Kyoto University, Graduate School of Economics Research Project Center Discussion Paper Series

The Safer, the Riskier:

A Model of Bank Leverage and Financial Instability

Ryo Kato and Takayuki Tsuruga

Discussion Paper No. E-10-014

Research Project Center Graduate School of Economics

Kyoto University Yoshida-Hommachi, Sakyo-ku Kyoto City, 606-8501, Japan

February, 2011 (revised March, 2013)

The Safer, the Riskier: A Model of Financial Instability and Bank

Leverage�

Ryo Katoy and Takayuki Tsurugaz

March 25, 2013

Abstract

We examine the role of bank leverage in an attempt to explain why �nancial crises unfold

at a time when the economy appears to be less fragile to crisis risks. To this end, we extend the

model introduced by Diamond and Rajan (2012) to a variant where the probability of �nancial

crises varies endogenously. In our model, households� liquidity preference, modeled following

Allen and Gale (1998), plays a key role in precipitating a crisis because a high liquidity demand

under a highly leveraged banking system are likely to expose the economy to higher crisis

risks. We consider two examples of a �safe�environment: (i) the households�liquidity demand

tends to be low on average and (ii) the probability of bank bailouts is relatively high. Using

numerical analysis, we show that the �safer�environment could incentivize banks to raise their

leverage, resulting in more vulnerable banking system to liquidity shocks that endanger the

entire economy.

JEL Classi�cation: E3, G01, G21

Keywords: Bank run; Financial crisis; Maturity mismatch

�We thank Robert De Young, Koichi Futagami, Katsuya Ue, Dan Sasaki, Kenta Toyofuku, Noriyuki Yanagawa, andseminar and conference participants at the Bank of Japan, Kobe University, Kyoto University, University of Tokyo,the 2011 Autumn Annual Meeting of the Japan Society of Monetary Economics, and the 2011 Annual Meeting of theJapanese Economic Association for helpful comments and discussions. We are also indebted by helpful discussion withDaisuke Oyama and other members of the Banking Theory Study Group. Takayuki Tsuruga gratefully acknowledgesthe �nancial support of the Grant-in-aid for Scienti�c Research, Japan Securities Scholarship Foundation and NomuraFoundation for Social Science. Views expressed in this paper are those of the authors and do not necessarily re�ectthe o¢ cial views of the Bank of Japan.

yBank of Japan; e-mail: ryou.katou@boj.or.jpzGraduate School of Economics, Kyoto University; e-mail: tsuruga@econ.kyoto-u.ac.jp

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1 Introduction

The 2007-08 global �nancial crisis directed renewed attention to the anatomy of �nancial sector

disruptions. Maturity mismatch and the anticipation of bailouts (e.g., Greenspan put), among other

factors, are broadly considered to have played critical roles. A maturity mismatching structure of

bank assets and liabilities to provide the liquidity insurance for depositors (and other types of

short-term creditors) has long been emphasized as a crucial element in understanding �nancial

crises since Diamond and Dybvig (1983). More recently, Diamond and Rajan (2012, hereafter DR)

argued that the so-called Greenspan put may have encouraged banks to increase their illiquidity,

making them highly exposed to crisis risks.

In this paper, we aim to explore what may give rise to a highly leveraged banking system,

resulting in a high probability of a crisis. Among others, we solely focus on changes in macroeco-

nomic �fundamentals� rather than unmodeled sunspots. We integrate the benchmark model by

DR with the utility function employed in Allen and Gale (1998) and derive the endogenous prob-

abilities of �nancial crises. Using the framework, we explore how banks�risk taking interacts with

changes in the macroeconomic factors including policy interventions and how the crisis probability

is determined as a consequence of such interactions.

The changes in macroeconomic fundamentals that we consider in this paper are twofold. The

�rst is changes in the distribution of liquidity preference shock. It is widely acknowledged that, in

the run-up to the 2007-08 crisis, the global �nancial markets as well as the real economy appeared

to be increasingly stable in the era of Great Moderation. In our model, we translate this stability in

the �nancial market into changes in the distribution of liquidity preference shock. More speci�cally,

we assume that, during the period of the Great Moderation, the liquidity demand by creditors (i.e.,

households in our model) decreased on average, re�ecting the fact that investors preferred illiquid

investment (e.g., mortgage-backed securities) to liquid assets. Such an underlying change in the

liquidity preference appears to make the banking sector less fragile, or �safer,�because the banking

sector is faced with smaller upper tail risk of liquidity demand that precipitates a crisis.

As the second change in macroeconomic fundamentals, we consider a shift in the policy stance

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of the government (or the central banks). The experience of the 2007-08 �nancial crisis provokes

discussions on a variety of policy measures that have been taken to enhance the resilience of the

banking sector. Among numerous practices, we focus on emergency liquidity provisions (ELP), one

of the most widely undertaken policy measures across the economies during the crisis period. In

the speci�c context of our model, we consider an increase in probability that the government (or

a central bank) embarks on ELPs in an attempt to forestall a �nancial crisis. This probability of

ELPs is interpreted as the government�s policy stance for maintaining the �nancial stability. We

analyze how the crisis probabilities are a¤ected if the government becomes more concerned about

�nancial stability with more frequent interventions to bailout banking system. In other words, we

ask whether a change in the government�s policy stance represented by an increased probability of

ELPs would reduce the probability of a �nancial crisis.

We �nd that both changes in the macroeconomic fundamentals toward �nancial stability could

expose the economy to higher risks of �nancial crises. A key to understand these �the safer, the

riskier�cases is the banks�endogenous systemic risk taking. We show that the banks�risk taking

with a higher leverage o¤sets or even dominates the exogenous improvement of the macroeconomic

fundamentals in terms of the probability of �nancial crises. In particular, when the liquidity demand

is expected to be low, banks feel at ease and then raise their leverage. The increased leverage can

result in the higher risk of �nancial crises. In a similar vein, the elevated probability of ELPs would

also incentivize banks to be more leveraged, resulting in higher crisis probabilities. These two

examples may help better understand why the 2007-08 �nancial crisis unfolded at the time when

the banking sector was broadly believed to be increasingly surrounded by improved fundamentals.

Our paper di¤ers from related works on �nancial (in)stability in several aspects. Our �rst

example associated with the era of Great Moderation is related to Gertler, Kiyotaki, and Queralto

(2012). They calibrate the model to the �low risk� economy where the economy is in the period

of the Great Moderation and show that banks issue more short-term debt than the �high risk�

economy calibrated to the economy after the Great Moderation. While their focus lies in the

ampli�cation mechanism of the model as a result of higher leverage of the banking sector, ours

points to the e¤ect of bank leverage on the probability of �nancial crises. Our second example

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that higher expectations of bank bailouts may expose the economy to higher risks of �nancial

crises is also related to in�uential works by DR and Fahri and Tirole (2012). DR argue that direct

support to insolvent banks undermines the disciplinary e¤ect of deposits and suggest that interest

rate intervention dominates unconstrained directed bank bailouts. They further discuss that, even

under interest rate intervention, banks take on more risks through higher leverage. Fahri and

Tirole (2012) focus on strategic complementarities on the determination of banks�leverage under

anticipated bailouts and explore the optimal macro-prudential regulation. Our result on bank

bailouts basically relies on the framework by DR and is obtained in a di¤erent and simpler way. Our

scope is limited to a consequence of anticipated bailouts but we assess the frequency of �nancial

crises numerically under seemingly desirable changes in the government�s policy stance towards

�nancial stability.

The rest of the paper is organized as follows. Section 2 introduces the model of banking system,

which is an extension of DR. In Section 3, we calibrate the model and discuss the e¤ect of the

changes in distribution of the liquidity preference on the probability of �nancial crises. Section

4 further extends the DR�s model to incorporate the ELP by the government and discusses our

results that predict higher crisis probabilities under such interventions. Section 5 concludes.

2 The Model

2.1 Agents, Endowment, Technology and Preference

We consider a variation of the economy described by DR. Most of the assumptions are maintained

in line with the original DR model except for the households�preference, which we replace with that

in Allen and Gale (1998) to focus on the aggregate liquidity shock. In DR, the random shock arises

from the uncertainty over future income and DR consider unobservable �nite discrete aggregate

states where households expect either high income or low income. By contrast, we eliminate

uncertainty with respect to households�income while incorporating a more straightforward random

shock regarding liquidity preference into our model. Speci�cally, the household utility function is

given by U (C1; C2) = � log (C1) + (1� �) log (C2), where Ct is consumption at date t and � is a

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continuous random variable with a support � 2 [0; 1]. Here, � represents households�preference for

date-1 consumption, while e¤ectively it signals how much liquidity is needed at date 1.1 This Allen-

Gale utility provides the advantage that we can focus on aggregate uncertainty in a straightforward

manner. In contrast to the risk averse households with the Allen-Gale utility, entrepreneurs and

bankers are risk-neutral.

The economy lasts for three periods. At date 0, households are born with a unit of a good.

By assumption, no household consumes at date 0. They deposit all the date-0 endowments into

banks. Banks are competitive in raising and lending funds. They o¤er demand deposits, of which

face values are D, to households and lend the households� endowment to entrepreneurs.2 Each

entrepreneur invests a unit of the good to launch a long-term project at date 0. These transactions

are settled before the realization of the liquidity shock.

At date 1, the liquidity shock � is realized. Given the realized � (and their �xed endowment at

dates 1 and 2), households determine the date-1 withdrawal w1 to smooth consumption. Turning

to entrepreneur�s project, each of the projects yields a random output ~Y2 at its completion at date

2, but the outputs are reduced to X1 (< 1) if the projects are prematurely liquidated at date 1.3

Outcomes of projects follow a uniform distribution with a support [Y 2; �Y2]. In this model, there

is no aggregate uncertainty in ~Y2, and thus the �nancial stability entirely relies on the aggregate

uncertainty in �. At date 2, households can enjoy the rest of deposits together with the date-2

endowment on the condition that a bank run is not taking place.

We focus on the endogenous crisis probability subject to aggregate preference/liquidity shocks.

In what follows, we �rst describe the agent�s decisions after the realization of � and then the banks�

choice of D before the realization of �.

1While we describe the utility function as an Allen-Gale type here, in principle the main results remain unchangedif we use a Diamond and Dybvig (1983) utility. In essence, in light of the purpose of this paper, the two types ofutility functions work identically.

2While we model banks that make loans funded from demand deposits, we do not necessarily limit our attentionto commercial banks. Rather, the bank in our paper can broadly refer to �nancial intermediaries that raise funds viashort-term debts (e.g., a repo) and invest them into longer-term assets, by maturity transformation.

3For simplicity, we assume that storage technology is not available to economic agents in the model. Hence, theinvestment project by entrepreneurs is only a way of transferring goods intertemporally.

5

2.2 Households

A household chooses its withdrawal w1, given deposit face value D, the one-period gross interest

rate r12 (from date 1 to 2), and the preference shock �. The interest rate r12 represents the price

for liquidity which equates the withdrawal with the value of liquidated project. Given that a bank

run is not taking place, the households�maximization problem is given by

maxw1

� logC1 + (1� �) logC2;

s:t: C1 = e1 + w1 (1)

C2 = e2 + r12 (D � w1)

where et is the household�s endowment at date t.4 As discussed, � can be interpreted as a �liquidity

shock� because � determines the need for liquidity for each period. When � is low, households�

deposits are likely to be fully repaid by banks over the two periods, which means the households

fully smooth out their consumption at normal times. As we will show later in detail, however, �

exceeding a threshold value of the liquidity preference shock, the households�deposits are not fully

repaid at date 1. Then, a bank run takes place and the household receive only X1 at date 1 and

nothing from banks. Thus, the households fail to smooth out their consumption and end up with

C1 = e1 + X1 and C2 = e2. In what follows, we denote � as the probability of bank runs (i.e.,

�nancial crises).

When the households can smooth out their consumption, the intertemporal �rst-order condition

for consumption�

1� �

�C1C2

��1= r12 (2)

is satis�ed. Meanwhile, the budget constraint holds with equality. Hence, the withdrawal can be

written as

w1 = �

�e2r12

+D

�� (1� �) e1: (3)

4We implicitly assume the information structure where bank runs are precipitated as a Nash equilibrium whenbanks are revealed to be insolvent.

6

This extension of the preference enables us to assess the probability of �nancial crises endoge-

nously and continuously. Furthermore, it is convenient to de�ne the households�lifetime income at

normal times by m:

m = e1 +D +e2r12: (4)

The log-utility implies that consumption at normal times is proportional to m, namely C1 = �m

and C2 = (1� �) r12m.

2.3 Entrepreneurs and Bankers

Entrepreneurs and bankers in our model replicate those in DR. As discussed in Diamond and Rajan

(2001), each banker is a relationship lender that has obtained special knowledge of the entrepreneurs�

business, and this knowledge assures the banker�s collection skill to acquire a fraction ~Y2(< ~Y2)

of the output from the entrepreneurs. The collection skill is assumed to be not transferable to

other lenders. Let the banker�s assets be A (r12). As assumed in DR, at date 1, bankers receive

signals Y2 that perfectly predict each realization of ~Y2.5 Each banker is assumed to attract many

entrepreneurs through the competitive o¤er on the loan, resulting in the identical portfolio shared

by all the symmetric banks. Then, each banker�s assets A (r12) can be expressed as

A (r12) =1

�Y2 � Y 2

Z Y2(r12)

Y 2

X1dY2 +1

�Y2 � Y 2

Z �Y2

Y2(r12)

Y2r12

dY2; (5)

where the �rst and second terms of the equation indicate the values of liquidated projects and

completed projects evaluated at t = 1, respectively. In (5), Y2 (r12) denotes the threshold return of

projects satisfying Y2 (r12) = r12X1= . Bankers liquidate a project, whose return falls short of the

opportunity cost, to meet the households�liquidity demand. Furthermore, it can be easily shown

that A0 (r12) < 0.

Banks turn to be insolvent if the solvency condition D � A (r12) is violated. In this case, a

bank run is precipitated: The bankers liquidate all of the entrepreneurs� projects, repay X1 to

households, and lose all their assets. We then de�ne the threshold interest rate r�12 which satis�es

5Allen and Gale (1998) make a similar assumption for this interim signal.

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the solvency condition with equality:

D = A (r�12) ; (6)

where r�12 strictly decreases with D since A0 (�) < 0. In other words, a higher level of D requires a

lower level of r�12 for the bankers to be solvent.

2.4 Liquidity Market

As far as a run is not taking place, the liquidity market clearing condition holds:

�

�e2r12

+D

�� (1� �) e1 =

1�Y2 � Y 2

�r12X1

� Y 2�X1; (7)

where the equilibrium interest rate r12 is uniquely determined. The left-hand side of the equation

points to liquidity demand (3) while the right-hand side indicates supply from the project liquidation

shown in (5).

Along with (7), the threshold interest rate r�12 allows us to de�ne �� as the threshold value of �

that precipitates bank runs if and only if � > ��:

�� =(r�12X1= � Y 2)X1=

��Y2 � Y 2

�+ e1

e1 +D + e2=r�12: (8)

Since r�12 is strictly decreasing in D, (8) indicates that �� is strictly decreasing in D. When we

emphasize this relationship between �� and D, we express �� as �� (D) and express its derivative

as ��0 (D). We also note that, because a larger liquidity shock increases households�withdrawal, by

de�nition, a smaller �� points to a higher crisis probability. Using ��, we can express the endogenous

probability of a crisis as � (��) = 1�F (��), where F (�) denotes the cumulative distribution function

of �.

2.5 Bank�s Choice of Leverage

We also follow Diamond and Rajan�s (2001) argument for the reason why bankers issue demand

deposits D: as a commitment device to compensate for the lack of transferability of their collection

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skill and to promote liquidity creation. In line with this argument, the bankers need to determine

the face value of deposits before observing � in a competitive banking sector. As a result of

competition, the bankers are forced to make a competitive o¤er of deposits for households. The

competitive o¤er maximizes the household welfare (Allen and Gale, 1998), taking the distribution

of � as given.6 Here, the choice of D has a one-to-one relationship with the bank�s leverage. The

bank�s leverage in our model can be de�ned as D= [A (r12)�D] and is determined once D is chosen.

Therefore, the optimal choice of bank leverage always coincides with the optimal choice of D in our

model.

Formally, the banks�maximization problem is given by

maxD

Z ��(D)

0f� log (�m) + (1� �) log [(1� �) r12m]g dF (�)

+

Z 1

��(D)[� log (e1 +X1) + (1� �) log (e2)] dF (�) ; (9)

subject to (4) and (8), where the �rst term of (9) corresponds to the utility from consumption

under no crisis while the second term points to the utility from consumption under a crisis. The

integral is taken over � 2 [0; ��] for the �rst term, because any � that is lower than the threshold

value does not precipitate crises. In contrast, the second term indicates that banks recognize that

consumption smoothing is impossible for � > ��.

The �rst-order condition for D is given by

��� log

���m�

e1 +X1

�+ (1� ��) log

�(1� ��) r12m�

e2

���0 (��) ��0 (D)

=

Z ��

0

�1

m

�1� e2

r212

@r12@D

�+1� �r12

@r12@D

�dF (�) ; (10)

where m� = e1 + D + e2=r�12 and f (�

�) is the probability density function evaluated at � = ��,

de�ned by (8). The partial derivative @r12=@D can be implicitly de�ned by the liquidity market

clearing condition (7).

In choosing the optimal D, banks strike a right balance between the marginal bene�t and cost

6 In the model, the bankers in fact are maximizing their own pro�ts by household welfare maximization.

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of increasing D on behalf of the households. The right-hand side of (10) can be interpreted as the

marginal bene�t of increasing D through changes in households�lifetime income and interest rate.

Due to the log period utility, the households�utility can be expressed as logm + (1� �) log r12 +

� log � + (1� �) log (1� �). A higher D allows households to receive higher income from their

deposits and to enjoy more consumption at both dates.7 Hence, as far as � is smaller than the

threshold value ��, households obtains higher returns from increasing D.

The left-hand side of (10) represents the marginal cost of increasing D. The term in the

curly brackets compares households�utility on the brink of the crisis with that under the crisis,

representing the households�loss of crises. The term outside the curly brackets assesses how the

crisis probability changes in response to an increase in D, indicating the marginal changes in the

crisis probability of increasing D. Hence, putting them altogether, we can interpret the left-hand

side as the marginal cost of increasing D.

3 Simulation of the Benchmark Model

3.1 Calibration

The numerical example that we consider here broadly follows the parameter set chosen by DR. Let

e1 = 0:65, X1 = 0:95, Y 2 = 0:0, �Y2 = 3:5, and = 0:9. Instead of the discrete states for endowment

at t = 2 in DR, we take a single constant value e2 = e1 and assume that a liquidity shock � is

generated from the beta distribution with two parameters that give us �� = 0:5 and �� = 0:07,

where �� and �� denote the mean and the standard deviation of �, respectively. The marked

di¤erence in our model from DR lies in the fully endogenous probability of a crisis, � = 1�F (��),

which e¤ectively replaces the exogenously given probability for an �exuberant�state to materialize

in DR.

Under the parameter set with a mean of 0.50 and a standard deviation of 0.07 for �, the banks

set the level of the deposit face value D at 1.08, striking a proper balance between the return from

7More technically, (4) suggests that m could decrease with D if the e¤ect of r12 that discounts households�endowment at date 2 exceeds the outright e¤ect of an increase in D. Here, we limit our attention to the case wherethe outright e¤ect exceeds the discounting e¤ect of r12. This ensures the necessity of the �nancial transaction.Otherwise, there would be no point in discussing banks.

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high leverage and the risk of a run. The resulting probability of a crisis is 11.0 percent. Figure

1 plots the households�utility over a variety of deposit face values D. The �gure also articulates

the sub-components of the utility. The smooth bell shape of the utility can be understood as

the weighted average of the two sub-components, (i) the expected utility in the absence of a run

E (U jno run) and (ii) the expected utility under a run E (U jrun). In the �gure, the probability of

a crisis is represented by the ratio of the distance along the vertical axis between the solid and the

upper dashed lines to that between the upper and lower dashed lines.

3.2 The Safer, the more secured?: Structural Changes in the Distribution of

Liquidity Preference

It came as surprise to many people at a time when the 2007-08 �nancial crisis followed the Great

Moderation. The model discussed in this paper suggests a possible explanation of why the �nancial

crisis unfolded at a time when the banking sector appeared to be increasingly more secured in the

era of Great Moderation.8

The experiment that we perform here investigates changes in the distribution of the underlying

shock �. Table 1 compares the crisis probabilities for a few cases where we change �� while keeping

�� unchanged. A decrease in �� implies that banks �nd a smaller upper tail risk (i.e., a risk of a

large �) �put di¤erently, they recognize that the fundamentals are safe. Recall that the probability

of a crisis was 11.0 percent in the initial �risky�distribution with the mean of 0.50 (Case 1 in Table

1). With the lower mean of 0.35 in the �safer� distribution, however, the probability of a crisis

increases from 11.0 to 12.4 percent (Case 2 in Table 1). Therefore, it is not always true that the

�safer�the economy is, the more secure the banking system is.

Why does this �the safer, the riskier� happen? The key to understanding this result lies in

banks�risk taking. Table 1 also reports that the banks�leverage increases when �� declines. As

discussed so far, banks in our model have a strong incentive to raise D when they face a smaller

upper tail risk. Though a higher leverage of banks gives rise to higher returns to households, it

also increases the risk of the insolvency of banks. This elevated risk is re�ected in a decline in ��.

8See Bernanke (2004), for example.

11

As shown in the last row in the table, �� declines from 0.58 in Case 1 to 0.43 in Case 2. Thus,

despite the reduction in the ex-ante crisis probability re�ecting the safer fundamentals, the banks�

risk taking e¤ect can result in a higher ex-post crisis probability.

Figure 2 shows how banks�leverage a¤ects the ex-post crisis probability through ��. If banks

do not react to the change in the distribution of �, �� remains unchanged and the resulting ex-ante

crisis probability sharply drops to nearly zero. In contrast, if banks react to the macroeconomic

changes correctly, �� decreases from ��R to ��S , giving rise to a higher ex-post crisis probability (region

A in Figure 2). The economic interpretation of the decrease in �� is that the �safer�distribution

incentivizes banks to raise leverage, taking on more risks. The higher leverage makes banks more

vulnerable to liquidity shocks, and the elevated vulnerability can endanger the economy. As a

result, the more secure fundamentals can leave the economy exposed to higher crisis risks.

Figure 3 con�rms that the crisis probability can be plotted as a U-shaped curve against ��. The

fact that the crisis probability is downward-sloping over a wide range of �� indicates that our result

is not an artifact of our arbitrary choice of parameters but can be widely observed in our model.

4 The Model with Emergency Liquidity Provisions

4.1 Characterization

We next extend the model introducing the emergency liquidity provision (ELP) by the govern-

ment/the central bank (GC, hereafter) who aims to rescue troubled banks and to prevent a crisis.

It is widely argued that the ELP could help mitigate or prevent crises when the banking sector

is poised on the brink of a crisis. At the same time, however, policy makers are concerned about

an apparent trade-o¤ arising from moral hazard. If the GC makes announcement of such liquidity

provisions, banks come to realize that they will be bailed out at a time of �nancial distress. A

consequence is that they can be more leveraged than otherwise, resulting in higher risks of crises.

In practice, the liquidity provisions during the time of a �nancial distress of the banking sector

may not always be carried out. ELPs are, more or less, at the discretion of the GC, depending on

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the extent of �nancial distress and on other factors that are not modeled here.9 For this reason,

this section assumes that ELPs are activated as an random event, rather than assuming that the

GC always acts for maintaining the �nancial stability by bailing out banks near crisis. In this

context, we incorporate the exogenous probability of the ELP into our analysis. Interpreting the

probability as the GC�s policy stance on pursuing the �nancial stability, we ask how the GC�s policy

stance on the �nancial stability a¤ects the crisis probability. We further investigate how the banks�

over-optimistic anticipation on the policy stance could amplify the �nancial instability.10

4.2 Implementation

Suppose that the GC imposes a levy on bank size measured by banks�liability to �nance their ELP

for bailing out banks near crisis. Let � be the tax rate imposed by the GC on banks�liability D.

The GC collects the levy of r12�D at date 2 from banks. This amount is �D, if it is measured by

the consumption goods at date 1. Accordingly, the banks�solvency condition can be rewritten as

(1 + �)D � A (r12) :

Thus, this levy restricts the bank�s liability in its size, reining in banks�leverage.

The to-be-collected levy can be used for the ELP when banks are faced with the liquidity

shortage at date 1. More speci�cally, in terms of operation, the GC can collect a tax from households

at date 1 after the realization of � and step into the liquidity market to provide the liquidity M .

Then, at date 2, the GC can fully compensate for the tax, which households paid at date 1,

by transferring income from banks to households together with interest payment. As a result,

households�lifetime income can be kept unchanged. This ELP is feasible as far as the supply of

liquidity M does not exceed the total amount of bank levy equal to �D.

With this operation in mind, suppose, as an experiment, that the GC makes an ELP for a

9An immediate factor that needs to be taken into account in the real world is the �scal condition (e.g., levels ofpublic debt). We leave such factors out of the scope of this paper.10We perform this analysis for the purpose of positive analysis rather than normative analysis. As suggested by

Allen and Gale (1998, 2007), the laissez-faire banks would choose the socially optimal decision on their leveragebecause the current model does not include any welfare-reducing externalities. The purpose of our analysis here issolely for evaluating policy measures that have been taken by the GC.

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constant probability p. For the remaining probability 1� p, the GC wastes the levy for nothing or

uses it for the GC�s own consumption that has no bene�t for the banking system. Our interpretation

is that the GC tends to make e¤orts more proactively to keep the economy safer for a larger p,

because the ELP is perceived to help stabilize the banking system. Accordingly, this probability

p is considered the policy stance indicating to what extent the GC is proactive in maintaining the

�nancial stability. We assume that the probability of ELPs is independent of � and is predetermined

before � is realized. Hereafter, unless otherwise noted, the probability is publicly known to banks

and households.

Due to the GC�s randomized bailout, we need to consider two cases: with and without the

activation of the ELP. In the case of no intervention, which occurs at the probability 1 � p, the

solvency constraint with equality rede�nes the threshold interest rate beyond which a �nancial

crisis is precipitated.

(1 + �)D = A (r�12) : (11)

Together with this newly de�ned r�12, we de�ne ��L as

��L =(r�12X1= � Y 2)X1=

��Y2 � Y 2

�+ e1

e1 +D + e2=r�12; (12)

from the liquidity market clearing condition (7). We emphasize that ��L is the threshold value of

� that precipitates a crisis if � > ��L and if the GC does not provide emergency liquidity for the

market. In parallel with ��, ��L remains to be a decreasing function of D. We express ��L as �

�L (D)

and its derivative as ��0L (D) to clarify the relationship between ��L and D.

In the second case, the GC carries out the ELP. With the probability p, the GC provides

liquidity M � �D to bailout the banks if they are faced with � > ��L. In this case, the GC can

forestall a crisis by supplying M and keeping the market interest rate at r�12, at which the banks�

solvency stands marginally solvent. With the intervention underway, the liquidity market clearing

condition is

�

�e2r�12

+D

�� (1� �) e1 =

1�Y2 � Y 2

�r�12X1

� Y 2�X1 +M; (13)

14

whereM is set such that the interest rate is kept at the level of r�12. The ELP can be also interpreted

as a low interest rate policy, since, for � > ��L, the GC can place a cap on the market interest rate

at r12 � r�12. However, for a su¢ ciently high � exceeding a certain level of the preference shock,

the GC cannot keep the market interest rate at the level of r�12 because the maximum fund for the

ELP is constrained at �D. If the GC fails to provide enough liquidity at r�12, a �nancial crisis is

precipitated. We de�ne the threshold level of the liquidity preference shock, beyond which banks

cannot remain solvent even with the ELP, by ��H :

��H =(r�12X1= � Y 2)X1=

��Y2 � Y 2

�+ e1 + �D

e1 +D + e2=r�12: (14)

In line with the notation for ��L, we express ��H as ��H (D) and its derivative as �

�0H (D) to clarify

the relationship with D.

Now, we formally state the banks�problem. In an economy with non-zero probability of ELPs,

banks maximize

maxD

Z ��L(D)

0f� ln (�m) + (1� �) ln [(1� �) r12m]g dF (�)

+ (1� p)(Z ��H(D)

��L(D)[� ln (e1 +X1) + (1� �) ln (e2)] dF (�)

)

+p

Z ��H(D)

��L(D)f� ln (�m�) + (1� �) ln [(1� �) r�12m�]g dF (�)

!

+

Z 1

��H(D)[� ln (e1 +X1) + (1� �) ln (e2)] dF (�)

subject to (4), (12), (14), and m� = e2=r�12 +D + e1. The �rst line corresponds to the utility for

� 2 [0; ��L] and is unrelated to the bailout probability p because the economy is at a normal time.

The fourth line points to the utility for � 2 (��H ; 1] and is also unrelated to the bailout probability

because � is so high that banks are insolvent regardless of the GC�s bailouts. As indicated in

the second and third lines, however, the utility for � 2 (��L; ��H ] varies conditionally on the GC�s

intervention to provide liquidity. In the second line, the utility is evaluated at the consumption at

a crisis because the GC does not intervene. By contrast, the third line corresponds to the utility

15

where the GC carries out the ELP. In the latter case, households attain the level of their utility

on the brink of the crisis because the GC keeps the market interest rate exactly at r12 = r�12 to

maintain the banks�solvency.

Taking into account the GC�s random attempts to prevent a crisis, the crisis probability is now

given by

� (��L; ��H) = (1� p) [F (��H)� F (��L)] + 1� F (��H) : (15)

Clearly, the crisis probability depends negatively on the GC�s policy stance on pursuing the �nancial

stability. Other things being equal, a large p implies a low crisis probability.

4.3 The Safer, the More Secured?: Changes in the Policy Stance on the Finan-

cial Stability

Using the framework introduced in the previous section, we assess how the GC�s policy stance on

the �nancial stability a¤ects the probability of �nancial crises. As we discussed, the GC�s policy

stance on the �nancial stability - the economy�s safety - is translated into p. In fact, as (15)

indicates, an increase in p would suggest a lower probability of crises, if banks do not react to the

change in p and keep the values of ��L and ��H with their leverage D held unchanged. However,

banks would react to the increase in p, most likely by raising their leverage D. Hence,the ultimate

question posed here is, how much net reduction in the crisis probability can be expected by the

ELP when the banks�endogenous response is taken into account.

To answer this question, we calibrate the model with the ELP as follows. New parameter in

this experiment is � , the tax rate imposed on the banks liability. This � is set to 0.03. Regarding

�� and ��, we set them to the benchmark calibration, i.e., �� = 0:50 and �� = 0:07.

Table 2 reports the results of this experiment. In our experiment, Case 1 with p = 0 can be

interpreted as the �risky� economy where the troubled banks have no chance to be rescued. In

Case 3 with p = 1:0, the economy could be perceived to be �safer,� because it is assumed that

the GC always steps in the liquidity market to bailout the troubled banks. But, such perception

is prima facie case of safety. In fact, the crisis probability slightly increases, rather than decreases,

16

from 12.0 percent in Case 1 to 12.3 percent in Case 3. The GC�s attempt to increase the safety of

the economy does not make the economy safe, but, conversely, such an attempt makes the economy

riskier. Figure 4 con�rms that the crisis probability is upward-sloping over a broad range of p.

This �the safer, the riskier� case can be again understood through banks�endogenous choice

of their leverage. As the third row in the table indicates, banks raise their leverage in response to

changes in the policy stance toward the �nancial stability (i.e., larger values for p). If banks do not

react to the changes in the GC�s policy stance, ��H and ��L remain unchanged. Then, the ex-ante

crisis probability can be computed from (15) with ��H and ��L in Case 1 (i.e., �

�H = 0:598, �

�L = 0:583)

and p in Case 3 (i.e., p = 1:0): � = 1�F (0:598) = 0:081, 8.1 percent. In this numerical example, the

change in the GC�s policy stance improves the economy�s safety measured by the crisis probability

by 3.9 percentage point. Meanwhile, banks�endogenous risk taking, conversely, undermines the

economy�s resilience with 4.2 percentage point higher crisis probability. Consequently, the initially

intended policy e¤ect is fully o¤set by the banks�risk taking and the economy is left with the net

increase in the crisis probability.

While these changes in the probability appear to be small in the magnitude measured by the

crisis probability, the �nancial instability could be ampli�ed if the banks� endogenous choice of

leverage is in�uenced by their optimistic perception on the GC�s policy stance. As an extreme

case, suppose that the GC are extremely worried about banks�moral hazard (p = 0:0) but banks

incorrectly anticipate that the GC always helps the troubled banks (p = 1:0). Then, we again use

a back-of-the-envelope calculation with the numbers in Table 2 to calculate the crisis probability.

We take ��H and ��L from Case 3 but evaluate � given by (15) at p = 0:0. The consequence of

the misperception is banks�overleverage and the excessively elevated crisis probability. In fact,

the crisis probability is now calculated by � = F (��H)� F (��L) + 1� F (��H) = 1� F (��L), which is

17.5 percent (i.e., � = 1� F (0:566) = 0:175). Therefore, when banks are overoptimistic about the

bailout probability, the �nancial instability could be ampli�ed easily and considerably.

17

5 Concluding Remarks

We developed a model of �nancial instability with endogenously determined bank leverage to explore

how changes in macroeconomic fundamentals a¤ect the probability of �nancial crises. Aggregate

liquidity preference shock is useful in considering the endogenously changing crisis probability

because such shocks can generally mimic broad types of other shocks (e.g., income shock and

productivity shocks) that result in systemic bank runs via the same channel. Using our framework,

which basically follows Diamond and Rajan (2012) with the above-mentioned aggregate shock, we

perform the two experiments where the macroeconomic fundamentals change: (i) average declines

in the liquidity preference (i.e., lower demand for liquidity) and (ii) rises in the probability of the

government�s ELP to bailout banks. We showed that, despite these exogenous improvements of the

macroeconomic fundamentals, the banking system could expose the economy as a whole to higher

risks of crises due to the banks�endogenous risk taking in response to the �safer� fundamentals.

We argue that these two experiments may help better understand the 2007-08 �nancial crisis which

took place amid the allegedly sound and improved macroeconomic fundamentals.

Our analysis can be extended in a number of directions. First, the rationale of the government

intervention was not fully speci�ed in our model. To better motivate the government intervention,

the model may need to include welfare-reducing distortions, such as externalities and/or coordina-

tion failure, that have been emphasized in the recent studies.11 Second, developing a full-�edged

in�nite horizon dynamic model can promote better understanding of dynamics of �nancial crises

and real economy.12

References

[1] Allen, F., and D. Gale (1998), �The Optimal Financial Crises,� Journal of Finance, 53 (4),

pp. 1245-1284.

[2] Allen, F., and D. Gale (2007), Understanding Financial Crises, Oxford University Press.

11A few of examples include: Lorenzoni (2008), Jeanne and Korinek (2010) and Bianchi (2011) and de Groot (2011)12Angeloni and Faia (In press) and Gertler and Kiyotaki (2011, 2012), who introduce banks in this class of models,

are marked examples of this direction.

18

[3] Angeloni, I., and E. Faia (In press),�Capital Regulation and Monetary Policy with Fragile

Banks,�Journal of Monetary Economics, Forthcoming.

[4] Bernanke, B. (2004), �The Great Moderation,� remarks at the meetings of the Eastern Eco-

nomic Association, Washington, D.C.

[5] Bianchi, J. (2011), �Overborrowing and Systemic Externalities in the Business Cycle,�Amer-

ican Economic Review, 101 (7), pp. 3400-3426.

[6] de Groot, O. (2011), �Coordination Failure and the Financial Accelerator,�mimeo.

[7] Diamond, D. W., and P. Dybvig (1983), �Bank Runs, Deposit Insurance, and Liquidity,�

Journal of Political Economy, 91 (3), pp. 401-419.

[8] Diamond, D. W., and R. Rajan (2001), �Liquidity Risk, Liquidity Creation, and Financial

Fragility: A Theory of Banking,�Journal of Political Economy, 109 (2), pp. 287-327.

[9] Diamond, D. W., and R. Rajan (2012), �Illiquid Banks, Financial Stability, and Interest Rate

Policy,�Journal of Political Economy, 120 (3), pp. 552-591.

[10] Farhi E., and J. Tirole (2012), �Collective Moral Hazard, Maturity Mismatch, and Systemic

Bailouts,�American Economic Review, 102 (1), pp. 60-93.

[11] Gertler, M., and N. Kiyotaki (2011), �Financial Intermediation and Credit Policy in Business

Cycle Analysis,�Handbook of Monetary Economics, B. Friedman and M. Woodford (ed.), 3A,

pp. 547-599.

[12] Gertler, M., and N. Kiyotaki (2012), �Banking, Liquidity and Bank Runs in an In�nite Horizon

Economy,�mimeo.

[13] Gertler, M., N. Kiyotaki, A. Queralto (2012), �Financial Crises, Rank Risk Exposure and

Government Financial Policy,�Journal of Monetary Economics, 59, Supplement, S17-S34.

[14] Jeanne, O., and A. Korinek (2010), �Managing Credit Booms and Busts: A Pigouvian Taxation

Approach,�NBER Working Paper No. 16377.

[15] Lorenzoni, G. (2008), �Ine¢ cient Credit Booms,�Review of Economic Studies, 75 (3), pp.

809-833.

19

Figure 1: Banks�leverage and utility

1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2

0.1

0.15

0.2

0.25

0.3

Deposits

E[U

], E

[U|N

o ru

n], E

[U|R

un]

E[U]E[U|No run]E[U|Run]

Note: The solid line represents the utility level against the face value of deposits. The upward-sloping dashed line is the

expected utility conditional on no bank run, and the downward-sloping dashed line is the expected utility conditional

on a bank run. The calibration is based on the assumption that a liquidity shock follows a beta distribution with a

mean of 0.50 and a standard deviation of 0.07.

20

Figure 2: Comparison of distribution for �

0.1 0.2 0.3 0.4 0.5 0.6 0.70

1

2

3

4

5

6

Pro

babi

lity 

dens

ity

θS* θR

*

A

Case 2

Case 1

Note: The solid line represents the probability density function based on a beta distribution with a mean of 0.50 and

a standard deviation of 0.07 (Case 1). The dashed line is the probability density function of a beta distribution with

a smaller mean of 0.35 but with the same standard deviation (Case 2). Here ��R is the threshold value of a liquidity

shock that precipitates a bank run under Case 1, while ��S is the threshold value corresponding to Case 2.

21

Figure 3: Financial crisis probabilities against ��

0.35 0.4 0.45 0.5 0.55 0.6 0.650.11

0.112

0.114

0.116

0.118

0.12

0.122

0.124

0.126

0.128

0.13

µθ

Pro

babi

lity 

of fi

nanc

ial c

rises

Note: The curve represents the probabilities of bank runs against the mean of �. The standard deviation of � is set

to 0.07.

Figure 4: Financial crisis probabilities against p

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.12

0.1205

0.121

0.1215

0.122

0.1225

0.123

0.1235

0.124

Probability of emergency liquidity provision

Pro

babi

lity 

of fi

nanc

ial c

rises

Crisis probability against p

Note: The crisis probability � given by (15) against the probability of the government/central banks� emergency

liquidity provision when the banking sector is near a crisis. The mean of � is set to 0.5 and the standard deviation

of � is 0.07.

22

Table 1: Numerical examples of crisis probability (�) against the mean of �

Case 1 Case 2 Case 3�� 0.500 0.350 0.350�� 0.070 0.070 0.065� 0.110 0.124 0.115D 1.084 1.322 1.328�� 0.586 0.432 0.429

Note: Each column calibrates the mean (��) and standard deviation (��) of � under the assumption that � followsthe beta distribution. The probability of bank runs �, the threshold level of ��, and the level of deposits D arecomputed from the calibrated moments.

Table 2: Numerical examples of crisis probability (�) against the policy stance on the �nancialstability (p)

Case 1 Case 2 Case 3p 0.00 0.50 1.00� 0.120 0.122 0.123D 1.066 1.074 1.083��H 0.598 0.590 0.582��L 0.583 0.575 0.566

Note: In each simulation, the mean and the standard deviation of � is set to 0.5 and 0.07. The tax rate for bank levyis 0.03.

23