the lender of last resort

Journal of Banking & Finance 29 (2005) 1059–1082

www.elsevier.com/locate/econbase

The lender of last resort

Charles A.E. Goodhart a,*, Haizhou Huang b

a Financial Markets Group, London School of Economics, Houghton Street,

London WC2A 2AE, UKb Research Department, International Monetary Fund, Washington, DC 20431, USA

Received 20 December 2002; accepted 13 November 2003

Available online 13 August 2004

Abstract

This paper develops a model of the lender of last resort (LOLR) from a Central Bank (CB)

viewpoint. The model in a static setting suggests that the CB would only rescue banks which

are above a threshold size, consistent with the insight of ‘‘too big to fail’’. In a dynamic setting,

CB�s optimal policy in liquidity support depends on the trade off between contagion and moralhazard effects. Our results show that contagion is the key factor affecting CB�s incentives inproviding LOLR and they also provide a rationalization for ‘‘constructive ambiguity’’.

� 2004 Elsevier B.V. All rights reserved.

JEL classification: E50; E58; G21; G28

Keywords: Lender of last resort; Banking crises; Financial contagion; Moral hazard

1. Introduction

Although lender of last resort (LOLR) services to individual commercial banks

have been a regular, albeit often contentious, part of a central bank�s armorysince Bagehot (1873) and they have been discussed and debated in the literature

(Goodhart, 1988, 1995), there have been few formal models seeking to analyze

0378-4266/$ - see front matter � 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.jbankfin.2003.11.003

* Corresponding author. Tel.: +44 20 7955 7555; fax: +44 20 7371 3664.

E-mail address: [email protected] (C.A.E. Goodhart).

mailto:[email protected]

1060 C.A.E. Goodhart, H. Huang / Journal of Banking & Finance 29 (2005) 1059–1082

how and why central banks have provided such acts. One reason why there have

been few formal models of LOLR is that some economists in this field believe that

providing LOLR to individual banks, rather than to the market as a whole (via open

market operations (OMO)), is fundamentally misguided.

As a generality, such economists believe that central banks should not lend toindividual banks, e.g., through a discount window; the market is as well or better in-

formed than the central bank (CB) about the relative solvency of a bank short of

liquidity. 1 Given an aggregate sufficiency of high-powered money, illiquid (but sol-

vent) banks will be able to borrow in the interbank market, whereas potentially

insolvent banks will be driven out of the system. Moreover, the monetary authorities

will have incentives to exercise forbearance (Kane, 1992) and rescue banks that

should have been closed; and the pursuit of financial stability by direct intervention

may divert the CB from achieving its primary goal of controlling the monetaryaggregates so as to achieve price stability.

There are two ripostes to this position. The first, though not the subject of this

paper, is the potential for ‘‘market failure’’. 2 For example, when the Bank of

New York computer malfunctioned in 1985 and would not accept incoming pay-

ments for bond market dealings, the resultant illiquidity position soon ballooned

to a point where no one counterparty bank could take on the risk of making a suf-

ficiently large loan. It would have required a coordinated syndicate, but such syndi-

cates take time to organize, and time was scarce. An even more dramatic example isgiven by the recent events of September 11, 2001. The functioning of many markets

had been severely disrupted. In this crisis, the Federal Reserve System hugely ex-

panded its discount window lending to many individual banks (McAndrews and Pot-

ter, 2002). Most central banks would also argue that their supervisory role – or their

ready access to supervisory information – should give them additional information,

not available in the market. Moreover, as in the case of the Bank of New York, when

there is any large-scale need to redirect reserves, there must be a coordination prob-

lem. No one commercial-counterparty can single-handedly assume the credit risk,and there is no incentive for a single commercial bank to take on the time, effort

and cost of coordinating the exercise of sorting out the problem. 3

A co-ordination failure 4 may be defined as a condition where a bank, (or, as in

the case of 9/11, a set of banks), is solvent, but illiquid, but the market cannot resolve

this difficulty, which would be temporary if resolved quickly. This may be because

credit counterparty limits prevent any single institution doing the necessary lending,

1 See Bordo (1990), Goodfriend and King (1988), Humphrey (1989), Kaufman (1991), Schwartz (1988),

amongst others.2 Focusing on the micro-aspects of central banks� intervention in dealing with market failure, Freixas

et al. (1998) build one of the rare formal models of LOLR. Using the framework of Diamond and Dybvig

(1983), they analyze the moral hazard problem caused by bank managers� incentive to choose an inefficienttechnology that gives them some private benefit. This moral hazard problem, as in Holmstrom and Tirole

(1998), sets an upper limit to the finance that would be provided at interim dates by outside investors.3 Although financial innovations may make some aspects of coordination easier, they may increase the

difficulty of coordination in other aspects, as argued in Merton (1995) and evidenced by the LTCM crisis.4 See Rochet and Vives (2003) for an analysis of co-ordination failure from the perspective of LOLR.

C.A.E. Goodhart, H. Huang / Journal of Banking & Finance 29 (2005) 1059–1082 1061

and thus requires co-ordinated lending, and no private sector body is willing to

undertake the transaction costs of acting as co-ordinator; or because markets them-

selves are shut, or malfunctioning, as on 9/11; or for a variety of other potential

reasons.

In the case of a co-ordination failure, the problem of illiquidity may affect onlyone bank, and, especially if rapidly resolved, may have no implications at all for

other banks, as with Bank of New York in 1985. If that illiquidity problem is

not rapidly resolved, however, the attempts of the bank involved to sell assets,

and to withdraw loans, to raise cash, and/or the implications of its enforced clo-

sure on depositor and market confidence, and on market prices, (see, for example,

Gorton and Huang, 2002) can then affect both the liquidity, (e.g., via deposit with-

drawals), and the solvency of other banks. The definition of contagious risk is an

occasion where adverse developments in one bank, whether due to illiquidity, orinsolvency, or both, then threaten the viability of some other bank or group of

banks.

It is possible to have co-ordination failure without contagious risk. This would

occur, for example, when a financial institution, or bank, which was illiquid, but

not insolvent, was forced into closure while still solvent, without any significant ad-

verse implications for other banks, or financial institutions. This is most likely for

small banks without important connections with other parts of the banking, or

financial, system. Such events are rarely memorable. It is certainly possible to havecontagious risk without any co-ordination failure. This occurs whenever a big bank

fails as a result of publicised solvency problems. Frequently, however, co-ordination

failure can spark contagious risk, and represents a reason why the CB is keen to

eliminate such co-ordination failures at source.

Our focus will be on a second concern that arises at the macro-level, and is af-

fected by both contagious risks and moral hazard. Contagious risks provide a strong

and compelling call for CB to play the role of LOLR. Recent financial crises, such as

Mexico (1995), Asia (1997–1998), Russia (1998), and Argentina (2000–2001), remindeconomists that contagious risks can cause banking panics, with depositors seeking

to switch out of the deposits of banks perceived as riskier into currency, foreign ex-

change, or into those banks perceived as safer. This history has led to a growing lit-

erature on banking crises and contagion, which identifies several channels for

transmitting financial contagion, including payments/settlements systems (e.g.,

Bankhaus Herstatt), interbank deposit and correspondent systems (e.g., Continental

Illinois), and effects via asset/liquidity prices. 5

5 For contagious risks in payment/settlements system, see Rochet and Tirole (1996); in interbank

market, see Aghion et al. (2000) and Huang and Xu (2000); and in asset/liquidity markets, see Caballero

and Krishnamurthy (1999), Diamond and Rajan (2001), Kodres and Pritsker (2002), among others. In

Allen and Gale (1998, 2000) financial contagion arises from a combination of incomplete markets and

interconnectedness between a failing institution and other institutions. See Huang (2000) for a survey on

the growing literature.


While contagious risks provide a main reason for CB to play the role of LOLR,

moral hazard effects and uncertainty at the macro-level pose important challenges to

CB in conducting LOLR. 6 At the macro-level, even if the CB is aware of particular

examples of ‘‘market failure’’ at micro-level, there is still an important judgement to

be made about the appropriate balance between ‘‘market failure’’ on the one handand ‘‘official intervention failure’’ on the other hand. See, for example, Bernanke

(1983), Goodhart and Schoenmaker (1995), Gorton (1985), among others for related

discussions.

Our aim is to provide a model of central bank provision of LOLR. To the best of

our knowledge, our model of LOLR is the first attempt to address both effects of

contagious risks and moral hazard at the macro-level. To handle this challenging

task and focus our attention on the key issues, we choose not to complicate our mod-

el with other issues, 7 for example, the possible adoption of a broader coverage ofdeposit guarantees. 8

Given the concern in the euro-zone about how to deal with bank crises that have

spill-over effects between countries; and the establishment of a standing committee,

consisting of the Financial Services Authority (FSA), Bank of England and Treas-

ury, to handle potential financial crises in the UK, under the umbrella of a

Memorandum of Understanding (MOU), we fully understand that the main focus

for co-ordination problems nowadays should be, not so much on private co-ordina-

tion, but on co-ordination between different arms of government, or bureaucracies.In due course we would, indeed, hope to model such interactions.

But the trend towards separating banking supervision at the micro-level from the

CB, (or in the case of Japan from the Ministry of Finance), and giving it to a spe-

cialized institution, (while leaving responsibility for macro-level financial stability

with the CB), is still relatively recent. The FSA in the UK was established in

1997, and the Japan Financial Services Agency in 2000. After such a development,

financial crises, more or less, have to be handled by committee. It has yet to be seen

whether such crises would be well handled by committees of bureaucracies. One of ushas written extensively on these developments (see Goodhart, 2000a,b). We have

6 While deposit insurance, or to some extent subordinated debt, can help to prevent bank runs,

contagious risks can come from channels in addition to or in conjunction with deposit withdrawals. For

example, Flannery (1996) shows that banks may rationally withdraw from new loans in a financially

difficult time. See Blum (2002) for an analysis of subordinated debt.7 For a broader discussion on how to resolve systemic financial crises efficiently, including using

disaster planning to optimize expenditures on financial safety nets, imposing blanket guarantees and

dealing with intersectoral politics and agency conflicts, see Kane (2001, 2002). Amongst the several reasons

why we did not include any discussion of deposit guarantees or deposit insurance in this paper is that these

latter are typically provided by a different arm of government, usually the Ministry of Finance or

Treasury. A decision to introduce such guarantees/insurances may be taken under the advice of the central

bank, but involves a different locus of decision making.8 In some recent financial crises, ad hoc expansion in deposit guarantees were employed after CB failed

to provide prompt last resort lending. Had such lending been provided promptly, contagion might have

been contained and, if so, there would have been no need for the use of blanket guarantees. Such blanket

guarantees, moreover, intensify moral hazard even more than LOLR.


doubts whether the time is yet ripe to model this more formally, although attempts

have begun, see for example Dell�Ariccia and Marquez (2003).Our main claim is that most studies of LOLR implicitly involve a certainty equiv-

alent postulate and thus ignore financial contagious risks. That is, the CB is just as

confident and knowledgeable about the optimal level of open market operations,high-powered money and aggregate money stock after the onset of bank failures

and panic, as it would have been if such a panic was prevented at the outset. We ar-

gue that when failures occur, it can lead to substantial financial contagious risks, as

people start to panic and their behavior is likely to become far less predictable, and

financial markets become much more volatile. Policy mistakes become much more

likely in such an environment. 9

In this respect, LOLR can be perceived as a back-up instrument that helps to pro-

tect the economy and the CB from OMO mistakes, at a time when such mistakesmay be particularly easy to make. A distinction, perhaps, should also be made be-

tween arrangements in the Federal Reserve System, where the initiative to borrow

at the Discount Window lies more with the borrowing commercial bank, subject

to various non-pecuniary disincentives, and conditions in Europe, where LOLR is

decided rather by the CB. For descriptions of the US system during the first half

of its life, see Meltzer (2003). Our own focus has been primarily on the European

mechanism, but we believe that it is applicable to US conditions as well.

Our model offers several interesting results. The model in a static setting suggeststhe CB would only rescue banks which are above a threshold size, consistent with the

insight of ‘‘too big to fail’’. This is because when a failure of a larger bank occurs, it

causes more uncertainty about subsequent changes in deposits in the banking system

due to financial contagion. Although the CB can immediately take open market

operations to offset the deposit change, it can still suffer a bigger loss, as a result

of more likely getting macro-policy wrong in the more uncertain macro-economic

environment involving financial contagion.

In a dynamic setting both the probability of a further failure, and the likelihood ofa bank requiring LOLR being insolvent, in each period are a function of CB�s prioractions, which then through moral hazard effects influence the actions of banks and

depositors. The optimal policy, based on the trade-off between contagion and moral

hazard effects, may be non-monotonic in bank size and is time varying, and it is also

contingent on the probability of a failure, the likelihood of a bank requiring LOLR

being insolvent, and random shocks. Our results also provide a rationalization for

‘‘constructive ambiguity’’.

9 For example, the bank failures in the USA in the 1930s shifted the high-powered money (H) to

aggregate money (M) ratio. Although, as Kaldor (1958) noted, the Fed�s actions led to a much faster, thanprevious, growth in H during these years, M still fell. Moreover, after the 1987 stock market collapse,

central banks lowered interest rates aggressively, scared of a replay of 1929, only to discover a couple of

years later that they had overdone such ease. Goodhart and Huang (1999) contains further empirical

evidence of bank failures causing macro-uncertainty.


The remainder of the paper is organized as follows. Section 2 sets up and analyzes

the static model, which is followed by the dynamic model in Section 3. Concluding

remarks are in Section 4.

2. The static model

2.1. The basic setup

We start with a commercial banking system with many banks of varying sizes, and

assume that all of them choose a similar risk profile (h). We shall try to justify this

assumption as a reasonable approximation to reality shortly. Given this preferred

risk profile there is a probability (p) of a bank holding j of the system�s deposits com-ing to the CB asking for LOLR assistance in any period. We set the initial volume of

deposits at D*, a pre-determined desired level, 0 < j < D*.

If no bank seeks LOLR, as occurs with probability (1 � p), CB takes no action, D

remains equal to D*, and the CB suffers no loss. If p occurs, with j > 0, the CB has to

decide whether to say yes to the request for LOLR (St = 1), or no (St = 0). There is a

probability, x, where x = f(h), that the bank (or banks) coming to the CB will also be

insolvent. This is not revealed to the CB at this stage, 10 and the CB also at this stage

has to decide on its OMO. 11 By OMO the CB is assumed to be able to change D byany desired amount, OMO being unlimited in size and direction, so the CB can

always achieve its desired expected value of D, i.e., it can make E[D] = D*, on the

strong assumption that the demand for money function itself remains unchanged

by bank failures. 12,13

If the CB does not provide LOLR, the illiquid bank(s) will shut. This will cause

the public to move out of deposits into cash, via the linear relationship B1j + j�,

10 There are, of course, many reasons for individual commercial bank risk profiles to differ, for

example, the risk aversion of its managers, its initial capital ratio, etc. The CB will have (partial)

information on several of these characteristics through its access to supervisory reports. Even so, it takes a

long time, often measured in years rather than months, to evaluate the final balance sheet position of

banks that are liquidated, or taken over by authorities with a view to subsequent resale. So, the CB has

some prior knowledge of xi and hi in each case. As one referee noted, the recent trend towards the

separation of banking supervision from the CB further supports the assumption the CB is only partially

informed on xi and hi. We abstract from this (incomplete) information, partly because the implications are

obvious and partly because dealing with this makes the model more complex. Our only assumption is that

x, and h, are not systematically related to size.11 All practical descriptions of LOLR activities reveal that the CB is under tremendous time–pressure

to take decisions before the market reopens and has to do so when in possession of only sketchy details of

the financial position of the commercial bank needing help.12 This biases the analysis against LOLR, since in a panic the demand for money function is also liable

to become unpredictable.13 Panics also act to increase uncertainty about the appropriate selection of interest rates, and the

analysis can be done similarly. During panics there is a rush to safety, so interest rates on high-quality,

short-term government paper frequently fall. Since risk-premia and spreads then typically widen, low

quality debt yields may simultaneously increase.

Fig. 1. Time line.


where B1 is a positive coefficient known to the CB, and � is a stochastic variable, withE[�] = 0 and Var(�) = k, where k is also known to the CB. The loss from getting

macro-policy wrong is assumed to be quadratic, and for simplicity as (D � D*)2. 14The identity of the illiquid bank, which has been supported by LOLR, is then

made known. If it is insolvent, with probability x, the CB will face a cost Z, where

Z = n + B2j (n > 0,B2 > 0).

Thus, in the static model, with h, f and x all given and constant, the CB wishes to

minimize the loss functions,

min½EðD� D�Þ2;EZ�: ð1ÞThe sequence of events is as in Fig. 1.

2.2. The analysis

By using OMO to off-set the expected amount that the public moves out of depos-

its into cash, B1j, the CB can always achieve its desired expected value of deposits,

i.e., E[D] = D*. But this is the best the CB can achieve by using OMO, as the CB

cannot eliminate the variance item in E(D � D*)2, which is kj2, thus

min[E(D � D*)2] = kj2. Comparing kj2 with (n + B2j)x, LOLR is preferable if and

only if:

EZ ¼ ðnþ B2jÞx 6 kj2 ¼ min½EðD� D�Þ2�:That is 15

j P �j B2xþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB22x

2 þ 4knxq2k

: ð2Þ

Thus we reach our first result regarding the comparison between LOLR and OMO ina static setting.

Proposition 1. In a static setting, LOLR is preferable to OMO if and only if the size of

the bank seeking assistance is above a threshold level �j; otherwise OMO is preferable.

14 Since this variable, (D � D*)2, drives the deviation of nominal demand from its desired level,

including it in the objective function is consistent with an objective for price and output stability, the

conventional monetary policy objectives, although the latter are not explicitly entered.15 The technical condition for �j < D� is assumed to be satisfied.


According to the above proposition, �j only depends on x, k, n and B2, but not onD* or B1. Some comparative statics on �j further reveal that �j increases as x increases,k decreases, n increases, or B2 increases. These results have the following intuition:

The CB should raise the threshold level of bank size and thus only rescues bigger size

banks, when the probability of insolvency x increases, when the risk of deposits mov-ing out of the banking system k decreases, or when either the fixed cost (n) or var-

iable cost (B2) of rescuing banks that turn out to be insolvent increases.

Moreover, if the risk of deposits moving out of banking system k is very small,

then �j can be very large; thus OMO becomes optimal even for the largest bank; ifthe probability of insolvency x is very small, then �j can be very small; thus LOLRbecomes optimal even for very small banks.

Thus Proposition 1 captures the insight that when a failure of a big bank gener-

ates a crisis in trust and creates systemic uncertainty (e.g., bank panics), it is very dif-ficult for the CB to calibrate exact measures (like OMO) to offset the net effective loss

of liquidity in the banking system. In particular, when (1) volatility in financial mar-

kets increases, (2) banking contagion becomes a greater danger, and (3) the behavior

of the agents (depositors) is less predictable, each of these factors lead to an increase

in the risk of making a mistake on account of not providing LOLR. Since the risk

can be used to define a threshold level of bank size, beyond which the risk would

be too high for the CB, Proposition 1 provides a rationale for the well known

too-big-to-fail (TBTF) doctrine. 16

The crucial feature of this model is that the costs of allowing a bank to fail (quad-

ratic in j) rise at a faster rate with respect to the size of bank than the costs of res-

cuing a bank that may turn out to be insolvent (proportional to j). But so long as

the costs of failure rise consistently faster than the costs of rescue with respect to size,

the same qualitative results will hold. Models of financial contagion, such as those

cited in footnote 5, support this crucial feature, in the sense that contagious risks

are amplified in a nonlinear fashion. Historical examples of banking system failures,

e.g., Australia 1892, USA 1931–33, involved losses well in excess of any plausiblecosts of rescue.

There are several reasons for this asymmetry. Let us start with the quadratic loss

function from policy error, kj2. In virtually all the CB independence literature, policy

errors, e.g., deviations of inflation from target, are taken to be quadratic. But the jus-

tification for quadratic loss functions is rarely profound, and often just based on

mathematical convenience. More convincingly, the literature of banking crises and

financial contagion as described in the introductory section shows that the damage

can spill over from the original bank to many other banks, which can further spillover to many more banks. The total damage is thus a nonlinear function of the size

of the original troubled bank. What we claim is that the risks arising from such con-

tagion is a nonlinear convex function of the size of the original troubled bank, and

the exact quadratic form is just a mathematical convenience.

16 If many, say m, similar-size banks seek CB help simultaneously, then the cutoff level becomes �j=m.Thus our model can readily explain a policy of ‘‘too many to fail’’.


Moreover, the costs of rescue rise less fast. Here there are two costs, reputational

and financial costs, when the rescued bank is insolvent, with the latter falling on

some combination of surviving banks, CB 17,18 and taxpayers. There is a significant

fixed element in reputational costs. The CB may, or may not, make an obvious, pub-

licly observable, error of judgement. Again, there is a fixed cost in making the tax-payer, or other surviving banks, face the reality of having to contribute to an ex post

bailout at all (e.g., see the problem in Japan of sharing out the costs of rescue). That

fixed cost may be large, perhaps even so large that no banks, even the largest, will

actually be supported. But, once taxpayers have come to accept that they must bear

the costs of rescue, then our assertion is that the disutility is just proportional to the

cash burden.

The assumption that we make here is that, once the CB finds out that the commer-

cial bank, to which it has granted LOLR, is insolvent it can, and will, with thegovernment�s assistance, so restructure the commercial bank that there is a once-and-for-all cost on the taxpayer. Too often, however, potentially insolvent banks

are allowed to continue, without effective restructuring, thereby in many cases

increasing the present cost of future unbooked losses. In so far as the CB appreciates

that this latter is likely to happen, then EZ and the coefficient B2 should be equiva-

lently much increased. If the CB realizes that the government has no stomach for

handling bank insolvencies effectively, then it should be much less forward in offering

LOLR support. This can be handled in our model by an appropriate recalibration ofEZ.

We have now formally explained how a policy of ‘‘too big to fail’’, or ‘‘too many

to fail’’, minimizes the CB�s loss function in a static setting. But if decisions whetherto provide LOLR are a function of the size of the commercial bank in difficulties, one

may wonder whether these decisions will make individual choice of risk profile (h) a

function of size, and thus contradicting our initial simplifying assumption. Not nec-

essarily so. Because the benefits that a manager obtains, both pecuniary and non-

pecuniary, from his (continued) position are a positive function of the size of hisbank as well as a function of its profitability, the larger the bank, the less that the

manager would want to put his status at risk. So greater size may on the one hand

be predicted to lead to more probable CB intervention ex post, but on the other

hand to make managers more unwilling to put their own position at risk ex ante.

17 CB losses may be minimized by an appropriate requirement for collateral. But as Goodfriend and

King (1988, p.12), note, ‘‘Fully collateralizing a loan with prime paper such as US Treasury bills would

make the value of a central bank�s line of credit minimal, since a bank could acquire the funds by simplyselling the bills on the private market’’.

Even if a commercial bank seeking help from a CB will usually have used up its best collateral already

(to borrow on finer terms from the market), the CB may be able to extract such tough terms for its LOLR

lending that its own resources are largely protected in the case of an insolvency. But some (junior) creditors

would then be hit all the harder, and there would still be a reputational loss to the CB, perhaps the more

severe if it was perceived as refusing to take its ‘‘share’’ of the losses – especially if it was also responsible

for bank supervision.18 As Fry (1997) has pointed out, CBs have on some occasions after crises, e.g., in Chile, had liabilities

greater than their assets. In such cases they too have been bailed out by taxpayers.


Moreover, large banks may be subject to a tighter monitoring and regulation by CB.

Thus the ‘‘too big to fail’’ is apparently realized by both the CB and banks ex post,

but not by banks ex ante. 19

Moreover, a CB might be able to make its LOLR assistance conditional on a

management shake-up. There is an, unfortunately apocryphal, story of a commercialbank manager enquiring of Paul Volcker how Paul would react if the manager came

to him for help. ‘‘I would be very pleased to discuss the issue with your successor’’

was the supposed reply. ‘‘Too big to fail’’ does not necessarily imply that the man-

agement of as assisted bank escapes sanction unscathed.

Of course, if the value of �j was public knowledge, there would be a discontinuity.Banks just larger than the cut-off point would have an incentive to increase risk,

whereas banks just below the cut-off point would become very risk averse. But �j isnot generally publicly observable. 20 This is partly because individual banks have idi-osyncratic features making the CB more, or less likely, to intervene on their be-

half, 21 partly because �j varies over time (and in a somewhat unpredictablefashion, as we will show in subsequent sections), and partly because the monetary

authorities maintain a policy in this respect of ‘‘constructive ambiguity’’. Indeed

our model enables us to interpret this latter as a rational response by the CB pre-

cisely to prevent commercial banks� chosen risk profiles becoming a function ofsize. 22

In summary, we believe that even though the CB response to a commercial bankin difficulties will be a function of the size of that bank, a bank�s chosen risk profileneed not necessarily be a function of size. ‘‘Too big to fail’’ is widely accepted to be

an almost universal phenomenon, yet we are not aware of any empirical finding that

risk preference among bankers is a positive function of size. This is not to suggest

that commercial banks� risk profile choices do not respond to the actions and signalsof the CB. Indeed the next Section focuses directly on that. Rather we claim that our

model simplification, whereby all banks react similarly to such CB signals, despite

being of differing initial sizes, is sufficiently close to reality to make our model resultsinteresting.

19 One of our referees noted that, in so far as the closure decision was taken by a separate bureaucracy,

e.g., a specialised FSA or a Ministry of Finance, banks might also be less likely to rely on greater size as a

basis for securing LOLR support from the CB.20 There are some exceptions to this dictum. The Comptroller of the Currency, at the time of the

Continental Illinois crisis, stated that all larger banks would also be automatically rescued. The Japanese

monetary authorities in recent years have made it publicly known that the large City banks are ring-fenced

against failure.21 For example Johnson Matthey�s involvement in the gold market in London. Again, BCCI and

Drexel Lambert could be let go, because their interconnectedness was low. Per contra, the interconnec-

tedness of LTCM was high.22 See Enoch et al. (1997) for empirical evidence.


3. The dynamic model

In the static model above, the probability of a commercial bank needing LOLR

(p), the probability of it then also being insolvent (x) and its risk profile (h) were

all taken as given. In this section we consider the dynamic equilibrium in which,p, x, and h will become time-varying in response of the CB�s actions and signals.There are two main channels of intertemporal interactions that we identify here.

The first is moral hazard, whereby a rescue (St = 1), is likely to cause commercial

banks to increase their risk profiles, thereby raising both p and x. In reality, of

course, moral hazard will be much worse if the CB provides LOLR assistance to

an insolvent bank than to a solvent, but illiquid, bank (when there might be no moral

hazard). The second is contagion, whereby failure now, when the CB refuses the re-

quest of LOLR (St = 0), is likely to lead to more failures subsequently.In the dynamic setting, this problem the CB faces generalizes to:

minSt

E0X1t¼0

dtpt½j2kð1� StÞ þ ðnþ B2jÞxtSt�( )

; ð3Þ

where 0 < d < 1 is discount factor, and 06St61 is the CB�s control variable. This issubject to the equation of motion of pt or/and xt. We choose to solve the dynamic

programming problem by using the Lagrange method. 23

In order to derive basic economic intuitions out of clear-cut closed form solutions,

we start our analysis by allowing one of the two stochastic variables, p or x, to vary

over time, holding the other fixed. We present the analysis with xt time-varying (p

constant) as focussing on moral hazard in Section 3.1, and then we treat the analysis

with pt time-varying (x constant) as primarily about contagion and present it in Sec-tion 3.2. Then in Section 3.3, we allow both contagion and moral hazard to operate,

i.e., with both pt and xt stochastically time-varying.

3.1. Moral hazard in dynamic model

Moral hazard effects arise when the nature of future ht and xt regimes depend on

the expected sequence of St decisions. The time paths of ht and xt are now influenced

both by policymakers� current choices and by their commitments to particular futurestrategies of insolvency resolution. To focus on moral hazard effects, we switch off

contagion (i.e., holding p constant).

23 Our problem in the case of contagion, or of moral hazard alone, is a standard dynamic programming

problem and thus can be solved by solving the standard Bellman equation, or employing the Lagrange

method, as shown by Chow (1993, 1997). See in particular pages 7–10 in Chow (1997) for a comparison of

the two methods, and why the Lagrange method can also be used to solve the dynamic programming

problem, even more efficiently. We chose the Lagrange method because it appears more efficient in dealing

with non-quadratic objective functions (the joint case of contagion and moral hazard in Section 3.3). For

consistency in approach, we use the Lagrange method for the whole paper.


We assume that x is a linear function of h, and without loss of generality we set

the coefficient equal to unity, so xt = ht. With p constant, we can drop that from the

CB�s objective function, which involves setting St so as to minimize:

minSt

E0X1t¼0

dt½j2kð1� StÞ þ ðnþ B2jÞhtSt�( )

; ð4Þ

subject to the equation of motion for h, whereby:

htþ1 ¼ a0 � ða1 þ b1jÞð1� StÞ þ ða2 þ b2jÞSt þ a3ht þ etþ1; ð5Þwhere et+1 is a stochastic term with mean zero and a constant variance, and a1 > 0,

b1 > 0, a2 > 0, b2 > 0, with a1 < a2 and b1 < b2. With any CB action, in response to a

call for assistance from a commercial bank, which leads to further difficulties (i.e., ptrising), there is a question whether the banking system is globally stable, and past

history suggests that such worries could have some foundation. We show the neces-

sary stability conditions and discuss their economic intuition below.

The Lagrange function of this question is 24

L ¼ E0X1t¼0

dt½j2kð1� StÞ þ ðnþ B2jÞhtSt�(

� dtþ1ltþ1½htþ1 � a0 þ ða1 þ b1jÞð1� StÞ�ða2 þ b2jÞSt � a3ht � etþ1�):

The first order conditions (FOCs) of this Lagrange with respect to St and ht yield:

j2k � ðnþ B2jÞht � dða1 þ b1jþ a2 þ b2jÞEtltþ1 ¼ 0; ð6Þ

ðnþ B2jÞSt þ da3Etltþ1 ¼ lt: ð7ÞFor a quadratic objective function like ours, we can conjecture lt as a linear function:

lt ¼ r0 þ r1ht;

thus,

Etltþ1 ¼ r0 þ r1Ethtþ1 ¼ r0 þ r1½a0 � a1 � b1jþ ða1 þ b1jþ a2 þ b2jÞSt þ a3ht�:

Substituting this Etlt+1 in (6) and (7) respectively, solving for r0 and r1, we finally ar-

rive at the following proposition regarding the solution for St in the case of moral

hazard, denoted as Smt .25

Proposition 2. In dynamic setting, the optimal monetary policy dealing with moral

hazard case is:

24 Technically, we should have a further Lagrange multiplier, st, and include st(1 � ht) in the Lagrangefunction to take fully into consideration the case of 06ht61. For simplicity, we assume 0 < ht < 1 (and

0 < pt < 1) thus st = 0, and thus ignore it in the Lagrange function here (and below). We also assume thatthe technical condition for 06St61 is satisfied.25 It is easy to check that the sufficient conditions for global optimization in this case of moral hazard,

and contagion below, are satisfied.


Smt ¼ g0 þ g1ht; ð8Þ

whereby:

g0 ¼1

1� da3 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2a23 � d

q �a0 þ a1 þ b1ja1 þ b1jþ a2 þ b2j

�ðda3 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2a23 � d

qÞð1� da3Þj2k

dða1 þ b1jþ a2 þ b2jÞðnþ B2jÞ

24 35;g1 ¼ �


qdða1 þ b1jþ a2 þ b2jÞ

:

This result has the following implications. Notice first that g1 < 0 holds for all

given parameter settings. Therefore, if ht rises, i.e., there is a structural shift makingthe risk level in the banking system higher, then the CB will accommodate less often,

i.e., oSmt =oht < 0, regardless of the banks� sizes; and vice versa. This is obviously

the opposite case to contagion case (below), whereby oSct =opt > 0, and thus theCB always wants to accommodate banks� calls for assistance more often.The system will tend to a long-run equilibrium value for h, set as hm, where hm is

given by the following equation:

hm ¼ 1

2da3 � ð1þ dÞ dð�a0 þ a1 þ b1jÞ �ð1� da3Þj2k

nþ B2j

�:

For 2da3 > 1 + d, which is the stability condition to be shown below, comparativestatics analysis on hm indicates that hm goes down if a0 or j

2k goes up, or a1 + b1j

or n + B2j goes down.Although the equilibrium risk level does depend on the average size of failing

banks, in particular the higher is that average size, the higher is the equilibrium risk

level for a1 + b1j > a0, the difference is much smaller than in contagion. These results

are consistent with intuition, because when moral hazard is the sole concern of the

CB, then it should always have strong incentive to reject the request from the trou-

bled banks for LOLR, and as a result of such an extreme policy, the equilibrium risk

level in the banking system should be quite low, for small and large banks as well.

Furthermore, from the equation of motion for ht above and substituting St,we get:

htþ1 ¼ a0 � a1 � b1jþ ða1 þ b1jþ a2 þ b2jÞg0 þ a3 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia23 � 1=d

q� ht þ etþ1:

Thus, the stability condition is:

dða3 � 1Þ <ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2a23 � d

q:

Combining this condition with 1=ffiffiffid

p< a3 (so that d2a23 � d > 0Þ and noticing that

1þd2d > 1ffiffi

dp for 0 < d < 1, we thus have:

a3 >1þ d2d

: ð9Þ


In the case of moral hazard, bank size does not play a key role. If moral hazard is

the main concern of the CB, CB�s rescuing policy is more uniform across banks withdifferent sizes, and the banking system can be more easily stabilized (than that in the

case of contagion alone). The stability conditions do not depend on the size of the

illiquid banks.The threshold (j) for LOLR with moral hazard is time-varying. Setting Sm

t ¼ bS , weget:

g0ðjÞ þ g1ðjÞht ¼ bS : ð10ÞMultiplying (a1 + b1j + a2 + b2j) on both sides of Eq. (10), it becomes a linear

increasing function of j. Thus there exists a real-number root, which is a function

of all the parameters and variables including ht. Therefore, a same size bank may

be rescued by the CB for a low value of ht, but may not be rescued when ht + 1 in-

creases. This changing of CB�s policy reflects the stochastic dynamics and is causedby the changes in time-varying state-variable, ht.As the left-hand-side of Eq. (10) is a monotonic increasing function of j, and thus

a cut-off point in j, bj, exists for a given ht and a rescue threshold bS , the CB will rescueany bank with size above bjðht; bSÞ. Therefore, ‘‘too big to fail’’ holds even in the dy-namic setting with moral hazard being the main concern.

3.2. Contagion in dynamic model

To focus on the contagion problem, we can set ht = x as constant. Thus, the objec-tive function is:

minSt

E0X1t¼0

dtpt½j2kð1� StÞ þ ðnþ B2jÞxSt�( )

: ð11Þ

This is subject to the equation of motion for pt, which will depend on the CB�s ac-tions, that is either to grant or refuse LOLR (St = 1 or 0), so that

ptþ1 ¼ a0 þ ða1 þ b1jÞð1� StÞ � ða2 þ b2jÞSt þ a3pt þ etþ1; ð12Þwhere et + 1 is a stochastic term with mean zero and a constant variance. Note thatallowing a bank to fail will raise the probability of future failures, i.e., contagion, so

that a1 > 0 and b1 > 0, so providing LOLR assistance can reduce the probability offailure by containing the contagious risk. 26

The Lagrange function of this question is:

L ¼ E0X1t¼0

dtpt½j2kð1� StÞ þ ðnþ B2jÞxSt�(

�dtþ1ktþ1½ptþ1 � a0 � ða1 þ b1jÞð1� StÞ þ ða2 þ b2jÞSt � a3pt � etþ1�):

26 As discussed in footnote 5, contagious risks can arise via one or several channels, including payment/

settlements system, interbank market, and asset/liquidity markets, and they could be triggered by either

fundamental runs or speculative runs.


The FOCs of this Lagrange with respect to St and pt yield:

pt½j2k � ðnþ B2jÞx� þ dða1 þ b1jþ a2 þ b2jÞEtktþ1 ¼ 0; ð13Þ

j2kð1� StÞ þ ðnþ B2jÞxSt þ da3Etktþ1 ¼ kt: ð14ÞAgain, we can conjecture kt as a linear function:

kt ¼ q0 þ q1pt; ð15Þthus,

Etktþ1 ¼ q0 þ q1Etptþ1

¼ q0 þ q1½a0 þ a1 þ b1j� ða1 þ b1jþ a2 þ b2jÞSt þ a3pt�:

Substituting this Etkt+1 in (13) and (14) respectively, solving for q0 and q1, we fi-nally arrive at the following proposition regarding the solution for St in the case of

contagion, denoted as Sct .

Proposition 3. In dynamic setting, the optimal monetary policy dealing with contagion

case is:

Sct ¼ c0 þ c1pt; ð16Þwhereby:

c0 ¼1

1� da3 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2a23 � d

q a0 þ a1 þ b1ja1 þ b1jþ a2 þ b2j

�ðda3 �


qÞj2k

j2k � ðnþ B2jÞx

24 35;c1 ¼


qdða1 þ b1jþ a2 þ b2jÞ

:

This result has the following implications. Notice first that c1 > 0 holds for all thegiven parameter settings, as long as da23 > 1, i.e., a3 > 1=

ffiffiffid

p> 1, which we will fur-

ther discuss in connection with the stability condition below. Therefore, if pt rises,

i.e., there is a structural shift making the banking system less stable, then the CB al-

ways wants to accommodate banks� calls for assistance more frequently, i.e.,oSct =opt > 0, regardless whether their sizes are above the threshold level or not. Thusit will accommodate banks� calls for assistance more frequently than in the static caseand rescue ‘‘smaller’’ as well as all big banks. This should not be surprising whencontagion is the main concern of the CB.

Moreover, the system will tend to a long-run equilibrium value for p, set as pc,

where pc is given by the following equation:

pc ¼ d2da3 � ð1þ dÞ �a0 þ a2 þ b2jþ

ða1 þ b1jþ a2 þ b2jÞðnþ B2jÞxj2k � ðnþ B2jÞx

�:

If 2da3 > 1 + d, which is the stability condition to be shown below, the value of pc

critically depends on whether j2k � (n + B2j)x > 0 or not. If j2k � (n + B2j)x > 0, that


is the cost of LOLR is smaller than that of OMO for a given j in the static model,

then ðnþB2jÞxj2k�ðnþB2jÞx

> 0. Comparative statics analysis on pc in this case indicates that pc

goes down if a0 or j2k goes up, or (a1 + b1j), (a2 + b2j) or (n + B2j)x goes down. These

results are consistent with intuition, because when j2k�(n + B2j)x > 0, the CB hasmore incentive to provide LOLR. A higher a2 + b2j implies a stronger effect ofOMO on the risk level of the banking system; a higher (n + B2j)x implies higher costs

of LOLR, and a lower j2k implies lower costs of OMO. The system will fluctuate

around pc as et varies stochastically.If j2k � (n + B2j)x < 0, that is the cost of LOLR is bigger than that of OMO for a

given j in the static model, however, then ðnþB2jÞxj2k�ðnþB2jÞx

< 0 and

pc ¼ d2da3 � ð1þ dÞ �a0 þ a2 þ b2j�

ða1 þ b1jþ a2 þ b2jÞðnþ B2jÞxðnþ B2jÞx� j2k

�:

From this equation it is clear that the long-run equilibrium value pc is lower when the

cost of LOLR is bigger than that of OMO for a given j in the static model. Moreover,

comparative statics analysis on pc in this case indicates that pc goes up if a0, a1 + b1jor j2k goes down, or (n + B2j)x goes up. These results are also consistent with intu-

ition, because when j2k�(n + B2j)x < 0, the CB has more incentive to provide OMOrather than LOLR. A lower a1 + b1j implies a weaker effect of providing LOLR onthe risk level of the banking system. 27

Furthermore, from the equation of motion for pt above and substituting St, we

get:

ptþ1 ¼ a0 þ a1 þ b1j� ða1 þ b1jþ a2 þ b2jÞc0 þ a3 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia23 � 1=d

q� pt þ etþ1:

Thus in addition to the required stability condition from pc, j2k5 (n + B2j)x, the sta-

bility condition (on a3 and d) is a3 � 1 <ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia23 � 1=d

p. Combining this condition with

1=ffiffiffid

p< a3 and noticing that 1þd

2d > 1ffiffid

p for 0 < d < 1, we thus have the stability condi-tion for pt:

a3 > 1þd2d ;

j2k 6¼ ðnþ B2jÞx:

�ð17Þ

Finally, notice that the result of the above proposition can also be interpreted in

terms of bank size j. Setting Sct ¼ bS in (16), where bS is close to the upper bound of Stand the CB will accommodate bank j�s request for LOLR if S P bS , we get:
27 Notice further that, if failures have occurred in a series of large banks, i.e., j2k > (n + B2j)x, then p
c

may even be close to its upper bound, 1, for some parameter configurations. The intuition is that when the

CB is only concerned with contagious risk, which is more severe for large banks, it would rescue too many

banks so that the banking system risk level would then become extremely high. Similarly, if the only

failures to have occurred were among small banks, i.e., j2k < (n + B2j)x, pc may even be close to its lower

bound, 0, for some parameter configurations. The intuition is that even when the CB is concerned with

contagious risk, yet the troubled banks have been small, it would limit access to LOLR and rather use

OMO, so that the banking system risk level would be extremely low.


c0ðjÞ þ c1ðjÞpt ¼ bS : ð18ÞEq. (18) becomes a cubic function of j after we multiply [j2k�(n + B2j)x] (a1 + b1j +a2 + b2j) on both sides. Thus there may exist either one or three real-number roots,which are functions of all the parameters and variables including pt. Therefore, a

same size bank may be rescued by the CB for a given pt, but may not be rescuedfor another value of pt + 1. This shifting of CB�s policy reflects the stochastic dynam-ics and is caused by the changes in the state-variable pt.

Even for the same value of pt, the CB may optimally rescue only those banks that

have the ‘‘right sizes’’. As the above cubic equation (18) may have three real-number

roots, which are non-negative yet smaller than D*, these roots also clearly define use-

ful bank size categories and the CB should rescue these banks within such categories,

but not to do so for those banks which are not in the categories.

For a variety of possible parameter settings, some small or large banks may berescued, while banks whose sizes are intermediate may not be. Consequently, the

CB�s optimal behavior may appear non-monotonic in bank size and thus ambiguousif viewed from outside, which will further enhance the constructive ambiguity which

we described in the above section.

Next we turn to the joint case of moral hazard and contagion.

3.3. Contagion and moral hazard in dynamic model

When both p and x are allowed to be time-varying, we assume (as above) that x is

a linear function of h, and we set the coefficient equal to unity, so xt = ht. Thus the

new problem is to set St optimally so as to minimize:

minSt

E0X1t¼0

dtpt½j2kð1� StÞ þ ðnþ B2jÞhtSt�( )

; ð19Þ

subject to the equations of motion for ht and pt, whereby:

htþ1 ¼ a0 � ða1 þ b1jÞð1� StÞ þ ða2 þ b2jÞSt þ a3ht þ etþ1; ð20Þ

ptþ1 ¼ a0 þ ða1 þ b1jÞð1� StÞ � ða2 þ b2jÞSt þ a3pt þ etþ1; ð21Þ

where again et + 1 and et + 1 are stochastic terms with mean zero and constant vari-ances, and all other coefficients are positive.

Again, the Lagrange function of this question is:

£ ¼ E0X1t¼0

dtpt½j2kð1� StÞ þ ðnþ B2jÞhtSt�(

�dtþ1ltþ1½htþ1 � a0 þ ða1 þ b1jÞð1� StÞ � ða2 þ b2jÞSt � a3ht � etþ1�

�dtþ1ktþ1½ptþ1 � a0 � ða1 þ b1jÞð1� StÞ þ ða2 þ b2jÞSt � a3pt � etþ1�):


The FOCs of this Lagrange with respect to St, ht and pt yield:

0 ¼ ½j2k � ðnþ B2jÞht�pt � dDaEtltþ1 þ dDaEtktþ1;

lt ¼ ðnþ B2jÞptSt þ da3Etltþ1;

kt ¼ j2kð1� StÞ þ ðnþ B2jÞhtSt þ da3Etktþ1;

where we denote:

Da ¼ a1 þ b1jþ a2 þ b2j;

Da ¼ a1 þ b1jþ a2 þ b2j:

Intuitively, the joint case of contagion and moral hazard is some form of convexcombination of each case alone, with both ht and pt changing over time. As discussed

above, the contagion case is a special case of the joint case, with ht fixed; and simi-

larly, the moral hazard case is a special case of the joint case, with pt fixed.

To solve a problem with a quadratic first order conditions like ours, we shall start

with linearization of the FOCs around the steady states. 28 Doing so leads to:

0 ¼ ðnþ B2jÞ�h�p � ðnþ B2jÞ�pht þ ½j2k � ðnþ B2jÞ�h�pt � dDaEtltþ1 þ dDaEtktþ1;

lt ¼ �ðnþ B2jÞS�p þ ðnþ B2jÞSpt þ ðnþ B2jÞ�pSt þ da3Etltþ1;

kt ¼ j2k � ðnþ B2jÞhS þ ðnþ B2jÞSht þ ½�j2k þ ðnþ B2jÞ�h�St þ da3Etktþ1;

where �h and �p denote the steady state for ht and pt respectively, and S denotes St eval-uated at �h and �p.With a linear FOCs, we can conjecture

ltkt

� ¼

h0q0

� þ

h1 h2q1 q2

� htpt

� :

Following the same procedure as in the case of contagion or moral hazard, we can

use this conjecture in the FOCs to solve for the unknowns (h0,h1,h2,q0,q1,q2), andthen for St in the joint case of contagion and moral hazard, denoted as S

�t .

Proposition 4. In the dynamic setting, the optimal monetary policy dealing with both

contagion and moral hazard is:

S�t ¼ l0 þ lhht þ lppt; ð22Þ

whereby:

28 Since both the objective function and two constraints are convex in St, ht and pt, global optimality is

warranted. Consequently we can linearize the first order conditions around the steady states to solve the

problem. If the stability conditions are not met, however, then presumably other measures, such as blanket

guarantees, would be needed instead.


l0 ¼ S � lh�h� lp�p;

lh ¼ � ðnþ B2jÞS þ ðda3a3 � 1Þq1j2k � ðnþ B2jÞ�hþ da3ðDaq2 � Daq1Þ

;

lp ¼ðda23 � 1Þq2

j2k � ðnþ B2jÞ�hþ da3ðDaq2 � Daq1Þ:

This result has the following implications. Notice first that lh < 0 holds, exceptfor the case in which j2k < ðnþ B2jÞ�h� da3ðDaq2 � Daq1Þ and (n + B2j)S <�(da3a3�1)q1. In general, we shall expect lh < 0, thus similar to the case of moralhazard alone, if ht rises, i.e., there is a structural shift making the risk level in the

banking system higher, then the CB will accommodate less often, i.e., oS�t =oht < 0.

Notice that, unlike in the case of moral hazard alone, lh becomes dependent onmany parameters, including the equilibrium states, �h and �p, which were not in themoral hazard case alone. The CB is still reluctant to provide LOLR even if conta-

gious risks are considered, but its reluctance to provide LOLR is not as strong asin the case of moral hazard alone.

Moreover, for da23 > 1, q2 > 0 and Daq2 > Daq1, a strong condition for lp > 0 is

j2k > ðnþ B2jÞ�h;

that is the cost of LOLR is smaller than that of OMO when the risk level is in equi-

librium. More precisely,

j P �j� B2�hþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB22�h

2 þ 4kðn�h� da3ðDaq2 � Daq1ÞÞq

2k;

thus �j� < �jðx ¼ �hÞ and it is smaller than the cutoff size in the static model.We may compare this result with that for contagion alone. In the case of conta-

gion alone, c1 > 0 holds regardless of bank size, and so the CB always had an incen-tive to rescue banks. In the joint case, the CB will only provide LOLR for large

banks. The CB is still willing to provide LOLR even if moral hazard effects are con-

sidered, but its incentives to provide LOLR are not as strong as in the case where

moral hazard is excluded. Indeed, the CB�s incentives to provide LOLR in the jointcase of contagion and moral hazard are a convex combination of its incentives in

each case alone.

Similarly, the system will tend to a long-run equilibrium value for both h and p,set as �h and �p, which are more complicated than in the case of moral hazard or con-tagion alone. But it is easy to see that both the equilibrium risk levels, �h and �p, crit-ically depend on the average size of failing banks. Both �p and �h critically depend onbank size, because lp critically depends on bank size. A higher equilibrium �p impliesa higher �h, and hence a higher x. In words, a system in which banks are regularlycoming to the CB to seek LOLR assistance is likely at the same time to have

adopted, and to exhibit, a riskier choice of asset portfolios, and thereby to be more

subject to insolvencies.


The dynamics of the joint case of contagion and moral hazard are more interest-

ing and complicated. From our above analysis, we can see that the system will, as

before, fluctuate with the stochastic shocks, but also both contagion and moral haz-

ard provide an inbuilt cycling mechanism. When ht is (temporarily) low, the CB will

be induced to accommodate more, but that will signal the commercial banks to raiseht and raise the contagion risk pt, which will cause the CB to refuse accommodation

more often, and so on. For plausible values of the coefficients, however, we would

expect this cycle to be damped. But the combination of stochastic shocks to riskiness

together with this inbuilt damped cycling mechanism will always leave the choice of

whether to accommodate, or not, fluctuating around its long-run equilibria.

Finally, there may be multiple values of j around which CB�s policy should shift,and such values will always be changing over time, more often and perhaps more

strongly than in the contagion case alone, as can be seen more clearly below.If we set S�

t ¼ bS in (22), we get:l0ðjÞ þ l1ðjÞht þ l2ðjÞpt ¼ bS : ð23Þ

This is a high-order equation in j, hence there may exist several real-number roots

which are obviously functions of all the parameters and variables including ht and

pt. Thus, when ht and/or pt change, CB should adjust its rescuing policy accordingly.

Therefore, a bank of equal size may be rescued by the CB for a set of values for htand pt, but may not be rescued when ht+1 and pt+1 change. Such a shift in CB�s policyreflects the time dynamics and is caused by the changes in time-varying variables, htand pt.Moreover, even for the same value of ht and pt, the CB may optimally rescue only

those banks that have the ‘‘right sizes’’. For the above high-order equation (23),

there may be several real number roots which are non-negative yet smaller than

D*. These roots also clearly define bank size categories and the CB should rescue

these banks within such categories, but not rescue banks outside these categories.

4. Conclusion

This paper has developed a model of the lender of last resort. In a simple static

setting, the CB will only rescue banks above a threshold size. This result provides

an analytical basis for the well known ‘‘too big to fail’’ syndrome. If that key thresh-

old size were known to commercial banks, it would influence their risk preferences.

To avoid this the regulatory authorities should, and do, use ‘‘constructive ambigu-

ity’’ to make their decisions on which banks they are likely to rescue.

In a dynamic setting, wherein both the probability of a failure and the likelihoodof a bank requiring LOLR being insolvent in each period are a function of CB�s prioractions, which then influence the actions of banks and depositors, we focus our anal-

ysis on the effects of contagion and/or moral hazard. We show that CB�s optimal res-cuing policy, whether to support or not, depends not only on bank size, but also on

the time-varying variables, such as the probability of further contagious failures and

the likelihood of a bank requiring LOLR being insolvent.


Unlike the static setting wherein the CB only rescues banks above a single thresh-

old size, in a dynamic setting with concerns about contagion, the CB�s optimalrescuing policy may be non-monotonic in bank size, and its optimal policy is

time-varying. More importantly, we have found that if contagion is the main con-

cern, then the CB in general would have an excessive incentive to rescue banksthrough LOLR, though its incentives to rescue big (small) banks are very strong

(weak) and thus the equilibrium risk level is high (low). If moral hazard is the main

concern, then the CB in general would have less incentive to rescue banks through

LOLR than in the dynamic setting with contagion alone, and it should only rescue

banks above a threshold size. When both contagion and moral hazard are included

as main concerns, then the CB�s incentives to rescue through LOLR are strongerthan in the static setting but weaker than in the dynamic setting with contagion

alone, and the CB�s optimal policy in handling moral hazard is similar to the dy-namic setting with moral hazard alone.

A key result coming out of our model is that the likelihood of contagion is the key

factor affecting CB�s incentive in providing LOLR, while moral hazard is not. Whencontagion is the main concern, the CB has a very strong incentive to provide LOLR.

When moral hazard is also included, in the joint case, even though it weakens the

CB�s incentive to rescue in general, its effects are quite weak, and the qualitative fea-tures of CB�s incentive remain the same as when contagion is the main concern. Thisis so because moral hazard can be viewed as an unpleasant by-product of contagion.If it was not for worries about contagion, then CB�s incentive to provide LOLRwould be strictly limited, and consequently there would be very little moral hazard.

This conclusion has some implications for the ongoing debates over CB�s, and inparticular the IMF�s, rescuing policy in financial crises. When contagion becomes amain concern, even if moral hazard is also present, LOLR becomes (perceived as) nec-

essary and justified. Attacks on such LOLR policies, largely based on arguments from

moral hazard, are insufficient unless they also address the possibility of contagion. 29

Turning finally to further possible research, in our model we have assumed thatcommercial bank managers� appetite for risk remains constant as size varies. Thiswas because we assumed that the incentive for risk-taking inherent in too-big-to-fail

was (roughly) balanced by the greater risk aversion of managers in high status large

banks. This assumption can be examined. In future work, we intend to model the

incentives on commercial bank managers, in such a game, more rigorously. We also

wish to examine the effects of contagion and moral hazard on the optimal decision of

the international lender.

Moreover, institutional arrangements for handling financial crises have beenchanging. Once upon a time the CB handled such crises in most countries as a sole,

or at least a dominant, agent. Nowadays crises are increasingly managed by a com-

mittee, or consortium, of public sector bureaucracies. We look forward to the chal-

lenge of trying to formally model such a development in future.

29 For international perspectives on the LOLR function, see Fischer (1999), Giannini (1999), Goodhart

and Huang (2000), and Clark and Huang (2001).


Acknowledgements

For helpful comments and conversations, we would like to thank William Alex-

ander, Peter Clark, Tito Cordella, Fabrizio Coricelli, Gianni de Nicolo, Carsten Det-

ken, Stanley Fischer, Michele Fratianni, Xavier Freixas, Frank Heinemann, HeinzHerrmann, Gerhard Illing, Barry Johnston, Se-Jik Kim, Allan Meltzer, Hyun Song

Shin, Charles Siegman, Mark Spiegel, Jukka Vesala, Geoffrey Wood, and partici-

pants in seminars at IMF and ECB and conferences in Stanford, LSE, Siena, and

Frankfurt. We are especially grateful to Giorgio Szego and two anonymous referees

for useful suggestions. The views are our own and do not necessarily represent those

of the authors� affiliated institutions.

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