NBER WORKING PAPER SERIES
BANKING, LIQUIDITY AND BANK RUNS IN AN INFINITE-HORIZON ECONOMY
Mark Gertler
Nobuhiro Kiyotaki
Working Paper 19129
http://www.nber.org/papers/w19129
NATIONAL BUREAU OF ECONOMIC RESEARCH
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June 2013
Gertler and Kiyotaki acknowledge the support of the National Science Foundation The views expressed
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Banking, Liquidity and Bank Runs in an Infinite-Horizon Economy
Mark Gertler and Nobuhiro Kiyotaki
NBER Working Paper No. 19129
June 2013
JEL No. E44
ABSTRACT
We develop a variation of the macroeconomic model with banking in Gertler and Kiyotaki (2011)
that allows for liquidity mismatch and bank runs as in Diamond and Dybvig (1983). As in Gertler
and Kiyotaki, because bank net worth fluctuates with aggregate production, the spread between the
expected rates of return on bank assets and deposits fluctuates counter-cyclically. However, because
bank assets are less 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 endogenously determined 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 that synthesizes the macroeconomic and microeconomic approaches to banking and banking
instability.
Mark Gertler
Department of Economics
New York University
269 Mercer Street, 7th Floor
New York, NY 10003
and NBER
mark.gertler@nyu.edu
Nobuhiro Kiyotaki
Department of Economics
Princeton University
Fisher Hall
Princeton, NJ 08544-1021
and NBER
kiyotaki@princeton.edu
1 Introduction
There are two complementary approaches in the literature to capturing the
in teraction betw een banking distress and the real eco nomy. The first, summ a-
rized r ecently in Gertler and K iyotaki (2 011), emph asizes ho w th e depletion
of bank capital in an economic do w nturn hinders banks ability to inter m ed i-
ate fun ds. Due to agency problems (and possibly also regulatory constraints)
a b ank ’s ability to raise funds depends on its cap ital. Portfolios losses ex -
perienced in a down tu rn accordingly lead to losses of bank capita l that are
increasing in the degree of leverage. In equilibrium, a contraction of bank
capital and ba n k a ssets raises the cost of bank cred it, slo w s th e econo my and
depresses asset prices and bank capital further. The second approach, pio-
neered by D iam ond an d Dybvig (1983), focuses on how liquidity mism atch
in bank in g, i.e. the combin ation of short term liabilities and partially illiq-
uid long term assets, opens u p the possibility of bank runs. If they occur,
runs lead to inefficient asset liquidation along with a general loss of banking
services.
In the recent crisis, both pheno m en a were at w ork . Depletio n of capi-
tal from losses on sub-prime loans and related assets forced many financial
institutions to contract lending and raised the cost of credit they did offer.
(See, e.g. Ad rian , Colla and Shin, 2012, for example.) Event ua lly, how ever,
weakening fin an cial positions led to classic run s on a number of the invest-
men t banks and money mark et funds, as emphasized by Gorton (2010) and
Bernan ke (2010). The asset firesale in du ced b y the runs amplified the over a ll
fin an cial distress.
To date, m ost macroeconomic models which have tried to capture the
effects of banking distress have emphasized financial accelerator effects, but
not adequately captured bank runs. Most models of bank runs, how ever, are
t y p ically quite sty lized and not suitable for qu antitative ana lysis. Further,
often the runs are not connected to fundamen tals. That is, they ma y be
equallylikelytooccuringoodtimesaswellasbad.
Our goal is to develop a simple macroeconomic model of banking insta-
bility that features both financial accelerator effects and bank runs. Our
approach emphasizes the comp leme ntary na ture of these mechanisms. Bal-
ance sheet conditions not only affect the cost of bank credit, they also affect
whe ther runs a re possible. In this respect on e can relate the possibility of
runs to macroeconom ic conditions and in turn characterize how runs feed
back in to the macroeconomy.
2
For simplicity, we consider an infinite horizon economy with a fixed supply
of capital, along with households and bank ers. It is not difficult to map
the framework into a more con ven tion al macroeconomic model with capital
accumulation. The economy with a fixed supply of capital, howev er, allows
us to characterize in a fairly tractable w a y ho w banking distress and bank
runs affect the behavior of asset prices.
As in Gertler and Karadi (2011) an d G ertler and K iyotaki (2011), endoge-
nous pr ocyclical mo vements in bank balan ce sh eets lead to countercyclical
mo vements in the cost of bank credit. At the same time, due to liquidity mis-
matc h, bank runs may be possible, following Diamond and Dyb vig (1983).
W heth er or not a bank run equilibriu m exists will depend on tw o k ey fac-
tors: the condition of bank balance sheets and an endogenously determined
liquidation price. Thus, a situation can arise where a bank run cannot occur
in normal times, but where a severe recession can open up the possibilit y.
Critical to the possibilit y of runs is that banks issues dema nd ab le short
term debt. In our baseline model w e simply assum e this is the case. We
then pro vide a stronger motivation for this scenario by in tr oducing household
liquidity risks, in the spirit of Diamond and Dybvig.
Some other recen t examples of macroeconomic models that consider bank
runs include Enn is and Keister (2003), Martin, Skeie, and Von Thadden
(2012) and Angeloni and Faia (2013).
1
These papers typically incorporate
banks with short horizons (e.g. two or three periods)
2
.Wediffer by mod-
eling banks that optimize over an infinite horizon. In addition, bank asset
liquidation prices are endogen ous and affect whether a sunspot bank run
equilibrium exists. We further use our dynamic framework to evaluate the
effect of an ticipated bank runs. Our paper is also related to the literature on
so vereign debt crises (e.g Cole and Kehoe, 2 000) whic h similarly character-
izes how self-fulfilling crises can arise, where the existence of these kinds of
equilibria depends on macroeconomic fundam e ntals.
Section 2 presents the model, including both a no-bank run and a bank
run equilibria, along with the extension to the economy with household liq-
1
See Boissay, Collard, and Smets (2013) for an alternative way to model banking crises
that 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 Ti-
role (2011) and the reference within.
2
A very recent exception is Robatto (2013) who adopts an approach with some simi-
larities to ours, but with an emphasis instead on money and nominal contracts.
3
uidity risks. Section 3 presents a num ber of illustrativ e n u m erica l experi-
ments. While in our ba seline model w e restrict atten tion to unanticipated
bank runs, w e 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 lik elihood of bank runs.
2BasicModel
2.1 Key Features
The framework is a variation of the infinite h orizon macroeconom ic model
with a banking sector and liquidity risks developed in Gertler and Ka radi
(2011) and Gertler and Kiy otaki (2011).
3
There are two classes of agents -
households and bank ers - w ith a con tin uum of measure unity of each t ype.
Bankers in term edia te funds between househ olds and productive assets.
There are two goods, a nondurable product and a durable asset "capital."
Capital does not depreciate and is fixe d in total supply which we nor m alize
to be unity. Capital is held by banks as w ell as households. Their total
holdings of capital is equal to the total supply,
+
=1 (1)
where
is the total capital held b y banks and
be the amount held by
households.
Wh en a bank er intermediates
units of capital in period there is
apayoff of
+1
units of the nondurable good in period +1 plus the
undepreciated leftover capital:
date t
capital
ª
→
date t+1
½
capital
+1
output
(2)
where
+1
is a m ultiplicative aggregate shock to productivit y.
By con trast, we suppose that households that directly hold capital at
for a pa yoff at +1 m ust pay a management cost of (
) units of the
3
See also He and Krishnamurthy (2013) for a dynamic general equilbirum model with
capital constrained banks.
4
