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

Bank Overleverage and Macroeconomic Fragility
Ryo Kato and Takayuki Tsuruga

Discussion Paper No. E-12-002

Research Project Center
Graduate School of Economics
Kyoto University
Yoshida-Hommachi, Sakyo-ku
Kyoto City, 606-8501, Japan

April, 2012 (revised March, 2013)

Bank Overleverage and Macroeconomic Fragility
Ryo Katoyand Takayuki Tsurugaz
March 25, 2013

Abstract
We incorporate the banks, de…ned as maturity-mismatching …nancial intermediaries
by Diamond and Rajan (2001a, 2012), into an overlapping-generations model where
capital good is reproducible. We show that, in our model, the laissez-faire banks takeon undue risks, compared to the social optimum, owing to the pecuniary externalities.
Further, the model replicates “rare but severe crises” without assuming any large exogenous shocks because systemic bank runs take place endogenously followed by sharp
contractions in output. We also make policy assessments based on the model. The
assessment favors some macro-prudential measures over pre-committed bank bailouts.
JEL Classi…cation: E3, G01, G21
Keywords: Financial crisis, Liquidity shortage, Maturity mismatch, Credit externalities, Financial regulation

We thank Kosuke Aoki, Xavier Freixas, Shinichi Fukuda, Gary Gorton, Fumio Hayashi, Keiichiro
Kobayashi, Pok-sang Lam, Hidehiko Matsumoto, Kiyohiko G. Nishimura, Maurice Obstfeld, Akihiko Senoo,
Alp Simsek, Frank Smets, Javier Suarez, and Aleh Tsyvinski for comments and discussions. In addition, we
received helpful comments from participants at numerous conferences and seminars. We are also indebted
to Daisuke Oyama and other members of the Banking Theory Study Group for helpful discussion. Takayuki
Tsuruga gratefully acknowledges the …nancial support of the Grant-in-aid for Scienti…c Research and theJapan Securities Scholarship Foundation. Views expressed in this paper are those of the authors and do notnecessarily re‡ect the 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.jp1

1

Introduction

Since the global …nancial crisis for 2007-08, one of the challenges posed for macroeconomists
has been how to replicate systemic …nancial crises and ensuing sharp contraction in macroeconomic activity in dynamic stochastic general equilibrium (DSGE) models. To generatesuch large declines in output, many preceding macroeconomic models with …nancial frictions
assume relatively large shocks to economic fundamentals as a rare event and examine howsuch a “rare but large” shock ampli…es macroeconomic ‡uctuations.1 From this viewpoint,
the 2007-08 systemic …nancial crisis could be interpreted as an unavoidable and unfortunate
accident arising from tail risks. In the meantime, Bernanke (2012) argues that the triggers ofthe 2007-08 crisis were “quite modest”in size while heavy dependence on short-term funding,high leverage, and inadequate risk management in the (shadow) banking sector that engaged
in maturity transformation was key vulnerability of the system for the devastating outcomesof the crisis. In a related context of the underlying vulnerabilities, others argue that there
might have been erosion of discipline owing to the anticipated bank bailouts.2
This paper develops a dynamic general equilibrium model with maturity-mismatchingbanks and explores how individual banks could take on excessive systemic risks that could
result in a devastating contraction in macroeconomic activity. We incorporate the banksmodeled by Diamond and Rajan (2012, DR) into a two-good overlapping-generations (OLG)
model where the capital good is reproducible. In this model, bank overleverage arises due
to pecuniary externalities. Recent studies on the pecuniary externalities have reached abroad consensus that pecuniary externalities, or credit externalities, tend to prevent thelaissez-faire (LF) economy from achieving the social optimum (e.g., Benigno, Chen, Otrok,
Rebucci, and Young 2012, Bianchi 2010, 2011, Bianchi and Mendoza 2010, Jeanne andKorinek 2012a, Korinek 2010, Lorenzoni 2008, Mendoza 2010, Nikolov 2010, and Stein 2012).
In line with these previous studies, our result of the bank overleverage is juxtaposed withthe second-best allocations which a social planning (SP) agent achieves subject to the same…nancial constraint as the LF economy. In our model, however, the source of the pecuniary
externalities di¤ers from those of the earlier studies. Many of the previous studies include
borrowers’collateral constraints a¤ected by price changes that the agents in the LF economy
do not internalize. By contrast, the pecuniary externalities in our model operate on the
1

A few examples include: Gertler and Karadi (2011), Gertler and Kiyotaki (2011), and Christiano, Mottoand Rostagno (2010).
2
See Rajan (2010), Diamond and Rajan (2012), and Fahri and Tirole (2012). They argue the idea using
the widely acknowledged term, “the Greenspan put.”2

solvency constraint a¤ected by changes in the price of the banks’ illiquid assets (i.e., thepro…tability of entrepreneurs’capital investment). Notably, the solvency constraint arises inour model as a natural consequence of incorporating the maturity-mismatching banks thatissue non-state-contingent short-term debt and invest them in illiquid asset.3 Based on themodel, we discuss how the bank overleverage leads to ine¢ ciently high crisis probabilities
and sharp declines in economic activity after a crisis. We also explore how existing policy
measures to enhance the resilience of banking systems a¤ect crisis probabilities.
Key elements of the model are (i) issuance of non-state-contingent debt by banks and(ii) the banks’ risk exposure to capital prices which a¤ects the banks’ solvency. The …rst
element enables banks to raise funds and to promote liquidity creation in the absence ofthe complete markets, as discussed in Diamond and Rajan (2001a, b). But this bene…tof non-state-contingent debts comes with the cost of potential insolvency of banks, and in
extremis of …nancial crises. Facing this risk of insolvency, banks need to strike the rightbalance between marginal cost and bene…t of increasing their leverage. When a …nancial
crisis is precipitated, banks are required to liquidate all illiquid loans to entrepreneurs to
repay depositors, which discontinues the capital goods production and subsequently resultsin large contraction in output. The second element is incorporated with our OLG frameworkthat explicitly includes factor markets. In our model, banks’balance sheet depends on thecapital price for which entrepreneurs sell their own produced capital in the capital goodmarket. This dependence of banks’ balance sheet on capital price implies capital-pricedependent banks’ solvency which leads to excessive risk-taking in the banking sector as awhole.
The intuition behind banks’excessive risk-taking can be understood as follows. Financial
crises in our model are precipitated only if a liquidity shock exceeds a certain thresholdbecause of the non-state-contingency of their debt. As in DR, a highly leveraged bankholding illiquid assets is more vulnerable to liquidity shocks. If a single bank …nds it optimal
to increase their leverage, the bank needs to keep more illiquid asset until maturity toconsolidate their balance sheet. In the absence of heterogeneity, however, every other banktakes the same action based on the same idea, holding on to illiquid asset in the economy asa whole. Therefore, the systemic risk-taking based on the same idea promotes entrepreneurs
to continue long-term capital producing projects. As a result, a larger supply of capital3

Our model assume that banks issue demand deposits following DR. The banks in our paper, however,can be broadly interpreted as …nancial intermediaries that raise funds via short-term debt such as repo andcommercial paper, and transform maturities on their balance sheet. Demand deposits are an extreme caseof short-term debt.

3

reduces the capital price and, accordingly, the returns of illiquid assets in a banking system
as a whole. This reduction in the capital price further erodes the individual banks’solvency.
Even though all the LF banks understand a further erosion of their solvency due to thecapital price declines, they do not internalize this pecuniary externality. Each atomic LF
bank is simply sel…sh and lack incentives to coordinate with each other to a¤ect the capital
prices. Put another way, pecuniary externalities create the wedge between the social andprivate marginal cost of increasing the leverage and ill-incentivizes banks to take on excessive
risks systemically. In our benchmark simulations, the crisis probability in the LF economy
is 6.6 percent, compared to 4.5 percent in the constrained social optimum.
We further suggest several policy implications from our model. Given that a variety ofpolicy measures to enhance the resilience of banking systems are currently being explored in
the real world, we assess the realistic policy measures in terms of crisis probabilities. One
highlight is that pre-announced bank bailouts may ill-incentivize banks to take on even higher
risks, adding to the even higher crisis probabilities. A similar argument is likely to be applied
to pre-commitment to a low interest rate policy that aims to prop up a banking systemnear crisis for the purpose of emergency liquidity provision (ELP). In principle, if banks
are informed in advance that they would be bailed out by, for example, publicly suppliedliquidity at a time of elevated market distress, they have little reason to be better prepared
for an extremity through precautionary deleveraging. While public supply of liquidity per
se has the e¤ect of curbing the crisis probability, banks’anticipation of such a bailout canreverse the expected outcome, resulting in even more frequent crises. Finally, we argue thatbanks’capital requirement with prompt corrective action (PCA) may be less exposed to suchrisks of exacerbation of excessive risk-taking in the banking system.
Our paper is related to previous studies in at least three strands of literature. First,
our model relies on the theory of banking in terms of the micro-foundation of banks. Ourmodel is a straightforward extension of Allen and Gale (1998) and DR in terms of thebasic modeling approach of a banking system. In our model, however, the socially optimal
allocation cannot be achieved in the LF economy in contrast to their model. Their model
includes a single liquidity market where the banks can internalize all the e¤ects throughthe single price changes. In contrast, our model includes multiple markets (e.g., the capital
market), in which banks take the price as given and fail to internalize the general equilibriume¤ect of the price.
Second, our paper contributes to the literature on macroeconomic models with banks.
Macroeconomic models with banks have primarily focused on how …nancial frictions amplify
4business cycles (e.g., Gertler and Karadi 2011, Gertler and Kiyotaki 2011, Meh and Moran
2010) rather than how and why devastating …nancial crises could take place sporadicallybeyond the business cycle frequency. The focus of our model starkly contrasts with thesestudies in the following ways: we aim to (i) explore a rationale for government interventionand (ii) account for the vast standstill in …nancial intermediations and subsequent sharp
declines in output and investment, both of which the global economy recently experienced.
We also stress that our model with banks investigates how frequently …nancial crises are
precipitated as a result of banks’ insolvency. Angeloni and Faia (2012) incorporate banks
à la Diamond and Rajan (2000, 2001a) into the DSGE model. Whereas their model has
the endogenous probability of bank insolvency, the probability of the bank insolvency canbroadly be interpreted as a measure of individual bank fragility rather than the probabilityof …nancial crises in which the great majority of the banking system comes to a standstill.
Finally, as we mentioned, our paper is closely related to the literature on pecuniary
externalities which explores over-borrowing or over-credit based on the borrowers’collateral
constraint. Our model is based on the theory of banking and emphasizes that the solvency
constraint plays a crucial role in generating overleverage of the LF banking sector. In thisregard, Stein (2012) introduces the macroeconomic model with banks that are faced with thepecuniary externalities. While he focuses more on how to contain risks of …re sales, we assess
the probability of ine¢ cient …nancial crises by explicitly modeling banks that are subject torisks of systemic …nancial crises.
Perhaps, the models that are the closest to ours is Gertler and Kiyotaki (2012) andBoissay, Collard and Smets (2012). In Gertler and Kiyotaki (2012), whether an equilibriumwith bank runs exists depends on macroeconomic fundamentals, but …nancial crises per seare precipitated by self-ful…lling expectations as in Diamond and Dybvig (1983). In ourmodel, …nancial crises are precipitated by fundamental shocks (liquidity preferences) rather
than self-ful…lling expectations. In this regard, our model seems to be broadly in line with
empirical …ndings of the business cycle view by Gorton (1988) and Allen and Gale (1998).
In other words, …nancial crises in our model are not unpredictable, entirely random events,
but the consequence of excessive risk-taking of banking systems. Boissay, Collard and Smets(2012) focus on the moral hazard in the interbank markets and successfully replicate systemic…nancial crises followed by deep recessions. On the other hand, we stress the fragility of thematurity mismatching banks as highlighted by the Gorton and Metrick’s (2012) “run-onrepo”view.
The rest of the paper proceeds as follows. Section 2 illustrates the macroeconomic model5

with maturity-mismatching banks and characterizes the banks’optimal leverage in the competitive equilibrium. In Section 3, we compare the LF equilibrium in Section 2 with theallocation achieved by the SP banks and explain why a competitive banking sector tends tobe overleveraged. Section 4 discusses numerical results. Section 5 assesses policy measures
aimed at reducing crisis probabilities and discusses policy implications. Section 6 concludes.

2

A Macroeconomy with Banks

2.1

Agents, Endowment, Preferences, and Technology

We consider an in…nite-horizon OLG model incorporating banks with a maturity mismatch.
Each generation of agents consists of households, entrepreneurs, and bankers. Each period,
generation t is born at the beginning of period t and lives for two periods, t and t + 1. Eachagent is identical and constant in the population. Furthermore, an initial old generation
lives for one period and the subsequent generations live for two periods.
Households are risk averse and subject to a liquidity shock that a¤ects their preference
for consumption over the two periods. The liquidity shock is an aggregate shock and the only
source of the uncertainty in the model. The households aim to smooth their consumption
intertemporally. Following DR, households are endowed with a unit of consumption goods
at birth and do not consume the initially endowed consumption goods at the beginning of
period t. The households deposit all initial endowments at banks operating in the samegeneration.4 They receive wages wt in the competitive labor market by supplying one unitof labor in both periods, t and t + 1.
Entrepreneurs are risk neutral and have access to capital-producing technology. They
launch long-term investment projects at the beginning of period t, by borrowing households’
endowments via the banks in the same generation. The investment project requires one
period for gestation, and capital goods are produced in period t + 1. We call this capital
producing technology a “project.” Entrepreneurs sell the capital goods in the competitivemarket for the capital goods price qt+1 .
Banks raise funds from households and lend them to entrepreneurs at the beginning of
period t.5 In principle, we follow Diamond and Rajan (2001a) to model banks. Banks
4

We implicitly assume intra-period perishability of endowments. More precisely, all endowments perish
before the realization of the liquidity shock.
5
We assume intra-generational banking, which e¤ectively means that all bankers of generation t die outat the end of period t + 1.

6

are risk neutral and competitive at raising and lending funds in the markets. They issuedemand deposits (short-term debt) and commit to repaying the households. In the nature ofdemand deposits, banks can provide insurance against depositors’liquidity shocks. However,when households demand repayment before the completion of the entrepreneurs’ projects,
banks must liquidate premature projects to meet the demand for repayment. This maturity
mismatch, represented by the combination of long-term assets and short-term liabilities,
leaves banks exposed to risks of a default because, depending on the amount of withdrawals
in the interim period, the banks’solvency is endangered.
The technology to produce consumption goods Yt is represented by a standard constantreturns-to-scale Cobb-Douglas production function:
Yt = F (Kt ; Ht ) = Kt Ht1;

where Kt and Ht denote the capital stock and hours worked, respectively. Demand for laborand capital satis…es
wt = FH;t = (1

)
Kt
Ht

qt = FK;t =

Kt
Ht

(1)

1

:

(2)

Accordingly, the second derivatives are denoted by FKK;t ; FHH;t , and FHK;t .
In what follows, we describe each agent’s decisions (consumption, withdrawal, and liquidation of the entrepreneurs’projects) after the liquidity shock is realized. Then, we moveon to the bank’s decision on its leverage before the realization of the liquidity shock. Table
1 summarizes the sequence of events in each generation.

2.2

Households

Under the competitive banking sector, each household accepts the banks’ o¤er on deposit
face value Dt at the beginning of period t, and observes the liquidity shock t in the middleof period t. The liquidity shock is common across all households in the same generation andhas the probability density function f ( t ) with a support of [0; 1]. This shock representshouseholds’ preference for consumption when young and signals the need for liquidity in7period t.6
After the realization of t , households make their decisions for consumption smoothing without uncertainty. Given that a crisis does not take place, households then choose
withdrawal amount gt to maximize
U (C1;t ; C2;t+1 ) =t log C1;t + (1t ) log C2;t+1

s.t. C1;t = wt + gt

(3)

C2;t+1 = wt+1 + Rt (Dtgt ) ;

where C1;t and C2;t+1 denote the consumption of households born in period t when young
and old, respectively. Each household supplies a unit of labor in each period and receives
wage income wt in period t and wt+1 in period t + 1. Here Rt denotes the one-period grossinterest rate from period t to t + 1.
In our model, a …nancial crisis takes place with the endogenous probability t , dependingon the realization of t : With the probability 1
t , a …nancial crisis is not taking place andhouseholds can withdraw gt in period t and all the remaining deposits in period t + 1.7 With
the probability t , however, a …nancial crisis arises and households’withdrawals amount to
the liquidation value of premature projects, X (< 1), in period t and nothing is left in periodt + 1. In the case of a crisis, households fail to smooth out their consumption and end up
with C1;t = wt + X and C2;t+1 = wt+1 .
When the households can smooth out their consumption, the intertemporal …rst-ordercondition for consumption is satis…ed:
t

1

t

C1;t
C2;t+1

1(4)= Rt :

Given the Euler equation (4), the withdrawals in the absence of a crisis can be written as
gt =

t

wt+1
+ Dt
Rt

6

(1

t ) wt :(5)

Although in fact all households are subject to the same aggregate shock, we assume that an in…nitesimallysmall number of households are believed to face a di¤erent t from other households. This assumption ensuresthe existence of a Nash equilibrium, in which all households run to the banks when households believe thatthe banks are insolvent under the observed t :7
In the maximization problem of households, we assume that wage income in period t is low relative tothe initial endowment, ensuring a non-negative withdrawal gt in the equilibrium.

8The withdrawal function implies that large
crisis.

2.3

t and Dt are likely to precipitate a …nancial

Entrepreneurs

Entrepreneurs are risk neural and maximize their expected lifetime utility represented byeeeedenote entrepreneurs’consumption when young andand C2;t+1; where C1;t
+ C2;t+1
E C1;told. They use a unit of consumption goods …nanced from banks for their capital goods production, and this production technology takes one period for gestation before its completion.
In period t + 1, the project yields a random capital goods output !, which is uniformly
distributed over [! L ; ! H ] with the probability density function h (!).8 If this project is prematurely liquidated in period t, the transformation from the consumption goods into capitalis incomplete. As a result, the output is reduced to X units of consumption goods and isrepaid fully to banks in period t. When the project is completed in period t + 1, however,entrepreneurs can sell their output in the capital goods market for the capital price qt+1 .
Each entrepreneur can borrow from a bank who has, or can learn until the project tomature, knowledge about an alternative, but less pro…table, method to operate the project.
The bank’s speci…c knowledge allows it to generate qt+1 ! from a project outcome with< 1. Once a bank has lent, no one else (including other banks) can learn this alternative
way to operate the project. As a result, entrepreneurs accept the …nancing contract with eachbank and repay qt+1 !. They are left with 1of the share of their pro…t and enjoy theirown consumption based on their linear utility. We assume that entrepreneurs are endowedwith I units of capital goods at the beginning of period t + 1.9 They sell this endowment
capital together with the newly created capital made from the consumption goods.

2.4

Banks

b
b
Banks are also risk neutral and maximize their expected lifetime utility E C1;t
+ C2;t+1,bbwhere C1;tand C2;t+1
denote consumption of banks when young and old. We borrow the
micro-foundation of the banking business from Diamond and Rajan (2001a, b, 2012). Banks
have no initial endowment at birth and thus they need to raise funds from households. As
8Following the literature, we take the assumption that there is no aggregate uncertainty in the project
outcome.
9
For simplicity, we assume a 100 percent depreciation rate in the law of motion for capital. The introduction of the endowment of capital goods here guarantees a …nite capital price in the aftermath of a …nancial
crisis in which all projects are scrapped due to full liquidation.

9the relationship lender, a bank has the knowledge to operate the entrepreneurs’project thatcannot be transferred to households. Banks issue demand deposits (short-term debt) as acommitment device to compensate for the lack of transferability of their knowledge (i.e.,collection skills).10 As discussed by a number of early studies, a collective action problem fordepositors is created by this demandable nature of deposit contracts: depositors run to thebanks whenever depositors anticipate that the banks cannot honor the debt. In the theoryof banking, the deposit contract is predetermined before observing the liquidity shock. In
our model, Dt is predetermined at the beginning of period t, and a liquidity shock is realized
in the middle of period t.
Each bank attracts many entrepreneurs through a competitive o¤er on the loan, resulting
in an identical portfolio shared by all the symmetric banks. This setup e¤ectively leads toa convenient outcome in the model: each bank and the aggregate economy face an identical
distribution of entrepreneurs. In period t, the banks receive signals ! that perfectly predictthe realized value of ! in period t + 1. With this information ! and the households’liquidity
demand observed in period t, each bank chooses one of the options: (i) to liquidate projects
in period t, obtaining X of consumption goods per project; or (ii) to collect a fraction qt+1 !
from a completed project in period t + 1. The bank liquidates the project if the outcome ofa project falls short of !
~ t+1 , de…ned as a function of Rt =qt+1 :11!
~ t+1 =

X Rt
:
qt+1

(6)

Otherwise, the bank continues the project, and then receives repayment of qt+1 ! and entrepreneurs consume the remaining fraction of outcome, (1) qt+1 !, per project. After
repaying the full amount of the households’withdrawals, the banks consume their own capital.
Let the banks’asset be A (Rt =qt+1 ). The banks’asset at the beginning of period t (i.e.,
10

Diamond and Rajan (2001b) discuss the micro-foundation of maturity mismatching banks and explain
why this demand deposits can promote liquidity creation under the lack of transferability of their collection
skill.
11
Equation (6) can be reinterpreted as follows: qt+1 !=X corresponds to the marginal rate of transformation (MRT) between the period-t consumption goods (i.e., liquidation) and the period-t + 1 consumption
goods (i.e., continuation of projects). The MRT is here compared with the marginal rate of substitution of
the households that is observed as the interest rate, Rt .

10

prior to the withdrawals) can be expressed as
A

Rt
qt+1

Z !~ t+1

Z
qt+1 !H=Xh (!) d! +!h (!) d!
Rt !~ t+1
!L
Rt
qt+1
Rt
= L
+
I
:
qt+1
Rt
qt+1

(7)

Note that h (!) is interchangeable with h (!) owing to the perfect signaling. The banks’
asset denoted in (7) can be decomposed into two components: the values of the premaR !~ t+1
turely liquidated projects denoted as Lt = L (Rt =qt+1 )Xh (!) d!; which is used to!Lmeet the liquidity demand (i.e., withdrawals) from the households, and the banks’share of
the investment output (measured in the present value of consumption goods) denoted as
R !H
qt+1 It =Rt ; where It = I (Rt =qt+1 ) = !~ t+1!h (!) d!. The banks are subject to the solvency
constraint Dt
A (Rt =qt+1 ). Owing to the uniform distribution assumption for !, it canbe easily shown that A ( ) monotonically decreases with Rt =qt+1 . We can then de…ne therelative price Rt =qt+1 that satis…es the solvency constraint with equality
Dt = A

Rt
qt+1

(8)

:

We refer to Rt and qt+1 as the threshold interest rate and capital price, respectively. Hereafter, we denote a variable with an asterisk as the variable on the threshold. For the purposeof subsequent discussion, we note that given A (Rt =qt+1 ), the bank leverage Dt =(At Dt ) is
uniquely determined once Dt is chosen, and hence we refer to Dt as leverage hereafter. We
will discuss this issue in Section 4 in terms of numerical interpretation.

2.5

Market Clearing Conditions

Four markets need to clear in the competitive equilibrium: (i) liquidity; (ii) consumption
goods; (iii) capital goods; and (iv) labor. The liquidity market clearing condition is given by
L

Rt
qt+1

=

t

wt+1
+ Dt
Rt

t ) wt :(9)eb= C1;t + C2;t + C2;t
+ C2;t:(10)(1

Next, the market clearing condition for consumption goods is
Yt + L

Rt
qt+1

11

The left-hand side of (10) includes the supply of goods from the liquidated projects. On thebe
are consumption when generation t 1 is old.
, and C2;tright-hand side of (10), C2;t , C2;tThe capital goods market clearing condition is
Kt+1 =(

I + I (Rt =qt+1 ) at normal times
Iat crises.

(11)

Here the equation suggests that the capital goods supply sharply declines, in the aftermath ofa crisis. Throughout the paper, we use w and F H to denote the wage rate and the marginal
product of labor evaluated at Kt+1 = I.
Finally, both young and old generations supply a unit of labor in each period. Therefore,
Ht equals two for all t.

2.6

Optimal Bank Leverage

We now consider the banks’optimal leverage, which is chosen before the realization of theliquidity shock. We focus on the laissez-faire (LF) banks in this subsection, and will discuss
the social planning (SP) banks in Section 3.
The banks are competitive at issuing demand deposits, and we assume that households’
endowments are scarce in comparison to entrepreneurs’projects. As a result of competition,
the banks make a competitive o¤er of deposits for households, aiming to maximize thehousehold welfare (Allen and Gale 1998, 2007), while in fact they are maximizing their own
pro…ts. Maximizing the household utility via the deposit o¤ers means that banks internalize
the liquidity market clearing condition in determining the o¤er. Through this internalization,
the banks take into account possible changes in the crisis probability t . On the other hand,outside the liquidity market, they take the capital prices and wages as given.
To understand how the banks’ choice of Dt a¤ects t ; we take three steps. First, wede…ne a function RLF as
RLF (Dt ) = qt+1 A 1 (Dt ) ;(12)from (8). We emphasize that, in RLF (Dt ) ; the threshold capital price qt+1 is treated as aparameter, re‡ecting the price-taking behavior of the LF banks. Second, using (9), we de…nea function LF as
L Rt =qt+1 + wt;
(13)
LF (Dt ) =wt + Dt + wt+1 =Rt
where Rt = RLF (Dt ) while wt , wt+1 , and qt+1 are given parameters for the LF banks. We
12reemphasize that these prices are not constant parameters but in fact vary according to (1)and (2). But for the LF banks, when they determine the optimal Dt , they just take them asgiven. With Rt = RLF (Dt ) and other threshold variables, the threshold level of the liquidity
shock t = LF (Dt ) clears the liquidity market with the LF banks as shown in (13).
The …nal step is to connect t to the crisis probability t : The above-de…ned LF (Dt )means that any changes in Dt always give rise to changes in t for the liquidity market toclear. By the solvency constraint with equality (8), any level of Dt , once chosen, determines
the threshold relative price, Rt =qt+1 . Hence, t can be interpreted as the liquidity shock on
the brink of a …nancial crisis. Namely, when t is strictly greater than t , the banks turn out
to be insolvent and a crisis is precipitated. Thus, the crisis probability t has a one-to-one
relationship to t via the probability density function f ( t ):
t =

Z 1

(14)

f ( t) d t:t

In sum, the banks’choice of the leverage speci…es Rt =qt+1 and this threshold relative pricedetermines the threshold level of the liquidity shock t , completing the link between thebank leverage and the crisis probability.
We are now ready to set up the optimization problem for the banks to determine the sizeof their leverage. In the problem, as discussed, banks take into account the endogenously
changing t .
Problem LF In a laissez-faire economy, banks maximize the household expected utility
Z

t

max
f t ln (wt + Lt ) + (1t ) ln [wt+1 + Rt (Dt
Dt
0
Z 1
+
[ t ln (wt + X) + (1t ) ln (w)] f ( t ) d t ;Lt )]g f ( t ) d t(15)t

subject to (9) and (13).
The banks choose their leverage according to the following …rst-order condition:
t log

=

Z

0

t

1
mt

t mt

wt + X1+ (1t ) log

Rt (1wt+1 0R (Dt ; t ) + (1Rt2 LF13t ) mt

w
t)

d t 0
(Dt )
d t LF

0RLF(Dt ; t )f ( t) d t;
Rt

(16)

wt + Dt +where mt
wt + Dt + wt+1 =Rt is the lifetime income of households and mt0
wt+1 =Rt , accordingly. More importantly, LF (Dt ) is calculated from (13), taking capital
prices and wages as given in line with the behavior of the LF banks. Likewise, with a slight0
abuse of notation, RLF(Dt ; t ) denotes marginal changes in Rt with respect to Dt , given t :By (9),
0(Dt ; t )RLF

tL0 =qt+1 +2
t wt+1 =Rt

>0

(17)

can be obtained where L0 is the derivative of Lt with respect to Rt =qt+1 .
Equation (16) provides an economic interpretation that is in line with broad intuition.
The terms in brackets on the left-hand side of (16) represent the loss of utility in a crisis
compared to the threshold. From (14), the term outside the brackets indicates the marginalchanges in a crisis probability with respect to bank leverage. The left-hand side of the equation consists of the expected loss of utility and the marginal change in the crisis probability.
Simply put: the left-hand side of (16) is the marginal cost of increasing Dt .
The right-hand side of (16) consists of the e¤ects of increasing leverage on the expectedhouseholds’utility through their lifetime income. On the one hand, the increase in Dt has
an outright positive e¤ect on the households’income: the higher the leverage, the larger thewithdrawal, allowing households to enjoy more consumption. On the other hand, the increasein Dt leads to a higher interest rate via liquidity shortage, discounting the households’labor
income in period t + 1 and reducing returns on forgoing withdrawal until period t + 1. Hence,as far as the outright e¤ect on the lifetime income exceeds the e¤ect on the interest rate, thehigher leverage is bene…cial to households. Simply put: the right-hand side of (16) is themarginal bene…t of increasing Dt .
We …nally de…ne the equilibrium in the LF economy as follows.
De…nition (Laissez-faire economy) A competitive equilibrium consists of allocations
and prices fgt ; Dt ; Lt ; Kt ; It ; Ht ; Rt ; qt ; wt g1
t=0 such that (i) withdrawal decisions are given by(5) for tt ; (ii) banks’leverage satis…es (16); (iii) banks’liquidity supply is determinedby (6); and (iv) all markets clear.

14

3

Systemic Risks and Welfare

3.1

Social Planning BanksThis section introduces the SP banks which choose their leverage as the constrained social
planner. We characterize the allocations for the SP banks as the constrained social optimum
and compare the allocations with those in the LF economy. To lead o¤ the analysis, weclarify the constraint to which the SP banks are subject. We assume that the SP banks
must make all their decisions before observing t . After realizing t , they are left withno options. In other words, the SP banks are subject to the constraint that they canneither control households’behaviors nor choose their outright consumption levels becausehouseholds can react to any realized value of t . The allocations chosen by the SP banks must
be distinguished from the unconstrained, …rst-best optimum. Under the …rst-best optimum,
the Arrow securities that pay o¤ contingent on all possible realizations of t are available andhence households can enjoy the maximum utility without experiencing any …nancial crisis.
But we cannot disregard a maturity mismatch and resulting …nancial crises. As such, we
assume that the SP banks are entities engaged in a maturity mismatch and pre-commit to
payment on their debt regardless of the states realized following their commitment. The
extra ability given to the SP banks compared to the price-taking competitive banks is thatthe former can internalize all price e¤ects in all markets when they make decisions regarding
their leverage.
The SP banks do not take the factor prices as given, but take into account their changes
re‡ecting the marginal product. Formally, we replace qt and wt with FK;t and FH;t , respectively. Note that, nonetheless, the SP banks take the households’behaviors as given, as theycannot make their contract contingent on t . In other words, households always choose theirconsumption and withdrawal given Dt pre-committed by the SP banks.
To specify the problem for the SP banks, we clarify the solvency constraint with which
the SP banks are faced, Dt A (Rt =FK;t+1 ) : We note that the constraint e¤ectively remains
the same as in Problem LF because qt+1 = FK;t+1 from (2). The newly introduced solvency
constraint for the SP banks, however, has di¤erent e¤ects on the threshold because (8) and(12) are now replaced with
Dt = ARt
FK;t+1!

RSP (Dt ) = FK;t+1 A 1 (Dt ) ;
15

(18)
(19)

respectively, in the problem for the SP banks where the SP banks can internalize generalequilibrium e¤ects of the factor prices.
We summarize the SP banks’problem as follows:
Problem SP The social planning banks maximize the household expected utility,Ztf t ln (FH;t + Lt ) + (1maxt ) ln [FH;t+1 + Rt (Dt
Dt
0
Z 1
+
[ t ln (FH;t + X) + (1t ) ln F H ] f ( t ) d t ;Lt )]g f ( t ) d tt

subject to
L

Rt

=

FK;t+1SP (Dt )FH;t+1+ Dt(1Rt
L Rt =FK;t+1 + FH;t;
FH;t + Dt + FH;t+1 =Rttt ) FH;t

(20)
(21)

where Rt = RSP (Dt ) from (19) and t = SP (Dt ) from (21).
Note that all the factor prices, including qt+1 and wt+1 in Problem LF, are replaced with
marginal products in Problem SP. More importantly, because the SP banks factor in all0
general equilibrium e¤ects, SP(Dt ) can be denoted as d t =dDt . The solution of Problem
SP is conceptually comparable to the constrained optimum as discussed in Allen and Gale
(1998).
In Problem SP, all the factor prices in period t + 1 are functions of Kt+1 or Kt+1 . In thiscontext, the capital goods market clearing condition in Problem SP needs extra attention
when we replace the capital price in (11) with the marginal product. We note that Kt+1
depends solely on the market interest rate Rt and this relationship is denoted by a function(Rt ). Provided that a crisis does not take place, Kt+1 evolves according toKt+1 = I + I

Rt
FK;t+1(22)(Rt ) ;

where 0 < 0 represents the derivative of Kt+1 with respect to Rt .12 To move on, along withthe labor market clearing condition Ht = H = 2 for all t, we rea¢ rm that FK;t+1 and FH;t+1
12

Solving

0

= (1

0

Rt FKK;t+1 =FK;t+1 ) I 0 =FK;t+1 for

16

0

ensures that0is negative.

in (18), (19), and (21) can be written as
FK;t+1 = FK Kt+1 ; H = FK [ (Rt )]FH;t+1 = FH Kt+1 ; H = FH [ (Rt )] :Our …rst main result is as follows. With the factor prices replaced by marginal products
in Problem SP, the allocations that the SP banks achieve di¤er from those achieved by the
LF banks because of the extra ability given to the SP banks. Comparison between Problems
LF and SP con…rms that the two problems are subject to exactly the same constraints.
With identical constraints, any discrepancy in the …rst-order conditions generally resultsin di¤erent allocations across the two problems. To see this, for example, we focus on00
SP = d t =dDt and LF , both of which are the part of the …rst-order conditions in eachproblem:0
SP (Dt )=0
LF (Dt )

where

"
0

(

0

=Rt )

h

Rt =FK;t+1

1
mt
1
mt
2

@
(L
@Rt t
@
(L
@Rt t

L 0 FKK;t+1 +

gt )

"

dRt
dDt

0
gt ) RLF
(Dt )

t

mt
t

mt

;

(23)

;

(24)

i
0
F
t HK;t+1 and dRt =dDt = RSP (Dt ). Along-side,
and L 0 in " represent 0 (Rt ) and L0 Rt =qt+1 , respectively.
In general, non-zero " ensures the di¤erence between the two equilibria, while in factcomparison of the two …rst-order conditions reveals further discrepancies in addition to " .
0(Dt ) and dRt =dDt =The next subsection focuses on the key discrepancy between RLF
0RSP(Dt ), which provides a clear economic interpretation.

3.2

Crisis Probabilities and Marginal Systemic Risk (MSR)

To facilitate assessment of the systemic risk of an economy, we de…ne marginal systemicrisk (MSR) as the marginal increase in the crisis probability against a unit change in bankleverage at and around the equilibrium.13 Speci…cally, let DLF;t be the level of bank leverage
13

MSR can be applied in broad models where a …nancial crisis takes place as a non-zero probability event.
Depending on the focus of studies, MSR can be de…ned vis-à-vis bank leverage, aggregate credit, banklending, or potentially, asset prices.

17chosen in the LF economy. Then,d t 0(DLF;t )d t k=f ( t ) k0 (DLF;t ) for k = LF; SP:

M SRk;t =(25)

Recall that a bank in the LF economy takes other banks’ decisions as given, but in fact
the crisis probability is a¤ected by the synchronized decisions by the banking sector as a0
(DLF;t ) d t =d t can be interpreted as the marginal risks perceived
whole. In this regard, LF
by the individual price-taking banks, which can be contrasted with the true marginal risk,0
SP (DLF;t ) d t =d t at the chosen Dt = DLF;t . Technically, MSR is applicable for any levelof Dt . We utilize MSRs evaluated at Dt = DLF;t because this allows us to directly compare
the MSRs across di¤erent problems under the same allocations and prices.
With this interpretation in mind, we stress that, if M SRLF;t is smaller than M SRSP;t ,the gap indicates that the banking sector in the LF economy underestimates the marginal
cost of higher leverage by not taking into account the systemic risks. As a result, the LF
banks are likely to be overleveraged at (and around) the LF equilibrium. Instead, if M SRLF;ttakes a larger value than M SRSP;t , the gap points to underleverage in the LF economy. In
principle, an undervaluation of M SRLF;t compared to M SRSP;t would provide the groundfor government intervention to rein in excessive leverage of banks, because such regulatory
risk reduction can improve welfare.
Our second main result is that M SRSP;t exceeds M SRLF;t , relying on the concept of theMSR. For the formal proof, our second main result requires one technical condition:
Condition 1 "g;t (1) "L;t for Dt = DLF;t , where "g;t = @ log gt =@ log Rt and "L;t =@ log Lt =@ log Rt are the elasticity of the liquidity demand and supply with respect to Rt in
the liquidity market, respectively.
Condition 1 ensures that "0 for a chosen Dt . This condition is likely to be satis…ed
because, in general, the capital share takes a substantially smaller value than the labor
share (e.g., = 1=3). With the Cobb-Douglas production function, these shares, and 1,
can also be interpreted as the elasticities of wages and the capital price with respect to Kt+1 ,respectively, both of which translate into demand and supply in the liquidity market. The
SP banks internalize factor price changes via the production technology but the LF banks donot. Due to this internalization by the SP banks, a discrepancy " arises. This " is positive
under Condition 1 since we assume that the capital price is more sensitive to changes in18

capital than wages and/or the supply curve is relatively ‡atter than the demand curve withrespect to Rt in the liquidity market. Now, we are ready to state Proposition 1.
Proposition 1 Under Condition 1, M SRSP;t is strictly larger than M SRLF;t .
Proof. See Appendix A.1.
Proposition 1 provides a foundation for understanding why the crisis probability is higher
in the LF economy than in the social optimum. From (23) and (24),
M SRSP;t0
0(DLF;t )M SRLF;t = f ( t ) [ LF
SP (DLF;t )]f ( t)@
0=(Lgt ) [RLF(DLF;t )
mt
@Rt t

0
RSP
(DLF;t )] +

"

:(26)

Note that all functions are evaluated at the LF equilibrium (i.e., Dt = DLF;t ). In (26),f ( t ) =mt and the slope of the excess liquidity supply function denoted as @ (Lt gt ) =@Rt
are both positive. Condition 1 ensures that we are left with the deviation of the changes in
Rt with respect to Dt . From (12) and (19), the inverse function theorem yields
0
RLF(DLF;t )0
RSP(DLF;t ) =

=

1

1
1

FKK;t+1

Rt
F
qt+1 KK;t+1

0 R =q
t
t+1

0 dRt

dDt

> 0;

!

qt+1
A0
(27)

0
where A 0 = A0 Rt =qt+1 . The sign of dRt =dDt = RSP(DLF;t ) is ensured to be negative asshown in Appendix A.1.
Equation (27) indicates that an individual price-taking bank underestimates the marginalchanges in Rt due to the pecuniary externalities. An increase in leverage reduces Rt , because,in general, highly leveraged banks would be more likely to default under a lower thresholdinterest rate. But the perceived reduction in Rt di¤ers between the LF and SP banks andthis gap creates the wedge in the two MSRs in our model. In this regard, this wedge re‡ects
the LF banks’ underestimation of the marginal cost of higher leverage. As a result of theunderestimation of the marginal cost, the LF banking system …nds itself insolvent more
frequently than expected.
To better understand the gap in the MSRs, we can focus on FKK;t+1 0 , included in (27).
This term points to a side e¤ect arising from higher leverage: in general equilibrium, thereduction in Rt increases Kt+1 = (Rt ). That is, the lower Rt stimulates capital supply

19

on the threshold and this increase in Kt+1 triggers the decline in the threshold capital priceqt+1 via the lower marginal product of capital.14 With this side e¤ect, the lower capital
price further undermines the bank’s solvency, compared to the case without the side e¤ectof increasing the leverage. Because the atomistic banks do not take into account this sidee¤ect, the lower-than-expected capital price and the undermined banks’solvency raise theprobability of a …nancial crisis compared with the economy with the SP banks.
Looking at the real-world experience of past …nancial crises, it may be pointed out that,with hindsight, outlooks regarding asset prices frequently tended to be overly optimistic in
the run-ups to crises. Some argue that such over-optimism arises from irrationality. While
we do not claim that irrational behavior is irrelevant, our model suggests that despite the fullrationality, pecuniary externalities can result in seemingly irrational over-optimism. The key
to understanding the externalities lies in the synchronized decisions made by the individual
banks in a competitive sector. For each bank, capital prices are given but the given pricesa¤ect the solvency of the banking system as a whole. In general, maturity-mismatchingbanks face solvency constraints because they issue non-state-contingent debt. As long as
the e¤ects of the asset prices on their solvency are not internalized, distortion arises. In ourmodel, the distortion shows up as the overleveraged banking sector with a higher crisis risk
because of the side e¤ect as shown in (27), which can be interpreted broadly in line with thereal-world observations.

4

Numerical Results

4.1

Solving the Model

We provide numerical solutions of the model in this section to address the following quantitative questions: (i) How frequently does a …nancial crisis arise? (ii) To what extent do the
LF banks deviate from the social optimum? And (iii) How can we compare the numericalresults with existing empirical studies on the probability of crises?
Our calibration mostly follows DR. We set the value of prematurely liquidated project X
at 0.95. The parameters for the distribution of ! are set to [! L ; ! H ] = [0:5; 3:5], similar to
14

Here, the actual capital supply Kt+1 observed for normal times and the threshold level capital Kt+1
need to be distinguished. When the LF banks are about to choose on Dt , they fail to take into account theoversupply of capital represented by Kt+1 . Once Dt is chosen at a higher level in the LF economy, however,the LF economy overconsume and underinvest compared to the SP economy because the higher leverage
allows households higher consumption on average for normal times.

20the original calibration of DR. The degree of banks’special collection skills is set at 0.9. In
addition to parameterization of DR, we need to set several other parameters. We calibrate
the capital share in the production function, , to be 1/3. The capital goods endowment
received by entrepreneurs, I, and the level of total factor productivity, which is suppressed in(1) and (2), together e¤ectively determine the size of the scarring e¤ect of a …nancial crisis.
We parameterize them so that the post-crisis contraction in output matches the estimatedsize of scarring e¤ects from past empirical studies.15 More importantly, we assume that theliquidity shock t follows the beta distribution with a mean of 0.50 and a standard deviationof 0.07. This parameterization indicates a symmetric bell-shaped distribution. To simulate
the model, we numerically solve the nonlinear system of the equations consisting of the…rst-order conditions and resource constraints.
Before interpreting the numerical results, we recon…rm the economic interpretations of
Dt . In the context of our model, Dt represents the pre-committed gross return on bankdeposits. On the other hand, Dt cannot be translated into an annual percentage rate or aninterest rate per annum, because the model does not specify the length of each period oftime (e.g., one year or one quarter). To focus more clearly on the economic interpretations,
Dt needs to be translated into a timeless measure such as the bank leverage. It is the exact
reasoning that we have relied on in this interpretation of Dt , that is, bank leverage.

4.2

The Endogenous Crises Probabilities and Model Dynamics

Our benchmark results are summarized in Table 2. The upper panel of the table reports that
the LF banks take on more risks than the SP banks, indicating a higher crisis probability.
We note that the two economies share the same state variable (i.e., the initial capital stockKt ) in our comparisons. Our calibration points to a 6.59 percent crisis probability in the
LF economy compared to 4.50 percent in the social optimum. Hence, the results suggestthat in an arbitrary period out of 1,000 simultaneous attempts, about 66 attempts would
trigger crises in the LF economy. The overleverage can be con…rmed by the values of Dt in
the upper panel of the table. In fact, Dt is 1.2 percentage points higher in the LF economy
than in the social optimum.
The prediction that the LF banking sector is overleveraged implies that the LF economy
undergoes crises created by pecuniary externalities, some of which could be avoided under
the social optimum. To illustrate this, we run the model over 100 periods by generating
15

See Barro (2009) as discussed in Section 4.5.

21liquidity shocks randomly. The upper panel of Figure 1 plots the dynamic paths of output
Yt generated by the same random shock sequences across the two economies. The red lines
correspond to the case of the LF economy, while the blue lines point to the case of the social
optimum.
Notably, this random shock follows an i.i.d. process and no “rare but large shock” isprompting a crisis as con…rmed by the lower panel of Figure 1. Comparisons between thetwo economies reveal that the dynamic paths of the output are almost identical exceptthat the production under the LF economy sharply declines more frequently. Under the
LF economy, crises take place in periods 5, 16, and 94, and output falls sharply in eachsubsequent period. This simulation result indicates that the latter two crises could havebeen prevented if the banks had taken the risks at the optimal level, implying a need forgovernment intervention to forestall the crises. However, the …rst crisis takes place even in
an economy in which the SP banks strike the right balance between the costs and bene…tsof increasing the leverage. Therefore, this crisis should not be avoided, as discussed in Allenand Gale (1998) in the context of the optimal …nancial crises.
The overleveraged LF banking system, which results in ine¢ cient …nancial crises, remains
a robust outcome across a range of calibrated parameters (e.g., X and distributions of ! and16
In addition to the robustness, the sensitivity analysis regarding the volatility of t has
t ).
noteworthy implications for the inherent fragility of the banking system in our model. Table
3 examines how bank leverage and crisis probabilities are a¤ected by the volatility of t .
In each simulation, the standard deviation of t changes from the benchmark value of 0.07
to either 0.02 or 0.10 with other parameters held unchanged at the benchmark calibration.
The table, together with the benchmark results, con…rms that the crisis probability rises
monotonically along with the increase in the standard deviation of t . We emphasize thatany extremely small volatility of the liquidity shock does not fully wipe out …nancial crisis
because the banks raise their leverage if they …nd liquidity demand less volatile. As a result,a …nancial crisis always remains a non-zero probability event.

4.3

The Size of Distortions

We next examine the extent to which the LF banks deviate from the allocation achieved by
the SP banks. More broadly, we discuss the quantitative implications of the higher leveragein the LF economy for welfare.
16

The tables of the sensitivity analysis are available upon request.

22

The upper panel of Table 2 reports the MSRs evaluated under the allocation in the LF
economy. The deviation of the two MSRs is 0.45, indicating a higher risk in the LF economy
and the extra increase in the crisis probability arises solely from pecuniary externalities. In
particular, if leverage is increased by one percent from the allocation in the LF economy,each LF bank expects that the crisis probability increases up to 8.22 percent, but they are,in fact, exposed to a higher crisis probability of 8.70 percent.17
The lower panel of Table 2 compares the bank capital ratio de…ned as (At Dt ) =At andthe output of the consumption goods Yt+1 under the LF economy and the social optimum,
when the realized value of t takes the mean of 0.5. The LF banks are undercapitalized by
1.1 percentage points compared to the social optimum. Nevertheless, it may be surprisingthat the production does not substantially di¤er across the two allocations, provided that a…nancial crisis does not take place. We also compute the levels of consumption for households
in a generation. The household’s consumption is (C1;t ; C2;t+1 ) = (2:21; 2:61) in the LF
economy in comparison to (C1;t ; C2;t+1 ) = (2:21; 2:59) in the social optimum.
The above exercise indicates that the welfare loss primarily arises from the ine¢ ciently
elevated crisis probability. Given that crises are considered rare events that we cannot
observe frequently, the ine¢ ciency or welfare loss may not be easily detected by looking
at the volatility of consumption or output in normal times. In this sense, assessing theine¢ ciency with crisis probabilities or MSRs appears to be more appropriate than using thevolatility of consumption or output.

4.4

Comparison with Empirical Studies

The numbers included in Table 2 may be compared to some recent empirical studies oncrises. Among a number of works on catastrophe risks, Barro (2009) provides a comparable
benchmark. He sets the disaster probability at two percent per year, arguing that a disaster
could reduce GDP by 30 percent on average. Another notable example is Reinhart andRogo¤ (2008, 2009). Based on their extensive data going back to the 1800s, they report thatthe frequencies are 7.2 percent for the advanced economies. In a similar context, the BaselCommittee on Banking Supervision (2010, BCBS) provides a broader perspective on thefrequency of banking crises based on multiple datasets, such as Laeven and Valencia (2008)
and Reinhart and Rogo¤ (2008), and summarizes that the frequency lies in the range of 3.6
to 6.8 percent.
17

These increases in the probability are obtained by transforming the MSRs into the semi-elasticity of theprobability: M SRk;t DLF;t for k = LF; SP .

23

MSR can be applied in line with a broad empirical exercise as demonstrated by the BCBS.
They report empirical measures that are comparable to the MSR in assessing the impact ofchanges in bank capital on the probability of systemic banking crises. Using reduced-form
econometric models, they estimate that a one percent increase in bank capital from the prereform cross-country average level could reduce the crisis probability by 1.0-1.6 percentage
points. Their estimates roughly match our simulated value (8:22 6:59 = 1:63) of the extra
increase in the crisis probability from raising the leverage computed from the MSR for the
LF economy. We underscore the proximity of the BCBS estimates and the numerical results
included in Table 2.18

5

Policy Intervention

This section discusses a variety of policy measures that have been implemented and areabout to be enacted by both national and international regulatory bodies. In this regard,
rather than examining the maximization of the social welfare, whose function is not speci…ed,
we focus on assessing the policies that aim to reduce the crisis probability by curbing bankleverage Dt .

5.1

Bank Levy

Suppose that, in an attempt to decrease the crisis probability, a government/central bank(GC) introduces a levy on bank size measured by its liability (Dt ).19 In practice, the levy canbe used to rescue troubled banks by means of a bailout and, in fact, in the next subsection
we will consider such interventions with the intentions of a bank bailout. For the moment,in this subsection we do not specify the purpose for which the GC spends the funds earned
from the levy, but we simply assume that the GC consumes the levy (i.e., they just wasteit) to focus on the impact of such a levy per se on banks’risk-taking behavior.
18

In addition to the evident proximity of the marginal changes in the crisis probabilities, the level of bankcapital reported in BCBS (2010) does not substantially di¤er from that in our model. BCBS (2010) argues
that the pre-reform cross-country average of the TCE/RWA (tangible common equity divided by Basel II
risk-weighted assets) ratio is 7 percent. The TCE is an extremely narrow de…nition of bank capital that, by
and large, could be doubled or even tripled (i.e., to 14-21 percent) if measured in more conventional measuresfor bank capital, such as the Tier I ratio.
19
For example, the U.K. government enacted a bank levy as of January 2011 in an attempt to “encouragebanks to move away from risky funding models”and to share the burden of …nancial crises with the banking
sector. See HM Revenue and Customs (2010). Another example is the Volcker plan, which includes a
proposal to restrict the size of banks’liabilities.

24

The GC imposes percent of the levy on banks’liabilities. Under the levy in place, Dtof the tax burden, which is measured by period-t consumption goods, falls on the banks ofgeneration t. In terms of the operation, we assume that the GC collects Rt Dt of consumption goods at t + 1 from banks. In period t + 1, the GC consumes the collected consumptionggoods for itself, which is denoted by Ct+1
= Rt Dt . Because there is no uncertainty in theeconomy after the realization of t , both banks and households correctly recognize that thebanks’ solvency is undermined by Dt as of period t. Under the bank levy, the solvency
constraint is written as(1 + ) Dt A (Rt =qt+1 ) :(28)
Accordingly, we replace (8) with
(1 + ) Dt = A Rt =qt+1 ;(29)

where all variables with an asterisk are rede…ned in line with the new constraint (28). From(29), we can de…ne the threshold interest rate function RBL ,RBL (Dt ) = qt+1 A 1 [(1 + ) Dt ] :(30)

0In a similar manner to Problem LF, note that by the inverse function theorem, RBL(Dt ) =
qt+1 (1 + ) =A 0 . Using (9), a function BL can be de…ned as

BL (Dt ) =L Rt =qt+1 + wt;
wt + Dt + wt+1 =Rt(31)

where Rt = RBL (Dt ) from (30) while wt , wt+1 , and qt+1 are given parameters for thecompetitive banks.
We now formally state the banks’problem with the levy as follows:
Problem BL Let t = BL (Dt ) : In an economy with a bank levy (BL), banks maximize(15) subject to (9) and (31).
Table 4 reports the allocation and the crisis probabilities under a reasonable range of= 0:01; 0:02; and 0:03: In comparison to the leverage of 1.061 in the LF economy (Table2), banks’ overleverage is reined in to 1.052, 1.044, and 1.036 for each case as con…rmedin the …rst column (“Bank levy”) in Table 4. But the crisis probabilities rise, rather thandecline, contrary to the expected outcome. The results can be interpreted easily. The e¤ects
25

of the levy act via two channels: (i) the banks deleverage in response to a levy and thischannel in fact reduces crisis probabilities because the bank’s active, intentional risk-taking
subsides; and (ii) on the other hand, the levy erodes the banks’ pro…tability and capital,
which are tabulated in the fourth row of each panel. By law, the banks pay out money totheir depositors or creditors out of the after-tax pro…t and assets. This simply exposes thebanks to a higher risk of insolvency, because they are left with fewer resources that can be
paid out to their creditors. On balance, the latter e¤ect dominates the former one, resulting
in the higher crisis probabilities despite the banks’lower leverage. The point is that the GC
does not rescue banks near crisis by using the collected tax, and this assumption regarding
how the levy is spent is admittedly less realistic in the light of past experience and current
practice.

5.2

Public Liquidity Provision as a Bank Bailout

5.2.1

Characterization

In the previous case, we assumed that the GC consumes the collected tax for its own bene…t.
But, presumably and more realistically, the bank levy would be used for the particularpurpose of bailing out troubled banks. As assumed in the previous case (BL), likewise,
the GC collects Rt Dt of the levy in period t + 1. Furthermore, in this case, the GC precommits to stepping into the liquidity market to provide the liquidity Mt to rescue a banking
system near crisis in period t when such intervention is possible and needed. This emergency
liquidity provision (ELP) can prop up the banking system near crisis if properly designedwith feasible …nancing.
In this case, we assume that the ELP by the GC is funded by tax collected from households
after the realization of t in period t. We further assume that the tax burden on households
at t will be compensated by the income transfer from the GC at t + 1 together with theinterest payment.20 Because, as noted, the GC collects Rt Dt of consumption goods frombanks as the bank levy in period t + 1, the GC can make the income transfer that keeps the
lifetime income of households unchanged as long as the total supply of public liquidity does
not exceed the amount of the bank levy, that is, Dt Mt . Put di¤erently, as long as thefunding resource of the GC is ensured, the income transfer can increase the total supply of
liquidity by the amount of Mt in the period-t liquidity market.212021This assumption rules out inter-generational income transfers from the GC.
An alternative way to validate the same operation is to simply assume that the GC has a storage

26

5.2.2

Implementation: commitment to a low interest rate policy

Because this intervention helps increase liquidity supply in the market, the interest ratewould decline to a lower level than would have been the case otherwise. Consequently, the
ELP at a time of elevated …nancial tension can be reinterpreted as a commitment to a lowinterest rate policy by a central bank.
The sequence is as follows. First, the GC announces its commitment to forestalling banks’insolvency by stepping into the period-t liquidity market to provide public liquidity in a casewhere the banking system cannot remain solvent without such an operation. Then, by fullyrecognizing the GC’s commitment, the banks determine their leverage. Subsequently, theGC (and everyone else) recognizes the maximum interest rate (Rt ) in line with (29) above
which the banking system fails to remain solvent. Then, t is realized. Suppose that thematerialized t exceeds a certain level ct . The high t accordingly would give rise to a highinterest rate, Rt > Rt , if the intervention did not take place. In line with the pre-announced
commitment, however, the GC provides liquidity to prop up a banking system near crisis by
cutting the market interest rate.22 Such ELP can e¤ectively be reinterpreted as placing acap on the market interest rate at Rt Rt . At normal times, the liquidity market clearingcccondition in period t is given by (9) for t
t , where t is formally de…ned as the level ofliquidity preference shock that requires the GC to intervene in the market to supply extra
liquidity and keep the interest rate at Rt :ct =L Rt =qt+1 + wt:
wt + Dt + wt+1 =Rt(32)

By contrast, similar to the BL economy, we de…ne t = BB (Dt ) that precipitates a…nancial crisis:
L Rt =qt+1 + wt + Dt;
(33)
BB (Dt ) =wt + Dt + wt+1 =Rt
where Rt = RBB (Dt ) : Here, RBB (Dt ) is identical to RBL (Dt ) because the banks’solvency
constraint remains the same as (28). Note that if t > t ; a …nancial crisis cannot be foretechnology for consumption goods between before and after the realization of t . If this is the case, the GC
can collect Dt of consumption goods from banks of generation t before t is realized and uses the resource
to …nance the intervention when possible and needed after t is realized. If a high t is realized, the GC
provides Mt (Dt ) of liquidity out of the Dt of funds that is wasted.
22
From the viewpoint of the implementation, the GC makes a commitment to a level of Rt while thecommitment is e¤ectively equivalent to a level of the relative price Rt =qt+1 . This is because qt+1 has a
one-to-one relationship with Rt , through qt+1 = FK [ (Rt )].

27

stalled by the GC’s intervention. For t 2 ( ct ; t ], the GC steps in and rescues a banking
system near crisis by ELP of Mt . Accordingly, the liquidity market clearing condition when
the GC’s intervention is underway is given by
t

wt+1
+ Dt
Rt

(1

t ) wt = L(34)

Rt =qt+1 + Mt :In other words, the GC’s liquidity provision is performed subject to the response function:
Mt = gt(35)

L Rt =qt+1 ;with Mt
Dt , because the GC can …nance this intervention only with the funds raised via
the bank levy.
We formally state the banks’problem under this policy intervention:
Problem BB Let
mize
Z

t =

BB (Dt ) : In an economy with the bank bailout (BB), banks maxi-ct

f t ln (wt + Lt ) + (1
max
Dt
0
Z t
+
t ln (wt + Lt ) + (1t ) ln [wt+1 + Rt (Dtt ) ln

c
t

+

Z 1

[ t ln (wt + X) + (1wt+1 + Rt (Dtt ) ln (w)] f ( t ) d t ;Lt )]g f ( t ) d t
Lt )f ( t) d t(36)t

subject to (9), (34), (32), and (33).
In this problem, the competitive banking sector takes capital prices, wages, and the GC’s
response function (35) as given. The objective function includes three terms. The newly
included term re‡ects the household’s expected utility with a banking system near crisis forct 2 ( t ; t ] while the GC successfully bails out the system with the ELP. Re‡ecting the
new term in the objective function, the e¢ ciency condition for the banks with respect to Dt
accordingly has the new term for the range of t .23
Before moving on to the main results of this section, we emphasize that if = 0; theallocations and prices both in the BL and BB economies are identical to the LF equilibrium.
Bearing this fact in mind, the following proposition summarizes the main results regarding
23

The detail of the derivation is available upon request.

28

crisis probabilities across the BL and BB economies.
Proposition 2 Let M SRBL;t and M SRBB;t be the marginal systemic risks in Problems
BL and BB, respectively. Then, around the LF allocation, M SRBL;t is strictly larger than
M SRBB;t .
Proof. An in…nitesimally small
M SRBL;t ==M SRBB;t ==> 0 a¤ects each MSR as follows:

0(DLF;t )f ( t ) BL
1@f ( t)(L
mt @Rt t
0f ( t ) BB
(DLF;t )1@
f ( t)(L
mt @Rt t0gt ) RBL(DLF;t )0gt ) RBB(DLF;t )

t

mt
t

mt

:

0
0
Recall RBL(DLF;t ) = RBB(DLF;t ) = (1 + ) qt+1 =A 0 . With Dt = DLF;t , the allocations Lt ,gt , mt , and t are the same across the two economies. Hence,

M SRBL;t

M SRBB;t =f ( t)> 0;mt

which proves the proposition.
As discussed in the comparison between the SP and the LF economies, likewise, a smaller
MSR in the BB economy suggests that the crisis probability would be higher than in theBL economy. This can be interpreted in line with economic intuition: the GC intends toforestall a crisis by reining in the otherwise rising interest rate to a low level, that is, Rt . If theintervention succeeds, the banking system remains solvent even if it faces a high t 2 ( ct ; t ]with the aid of the GC. But this is not the end of the story. When the banks determine
their Dt , they fully anticipate that the bailout will be enacted at a time of …nancial distress.
By correctly taking into account the increased safety owing to the bailout (i.e., ELP), thebanks take on more risks, resulting in a higher leverage, given the same burden levied on thebanks. The ultimate outcome would be an even higher probability of crisis compared to an
economy without such a bailout. Public liquidity provision as a bank bailout can thus raise,rather than reduce, the probability of a crisis as articulated in Table 4.24
24
In a similar context, repercussions of the authorities’ commitment to a low interest rate policy via
liquidity provision are pointed out by DR and Fahri and Tirole (2012) and are discussed from a broader
perspective in Rajan (2010).

29

5.3

Capital Requirement with Prompt Corrective Action

In an attempt to reduce the crisis probability, the GC may choose another option: in fact,
the capital requirement with prompt corrective action (PCA) has been up and runningas the primary banking sector regulatory tool, as typically represented by Basel II. The
regulation requires banks to hold a certain minimum level of capital. In this context, thisregulation can be translated into the temporary transfer of the required bank capital frombanks to the GC.25 If a bank fails to maintain the required level of capital, it would be taken
into receivership by the GC. Because of the capital requirement, the banks are faced withthe constraint,[A (Rt =qt+1 ) Dt ] =A (Rt =qt+1 ), where points to the required minimum
capital ratio. The requirement can equivalently be rewritten as the PCA activation condition,
Dt

(1

(37)

) A (Rt =qt+1 ) ;which appears, on surface, similar to (28) although it is e¤ectively quite di¤erent. Basically,(37) is not a solvency constraint for banks because, even if the constraint is violated, banks
may still hold positive capital and remain solvent. If a bank fails to hold A (Rt =qt+1 ) of
capital, it is taken into receivership by the GC. Under the PCA, the bank can continue tooperate but it is run by new management, typically appointed by the GC.
An issue that emerges in such cases is that the new management appointed by the GC may
have inferior skills in fostering the remaining long-term projects because the new management
consists of less experienced bankers who inherited unfamiliar projects. Re‡ecting the inferior
skills, we assume that, once the bank is taken into receivership, the new management of thebank can obtain!
! from the completed project. If is equal to one, the bank’s
ability as a relationship lender is fully retained while, by contrast, if is strictly lower thanone, it points to a loss of human capital in the banks because of the receivership.
The banks’assets under the PCA can be expressed as
A~ (Rt =qt+1 ; ) =Z !at+1

Xh (!) d! +

!L

qt+1
Rt

Z !H

!h (!) d!;!at+1

where we denote A~ (Rt =qt+1 ; 1) = A (Rt =qt+1 ) and ! at+1 = XRt = (qt+1 ). Using the new25
While the ownership of the required capital remains with the bankers legally in the real world, ourinterpretation is compatible with a variety of cases where the required capital is not at the bankers’disposal
because of the regulation.

30

notations, the bona …de solvency constraint of the bank is
Dt(1) A~ (Rt =qt+1 ; ) + Tt+1 ;(38)

where the required bank capital is represented by Tt+1 = A~ (Rt =qt+1 ; ) that the banktakes as given. The solvency constraint remains broadly unchanged compared to the LF
case, except for ; because the required capital is possessed by the new management of thebank. Although the capital requirement generates the incentives for banks to deleverage,
this requirement is much less stringent on the bank’s solvency than the bank levy because
Tt+1 is left with banks as usable funds for payout.
In parallel with ct de…ned in the previous subsection, a threshold value for the PCA
activation needs to be introduced. Suppose that the realized t is larger than a certain
level at . This condition de…nes the interest rate and capital price on the brink of the PCA
activation such that1a
Dt = A Rta =qt+1:
1a, the PCA is activated. In this case, the banks
If the relative price Rt =qt+1 exceeds Rta =qt+1
can remain solvent but are taken into receivership due to undercapitalization. As we de…ne(32) in the BB case, at is written as

a
t =

a
+ wt
L Rta =qt+1:
a
wt + Dt + wt+1
=Rta

(39)

By contrast, if t > t , the bona …de solvency constraint (38) is violated and a crisis isprecipitated. This condition reintroduces the interest rate and capital price on the brink of…nancial crises:
1
1
Dt = A~ Rt =qt+1 ; +Tt+1 :(40)
1
1
If the relative price Rt =qt+1 is greater than Rt =qt+1 , the banking system is precipitated intoa crisis, and this is likely to take place for low values of . Accordingly, we de…ne a function
RP CA as
Tt+1
Dt
;
;
(41)
RP CA (Dt ) = qt+1 A~ 1
1
1
in parallel with the hpractice in previous
cases. Note that the inverse function theorem assuresi00
RP CA (Dt ) = qt+1 = (1
) A~ < 0, where A~ 0 A~0 Rt =qt+1 ; .
Under the PCA, the threshold level of the liquidity shock for banks’ solvency takes a31form similar to (13) in the LF economy. We de…ne a function

P CA as~ Rt =qt+1 ; + wt
L;wt + Dt + wt+1 =Rt

P CA (Dt ) =

(42)

R a
~t = L~ (Rt =qt+1 ; ) = !t+1 Xh (!) d! is the liquidity supply under the PCA, which iswhere L!L
also a function of and Rt = RP CA (Dt ) from (41). The liquidity market clearing condition
under the PCA is~ (Rt =qt+1 ; ) = t wt+1 + Dt
L
(1
(43)
t ) wt :Rt
We then state the banks’problem under the capital requirement with the PCA:
Problem PCA Let t = P CA (Dt ) : In an economy with the capital requirement withprompt corrective action (PCA), banks maximize
Zatf t ln (wt + Lt ) + (1
max
Dt
0
Z tn
~
+
t ln wt + Lt + (1
a
t

+

Z 1

[ t ln (wt + X) + (1t ) ln [wt+1 + Rt (Dth)
ln
wt+1 + Rt Dttt ) ln (w)] f ( t ) d t :Lt )]g f ( t ) d t
~t
L

io

f ( t) d t(44)t

subject to (9), (43), (39), and (42).
This policy intervention can reduce the crisis probability compared with that in the LF(and BL) economies, as shown in the …rst column of Table 5. Taking an example of Panel
A where the required minimum capital ratio is 4 percent, the probability is 6.32 percent
when = 1, lower than 6.58 percent in the LF economy. The capital requirement per se
discourages banks’risk-taking while keeping unchanged the banks’resources that are payable
to creditors. This improved resilience of the banking system, however, may not be achieved
under alternative conditions. As the third column in Table 5 ( = 0:75) indicates, the crisis
probability is 7.30 percent, which is higher than that in the LF economy. If the banks under
PCA are run by low-skilled bankers, the risk-reduction e¤ect through deleveraging could
be dominated by the perils of lower solvency owing to the loss of resources. The overall
assessment regarding the capital requirement with the PCA suggests a straightforward point:
the success of this policy option largely depends on how e¢ ciently the GC can manage thetroubled banks under receivership.
32

Finally, we reiterate that this result should not be interpreted as either welfare improvement or deterioration. Identifying the optimal policy design requires a full-‡edged assessmentof policy alternatives with a well-de…ned social welfare function.

6

Conclusions

Our model demonstrated that a competitive banking sector cannot always achieve thesecond-best allocation. The banks fail to internalize the side e¤ects of changes in the illiquid
asset prices on their own solvency because of pecuniary externalities. In the light of realworld experience, our model could serve as a foundation for a better understanding of therepeatedly observed …nancial and economic crises. Further, our policy experiments tend tofavor some macro-prudential policy measures over an anticipated bank bailout, because thebailout could ill-incentivize banks to take on even higher risks by undermining discipline in
the banking system.
The analysis demonstrated in this paper can be extended in a number of directions. First,
it may be necessary to examine how changes in a variety of economic environments (e.g.,changes in the stochastic process of the liquidity shock) or newly introduced aggregate shocks(e.g., shocks to the asset side of banks’balance sheets) a¤ect the economy’s exposure to crisis
risks and macroeconomic ‡uctuations. Second, the optimal macro-prudential regulations
could be fully considered. Third, our framework may be translated into an in…nitely-lived
agent model for integration with quantitative business cycle studies. All of these directions
would provide important avenues for future research.

33

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34

[13] Christiano, L., R. Motto, and M. Rostagno (2010), “Financial Factors in Economic
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availableat[26] Jeanne, O., and A. Korinek (2012a), “Managing Credit Booms and Busts: A Pigouvian
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[33] Rajan, R. (2010), Fault Lines, Princeton University Press.
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NBER Working Paper No. 14587.
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Folly, Princeton University Press.
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Journal of Economics, 127 (1), pp. 57-95.

36

A

Appendix

A.1

Proof of Proposition 1

We …rst show Lemma 1 which indicates that dRt =dDt < 0 and then proves the proposition.
Lemma A.1 dRt =dDt < 0.
Proof By taking total derivatives of (18) with respect to Dt , we obtain
dRt
=
dDt

1
1
0

Then it su¢ ces to show FKK;t+1(22) yields
0=0 R =qt
t+1

FKK;t+1

!

qt+1
:
A0

Rt =qt+1 < 1. Applying the implicit function theorem to1;
qt+1 =I 0 + Rt FKK;t+1 =qt+1

where I 0 = I 0 Rt =qt+1 < 0. Hence,
FKK;t+1

0 Rt

qt+1

=

1
1 + qt+1

2= Rt I 0 FKK;t+1< 1;

which proves Lemma A.1.
0
0
Recall that the discrepancy between RLF(DLF;t ) and RSP(DLF;t ) is given by (27)0
RLF
(DLF;t )0
RSP(DLF;t ) =RtF
qt+1 KK;t+1

The sign of the discrepancy is ensured by Lemma A.1.

37

0 dRt

dDt

> 0:

Table 1: Sequence of events for generation t

Period t
1.
2.
3.
4.
5.
6.
7.
(i)
(ii)
8.
1.
2.
3.
4.
5.

Households receive endowments.
Banks o¤er deposits to households and loans to entrepreneurs.
Entrepreneurs launch their projects.
Households supply labor and receive wages wt determined by the labormarket conditions along with the old generation’s labor supply.
Liquidity shock t is realized, and banks receive signals of project
outcomes.
Households decide the withdrawal amount gt .
Banks decide which projects to discontinue and supply liquidity Lt .
If gt > Lt , a …nancial crisis is precipitated and households receive
repayment of X.
Otherwise, the households can transfer their wealth into the period t + 1.
All agents consume.
Period t + 1
Entrepreneurs receive endowments.
Entrepreneurs’projects are completed, and they sell their capital
goods for qt+1 and make repayment to banks.
Households supply labor and receive wages wt+1 determined by the labormarket conditions along with the young generation’s labor supply.
Households fully withdraw deposits, if any.
All agents consume.

38

Table 2: Crisis probabilities and allocations
under laissez-faire banking sector and social planning banks

SP banks

LF banks

Leverage and crisis probabilities
Dt
t (%)
MSR

1.049
4.499
1.993

1.061
6.585
1.544

Bank capital and output
Bank capital ratio (%)
Yt+1

15.0975.459

13.952
5.457

Note: Simulation results based on the assumption that the liquidity shock t follows the beta distribution.
The level of bank leverage Dt and the probability of a …nancial crisis t are obtained from Problems LF andSP, respectively. The marginal systemic risk, MSR, is given by (25). The bank capital ratio is (At Dt ) =At .

Table 3: Financial crisis probabilities and bank leverage
SP banks
Leverage and probabilities
Dt
t (%)

LF banks

std( t ) = 0.02
1.129
1.132
1.275
1.703

SP banks

LF banks

std( t ) = 0.10
1.014
1.030
5.997
9.433

Note: Simulation results based on various values of the standard deviation of the liquidity shock
other details can be seen in the note for Table 2.

39

t.

The

Table 4: Bank levy and bank bailout

Bank levy
A:

Bank bailout= 0:01

Leverage and probabilities
Dt
t (%)
MSR

1.052
6.670
1.704

1.055
6.740
1.613

Bank capital and GDP
Bank capital (%)
Yt+1

13.9485.458

13.6605.458

B:= 0:02

Leverage and probabilities
Dt
t (%)
MSR

1.044
6.755
1.855

1.050
6.890
1.673

Bank capital and GDP
Bank capital (%)
Yt+1

13.943
5.460

13.3655.459

C:= 0:03

Leverage and probabilities
Dt
t (%)
MSR

1.036
6.840
1.996

1.044
7.034
1.725

Bank capital and GDP
Bank capital (%)
Yt+1

13.937
5.462

13.0685.460

Note: The bank leverage Dt and the probability of a …nancial crisis t are obtained from di¤erent values ofin (28). Panels A, B, and C correspond to the case of = 0.01, 0.02, and 0.03, respectively. The bankcapital ratio excludes the surcharge and is de…ned as [At = (1 + ) Dt ] = [At = (1 + )]. The other details canbe seen in the note for Table 2.

40

Table 5: Capital requirement with prompt corrective action

A:
Dt
t (%)
Yt+1

= 1:00= 0:95= 0:75

1.060
6.321
5.457

1.057
6.457
5.457

1.043
7.298
5.460

B:
Dt
t (%)
Yt+1

= 0:04= 0:06= 1:00= 0:95= 0:75

1.059
6.188
5.457

1.056
6.321
5.458

1.042
7.066
5.461

Note: Numbers in the table are obtained under the capital requirement with prompt corrective action acrossvarious capital requirement ratios ( ) and various degrees of reduction in the value of investment projects( ) arising from the new management taken by the government/central bank. Panels A and B correspondto the cases of = 0.04 and 0.06. Each column shows the degrees of reduction in the value of investment
projects ( ). The level of output Yt+1 is obtained under the assumption that t takes the mean of 0.5.

41Figure 1: Simulated paths of output and the liquidity shock
Output
6

Y

t

5.5

5

LF

4.5

4

0

10

20

30

40

50

60

SP

70

80

90

100

80

90

100

periods
Liquidity shock and its threshold levels
0.7
0.65

θt and θt

*

0.6
0.55
0.5
0.45
0.4
0.35

0

10

20

30

periods

40

50*
θt (LF)

60

70*
θt (SP)

Realized θtNote: In both panels, the blue lines are for the economy with the social planning (SP) banks and the redlines are for the economy with the laissez-faire (LF) banks. The upper panel shows the simulated dynamic
paths of output Yt . The liquidity shock plotted as the dashed black line in the lower panel is generated fromthe beta distribution with a mean of 0.50 and a standard deviation of 0.07. The solid blue and red lines
in the lower panel are the threshold level of the liquidity shock that satis…es the solvency constraint withequality.

42