ISSN (Print) 0473-453X

Discussion Paper No. 1078

ISSN (Online) 2435-0982

PECUNIARY EXTERNALITIES,BANK OVERLEVERAGE,AND
MACROECONOMIC FRAGILITY

Ryo Kato
Takayuki Tsuruga

March 2020

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, JapanPecuniary Externalities, Bank Overleverage, andMacroeconomic Fragility
Ryo Katoyand Takayuki Tsurugaz
March 2020

Abstract
Pecuniary externalities in models with …nancial friction justify macroprudential
policies for preventing economic agents’excessive risk taking. We extend the Diamondand Rajan (2012) model of banks with the production factors and explore how a pecuniary externality a¤ects a bank’s leverage. We show that the laissez-faire banks in
our model take on excessive risks compared with the constrained social optimum. Our
numerical simulations suggest that the crisis probability is 2–3 percentage points higher
in the laissez-faire economy than in the constrained social optimum.
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. The
authors gratefully acknowledge the …nancial support of Joint Usage/Research Center at ISER, Grants-in-aid
for Scienti…c Research (15H05729, 15H05728 and 19K23219). This paper is previously circulated as “BankOverleverage and Macroeconomic Fragility.”yGraduate School of Public Policy, The University of Tokyo; e-mail: rkato@pp.u-tokyo.ac.jpzInstitute of Social and Economic Studies, Osaka University e-mail: tsuruga@iser.osaka-u.ac.jp1

1

Introduction

Since the global …nancial crisis of 2007–08, one of the challenges for policymakers has been
how to avoid systemic …nancial crises. Policymakers and researchers have discussed a pecuniary externality as a rationale for preventing economic agents’ excessive risk taking. For
example, Mendoza (2010), Korinek (2018), and Jeanne and Korinek (2019) use the small
open economy model with the collateral constraint. They show that households’ overborrowing takes place because households fail to internalize the general equilibrium e¤ect ofthe price of the collateral on the collateral constraint.1 However, as argued in Dávilla andKorinek (2017), it is “remarkably di¢ cult to provide general results on the direction of ine¢ ciency.”In other words, even if a model of …nancial crises has a pecuniary externality, it
does not guarantee that economic agents will engage in excessive risk taking. An example isthe small open economy model with tradables and nontradables by Benigno, Chen, Otrok,
Rebucci, and Young (BCORY, 2013). They show that, once they change the endowment
economy model into the production economy model, the model of overborrowing predicts
underborrowing in normal times. This example suggests that it is important to check thedirection of ine¢ ciency, namely, whether a pecuniary externality creates excessive risk taking
in a model of a …nancial crisis.
In this paper, we take a model of bank runs to investigate the direction of ine¢ ciency.
We develop a production economy model with maturity-mismatching banks. Here the modelwith banks is a variant of Diamond and Rajan (2012, DR) and is discussed in Kato and
Tsuruga (2016) in which a liquidity preference shock precipitates a banking crisis. We ask
how a pecuniary externality a¤ects allocations under the competitive equilibrium with thelaissez-faire (LF) banks. In particular, we introduce the prices of production factors (e.g.,
the capital price and wages) to ask how pecuniary externalities generate ine¢ ciency. Thisextension is similar to that in BCORY because we also change the endowment economy
model into the production economy model. In our model, banks fail to internalize the e¤ectof their risk taking on factor prices. Our question is whether the pecuniary externality leads
to “underleverage”or “overleverage”in a model of banks.
The pecuniary externality in our model leads to ine¢ cient allocations compared withthe constrained social optimum that a social planning (SP) agent can achieve. However,
our model di¤ers from the abovementioned small open economy model with the collateral1

See also other related studies such as Lorenzoni (2008), Bianchi (2010, 2011), Stein (2012), Benigno,
Chen, Otrok, Rebucci, and Young (2016), and Bianchi and Mendoza (2018).

2

constraint in a few aspects. First, the economic agent that engages in ine¢ cient borrowing
is a bank rather than a household. Second, the pecuniary externality in our model operates
through the constraint on the bank’s solvency rather than the collateral. The solvency
constraint arises as a natural consequence of the maturity-mismatching banks that issue
non-state-contingent short-term debt and invest them in illiquid assets.
Key elements of the model are (i) the bank’s issuance of non-state-contingent debt and(ii) the dependence of the banks’ asset value on factor prices. The …rst element enables
banks to raise funds and to promote liquidity creation in the absence of complete markets,as discussed in Diamond and Rajan (2001a, b). However, this bene…t of non-state-contingent
debt comes at the cost of potential insolvency of banks. The second element is incorporated
with our overlapping generations (OLG) framework that explicitly includes factor markets.
In our model, banks’ balance sheet directly depends on the price of capital goods. Under
this dependence of banks’ balance sheet on capital price, the banks fail to internalize thee¤ect of capital prices on their solvency.
Our main …nding is that banks leverage to an excessive level because of a pecuniary
externality. Using numerical analysis, we compute the crisis probabilities in the competitive
equilibrium. The crisis probability is ine¢ ciently high and the magnitude of the ine¢ ciency
is not negligible. In our benchmark simulations, the crisis probability is 6.6 percent under thecompetitive equilibrium, compared with 4.5 percent under the constrained social optimum.
Our OLG framework also generates sharp declines in economic activity in the aftermath of acrisis. When a liquidity shock precipitates a crisis, banks are required to liquidate all illiquidloans to entrepreneurs to repay depositors. The liquidation stops capital goods production
and subsequently results in a signi…cant contraction in consumption and output.
Our numerical analysis suggests that the primary source of overleverage is the underestimation of the crisis probability by the price-taking banks. The crisis probability in generalequilibrium depends not only on the price of the liquidity (i.e., the interest rate) but also on
all the other prices in the economy. In our model, price-taking banks miscalculate the crisis
probability because they fail to internalize the general equilibrium e¤ects stemming fromfactor prices. More speci…cally, when the LF banks raise their leverage, they underestimate
the marginal increase in the crisis probability.
We quantify the importance of the pecuniary externality stemming from the banks’solvency because the banks’ solvency directly in‡uences the crisis probability. The banks’solvency depends on the capital price, which the LF banks take as given. We perform acounterfactual experiment. In the experiment, the LF banks internalize changes in the cap3

ital price that appear in the solvency constraint while ignoring all other general equilibriume¤ects stemming from factor prices. The result suggests that the LF banks would keep
their leverage close to the constrained social optimum if they could internalize the pecuniary
externality stemming from the solvency constraint.
Our paper makes a contribution to the literature by incorporating the pecuniary externality into the theory of banking. Our model extends Allen and Gale (1998) and DR inmodeling banks.2 At the same time, our model borrows the idea of the pecuniary externality from the model with collateral constraint such as Lorenzoni (2008), Bianchi (2011), andJeanne and Korinek (2019) among others. In our model, the banks’solvency plays a crucialrole in generating bank runs or …nancial crises. Our model further explores the possibilityof ine¢ cient …nancial crises via the dependence of the banks’solvency on factor prices.3
Our paper is also related to the macroeconomic literature on models with …nancial constraints. Macroeconomic models with banks have focused on how …nancial frictions amplify
business cycles (e.g., Gertler and Karadi 2011, Gertler and Kiyotaki 2011, and Meh andMoran 2010). Our focus contrasts with these studies because we examine the e¢ ciency ofthe competitive equilibrium. Angeloni and Faia (2012) incorporate banks à la Diamond andRajan (2000, 2001a) into the dynamic stochastic general equilibrium model. While theirmodel has the endogenous probability of bank insolvency, the probability of bank insolvency
can be interpreted as a measure of individual bank fragility rather than the probability ofsystemic …nancial crises. Our model is also close to Gertler and Kiyotaki (2015) and Boissay,
Collard, and Smets (2016). In Gertler and Kiyotaki (2015), whether an equilibrium withbank runs exists depends on macroeconomic fundamentals. However, …nancial crises per
se are precipitated by self-ful…lling expectations as in Diamond and Dybvig (1983). In ourmodel, fundamental shocks (liquidity preferences) precipitate …nancial crises. In this regard,
our model is broadly in line with empirical …ndings of the business cycle view of Gorton(1988) and Allen and Gale (1998). Boissay, Collard, and Smets (2016) focus on the moral
hazard in the interbank markets and successfully replicate systemic …nancial crises. On theother hand, we stress the fragility of the maturity mismatch of banks, as highlighted by the
Gorton and Metrick (2012) “run-on-repo”view.
2

Our model assumes that banks issue demand deposits following DR. In our model, however, we broadlyinterpret banks as …nancial intermediaries that raise funds via short-term debt such as repo and commercial
paper and transform maturities on their balance sheet. Demand deposits are an extreme case of short-term
debt.
3
Stein (2012) introduces pecuniary externalities into the macroeconomic model with banks to focus on
the risks of …re sales. Ikeda (2018) studies banks in a global game context and explores banks’excessive risktaking in the general equilibrium model.

4

The remainder of the paper proceeds as follows. Section 2 illustrates the model. In
Section 3, we compare the competitive equilibrium in Section 2 with the constrained social
optimum achieved by the SP banks. Section 4 discusses the numerical results and explains
why the banking sector tends to be overleveraged. Section 5 concludes.

2

A Macroeconomy with Banks

2.1

Agents, Endowment, Preferences, and Technology

We consider an in…nite-horizon OLG model with banks. Each generation of agents consists ofhouseholds, entrepreneurs, and bankers. Each period, generation t is born at the beginningof period t and lives for two periods, t and t + 1. Each agent is identical and constant in the
population.4
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 theonly source of uncertainty in the model. Following DR, households are endowed with aunit 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 same generation.5 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
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 capitalproducing technology a “project.” Entrepreneurs sell the capital goods in the competitivemarket for the capital price qt+1 .
Banks raise funds from households and lend them to entrepreneurs at the beginning of
period t.6 In principle, we follow Diamond and Rajan (2001a) to model banks. Banks
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. As the nature of4

We assume that an initial old generation lives for one period.
We implicitly assume intraperiod perishability of endowments. More precisely, all endowments perish
before the realization of the liquidity shock.
6
We assume intragenerational banking, which e¤ectively means that all bankers of generation t die outat the end of period t + 1.
55

demand 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 ; ZHt ) = Kt (ZHt )1;

where Kt , Ht , and Z denote the capital stock, hours worked, and labor-augmenting technological progress, respectively. Demand for labor and capital satis…es
wt = FH;t = (1)

qt = FK;t =

Kt
ZHt

Kt
ZHt

Z

(1)

1

:

(2)

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 inperiod t.7
7

Although 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 theexistence of a Nash equilibrium, in which all households run to the banks when they believe that the banks
are insolvent under the observed 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.
The intertemporal …rst-order condition for consumption is
t

1

t

C1;t
C2;t+1

1(4)= Rt :

The withdrawals can be written as
gt =

t

wt+1
+ Dt
Rt

(1

t ) wt ;(5)which implies that large t and Dt increase their withdrawals.
In our model, a …nancial crisis takes place endogenously, depending on t and Dt . Ift and Dt are su¢ ciently low, a …nancial crisis does not take place and households can
withdraw gt in period t and all the remaining deposits in period t + 1.8 With the endogenous
probability t , however, a …nancial crisis arises. When the banks cannot honor the debt,
households run to the banks and their withdrawals equal the liquidation value of premature
projects, X (< 1), in period t and nothing is left in period t + 1. In the case of a crisis, theirconsumption ends up with C1;t = wt + X and C2;t+1 = wt+1 .

2.3

Entrepreneursee
Entrepreneurs maximize their expected lifetime utility represented by E C1;t
+ C2;t+1;eewhere C1;tand C2;t+1
denote entrepreneurs’ consumption when young and old. They use
8

In the maximization problem of households, we assume that wage income in period t is low relative tothe initial endowment, ensuring a nonnegative withdrawal gt in the equilibrium.

7

a unit of consumption goods …nanced by banks for their capital goods production, and thisproduction technology takes one period for gestation before its completion. In period t + 1,
the project yields a random capital goods output !, which is distributed over [! L ; ! H ] withthe probability density function h (!).9 If this project is prematurely liquidated in period t,
the transformation from consumption goods into capital is incomplete. As a result, output
is reduced to X units of consumption goods and is repaid fully to banks in period t. Whenthe project is completed in period t + 1, however, entrepreneurs can sell their output in thecapital goods market for the capital price qt+1 .
Each entrepreneur can borrow from a bank that has, or can learn until the project
matures, 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 + 1.10 They sell this endowment
capital together with the newly created capital made from the consumption goods.

2.4

Banks

b
b
b
b
denoteand C2;t+1+ C2;t+1, where C1;tBanks maximize their expected lifetime utility E C1;tconsumption of banks when young and old. We borrow the microfoundation of modelingbanks from Diamond and Rajan (2001a, b, 2012). Banks have no initial endowment atbirth, and thus they need to raise funds from households. As the relationship lender, a bankknows how to operate the entrepreneurs’project, knowledge that cannot be transferred tohouseholds. Banks issue demand deposits (short-term debt) as a commitment device tocompensate for the lack of transferability of their knowledge (i.e., collection skills).11 The
demandable nature of deposit contracts creates a collective action problem for depositors:
depositors run to the banks whenever depositors anticipate that the banks cannot honor
9

Following DR, we assume that there is no aggregate uncertainty in the project outcome.
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.
11
Diamond and Rajan (2001b) discuss the microfoundation of maturity-mismatching banks and explain
why these demand deposits can promote liquidity creation under the lack of transferability of their collection
skill.
108

the debt. The deposit contract is predetermined before observing the liquidity shock. In
particular, 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 loan o¤er, resulting in an
identical portfolio shared by all the symmetric banks. This assumption implies that eachbank and the aggregate economy face an identical distribution of entrepreneurs. In periodt, the banks receive signals ! that perfectly predict the realized value of ! in period t + 1.
With this information ! and the households’ liquidity demand observed in period t, eachbank chooses one of the following 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 of a project falls shortof !
~ t+1 , de…ned as a function of Rt =qt+1 :12!
~ 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 capital.
Let the banks’asset be A (Rt =qt+1 ). The banks’assets at the beginning of period t (i.e.,
prior to the withdrawals) can be expressed as
A

Rt
qt+1

Z
qt+1 !H=Xh (!) d! +!h (!) d!
Rt !~ t+1
!L
Rt
qt+1
Rt
= L
+
I
:
qt+1
Rt
qt+1
Z !~ t+1(7)Note that h (!) is interchangeable with h (!) owing to perfect signaling. The banks’assets
denoted in (7) can be decomposed into two components: the values of the prematurely
R !~ t+1
liquidated projects Lt = L (Rt =qt+1 )Xh (!) d!; which is used to meet the liquidity!Ldemand (i.e., withdrawals) from the households, and the banks’ share of the investment
output (measured by the present value of consumption goods) denoted as qt+1 It =Rt ; where
R !H
It = I (Rt =qt+1 ) = !~ t+1!h (!) d!.
12

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 .

9

The banks are subject to the solvency constraint:
Dt(8)

A (Rt =qt+1 ) :If ! follows a uniform distribution, A ( ) monotonically decreases with Rt =qt+1 . We thende…ne the relative price Rt =qt+1 that satis…es the solvency constraint with equality:
Dt = A

Rt
qt+1

(9)

:

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.

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 :(10)eb= C1;t + C2;t + C2;t
+ C2;t:(11)(1

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

The left-hand side of (11) includes the supply of goods from the liquidated projects. On theebright-hand side of (11), C2;t , C2;t, and C2;tare consumption when generation t 1 is old.
The capital goods market clearing condition is
Kt+1 =(

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

(12)

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.

10

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. The banks are competitive at issuing demand
deposits, and households’ endowments are scarce compared with entrepreneurs’ projects.
As a result of competition, the banks make a competitive o¤er of deposits for households,
aiming to maximize household welfare (Allen and Gale 1998, 2007), while, in fact, they
are maximizing their pro…ts. Maximizing household utility via the deposit o¤ers meansthat banks determine the o¤er taking into account the optimal behavior of the price-taking
households.
In making the competitive o¤er to households, the banks take into account possible
changes in the crisis probability t . To understand how the banks’choice of Dt a¤ects t ,we take three steps. First, we de…ne a function RLF as
RLF (Dt ) = qt+1 A 1 (Dt ) ;from (9). Second, using (10), we de…ne a function
LF (Dt ) =

(13)

LF as

L Rt =qt+1 + wt;
wt + Dt + wt+1 =Rt(14)

where Rt = RLF (Dt ) and wt , wt+1 , and qt+1 are taken as given for the LF banks. Third,
we connect t to the crisis probability t : Equation (13) means that any changes in Dt
give rise to changes in Rt =qt+1 . Once Rt =qt+1 is determined, we can calculate the liquidity
demand L Rt =qt+1 in which the solvency constraint is satis…ed with equality. For theliquidity market to clear, we obtain the threshold level of the liquidity shock on the brinkof a …nancial crisis. Namely, when t is strictly greater than t , the banks turn out tobe 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

f ( t) d t:

(15)

t

We are now ready to set up the optimization problem for the banks to determine the11level of their leverage.
Problem LF The laissez-faire banks maximize the household expected utility
Ztf t ln (wt + Lt ) + (1maxt ) ln [wt+1 + Rt (DtDt0
Z 1[ t ln (wt + X) + (1+t ) ln (w)] f ( t ) d t ;Lt )]g f ( t ) d t(16)t

subject to
L

Rt
qt+1

wt+1
+ Dt
(1
Rt
L Rt =qt+1 + wt;
LF (Dt ) =wt + Dt + wt+1 =Rt=tt ) wt ;where Rt = RLF (Dt ) from (13) and t = LF (Dt ) from (14).
The …rst-order condition for their leverage isf[ t ln ( t mt ) + (1t ) ln (Rt (1t ) mt )]
d t 0
(Dt )
[ t ln (wt + X) + (1t ) ln (w)]gd t LFZ t
01
wt+1 0RLF(Dt ; t )
=
1
R
(D
;
)
+
(1
)
f ( t) d t;
t
t
t
LF
2
mt
Rt
Rt
0

(17)

where mt
wt + Dt + wt+1 =Rt is the lifetime income of households and mt
wt + Dt +0
wt+1 =Rt , accordingly. More importantly, LF (Dt ) is calculated from (14), taking capital
0
prices and wages as given. Likewise, RLF(Dt ; t ) is calculated from (10), taking capital
prices, wages, and t as given. Here we slightly abuse notations. While we use the prime for00
LF (Dt ) and RLF (Dt ; t ) to denote marginal changes in t and Rt with respect to Dt , they
are not the total derivative that internalizes the general equilibrium e¤ect of Dt on factor
prices. However, these notations will be compared in the next section.
Equation (17) provides an economic intuition. The terms in curly brackets on the lefthand side of (17) represent the loss of utility in a crisis compared with the threshold. From(15), the term outside the curly brackets indicates the marginal changes in a crisis probability
with respect to bank leverage. The left-hand side of the equation is the expected loss of utility
multiplied by the marginal change in the crisis probability. Simply put, the left-hand side of12

(17) is the marginal cost of increasing Dt .
The right-hand side of (17) 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
a positive e¤ect on households’income: the higher the leverage, the larger the withdrawal,
allowing households to enjoy more consumption. On the other hand, the increase in Dt leads
to a higher interest rate via liquidity shortage, discounting the households’labor income inperiod t + 1, and reducing returns on forgoing withdrawal until period t + 1. Therefore,
as long as the positive e¤ect on lifetime income exceeds the e¤ect on the interest rate, thehigher leverage is bene…cial to households. Simply put, the right-hand side of (17) is themarginal bene…t of increasing Dt .
We de…ne the competitive equilibrium as follows.
De…nition (A competitive equilibrium) 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 (17); (iii) banks’ liquidity supply isdetermined by (6); and (iv) all markets clear.

3

Social Planning Banks

We next consider the SP banks that choose their leverage as the constrained social planner.
We call the allocations for the SP banks the constrained social optimum and compare theallocations with those for the LF banks. To lead o¤ the analysis, we clarify the constraint
to which the SP banks are subject. The SP banks must make all their decisions before
observing t . After realizing t , they are left with no options. In other words, the SP banks
are subject to the constraint that they cannot control households’consumption in response
to a realized value of t . The allocations chosen by the SP banks must be distinguished fromthe unconstrained, …rst-best optimum. The SP banks precommit to payment on their debt
regardless of the states realized following their commitment. The extra ability given to the
SP banks compared with the LF banks is that the former can internalize all price e¤ects inall markets when they make decisions regarding their leverage.
Formally, Problem SP replaces qt and wt in Problem LF with FK;t and FH;t , respectively.
SP banks are faced with the solvency constraint Dt A (Rt =FK;t+1 ). The newly introduced
solvency constraint for the SP banks has di¤erent e¤ects on the threshold because (9) and13

(13) are now replaced with
Dt = ARt
FK;t+1!

(18);

RSP (Dt ) = FK;t+1 A 1 (Dt ) :(19)

We summarize the SP banks’problem as follows.
Problem SP The social planning banks maximize 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 toL

Rt
FK;t+1

FH;t+1+ Dt(1Rt
L Rt =FK;t+1 + FH;t;
FH;t + Dt + FH;t+1 =Rt=t

SP (Dt )t ) FH;t

(20)
(21)

where Rt = RSP (Dt ) from (19) and t = SP (Dt ) from (21).
The …rst-order condition for the SP banks’leverage isf[ t ln ( t mt ) + (1t ) ln (Rt (1

d t 0
(Dt )
d t SP0
wt+1 0RSP(Dt ; t )
(D
;
)
+
(1
)
R
f ( t) d t;
t
t
t
SP
2
Rt
Rt

[ t ln (FH;t + X) + (1
=

Z

0

t

1
mt

1

t ) mt )]t ) ln (F H )]g(22)

where mt
FH;t + Dt + FH;t+1 =Rt and mt
FH;t + Dt + FH;t+1 =Rt . Importantly, the SP00banks factor in all general equilibrium e¤ects when they calculate SP(Dt ) and RSP(Dt ; t ).
We de…ne the allocations chosen by the SP banks as follows.
De…nition (Constrained social optimum) Allocations and prices under the constrainedsocial optimum are fgt ; Dt ; Lt ; Kt ; It ; Ht ; Rt ; qt ; wt g1
t=0 such that (i) withdrawal decisions are14

given by (5) for tt ; (ii) banks’ leverage satis…es (22); (iii) banks’ liquidity supply isdetermined by (6); and (iv) all markets clear.
In general, the allocations that the SP banks achieve di¤er from those achieved by the000(Dt ) and RSP(Dt ) andLF banks because SP(Dt ; t ) are not necessarily the same as LF
0RLF(Dt ; t ), respectively. In fact, because the two problems are subject to the same constraints, any discrepancy in the …rst-order conditions results in di¤erent allocations between
0(Dt ), taking capitalthe two problems. Moreover, as long as the LF banks calculate LF
prices and wages as given, the LF banks cannot achieve the constrained social optimum that
the SP banks can achieve. We summarize the result in the following proposition.
Proposition 1 The allocations under the competitive equilibrium are less e¢ cient than the
allocations under the constrained social optimum.
While the presence of the ine¢ ciency in the competitive equilibrium is clear, the above
result does not necessarily imply overleverage by the LF banks. The next subsection focuseson the direction of the ine¢ ciency, relying on the numerical simulations.

4

Numerical Results

4.1

Calibration

This section provides numerical simulations of the model to explore the ine¢ ciency in thecompetitive equilibrium. We numerically consider the following questions: (i) Do the LF
banks take on excessive risks? (ii) If yes, what is the primary source of overleverage? (iii)
How much is the solvency constraint quantitatively important for overleverage?
Our calibration mostly follows DR. We set the value of prematurely liquidated project
X at 0.95. We assume that ! follows the uniform distribution over a range of [! L ; ! H ] =[0:5; 3:5], similar to the original parameter values in 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 1/3.
The capital goods endowment received by entrepreneurs, I, and technological progress Z
determine the size of the scarring e¤ect of a …nancial crisis. Here, we set I = 1 and Z = 4.
More importantly, we assume that the liquidity shock t follows the beta distribution with amean of 0.50 and a standard deviation of 0.07. This parameterization implies a symmetricbell-shaped distribution.

15

To simulate the model, we numerically solve the nonlinear system of the equations consisting of the …rst-order conditions and resource constraints.13 The system of equations
includes equations that de…ne the threshold variables, as well as the equations describing
normal times. For example, the system includes the solvency constraint (9), which de…nes
the relative price Rt =qt+1 on the brink of a crisis.
Before interpreting the numerical results, we recon…rm the economic interpretations of
Dt . In the context of our model, Dt represents the gross return to bank deposits. On theother hand, Dt cannot be translated into an annual percentage rate or an interest rate perannum, because the model does not specify the length of each period of time (e.g., one
year or ten years). To focus more clearly on the economic interpretations, Dt needs to betranslated into a timeless measure such as bank leverage.

4.2

Excessive Risk Taking by LF Banks

We summarize the benchmark results in Table 2. The results suggest that the LF banks
take on excessive risks in determining their leverage. Thus, the direction of ine¢ ciency can
support macroprudential policy. More speci…cally, the upper panel of the table reports that
the LF banks choose a higher level of leverage Dt than the SP banks. As the …rst row shows,the LF banks set the level of Dt at 1.061, 1.2 percentage points higher than the constrainedsocial optimum of 1.049. This overleverage of the LF banks leads to a higher crisis probability.
Our calibration points to a 6.59 percent crisis probability under the competitive equilibriumcompared with 4.50 percent under the constrained social optimum. These numbers may
be comparable with some recent empirical studies on crises.14 The lower panel of Table 2
compares the bank capital ratio de…ned as (At Dt ) =At and the output of consumption
goods Yt+1 under the competitive equilibrium and the constrained social optimum when therealized value of t takes a mean of 0.5. The LF banks are undercapitalized by 1.1 percentage
points compared with the constrained social optimum. Nevertheless, production does notsubstantially di¤er between the two allocations, provided that a …nancial crisis does not takeplace. The above exercise indicates that the welfare loss primarily arises from ine¢ ciently
elevated crisis probability.
It would be helpful to see our results graphically. The higher t in the competitive13

For details, see the Appendix.
Based on data going back to the 1800s, Reinhart and Rogo¤ (2013) report that the frequencies are
7.2 percent for the advanced economies. The Basel Committee on Banking Supervision (2010, BCBS) uses
multiple datasets on banking crises. In BCBS, the frequency ranges from 3.6 to 6.8 percent.
14

16

equilibrium means that the economy undergoes too frequent crises created by pecuniary
externalities. To illustrate this, we run the model over 100 periods by generating liquidity
shocks randomly. The upper panel of Figure 1 plots the dynamic paths of output Yt generated
by the same random shock sequences across two economies. The red lines correspond to thecompetitive equilibrium, while the blue lines point to the constrained social optimum.
Comparisons between the two economies reveal that the dynamic paths of the productionof consumption goods are almost identical except for more frequent crises under the competitive equilibrium. Under the competitive equilibrium, crises take place in periods 5, 16,
and 94, and Yt falls sharply in each subsequent period. This simulation result indicates thatthe last two crises should have been prevented if the banks had taken risks at the optimallevel. However, the …rst crisis takes place even in the constrained social optimum with the
SP banks. The crisis in this allocation should not be avoided in the context of the optimal…nancial crises (see Allen and Gale 1998).15
In the model, the liquidity shock t precipitates …nancial crises. As the lower panel of
Figure 1 shows, …nancial crises in our model take place only when t exceeds the thresholdt . For example, the liquidity shock t in period 5 exceeds the curves of t for both the LF
banks and the SP banks, and the output of consumption goods falls in period 6. However,t for the LF banks is always lower than t for the SP banks. It implies that the …nancial
system under the competitive equilibrium is more vulnerable to liquidity shocks than under
the constrained social optimum.
The overleveraged LF banking system that results in ine¢ cient …nancial crises remains arobust outcome across a range of calibrated parameters (e.g., X and distributions of ! and16
In addition to robustness, sensitivity analysis regarding the volatility of t has notet ).
worthy implications for the inherent fragility of the banking system. Table 3 examines howbank 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 alongwith the increase in the standard deviation of t . Note that smaller volatility of the liquidity
shock does not fully wipe out the …nancial crisis because the banks raise their leverage under
the less volatile liquidity demand. As a result, a …nancial crisis always remains a nonzero
15

Their model includes a single liquidity market where the banks internalize all the e¤ects through thesingle price changes. In contrast, our model includes multiple markets (e.g., the capital market), in whichbanks fail to internalize their general equilibrium e¤ect.
16
The tables of the sensitivity analysis are available upon request.

17

probability event.

4.3

The Primary Source of Overleverage

To understand the LF banks’overleverage in more detail, we numerically compare the optimality condition for Dt under the two allocations. Figure 2 plots the marginal cost andbene…t of increasing the leverage under the two allocations. In the …gure, we compute thecurves based on (17) and (22).17 The solid blue line represents the marginal cost for the SPbanks (the right-hand side of (22)) while the blue dashed line is the marginal bene…t for the
SP banks (the left-hand side of (22)). The red solid and dashed lines are the marginal costand bene…t for the LF banks, respectively, and calculated from (17). Under the competitive
equilibrium, Dt is 1.061 at the intersection of the two red lines. In Figure 2, however, themarginal bene…t for the SP banks is not as high as that for the LF banks. Numerically, if we
evaluate the SP banks’marginal bene…ts at Dt in the competitive equilibrium, the marginal
bene…t for the SP banks is 4.5 percent lower than that for the LF banks. By contrast, thedistortion in the marginal cost is much larger than in the marginal bene…t. The marginal
cost for the SP banks is not as low as the marginal cost for the LF banks. Numerically, if
we evaluate the SP banks’marginal cost at Dt in the competitive equilibrium, the marginal
cost for the SP banks is 29.2 percent higher than that for the LF banks.
Given that the distortion is substantially large in the marginal cost, it would be useful
to focus on the marginal cost as the primary source of overleverage. We are interested in
the di¤erence between the left-hand side of the equations (17) and (22). At a given level of
Dt in the …gure, our observation is as follows.
f[ t ln ( t mt ) + (1t ) ln (Rt (1t ) mt )]

d t 0
(Dt )
d t SP> f[ t ln ( t mt ) + (1t ) ln (Rt (1t ) mt )]
d t 0
(Dt )
[ t ln (FH;t + X) + (1t ) ln (F H )]gd t LF[ t ln (wt + X) + (1t ) ln (w)]g

All the di¤erence between the marginal cost for the SP banks and the LF banks arises fromthe terms outside of the curly brackets because the marginal products replace all factor
prices. Because the expression inside the curly brackets is naturally positive,18 the above
17
18

For the details of computations, see the Appendix.
For the expression inside the curly brackets, note that18t ln ( t mt ) + (1t ) ln (Rt (1t ) mt ) is theexpression can further be simpli…ed to
d t 0
d t 0
(Dt ) :SP (Dt ) >d t
d t LF(23)

The above equation indicates that the primary source of the overleverage in our modelis the banks’evaluation of the marginal changes in the crisis probability with respect to Dt .
The LF banks take other banks’decisions as given. However, the crisis probability is a¤ectedby the synchronized decisions by the banking sector as a whole. Equation (23) means that
the LF banks underestimate the increase in the crisis probability when they increase theirleverage. As a result, the LF banks are likely to be overleveraged. In our numerical simula00
tions, (d t =d t ) SP
(Dt ) = 1:993 and (d t =d t ) LF
(Dt ) = 1:544. If leverage is increased by
one percent from the allocation in the competitive equilibrium, each LF bank expects thatthe crisis probability increases up to 8.22 percent, but they are, in fact, exposed to a higher
crisis probability of 8.69 percent.19

4.4

The Importance of the Pecuniary Externality Stemming fromthe Solvency Constraint

The tightness of the solvency constraint is closely related to the crisis probability. Recall thatthe solvency constraint of the LF banks is Dt = A Rt =qt+1 , or equivalently RLF (Dt ) =qt+1 A 1 (Dt ). When the LF banks determine the leverage, they internalize Rt . However, theytake qt+1 as given because qt+1 is the price outside the liquidity market. The overestimationof the banks’solvency leads to their overleverage and the ine¢ ciently high crisis probability.
To assess the importance of the pecuniary externality stemming from the solvency constraint, we perform a counterfactual experiment. Suppose that hypothetical banks internalizechanges in qt+1 in the solvency constraint Dt = A Rt =qt+1 while they do not internalize
other wedges caused by the pecuniary externalities (e.g., wages). In particular, the hypothetical banks calculate the direct e¤ects of changes in qt+1 on their solvency as the SP banks
lifetime utility on the brink of a …nancial crisis while [ t ln (wt + X) + (1t ) ln (w)] is the lifetime utility
at crisis evaluated at t = t . The …rst-order conditions are satis…ed at t = t in the former but not in thelatter. As long as X < 1, the expression inside the curly brackets is strictly positive.
19
These increases in the probability are obtained by transforming the marginal changes in the probabil0
ity into the semielasticity of the probability: (d t =d t ) LF
(Dt ) DLF;t = d t = (dDt =DLF;t ) = 1:64 and0(d t =d t ) SP (Dt ) DLF;t = d t = (dDt =DLF;t ) = 2:11. Together with 6.58 percent of the crisis probability, the crisis probability perceived by the LF banks is 8:22 (= 6:58 + 1:64) but the true probability is
8:69 (= 6:58 + 2:11).

19

do. Apart from the e¤ects on their solvency, they take other prices as given, as in the LF
banks’ decision-making. While the allocations implied by the hypothetical banks are notsustainable as an equilibrium, this counterfactual experiment helps assess the importance ofthe pecuniary externality stemming from the solvency constraint.
The pecuniary externality stemming from the solvency has a substantial impact on allocations. In the simulations, the banks choose their leverage at Dt = 1:047. The leveragein the counterfactual allocation is lower than the leverage in the competitive equilibrium(1.061). As a result, the crisis probability is also low (4.18 percent) compared with 6.58
percent in the competitive equilibrium. This level of the leverage (Dt = 1:047) is close tothat in the constrained social optimum (1.049). Consequently, the crisis probability of 4.18
percent is close to the probability under the constrained social optimum (4.50 percent). The
proximity to the constrained social optimum implies that the welfare level is also close to
the constrained social optimum.
Figure 3 additionally plots the marginal cost against Dt for the hypothetical banks. For
comparison, we replicate the curves for the marginal cost and the marginal bene…t shownin Figure 2. The marginal bene…t for hypothetical banks is not shown in the …gure because
it remains the same as the marginal bene…t for the LF banks. All the di¤erence from thecompetitive equilibrium is the internalization of the pecuniary externality stemming from thesolvency constraint. In the …gure, the internalization leads to the upward shift of the curve
for marginal cost. The marginal cost for the hypothetical banks is 41.6 percent higher thanthe marginal cost for the LF banks, if we evaluate them at Dt in the competitive equilibrium.
We emphasize that our experiment is counterfactual. The LF banks have no incentive
to internalize this pecuniary externality in the solvency constraint. From the households’
viewpoint, Dt = 1:047 is less attractive to households than Dt = 1:061 in the competitive
equilibrium. Each LF bank is sel…sh and lacks incentives to coordinate with each other. The
competition across the LF banks ill-incentivizes banks to take on excessive risks systemically.

5

Conclusions

We extended the Diamond and Rajan (2012) model of banks with the production factors
and explore how a pecuniary externality a¤ects a bank’s leverage. This extension generates
a pecuniary externality because the banks do not internalize their general equilibrium e¤ectof the factor prices on the economy. As argued in the existing literature on the pecuniary
externality, however, the direction of ine¢ ciency is model dependent. In other words, the20

pecuniary externality itself does not justify the macroprudential policies.
Using numerical simulations, we showed that banks in the competitive equilibrium could
take on excessive risks, compared with the constrained social optimum. In the benchmark
calibration, the crisis probability is 6.58 percent in the competitive equilibrium, higher than
the 4.50 percent under the constrained social optimum. We numerically showed that overleverage mainly comes from the underestimation of the crisis probability when banks choose
their leverage. In our simulation, when they increase their leverage by one percent, they
underestimate the crisis probability to be 8.22 percent in contrast to the actual crisis probability of 8.69 percent. Our …ndings include the importance of the pecuniary externality
stemming from the bank’s solvency for generating their overleverage. We performed counterfactual analysis in which banks internalize only the pecuniary externality stemming fromthe solvency constraint. In this case, the chosen leverage is very close to the constrainedsocial optimum.
The analysis in this paper can be extended in a number of directions. First, it would
be useful to examine how newly introduced aggregate shocks (e.g., shocks to the asset sideof banks’balance sheets) a¤ect the economy’s exposure to crisis risks. Second, the optimal
macroprudential policy could be analyzed. Third, our framework may be translated into an
in…nitely-lived agent model for integration with quantitative business cycle studies. All ofthese directions would provide important avenues for future research.

21

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[2] Allen, F. and D. Gale (2007). Understanding Financial Crises, Oxford University Press.
[3] Angeloni, I. and E. Faia (2012). “Capital Regulation and Monetary Policy with Fragile
Banks,”mimeo.
[4] Basel Committee on Banking Supervision (2010). “An Assessment of theLong-Term Economic Impact of Stronger Capital and Liquidity Requirements,”
http://www.bis.org/publ/bcbs173.htm[5] Benigno, G., H. Chen, C. Otrok, A. Rebucci, and E. R. Young (2013). “Financial Crises
and Macro-prudential Policies,” Journal of International Economics, 89 (2), pp. 453470.
[6] Benigno, G., H. Chen, C. Otrok, A. Rebucci, and E. R. Young (2016). “Optimal Capital Controls and Real Exchange Rate Policies: A Pecuniary Externality Perspective,”
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[9] Bianchi, J. and E. G. Mendoza (2018). “Optimal Time-Consistent Macroprudential
Policy,”Journal of Political Economy, 126 (2), pp. 588-634.
[10] Boissay, F., F. Collard and F. Smets (2016). “Booms and Banking Crises,” Journal of
Political Economy, 124 (2), pp. 489-538.
[11] Dávila, E. and A. Korinek (2017). “Pecuniary Externalities in Economies with Financial
Frictions,”Review of Economic Studies, 85 (1), pp. 352-395.
[12] Diamond, D. W. and P. Dybvig (1983). “Bank Runs, Deposit Insurance, and Liquidity,”
Journal of Political Economy, 91 (3), pp. 401-419.

22

[13] Diamond, D. W. and R. G. Rajan (2000). “A Theory of Bank Capital,” Journal of
Finance, 55 (6), pp. 2431-2465.
[14] Diamond, D. W. and R. G. Rajan (2001a). “Liquidity Risk, Liquidity Creation, andFinancial Fragility: A Theory of Banking,” Journal of Political Economy, 109 (2), pp.
287-327.
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Review, 91 (2), pp. 422-425.
[16] Diamond, D. W. and R. G. Rajan (2012). “Illiquid Banks, Financial Stability, andInterest Rate Policy,” Journal of Political Economy, 120 (3), pp. 552-591.
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24

A

System of Equations

Given Kt , the system of equations consists of 15 equations and 15 unknown variables Dt ,t , Rt , Kt+1 , qt+1 , wt , wt+1 , mt , H, Rt , t , Kt+1 , qt+1 , wt+1 , and mt . Of the 15 equations, 9
specify the economy in normal times, and 6 de…ne the threshold variables. Below, we solve
the system of equations given Kt . After solving the system of equations, we can computeother endogenous variables such as output Yt = F (Kt ; ZHt ) and the bank capital ratio(At Dt ) =At .
Equations at normal times– The …rst-order condition for the leverage:20
f[ t ln ( t mt ) + (1t ) ln (Rt (1

d t 0
(Dt )
d t LF0
wt+1 0RLF(Dt ; t )
R
(D
;
)
+
(1
)
f ( t) d t;
t
t
t
LF
2
Rt
Rt

[ t ln (wt + X) + (1
=

Z

0

t

1
mt

1

(24)

t ) mt )]t ) ln (w)]g) [I= (ZH)] Z. The probability
where w is the wage in the crisis: w = (1density function f ( t ) is computed from the beta distribution. We use numericalintegration for the right-hand side of (24).
– The crisis probability:

Z 1

t =

(25)

f ( t) d t;twhich implies that the marginal change in the crisis probability with respect tof ( t ).
t is d t =d t =– Liquidity market clearing condition:
L

Rt
qt+1

=

t

wt+1
+ Dt
Rt

– Capital market clearing condition fort

Kt+1 = I + I
20t ) wt :

(26)

t:

Rt
qt+1

We use the numerical integration for the right-hand side of (24).

25

(1

:

(27)

If

t >

t , Kt+1 =I.

– Capital demand from …rms:
1

Kt+1
ZH

qt+1 =(28):– Labor demand from …rms in period t and t + 1:
wt = (1)

wt+1 = (1

)

Kt
ZH
Kt+1
ZH

– Lifetime income:
mt = wt + Dt +

Z;

(29)

Z:

(30)

wt+1
:
Rt

(31)

– Labor supply:(32)

H = 2:
Equations for the threshold variables– The solvency constraint with equality:
Dt = L

Rt
qt+1

+

qt+1
I
Rt

Rt
qt+1

:

(33)

(1

t ) wt :(34)– Liquidity market clearing condition:
L

Rt
qt+1

=

wt+1
+ Dt
Rt

t

– Capital market clearing condition:
Kt+1 = I + I

Rt
qt+1

:

(35)

:

(36)

– Capital demand from …rms:
qt+1 =
26

Kt+1
ZH

1

– Labor demand from …rms:
wt+1 = (1

)

Kt+1
ZH

– Lifetime income:
mt = wt + Dt +

wt+1
:
Rt

Z:

(37)

(38)

Leverage and the crisis probability in Tables 2 and 3 are calculated under the steadystate value of Kt = Kt+1 for any t. To generate the dynamic paths of Yt in Figure 1, we use
the steady-state value of capital under the competitive equilibrium as the initial value andsimulate t . Once t is realized, we can update the capital stock in the next period and solve1
where
the system of the equations consisting of (24)–(38). If t
t , Yt+1 = Kt+1 (ZH)1
Kt+1 is given by (27). Otherwise, Yt+1 = I (ZH) . To plot the marginal cost and themarginal bene…t in Figures 2 and 3, we compute the threshold values of Kt+1 over di¤erent
values of Dt . The initial value of capital is the steady-state value of Kt under the competitive
equilibrium. After computing the sequence of Kt+1 that corresponds to the value of Dt , wecalculate qt+1 ; wt+1 , and t from (36), (37), and (34), respectively. These threshold variables
allow us to compute the marginal cost given by the left-hand side of the equations (17) or(22). Regarding the marginal bene…t, we log-linearize the equation for Kt+1 around thesteady state and compute Rt and wt+1 along with the partial derivative of Rt with respect
to Dt . Given t , we take the numerical integration over t .

27

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 that 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, 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 that labormarket conditions along with the young generation’s labor supply.
Households fully withdraw deposits, if any.
All agents consume.

28

Table 2: Crisis probabilities and allocations under LF banks and SP banks

SP banks

LF banks

Leverage and crisis probabilities
Dt
t (%)

1.049
4.499

1.061
6.585

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

15.0975.459

13.952
5.457

Note: Simulation results are based on the assumption that the liquidity shock t follows the beta distribution. The level of bank leverage Dt and the crisis probability t are obtained from Problems LF and SP,
respectively. 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 are based on various values of the standard deviation of the liquidity shock
other details can be seen in the note for Table 2.

29

t . The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: The blue lines (denoted by SP) are the constrained social optimum with the SP banks. The red lines(denoted by LF) are the competitive equilibrium with the 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 isgenerated from the 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 with equality.

30

Figure 2: Marginal cost and bene…t against Dt
0.8

MC
MC

0.7

LF
SP

MB LF
MB SP

0.6

MB & MC

0.5

0.4

0.3

0.2

0.1

0
1

1.05

1.1

1.15

D

Note: The blue and red solid lines represent the marginal cost under the SP banks calculated from theleft-hand side of (22) and the marginal cost under the LF banks calculated from the left-hand side of (17),respectively. The blue and red dashed lines refer to the marginal bene…t under the SP banks calculated fromthe right-hand side of (22) and the marginal bene…t under the LF banks calculated from the right-hand sideof (17), respectively.

31

Figure 3: Marginal cost for the hypothetical banks
MC

0.8

MC

LF
SP

MC counterfactual

0.7

MB LF
MB

SP

MB & MC

0.6

0.5

0.4

0.3

0.2

0.1

0
1

1.05

1.1

1.15

D

Note: The blue and red lines replicate the curves for marginal cost and marginal bene…t in Figure 2. The
newly added curve represents the marginal cost for the banks in the counterfactual experiment. The marginal
bene…t in the counterfactual experiment coincides with the marginal bene…t for the LF banks.

32