Welfare and Optimal Bank Capital Structure: AMacro-finance Approach

Paul Luk∗

Hong Kong Baptist University

December 1, 2017

Abstract

Commercial banks issue deposits and wholesale debt to finance their assets. Thispaper studies the optimal bank capital structure and computes the welfare cost ofdeviating from it. I present a model in which banks have risk-taking incentives,deposits are senior debt, and a regulator monitors and liquidates banks when thisbenefits depositors. In the optimal contract banks accept deposits up to its liqui-dation value, because this induces the regulator to monitor banks and minimizesbank risk-taking. I embed the optimal financial contract in a standard real businesscycle model. Using US data, the welfare cost of bank risk-taking is equivalent toa permanent loss in consumption of around 2.7% under the optimal bank capitalstructure. If banks can only issue deposits, the loss increases by around 0.9%.

Keywords: Bank capital structure; welfare; risk-taking; bank regulation.

1 Introduction

One major shift in US commercial bank’s capital structure over the last three decades is

an increasing reliance on wholesale borrowing and a decreasing reliance on core deposits

(Feldman and Schmidt, 2001). Figure 1 shows the US commercial bank’s liability struc-

ture using Federal Reserve flow of fund data. It shows that the deposit to asset ratio

declined between 1980 up to the global financial crisis in 2008, and has recovered since

then. Deposits refer to checking accounts, saving accounts, monetary market deposit

accounts and small time deposits. Banks also use wholesale borrowing, which includes

large time deposits (jumbo deposits that are not covered by depository insurance) by

∗Mailing address: Department of Economics, Hong Kong Baptist University, WLB 530, The WingLung Bank Building for Business Studies, 34 Renfrew Road, Kowloon Tong, Kowloon, Hong Kong, China.Phone number: +852 3411 5512. Corresponding email: paulskluk@hkbu.edu.hk.

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1980 1985 1990 1995 2000 2005 2010 20150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1US commercial bank capital structure

depositwholesale borrowingbank equity

Figure 1: US Commercial bank’s liability structure (FRED Flow of fund data)

households and non-financial corporations, issuance of other debt securities, interbank

liabilities for short-term liquidity management and central bank liquidity.

Given these variations in bank capital structure over time, a natural question is if

there is an optimal capital structure for commercial banks. If so, how should the optimal

capital structure vary over the business cycle? Moreover, what are the costs of deviating

from the optimal capital structure?

Of course, in a world without financial frictions and other institutional restrictions,

Modigliani and Miller (1958)’s theorem applies, and the mix of deposits and wholesale

borrowing does not matter. But banks are prone to information asymmetry and moral

hazard. In this paper, I focus on a risk-shifting problem in which a bank may invest in

a safe project or a risky and socially-inefficient project and shift the risks to the lenders.

Without proper monitoring, banks cannot credibly commit to investing in safe projects,

lenders do not lend and the financial market shuts down.

There are several institutions in reality to protect depositors from risk-shifting be-

havior on banks’ part. First, depositors are senior creditors in many major economies

including the US (Lenihan, 2012; Wood, 2011). These rules are called depositor prefer-

2

ence rules (henceforth DPR). In the US, for instance, when a bank liquidates, deposit

claims rank behind secured claims and other preferred claims, but, among unsecured

creditors, ahead of tax and employee compensation. Although DPR is less common in

EU countries, Greece, Portugal, Hungary, Latvia and Romania have introduced depositor

preference regimes. The Vickers report (Edmonds, 2013) in the UK also advocates the

introduction of DPR.

Second, most countries have regulatory bodies which conduct costly monitoring to

ensure the stability of the banking sector and protect the depositors. Regulators have

legal rights to punish or even force banks to shut down when they do not comply with

regulatory requirements.1 However, banking crises still happen. Calomiris (1999) argues

that regulators and politicians who control banking safety nets have little incentive to

monitor and enforce prudential guidelines. One interpretation, as given by Kane (1989,

1992, 1993) is that financial institutions may bribe or cajole regulators when the stakes

are high. There were numerous anecdotal stories of such acts during the Global Financial

Crisis in 2008, and some banks were fined for the misconduct. Another possible reason

is that shutting banks down involves large liquidation costs which may hurt the credi-

tors such as depositors and disrupt financial intermediation. Given these considerations,

Calomiris (1999) argues that the design of banking regulations has to take into account

the political viability of the regulations.

This paper models banking frictions and the institutional features described above as

a perfect Bayesian game between a bank and a regulator. In this environment, I derive

the optimal bank capital structure which induces the regulator to monitor responsibly so

that bank risk-taking is minimized. The optimal capital structure is one in which deposits

equal the liquidation value of the bank; wholesale lenders finance the rest. The intuition

is similar to Park (2001)’s model of debt seniority. Consider a bank which has $10 worth

of equity and $100 assets, and the bank’s liquidation value is $70. For simplicity, assume

the interest rate is zero. Suppose costly monitoring by the regulator reveals that the bank

engages in risk-taking and return is only $80. If deposit is $60, then the regulator simply

liquidates the bank because the liquidation value is higher than the value of deposits, and

DPR ensures that depositors are repaid in full. But this means that the regulator has

no incentive to monitor because the liquidation decision does not depend on monitoring.

And the unprotected wholesale lenders have no incentive to lend. Alternatively, if the

1In the US, a bank’s chartering authority, which is either the bank’s state banking department or theUS Office of the Comptroller of the Currency closes a bank.

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bank takes only deposits so deposits are $90, the regulator is reluctant to liquidate the

bank (and recovers $70) because this hurts depositors more than keeping the bank alive

(deposits get back $80). Again, the regulator has little incentive to monitor banks. When

deposits are $70, and wholesale borrowing is $20, the regulator has the strongest incentive

to monitor and liquidate the banks prudently. This capital structure reduces risk-taking

by banks.

I embed the banking sector into a standard real business cycle model. The general

equilibrium environment is a tractable and coherent framework to compare welfare across

different financial regulatory regimes. Under my calibration with US data, the presence of

bank risk-taking incentives in the steady state has a welfare cost equivalent to 2.7% lower

permanent consumption. If wholesale lending in the banking sector is completely abol-

ished, then in steady state there will be more risk-taking and the regulator will have to

perform more costly monitoring, leading to a further reduction of consumption by about

0.9% permanently. Moreover, my model suggests that recent introduction of complex

financial instruments and financial deregulation might have induced more risk-taking and

increased welfare loss.

The model sheds light on financial regulation policies over the business cycle. I find

that the optimal deposit to bank asset ratio does not vary over the business cycle. More-

over, the regulator should increase monitoring effort strongly in a downturn. A negative

shock reduces a bank’s net worth and increases its expected return on capital as well as

bank leverage. In this case, banks have stronger incentives to take on more risks. There-

fore, the regulator should step up monitoring effort to deter excessive risk-taking in the

banking sector.

There is a large literature in finance which studies the moral hazard problem of banks.

For instance, Calomiris and Kahn (1991) argue that demandable deposits are optimal

because better informed depositors can withdraw deposits, thus enforcing market disci-

pline. Rochet and Tirole (1996) suggest that interbank loans provide the lending bank

with an incentive to monitor other banks. Given the intrinsic moral hazard problem of

banks, a large literature studies prudential regulations for the banking sector. Hellmann

et al. (2000) argue that capital-requirement regulation alone is not optimal unless it is

coupled with deposit-rate controls. Van den Heuvel (2008) embeds bank capital require-

ments in a general equilibrium framework and finds that capital requirements have large

welfare costs. Allen et al. (2015) derive the optimal bank capital structure in a model

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in which deposit and equity markets are segmented and liquidation costs exist for banks.

The issue of bank borrowing using deposit versus wholesale borrowing remains relatively

unexplored. Park (2001) studies the optimal debt structure with senior and junior debt

in a corporate finance model. My model is an adaptation of Park (2001)’s result into a

model of the banking sector.

This paper is related to a growing literature which studies the role of financial in-

stitutions in the macroeconomy in a DSGE framework. This literature emphasizes the

balance sheet channels of the financial institutions. Gertler and Karadi (2011), Gertler

and Kiyotaki (2015), Hirakata et al. (2017) and Luk (2015) show that high leverage in

the financial sector results in large amplifications of shocks in the business cycle through

a financial accelerator mechanism. Boissay et al. (2016) and Gertler and Kiyotaki (2010)

study the banking sector with interbank markets. An interbank market exists in Boissay

et al. (2016) because banks have different intermediation efficiencies; it exists in Gertler

and Kiyotaki (2010) because the interbank market opens after investment opportunities

available to each bank is realized. In this paper, I consider bank borrowing using deposits

and wholesale funding, which differ because deposits are senior and there is a depositor

preference rule. Meh and Moran (2010) consider a moral hazard–hidden action problem

similar to this paper. But they focus on the interactions amongst depositors, financial

intermediaries and firms, without explicitly considering bank capital structures. Similar

to this paper, Nuno and Thomas (2017) study bank risk-taking incentive in a Bernanke,

Gertler and Gilchrist (1999) costly-state-verification framework. However, they assume

that depositors restrict lending in equilibrium so banks do not engage in socially-inefficient

risk-taking activities in equilibrium. By contrast, in my model banks do take risk in equi-

librium because bank monitoring is also imperfect, and I consider the incentive problems

associated with bank monitoring explicitly.

The rest of the paper is organized as follows. Section 2 explains in detail the credit

frictions and institutional structure faced by the banking sector and solves for the optimal

contract with deposits and wholesale borrowing. Section 3 embeds the banking sector into

a real business cycle model. Section 4 discusses model calibration. In Section 5 I study

how bank capital structure affects risk-taking incentives and social welfare. Section 6

compares the dynamic behavior of my model with alternative models and discusses how

optimal bank monitoring effort should vary over the business cycle. Section 7 concludes.

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2 Model – banks

This section discusses the partial equilibrium in the banking sector, which is the novel

part of the model. The rest of the model is described in the next section.

Banks borrow from investors by means of one-period debt contracts and invest in

goods-producing firms. This paper focus on financial frictions faced by the banking sector.

For simplicity I assume there is no friction between a bank and a firm. This assumption is

common in models with banking frictions (such as Gertler and Karadi (2011) (henceforth

GK) and Gertler and Kiyotaki (2010)).

There is a continuum of banks i ∈ [0, 1]. In the beginning of a period, bank i has

net worth Nit. It borrows from perfectly competitive investors. There are two type of

debt: bank deposits and wholesale borrowing. Bank i borrows Bdit unit of deposits, Bw

it

unit of wholesale borrowing and use the proceed to purchase capital Kit at price Qt for

production. Bank i’s balance sheet is given by:

Nit +Bdit +Bw

it = QtKit. (1)

When a bank borrows, it signs contracts with depositors and wholesale lenders. A

deposit contract specifies a non-state-contingent deposit rate Rdit; a wholesale borrowing

contract specifies a non-state-contingent wholesale interest rate Rwit. According to DPR,

in the debt repayment or liquidation phase, a bank always repays depositors first, and

then wholesale lenders. If there is capital left behind, the bank retains it as net worth for

the next period.2 Suppose a bank has Xit+1 in the repayment or liquidation phase, the

priority structure can be summarized as follows:

Depositors’ share : minXit+1, RditB

dit,

Wholesale lender’s share : minmaxXit+1 −RditB

dit, 0, Rw

itBwit,

Bank’s share : minXit+1 −RditB

dit −Rw

itBwit , 0. (2)

Banks are subject to two types of financial frictions. First, a bank may choose a risky

business strategy and shift the risk to the lenders. Second, as in GK (2011) a bank may

divert funds. The following two subsections discuss these frictions.

2I abstract from outside equity injection in this model, except for new banks.

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2.1 Friction due to bank risk-taking incentives

I first describe the financial frictions due to bank risk-taking. A bank faces a shock to

capital quality, ωit+1, so the actual return on capital is ωit+1RKt+1, where RK

t+1 is exogenous

to the bank. A bank’s strategy can influence ωit+1. In particular, a bank has access to a

safe project and a risky project. A safe project always yields ωit+1 = 1. In a risky project

ωit+1 is a random variable given by:

ωit+1 =

ωH > 1 Pr = π,

0 Pr = 1− π.(3)

The random variable ωit+1 is independent across banks and time and is uncorrelated with

aggregate shocks. Assume ωH > 1, and πωH < 1, so risky projects are socially inefficient.

I assume that only the success or failure of the bank investment is verifiable, and so the

most a lender can get from the bank is Et(RKt+1)QtKit. A bank chooses a mixed business

strategy pit ∈ [0, 1] in period t which is the probability of a project being safe. The project

type is not known when it is chosen; the type and ωit+1 realizes in the beginning of period

t+ 1.

To prevent the bank from taking excessive risks, depositors delegate the regulatory

task to a bank regulator who conducts costly monitoring after contracts are signed in pe-

riod t. The regulator chooses a monitoring effort λit ∈ [0, 1] without knowing the bank’s

choice of pit. In period t + 1, the bank’s strategy pit becomes public information. Moni-

toring may yield additional information: with probability λit the regulator observes the

type of project (safe or risky) in the beginning of period t+ 1; with probability (1− λit)no information is revealed, and the regulator can only observe pit. The regulator’s infor-

mation status is common knowledge.

In addition to specifying a contractual interest rate Rdit, a deposit contract can be

contingent on the publicly observable pit. In particular, I consider contracts that specify

a covenant pit ≥ pcit. If a bank chooses a strategy pit < pcit which violates the covenant,

the regulator has the right to liquidate the bank in period t + 1. However, liquidation is

costly. The regulator only liquidates a bank when it is in depositors’ interests to do so.

For this reason, a bank may choose a strategy pit < pcit if it expects the covenant can be

7

Figure 2: Timeline of credit contract events

waived through renegotiation.34

For simplicity, I assume that loan repayments, bank monitoring costs and liquidation

payments are settled in period t + 1. If a bank is liquidated, the liquidation value in

t+ 1 is given by (1−µ)RtQtKit, where µ represents linear liquidation costs, and Rt is the

risk-free rate in the economy. Monitoring costs are increasing in monitoring effort, given

by λitΨmRtQtKit.

I assume that a bank’s borrowing is subject to an additional type of financial friction

as in GK (2011), in which a bank may divert funds. The details of this friction will be

discussed in the next subsection, but it is useful to state that given this friction, all banks

will choose the same leverage φt ≡ QtKit/Nit.

Figure 2 shows the events in the banking sector. I consider a perfect Bayesian equi-

librium as follows. In stage 1, contracts (Bdit, R

dit, p

cit) and (Bw

it , Rwit) are signed. Each

3Of course, wholesale lenders can sign contracts with covenants with the banks. But later on I willshow that in equilibrium wholesale lenders have no incentive to liquidate the banks, and therefore I donot consider wholesale lending contracts with covenants.

4In practice, deposit insurance has a coverage limit, so deposits are not risk-free. For instance, thedeposit insurance in the US covers only small time deposits with a face value less than USD 250,000.

8

contract generates a sequential game in stage 2 in which the strategic pair (pit, λit) are

chosen by bank i and the regulator. Expected equilibrium payoffs for the bank, depos-

itors and wholesale lenders are determined, as functions of (pit, λit). In the last stage,

the project type is realized, pit and monitoring information are observed. The regulator

decides to do nothing (waive the covenant), liquidate, or renegotiate with the bank.5 An

optimal contract is one for which there is a sequential equilibrium that maximizes the

bank’s expected payoff, and ensures that depositors and wholesale lenders yield at least

Rt in expected returns.

I state three assumptions regarding the macro-variables and parameter values:

Assumption 1 For any period t,

Et(RKt+1)

Rt

>φt − 1

φt> (1− µ) > πωH

Et(RKt+1)

Rt

. (4)

I assume that the return on a safe project is higher than the opportunity cost of lending

(product of the risk-free rate and the loan amount), which is in turn bigger than the

liquidation value of the bank, which is bigger than the expected return of a risky project.

The last inequality implies that if monitoring reveals that a bank has a risky project,

lenders always prefer liquidating the bank.

Assumption 2 For any period t,

Et(RKt+1)

Rt

− φt − 1

φt< π

[ωH

Et(RKt+1)

Rt

− φt − 1

φt

]. (5)

This assumption states that if a bank is unregulated it has an incentive to choose a risky

strategy. A bank can shift its risks to its borrowers to make the expected return to the

bank of a risky project higher than that of a safe project.

Assumption 3 For any period t,

(1− pt)[(1− µ)− πωH

Et(RKt+1)

Rt

]> Ψm, (6)

where pt satisfies [pt + (1− pt)π]Et(RKt+1)/Rt = (φt − 1)/φt.

5Renegotiation may take place when a bank fulfills the covenant pit ≥ pcit but monitoring reveals thatit has a risky project. As liquidation return is higher than the return of a risky project (Assumption 1),the regulator can renegotiate with the bank, liquidate it and split the surplus. However, as I will show,Proposition 1 implies that renegotiation never occurs on the equilibrium path.

9

This assumption requires that monitoring costs Ψm are not too large so that monitoring

provides returns high enough to cover the costs. If a bank is expected to use a strategy

pit < pt, no lender is willing to lend because the lender cannot break even. If pit = pt, the

left-hand side of (6) is the marginal benefit of monitoring – a risky project is identified

and liquidated, and the benefit has to be bigger than the marginal cost of monitoring. If

(6) is violated, lenders will not monitor for pit ≥ pt.

In the following, I work with quantities as a ratio of total capital intermediated. I

rescale bank debts as follows:

bdit ≡Bdit

QtKit

, bwit ≡Bwit

QtKit

.

The game is solved by backward induction. I state Park (2001)’s result that in equi-

librium a bank always chooses pit ∈ [pit, pcit), where pit satisfies:6

[pit + (1− pit)π]Rditb

dit = (1− µ)Rt. (7)

In the last stage, the regulator’s action depends on monitoring information, which may

reveal the project is safe, or risky, or may reveal no additional information. If a safe

project is revealed, the regulator has no incentive to liquidate the bank. If no additional

information is revealed so the regulator only observes pit, condition (7) ensures that the

expected return of deposits exceeds the liquidation value. Again, the regulator has no

incentive to liquidate the bank. If monitoring reveals a risky project, the bank is liqui-

dated. Note that, since Rditb

dit ≥ [pit + (1− pit)π]Rd

itbdit = (1− µ)Rt, wholesale lenders are

wiped out according to DPR (2).

Given the choices (bdit, Rdit, p

cit) and (bwit, R

wit) and the regulator’s action in stage 3, one

can write down the expected profit of the bank EωΠi(pit, λit), depositors EωΠd(pit, λit) and

wholesale lenders EωΠw(pit, λit) in stage 2 where the expectation Eω(.) is taken conditional

6A complete proof of optimality requires one to check other bank strategies pit /∈ [pit, pcit) as well. The

proof is in Park (2001) and is not repeated here.

10

on the distribution of ωit+1. These expected profits are given by:

EωΠi(pit, λit) =

pit[Et(R

Kt+1)−Rd

itbdit −Rw

itbwit]

+(1− λit)(1− pit)π[ωHEt(RKt+1)−Rd

itbdit −Rw

itbwit]

, (8)

EωΠd(pit, λit) =

[pit + (1− pit)π]Rd

itbdit

+λit(1− pit)[(1− µ)Rt − πRditb

dit]−ΨmRt

, (9)

EωΠw(pit, λit) = [pit + (1− λit)(1− pit)π]Rwitb

wit. (10)

The break-even conditions for depositors and wholesale lenders are given by EωΠd(pit, λit) =

Rtbdit and EωΠw(pit, λit) = Rtb

wit respectively.

The following proposition states the solution of the optimal contract:7

Proposition 1 Given Assumptions 1,2 and 3 and the same leverage ratio across banks

φt determined outside the perfect Bayesian game, there exists a unique optimal two-class

contract characterized by the following features:

1. For any bank i and time t, the senior debt contract has a covenant pc ≡ pcit = 1.

2. Banks always violate the covenant by choosing a constant p ≡ pit < 1 where p is the

larger root of the following equation:

(1− p)[1− π

p+ (1− p)π

]=

Ψm

1− µ. (11)

3. The wedge between depositor’s contractual rate and the risk-free rate is a constant,

given by:Rdt

Rt

=1

p+ (1− p)π. (12)

4. The wedge between wholesale financing contractual rate and the risk-free rate is

common across banks, given by:

Rwt

Rt

=1

p+ (1− λt)(1− p)π. (13)

5. For any bank i and time t, the deposits to capital ratio is a constant, given by:

bd ≡ bdit = 1− µ, (14)

7All proofs are given in Appendix B.

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and the wholesale financing to capital ratio is equal across banks, given by:

bwt ≡ bwit =φt − 1

φt− bd. (15)

6. In a given period t, the regulator chooses the same λt = λit for all banks where λt

solves:

(1− λt) =Et(R

Kt+1)−Rd

t bd −Rw

t bwt

π[ωHEt(RK

t+1)−Rdt bd −Rw

t bwt

] . (16)

The regulator always has the strictest covenant. A contract with a weaker covenant

pc < 1 is dominated by pc = 1 because it reduces the regulator’s incentive to monitor.

The optimal choice for (pit, λit) are mutual best responses. Figure 3 shows the best

response functions, pit(λit) for bank i, and λit(pit) for the regulator. Consider the best

response function for the regulator (red line) first. If the bank chooses a low pit, then

monitoring is likely to detect a risky project. By liquidating the risky project depositors’

profit increases by (1 − pit)[(1− µ)Rt − πRd

itbdit

], which exceeds the monitoring effort

ΨmRt. Therefore, the regulator chooses the highest monitoring effort λit = 1. Contrarily,

if the bank chooses a high pit, the benefit of monitoring is smaller than the monitoring

costs, so the regulator chooses a low monitoring effort. There is an intermediate value of

pit such that the regulator is indifferent between different monitoring efforts:

(1− pit)[(1− µ)− πR

dit

Rt

bdit

]= Ψm, (17)

Similarly, the best response of bank i’s business strategy pit (blue line) depends on the

regulator’s monitoring effort. If monitoring is tight (λit is large), then by choosing a risky

business strategy (low pit) the bank gets caught too often and profit is low. In this case,

the bank’s best response is to choose pit = 1. Contrarily, if monitoring is relaxed (λit is

small), a bank chooses the riskiest strategy pit. There is an intermediate value of λit such

that the bank is indifferent between different business strategies:8

(1− λit)π[ωHEt(R

Kt+1)−Rd

t bdit −Rw

t bwit

]= Et(R

Kt+1)−Rd

t bdit −Rw

t bwit. (18)

The intersection of these two conditions pins down the optimal choice for (pit, λit).

8Mathematically, (17) is derived by setting to zero the coefficient of λit in depositors’ profit function(9); and (18) is derived by setting to zero the coefficient of pit in bank i’s profit function (8).

12

Figure 3: Best response functions

Deposit and wholesale interest rates in stage 1 are determined by the participation

constraints of the perfectly competitive depositors and wholesale lenders whose expected

return is the risk-free rate Rt:

Rtbdit = pitR

ditb

dit + (1− pit)πRd

itbdit, (19)

Rtbwit = pitR

witb

wit + (1− λit)(1− pit)πRw

itbwit. (20)

I conjecture that the bank’s strategy is a constant pit ≡ p and verify it subsequently.

Rearranging the participation constraint of depositors, I obtain (12), which implies that

the deposit rate is common across banks.

Equation (14) states that it is optimal for depositors to lend only up to the liquidation

value of the bank, which is the smallest amount that will keep the depositors impaired

by liquidation. The following explains why a smaller or larger amount is not optimal. If

there are less deposits, say bd < (1 − µ), using the equilibrium monitoring strategy the

return on deposit is bdR. If instead the regulator does not monitor and liquidates the

bank, depositors get back min(1−µ)R, bdRd according to DPR (2), but this amount is

bigger than bdR. So, the regulator has no incentive to monitor and always liquidate the

13

bank. However, wholesale lenders will not accept such a contract.

If depositors lend more (bd > 1 − µ), the marginal return of monitoring is reduced

because the stake is larger but the liquidation value remains the same. The regulator has

weaker incentive to liquidate a bank with a risky project or a project with unknown type.

Equation (17) then implies that the bank chooses a riskier strategy and default more often

in equilibrium, which in turn increases the interest rate spreads for bank borrowing, and

is suboptimal.9 This argument explains why an optimal two-class contract with deposits

and wholesale lending dominates a deposits-only debt structure.

Recall that p is determined in (17) by equating depositors’ marginal return and

marginal costs of monitoring. This tradeoff does not change over the business cycle.

In particular, when the regulator liquidates a bank, depositors lose the discounted return

of deposits (Rdt /Rt) × bdt , which is a constant given the conjecture. This confirms that

p is a constant, and is given by the solution of (11). Clearly, p = 1 is not a solution of

(11), so all banks violate the covenant by choosing p < 1. Since there is risk-taking in

equilibrium,∫iωitdi < 1. This is a source of inefficiency in the economy.

Equation (15) pins down the wholesale borrowing to bank capital ratio. As all banks

have the same leverage ratio φt, and the deposit share is a constant, the balance sheet of

the banks implies that all banks have the same wholesale borrowing share too.

Finally, equations (18) and (20) jointly solve for the monitoring effort λit and the

wholesale lending rate Rwit. Since all other variables are the same across banks, λt and

Rwt are the same across banks as well. Comparing (12) and (13), I get Rw

t > Rdt >

Rt. The first inequality holds because when the regulator reveals a risky bank through

monitoring and liquidates it, depositors are protected by receiving the liquidation value

of the bank but wholesale lenders receive nothing. Therefore, wholesale lenders require a

higher contractual rate. The second inequality holds because when a risky project is not

revealed by monitoring, and so the project proceeds depositors may get nothing (with

probability (1− π)), and the wedge between Rdt and Rt compensates for this risk.

9To see why p falls when bd rises, substitute the deposit spread (12) into the marginal return ofmonitoring equation (17) and rearrange to obtain:

f(p, b) = (1− p) (1− µ)[p+ (1− p)π]− πb −Ψm[p+ (1− p)π] = 0.

Here, f is continuous and quadratic in p. Since the equilibrium p is the larger root of f(p, 1 − µ) = 0,fp < 0 around the equilibrium. Also, fb < 0. So, dp

db < 0 around the equilibrium.

14

2.2 Friction due to divertible bank assets

In the following, I introduce a second moral hazard problem following GK (2011), and

show that all banks choose the same leverage ratio. Specifically, after a bank borrows from

the financial market, the bank has an option to divert a fraction γ of funds. If this happens

the bank will shut down. Assume that the decision to divert funds takes place before

monitoring takes place (see points 3 and 4 in Figure 2), so monitoring cannot mitigate

this problem. A bank manager has an incentive to divert funds when the continuing value

of the bank is less than the value of divertible capital. To prevent the bank from diverting

funds, wholesale lenders must restrict their lending so that banks do not divert funds in

equilibrium. Assuming the value of a bank is Vt(Nit). The incentive constraint for the

bank is:

Vt(Nit) ≥ γQtKit. (21)

The value of the bank is given by

Vt(Nit) = maxKit

EtΛt,t+1[(1− θ)Nit+1 + θVt+1(Nit+1)], (22)

where Λt,t+1 is the stochastic discount factor for households. I assume that in each period,

after a bank settles its debt repayment, there is an exogenous probability (1−θ) it exits.10

Following GK (2011), I conjecture Vt(Nit) = ψtNit, where ψt is the marginal value of net

worth common across banks. The above equation can be re-written as:

ψt = maxbwit,φit

Et

[Λt,t+1(1− θ + θψt+1)

EωNit+1

Nit

], (23)

where EωNit+1 = Πi(p, λt)QtKit.

Assuming that the incentive constraint (21) is binding, then all banks choose a common

leverage QtKit/Nit ≡ φt = ψt/γ, and the marginal value of bank net worth is ψt = ξtφt+νt,

according to (23), where:

ξt ≡ Et

Λt,t+1(1− θ + θψt+1)[RKt+1 − (1− bd)Rw

t − bdRdt ], (24)

νt ≡ Et[Λt,t+1(1− θ + θψt+1)]Rwt . (25)

If a bank has an additional unit of net worth it can save the cost of wholesale borrowing

10Following Carlstrom and Fuerst (1997), Bernanke et. al. (1999) and GK (2011), this assumptionprevents banks from growing out of their financial constraints.

15

νt and can increase assets by the leverage ratio φt, and ξt measures the excess marginal

value of assets over borrowing. The marginal value of net worth ψt does not depend on

bank characteristics, which verifies the guess. A binding constraint (21) requires that

γ > ξt, that is the marginal gain from diverting γ is bigger than the marginal value of

operating honestly with an additional unit of capital. Furthermore, we require ψt > 1, or

the franchise value of a bank is bigger than unity so banks prefer operating than exiting.11

3 The rest of the model

I embed the above banking sector into a standard DSGE model. The model has four other

types of agents, namely households, investors, goods producers and capital producers.

3.1 Households

Infinite-lived representative households derive utility from consumption and disutility

from supplying labor. Their preferences are given by:

E0

∞∑t=0

βt(

lnCt − χL1+ϕt

1 + ϕ

). (26)

In each period, a representative household receives wage income, saves to the investors

who guarantee them a risk-free return Rt and receives transfers from firms and banks. A

representative household faces the following budget constraint:

wtLt + Πkt + trt +Rt−1Bt−1 = Ct +Bt. (27)

The intratemporal labor supply conditions are given by the following:

wt = χLϕt Ct. (28)

The consumption Euler equation is given by:

1 = RtEt(Λt,t+1), (29)

where the stochastic discount factor is Λt−1,t = βUC,t/UC,t−1.

11I check that ψt is bigger than unity in the neighborhood of the steady state.

16

3.2 Investors

The representative investor collects savings from households and lend to the banks. They

invest in both bank deposits and wholesale lending to banks. Both require a risk-free

return Rt in every state of the world. Investors make sure households hold a diversified

loan portfolio across the banks.

3.3 Firms

There is a unit measure of firms i ∈ [0, 1], each owned by a bank. A firm produces with

the following Cobb-Douglas production function:

Yit = AtKαitL

1−αit ,

where At denotes the aggregate productivity of the firm sector, and Kit effective capital.

The aggregate productivity shock follows an AR(1) process:

logAt = ρA logAt−1 + εAt, εAt ∼ N(0, σ2A). (30)

After bank-firm i purchases its capital Kit−1, it faces an idiosyncratic productivity

shock which turns the firm’s capital into effective capital according to Kit ≡ ωitKit−1.

The firm then hires labor, produces, and sells undepreciated capital to capital producing

firms. The marginal product of capital rKt is defined as:

rKt Kit ≡ maxLit

Yit − wtLit (31)

The optimal choice of labor requires wtLit = (1 − α)Yit, and this implies that all firms

have the same labor to output ratio.12 The return on capital is given by:

[rKt + (1− δ)Qt]Kit

Qt−1Kit−1= ωit

[rKt + (1− δ)Qt

Qt−1

]≡ ωitR

Kt . (32)

3.4 Capital goods producers

A representative capital good producer buys previously installed capital and combines

with investment good It from final goods producers to produce new capital. Newly pro-

12rKt is given by:

rKt ≡ αAt

[(1− α)At

wt

] 1−αα

.

17

duced capital is sold back to the firms within the same period. Production of new capital

is subject to convex investment adjustment costs. The evolution of capital is given by:

Kt = (1− δ)Kt + (1− Adjt)It, (33)

where Adjt = 0.5ΨI (It/It−1 − 1)2 are investment adjustment costs. Capital goods pro-

ducers maximize discounted sum of expected future profit, Et∑∞

s=0 Λt,t+sΠKt+s, where

ΠKt = Qt[Kt− (1− δ)Kt]− It. The first order condition for the optimal investment choice

is:

1 = Qt

[1− Adjt −ΨI It

It−1

(ItIt−1− 1

)]+ Et

[Λt,t+1Qt+1Ψ

I

(It+1

It

)2(It+1

It− 1

)].(34)

3.5 Aggregation and accumulation of net worth

Since all banks have the same capital structure and leverage ratio, and all firms have the

same capital to labor ratio, I only need to keep track of the sector-level quantities. For

variables Z ∈ Y,K, K, L,N,Bd, Bw, I define Zt ≡∫iZitdi.

Total capital in the economy is given by Kt =∫ 1

0Kitdi. The total leverage is:

φt = QtKt/Nt. (35)

The net worth of existing banks evolves as follow:

Nt = [p+ (1− λt−1)(1− p)πωH ]RKt Qt−1Kt−1

−[p+ (1− λt−1)(1− p)π](Rdt−1B

dt−1 +Rw

t−1Bwt−1). (36)

The first term on the right hand side is the aggregate revenue of the banking sector and

the second term the repayment to depositors and wholesale lenders if the bank is not

liquidated. With an exogenous probability (1 − θ) a bank quits at the end of a period.

New entrants receive a small startup fund τNt−1 from households to kick-start their banks.

So, the aggregate bank net worth evolves as follows:

Nt = θ

[p+ (1− λt−1)(1− p)πωH ]RK

t Qt−1Kt−1

−[p+ (1− λt−1)(1− p)π](Rdt−1B

dt−1 +Rw

t−1Bwt−1)

+ τNt−1. (37)

The goods market clearing condition states that output equals the sum of consumption

18

and investment:

Yt = Ct + It. (38)

The aggregate production function is given by:

Yt = AtKαt L

1−αt , (39)

where aggregate effective capital is:

Kt =

∫i

ωitKit−1di = [p+ (1− λt−1)(1− p)πωH ]Kt−1 (40)

The credit market clearing condition is

Bt = Bwt +Bd

t . (41)

This completes the description of the model.

4 Calibration

In the following I study the model numerically and this section discusses the calibrations.

Each period is a year. Parameters in production and household sectors are relatively

standard in the macroeconomic literature. These are given in Table 1. I set β = 0.96,

which generates a steady state annualized interest rate around 4%. I set ΨL = 5, so

that households devote 41 percent of their time to work in the steady state. The Frisch

elasticity of labor supply is set to χ = 1. For production, the capital share is set to

α = 0.36, and the depreciation rate to δ = 0.1. The curvature of investment adjustment

costs ΨI is set to 2.5. These parameter values are within acceptable range in the literature.

Parameters related to the financial market are calibrated as follows. I use θ = 0.9,

which implies that an average horizon of a bank is a decade. As for bank liquidation

costs, James (1991) finds costs of 30% of total assets. More recently, data from the US

Federal Deposit Insurance Corporation (FDIC) suggests that the liquidation costs are

around 25% of bank assets (Mason, 2005; Bennett and Unal, 2014). I set µ = 0.25. The

remaining parameters are calibrated to jointly match the following five targets: (i) The

steady-state bank leverage ratio is φ = 8. This is lower than a commercial bank leverage

19

Table 1: Calibrated parameters.Parameter Value Meaning

β 0.96 Subjective discount factorα 0.36 Capital share in productionδ 0.1 Capital depreciation rate

ΨL 5 Labor disutilityϕ 1 Inverse of Frisch labor elasticityΨI 2.5 Convexity of investment adjustment costsθ 0.9 Bank survival probabilityµ 0.25 Bank liquidation costsπ 0.60 Risky project success probabilityωH 1.12 Idiosyncratic productivity in risky projectΨm 0.0042 Bank monitoring costsγ 0.2 Fraction of bank capital that can be divertedτ 0.026 Initial transfer to banksρA 0.85 Persistence of productivity shocksA 0.01 Std. dev of productivity shock innovation

of 10 observed in data (See Figure 1), but non-financial firms have leverage close to two in

the aggregate. For similar models, GK (2011) target a leverage of 4 and Gertler, Kiyotaki

and Prestipino (2015) and Gertler and Kiyotaki (2015) target 10. (ii) I target an annual

spread between the expected return on bank assets and the risk-free rate of 100 basis

points, as in GK (2011) and Nuno and Thomas (2017). (iii) Rd/R = 1.0057; and (iv)

Rw/R = 1.0068. These are the average spread between the deposit rate and the risk-free

rate and the average spread between the wholesale rate and the risk-free rate, obtained

using monthly interest rate data from St. Louis Fed between 1986M1-2008M12.13 The

deposit rate spread is proxied by the difference between the 3-month certificate of deposit

rate and 3-month treasury bill rate. The wholesale rate spread is proxied by the differ-

ence between the 3-month interbank rate and 3-month treasury bill rate.14 (iv) I set the

success probability of a risky project to π = 0.6. I conduct sensitive analysis and show

that the main results are robust to alternative parameters.

These targets identify ωH ,Ψm, γ, p and the steady state value of λt uniquely. I obtain

ωH = 1.12. It is straightforward to verify that Assumptions 1-3 hold in the steady state.

With the optimal contract, the banks’ equilibrium business strategy is p = 0.986, or 1.4%

13I do not use subsequent data because starting from 2009 US interest rates are constrained by thezero lower bound.

14I cannot use large time deposit interest rate data, which is available only after 2009.

20

of banks have a risky project. The fraction of divertible banks capital is γ = 20%, close

to the calibrated value in Gertler and Kiyotaki (2015) (who use 19%).

Finally, in the dynamic simulation of the model, I set ρA = 0.85, σA = 0.01.

5 Steady-state analysis

An advantage of this general equilibrium model is that it provides a coherent framework to

conduct welfare analysis. In this section I study the steady-state welfare under alternative

assumptions. I first compare my model with a model in which banks do not engage in

risk-taking behavior and a model in which banks can only borrow by issuing deposits.

Next, I study how bank deregulation and innovations in complex financial instruments

prior to the 2008 global financial crisis may affect risk-taking behavior and welfare. To

make the comparison fair, in the numerical exercises I keep all parameters the same unless

otherwise specified. Alternative models are presented in full in Appendix A.

5.1 Model comparison

To see how risk-taking affects the banking sector in my model, it is instructive to compare

with GK (2011). In GK (2011) all projects are safe (ωit ≡ 1), so the risk-taking channel

is shut down. The regulator need not monitor. This makes the GK (2011) system more

efficient than the benchmark model.15 Table 2 shows the steady-state value of selected

variables (percentage change relative to the benchmark model in brackets). Comparing

the second and third column, we see that economic activity in GK (2011) model is higher.

Households enjoy 2.4% more steady-state consumption and supply 0.35% less labor than

in the benchmark model. Without any risk of bank default, the deposit rate is simply the

risk-free interest rate, 1/β.

I compute the welfare gain/loss in terms of the change in permanent consumption

relative to the benchmark model, x, where x solves:

1

1− β

ln[Cbm(1− x)]− χ(Lbm)1+ϕ

1 + ϕ

= Welfalt, (42)

15An alternative interpretation is that the regulator can commit to time-inconsistent policy of liqui-dating every bank which violates the covenant. In this case, all banks choose p = 1 and no bank isliquidated.

21

Table 2: Comparison of steady statesVariable SS in model SS in Gertler- SS in deposits-

benchmark model Karadi (2011) model only model(1− p)(%) 1.42 0 1.91 (34.61)λ 0.128 – 0.128 (0.52)Rd(%) 4.760 4.167 (-12.46) 4.967 (4.35)Y 0.671 0.682 (1.66) 0.668 (-0.47)K 1.598 1.674 (4.74) 1.576 (-1.35)C 0.503 0.515 (2.40) 0.499 (-0.75)L 0.413 0.412 (-0.35) 0.414 (0.14)Welfare-eq. cons, x (%) – -2.71 0.87

where a variable with superscript bm denotes the value in the benchmark model, and

alt denotes an alternative model. I report x in the last row of Table 2. With optimal

bank capital structure and optimal monitoring, the steady-state welfare loss due to bank

risk-taking amounts to 2.71% of permanent consumption.

To see why the benchmark model has the optimal bank capital structure, I compare

the benchmark model with one in which banks can only borrow by issuing deposits. I call

this economy a deposits-only economy. In this economy the game between a bank and

lenders is the same except that in stage 1 of the game the bank does not borrow from

wholesale lenders. Hence, in this economy the expected profits of bank i and of depositors

for pit ∈ [pit, pcit) are given by:

EωΠi(pit, λit) =

pit[Et(R

Kt+1)−Rd

itbdit]

+(1− λit)(1− pit)π[ωHEt(RKt+1)−Rd

itbdit]

, (43)

EωΠd(pit, λit) =

[pit + (1− pit)π]Rd

itbdit

+λit(1− pit)[(1− µ)Rt − πbditRdit]−ΨmRt

. (44)

The following proposition presents the solution of the optimal contract in the deposits-

only economy.

Proposition 2 Given Assumptions 1,2 and 3 and the same leverage ratio across banks φt

determined outside the perfect Bayesian game, there exists a unique optimal deposits-only

contract characterized by the following features:

1. For any bank i and time t, a debt contract has a covenant pc ≡ pcit = 1.

22

2. Banks always violate the covenant by choosing pt ≡ pit < 1 where pt solves:

(1− pt)[1− µ− π

pt + (1− pt)πbdt

]= Ψm. (45)

3. All deposits have the same contractual interest rate Rdt ≡ Rd

it and that the wedge

between the deposit rate and the risk-free rate is:

Rdt

Rt

=1

pt + (1− pt)π. (46)

4. The deposits to capital ratio is equal across banks, given by:

bdt ≡ bdit =φt − 1

φt. (47)

5. In a given period t, the regulator chooses the same λt = λit where λt solves:

(1− λt) =Et(R

Kt+1)−Rd

t bdt

π[ωHEt(RK

t+1)−Rdt bdt

] . (48)

The optimal deposits-only contract has a similar structure to the optimal two-class

contract in the benchmark model. Equations (45) and (48) are the marginal conditions

which determine the optimal business strategy for banks and the optimal monitoring pol-

icy for the regulator. For a given leverage ratio of banks, the budget constraint requires

that banks accept deposits to achieve the leverage ratio (47). Lastly, the break-even

condition of depositors determines the spread between the deposit rate and the risk-free

rate, given by (46). Without wholesale borrowing, banks take more deposits. But, ce-

teris paribus, this reduces the regulator’s marginal benefit of monitoring a bank and his

incentive to monitor. In turn, banks choose a riskier business strategy (lower p). Another

difference from the benchmark model is that since the leverage ratio varies over the busi-

ness cycle, banks’ business strategy pt varies over time.

Columns 2 and 4 of Table 2 compare the steady state of the benchmark model and

the deposits-only model. Relative to 1.4% in the benchmark model, the steady-state

risk-taking probability (1 − p) in the deposits-only model rises to 1.9%. Even though

the regulator is less incentivized to monitor, he expends higher effort to monitor in the

general equilibrium because banks take more risks. The deposit rate is higher because

depositors require a higher return to compensate for the increase in bank risk. Both low

23

p and high λ (which raise monitoring costs) make the economy more inefficient. As a

result, the steady-state output is 0.47% lower and capital stock is 1.35% lower than the

benchmark model. The associated welfare loss is equivalent to a 0.87% fall in consump-

tion permanently.

Figure 4 shows the welfare-equivalent permanent consumption loss for intermediate

deposit to bank asset ratios Bdt /(QtKt) = bdt ∈ [(1 − µ), (φ − 1)/φ]. A bank finances the

remaining fraction bwt = (φt−1)/φt− bd of bank assets by wholesale funding. In this case,

the optimal bank strategy p is given by the solution of the following equation:

(1− p)[(1− µ)− π

p+ (1− p)πbd]

= Ψm. (49)

Clearly, a higher deposit to bank asset ratio bd induces banks to take more risk, which

reduces p and the economy is more inefficient. To put things into perspective, Van den

Heuvel (2008) finds that capital requirement regulations in the US have a welfare cost

of about 0.1-1% permanent loss in consumption. I find that restricting wholesale lend-

ing by deposit ratio requirements has welfare costs of a similar magnitude. This result

demonstrates that having a suboptimal bank capital structure can be costly. The welfare

loss of suboptimal bank capital structure estimated here is large relative to estimates of

welfare costs of business cycles or the welfare improvement of implementing the optimal

monetary policy.

Many economies have some form of regulations on minimum deposit ratios for retail

banks. The welfare analysis above suggests, however, that having a high deposit to bank

asset ratio for commercial banks can have drawbacks and is socially inefficient. Calomiris

(1999) argues for bank regulations that require banks to maintain minimum ratios of

subordinated debt relative to risky assets. Indeed, such proposals for bank regulation

reform were given serious consideration in the US. The Chicago and Atlanta Federal Re-

serve Banks drafted detailed proposals to implement such rules (Keehn, 1989, Wall 1989).

However, the argument supporting such regulation then was one related to market disci-

pline: if banks take on excessive risks, uninsured wholesale borrowers will either withdraw

their funding or require a higher return. The subordinated debt ratio requirement would

then force banks to reduce their balance sheet. In my model, maintaining a minimum

subordinated debt to risky assets ratio is optimal because it incentivizes the regulator

to use higher effort to monitor the banks which reduces the benefits from taking exces-

24

0.76 0.78 0.8 0.82 0.84 0.86

Deposit ratio, b d

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Perm

anen

t con

sum

ptio

n lo

ss, x

(%)

Figure 4: Welfare loss with different deposit ratios

Table 3: Comparison of steady statesParameter Benchmark model

π ωH Ψm µ γ (1− p) (%) λ Welfare loss x (%)0.6 1.12 0.0042 0.25 0.2 1.42 0.128 00.65 1.62 0.195 0.28

1.20 1.42 0.348 0.520.0063 2.13 0.129 1.31

0.3 1.52 0.128 0.20.18 1.42 0.172 0.16

sive risks. The resulting steady-state spread between the wholesale lending rate and the

risk-free rate is indeed minimized.

5.2 Banking sector characteristics

In this subsection, I conduct sensitivity analysis to study how risk-taking behavior by

banks and the social welfare depend on the characteristics of the banking sector. In the

following, one parameter is changed at a time while other parameters are kept unchanged.

I link these changes to recent developments in the banking sector prior to the global fi-

nancial crisis in 2008.

Table 3 presents the results of the sensitivity analysis. The first row of Table 3 shows

25

the benchmark calibration. In the second row of Table 3, I increase the success probabil-

ity of risky project to π = 0.65. Driven by higher expected returns of the risky projects,

banks have a stronger incentive to engage in risk-taking, and so the welfare loss increases

by 0.28% of permanent loss in consumption, relative to the benchmark model. The third

row of Table 3 shows what happens if the return on risky investment ωH increases to

1.2. Although the equilibrium risk-taking by the bank remains unchanged, the regulator

monitors more intensively to deter banks from taking risk, which increases welfare loss to

0.52% of permanent consumption.

Rows 4 and 5 show what happens when monitoring is more difficult. The fourth row

studies an exogenous 50% rise in monitoring costs to Ψm = 0.0063. With a rise in mon-

itoring costs, in equilibrium banks choose a much lower p, which necessitates a slightly

higher monitoring effort by the regulator. The welfare costs are large. The fifth row

studies what happens when the liquidation value of the bank is lower (µ increases). In

this case the benefit of monitoring falls, and in equilibrium banks take more risks, which

again results in lower welfare.

The last row of Table 3 studies a fall in the fraction of divertible assets γ to 0.18.

With a looser borrowing constraint, lenders are willing to lend more to the banks which

raise the steady-state leverage ratio of the banks increases to 8.89. When banks take on

a higher leverage, their own stake in the risky project is smaller, so they have stronger

risk-shifting incentives and choose risky projects more often. In equilibrium, the regulator

monitors banks with a higher intensity, leading to a larger welfare loss.

These changes in parameters in the banking sector may reflect the effects of banking

deregulation and the advent of complex financial instruments prior to the global financial

crisis in 2008. Financial innovations increase the attractiveness of risky investments,

which can be captured by a rise in the success probability of a risky project π or the

return on risky investment ωH . Financial deregulations made bank capital structure

more opaque. Acharya et al. (2013) document that banks increase their holding of off-

balance-sheet items in the run-up of the crisis, by setting up special purpose vehicles.

Moreover, complex financial instruments such as mortgage-backed securities (MBS) and

collateralized debt obligations (CDOs) became popular prior to the crisis. The values of

these financial instruments are hard to evaluate. These developments could mean a rise

in monitoring costs and lower liquidation value of banks. Finally, banking deregulations

increased the effective leverage ratio of the banking sector. My model suggests that these

26

0 5 10-12

-10

-8

-6

-4×10-3 Output, Y

0 5 10-20

-15

-10

-5

0

5×10-3Price of capital, Q

0 5 10-0.15

-0.1

-0.05

0Bank net worth, N

0 5 10-0.02

-0.015

-0.01

-0.005

0Investment, I

0 5 10-0.01

-0.008

-0.006

-0.004

-0.002

0Capital stock, K

benchmarkdeposit-onlyGK

Figure 5: Impulse responses of macroeconomic variables to a negative productivity shock

developments in the banking sector had an effect of increasing the risk-taking activity of

the banking sector, and is associated with a large welfare loss.

6 Dynamic behavior of the system

This section studies the dynamic behavior of my model. I simulate the benchmark model

when it is subject to a 1% negative aggregate productivity innovation with persistence

ρA = 0.85, and compare the dynamic responses with the GK (2001) model and the

deposits-only model.16 Figures 5 and 6 show the responses of key macro and financial

variables. Each economy starts from the corresponding steady state and is hit by the shock

at time 0. All variables other than risk-taking probability and the interest rate spreads

are expressed in terms of percentage deviation from steady state. The risk-taking prob-

ability (1−p) is expressed in levels, and interest rate spreads are expressed in basis points.

Figure 5 shows that a negative productivity shock reduces the realized return on cap-

16I check that Assumptions 1,2 and 3 always hold over the business cycle.

27

0 5 101.4

1.5

1.6

1.7

1.8

1.9

2risk-taking prob., (1-p) (%)

0 5 10-0.05

0

0.05

0.1

0.15

0.2

0.25Monitoring intensity, λ

0 5 1055

60

65

70

75

80Deposit rate spread, R d-R, (b.p)

0 5 1067.5

68

68.5

69

69.5

70

70.5Wholesale rate spread, R w-R, (b.p)

0 5 10-0.05

0

0.05

0.1

0.15Bank leverage, φ

benchmarkdeposit-onlyGK

Figure 6: Impulse responses of financial variables to a negative productivity shock

28

ital which triggers a large fall in bank net worth. With impaired balance sheets banks

have a lower ability to raise funds so the price of capital falls, which leads to a sharp fall in

investment and output. The amplification of a shock is mainly driven by the financial ac-

celerator mechanism of GK (2011). Our benchmark model with bank risk-taking, as well

as the deposits-only model, produces aggregate responses of similar magnitude as the GK

(2011) model. If any, models with bank risk-taking yield a slightly larger amplification,

with a bigger drop in output and capital. Investment in the benchmark model drops less

than GK (2011) but capital drops more. This is because with risk-taking activity is in-

efficient and so a unit of capital stock today translates into less than (1−δ) unit tomorrow.

Figure 6 shows that the optimal monitoring effort by the regulator increases strongly

in a downturn, by more than 20% above its steady-state value. This is true for both

the benchmark model and the deposits-only model. To understand this, recall that the

optimal monitoring effort is determined by banks’ incentive to choose a risky strategy

(equation (18)). In response to a negative productivity shock, the expected return on

capital is high, so banks have an incentive to pursue a riskier strategy (they also take on

a higher leverage). The regulator should strengthen their monitoring of banks in order to

deter excessive risk-taking in the banking sector. The finding that banks have incentives

to increase risk-taking in the wake of adverse shocks and when bank capital is low is

common in a large finance literature (Merton, 1977; Barth and Bartholomew, 1992; Boyd

and Gertler, 1993; English, 1993).

In the benchmark model, bank business strategy p is a constant, and so the deposit

spread does not change over time. However, since monitoring effort is higher, there are

more liquidations and so the wholesale interest rate spread increases slightly. This find-

ing supports the notion that uninsured lenders have an incentive to assess risk. Because

wholesale lenders are not protected by DPR, they require a higher return to compensate

for the increased risk of bank liquidations. Feldman and Schmidt (2001) and Calomiris

(1999) argue that, because of this, wholesale investors provide market discipline and that

the regulator can observe such spread as an additional market indicator to gauge and

monitor banks’ risk-taking behavior.

The results that the optimal monitoring effort should increase when expected returns

and bank leverage are high is robust to the deposits-only model. By contrast, in the

deposits-only model, banks indeed take more risk. In response to a 1% negative produc-

tivity shock, bank risk-taking probability increases by around 3.5%. The actual increase in

29

bank risk-taking makes the deposits-only model more volatile which also reduces welfare.

7 Concluding remarks

In this paper I constructed a model which features bank risk-taking and the bank regula-

tor’s monitoring incentives and embedded the model in a DSGE framework. I showed that

by using an appropriate mix of deposits and wholesale borrowing, the banking system can

induce regulators to monitor banks effectively which reduces bank’s risk-taking behavior.

The optimal deposits to capital ratio is constant over the business cycle and equal to one

minus liquidation costs, and that regulator’s monitoring effort should increase strongly

in a downturn. Moreover, bank risk-taking has a substantial welfare cost which can be

reduced by using an appropriate mix of deposits and wholesale borrowing.

More recently, commercial banks have increased their reliance on short-term wholesale

borrowing to finance their assets. These liabilities, including asset-back securities and

short-term repo agreements, are different from traditional wholesale financing such as

large time deposits in two respects. First, these new instruments are secured. Second,

Huang and Ratnovski (2011) argue that these financial innovations reverse the effective

seniority of wholesale funding and retail deposits. Given the short maturity of complex

financial instruments, short-term wholesale financiers were able to exit ahead of retail

depositors without incurring much loss during the 2008 global financial crisis.17 I leave the

optimal bank capital structure in the presence of short-term secured financial instruments

to future research.

17Bolton and Oehmke (2015) provides an analysis of the implications of effective seniority of derivativesin a corporate finance model.

30

A Summary of models

A.1 Full system with bank depositors and wholesale lending

The macroeconomic part of the full benchmark system is given by:

Λt−1,t = βCt−1Ct

(50)

1 = RtEt(Λt,t+1) (51)

wt = χLϕt Ct (52)

wtLt = (1− α)Yt (53)

Yt = AtKαt L

1−αt (54)

Kt = (1− δ)Kt +

[1− ΨI

2

(ItIt−1− 1

)2]It (55)

Yt = Ct + It (56)

1 = Qt

[1− ΨI

2

(ItIt−1− 1

)2

−ΨI ItIt−1

(ItIt−1− 1

)]

+Et

[Λt,t+1Qt+1Ψ

I

(It+1

It

)2(It+1

It− 1

)](57)

RKt =

α(Yt/Kt) + (1− δ)Qt

Qt−1(58)

Kt = [p+ (1− λt−1)(1− p)πωH ]Kt−1 (59)

The credit contract part is given by:

QtKt = Nt +Bdt +Bw

t (60)

Ntφt = QtKt (61)

Bdt = bdQtKt (62)

Rdt =

1

p+ (1− p)πRt (63)

Rwt =

1

p+ (1− λt)(1− p)πRt (64)

π(1− λt) =Et(R

Kt+1)QtKt −Bd

tRdt −Bw

t Rwt

ωHEt(RKt+1)QtKt −Bd

tRdt −Bw

t Rwt

(65)

φt =νt

γ − ξt(66)

ξt = Et

Λt,t+1(1− θ + θγφt+1)[RKt+1 −Rw

t − bd(Rdt −Rw

t )]

(67)

νt = Et[Λt,t+1(1− θ + θγφt+1)]Rwt (68)

Nt = θ

[p+ (1− λt−1)(1− p)πωH ]RK

t Qt−1Kt−1−[p+ (1− λt−1)(1− p)π](Rd

t−1Bdt−1 +Rw

t−1Bwt−1)

+ τNt−1 (69)

31

where bd = 1 − µ. The above 20 equations solve for the dynamics of the following 20variables:

Λ, C,R,w, L, Y,K, K, I, Q,RK , λ, φ, Bd, Bw, Rd, Rw, ξ, ν,N

A.2 Full system with deposits only

The full deposits-only system has the same macroeconomic system, except that the risk-taking probability p is now an endogenous variable. The credit contract part is:

QtKt = Nt +Bdt (70)

Ntφt = QtKt (71)

Rdt =

1

pt + (1− pt)πRt (72)

π(1− λt) =Et(R

Kt+1)QtKt −Rd

tBdt

ωHEt(RKt+1)QtKt −Rd

tBdt

(73)

Ψm = (1− pt)[(1− µ)− π Bd

t

QtKt

](74)

φt =νt

γ − ξt(75)

ξt = Et

Λt,t+1(1− θ + θγφt+1)[RKt+1 −Rd

t ]

(76)

νt = Et[Λt,t+1(1− θ + θγφt+1)]Rdt (77)

Nt = θ

[pt−1 + (1− λt−1)(1− pt−1)πωH ]RK

t Qt−1Kt−1−[pt−1 + (1− λt−1)(1− pt−1)π]Rd

t−1Bdt−1

+ τNt−1 (78)

The above 19 equations solve for the dynamics of the following 19 variables:

Λ, C,R,w, L, Y,K, K, I, Q,RK , λ, φ, Bd, Rd, p, ξ, ν,N

A.3 Full system without monitoring problems

The full GK (2011) system has the macroeconomic system given by (50) – (59), exceptthat the effective capital is simply Kt ≡ Kt−1. The credit contract part is:

QtKt = Nt +Bdt (79)

Ntφt = QtKt (80)

Rdt = Rt (81)

φt =νt

γ − ξt(82)

ξt = Et

Λt,t+1(1− θ + θγφt+1)[RKt+1 −Rd

t ]

(83)

νt = Et[Λt,t+1(1− θ + θγφt+1)]Rdt (84)

Nt = θ[(RK

t −Rdt−1)Qt−1Kt−1 +Rd

t−1Nt−1]

+ τNt−1 (85)

32

The above 16 equations solve for the dynamics of the following 16 variables:

Λ, C,R,w, L, Y,K, I,Q,RK , φ, Bd, Rd, ξ, ν,N

B Proofs

Proof of Proposition 1 Although the model is dynamic, the credit contract is a staticproblem. In the following, I drop time subscripts. I show that the two-class credit con-tract is in fact identical to the problem in Park (2001). I then apply Proposition 6 inPark (2001) to prove the necessary results.

To show that my problem is same as Park (2001), I can rename the following quantitiesas:

DiS =Rdi

Rbdi , DiJ =

Rwi

Rbwi , Di = DiS +DiJ

IiS = bdi , IiJ = bwi , Ii = IiS + IiJ =φ− 1

φ,

and the following prices as:

G =E(RK)

R, B = ωH

E(RK)

R, L = (1− µ), c = Ψm.

Using these notations, one can write down the expected profit functions for banki, EωΠi(pi, λi), depositors EωΠd(pi, λi) and for wholesale lenders EωΠw(pi, λi) differ-ent ranges of pi. Define U(pi, λi) = EωΠi(pi, λi)/R, VS(pi, λi) = EωΠd(pi, λi)/R andVJ(pi, λi) = EωΠd(pi, λi)/R as discounted profit. Furthermore, the break-even conditionsfor depositors and wholesale lenders are given by VS(pi, λi) = IiS, and VJ(pi, λi) = IiJ .Note that these profit functions and break-even conditions for depositors and whole lendersare identical to those in Park (2011).

Applying Proposition 6 of Park (2011), we know that (i) there is a unique equilibriumand that pi satisfies condition (7); (ii) depositors’ contribution bdi must satisfy bdi ∈ [(1−µ), (φ− 1)/φ]; and (iii) the equilibrium is characterized by:

(1− pi)[(1− µ)− πR

di

Rbdi

]= Ψm, (86)

(1− λi)π[ωH

E(RK)

R− Rd

i

Rbdi −

Rwi

Rbwi

]=

E(RK)

R− Rd

i

Rbdi −

Rwi

Rbwi . (87)

33

The break-even conditions are re-written as:

Rdi

R=

1

pi + (1− pi)π, (88)

Rwi

R=

1

pi + (1− λi)(1− pi)π. (89)

Then, for given return on capital, expected bank profit EωΠi(pi, λi) is maximized whenrepayment is minimized. So, in stage one of the game, the bank’s problem is given by:

min(1−µ)≤bdi≤(φ−1)/φ

Rdi

Rbdi +

Rwi

Rbwi

subject to

bdi + bwi =φ− 1

φ, (90)

the equilibrium conditions (86), (87), and the break-even conditions (88), (89).

By total differentiating the equilibrium conditions and break-even conditions, andusing the fact that dbdi = −dbwi , I get:

Ω1 −1 0 −10 −1 Ω2 00 −1 Ω3 0

Ω4 0 Ω5 −1

dλi

d(Rd

i

Rbdi

)d(1− pi)d(Rw

i

Rbwi

) =

00

−Rdi

RRw

i

R

dbdi , (91)

where

Ω1 ≡π[ωH E(RK)

R− Rd

i

Rbdi −

Rwi

Rbwi

]1− (1− λi)π

> 0,

Ω2 ≡Ψm

π(1− pi)2> 0,

Ω3 ≡ (1− π)

(Rdi

R

)2

bdi > 0,

Ω4 ≡ (1− pi)π(Rwi

R

)2

bwi > 0,

Ω5 ≡ [1− (1− λi)π]

(Rwi

R

)2

bwi > 0.

This implies thatd(1− pi)dbdi

=1

(Ω2 − Ω3)

Rdi

R> 0,

34

anddλidbdi

=1

(Ω1 − Ω4)(Ω2 − Ω3)

[Rdi

RΩ5 −

(Rwi

R− Rd

i

R

)Ω2 +

Rwi

RΩ3

]> 0,

because(Rwi

R− Rd

i

R

)Ω2 = λi

Rwi R

di

R2

(1− µ− πR

di

Rbdi

)< λi (1− π)

Rwi (Rd

i )2

R3bdi < (1− π)

Rwi (Rd

i )2

R3bdi =

Rwi

RΩ3

But, (87) implies that λi is increasing in total repaymentRd

i

Rbdi +

Rwi

Rbwi . These mean

that bank i’s profit is maximized at bdi = (1− µ) ≡ bd.

We substitute bdi = (1− µ) and (88) into (86) to obtain:

(1− pi)[1− π

pi + (1− pi)π

]=

Ψm

1− µ. (92)

This equation solves for pi. But since pi only depends on aggregate parameters, all bankswill choose the same, constant p ≡ pi for all t. We rearrange the above equation to get:

f(p) = p(1− p)(1− π)− Ψm

1− µ[p+ (1− p)π] = 0. (93)

Clearly, f(p) is continuous in p for p ∈ [0, 1], quadratic in p and f(0) = f(1) < 0. Forsmall monitoring cost parameter Ψm, there exist p ∈ [0, 1] such that f(p) > 0, so thereare exactly two solutions for f(p) = 0 for p ∈ [0, 1], one close to 0 and the other close to1. However, recall that if p < pt, where pt satisfies

[pt + (1− pt)π]Et(RKt+1) =

φt − 1

φtRt

then no lender is willing to lend because the lender cannot break even. Rearranging this

equation, we know [pt + (1 − pt)π] = φt−1φt

Rt

Et(RKt+1)

>[ωHπ

Et(RKt+1)

Rt

]Rt

Et(RKt+1)

= ωHπ, so

pt > π(ωH − 1)/(1 − π). The smaller solution of p does not satisfy this condition, givenreasonable parameterization.

Rearranging the balance sheet of bank i, one obtains bw = bwi = φ−1φ− bd is also iden-

tical across banks. We combine equations (87) and (89) to solve for Rwi and λi. Clearly,

all banks choose the same Rw ≡ Rwi and depositors choose the same λ ≡ λi for all banks.

Proof of Proposition 2 As all banks face the same leverage ratio φ, the balancesheets of the banks imply all banks will choose the same deposits to capital ratio suchthat bd = bdi = φ−1

φ.

35

We continue to use the definition (86), but now define

Di =Rdi

Rbd. (94)

We can write down the profit functions for bank i, EωΠi(pi, λi) and depositors EωΠd(pi, λi),for different ranges of pi. Define U(pi, λi) = EωΠi(pi, λi)/R, and V (pi, λi) = EωΠd(pi, λi)/Ras discounted profit per unit of fund intermediated. Moreover, the break-even conditionfor depositors is given by V (pi, λi) = Ii. The optimal contract problem is then identicalto the optimal single-lender contract problem in Park (2001).

Using Proposition 2 of Park (2001), one can show that (i) the bank’s optimal choiceof pi lies in [pi, p

ci), where [pi + (1− pi)π]Di = L. For pi ∈ [pi, p

ci); (ii) the solution of the

optimal single-class contract is given by:

pci = 1, (95)

(1− pi)[(1− µ)− πR

di

Rbdi

]= Ψm, (96)

(1− λi)π[ωH

E(RK)

R− Rd

i

Rbdi

]=

E(RK)

R− Rd

i

Rbdi . (97)

The break-even condition for depositors is given by:

Rdi

R=

1

pi + (1− pi)π. (98)

The balance sheet requires that bd ≡ bdi = (φ− 1)/φ. Then, the last three equations solvefor pi, R

di , λi. Since other terms in these equations do not depend on bank characteristics,

so p ≡ pi, Rd ≡ Rd

i , λ ≡ λi are common across banks.

36

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