Bank Overleverage and Macroeconomic Fragility(Preliminary and incomplete)

Ryo Katoyand Takayuki TsurugazThis version: May 8, 2011
First draft: May 8, 2011

Abstract
This paper develops a dynamic general equilibrium model that explicitly includes a banking
sector with a maturity mismatch. We demonstrate that, despite the perfect competition in thebanking sector, rational banks take on excessive risks systemically, resulting in overleverage andine¢ ciently high crisis probabilities. The model accounts for the banks’rational over-optimism
regarding future capital prices which arises from pecuniary externalities on their own solvency.
Using the model as an example, we introduce MSR (marginal systemic risk) as a general measure
to assess the macroeconomic exposure to systemic risks.
JEL Classi…cation: E3, G01, G21
Keywords: Financial crisis, Liquidity shortage, Maturity mismatch, Pecuniary externalities

We thank seminar participants at Bank of Japan. We are also indebted to Daisuke Oyama and other membersof the Banking Theory Study Group for helpful discussion. Takayuki Tsuruga gratefully acknowledges the …nancial
support of the Grant-in-aid for Scienti…c Research (No. 21243027). Views expressed in this paper are those of theauthors and do not necessarily 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

Theories of banking and …nancial intermediation have come a long way, elaborating on the bene…ts and the inherent fragility of banking systems. In the meantime, macroeconomists have madea variety of attempts to incorporate realistic …nancial intermediaries — banks in particular— indynamic general equilibrium models. A juxtaposition of the standard models of banking (Diamondand Dybvig 1983, Allen and Gale 1998, 2004) with a few dynamic macroeconomic models (Calstrom
and Fuerst 1997, Gertler and Kiyotaki 2011) can clearly point to a yet-to-be …lled gap between
the micro-theory of banking and macroeconomics. The gap may be summarized as follows: Manymacroeconomic models have successfully incorporated the complicated lender-borrower relationship and analyzed its consequences and implications for the macroeconomy, while, conceivably, few
macroeconomic models have crystallized the roles and perils of banking systems in the light of anumber of real-world experiences of a …nancial crisis (Caballero 2010). This potential de…ciency inmacroeconomics manifests itself when the banks typically modeled in micro-theories of banking aswell as in the real world are compared to the over-simpli…ed “banks” in macroeconomic models.
Banks in the real world, as reasonably modeled by micro-theories, provide unique services for theircustomers. Banks typically raise funds via short-term liabilities (e.g., demand deposit, repos) andinvest them, in part, in illiquid assets. This notable business line undertaken by banks is widely
acknowledged as a maturity mismatch or maturity transformation. On top of this, as noted byAllen and Gale (2007), micro-theories of banking have broadly emphasized other special elements
in banking business, such as provision of liquidity insurance for depositors against liquidity (preference) shocks and inherent exposure to crisis risks. In contrast, however, these special elements inbanks are given short shrift in macroeconomics.
This paper develops a dynamic general equilibrium model with a realistic banking sector toaddress basic yet unsolved questions: Does a banking sector with maturity mismatch a¤ect macroeconomic ‡uctuations? If yes, how could it improve or undermine economic welfare? We lay out anoverlapping generations (OLG) model where a liquidity shortage in banks with maturity mismatch
precipitates ine¢ cient …nancial crises that can result in a devastating decline in macroeconomic
activity. As opposed to many existing models with “bubbles,” for example, where a bubble bursts
at a given (exogenous) probability, our model shows that, while not including asset price bubbles,
the probability of a crisis varies depending on the leverage that banks determine based on their rational decisions. More broadly, we argue that, regardless of bubbles taking place or not, the banking
sector in a laissez-faire economy generally cannot achieve the constrained optimum (a second-best
allocation) and tends to be overleveraged as a result of perfect competition.
We …nd as the main result, that a fully competitive banking sector tends to take on excessive
systemic risks and gives rise to an ine¢ ciently high crisis probability. The result may appear togo against the …rst theorem of welfare economics. To better understand the point, we introduce2

a new measure, marginal systemic risk (MSR) to assess the distortion as a fairly general concept.
MSR is de…ned as the deviation from the socially optimal marginal increases in crisis probability
vis-à-vis any target variable. If MSR is zero in an economy, this means that the economy achieves
the social optimum, whereas a positive MSR points to excessive risk-taking in the economy. In ourmodel, MSR is de…ned against bank leverage and is proved to be positive, which indicates that thebanking system is overleveraged. More speci…cally, MSR in our model depends on the marginalchanges in the shadow values of banks’solvency constraints as lower solvency gives rise to a higher
crisis probability. In general, the value of bank assets typically depends on asset prices. In ourmodel, the banks’solvency constraint depends on expected capital prices. Because the constraint
itself depends on capital prices, price-taking banks do not internalize the marginal changes in the
shadow values of the constraint against their own decision on leverages. The pecuniary externality
regarding the solvency constraint creates the positive MSR and the overleveraged banking systemin our model.
More generally, the non-zero MSR in our model can be interpreted in line with the …rst welfare theorem: Maturity-mismatching banks normally provide non state-contingent debt, typically
represented by demand deposits, rather than fully state-contingent bonds (i.e., Arrow securities),
where otherwise bank runs need not be considered. Diamond and Rajan (2001a, 2001b) highlightthe advantage of the banking business by demonstrating that such issuance of non state-contingent
short-term debt streamlines …nancial intermediation and promotes liquidity creation. While theirargument that banking business can improve welfare applies fairly generally, the …nancial intermediation by banks cannot fully compensate for the market incompleteness. Given the incompleteness,there is no guarantee that a competitive equilibrium can achieve the (constrained) social optimum(i.e., the second-best allocation). With an incomplete …nancial market, pecuniary externality canarise in a variety of forms. Because of such externalities, price-taking banks cannot strike theright balance between the cost and the bene…t of a marginal increase in their own leverage. The
ine¢ cient allocations in our model appear to contrast with Allen and Gale (1998), in which thebanking sector achieves the second-best allocation. Their model includes a single market where
the banking sector, as the result of the perfect competition, internalizes all the e¤ects throughprice changes. In contrast, our model includes more than one market (i.e., the capital and labormarkets) in which banks take prices as given. The price-taking banks, given the incompletenessof the …nancial markets, give rise to a positive MSR, or equivalently, over-risk-taking in a manner
similar to that studied by a number of recent works on pecuniary externalities (Lorenzoni 2008,
Jeane and Korinek 2011, Korinek 2010, Bianchi 2010).
Because of the incomplete …nancial markets and the resulting pecuniary externalities, nonzero MSR (or an excessive risk-taking, equivalently) arises not surprisingly. In the aforementioned
models with pecuniary externalities, decentralized agents take the capital price (or other asset
prices) as given, and such price-taking behavior distorts the e¢ cient allocations as opposed to the3cases presumed by the …rst welfare theorem. While our model shares positive MSRs with earlyworks on pecuniary externalities that result in over-credit (e.g., Lorenzoni 2008), our perspective on…nancial crises varies remarkably from early studies. A notable di¤erence is that our model explicitly
includes the maturity-mismatching banks and …nancial crises are precipitated by insolvencies of thebanking system triggered by a liquidity shortage. Furthermore, it could be noted that the market
incompleteness in our model arises from the maturity mismatch in the banking sector itself rather
than from any other extra friction or constraint (e.g., asymmetric information or a borrowing
constraint).
The remaining question is why the competitive banking sector gives rise to a strictly positive
MSR, or overleverage, rather than a negative MSR (i.e., underleverage). Suppose that a bankchooses to lever up to a higher level. While the bank can attract more creditors/depositors byincreasing the leverage (i.e., by o¤ering a higher return on their debt), it would be exposed toa higher risk of insolvency. Therefore, each bank needs to recognize their solvency on the brinkof a default, to strike the right balance between the cost and the bene…t of increasing their sizeof leverage. To this end, banks try to precisely assess the values of their own assets to estimate
the marginal cost of increasing the leverage, taking the price of their illiquid assets as given.
Notably, when the banks calculate the value of the illiquid assets, the pecuniary externalities are
not factored in and this disregarded pecuniary externalities prompt the banks to be overleveraged.
In general, highly leveraged banks hold large assets in an attempt to enhance their solvency. Byholding more assets, they try to pursue higher returns to weather high liquidity shocks with highpro…tability. In general equilibrium, however, the banking sector as a whole overinvests owing
to the externalities and this overinvestment in illiquid assets pares down the price of the illiquid
assets. The decreased value of the illiquid assets undermines the banks’ solvency more quickly
on the brink of a systemic …nancial crisis compared to the case of a run on an individual bank.
When individual banks determine their size of leverage, each of them takes future capital prices
as given and perfect competition among banks does not allow them to o¤er a lower deposit rateeven though the overleverage leads to an ine¢ ciently high crisis probability. Put another way, thepecuniary externality ill-incentivizes banks to take on excessive risks systemically. In retrospect,
the banks’ decisions on their leverage always appear to be over-optimistic, accompanied by theirover-optimistic outlook of future capital prices. Broadly speaking, banks correctly recognize theirprivate cost and bene…t of a high leverage while, from a viewpoint of the social planner, they always
underestimate the marginal cost of raising leverage. This “rational over-optimism” entrenched inbanking systems can pave the way to better understand banks’roles and incentives that have been
observed repeatedly in preceding …nancial crises.
The rest of the paper proceeds as follows. Section 2 illustrates the macroeconomic modelwith maturity-mismatching banks. Using the model, we characterize the banks’optimal leverageand the competitive market equilibrium. In Section 3, we compare the laissez-faire equilibrium

4characterized in Section 2 with the allocation achieved by the social planning bankers. Meanwhile,
Section 3 introduces MSR to elucidate why a competitive banking sector tends to be overleveraged.
Section 4 discusses numerically the probability of …nancial crisis, the size of distortions, and the issueof crisis prevention along with the comparisons with a few empirical studies. Section 5 discusses
relations with the existing models. Section 6 concludes.

2

A Macroeoconomy with Banks

2.1

Agents, Endowment, Preferences, and Technology

We consider an in…nite-horizon OLG model incorporating banks with maturity mismatch. Eachgeneration 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. Each agent is identicaland constant in the population. Furthermore, an initial old generation lives for one period and thesubsequent generations live for two periods.
Households are risk averse and subject to liquidity shock, which a¤ects their preference of 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 out their consumption intertemporally. Following models in the theories of banking (e.g., Diamond and Rajan 2009), 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 atbanks operating in the same generation. They receive wages wt in the competitive labor market
by supplying one unit of labor in both periods, t and t + 1.
Entrepreneurs are risk neutral and have access to capital producing technology. They launchlong-term investment projects at the beginning of period t, by borrowing households’endowment
via the bankers in the same generation. The investment project needs one period for gestation, andcapital goods are produced in period t + 1. We call this capital producing technology a “project.”
They sell the capital good in the competitive market for the capital good price qt+1 .
Banks raise funds from households and lend them to entrepreneurs at the beginning of period
t.1 In principle, we follow Diamond and Rajan (2001a) to model banks. Banks are risk neutral andcompetitive in raising and lending funds in the markets. They issue demand deposits (short-termdebt) and commit to repaying households. As a nature of demand deposits, banks can provide
insurance against the depositor’s liquidity shocks. However, when households demand repayment
before the completion of the entrepreneur’s projects, banks must liquidate premature projects tomeet the demand for repayment. This maturity mismatch, represented by the combination of thelong-term assets and the short-term liability, leaves banks exposed to risks of a default because,1

We assume intra-generational banking, which e¤ectively means that all bankers die at the end of the second
period.

5

depending on the amount of withdrawals in the interim period, the banks’solvency is endangered.
The technology of consumption good production Yt is represented by a standard constantreturns-to-scale Cobb-Douglas production function:
Yt = F (Kt ; Ht ) = Kt Ht1;

where Kt and Ht denotes the capital stock and hours worked, respectively. Demands for labor andcapital satisfy
wt = FH;t = (1

)
Kt
Ht

qt = FK;t =

Kt
Ht

(1)

1

:

(2)

In what follows, we describe each agent’s decisions (consumptions, withdrawal, and liquidationof entrepreneur’s projects) after the liquidity shock is realized. Then, we move on to the bank’s
decision on the level of leverage before the realization of the liquidity shock. Table 1 summarizes
the sequence of events in each generation.

2.2

Households

Under the perfectly 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 shockt in the middle of periodt. The liquidity shock is common across all households and has the probability density functionf ( t ) with a support of [0; 1]. This shock represents households’preference for consumption when
young and signals the need for liquidity in period t.2
After these events, households make their decisions for consumption smoothing. In particular,households then choose withdrawal gt to maximize
U (C1;t ; C2;t+1 ) =s.t. C1;t =C2;t+1 =t log C1;t + (1((t ) log C2;t+1 ;wt + gt with probability 1
wt + X with probability
wt+1 + Rt (Dttgt ) with probability 1

wt+1 with probability

(3)

t
t

t;

where C1;t and C2;t+1 denote the consumption of households born in period t when young and old,
2

Although in fact all households are subject to the same aggregate shock, we assume that an in…nitesimally small
number of households are believed to face a di¤erent t from other households. This assumption ensures the existenceof a Nash equilibrium, in which all households run to the banks when households believe that the banks are insolvent
under the observed t :6

respectively. Each household supplies a unit of labor every period and receives wage income wt inperiod t and wt+1 in period t + 1. Here Rt denotes the one-period gross interest rate from periodt to t + 1. In addition,t is the ex postprobability of a …nancial crisis. With the probability

3t , households withdraw gt in period t and all the rest of deposits in period t + 1.

1

probability

With thet , a …nancial crisis arises and households’withdrawals amount to the liquidation valueof premature projects, X (< 1), in period t and consumption is not smoothed out over the two
periods.
When the households can smooth out their consumption, the intertemporal …rst-order condition
for consumption is satis…ed:
t

1

t

C1;t
C2;t+11= Rt ;(4)

while the intertemporal budget constraint holds with equality C1;t +C2;t+1 =Rt = wt +Dt +wt+1 =Rt
mt , where mt is the lifetime income for households. Given the Euler equation (4), the withdrawals
in the absence of crisis can be written as
gt =

t

wt+1
+ Dt
Rt

The withdrawal function implies that large

2.3

(1

t ) wt :(5)t and Dt are likely to precipitate a …nancial crisis.

Entrepreneurs

e + CeEntrepreneurs are risk neural and maximize their lifetime utility represented by E C1;t2;t+1 ;e and C e
where C1;t2;t+1 denote entrepreneurs’ consumption when young and old. They use a unitof consumption goods …nanced from banks for their capital good production, and this production
technology takes one period for gestation before its completion. In period t + 1, the project yields a
random capital good output !
~ , which takes a value distributed over [! L ; ! H ] with the probability
density function h (!).4 If this project is prematurely liquidated in period t, the transformationfrom the consumption good into capital is incomplete. As a result, the output is reduced to X
units of consumption goods and is repaid fully to banks in period t. When the project is completed
in period t + 1, however, entrepreneurs can sell their output in the capital good market for the
price of qt+1 . After the repayment to banks, they are left with 1of the share of their pro…tand enjoy their own consumption based on their linear utility. We assume that entrepreneurs areendowed with I units of capital goods in period t + 1.5 They sell this endowment capital together
3
In the maximization problem of households, we assume that wage income in period t is low relative to the initialendowment, ensuring a non-negative withdrawal gt in the equilibrium. Furthermore, we assume the perishability of
consumption goods, so any positive gross rate of interest rate can be an equilibrium interest rate.
4
Following the literature, we take the assumption that there is no aggregate uncertainty in the project outcome.
5
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 case of a …nancial crisis in which all
projects are scrapped due to full liquidation.

7

with the newly created capital made from the consumption goods transferred from the households
at the beginning of t + 1.

2.4

Banks

b + Cbb
Banks are also risk neutral and maximize their lifetime utility E C1;t2;t+1 , where C1;t andb
C2;t+1denote consumption of banks when young and old. We borrow the microfoundation of thebanking business from Diamond and Rajan (2001a, 2001b): banks issue demand deposits (shortterm debt) as a commitment device to compensate for the lack of transferability of their collection
skill and to promote liquidity creation. As discussed in a number of preceding works on banking(such as Diamond and Dybvig, 1983 and Allen and Gale, 1998), the banks determine Dt before
observing the liquidity shock, which is realized in the middle of period t. We stress that Dt has
a one-to-one relationship with bank leverage and, hence, we refer to Dt as leverage hereafter. We
will discuss this issue later in this section and in Section 4 in terms of numerical interpretations.
Each bank has no initial endowment at birth but has a special skill to acquire knowledge about
entrepreneurs’business. This skill enables a bank to act as the relationship lender that can develop
the alternative, but less e¢ cient, use of an entrepreneur’s project. As discussed in Diamond andRajan (2001a, 2001b, 2009), this knowledge enables the bank to acquire a fractionof the realized

project’s outcome in period t + 1.6
We also follow the preceding works on the assumption for the distribution of entrepreneurs towhich each bank makes loans. Each bank attracts many entrepreneurs through the competitiveo¤er on the loan, resulting in the identical portfolio shared by all the symmetric banks. This setupe¤ectively leads to a 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 perfectlypredict the realized value of !
~ in period t+1. With this information ! and the households’liquidity
demands observed in period t, each bank chooses one of the options: (i) to liquidate projects inperiod t, obtaining X of consumption goods per project; or (ii) to collect a fraction qt+1 ! from acompleted project in period t + 1. The bank liquidates the project if the outcome of a project falls
short of ! t+1 , de…ned as7! 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, (16) qt+1 !; — per project. After repaying

We also assume that once a bank has made a loan to entrepreneurs, no other lenders can replicate the collection
skill.
7
Equation (6) can be reinterpreted as follows: qt+1 ! t+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 thatis observed as the interest rate, Rt .

8the full amount of the households’withdrawals, the banks consume their own capital, denoted as
b + CbC1;t2;t+1 .

Let the bank’s asset be A (Rt =qt+1 ). The bank’s asset at the beginning of period t (i.e., prior
to the withdrawals) can be expressed as
A

Rt
qt+1

Z !

=

t+1

!L

qt+1
Xh (!) d! +
Rt

Z !H

qt+1
I
Rt

:

Rt
qt+1

= L

+

!h (!) d!

! t+1

Rt
qt+1

(7)

The bank 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 meet!Lthe 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 qt+1 It =Rt ; whereR!
It is given by It = I (Rt =qt+1 ) = ! H !h (!) d!.
t+1

The banks are subject to the solvency constraint Dt

A (Rt =qt+1 ). Because banks’ assets

decrease when they are discounted by a high Rt and advance on high expected capital prices qt+1 ,it can be easily shown that A ( ) monotonically decreases with Rt =qt+1 . We can then de…ne therelative price Rt =qt+1 which 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, wedenote a variable with an asterisk as the variable on the threshold. Similarly, we denote a function
with an asterisk as the function evaluated at the threshold variable. For the purpose of subsequentdiscussion, we note that given A (Rt =qt+1 ), the ex post bank capital and bank leverage are denotedas At

Dt and Dt =(AtDt ), respectively, both of which are determined uniquely once Dt is chosen.

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

(1

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

Next, the market clearing condition for consumption goods is
Yt + LRt
qt+1

9

The left-hand side of (10) includes the supply of goods from the liquidity market. This is because
consumption of households in period t consists of wage income and withdrawal. On the right-hande , and C b denote consumption when generation tside of (10), C2;t , C2;t2;t1 is the old.

The capital good market clearing condition is
Kt+1 =(

I + I (Rt =qt+1 ) with probability 1Iwith probability

t

(11)

t:

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

2.6

Optimal Bank Leverage

We now consider the banks’ optimal choice for the leverage. The banks choose the size of theirleverage before the realization of the liquidity shock. We focus on the laissez-faire banks in thissubsection, and will discuss the social planning banks in Section 3.
The banks are competitive in issuing demand deposits, and we assume that households’ endowments are scarce in comparison to entrepreneurs’ projects. This implies that banks make acompetitive o¤er of deposits for households, taking the capital prices and wages as given. As aresult of perfect competition, the banks appear to be maximizing the household welfare (Allen andGale 1998, 2007), while in fact they are maximizing their own pro…t (linear utility).
Maximizing the households 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
understand how the banks’ choice on Dt a¤ects
shock

t , which appears in (3) and (11).

t ; we de…ne the threshold level of the liquidity

t that clears the liquidity market at the threshold interest rate and capital prices, Rt

qt+1 . The market clearing condition implies that
t =

To
and

t is given by

L Rt =qt+1 + wt:
wt + Dt + wt+1 =Rt(12)

The above equation means that given Dt , changes in Rt andqt+1 always give rise to changes int

for the liquidity market to clear. By the solvency constraint (8), any level of Dt , once chosen,

determines the threshold relative price, Rt =qt+1 . Hence,shock on the brink of a …nancial crisis. Namely, whent

can be interpreted as the liquidity

t is strictly greater thanout to be insolvent and a crisis is precipitated. Thus, the crisis probability

10t , the banks turnt has a one-to-one

relationship tot via the probability density function f ( t ):

t =1

Z

t

f ( t) d t:

(13)

0

In principle, the banks’ choice of the leverage speci…es Rt =qt+1 and this threshold relative pricedetermines the threshold level of the liquidity shockt , completing the link between the bankleverage and the crisis probability. We are now ready to set up the optimization problem for thebanks to determine their size of the leverage. In the problem, as discussed, banks take into accountthe endogenously changing
Problem LFt.

In a laissez-faire economy, banks maximize the household expected utility
Z tmax
f t ln (wt + gt ) + (1t ) ln [wt+1 + Rt (DtDt0
Z 1[ t ln (wt + X) + (1+t ) ln (w)] f ( t ) d t ;gt )]g f ( t ) d tt

subject to (8), (9) and (12).
The banks choose their leverage levels according to the following …rst-order condition:
t log

=

Z

0

The marginal changes ingiven by1
mt

wt + X
t mt
1

+ (1

t ) log

wt+1 dRt
Rt2 dDt LF

w
Rt (1

+ (1

t)

t ) mtf ( t)

d t
dDt

(14)

1 dRt
f ( t) d t:
Rt dDt LF

t with respect to Dt can be derived from (12). Likewise, dRt =dDt jLF is

dRt
t
= 0
> 0;
dDt LFL =qt+1 + t wt+1 =Rt2

(15)

where L0 is the derivative of Lt with respect to Rt =qt+1 .
Equation (14) provides an economic interpretation that is in line with broad intuitions. Notethat the terms in brackets on the left-hand side of (14) represent the loss of utility in a crisis compared to the threshold. Using (13), we also note that the term outside the brackets, f ( t ) d t =dDt
equalsd t =dDt , which indicates the marginal changes in a crisis probability with respect to bankleverage. The left-hand side of the equation consists of the expected loss of utility and the marginalchange in the crisis probability. Simply put: the left-hand side of (14) is the marginal cost ofincreasing Dt .
The right-hand side of (14) consists of the e¤ects of increasing leverage on the expected household’s lifetime income and the interest rate in the liquidity market. On the one hand, the increase11

in Dt has an outright positive e¤ect on the household’s income: the higher the leverage, the larger
the withdrawal, allowing households to enjoy more consumptions. 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 of forbearing from 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, the higher
leverage is bene…cial to households: Simply put, the right-hand side of (14) is the marginal bene…tof increasing Dt .
The following proposition establishes the uniqueness of the choice of Dt under our assumptions.
Proposition 1 The optimal bank leverage is unique.
Proof. See the Appendix.
While we leave the formal proof to the Appendix, the intuition behind the proposition canbe explained as follows. Throughout this paper, we assume that the outright e¤ect of increasing
Dt exceeds the e¤ect on the interest rate, since otherwise no deposit o¤er provides bene…ts tohouseholds and thus there would be no point in discussing banks. On the other extreme, leveraging
up to an in…nitely large level is clearly not optimal, because such an in…nitely large Dt decreasestto zero, which means a …nancial crisis should certainly take place. Thus, once again there would be
no bene…t in issuing deposits. Then, the strict concavity of the log utility ensures the uniquenessof the optimal leverage within [X; 1).

Finally, we de…ne the equilibrium in the laissez-faire economy as follows.

De…nition An 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) fortt ; (ii) the banks’ leverage satis…es(14); (iii) the bank’s liquidity supply is determined by (6); and (iv) all markets clear.

3

Systemic Risks and Welfare

3.1

Social Planning Bankers

In the previous section, banks maximize households’ expected utility as a result of the perfectcompetition. In this section, however, we assume that banks can choose allocations as the social
planner. Under this assumption, we demonstrate that the market equilibrium, de…ned in theprevious section, cannot replicate the social optimum that will be characterized in this section. To
lead o¤ the analysis, we clarify the constraint to which the social planning bankers are subject.
They must make all their decisions before observingt.

After realization oft , they are left withno options. In other words, the planners are subject to the constraint that they cannot control
households’behaviors or choose their outright consumption levels because households can react toany realized value oft . This assumption is made for an explicit reason. While we examine the social

12

planner’s problem in this section, nonetheless, we aim primarily to see the constrained optimum
rather than the unconstrained optimum. The unconstrained, …rst-best optimum is conceptuallyeasy to understand. By assuming that banks (or anyone else) can issue the Arrow security thatpays o¤ contingent on all possible realizations oft , households can enjoy the maximum utility

without experiencing any …nancial crisis. But as we already discussed, banks are, by de…nition,
entities engaged in a maturity mismatch and issue non state-contingent precommitted debt (e.g.,demand deposits). Otherwise, banks are no longer banks and should be regarded as other types of…nancial intermediaries (e.g., private equity funds). Because, in our model, we cannot disregard thespecial business line of banks, a maturity mismatch, we assume that banks pre-commit to paymenton their debt regardless of the states realized following their commitment. The extra capacity
given to the social planning bankers compared to the price-taking competitive banks is that theformer can internalize all price e¤ects in all markets when they make their decisions regarding theirleverage.
The social planning bankers do not take the factor prices as given, but now take into accounttheir changes re‡ecting the marginal product. Formally, we replace qt and wt with FK;t and FH;t ,respectively. Note that, nonetheless, social planning bankers take the households’ behaviors asgiven, as they cannot make their contract contingent ont.

In other words, households always

choose their consumption and withdrawal levels given the precommitted Dt by banks.
We summarize the social planning bankers’problem:
Problem SP

Social planning bankers maximize the household expected utility,
Z

t

max
f t ln (FH;t + Lt ) + (1t ) ln [FH;t+1 + Rt (DtDt0
Z 1[ t ln (FH;t + X) + (1+t ) ln F H ] f ( t ) d t

Lt )]g f ( t ) d tt

subject to
Lt =t

Dt = At=

FH;t+1+ Dt
Rt!
Rt
FK;t+1(1

L Rt =FK;t+1 + FH;t
FH;t + Dt + FH;t+1 =Rtt ) FH;t

(16)
(17)

:

(18)

The capital good resource constraint (11) remains the same as in Problem LF except that we
replace qt+1 with FK;t+1 . We emphasize that Kt+1 depends on the market interest rate in periodt and de…ne(Rt ) as a function of Rt . Provided that a crisis is not taking place, Kt+1 evolves
13

according to,Rt

Kt+1 = I + I

FK;t+1(Rt ) ;
where

0 < 0 represents the total derivative of K

(19)

8
t+1 with respect to Rt .

The solution of Problem(SP) can be considered as the constrained optimum as discussed in Allen and Gale (1998). The
next proposition establishes our second main result.
Proposition 2 The laissez-faire banking sector cannot achieve the constrained optimum.
Proof. It su¢ ces to show that d t =dDt in (14) in Problem LF deviates from that in Problem
SP. By di¤erentiating (18) at

t , we can write out d t =dDt in Problem SP asd t
1=
dDt SP
mt
where

0

"

h

Rt =qt+1

2

d
(Lt
dRt

@
(Lt
@Rt

gt )

Thus, the proposition is proven by either (i) non-zeroof the excess liquidity supply functions Lt
problems. While in general"

dRt
dDt SP

t

mt

;

(20)

i
F
t HK;t+1 . Furthermore, we compare (20) with

L 0 FKK;t+1 +d t
1=dDt LF
mt

gt )

dRt
dDt LF

t

mt

:

(21)

" , (ii) discrepancies in the derivatives/slopes

gt , or (iii) discrepancies in dRt =dDt across the two" is non-zero, it may also be the case that" is zero, depending on theparameter set. For completeness, we highlight (iii), the discrepancy in dRt =dDt . By di¤erentiating
(17),
dRt
d
dRt
Rt
=
+
F
dDt SP
dDt LF FK;t+1 dDt K;t+1(22)

con…rms that the second term on the right-hand side of (22) is non-zero.

3.2

Marginal Systemic Risk (MSR)

As a general concept, we introduce a measure to assess the systemic risk of a …nancial crisis, thatis, marginal systemic risk (MSR) as the deviation of the marginal increase in the crisis probability
against a target variable in the social optimum from the laissez-faire economy. MSR can be appliedin 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, bank lending, or
potentially, asset prices. In our model, MSR is de…ned with respect to a 1 percent change in bank8

Solving

0

= (1

0

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

14

0

ensures that0is negative.

leverage. Speci…cally, let DLF be the level of bank leverage chosen in the laissez-faire economy.
Then,
M SRt =d (Dt )
d log Dt Dt =DLFd^ (Dt );d log Dt Dt =DLF

(23)

where (Dt ) and ^ (Dt ) are the functions of Dt that indicate the crisis probabilities for the social
planning bankers and for the laissez-faire banking sector. Recall that the latter takes other banks’decisions as given, but, in fact, the crisis probability is a¤ected by the synchronized decisions by thebanking sector as a whole. In this regard, ^ (Dt ) can be interpreted as the probability perceived by
the individual price-taking banks, which can be contrasted with the true probability, (Dt ). With
this de…nition in mind, a positive MSR indicates that the laissez-faire banking sector underestimatesthe marginal cost of higher leverage by not taking into account the systemic risks. As the result,the banking sector is likely to be overleveraged at (and around) the laissez-faire equilibrium. The
MSR can take a negative value if the laissez-faire banking sector is underleveraged. While we do notdiscuss policy-related issues in this paper, in principle a positive MSR would provide the groundfor government intervention to rein in excessive leverage of banks, because such regulatory risk
reduction can improve welfare. In our model, and presumably in general, a marginal increase inbanks’leverage raises the probability of a …nancial crisis. If the economy includes, for example, apecuniary externality, measuring MSR would indicate the extent, to which the laissez-faire economy
is exposed to an excessively high probability of a …nancial crisis. To better grasp what creates thenon-zero MSR, or equivalently, the ine¢ cient allocation, we can inspect the e¢ ciency conditions ofour model. As a preparatory step, we make one assumption," = 0, to specify the most importantsource of the systemic risk.
The following proposition characterizes our main result in this section.
Proposition 3 Under the assumption" = 0, MSR is strictly positive.

Proof. See the Appendix.
The assumption that"is zero is made solely to crystallize the main channel creating thepositive MSR and to underscore the channel analytically.9 In fact," tends to be fairly small
10
because of the two o¤setting e¤ects. Proposition 3 provides a foundation for understanding why

the crisis probability is higher in the laissez-faire economy than in the social optimum. We discuss
the interpretation of Proposition 3 based on a sketch of the proof set out in the Appendix.
9

In a later section, we will show that even in the absence of the assumption, the numerical results indicate thatthe probability of a crisis is higher in the laissez-faire economy than in the social optimum.
10
The distortion, " includes two o¤setting e¤ects on demand and supply for liquidity. In particular, a marginalincrease in Dt reduces Rt and the reduction in Rt a¤ects marginal products of capital and labor via the capital
accumulations on the threshold. On the one hand, such an increase in capital on the threshold generates an increasein the marginal product of labor, resulting in higher liquidity demand from households. On the other hand, theincrease in the threshold capital gives rise to a decrease in marginal product of capital on the threshold. This e¤ecttightens the solvency condition, resulting in the higher liquidity supply. These two e¤ects raise both demand andsupply for liquidity, respectively. Consequently, the net e¤ect on Rt tends to be small.

15

1

MSR in our model can be approximately written out as follows. Bearing in mind thatR t
0 f ( t) d t;
M SRt = f ( t )'Again,d^ (Dt )d log Dtf ( t) d(Lt
mt dRtd (Dt )
d log Dt!

Dt =DLF

dR
dDt LF

gt )

t =

dR
dDt SP

DLF :

(24)

(Dt ) and ^ (Dt ) are the functions of Dt and ^ (Dt ) refers to the tightness of the liquidity

market perceived by the laissez-faire banks. Note that all functions are evaluated at the laissez-faireequilibrium. In the above equation, f ( t ) =mt and the excess liquidity supply function denoted
as d (Lg ) =dR are both positive. We are left with the deviation of the changes in Rt withrespect to bank leverage. Evidently, (24) indicates that MSR is zero if the marginal changes in Rt ,where banks remain solvent, do not vary across the two economies. However, the discrepancy ofthe marginal changes in Rt can be shown to be strictly positive. To see this, we investigate how
banks in each economy recognize their own solvency constraints. In the laissez-faire economy, bydi¤erentiating the solvency constraint (8), we obtainqdRt= t+10< 0;
dDt LF
A LF(25)

which points exactly to the marginal change in Rt against bank leverage. Since A( ) is monotoricallydecreasing in the relative price, the total derivative A0 is always negative. Bearing this in mind,
we note that a marginal increase in banks’leverage reduces Rt . A higher leverage means a higher
likelihood of insolvency. The issue on which we focus here is how quickly banks fall short of theircommitted payment. This information is translated into the changes in Rt against banks’leverage.
If banks can behave as the social planner, taking into account the changes in capital prices, themarginal change in Rt is written as
dRt=
dDt SP1
1

0
Rt qt+1

0 =q

t+1

qt+1
dRt
<
< 0;
0
A
dDt LF(26)

0where qt+1
is the total derivative of qt+1 with respect to Kt+1 (i.e., dFK;t+1 =dKt+1 ). Comparing(26) with (25) indicates clearly that an individual price-taking bank underestimates the marginalchanges in Rt against its own leverage.
Equations (26) and (25), together with (24), con…rm that the positive MSR arises from thebanks’ “rational over-optimism” of the marginal changes in the threshold interest rate Rt withrespect to their own leverage. In general, an increase in leverage reduces Rt , because highly
leveraged banks would be more likely to defalt under a lower interest rate. The positive MSR in

16our model is closely associated with the size of reduction in Rt . The banks in our model makerational decisions with full information (while admittedly, they need to decide Dt before observing
the random shockt ). Despite their rational decisions, banks always look over-optimistic about themarginal cost of higher leverage and turn out to be more frequently insolvent than expected. To
0understand the bank’s over-optimism, note that qt+1

0 is included in (26). This term points to aside e¤ect arising from higher leverage: In general equilibrium, the reduction in Rt increases Kt+1 .
That is, the lower Rt stimulates capital accumulation on the threshold and this increase in Kt+1
triggers the decline in the threshold capital price qt+1 via the lower marginal product of capital.
With this side e¤ect, the lower capital price further undermines the bank’s solvency, compared tothe case without the side e¤ect of increasing the leverage. Because the price-taking competitivebanks, however, have no incentive to take into account this side e¤ect, the lower-than-expected
capital price and the undermined banks’solvency raise the probability of a …nancial crisis.
Looking at the real-world experience of past …nancial crises, it may be pointed out that, withhindsight, outlooks regarding asset prices frequently tended to be overly optimistic in the runups to crises. Some argue that such over-optimism arises from irrationality. While we do notclaim that irrational behavior is irrelevant, our model suggests that despite the full rationality, apecuniary externality can result in seemingly irrational over-optimism. The key to understanding
the externality lies in the synchronized decisions by the individual banks in a competitive sector.
For each bank, capital prices are given. Recall that the …nancial market is incomplete in theabsence of the Arrow securities. Incompleteness of the …nancial market means, not surprisingly,that a competitive market equilibrium can result in a distorted allocation rather than a Paretooptimal one. In the case of our model, the distortion shows up as the overleveraged banking sector
with a higher crisis risk because of the side e¤ect as shown in equations (26) and (24), which canbe interpreted broadly in line with the real-world observations.

4

Numerical Results

4.1

Solving the Model

Analytical results in the previous sections can be translated into numerical examples. 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 does the laissez-faire banking
sector deviate from the social optimum, and to what extent does the laissez-faire banking sector act
over-optimistically regarding the cost of a crisis?; (iii) What can the calibrated model tell us aboutcrises which should be prevented by government intervention in the banking sector?; and (iv) How
can we compare the numerical results with existing empirical studies on the probability of crises?
Our calibration mostly follows Diamond and Rajan (2009). We set the value of prematurely
liquidated project X at 0.95. The distribution of entrepreneurs’projects is assumed to be uniform17

over [! L ; ! H ] = [0:5; 3:5], similar to the original calibration of Diamond and Rajan (2009). The
degree of banks’special collection skill

is set at 0.9. In addition to parameterization of Diamondand Rajan (2009), we need to set some more parameters. We calibrate the capital share in the
production function,, to 1/3, the capital good endowment received by entrepreneurs, I, to one.

More importantly, we assume that the liquidity shockt follows the beta distribution with a meanof 0.50 and a standard deviation of 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 bank deposits. On theother hand, the model does not specify the length of each period of time (e.g., one year or one
quarter) and because of the absence of a time unit, Dt cannot be translated into an annual percentage rate (APR) or an interest rate per annum. To better focus on the economic interpretations,
Dt needs to be translated into a timeless measure such as the bank leverage. This is the exact
reasoning that we have relied on this interpretation of Dt , that is, bank leverage.

4.2

The Probability of Financial Crises

A notable feature of the model is that the probability of a …nancial crisis varies endogenously. A
primary quantitative question to be addressed here is how frequently a …nancial crisis can takeplace.
Our simulation results are summarized in Table 2, where the realization oft is assumed to takethe mean of 0.50. The upper panel of the table recon…rms that the laissez-faire banking sector takes
on more risks than the social planning banks, indicating a higher crisis probability. Our calibration
points to a 6.59 percent crisis probability in the laissez-faire economy compared to 4.50 percent
in the social optimum. The results should not be translated into a duration of non-crisis periods,
because of the absence of the time unit. Hence, we argue that, in an arbitrary period, out of 1,000
simultaneous attempts, about 66 attempts would trigger crises. The overleverage can be con…rmedby the values of Dt in the upper panel of the table. In fact, Dt is 1.2 percentage points higher inthe laissez-faire banking sector than in the social planning banks. Figure 1 plots the level of the
expected utility against Dt . As Proposition 1 indicates, laissez-faire banks choose their leverage
uniquely, shown at point B in the …gure. In line with Proposition 2, however, the laissez-fairebanking sector cannot achieve the constrained optimum. In fact, our computation results clearlyrea¢ rm that this is the case.

18

4.3

The Size of Distortions

We next examine to what extent the laissez-faire banking sector deviates from the allocations
achieved by the social planning banks. More broadly, we discuss the quantitative implications of
the higher leverage in the laissez-faire economy for the welfare.
We have argued that MSR is a useful measure in assessing the ine¢ ciency of the economy
and have shown in Proposition 3 that laissez-faire banks in our model generate a strictly positive
MSR under the assumption" = 0.

If we remove this assumption, it could be argued that the

proposition may not always be the case. Our simulation results con…rm that the MSR, in fact,
takes a positive value under a plausible parameter set. The lower panel of Table 2 reports the MSR
evaluated under the allocations in the laissez-faire economy. The value is 0.48 percentage point, andthis extra increase in crisis probability arises solely from pecuniary externalities. To evaluate thesize of pecuniary externalities, suppose that banks increase the leverage by 1 percent. The MSR
obtained implies that each bank underestimates the crisis probability by 0.48 percentage point.
Since the laissez-faire economy generates the semi-elasticity of a crisis probability, d t =d log Dt ,of 1.64 percent, this MSR means that each bank expects the crisis probability to increase by
only 1.16 (i.e., 1.64 - 0.48) percentage points. In other words, each bank makes the individually
rational assessment that, by increasing 1 percent of leverage, they are exposed to 7.75 percent crisis
probability. However, the true probability is 8.23 percent. Figure 2 also con…rms t he positive
MSR. The red and blue lines in the …gure plot ^ (Dt ) and(Dt ), respectively, around the value of

Dt chosen under laissez-faire banking. As discussed, the former corresponds to the case when banks
take factor prices as given and the latter corresponds to the case where pecuniary externalities areinternalized. The …gure shows that the red line is ‡atter than the blue line, re‡ecting the positive
MSR.
Figure 3 plots the marginal cost and bene…t in the two economies. The marginal cost is represented by solid lines, and the marginal bene…t by dashed lines. The intersections A and B represent
the choices of leverage under the social optimum and the laissez-faire economy, respectively. This
numerical computation recon…rms that a bank’s underestimation of the marginal cost leads to theoverleverage. In this computation, we also observe a relatively small shift in the marginal bene…t
curve across the two economies, which slightly exacerbates the bank’s overleverage.11 However, theshift in the marginal cost curve dominates the shift in the marginal bene…t curve.
The lower panel of Table 2 compares the bank capital ratio de…ned as (AtDt ) =At and theoutput of the consumption goods Yt+1 under the laissez-faire economy and the social optimum.
11

This shift results mainly from the laissez-faire bank’s overestimation of wages in period t + 1, conditional on a…nancial crisis not taking place. A marginal increase in leverage leads to greater consumption by households anda higher interest rate. A higher interest rate implies a lower amount of the completed project (i.e., lower capital
accumulation) in the laissez-faire economy. The lower capital accumulation also reduces the marginal product oflabor, resulting in lower wages in period t + 1. Since the laissez-faire banks disregard this income-reducing e¤ect, thebanks overestimate the marginal bene…t of increasing Dt .

19

The laissez-faire banking sector is undercapitalized by 1.1 percent compared to the social optimum.
Nevertheless, it may be surprising that the production does not substantially vary 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) for the laissez-faire banking sector in comparison to (C1;t ; C2;t+1 ) = (2:21; 2:59) inthe social optimum.
The above exercise indicates that the welfare loss primarily arises from the ine¢ ciently elevatedcrisis probability. Given that crises are considered to be 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 the ine¢ ciency or welfare loss with
MSR appears more appropriate than with the volatility of the consumption or output.

4.4

Crisis Prevention

The prediction that the laissez-faire banking sector is overleveraged implies that the laissez-faireeconomy undergoes crises created by banks’overoptimism, some of which could be avoided under
the social optimum. To illustrate this, we run the model over 100 periods by generating liquidity
shocks randomly. Figure 4 plots the dynamic paths of output Yt andt under the two allocations.

The red line corresponds to the case of laissez-faire economy, while the blue line points to the caseof the social optimum.
The upper panel of Figure 4 shows that the dynamic paths of the output are almost identical
except that the production under the laissez-faire economy sharply declines more frequently. Crises
take place in period 5, 16, and 94 and output sharply falls in each subsequent period. We note that
this simulation result indicates that the latter two crises could have been prevented if the banks
were taking the risks at the optimal level, implying a need for government interventions to forestall
the crises. However, the …rst crisis takes place even in an economy in which the social planningbanks strike the right balance between the costs and bene…ts of increasing the leverage. Therefore,
this crisis should not be avoided as discussed in Allen and Gale (1998) in the context of the optimal…nancial crises.
To better understand how di¤erences arise in the two economies, the lower panel of the …gureshows two di¤erent dynamic paths oft.

Recall thattis de…ned as the threshold value of theliquidity shock that satis…es the solvency constraint on the brink of a default. This threshold leveloftis always lower in the laissez-faire economy, implying that the solvency constraint is tighterand the economy is more exposed to the macroeconomic fragility. The dashed line represents therealizedt in the simulations, which is identical across the two economies. The realized

both high and low

t in the …rst crisis and reaches only lower

the di¤erence in the realized
volatility of

t in the last two crises.

t exceeds

Althought appears to be fairly small among the three crises, relative to thet , the small di¤erence in the liquidity shock a¤ects the economic performance, through20

the risk-taking of the banking sectors.

4.5

Comparison with Empirical Studies

The numbers included in Table 2 may be compared to some empirical works on (i) catastrophe(CAT) risks and on (ii) the optimal level of bank capitals. Among a number of works on CAT risks,Barro (2009) can provide a comparable benchmark. He sets the “disaster probability”at 2 percent
per year, arguing that a “disaster” could reduce GDP by 30 percent on average. By incorporatingsuch disaster risks into the Epstein-Zin type asset pricing model, he concludes that the 2 percent
CAT risk can account for an equity premium of around 4-6 percent. He also argues that the sizeof scarring e¤ects of a disaster can range from 15 to over-60 percent of GDP.
A more straightforward study by Basel Committee on Banking Supervision (2010, BCBS) assesses impacts of changes in bank capital on the probability of systemic banking crises; it aims to
explore the optimal level of bank capital in the context of the yet-to-be enacted Basel III, the new
global regulation framework for banking systems. BCBS estimates that a marginal increase in 1
percent bank capital from the pre-reform cross-country average level could reduce the crisis probability by 1.0-1.6 percentage points.12 We can underscore the proximity of the BCBS estimates andthe numerical results included in Table 2.13 Beyond such simple numerical comparisons, we emphasize that MSR can be applied in line with a broad empirical exercise as typically demonstrated
by BCBS. Using reduced-form econometric models, a marginal increase in the crisis probability
is evaluated at around a certain point in the data. But empirical estimates, in general, cannot
provide information on how distant the economy is located from the social optimum. Assessing towhat extent the observed equilibrium could be improved by policy interventions may be of moreinterest. To this end, comparison of empirically estimated marginal changes in crisis probabilities
with MSRs calculated using structural models would promote discussion on the desirable size anddesign of regulations in the banking sector down the road.

5

Related Literature (List only)

To be added.
12
Its estimates are based on multiple empirical methodologies, but it could be said that, primarily, reduced form
econometric models, such as probit/logit models, are used to yield those estimates.
13
In addition to the evident proximity of the marginal changes in the crisis probabilities, the level of bank capital in
BCBS does not substantially di¤er from that in our model. BCBS argues that the pre-reform cross-country averageof the TCE/RWA (tangible common equity divided by Basel II risk-weighted assets) ratio is 7 percent. The TCE isan extremely narrow de…nition of bank capital which, by and large, could be doubled or even tripled (i.e. 14 - 21
percent) if measured in more conventional measures for bank capital, such as the Tier I ratio.

21

5.1

Banks and macroeconomics
Diamond and Rajan (2001a, 2001b, 2009)
Diamond and Dybvig (1983),
Allen and Gale (1998, 2007)
Fahri, Golosov, and Tsyvinski (2009)
Carlstrom and Fuerst (1997),
Christiano, Motto and Rostagno (2010),
Angeloni and Faia (2010),
Gertler and Kiyotaki (2011)
Kato (2006)
Kato and Tsuruga (2011)5.2

Bubbles
Abreu and Brunnermeier (2003)
Martin and Ventura (2010, 2011),
Aoki and Nikolov (2010)
Reinhart and Rogo¤ (2009)

5.3

Pecuniary externalities: Applications of MSR
Bianchi (2010)
Bianchi and Mendoza (2011)
Jeanne and Korinek, (2010)
Lorenzoni (2008)
Acharya, Pedersen, Philippon, and Richardson (2010)22

6Conclusions

We developed a dynamic general equilibrium model that explicitly includes banks with maturity
mismatches. Using the model, we showed that, under the laissez-faire economy, ine¢ cient …nancial
crises are precipitated by a liquidity shortage in the overleveraged banking system. In general, aperfectly competitive banking sector cannot always achieve the …rst-best allocation because thebanking business per se (i.e., maturity transformation via issuance of non state-contingent debt
rather than the Arrow security) implies that …nancial markets are incomplete. Our model demonstrated that a perfectly competitive banking sector cannot achieve the second-best allocations,
resulting in ine¢ cient …nancial crises. In our model, pecuniary externalities arise, distorting thebanks’assessment on their own solvency. The banks fail to internalize the side e¤ects from changes
in the illiquid asset prices on their own solvency. From the viewpoint of a social planner, because ofthis failure, the rational banks overestimate their solvency and underestimate the cost of increasing the leverage systemically. This rational over-optimism exposes banks to ine¢ ciently elevatedsystemic risks. In the light of real world experience, our model could lay a foundation for better
understanding of the repeatedly observed …nancial and economic crises.
We also introduced MSR (marginal systemic risks) as a measure to assess the (in)e¢ ciency
of an economy. When MSR takes a positive value, the economy is considered being exposed
to systemically excessive risks of …nancial crises. While our model de…ned MSR vis-à-vis bankleverage, it can also be de…ned against some other …nancial variables, and thus can be applied toa wide class of models explaining …nancial crises. As discussed by Allen and Gale (1998), …nancial
crises can arise as a socially optimal outcome. In line with their argument, …nancial crises need to
be prevented, only if it is ine¢ cient. To assess the need for government intervention, MSR can beused as an informative measure.
The analysis demonstrated in this paper can be extended in a number of directions. First, we
may need to consider policy measures that can eliminates the positive MSR in our model. Second,
we may need to examine how changes in a variety of the economic environment (e.g., changes inthe stochastic process of the liquidity shock) or newly introduced aggregate shocks (e.g., shocks tothe asset side of the banks’balance sheet) a¤ect the economy’s exposure to crisis risks.14 Finally,
introducing price stickiness into the model would pave the way for reconstruction of the roles ofmonetary policy in comparison with similar models which does not include the possibility of a…nancials crisis. All of these directions would provide important steps for future research.

14

Kato and Tsuruga (2011) demonstrate that rational banks can take on more risks, resulting in a higher default
probability, in response to changes in the underlying distribution of shocks that reduce exogenous risks of bankdefaults.

23

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173-204.
[2] Acharya, V. V., L. H. Pedersen, T. Philippon, and M. Richardson (2010) “Measuring Systemic
Risk,” mimeo.
[3] Allen, F., and D. Gale (1998), “The Optimal Financial Crises,” Journal of Finance, 53 (4),pp. 1245-1284.
[4] Allen, F., and D. Gale (2004), “Financial Intermediaries and Markets,” Econometrica, 72 (4),pp. 1023-1061.
[5] Allen, F., and D. Gale (2007), “Understanding Financial Crises,” Oxford University Press.
[6] Angeloni, I., and E. Faia (2010), “Capital Regulation and Monetary Policy with Fragile Banks,”
mimeo.
[7] Aoki, K., and K. Nikolov (2010), “Bubbles, Banks and Financial Stability,” mimeo.
[8] Barro, R. (2009), “Rare Disasters, Asset Prices, and Welfare Costs,” American Economic
Review, 99 (1), pp. 243-264.
[9] Basel Committee on Banking Supervision (2010),
Term

Economic

ImpactofStronger

Capital“An Assessment of the Longand

Liquidity

Requirements,”

http://www.bis.org/publ/bcbs173.htm[10] Bianchi, J. (2010), “Credit Externalities: Macroeconomic E¤ects and Policy Implications,”
American Economic Review, 100 (2), pp. 398-402.
[11] Bianchi J., and E. Mendoza (2011), “Overborrowing, Financial Crises and Macro-Prudential
Taxes,” NBER Working Paper No. 16091.
[12] Caballero, R. (2010), “Macroeconomics after the Crisis: Time to Deal with the Pretense-ofKnowledge Syndrome,” Journal of Economic Perspectives, 24 (4), pp. 85-102.
[13] Carlstrom, C. T., and T. Fuerst (1997), “Agency Costs, Net Worth, and Business Fluctuations:
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[14] Christiano, L., R. Motto, and M. Rostagno (2010), “Financial Factors in Economic Fluctuations,” ECB Working Paper Series No. 1192.

24

[15] Diamond, D. W., and P. Dybvig (1983), “Bank Runs, Deposit Insurance, and Liquidity,”
Journal of Political Economy, 91 (3), pp. 401-419.
[16] Diamond, D. W., and R. Rajan (2001a), “Liquidity Risk, Liquidity Creation, and Financial
Fragility: A Theory of Banking,” Journal of Political Economy, 109 (2), pp. 287-327.
[17] Diamond, D. W., and R. Rajan (2001b), “Banks and Liquidity,”American Economic Review,
91 (2), pp. 422-425.
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Paper No. 15197.
[19] Fahri, E., M. Golosov, and A. Tsyvinski (2009), “Theory of Liquidity and Regulation of
Financial Intermediation,” Review of Economic Studies, 76 (3), pp. 973-992.
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Cycle Analysis,”Handbook of Monetary Economics, B. Friedman and M. Woodford (ed.), 3A,
pp. 547-599.
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Approach,” NBER Working Paper No. 16377.
[22] Kato, R. (2006), “Liquidity, In…nite Horizons and Macroeconomic Fluctuations,” European
Economic Review, 50 (5), pp. 1105-1130.
[23] Kato, R., and T. Tsuruga (2011), “The Safer, the Riskier: A Model of Bank Leverage andFinancial Instability,” Kyoto University Research Project Center Working Paper Series, E10014.
[24] Korinek, A. (2010), “Regulating Capital Flow to Emerging Markets: An Externality View,”
mimeo.
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809-833.
[26] Martin, A., and J. Ventura (2010), “Economic Growth with Bubbles,” NBER Working Paper
15870.
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25

[29] Reinhart, C., and K. Rogo¤ (2009), “This Time Is Di¤erent: Eight Centuries of Financial
Folly,” Princeton University Press.

26

A

Appendix: Proofs (TBA)

A.1

Proof of Proposition 1

To be completed. The proof broadly follows Kato and Tsuruga (2011).

A.2

Proof of Proposition 3

To be completed. The proof is available upon request.

27

Table 1: Sequence of events for generation t

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

Period t
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 condition along with the old generation’s labor supply.
Liquidity shock t is realized, and banks receive signals of project
outcomes.
Households decide the withdrawal gt .
Banks decide which projects to continue 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 fully withdraw deposits, if any.
Households supply labor and receive wages wt+1 determined by the labormarket condition along with the young generation’s labor supply.
All agents consume.

28

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

Laissez-faire banks

Social planning banks

1.06
6.59

1.05
4.50

Leverage and crisis probabilities
Dt
t (%)
M SRt

0.48

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

13.95
5.46

15.10
5.46

Note: Simulation results based on the assumption that the liquidity shock t follows the beta distribution and themean is materialized as a realized value of t . The level of bank’s leverage Dt and the probability of a …nancial crisist are obtained from Problems LF and SP, respectively. The marginal systemic risk, M SRt , is given by (23). The
bank capital ratio is (At Dt ) =At .

29

Figure 1: Optimal bank leverage

E[U ]
t

0.87

A

B

Expected utility

0.86

0.85

0.84

0.83

1.02

1.04

1.06

1.08

1.1

1.12

1.14

Dt

Note: The expected utility against Dt . Point A corresponds to the expected utility level evaluated at Dt chosen
by social planning banks, while point B corresponds to the expected utility evaluated at Dt chosen by laissez-fairebanking sector.

30

Figure 2: Marginal systemic risk (MSR)

π hat (Dt)π(Dt)π(Dt) and π hat (Dt)

0.1

0.08

0.06

0.04

0.02
1.04

1.05

1.06

1.07

1.08

1.09

Dt

Note: The blue line plots the crisis probability in the general equilibrium allocation. The red line plots the crisis
probability that laissez-faire banks recognize.

31

Figure 3: Marginal cost and bene…t of increasing Dt

0.8
MC|
MC|

0.7

MB|
MB|

LF
SP

LF
SP

Marginal cost and benefit

0.6

0.5

0.4

0.3

B
A
0.2

0.1

0

1.02

1.04

1.06

1.08

1.1

1.12

1.14

Dt

Note: The blue solid line (M CjSP ) and dashed line (M BjSP ) represent the marginal cost and bene…t under the social
optimum, respectively. The red solid line (M CjLF ) and dashed line (M BjLF ) are the marginal cost and bene…t under
the laissez-faire economy, respectively.

32

Figure 4: Simulated paths of output and the liquidity shock
Output
6

Y

t

5.5

5

Laissez-faire

4.5

4

0

10

20

30

40

50

60

Social planner

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

periods

30

40

50
*

60

θt (Laissez-faire)70*

θt (Social planner)

Realized θtNote: The upper panel shows the simulated dynamic paths of output Yt under the laissez-faire economy and thesocial optimum. The liquidity shock plotted as the black dashed line in the lower panel is generated from the beta
distribution with a mean of 0.50 and a standard deviation of 0.07. The blue and red solid lines in the lower panel
are the threshold level of the liquidity shock that marginally satis…es the solvency condition. The blue line is for thesocial planning banks and the red line is for the laissez-faire banks.

33