Lagrangian is:

Lrisky =Et

βλct+1

λct

[At+1K

αj,t+1H

1−αj,t+1 + εj,t+1Kj,t+1 + (1− δ)Qt+1Kj,t+1 −Wt+1Hj,t+1 −Rl

j,t+1lj,t]

+

λrHtEt

βλct+1

λct

[((1− α)At+1

Wt+1

)1/α

Kj,t+1 −Hj,t+1

]+ λrlt (lj,t −QtKj,t+1) .

Notice that there is no expectation operator on the Lagrange multipliers because

those constraints hold under every state of nature. The problem implies the following

161

rst-order conditions

∂Lrisky

∂lj,t= −Et

βλct+1

λctRlj,t+1

+ λrlt = 0,

∂Lrisky

∂Kj,t+1

= Et

βλct+1

λct

[αAt+1

(Kj,t+1

Hj,t+1

)α−1

+ εj,t+1 + (1− δ)Qt+1

]+

λrHtEt

βλct+1

λct

((1− α)At+1

Wt+1

)1/α− λrltQt = 0,

∂Lrisky

∂Hj,t+1

= (1− α)At+1

(Kj,t+1

Hj,t+1

)α−Wt+1 + λrHt [−1] = 0.

Combining ∂Lrisky∂Hj,t+1

= 0 with equation (B.6.2) yields λrHt = 0. Plugging ∂Lrisky∂lj,t

= 0 into

∂Lrisky∂Kj,t+1

for λrlt, we get

Et

βλct+1

λctRlj,t+1

Qt = Et

βλct+1

λct

[αAt+1

(Kj,t+1

Hj,t+1

)α−1

+ (1− δ)Qt+1 + εj,t+1

].

Combining equation (B.6.1) with equation (B.6.2) results in:

Ki,t+1

Hi,t+1

=Kj,t+1

Hj,t+1

=Kt+1

Ht+1

(B.6.3)

under all states of nature. But remember that

Et

βλct+1

λctRli,t+1

Qt = Et

βλct+1

λct

[αAt+1

(Ki,t+1

Hi,t+1

)α−1

+ (1− δ)Qt+1

]=

Et

βλct+1

λct

[αAt+1

(Kt+1

Ht+1

)α−1

+ (1− δ)Qt+1

]= Et

βλct+1

λctRlt+1

Qt.

Therefore

Et

βλct+1

λctRlj,t+1

Qt = Et

βλct+1

λct

[Rlt+1Qt + εj,t+1

].

Consider the zero-prot condition of the risky rm under all states of nature. Due

to equation (B.6.2) we have:

At+1Kαj,t+1H

1−αj,t+1 = αAt+1

(Kj,t+1

Hj,t+1

)α−1

Kj,t+1 +Wt+1Hj,t+1.

162

Substituting this result for Yj,t+1 into πj,t+1 = 0 , using QtKj,t+1 = lj,t and equation

(B.6.3) yield:

Rlj,t+1QtKj,t+1 + εj,t+1Kj,t+1 −Rl

j,t+1QtKj,t+1 = 0.

Thus, Rlj,t+1Qt = Rl

t+1Qt + εj,t+1 under all states of nature. This condition implies

Et

βλct+1

λctRlj,t+1

Qt = Et

βλct+1

λct

[Rlt+1 + εj,t+1

].

B.6.3 Aggregating across firms

Here we show that we can aggregate individual rms into two representative rms.

Let denote Kji,t the capital chosen by rm i that is nanced by borrowing from bank

j. In this notation, the equation (B.6.3) is written as

Kji,t+1

Hji,t+1

=Kt+1

Ht+1

,

for all j ∈ [0, 1] and i ∈ [0, 1].

Dene the following objects: Let KSS,t+1 =

1

µt

1

νtKji,t+1didj be the total capital

allocated to the safe technology and nanced by borrowing from the banks that

choose a fraction σ of risky projects. Let KRS,t+1 =

µt0

1

νtKji,t+1didj be the total

capital allocated to the safe technology and nanced by borrowing from the banks

that choose a fraction σ of risky projects. Thus,

KS,t+1 = KSS,t+1 +KR

S,t+1,

where KS,t+1 is the total capital allocated to the safe technology.

Let KSR,t+1 =

1

µt

νt0Kji,t+1didj be the total capital allocated to the risky

technology and nanced by borrowing from the banks that choose a fraction σ

of risky projects. Let KRR,t+1 =

µt0

νt0Kji,t+1didj be the total capital allocated to the

163

safe technology and nanced by borrowing from the banks that choose a fraction σ

of risky projects. Thus,

KR,t+1 = KSR,t+1 +KR

R,t+1,

where KR,t+1 is the total capital allocated to the risky technology. The same upper

and lower case notation applies to labor, i. e. HSS,t+1 =

1

µt

1

νtHji,t+1didj; H

RS,t+1 =

µt0

1

νtHji,t+1didj; H

SR,t+1 =

1

µt

νt0Hji,t+1didj; H

RR,t+1 =

µt0

νt0Hji,t+1didj.

Safe representative rm:

Y St =

1

0

1

νt−1

F(Kji,t, H

ji,t

)didj =

1

0

1

νt−1

At(Kji,t

)α (Hji,t

)1−αdidj =

1

0

1

νt−1

At

[FKj

i,t

(Kji,t, H

ji,t

)Kji,t + FHj

i,t

(Kji,t, H

ji,t

)Hji,t

]didj =

1

0

1

νt−1

At

[fKj

i,t

(Kji,t

Hji,t

)Kji,t + fHj

i,t

(Kji,t

Hji,t

)Hji,t

]didj =

1

0

1

νt−1

At

[fKt

(Kt

Ht

)Kji,t + fHt

(Kt

Ht

)Hji,t

]didj =

µt−1

0

1

νt−1

At

[fKt

(Kt

Ht

)Kji,t + fHt

(Kt

Ht

)Hji,t

]didj+

1

µt−1

1

νt−1

At

[fKt

(Kt

Ht

)Kji,t + fHt

(Kt

Ht

)Hji,t

]didj =

At

[fKt

(Kt

Ht

)KRS,t + fHt

(Kt

Ht

)HRS,t

]+ At

[fKt

(Kt

Ht

)KSS,t + fHt

(Kt

Ht

)HSS,t

]=

At

[fKt

(Kt

Ht

)(KRS,t +KS

S,t

)+ fHt

(Kt

Ht

)(HRS,t +HS

S,t

)]=

At

[fKt

(Kt

Ht

)KS,t + fHt

(Kt

Ht

)HS,t

]= At (KS,t)

α (HS,t)1−α ,

where KS,t = 1

µt−1(1− σ) lj,t−1dj +

µt−1

0(1− σ) lj,t−1dj.

164

Risky representative rm:

Y Rt =

1

0

νt−1

0

[F(Kji,t, H

ji,t

)+ εi,tK

ji,t

]didj = At (KR,t)

α (HR,t)1−α +

1

0

νt−1

0

εi,tKji,tdidj = At (KR,t)

α (HR,t)1−α+(−ξ)KR,t = At (KR,t)

α (HR,t)1−α−ξKR,t,

where KR,t = 1

µt−1σlj,t−1dj +

µt−1

0σlj,t−1dj.

B.7 The Government

The government levies the tax to fully compensate for the loss to the deposit

insurance fund due to rescue of defaulted banks.

Tt = −ε∗t

−∞

((Rlt +

σt−1εtQt−1

)lt−1 −Rd

t−1dt−1

)dG(εt) =

σt−1lt−1

Qt−1

τ√2πe−(Rdt−1(1−γt−1)Qt−1−R

ltQt−1+ξσt−1

σt−1√

2τ

)2

−

1

2

(Rltlt−1 −

σt−1ξ

Qt−1

lt−1 −Rdt−1dt−1

)[1 + erf

(Rdt−1 (1− γt−1)Qt−1 −Rl

tQt−1 + ξσt−1

σt−1

√2τ

)].

B.8 Household

To express x?t , substitute ξ = 0 into the expression derived in Appendix B.2. Thus,

x?t =1

2

[1 + erf

((Rlt −RdS

t

)Qt−1

σS√

2τS+

RdSt Qt−1

σSφSt−1

√2τS

)],

When a shadow bank defaults, it liquidates its assets to partially reimburse

depositors. In such situation, the deposit fund gets

ΠSt = (1− x?t )RdS

t dSt +

ε∗t

−∞

((Rlt + σSεt −RdS

t

)lSt−1 +RdS

t eSt−1

)dG(εt).

165

It comprises the earnings that the fund would get if the defaulted deposits would

pay in full plus the loss it makes due to non-repayment of the full deposit rate.

Rewriting it as follows:

ΠSt = (1− x?t )RdS

t dSt +

(RltlSt−1 −RdS

t dSt

)−∞

ε∗t

((Rlt + σSεt −RdS

t

)lSt−1 +RdS

t eSt−1

)dG(εt)

=

(1− x?t )RdSt d

St +Rl

tlSt−1 −RdS

t dSt − lSt−1σ

S τS

Qt−1

√2πe−(

(Rlt−RdSt )Qt−1

σS√

2τS+

RdSt Qt−1

φSt−1σS√

2τS

)2

−

(RltlSt−1 −RdS

t dSt

)x?t = Rl

tlSt−1 (1− x?t )− lSt−1σ

S τS

Qt−1

√2πe−(

(Rlt−RdSt )Qt−1

σS√

2τS+

RdSt Qt−1

φSt−1σS√

2τS

)2

.

B.9 Calibration

To calibrate the variance of the idiosyncratic shock τ 2C , we link the production

function of the risky rm to the production function of the safe rm that has a

preexisting debt. Remember that the next period returns to safe and risky loans are

given by

Rlt+1 =

αAt+1

Qt

(Kt+1

Ht+1

)α−1

+ (1− δ)Qt+1

Qt

,

Rlrt+1 = Rl

t+1 + σRFεt+1

Qt

,

respectively. The parameter σRF captures the exposure to the idiosyncratic shock.

The risky bank that nances the maximum share of risky projects earns

Ωriskyt+1 = Rlr

t+1QtKrt+1.

It comprises EBITDA and what the bank makes or loses by selling the capital to

capital producers. The safe bank with preexisting debt earns

Ωsafet+1 = Rl

t+1Qt (Kt+1 +Bt)−QtBtRBt =

(Rlt+1

(1 +

Bt

Kt+1

)− Bt

Kt+1

RBt

)QtKt+1,

166

where Bt is a predetermined debt, measured in units of capital, and RBt is a

predetermined interest rate. We equate the conditional variances of the returns

to loans

V art(Rlrt+1

)= V art

(Rlt+1

(1 +

Bt

Kt+1

)− Bt

Kt+1

RBt

)to nd the variance of the idiosyncratic shock that matches Debt

EBITDA= 7. Note that

V art(Rlrt+1

)= V art

(Rlt+1

)+

(σRFQt

)2

τ 2C ,

V art

(Rlt+1

(1 +

Bt

Kt+1

)− Bt

Kt+1

RBt

)=

(1 +

Bt

Kt+1

)2

V art(Rlt+1

),

where Kt+1 is the steady-state level of capital of the safe rms that are nanced by

commercial banks and Qt = 1.

The conditional variance of the returns on safe loans is given by

V art(Rlt+1

)= α2

(Kt+1

Ht+1

)2α−2

V art (At+1) + (1− δ)2V art (Qt+1) +

2α

(Kt+1

Ht+1

)α−1

(1− δ)Covt (At+1, Qt+1) .

There is a way to calculate the conditional variance of Qt+1 by picking up its

process from the optimization problem of capital producers. However, our approach

is meant to be suggestive, and we consider that the conditional variance of Qt+1 is

the same as the conditional variance of the aggregate shock. The covariance term is

expected to be positive but we drop it in our calculations because the terms that

multiply it are small. The model's counterpart for EBITDA is a total output net of

compensation for labor. Thus

DebtEBITDA

=Bt

Y safet −WtH

safet

=Bt

αY safet

.

The data analog of σRF is the share of leveraged loans held by commercial banks.

We calibrate σRF to 45% that comes from the Shared National Credit Report issued

by the Fed, OCC, and FDIC.

167

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