IMES DISCUSSION PAPER SERIES

Banking, Liquidity and Bank Runs
in an Infinite Horizon Economy

Mark Gertler and Nobuhiro Kiyotaki
Discussion Paper No. 2014-E-7

INSTITUTE FOR MONETARY AND ECONOMIC STUDIESBANK OF JAPAN
2-1-1 NIHONBASHI-HONGOKUCHOCHUO-KU, TOKYO 103-8660
JAPAN

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IMES Discussion Paper Series 2014-E-7
August 2014

Banking, Liquidity and Bank Runs
in an Infinite Horizon Economy
Mark Gertler* and Nobuhiro Kiyotaki**

Abstract

We develop a variation of the macroeconomic model of banking in Gertler andKiyotaki (2011) that allows for liquidity mismatch and bank runs as in Diamondand Dybvig (1983). As in Gertler and Kiyotaki, because bank net worth fluctuateswith aggregate production, the spread between the expected rates of return on bankassets and deposits fluctuates countercyclically. However, because bank assets areless liquid than deposits, bank runs are possible as in Diamond and Dybvig.
Whether a bank run equilibrium exists depends on bank balance sheets and an
endogenous liquidation price for bank assets. While in normal times a bank run
equilibrium may not exist, the possibility can arise in a recession. We also analyze
the effects of anticipated bank runs. Overall, the goal is to present a framework thatsynthesizes the macroeconomic and microeconomic approaches to banking andbanking instability.

Keywords: Financial Intermediation; Liquidity Mismatch; Financial Accelerator;
Rollover Risk
JEL classification: E44, G21*New York University (E-mail: mark.gertler@nyu.edu)**Princeton University (E-mail: kiyotaki@princeton.edu)
Thanks to Fernando Alvarez, Marios Angeletos, Wouter den Haan, Doug Diamond, Matthias
Kehrig, Aleh Tsyvinski, and Stephen Williamson for helpful comments and to Francesco
Ferrante and Andrea Prespitino for outstanding research assistance, well above the call of duty.
The views expressed in this paper are those of the authors and do not necessarily reflect theofficial views of the Bank of Japan.

1Introduction

There are two complementary approaches in the literature to capturing theinteraction between banking distress and the real economy. The …rst, summarized recently in Gertler and Kiyotaki (2011), emphasizes how the depletionof bank capital in an economic downturn hinders banks ability to intermediate funds. Due to agency problems (and possibly also regulatory constraints)
a bank’s ability to raise funds depends on its capital. Portfolios losses experienced in a downturn accordingly lead to losses of bank capital that are
increasing in the degree of leverage. In equilibrium, a contraction of bankcapital and bank assets raises the cost of bank credit, slows the economy anddepresses asset prices and bank capital further. The second approach, pioneered by Diamond and Dybvig (1983), focuses on how liquidity mismatchin banking, i.e. the combination of short term liabilities and partially illiquid long term assets, opens up the possibility of bank runs. If they occur,runs lead to ine¢ cient asset liquidation along with a general loss of banking
services.
In the recent crisis, both phenomena were at work. Depletion of capital from losses on sub-prime loans and related assets forced many …nancial
institutions to contract lending and raised the cost of credit they did o¤er.
(See, e.g. Adrian, Colla and Shin, 2012, for example.) Eventually, however,
weakening …nancial positions led to classic runs on a number of the investment banks and money market funds, as emphasized by Gorton (2010) andBernanke (2010). The asset …resale induced by the runs ampli…ed the overall…nancial distress.
To date, most macroeconomic models which have tried to capture thee¤ects of banking distress have emphasized …nancial accelerator e¤ects, but
not adequately captured bank runs. Most models of bank runs, however, aretypically quite stylized and not suitable for quantitative analysis. Further,often the runs are not connected to fundamentals. That is, they may beequally likely to occur in good times as well as bad.
Our goal is to develop a simple macroeconomic model of banking instability that features both …nancial accelerator e¤ects and bank runs. Our
approach emphasizes the complementary nature of these mechanisms. Balance sheet conditions not only a¤ect the cost of bank credit, they also a¤ect
whether runs are possible. In this respect one can relate the possibility ofruns to macroeconomic conditions and in turn characterize how runs feedback into the macroeconomy.
2For simplicity, we consider an in…nite horizon economy with a …xed supplyof capital, along with households and bankers. It is not di¢ cult to map
the framework into a more conventional macroeconomic model with capital
accumulation. The economy with a …xed supply of capital, however, allows
us to characterize in a fairly tractable way how banking distress and bankruns a¤ect the behavior of asset prices.
As in Gertler and Karadi (2011) and Gertler and Kiyotaki (2011), endogenous procyclical movements in bank balance sheets lead to countercyclicalmovements in the cost of bank credit. At the same time, due to liquidity mismatch, bank runs may be possible, following Diamond and Dybvig (1983).
Whether or not a bank run equilibrium exists will depend on two key factors: the condition of bank balance sheets and an endogenously determinedliquidation price. Thus, a situation can arise where a bank run cannot occur
in normal times, but where a severe recession can open up the possibility.
Critical to the possibility of runs is that banks issues demandable shortterm debt. In our baseline model we simply assume this is the case. We
then provide a stronger motivation for this scenario by introducing household
liquidity risks, in the spirit of Diamond and Dybvig.
Some other recent examples of macroeconomic models that consider bankruns include Ennis and Keister (2003), Martin, Skeie, and Von Thadden(2012) and Angeloni and Faia (2013).1 These papers typically incorporatebanks with short horizons (e.g. two or three periods)2 . We di¤er by modeling banks that optimize over an in…nite horizon. In addition, bank asset
liquidation prices are endogenous and a¤ect whether a sunspot bank run
equilibrium exists. We further use our dynamic framework to evaluate thee¤ect of anticipated bank runs. Our paper is also related to the literature on
sovereign debt crises (e.g Cole and Kehoe, 2000) which similarly characterizes how self-ful…lling crises can arise, where the existence of these kinds ofequilibria depends on macroeconomic fundamentals.
Section 2 presents the model, including both a no-bank run and a bankrun equilibria, along with the extension to the economy with household liq1

See Boissay, Collard, and Smets (2013) for an alternative way to model banking crisesthat does not involve runs per se. For other related literature see Allen and Gale (2007),
Brunnermeier and Sannikov (2012), Gertler and Kiyotaki (2011) and Holmstrom and Tirole (2011) and the reference within.
2
A very recent exception is Robatto (2013) who adopts an approach with some similarities to ours, but with an emphasis instead on money and nominal contracts.

3

uidity risks. Section 3 presents a number of illustrative numerical experiments. While in our baseline model we restrict attention to unanticipatedbank runs, we describe the extension to the case of anticipated bank runs
in section 4. In section 5 we conclude with a discussion of policies that can
reduce the likelihood of bank runs.

2

Basic Model

2.1

Key Features

The framework is a variation of the in…nite horizon macroeconomic modelwith a banking sector and liquidity risks developed in Gertler and Karadi(2011) and Gertler and Kiyotaki (2011).3 There are two classes of agents households and bankers - with a continuum of measure unity of each type.
Bankers intermediate funds between households and productive assets.
There are two goods, a nondurable product and a durable asset "capital."
Capital does not depreciate and is …xed in total supply which we normalize
to be unity. Capital is held by banks as well as households. Their total
holdings of capital is equal to the total supply,
Ktb + Kth = 1;

(1)where Ktb is the total capital held by banks and Kth be the amount held byhouseholds.
When a banker intermediates Ktb units of capital in period t; there isa payo¤ of Zt+1 Ktb units of the nondurable good in period t + 1 plus theundepreciated leftover capital:
date t+1

date tKtb capital

Ktb capital
Zt+1 Ktb output!

(2)

where Zt+1 is a multiplicative aggregate shock to productivity.
By contrast, we suppose that households that directly hold capital at t
for a payo¤ at t + 1 must pay a management cost of f (Kth ) units of the3

See also He and Krishnamurthy (2013) for a dynamic general equilbirum model withcapital constrained banks.

4

nondurable goods at t; as follows:
date t+1

date tKth capitalf (Kth ) goods!

Kth capital
Zt+1 Kth output(3)

The management cost is meant to re‡ect the household’s lack of expertise
relative to banks in screening and monitoring investment projects. We suppose that for each household the management cost is increasing and convex
in the quantity of capital held:
8h>(Kth )2 ; for Kth K ;<2f (Kth ) =
(4)
>
h
h
h
:
K
K (Kth); for Kth > K :2hwith > 0: Further, below some threshold K 2 (0; 1) ; f (Kth ) is strictlyhconvex and then becomes linear after it reaches K : We allow for the kinkin the marginal cost to ensure that it remains pro…table for households toabsorb all the capital in the wake of a banking collapse.
In the absence of …nancial market frictions, bankers will intermediate theentire capital stock. In this instance, households save entirely in the form ofbank deposits. If the banks are constrained in their ability to obtain funds,
households will directly hold some of the capital. Further, to the extent thatthe constraints on banks tighten countercyclically, as will be the case in ourmodel, the share of capital held by households will move countercyclically.
As with virtually all models of banking instability beginning with Diamond and Dybvig (1983), a key to opening up the possibility of a bankrun is liquidity mismatch. Banks issue non-contingent short term liabilities
and hold imperfectly liquid long term assets. Within our framework, thecombination of …nancing constraints on banks and ine¢ ciencies in household
management of capital will give rise to imperfect liquidity in the market forcapital.
We proceed to describe the behavior of households, banks and the competitive equilibrium. We then describe the circumstances under which bank runs
are possible. For expositional purposes, we begin by studying a benchmark
model where we simply assume that banks issues short term debt. Within
this benchmark model we can illustrate the main propositions regarding the5possibility of bank runs and the connection to bank balance sheet strength.
We then generalize the model to allow for household liquidity risks in thespirit of Diamond and Dybvig in order to motivate why banks issue demandable deposits.

2.2

Households

Each household consumes and saves. Households save by either by lending
funds to competitive …nancial intermediaries (banks) or by holding capital
directly. In addition to the returns on portfolio investments, each householdalso receives an endowment of nondurable goods, Zt W h , every period thatvaries proportionately with the aggregate productivity shock Zt :Intermediary deposits held from t to t + 1 are one period bonds that pay
the certain gross return Rt+1 in the absence of a bank run. In the event of abank run, a depositor may receive either the full promised return or nothing,depending on the timing of the withdrawal. Following Diamond and Dybvig(1983), we suppose that deposits are paid out according to a "sequential
service constraint." Depositors form a line to withdraw and the bank meetsthe obligation sequentially until its funds are exhausted. If the bank has
insu¢ cient funds to meet its withdrawal requests, a fraction of depositors
will be left with nothing. In Basic Model, we assume that bank runs arecompletely unanticipated events. Thus, we proceed to solve the household’s
choice of consumption and saving as if it perceives no possibility of a bankrun. Then in a subsequent section we will characterize the circumstances
under which people anticipate a bank run may occur with some likelihood.
Household utility Ut is given by
!
1
X
i
h
Ut = Et
ln Ct+ii=0where Cth is household consumption and 0 < < 1. Let Qt be the market
price of capital, The household then chooses consumption, and householdbank deposit Dt and direct capital holdings Kth to maximize expected utility
subject to the budget constraint
Cth + Dt + Qt Kth + f (Kth ) = Zt W h + Rt Dt 1 + (Zt + Qt )Kth 1 :(5)Again we assume that the household assigns a zero probability of a bankrun. The …rst order conditions for deposits and direct capital holdings are6

given byEt ( t;t+1 Rt+1 ) = 1(6)hEt ( t;t+1 Rkt+1
)=1

(7)

where
t;t+i =

h
i Ct
h
Ct+i

h
Rt+1
=

Zt+1 + Qt+1
Qt + f 0 (Kth )

h

h

(8)
h

and f 0 (Kth ) = Kth for Kth (0; K ] and = K for Kth [K ; 1]: t;t+i is thehousehold’s marginal rate of substitution between consumption at date t + ihis the household’s gross marginal rate of return from direct
and t, and Rt+1
capital holdings.
Observe that so long as the household has at least some direct capital
holdings, the …rst order condition (7) will help determine the market priceof capital. Further, the market price of capital tends to be decreasing inhthe share of capital held by households given that over the range (0; K ],the marginal management cost f 0 (Kth ) is increasing. As will become clear, abanking crisis will induce asset sales by banks to households, leading a drop
in asset prices. The severity of the drop will depend on the quantity of sales
and the convexity of the management cost function. In the limiting case of
a bank run households absorb all the capital from banks. Capital prices willreach minimum as the marginal cost reaches a maximum.

2.3

Banks

Each banker manages a …nancial intermediary. Bankers fund capital investments by issuing deposits to households as well as by using their own equity,or net worth, nt . Due to …nancial market frictions, bankers may be constrained in their ability to obtain deposits from households.
To the extent bankers may face …nancial market frictions, they will attempt to save their way out of the …nancing constraint by accumulating
retained earnings in order to move toward one hundred percent equity …nancing. To limit this possibility, we assume that bankers have …nite expected
lifetime. In particular, we suppose that each banker has an i.i.d probabilityof surviving until the next period and a probability 1of exiting. The
7

expected lifetime of a banker is then 1 1 : Note that the expected lifetime
may be long. But it is critical that it is …nite.
Every period new bankers enter: The number of entering bankers equals
the number who exit, keeping the total population of bankers constant. Eachnew banker takes over the enterprise of an exiting banker and in the processinherits the skills of the exiting banker. The exiting banker removes his
equity stake nt in the bank. The new banker’s initial equity stake consists
an endowment wb of nondurable goods received only in the …rst period of
operation. As will become clear, this setup provides a simple way to motivate"dividend payouts" from the banking system in order to ensure that banks
use leverage in equilibrium.
In particular, we assume that bankers are risk neutral and enjoy utility from consumption in the period they exit.4 The expected utility of acontinuing banker at the end of period t is given by"1#
Xi
Vt = Et(1) i 1 cbt+i ;
i=1

where (1
) i 1 is probability of exiting at date t + i; and cbt+i is terminal
consumption if the banker exits at t + i:
During each period t a bank …nances its asset holdings Qt ktb with deposits
dt and net worth nt :Qt ktb = dt + nt :(9)
We assume that banks cannot issue new equity: They can only accumulate net
worth via retained earnings. While this assumption approximately accords
with reality, we do not explicitly model the agency frictions that underpin it.
The net worth of "surviving" bankers is the gross return on assets net
the cost of deposits, as follows:
nt = (Zt + Qt )ktb 1

Rt dt 1 :(10)

For new bankers at t, net worth simply equals the initial endowment:nt = w b :4

We could generalize to allow active bankers to receive utility that is linear in consumption each period. So long as the banker is constrained, it will be optimal to defer all
consumption until the exit period.

8

Finally, exiting bankers no longer operate banks and simply use their net
worth to consume:
cbt = nt :To motivate a limit on the bank’s ability to obtain deposits, we introduce
the following moral hazard problem: At the end of the period the banker canchoose to divert the fraction of assets for personal use. (Think of the way
a banker may divert funds is by paying unwarranted bonuses or dividends to
his or her family members.) The cost to the banker is that the depositors canforce the intermediary into bankruptcy at the beginning of the next period.
For rational depositors to lend, it must be the case that the franchise valueof the bank, i.e., the present discounted value of payouts from operating thebank, Vt , exceeds the gain to the bank from diverting assets. Accordingly,
any …nancial arrangement between the bank and its depositors must satisfy
the following incentive constraint:
Qt ktbVt :(11)

Note that the incentive constraint embeds the constraint that nt must be
positive for the bank to operate since Vt will turn out to be linear in nt :We
will choose parameters and shock variances that keep nt non-negative in a
"no-bank run" equilibrium. (See the Appendix for details).5
Given that bankers simply consume their equity when they exit, we canrestate the bank’s franchise value recursively as the expected discounted valueof the sum of net worth conditional on exiting and the value conditional oncontinuing as:
Vt = Et [ (1)nt+1 + Vt+1 ]:(12)
The banker’s optimization problem then is to choose ktb ; dt each period tomaximize the franchise value (12) subject to the incentive constraint (11)and the ‡ow of funds constraints (9) and (10).
We guess that the bank’s franchise value is a linear function of assets anddeposits as follows
Vt = kt ktb
t dt ;
5

Following Diamond and Dybvig (1983), we are assuming that the payo¤ on deposits
is riskless absent a bank run, which requires that bank net worth be positive in this state.
A bank run, however, will force nt to zero, as we show later.

9and then subsequently verify this guess. Using the ‡ow-of-funds constraint(9) ; we can express the franchise value as:Vt =bt Qt kt +

with

t nt

kt

t:
Qt
We can think of t as the excess marginal dollar value of assets over deposits.
In turn, we can rewrite the incentive constraint ast

Qt ktbbt Qt kt +t nt :

It follows that incentive constraint (11) is binding if and only if the excess
marginal value from honestly managing assets t is positive but less than themarginal gain from diverting assets ; i.e.6
0<

t <

:

Assuming this condition is satis…ed, the incentive constraint leads to thefollowing limit on the scale of bank assets Qt ktb to net worth nt :Qt ktb
=
nt

t

(13)

t:
t

The variable t is the maximum feasible assets-to-net worth ratio. For
convenience (and with a slight abuse of terminology), we will refer to t
as the maximum leverage ratio, (since the assets-to-net worth ratio re‡ects
leverage). t depends inversely on : An increase in the bank’s ability todivert funds reduces the amount depositors are willing to lend. As the bankexpands assets by issuing deposits, its incentive to divert funds increases.
The constraint (13) limits the portfolio size to the point where the bank’s
incentive to cheat is exactly balanced by the cost of losing the franchise value.
In this respect the agency problem leads to an endogenous capital constraint.
From equations (9) and (10); the recursive expression of franchise value(12) becomesbt Qt kt + t nt =

Et

1

+

t+1 +

6

t+1 t+1b(Rt+1

Rt+1 )Qt ktb + Rt+1 nt

In the numerical analysis in section 3, we choose parameters to ensure that the condition 0 < t < is always satis…ed in the no bank-run equilibrium.

10

;

b
where Rt+1
is the realized rate of return on banks assets (i.e. capital intermediated by bank), and is given by
b
Rt+1
=

Zt+1 + Qt+1:
Qt

Using the method of undetermined coe¢ cients, we verify the conjecture thatthe franchise value is indeed linear in assets and deposits, withb
Et [(Rt+1

t =

t =

Rt+1 )

Rt+1 Et (t+1 ];

t+1 )

(14)
(15)

with
t+1

1

+ ( t+1 +t+1 t+1 ):The excess marginal dollar value of asset over deposit t equals the discounted excess return per unit of assets intermediated. The marginal costof deposits t equals the discounted cost of per unit of deposit. In eachcase the payo¤s are adjusted by the variable t+1 which takes into accountthat, if the bank is constrained, the shadow value of a unit of net worth
to the bank may exceed unity. In particular, we can think of t+1 as aprobability weighted average of the marginal values of net worth to exiting
and to continuing bankers at t+1. For an exiting banker at t + 1 (which
occurs with probability 1), the shadow value of an additional unit of
net worth is simply unity, since he or she just consumes it. Conversely, fora continuing banker (which occurs with probability ), the shadow value is@Vt =@nt = t + t (@Qt ktb =@nt ) = t + t t . In this instance an additional unitof net worth saves the banker t in deposit costs and permits he or she toearn the excess value t on an additional t units of assets, the latter being
the amount of assets he can lever with an additional unit of net worth.
When the incentive constraint is not binding, unlimited arbitrage bybanks will push discounted excess returns to zero, implying t = 0: Whenthe incentive constraint is binding, however, limits to arbitrage emerge thatlead to positive expected excess returns in equilibrium, i.e., t > 0: Notethat the excess return to capital implies that for a given riskless interest
rate, the required return to capital is higher than would otherwise be and,conversely the price of capital is lower. Indeed, a …nancial crisis in the model
will involve a sharp increase in the excess rate of return on assets along witha sharp contraction in asset prices. In this regards, a bank run will be an
extreme version of a …nancial crisis.
11

2.4

Aggregation and Equilibrium without Bank Runs

Given that the maximum feasible leverage ratio t is independent of individualspeci…c factors and given a parametrization where the incentive constraint isbinding in equilibrium, we can aggregate across banks to obtain the relation
between total assets held by the banking system Qt Ktb and total net worth
Nt :Qt Ktb = t Nt :(16)
Summing across both surviving and entering bankers yields the following
expression for the evolution of Nt :Nt = [(Zt + Qt )Ktb 1

Rt Dt 1 ] + W b(17)

where W b = (1)wb is the total endowment of entering bankers. The…rst term is the accumulated net worth of bankers that operated at t 1
and survived to t, equal to the product of the survival rate and the net
earnings on bank assets (Zt + Qt )Ktb 1 Rt Dt 1 : Conversely, exiting bankers
consume the fraction 1of net earnings on assets:
Ctb = (1)[(Zt + Qt )Ktb 1

Rt Dt 1 ]:(18)

Total output Yt is the sum of output from capital, household endowment
Zt W h and bank endowment W b :Yt = Z t + Z t W h + W b :(19)

Finally, output is either used for management costs, or consumed by households and bankers:
Yt = f (Kth ) + Cth + Ctb :

2.5

(20)

Bank Runs

We now consider the possibility of an unexpected bank run. (We defer ananalysis of anticipated bank runs to Section 4.) In particular, we maintainassumption that when households acquire deposits at t 1 that mature in t;
they attach zero probability to a possibility of a run at t: However, we now
allow for the chance of a run ex post, as deposits mature at t and households
must decide whether to roll them over for another period.
12

An ex post "run equilibrium" is possible if individual depositors believethat if other households do not roll over their deposits with the bank, thebank may not be able to meet its obligations on the remaining deposits.
As in Diamond and Dybvig (1983), the sequential service feature of deposit
contracts opens up the possibility that a depositor could lose everything by
failing to withdraw. In this situation two equilibria are possible: a "normal"
one where households keep their deposits in banks, and a "run" equilibrium where households withdraw all their deposits, banks are liquidated,and households use their residual funds to acquire capital directly.
We begin with the standard case where each depositor decides whether
to run, before turning to a more quantitatively ‡exible case where at any
moment only a fraction of depositors consider running.
2.5.1

Conditions for a Bank Run Equilibrium

In particular, at the beginning of period t; before the realization of returns
on bank assets, depositors decide whether to roll over their deposits with thebank. If they choose to "run", the bank liquidates its capital and turns theproceeds over to households who then acquire capital directly with their lesse¢ cient technology. Let Qt be the price of capital in the event of a forced
liquidation. Then a run is possible if the liquidation value of bank assets(Zt + Qt )Ktb 1 is smaller than its outstanding liability to the depositors:7(Zt + Qt )Ktb 1 < Rt Dt 1 :(21)

If condition (21) is satis…ed, an individual depositor who does not withdraw
su¢ ciently early could lose everything in the event of run. If any one depositor faces this risk, then they all do, which makes a systemic run equilibrium
feasible. If the inequality is reversed, banks can always meet their obligations
to depositors, meaning that runs cannot occur in equilibrium.8
The condition determining the possibility of a bank run depends on two
key endogenous variables, the liquidation price of capital Qt and the condition of bank balance sheets. Combining the bank funding constraint (9) with(21) implies we can restate the condition for a bank run equilibrium as(Zt + Qt

Rt Qt 1 )Ktb 1 + Rt Nt 1 < 0:

7

Since banks are homogenous in n; the conditions for a run on the system are the same
as for a run on any individual bank.
8
We assume that there is a small cost of running to the bank, and that households will
not run if their bank will pay the promised deposit return for sure.

13

We can rearrange this to obtain a simple condition for a bank run equilibriumin terms of just three variables:
Rtb

Zt + Qt< Rt (1
Qt 1

1

)

(22)

t 1

where t 1 is the bank leverage ratio at t 1: A bank run equilibrium exists if
the realized rate of return on bank assets conditional on liquidation of assets
Rtb is su¢ ciently low relative to the gross interest rate on deposits Rt andthe leverage ratio is su¢ ciently high to satisfy condition (22). Note that1
the expression (1) is the ratio of bank deposits Dt 1 to bank capitalt 1
b
Qt 1 Kt 1 , which is increasing in the leverage ratio.
Since Rtb ; Rt and t are all endogenous variables, the possibility of a bankrun may vary with macroeconomic conditions. The equilibrium absent bankruns (that we described earlier) determines the behavior of Rt and t : Thebehavior of Rtb is increasing in the liquidation price Qt ; which depends on
the behavior of the economy.
Finally, we now turn to a more ‡exible case that we use in the quantitative analysis, where at each time t only a fraction of depositors consider
running. Here the idea is that not all depositors are su¢ ciently alert tomarket conditions to contemplate running. This scenario is consistent withevidence that during a run only a fraction of depositors actually try to quickly
withdraw.
In this modi…ed situation, a run is possible if any individual depositor who
is considering a run perceives the bank cannot meet the obligations of thegroup that could potentially withdraw. Thus, assuming individual depositors
who might run know , a run equilibrium exists if(Zt + Qt )Ktb 1 < Rt Dt 1 :This can be expressed as
Rtb < Rt (1

1

):

(23)

t 1

Thus, in the condition for the possibility of a bank run equilibrium, thedeposit rate is adjusted by the fraction of depositors who could potentially
run.

14

2.5.2

The Liquidation PriceTo determine Qt we proceed as follows. A depositor run at t induces all banks
that carried assets from t 1 to fully liquidate their asset positions and goout of business.9 New banks do not enter. Given our earlier assumptionthat new bankers can operate only by taking over functioning franchises ofexisting bankers, the collapse of existing banks eliminates the possibility oftransferring the necessary skill and apparatus to new bankers. Note, however,that this assumption is for purely technical reasons: It makes the liquidation
price easy to calculate. If we allowed new banks to enter, the banking system
would eventually recover, but under any reasonable calibration, it would bea very long and slow process. Under either approach, the near term behaviorof liquidation prices would be similar.
Accordingly, when banks liquidate, they sell all their assets to households.
In the wake of the run at date t and for each period after, accordingly:h
1 = Kt+i; for all i

0;

(24)

where, again, unity is the total supply of capital. Further, since banks nolonger are operating, entering bankers simply consume their respective endowments:
Ctb = W b :The consumption of households is then the sum of their endowment and thereturns on their capital net of management costs:
Cth = Zt W h + Zt

f (1)

(25)

where the last term on the right is household portfolio management costs,
which are at a maximum in this instance given that the household is directly
holding the entire capital stock. We assume each household consists of acontinuum of members and that when a run occurs, the household can share
the risk of failing to withdraw early.
h
Let Rt+1
be the household’s marginal return on capital from t to t + 1
when banks have collapsed at t. Then the …rst order condition for household
direct capital holding is given byh
Et f t;t+1 Rt+1
g=1
9

See Uhlig (2010) for an alternative bank run model with endogenous liquidation prices.

15

with
h
Rt+1
=

Zt+1 + Qt+1
Qt + Khhwhere K is the marginal portfolio management cost when households areabsorbing all the capital (see equation (4)). Rearranging yields the following
expression for the liquidation price in terms of discounted dividends net themarginal management cost.
#
"1
X
h
h
K )
K :
(26)
Qt = Ett;t+i (Zt+ii=1Everything else equal, the higher the marginal management cost the lower
the liquidation price. Note Qt as well as t will vary with cyclical conditions.
Thus, even if a bank run equilibrium does not exist in the neighborhood of
the no-run steady state, it is possible that a su¢ ciently negative disturbanceto the economy could open up this possibility. We illustrate this point in
Section 3 below.
Finally, we observe that within our framework the distinction between aliquidity shortage and insolvency is more subtle than is often portrayed in
popular literature. If a bank run equilibrium exists, banks become insolvent,i.e. their liabilities exceed their assets if assets are valued at the …re-saleprice Qt . But if assets are valued at the price in the no-run equilibrium Qt ;the banks are all solvent. Thus whether banks are insolvent or not depends
upon equilibrium asset prices which in turn depend on the liquidity in thebanking system; and this liquidity can change abruptly in the event of arun. As a real world example of this phenomenon consider the collapse of
the banking system during the Great Depression. As Friedman and Schwartz(1963) point out that, what was initially a liquidity problem in the banking
system (due in part by inaction of the Fed), turned into a solvency problem
as runs on banks led to a collapse in long-term security prices and in thebanking system along with it.

2.6

Household liquidity risks

Up to this point we have simply assumed that banks engage in maturity
mismatch by issuing non-contingent one period deposits despite holding risky
long maturity assets. We now motivate why banks might issue liquid shortterm deposits. In the spirit of Diamond and Dybvig (1983), we do so by
16

introducing idiosyncratic household liquidity risks, which creates a desire byhouseholds for demandable debt. We do not derive these types of deposits
from an explicit contracting exercise. However, we think that a scenario withliquidity moves us one step closer to understanding why banks issue liquid
deposits despite having partially illiquid assets.
As before, we assume that there is a continuum of measure unity of households. To keep the heterogeneity introduced by having independent liquidity
risks manageable, we further assume that each household consists of a continuum of unit measure individual members.
Each member of the representative household has a need for emergency
expenditures within the period with probability . At the same time, because the household has a continuum of members, exactly the fraction has
a need for emergency consumption. An individual family member can only
acquire emergency consumption from another family, not from his or her own
family. Conversely, drawing from its endowment, the family sells emergency
consumption to individuals from other families.
In particular, let cm
t be emergency consumption by an individual member,mbeing
the total emergency consumption by the family. For=
Cwith cmtt
an individual with emergency consumption needs, period utility is given bylog Cth + log cmt ;where Cth is regular consumption. For family members that do not need tomake emergency expenditures, period utility is given simply by
log Cth :Because they are sudden, we assume that demand deposits at banks arenecessary to make emergency expenditures above a certain threshold.
The timing of events is as follows: At the beginning of period t; before
the realization of the liquidity risk during period t, the household chooses
Cth and the allocation of its portfolio between bank deposits Dt and directly
held capital Kt:h subject to the ‡ow-of fund constraint:
Cth + Dt + Qt Kth + f (Kth ) = Rt Dt 1 + (Zt + Qt )Kth 1 + Zt W h
m

m

Ct ;

where the last term C t is the sales of household endowment to the otherfamilies needing emergency consumption (which is not realized yet at the
beginning of period). The household plans the date-t regular consumption
17

Cth to be the same for every member since all members of the household
are identical ex ante and utility is separable in Cth and cm
t . After choosing
the total level of deposits, the household divides them evenly amongst its
members. During period t, an individual member has access only to his or
her own deposits at the time the liquidity risk is realized. Those having to
make emergency expenditures above some threshold cm must …nance themfrom their deposits accounts at the beginning of t;10
cm
t

cm

(27)

Dt :

Think of cm as the amount of emergency expenditure that can be arranged
through credit as opposed to deposits.11 After the realization of the liquidity
shock, individuals with excess deposits simply return them to the household. Under the symmetric equilibrium, the expected sales of householdmendowment to meet the emergency expenditure of the other households C t
is equal to the emergency expenditure of the representative household cmt ;and deposits at the end of period Dt0 are
Dt0 = (Dtcmt ) + (1m

)Dt + C t = Dt ;and equal to the deposit at the beginning of period. Thus the budget constraint of the household is given simply byhhhhCth + cmt + Dt + Qt Kt + f (Kt ) = Rt Dt 1 + (Zt + Qt )Kt 1 + Zt W : (28)The next sequence of optimization then begins at the beginning of periodt + 1.
We can express the formal decision problem of the household with liquidity risks as follows:
Ut (Dt 1 ; Kth 1 ) =maxhCth ;cmt ;Dt ;Ktf(log Cth +hlog cmt + Et [Ut+1 (Dt ; Kt )]g

subject to the budget constraint (28) and the liquidity constraint (27).
10

One can think each member carrying a deposit certi…cate of the amount Dt . Eachfurther is unable to make use of the deposit certi…cates of the other members of the family
for his or her emergency consumption because they are spacially separated.
11
We allow for cm so that households can make some emergency expenditures in a bankrun equilibrium, which prevents the marginal utility of cm from going to in…nity.

18

Let t be the Lagrangian multiplier on the liquidity constraint. Then the…rst order conditions for deposits Dt and emergency expenditures are givenby:Et f t;t+1 Rt+1 g +
cm
t

1
=
Cth

t

1=Cth

= 1;

(29)
(30)

t:

The multiplier on the liquidity constraint t is equal to the gap between themarginal utility of emergency consumption and regular consumption for ahousehold member who experiences a liquidity shock. Observe that if theliquidity constraint binds, there is a relative shortage of the liquid asset,
which pushes down the deposit rate, everything else equal, as equation (29)suggests.
The …rst order condition for the households choice of direct capital holding is the same as in the case without liquidity risks (see equation (7)). The
decision problem for banks is also the same, as are the conditions for aggregate bank behavior.
In the aggregate (and after using the bank funding condition to eliminate
deposits), the liquidity constraint becomes:
Ctm

cm(Qt Ktb

Nt ):Given that households are now making emergency expenditures, the relation
for uses of output becomes
Yt = Cth + Ctm + Ctb + f (Kth ):(31)

Otherwise, the remaining equations that determine the equilibrium withoutliquidity risks (absent bank runs) also applies in this case.
Importantly the condition for a bank run (equation 22) also remains unchanged. The determination of the liquidation price is also e¤ectively thesame (see equation 26). There is one minor change, however: The calculation of Qt is slightly di¤erent since households are now making emergencyhexpenditures cm
t ; in addition to consuming Ct :19

3

Numerical Examples

Our goal here is to provide some suggestive numerical examples to illustrate
the workings of the model. Speci…cally we construct an example where arecession tightens bank balance sheet constraints, which leads to an "excessive" drop in asset prices and opens up the possibility of a bank run. We
then illustrate the e¤ects of an unanticipated run. We …rst present resultsfor our baseline model and then do the same for the model with liquidity
risks.

3.1

Parameter Choices

Table 1 lists the choice of parameter values for our baseline model, while
Table 2 gives the steady state values of the endogenous variables. We takethe period length to be one quarter. Overall there are ten key parameters inthe baseline model and an additional three in the model with liquidity risks.
Two parameters in the baseline are conventional: the quarterly discountfactor which we set at 0:99 and the serial correlation of the productivityhshock Zt which we set at 0:95. Eight parameters ( ; ; W b ; ; K ; ; W h ; Z)
are speci…c to our model. Our choice of these parameters is meant to besuggestive. We set the banker’s survival probability equal to 0:93 which
implies an expected horizon of three and half years. We choose values for thefraction of assets the bank can divert and the banker’s initial endowment
W b to hit the following targets in the steady state absent bank runs: a bankleverage ratio of six and an annual spread between the the expected returnon capital and the riskless rate of 240 basis points.12 We set the parametershhof the "managerial cost" and K to ensure that (i) Kth is strictly below K
in the no bank run case (so we can use loglinear numerical methods in thiscase) and (ii) in the bank run equilibrium managerial costs are low enoughto ensure that households …nd it pro…table to directly hold capital in thebank run equilibrium. Finally, we set the fraction of depositors who may
run at any moment to 0:75, which makes it feasible to have a steady state
without a bank run equilibrium with the possibility of a run equilibrium inthe recession. We set the household steady state endowment ZW h (which
12

In practice the leverage ratio ranges in the vicinity of 8 to 12 for commercial banks
and 20 to 30 for investment banks. Because the market values of bank assets in our model
are more cyclical than practice (since banks are e¤ectively holding equity claims on …rms),
we choose a smaller leverage ratio so as not to exaggerate the volatility of bank net worth.

20

roughly corresponds to labor income) to three times steady state capitalincome Z: We also normalize the steady state price of a unit of capital Qt
at unity, which restricts the steady value of Zt (which determined output
stream from capital).
Finally, for the model with liquidity risks we use the same parametrization
as in our baseline case. There are, however, three additional parameters( ; ; cm ): We choose these parameters to ensure that (i) the steady spread
between the households net return on capital Rh and the deposit R rate istwenty basis points at an annual level, and (ii) households can still makessome emergency expenditures in an bank run equilibrium. For all otherparameters, we use the same values as in the baseline case.

3.2

Recessions, Banking Distress and Bank Runs: Some
Simulations

We have parametrized the model so that a bank run equilibrium does notexist in the steady state but could arise if the economy enters a recession.
We begin by analyzing the response of the economy to a negative shock to
Zt assuming the economy stays in the "no bank run" equilibrium. We thenexamine the e¤ects of an unanticipated bank run, once the economy entersa region where runs are possible. For each case we …rst examine the baselinemodel and then turn to the model with liquidity risks.
Figure 1 shows the response of the baseline model to an unanticipatednegative …ve percent shock to productivity, Zt : This leads to a drop in outputof roughly …ve percent, a magnitude which is characteristic of a major recession. Though a bank run does not arise in this case, the recession induces…nancial distress that ampli…es the contraction in assets prices and raises
the cost of bank credit. The unanticipated drop in Zt reduces net worth Nt
which tightens bank balance sheets, leading to a contraction of bank deposits
and a …resale of bank assets, which in turn magni…es the asset price decline.
Households absorb some of the asset, but because this is costly for them, theamount they acquire is limited. The net e¤ect is a substantial increase in
the cost of bank credit: the spread between the expected return to bank assets and the riskless rate increases slightly on impact. Overall, the recession
induces the kind of …nancial accelerator mechanism prevalent in Bernanke
and Gertler (1989) and Kiyotaki and Moore (1997) and other macroeconomic
models of …nancial distress.

21

Figure 2 repeats the same experiment as in Figure 1, this time examining
the model with liquidity risks. The overall impact on the asset price and thecost of bank credit is similar to the baseline case. One interesting di¤erence,
however, is that unlike the baseline, there is a substantial drop in the depositinterest rates. The contraction in bank deposits raises the liquidity premiumfor deposits, which increases the spread between the expected rate of returnhon the household’s capital Et (Rt+1) and the deposit rate, forcing the later
down.
We now allow for the possibility of bank runs. To determine whether
a bank equilibrium exists, we …rst de…ne Qt as the threshold value of theliquidation price below which a bank run equilibrium exists. It follows fromequation (23) that Qt is given by
Qt = Rt (1

1

)Qt 1

Zt :

(32)

t 1

Note that Qt is increasing in t 1 , which implies that everything else equal abank run equilibrium is more likely the higher is bank leverage at t 1. We
next construct a variable called "run" that is the di¤erence between Qt andthe liquidation price Qt :(33)
runt = Qt Qt :A bank run equilibrium exists i¤runt > 0:
In the steady of our model run < 0, implying a bank run equilibrium does
not exist in this situation. However, the recession opens up the possibility ofrunt > 0, by simultaneously raising Qt and lowering Qt .
Figure 3 revisits the recession experiment for the baseline model, thistime allowing for a bank run ex post. The …rst panel of the middle row showsthat the run variable becomes positive upon impact and remains positive forroughly ten quarters. An unanticipated bank run is thus possible at any
point in this interval. The reason the bank run equilibrium exists is that thenegative productivity shock reduces the liquidation price Qt and leads to anincrease in the bank’s leverage ratio t (as bank net worth declines relative
to assets). Both these factors work to make the banking system vulnerable
to a run, as equations (32) and (33) indicate.
In Figure 3 we suppose an unanticipated run occurs in the second period
after the shock. The solid line portrays the bank run while the dotted line
22

tracks the no-bank run equilibrium for reference. The run produces a complete liquidation of bank assets as Ktb drops to zero. The asset price falls
to it liquidation price which is roughly forty percent below the steady state.
Output net of household capital management costs drops roughly …fteen percent. The high management costs arise because in the absence of the banking
system, households are directly holding the entire capital stock. The reduction of net output implies that household consumption drops roughly sevenpercent on impact. Bankers consumption drops nearly to zero as existingbankers are completely wiped out and new bankers are only able to consume
their endowment.
Finally, Figure 4 repeats the experiment for the model with liquidity risks.
The behavior of the economy in the wake of a bank run is very similar tothe baseline case. One di¤erence is that the time interval over which a bankrun equilibrium exits is shorter. This occurs because the drop in the deposit
rate following the recessionary shock is greater than in the baseline case (seeFigure 2) which reduces the likelihood that the conditions for a bank run
equilibrium will be met.

4

Anticipated Bank Runs

So far, we have analyzed the existence and properties of an equilibrium witha bank run when the run is not anticipated. We now consider what happens
if people expect a bank run will occur with a positive probability in future.
De…ne the recovery rate in the event of a bank run next period, xt+1 ; as
the ratio of the realized return on the bank assets to the promised deposit
return in the event of bank-run, as follows:
xt+1 =(Qt+1 + Zt+1 )ktb
Rt+1 dt

where as before Qt+1 is the liquidation price of capital during the run. The
recovery rate can be rewritten as a function of the rate of return on bankassets during the run and the leverage ratio of the previous period:
xt+1

b
Rt+1
Qt ktb=
Rt+1 Qt ktb nt
b
Rt+1
t
=
:
1
Rt+1
t

23

(34)

The realized rate of return on deposit depends upon whether the run occurs
as well as the depositor’s position in during the run, as

Rt+1 =

8<

Rt+1 if no bank run
Rt+1 with probability xt+1 if run occurs:
0 with probability 1 xt+1 if run occurs

Because we assumed that depositors will not run if they always receive thesame return, the equilibrium with run exists if and only if the recovery rateis less than one.
Continue to assume that each household consists of a continuum of members and that when a run occurs, exactly the fraction xt+1 of the members
receives the promised return on deposits. Then, the …rst order conditions forthe household’s consumption and portfolio choices implies
1 = Rt+1 Et (1t+1 )t;t+1 + t+1 xt+1

t;t+1

(35)

Zt+1 + Qt+1
Zt+1 + Qt+1
+ t+1 t;t+1
;
(36)
h
0
Qt + f (Kt )Qt + f 0 (Kth )where t+1 is the indicator function which is equal to 1 if the run occurs andequal to 0 otherwise. t;t+1 and t;t+1 are the marginal rates of substitution
between consumption of dates t + 1 and t in the equilibrium without andhh. From (34) and (35) ;and t;t+1 = Cth =Ct+1
with a run: t;t+1 = Cth =Ct+1
we getbt
1 Et t+1 t;t+1 Rt+1
t 1
Rt+1 =:(37)
Et [(1t+1 ) t;t+1 ]Observe that the promised rate of return on deposits is a decreasing functionof the leverage rate t as the recovery rate is decreasing in the leverage rate.
The bank chooses its balance sheet ktb ; dt ; nt to maximize the objective
Vt subject to the existing constraints (9; 10; 11; 12) and the constraint on
the promised rate of return on deposits (37) : Because the objective andconstraints of the bank are constant returns to scale, we can rewrite thebank’s problem to choose the leverage ratio t to maximize the value perunit of net worth as
1 = Et (1

t

t+1 )

t;t+1

Vt
= max Et (1ntt= max Et f(1
+
=

+

nt+1
nt
b
t+1 )[(Rt+1

t+1 )

t+1 )(1t24

Rt+1 ) t + Rt+1 ]g

subject to the incentive constraint tt:Using (37) to eliminate Rt+1 from the objective, we have
(
!)
b
)
R
1
E
(
t
t t t+1 t;t+1 t+1b
t+1 (1t+1 ) Rt+1 t
t = max Et
E
[(1
t
t
t+1 ) t;t+1 ]= max ( t +t t) ;

t

where
t =

t =

Et

"

t+1 (1

Et [(1
Et [(1

t+1 )

t+1 )
t+1 )

1

b
Rt+1

t+1 ]
t;t+1 ]; andb
Et ( t+1 t;t+1 Rt+1)
Et [(1t+1 ) t;t+1 ]

(38)
!#

(39)

with t+1 = 1 + t+1 . Note that we need to check whether the discounted
excess return on assets t remains positive, in order to show the incentive
constraint binds. In this case
t =

t+

t t =

t =

t

or

t;

(40)

:
t

In the Appendix, we analyze the optimization problem in detail.
Summing across banks yields the aggregate condition for bank asset holdings
Qt Ktb = t Nt ;where the evolution of the net worth is
Nt = (1t)

b(Zt + Qt )Kt+1

R t Dt 1 + W b :(41)

To analyze anticipated bank runs, we begin with a situation the economy
is functioning "normally," where a bank run equilibrium also exists in theneighborhood of the steady state, i.e., condition (23) is satis…ed. We thenassume that the probability depositors assign to a bank run happening insubsequent periods follows an exogenous AR(1) process
P robt ( t+1 = 1 j Zt+1 ) = pt
pt =p pt 1 + "pt ;where "pt is i.i.d.

25

(42)

Under assumption (42) ; the household condition for deposits (35) becomes
1 = Rt+1 [(1 pt )Et ( t;t+1 ) + pt Et ( t;t+1 xt+1 )]:(43)
Given xt+1 < 1, as long as the marginal rate of substitution in the bank run
equilibrium satis…es
Ett;t+1 xt+1

< Et ( t;t+1 );Rt+1 is an increasing function of pt . Also the discounted excess value of bankassets per unit becomes
t =

Et f

0
b
t+1 [(Rt+1

Rt+1 )bpt (Rt+1b
Rt+1 Et ( t;t+1 Rt+1
)]g

(44)

where 0t+1 is the value of t+1 in the equilibrium without a bank run, andwhere as before Rt+1 is the risk-free rate. Thus t is a decreasing function ofpt .
An increase in the perceived likelihood of a bank run has harmful e¤ects
on the economy even if a bank run does not materialize. Because the deposit rate increases to compensate the possible loss to depositors, bank net
worth shrinks over time (see equation (41)), contracting bank credit ‡ows.
In addition, an increase in pt reduces the bank franchise value and the excess
value of bank assets (everything else equal) as in equation (44) : In turn, thedecline in t places in downward pressure on the bank leverage ratio t (seeequation (40)); further contracting the economy.
We now present several numerical simulations to illustrate the impact anrise in the probability of a run. We stick with the same calibration as in ourbaseline case (see Table 1), with one exception. We now suppose that all
depositors can run (i.e., = 1). This opens up the possibility of a run in
the steady state, given our overall calibration (see equation (32)). We also
impose that the serial correlation coe¢ cient governing the exogenous process
for the run probability, p , equals 0:95: Finally, we assume further that insteady state the perceived probability of a run is zero.
Figure 5 shows the impact of an anticipated one percent increase in theprobability pt of a run in the subsequent period. The increased riskiness ofbank deposits leads to an out‡ow of bank assets (kb declines by 14 percent)and an increase in the spread between the deposit rate and the risk freerate. Both the price of capital and bank net worth decline. Output also
declines 1 percent because the household is less e¢ cient at managing assets
26

than banks. Accordingly, even if a run does not occur, the mere anticipationof a run induces harmful e¤ects to the economy.
While the spread between the deposit rate and the risk free rate increases,
the deposit rate actually declines. This occurs because there is a sharp drop in
the risk free rate that arises because households face increasing (managerial)
costs at the margin of absorbing the out‡ow of bank assets. Missing from themodel are a variety of forces that could moderate movements in the risk freerate. Accordingly, we consider a variant of the model which keeps the risk
free rate constant. We do this by allowing households the additional option ofsaving in the form of a storage technology (or net foreign asset) that pays the1constant gross return R =: Figure 6 then repeats the experiment as in
Figure 5. In this case the increase in the perceived probability of a run raises
the deposit more than one hundred basis points. The spread increases more
and output shrinks by 1:5 percent with one percent increase of likelihood ofbank run as the risk free rate remains constant.
Note that for these exercises we parametrized the model so that a bankrun equilibrium is feasible in the steady state. Though we do not report
the results here, it is possible that the economy could start in a steady state
where a bank run equilibrium is not feasible, but the anticipation of a bankrun could move the economy to a situation where a run is indeed possible.
The increase in the deposit rate stemming from an anticipated possibility ofa run could leave the bank unable to satisfy all its deposit obligations in theevent of a mass withdrawal.
Of course, it would be desirable to endogenize the probability of a run,which is a nontrivial undertaking. An intermediate step might be to makethe probability of a run an inverse function of the recovery rate Et xt+1 (seeequation (34)) in the region where the recovery rate is expected to be lessthan unity. Doing so would enhance the …nancial propagation mechanismin the no-bank run equilibrium (see Figure 1), as the downturn would bemagni…ed by a rise in the expected probability of a run.

5

Conclusion

We end with some remarks about government …nancial policy. As in Diamondand Dybvig (1983) the existence of the bank run equilibrium introduces arole for deposit insurance. The belief that deposits will be insured eliminates
the incentive to run in our framework just as it does in the original Diamond27

and Dybvig setup. If all goes well, further, the deposit insurance need never
be used in practice. By being available, it serves its purpose simply by rulingout beliefs that could lead to a bank run.
Usually, however, deposit insurance is considered in the context of commercial banking. However, our model of bank runs pertains to any …nancial
institutions that rely heavily on short term that hold assets that are imperfectly liquid, including investment banks, money market mutual funds and soon. Clearly, extending deposit insurance to these kinds of institutions would
be problematic. In addition, as is well understood, moral hazard considerations (not present in our current framework) could produce negative sidee¤ects from deposit insurance.
The solution used in practice is to combine capital requirements with deposit insurance. As many authors have pointed out, capital requirements helpo¤set the incentives for risk-taking that deposit insurance induces. Within
our framework, capital requirements have an added bene…t: they reduce thesize of the region where bank run equilibria are feasible. This occurs because
the possibility of a bank run equilibrium is increasing in the leverage ratio.
If equity capital is costly for banks to raise, as appears to be the case inpractice, then capital requirements may also have negative side e¤ects. A
number of authors have recently pointed that the "stabilizing" e¤ects of capital requirements may be countered to some degree by the increased cost ofintermediation (to the extent raising equity capital is costly).
Our model suggests that a commitment by the central bank to lender oflast resort policies may also be useful. Within our framework the endogenously determined liquidation price of bank assets is a key determinant of
whether a bank run equilibrium exists. In this regard, a commitment touse lender of last resort policies to support liquidation prices could be useful. For example, central bank asset purchases that support the secondary
market prices of bank assets might be e¤ective in keeping liquidation pricessu¢ ciently high to rule out the possibility of runs. (Consider the recent purchases by the Federal Reserve of agency mortgaged-backed securities, in thewake of the collapse of the shadow banking system.) An issue to investigate
is whether by signalling the availability of such a tool in advance of the crisis,
the central bank might have been able to reduce the likelihood of runs on the
shadow banks ex post. Gertler and Karadi (2011) and Gertler and Kiyotaki(2011) explore the use of this kind of policy tool in a macroeconomic model28

of banking distress but without bank runs.13 It would be useful to extend
the analysis of this policy tool and related lender of last resort policies to asetting with bank runs.
13

See also Gertler, Kiyotaki and Queralto (2012) which takes into account the moral
hazard e¤ects of lender of last resort policies by endogenizing the risk of bank liability
structure.

29

6

Appendix

This appendix describes the bank’s optimization under anticipated bank runs
as analyzed in Section 4.
From Section 4, we can write the bank’s decision problem as choosing;t Rt+1 to maximize
t

nt+1
nt

=

Vt
= max Et (1ntt ; Rt+1

+

t+1 )

=

max

pt )Et f(1+bt+1 )[(Rt+1(1

t ; Rt+1

Rt+1 ) t + Rt+1 ]g(45)subject totand
Rt+1 =

1

(46)

t

pt Et

b
t;t+1 Rt+1

(1pt )Et ( t;t+1 )

t
t

1

:

(47)

Using (47) to eliminate Rt+1 from the objective, we have
!)
(
b
1
p
E
(
R
)
t
t t t
t;t+1 t+1
0
b
(48)
Rt+1
pt )Ett = max (1
t
t+1
(1
p
)E
(
t
t
t
t;t+1 )= max ( t +t t) ;

t

where

t =

Et f

0
b
t+1 [(Rt+1t =Et ( 0t+1 )Et ( t;t+1 )Rt+1 )bpt (Rt+1b
Rt+1 Et ( t;t+1 Rt+1)]g:with 0t+1 = 1+ t+1 is t+1 conditional on no run. Clearly the incentive
constraint is binding if t is positive. Given that we verify t to be positive
numerically, we have characterized the local optimum for the bank’s decision
problem.
Next, we need to check the local optimum is the global optimum. In
particular, we need to take into account whether an individual bank mightwant to avoid leverage in order to survive a run on the banking system. The
bene…t is that the bank will be able to earn large pro…ts if the other banks
have failed. The cost is that it will earn a very low return on its net worth in30

absence of a run. We show under our parametrization that this strategy is
not optimal, implying that our local optimum is in fact a global result. The
argument goes as follows.
Suppose that an individual bank does not issue deposits. It just invests
out of its net worth and thus is able to function in the event of a bank run.
It’s objective then becomes
o
t =

f(1

pt )Et [(1obt+1 )Rt+1 ] + pt Et [(1++bt+1 )Rt+1 ]g(49)

where 0t+1 and t+1 is value conditional on no run and a bank run. We
suppose that the bank is "small", so that conditional on a bank run, it isstill the case that Qt = Qt . Accordingly,no+ t )[(Rt+1 Rt+1 ) t+1 + Rt+1
(50)
t = Et (1
with

b
Rt+1
=

Zt + Qt+1
Qt

Rt+1 = 1=Et ( t;t+1 )t;t+1 =

Ct
Ct+1

Ct = Zt (W h + 1)

Qt = Et(1X

t;t+j [Zt+j

j=1

f (1)

f 0 (1)])f 0 (1)= Q (Zt )as Zt follows a Markov process with bounded support. Note that because Qt
is at its liquidation value Qt in this case, after the initial run another run
will not occur. Hence the surviving ban can attach a probability zero to thisevent.
b
Because Rt+1
will be large in expectation relative to its "pre-run" valueb
Rt+1 ; the surviving bank has extra incentive to borrow to earn excess returns.
However, in this case the requirement the deposits be riskless will limit theamount the bank can borrows. In particular, one can show numerically that31

in this case that the requirement that deposits be riskless imposes a binding
upper bound t on the leverage ratio for the bank surviving the run: :
t =

Rt+1
Rt+1

b
M in Rt+1

:

(51)

Zt+1

Note that in the pre-crisis non-bank run equilibrium we restricted the sizeof the shock to ensure that the constraint that deposits be riskless does notbind. Given the extra-normal excess returns available after a bank run, however, this constraint is binding under our parametrization (and for reasonablevariations of this parametrization.)
Though we do not report the results here, we verify numerically that isnever pays for a bank to avoid leverage in normal times in order to survivea bank run. Intuitively, though the bank can earn high returns in the wakeof the bank run, the low probability of this event makes it not worthwhile
to reduce its earning in normal times. We can also verify that this result
is robust to permitting the bank to issue some debt in normal times (below
the amount that would lead to it being wiped out in a bank run.). Whilethe bank earns more in normal times, it winds up with lower net worth in
the wake of the run due to the debt exposure. Finally, the result is robust
to allowing the bank to hold deposits in other banks as opposed to the risky
security.

32

References[1] Adrian, T., Colla, P., and Shin, H., 2012. Which Financial Frictions?
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[8] Cole, H. and T. Kehoe, 2000, Self-ful…lling Debt Crises, Review of Economic Studies 67, 91-161/[9] Cooper, R. and T. Ross, 1998, Bank Runs: Liquidity Costs and Investment Distortions, Journal of Monetary Economics 41, 27-38.
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[14] Gertler, M., and Kiyotaki, N., 2010. Financial Intermediation and Credit
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34

Table 1: Parameters
Baseline Model
0.99
Discount rate
0.93
Bankers survival probability
0.35
Seizure rate
0.02
Household managerial cost
0.48
Threshold capital for managerial cost
Kh
0.75
Fraction of depositors that can run
0.95
Serial correlation of productivity shock
0.0161 Steady state productivityZ
W b 0.0032 Bankers endowment
W h 0.045 Household endowment
Additional Parameters for Liquidity Model
38.65 Preference weight on cm
0.01
Threshold for cm
cm¯b!
0.0033 Bankers endowment
0.03
Probability of a liquidity shock
0.7
Fraction of depositors that can run
L1

Table 2: Steady State Values

K
Q
Ch
Cm
Cb
Kh
Kb
Rb
Rh
R

K
Q
Ch
Cm
Cb
Kh
Kb
Rb
Rh
R

No Bank-Run Equilibrium
Baseline
Liquidity
111
10.05480.026100.02820.00860.00890.29700.27230.70300.72776
6
1.06441.06241.04041.04041.04041.0384Bank-Run Equilibrium
Baseline
Liquidity
1
10.63400.58450.05380.053300.01
0.00320.003311
001.10161.0404

1.10681.04042

Figure 1: A Recession in the Baseline Model: No Bank Run Case1Figure 2: A Recession in the Liquidity Risk Model: No Bank Run Case2

Figure 3: Ex Post Bank Run in the Baseline Model3

Figure 4: Ex Post Bank Run in the Liquidity Risk Model4

Figure 5: Increase in the Probability of a Run

5

Figure 6: Increase in the Probability of a Run, Fixed Riskless Rate

6