Fabio Castiglionesi

nancial contagion. Financial systems seem to be particularly vulnerable to systemic riskgiven their characteristics: the structure of bank balance sheets, the network of exposure

E-mail address: fabio.castiglionesi@uvt.nl

Journal of Banking & Finance 31 (2007) 81101

www.elsevier.com/locate/jbf0378-4266/$ - see front matter 2006 Elsevier B.V. All rights reserved.JEL classication: E58; G20

Keywords: Contagion; Reserve requirements; Bank crisis

1. Introduction

Various theoretical interpretations have been proposed in order to give a rationale forCenter and Tilburg University, Finance Department, Box 90153, Tilburg 5000 LE, The Netherlands

Received 1 November 2004; accepted 31 March 2005Available online 12 June 2006

Abstract

We investigate the role of a central bank (CB) in preventing and avoiding nancial contagion.The CB, by imposing reserve requirements on the banking system, trades o the cost of reducingthe resources available for long-term investment with the benet of raising liquidity to face anadverse shock that could cause contagious crises. We argue that contagion is not due to the structureof the interbank deposit market, but to the impossibility to sign contracts contingent on unforeseencontingencies. As long as incomplete contracts are present, the CB may have a useful role in curbingcontagion. Moreover, the CB allows the banking system to reach rst-best allocation in all the statesof the world when the notion of incentive-eciency is considered. If the analysis is restricted to con-strained-eciency, the CB still avoids contagion without, however, reaching rst-best consumptionallocation. The model provides a rationale for reserve requirements without the presence of atmoney or asymmetric information. 2006 Elsevier B.V. All rights reserved.Financial contagion and the role of the central bankdoi:10.1016/j.jbankn.2005.03.025

among nancial institutions, and the character of nancial contracts. Although a generalparadigm has not yet emerged, we have a better understanding of the propagation ofshocks in the banking and payment system.1 In this paper, we focus the attention on nan-

82 F. Castiglionesi / Journal of Banking & Finance 31 (2007) 81101cial institution linkages, in particular the interbank deposit market, that are able to gen-erate the possibility of contagion. In this context, we analyze the possible role for thecentral bank (henceforth CB) in preventing and avoiding bank crises and contagion.

From the early contribution by Diamond and Dybvig (1983), there is a shared view ofbanks as providers of liquidity. Banks, in this context, are pools of liquidity and theirexistence is not rooted in the presence of information asymmetries in credit markets.2

However, the DiamondDybvig model treats the whole banking industry as a singleentity. In reality there are many banks in dierent regions, and problems arising in onebank can spread through the entire banking system. Financial contagion can be inducedby an information-based mechanism. Diculties in one bank may cause depositors to sus-pect that the whole bank industry is under pressure. For example, Jacklin and Bhattach-arya (1988) argue that bank runs are triggered by asymmetry between the banksknowledge about its depositors liquidity needs and the depositors information aboutthe banks asset. Chen (1999) argues that information externalities are important in caus-ing contagious bank runs, since they force depositors to respond to noisy information suchas failures of other banks. However, nancial contagion is possible even without the pres-ence of asymmetric information.

Allen and Gale (2000) provide an explanation of nancial contagion as a phenomenonthat emerges in the banking system of a multi-region economy. Contagion can be the equi-librium outcome in which, after the distress of a region due to an adverse shock on agentspreferences, there is the possibility of a spillover in other regions because of the presence ofcross-holding interbank deposits. The interbank deposit market is able to provide insur-ance to the dierent regions against asymmetric liquidity needs, thus allowing the economyto reach rst-best allocation in a decentralized setting. However, this arrangement is vul-nerable to nancial contagion if the unexpected liquidity shock occurs. Then we can askthe following question: Is there any instrument that is able to avoid contagion in the mostecient way? We show that the CB could oer a solution.

The intervention of the CB takes the form of reserve requirements, which are a fractionof the amount of the banks deposits. The reserve requirements imposed on the bankingsystem imply that less resources can be allocated in long-term productive activities. How-ever, the reserves give the opportunity for the CB to face the adverse liquidity shock. Thenthe problem of the CB is to choose the optimal fraction of reserve requirements that, onthe one hand, ensures enough liquidity in case of an aggregate liquidity shortage occursand, on the other hand, does not divert too many resources from protable investmentopportunities.

The intervention of the CB allows the decentralized banking system to reach rst-bestallocation when contingent contracts are considered. Indeed, when there is no shortage ofaggregate liquidity, the imposition of reserve requirements delivers the ecient consump-tion allocation, making the payment to early (late) consumers higher (lower) than what thedecentralized banking system alone would oer. When the unexpected shortage of aggre-

1 See Rochet and Tirole (1996), Allen and Gale (2000), Freixas et al. (2000), and Dasgupta (2004).2 The alternative view of banks as agents who provide delegated monitoring services is due to Diamond (1984)and Williamson (1986).

F. Castiglionesi / Journal of Banking & Finance 31 (2007) 81101 83gate liquidity occurs, and consequently the possibility of contagion arises, the CBintervenes declaring the insolvency of the banking system, and guarantees rst-bestconsumption level with the liquidity collected by means of reserve requirements. If werestrict attention to non-contingent contracts, consequently shifting from the notion ofincentive-eciency to constrained-eciency, reserve requirements alone allow the CB toavoid contagion when the aggregate liquidity shortage occurs. No declaration of insol-vency is needed. However, in this case it is impossible to achieve rst-best consumptionallocation.

The underlying assumption is that the CB has a dierent a-priori about the possibilityof the occurrence of the aggregate liquidity shortage. While depositors and commercialbanks give a probability zero to this event to happen, the CB attaches to the occurrenceof this event its true (small) probability. The reason to assume a dierent a-priori isbecause it makes depositors and banks to sign incomplete deposit contracts, that is theydo not take into account the liquidity shock, which appears to be a widespread character-istic of real world deposit contracts. The CB, observing this contract incompleteness, andbeing worried by the occurrence of an aggregate liquidity shortage, can take the appropri-ate action.

Allen and Gale (2000) claim that the possibility of contagion is related to the structureof the interbank deposit market. The more complete is the structure of the interbankdeposit market, the more dicult it is for contagion to occur. This may incorrectly sug-gest that a complete interbank deposit market may essentially eliminate contagion, thus,reducing the need for a CB. It will be argued that contagion is not rooted in the structureof the interbank deposit market, but in the impossibility for agents to sign contracts con-tingent on unforeseen contingencies. This result is consistent with Dasgupta (2004), whoalso argues the necessity of a CB despite a complete structure of the interbank depositmarkets.

In this paper, the CB does not nd its rationale on the failure of the ex-post loan mar-ket, as in Bhattacharya and Gale (1987). The CBs role is rooted in the bad distribution ofliquidity among dierent regions, and its intervention is designed to avoid a coordinationproblem in the same fashion as Diamond and Dybvig (1983), i.e., one that is caused by anaggregate liquidity shock. There is, however, a distinction between the two models. In Dia-mond and Dybvig (1983) the intervention of the CB takes the form of deposit insurancesince this is enough to avoid the bad equilibrium (i.e., the bank runs) in the presence ofmultiple equilibria. In this model, the intervention of the CB needs to be more explicitthrough reserve requirements.

There is an authoritative doctrine claiming that reserve requirements are useless, at leastfrom the point of view of monetary policy (Sargent and Wallace, 1982). However, variousmonetary models have provided a rationale for reserve requirements. For example, theyreduce the nominal instability generated by a monetary policy that targets interest rate sta-bilization. In this context, reserve requirements are a useful companion to interest ratestabilization (Lorenzoni, 2001). Moreover, reserve requirements make the money marketmultiplier more stable and predictable, thus helping to control money and credit expan-sion (Brunner and Meltzer, 1990). Alternatively, a scope for a regulatory interventionby means of reserve requirements can derive from the fact that banks are characterizedby costly state verication (Di Giorgio, 1999).

In our model, there is no need of at money injections in order to rationalize the pres-

ence of reserve requirements. Consequently, a rationale for the introduction of a legal

ences, that is,Uc1; c2 uc1 with probability xiuc2 with probability 1 xi:

The function u() is assumed to be increasing, strictly concave, twice continuously dier-entiable, and satises the Inada conditions. The probability xi is the random fraction ofearly consumers in region i, and it can take two possible values xH and xL, withxH > xL. Let the average fraction of early consumers be dened by c (xH + xL)/2.The realization of the liquidity preference shocks is state-dependent, and is given inTable 1.

There are two equally likely states, S1 and S2, with probability p1 and p2, respectively,and a low-probability state S with probability p3. The latter state represents a perturba-tion of the model, such that the aggregate demand for liquidity is greater than the systemssupply of liquidity. While the state of nature S is given a probability zero by commercialbanks and depositors, the CB attaches to it the true, small, probability. This implies thatreserve requirement is obtained in a real (non-monetary) model. Moreover, in ourmodel banks do not face asymmetric information problems on the activities side oftheir balance sheets since they invest directly in productive assets. Finally, since the CBsuccessful intervention does not rely on liquidity creation, no direct transfers from tax-payer are necessary in order to bail out the distressed region (contrary to Freixaset al., 2000). In our model, the cost is represented by the missed long-term investmentopportunity.

The paper is organized as follows. In Section 2, we present the model, which builds onAllen and Gale (2000). In Section 3, we characterize the optimal risk-sharing solutionobtained by the social planner. In Section 4, the decentralized economy with the CB is ana-lyzed. In Section 5, we compare the nancial fragility of the decentralized economy withand without the intervention of the CB. In Section 6, we restrict the analysis to non-con-tingent contracts, considering then the notion of constrained-eciency, rather than incen-tive-eciency, as the benchmark. Finally, we draw the conclusions.

2. The model

Consider an economy with one good, which serves as numeraire, and a continuum ofconsumers (depositors). There are three dates: a planning period (t = 0) and two consump-tion periods (t = 1, 2). All consumers are endowed with one unit of the good at period 0.There are two types of assets, which are represented by storage technologies. A liquid asset(the short asset) that takes one unit of the good at date t and converts it into one unit of thegood at date t + 1; and an illiquid asset (the long asset) that takes one unit of the good atdate 0 and transforms it into R > 1 units of the good at date 2. If the long asset is liqui-dated prematurely at date 1, then it pays 1 > r > 0 units of the good. Only banks can investin nancial assets, and consumers have to deposit their endowment in a bank in order totake advantage of the available investment opportunities.

There are four regions (labeled A, B, C, and D) that are ex-ante identical but dier inthe liquidity shock. Each region contains a continuum of ex-ante identical consumers anda continuum of identical banks. Agents are assumed to have DiamondDybvig prefer-

84 F. Castiglionesi / Journal of Banking & Finance 31 (2007) 81101commercial banks and depositors do not consider this event in the deposit contract, how-

can always imitate early consumers.

Table 1Regional liquidity shocks

A B C D

S1 xH xL xH xLS2 xL xH xL xHS c + e c c c

F. Castiglionesi / Journal of Banking & Finance 31 (2007) 81101 85The role of the banks is to make investments on behalf of consumers and to ensurethem against liquidity shocks. In t = 0 each bank oers a deposit contract (c1,c2), whichallows depositors to withdraw either c1 units of consumption in t = 1 or c2 units of con-sumption in t = 2, and invests the deposit in a portfolio (x,y), where x and y are theper capita amounts invested in long and short assets, respectively. The CB imposes int = 0 to each bank reserve requirements q. The timing of the contract in t = 0 is as follows.First, depositors and banks sign the deposit contract, and banks decide the investmentportfolio. Both the consumption level and the investment are contingent on the value ofthe reserve requirement. Second, the CB xes the reserve q. Third, deposit contractsand investments are implemented. This kind of deposit contract is like a deposit with var-iable interest rate, which are observed in reality. When depositors and banks sign thedeposit contract they take as a parameter the aggregate condition of the economy thathere is represented by q. Once reserve requirements are chosen by the CB, the contractsare implemented in t = 0, before the realization of the state of nature, and the consump-tion levels in t = 1 and t = 2 are determined.

3. Optimal risk-sharing

The optimal risk-sharing can be characterized by the solution of the planners problem,given that a central planner can easily transfer resources across regions. Suppose that theplanner can observe the agents type, so that we do not need to consider the incentive com-patibility constraints. Since consumers are ex-ante identical, the planner is assumed toever the CB is concerned with the consequences of the low-probability aggregate shortageof liquidity.3

Ex-ante, each region has the same probability of having a high liquidity preference.There is no aggregate uncertainty for commercial banks and depositors. All uncertaintyis resolved in t = 1, when the state of nature is revealed and each consumer learns whethershe is an early or late consumer. A consumers type is not observable, so late consumersmaximize the unweighted sum of consumers expected utility in all the possible states ofnature. The planners problem is

3 Through the paper we will assume that the CB knows both the probability of the occurrence of the aggregateliquidity shortage (p3) and the value of the shock e. The results that are obtained in what follows are robust if weconsider a stochastic value of the liquidity shock. That is, the CB does not know the value of the shock but onlyits distribution. This extension is available upon request from the author.

86 F. Castiglionesi / Journal of Banking & Finance 31 (2007) 81101subject to: x y 6 1;cc1 6 y;

c e4

c1 6 y;

1 cc2 6 Rx;1 c e

4

c2 6 Rx;

where c1 and c2 are the consumption levels in the expected states of nature S1 and S2, whilec1 and c2 are those in the low-probability state S. The problem of the central planner as-sumes that no early liquidation of the long-term asset is ever implemented. The implicitassumption is that the cost of early liquidation r, even if greater than zero, is sucientlysmall that liquidation of the long-term asset is never optimal. The solution to this problemis the rst-best solution.

The planner optimally chooses the investment in the nancial assets, adjusting con-sumption contingent on the state of nature that occurs. This implies that the constraintsare binding and the rst-best solution is characterized by the following allocation:

y cc1 c e4

c1;

x 1 cc1 1 c e4c1;

c1 cc1c e4

;

c2 R1 cc11 c ;

c2 R1 cc11 c e

4

:

On the one hand, the rst-best solution is characterized by a larger amount of consump-tion in t = 1 when there is no aggregate liquidity shortage (c1 > c1). If more than expectedearly consumers want to withdraw, then a lower amount of consumption should be given tothem. This captures the idea that when a shock occurs (able to cause a crisis) then it is opti-mal to reduce consumption, since it allows to avoid the costly liquidation of the long-termasset. On the other hand, the rst-best solution gives more consumption to late consumersin case of the liquidity preferences shock than in the normal states (c2 > c2). This becausethe same amount of resources is available for the less-than-expected late consumers.

Proposition 1. If the probability of the unexpected state S is sufficiently small, then the first-best allocation d y; x; c1; c2;c1;c2 is equivalent to the incentive-efficient allocation inall the states of the world. The first-best allocation can be achieved even if the planner cannotobserve the consumers type.

Proof. The FOCs of the planners problem lead to the following condition:0 0 0 0maxfx;y;c1;c2;c1;c2g

p1 p2cuc1 1 cuc2 p3 ce4

uc1 1 c e

4

uc2

h ip1 p2Ru c2 u c1 p3u c1 Ru c2: 1

F. Castiglionesi / Journal of Banking & Finance 31 (2007) 81101 87First of all, note that the optimal allocation gives c1 > c1 and c2 > c2. Therefore, if the incen-tive compatibility c2 > c1 is satised then the other incentive compatibility condition c2 > c1is also satised.Moreover, if the condition c2 > c1 is violated the condition c2 > c1 still holds.In fact, suppose c1 > c2. Then Ru

0(c2) > u 0(c1), which in turn implies, from condition (1),u0c1 > Ru0c2, which in turn implies c2 > c1. Therefore, we have to rule out c1 > c2, whichis in principle possible. From condition (1), it is clear that for p3 = 0 the rst-best allocationimplies Ru 0(c2) = u 0(c1), which in turn implies c2 > c1. Then, when p3 = 0 we have c2 > c2 >c1 > c1, or u0c1 > u0c1 Ru0c2 > Ru0c2. Therefore, in this case incentive compatibil-ity is satised in all the states of the world. The objective function is strictly concave, so thatfor each p3 there is a unique solution dp3 c1p3; c2p3; c1p3;c2p3. By the maximumtheorem, d(p3) is a continuous function of p3. Thus, as p3 ! 0, we have d(p3)! d(0). This inturn implies that, since the incentive compatibility inequalities are satised with strictinequality at p3 = 0, they will also be satised in an open neighborhood of zero. h

We consider as benchmark the notion of incentive-eciency until Section 6, where weanalyze the case in which the planner is restricted to use non-contingent contracts. In thatcontext, we will see how results change if the notion of constrained-eciency is consideredas the appropriate benchmark.

Example 1. Let us characterize the rst-best solution with the logarithmic utility functionu() = log(c). The planners problem can be written as

maxfc1g

p1 p2 c logc1 1 c logR1 cc1

1 c

p3 ce4

log

cc1

c e4

264

375 1 c e

4

log

R1 cc11 c e

4

264

375

8>:

9>=>;:

Taking the FOC, we get the rst-best consumption in the expected states:

c1 1 p3e4c

:

Accordingly, the rst-best allocation in case of logarithmic utility function is

y c 1 p3e4c

c p3e

4;

x 1 c p3e4

;

c1 c p3e

4

c e4

;

c2 R 1 c p3e

4

h i1 c ;

c2 R 1 c p3e

4

h i1 c e

:4

Note that c1 > 1 > c1 and c2 > c2. Moreover, at p3 = 0 we have c2 = R > 1 = c1 andc2 R 1c1ce4 > R > 1 >

cce4

c1; so that both incentive compatibility constraints hold.

x c > cc = y. The advantage of the planner is that he can move consumption between

88 F. Castiglionesi / Journal of Banking & Finance 31 (2007) 81101H 1 1

regions so (in the expected states) he needs to satisfy only the average constraintcc1 = y. As shown by Allen and Gale (2000), the way to overcome this problem is to intro-duce an interbank market for deposits that is, banks of dierent regions exchange depos-its among themselves at t = 0. Two possible structures of the interbank deposit market areconsidered: complete and incomplete (see Fig. 1).

In the case of complete markets, each region exchanges deposits with all the otherregions. Each region is negatively correlated with two other regions, therefore it is

4 For expositional purposes, in this section we analyze the decentralized allocation in states S1 and S2, dealingThe inequalities still hold for p3 low enough. Since c2 > c1 implies c2 > c1; the thresholdlevel p3 is given by

p3 6 p3 4c1 cR 1e ecR 1 :

For values of p3 higher than p3 the incentive compatibility constraint c2 > c1 does not holdanymore.

4. Decentralized economy

The rst-best allocation can be decentralized by a competitive banking system with thepresence of the CB. Since in each region there is a continuum of identical banks, it is pos-sible to focus on symmetric equilibria. The decentralized allocation is then characterized interms of the behavior of a representative bank in each region.4

Agents deposit their endowment in the representative bank of their region. Each repre-sentative bank oers them a deposit contract (c1,c2), and it invests the deposits in the port-folio (x,y). As already mentioned in Section 2, the timing of the contract in t = 0 is asfollows. First, depositors and banks sign the deposit contract, and banks decide the invest-ment portfolio. Both the consumption level and the investment are contingent on the valueof reserve requirements q. Second, the CB determines q. Third, deposit contracts andinvestment portfolio are implemented before the realization of the state of nature. Clearly,the deposit contract has to respect the incentive compatibility constraint.

The reserve requirements imposed by the CB represent the minimum amount of depositsthat the commercial banks have to invest in the short-term liquid asset. Then, theresources xed as reserve requirements pay one unit of consumption both in t = 1 andin t = 2. The CB xes reserve requirements for precautionary reasons, in order to preventaggregate liquidity shortages. Commercial banks correctly anticipate that the CB is abenevolent institution that maximizes social welfare, and they rationally expect to use int = 1 the amount of reserves to face early consumers.

In t = 0 the representative banks have the budget constraint x + y 6 1 q, which is sat-ised by all of them. However, the budget constraint in the second period, namely cc1 6 y,is not satised by the regions that have a high proportion of early consumers becausewith the analysis in the unexpected state in the next section.

assumed that each representative bank in region i holds zi = (xH c)/2 deposits in all theother three regions. In the case of incomplete markets each region holds deposits only inthe adjacent region. Each bank is then assumed to hold zi = (xH c) deposits in the adja-cent region. In both market structures, we have zi = z "i (all the regions hold the sameamount of interbank deposits). Note that the amount of deposits exchanged in the inter-bank deposit market does not depend on the reserve q.

With a competitive market for deposits, a decentralized banking system will oerdeposit contracts that maximize the expected utility of consumers. The deposit contractoered will solve the following problem:

maxfx;y;c1q;c2qg

cu c1q 1 cu c2q

subject to the following three feasibility constraints:

xq yq 6 1 q;q 6 cc1q 6 yq q;1 cc2q 6 Rxq:

The constraints are faced by the representative banks in t = 0, t = 1, and t = 2, respec-tively. The bank takes as given reserve requirements imposed by the CB to the bankingsystem, and invests in t = 0 an amount 1 q in the portfolio (x,y). Since, for given q(for each value of the reserve xed by the CB), the total amount of consumption provided

A B

CD

A B

CD

Complete Market Incomplete Market

Fig. 1. The interbank deposit market.

F. Castiglionesi / Journal of Banking & Finance 31 (2007) 81101 89in each period is constant, it is optimal to provide for consumption at t = 1 by holding theshort asset (represented now by the amount invested in y plus the reserve imposed by theCB), and to provide for consumption at t = 2 by holding the long asset.

The constraint in t = 1 can be explained as follows. Without the help of the CB (i.e.,with q = 0), commercial banks would provide an amount of liquidity in t = 1 equal tocc1(0). As long as the imposed reserve by the CB is less than cc1(0), commercial banks willinvest cc1(0) q in the liquid asset and the rest in the long-term asset. On the other hand,if the reserve is greater than cc1(0) the banks nd it optimal not to invest in y that is,y(q) = 0. This because commercial banks rationally expect the CB to let them use thereserve for liquidity need in t = 1. However, if cc1(0) 6 q less resources are available toinvest in the long asset, reducing consumption for late consumers. This represent the costof imposing reserve requirements. We indicate the solution to this problem as the alloca-tion d(q) that is characterized as follows:

yq 0 when qP cc10;

achieved by the banking system with the presence of the CB.Proposition 2. As long as the reserve is q 6 cR1ccR < 1, the decentralized allocationd(q) = {x(q), y(q), c1(q), c2(q)} is equivalent to the incentive efficient allocation c1(q) 6 c2(q).The allocation can be achieved even if the representative banks and the CB cannot observe

consumers type.

Proof. The representative banks problem can be rewritten as follows:

maxfc1qg

cuc1q 1 cu R1 cc1q1 c

subject to q 6 cc1(q). The solution satises the FOC:

u0 c1q u0c2qR l;where l is the Lagrangian multiplier of the constraint. If the CB xes a reserve q < cc1(0)then c1(q) = c1(0) and c2(q) = c2(0), and the constraint is slack, i.e., l = 0. The incentivecompatibility condition c2(q)P c1(q) is always satised since R > 1. When the CB imposesa reserve qP cc1(0) then representative banks are forced to overinvest in the short-termasset, reducing the resources allocated in the long-term asset. This changes consumptionallocation, and the incentive compatibility condition c2(q)P c1(q) now implies

R1 q1 c P

qc:

This condition is satised if and only if q 6 cR1ccR. h

Example 2. With utility function u() = log(c) we have c1(0) = 1 and c2(0) = R. Therefore,c1q yq q

c c10 when q < cc10

qc when qP cc10;

8>:

xq 1 cc10 when q < cc101 q when qP cc10;

c2q R1 cc10

1 c c20 when q < cc10R1 q1 c when qP cc10:

8>>>:

It is clear that, if the CB wants to aect the decentralized allocation decision made bythe commercial banks, it has to x reserve requirements greater than cc1(0). In this way theCB can raise consumption in t = 1 and reduce consumption in t = 2. A reserve require-ment less than cc1(0) would not change the asset and consumption allocation chosen byrepresentative banks. On the other hand, the CB cannot x a q too high otherwise incen-tive compatibility could be violated. Indeed, the incentive-ecient allocation can becc10 q when q < cc1090 F. Castiglionesi / Journal of Banking & Finance 31 (2007) 81101the allocation is

We have proved that the decentralized solution is incentive compatible, and it is possi-

F. Castiglionesi / Journal of Banking & Finance 31 (2007) 81101 91ble to show that the constraints faced by the representative banks in t = 1 and t = 2 arealways satised in the states S1 and S2. Consequently, the same decentralized solutiond(q) can be achieved by both interbank deposit market structures. We leave this easy taskto the reader.

Let us now deal with eciency, comparing the decentralized solution d(q) with the rst-best solution d*. The decentralized solution with the intervention of the CB delivers therst-best allocation when banks and depositors assign zero probability to state S.

Proposition 3. The decentralized economy, with the CB fixing a reserve requirement

qCB = y*, reaches the first-best allocation in states S1 and S2. The decentralized allocationd(q) coincides with the first-best allocation d*.

Proof. Recall that rst-best consumption allocation is c1 yc and c2 R1y

1c : In thedecentralized solution the contract oered by the banks depend on q and it is equal to

yq cc10 q when q < cc100 when qP cc10;

c1q yq q

cwhen q < cc10

qc

when qP cc10;

8>>>:

xq 1 cc10 when q < cc101 q when qP cc10;

c2q R1 cc10

1 c when q < cc10R1 q

when qP cc10:

8>>>:yq c q when q < c0 when qP c;

c1q 1 when q < cqc

when qP c;

8 c1, and c2 > c20 > c2(both for q < c ). This means that the decentralized solution does not reach the rst-bestallocation when the reserve requirement xed by the CB does not bite the investmentdecision made by the commercial banks in the liquid asset.1 c

allocation because banks and depositors sign incomplete deposit contracts, which would

c qCB R1 q R1 c 4 c :21 c 1 c 2

Then, with logarithmic utility function, we have the following condition:

0 < qCB c p3e4

y(q) = cc1(0). When q = y*, then also the investment in thelong-term asset is equal to the rst-best since x(q) = 1 q = 1 y* = x*. h

The intervention of the CB allows the decentralized banking system to reach rst-best

92 F. Castiglionesi / Journal of Banking & Finance 31 (2007) 81101(2000), the order of liquidation of the assets is as follows: rst, the short asset, then the

cross-holding deposits, and, nally, the long asset.5 In order to ensure that this peckingorder is observed, we assume that r is small enough, so that for each q in the relevantrange we have Rr >

c2qc1q > 1.

5.1. Financial fragility without the CB

Allen and Gale (2000) have analyzed the case in which q = 0 and the banks are forced

F. Castiglionesi / Journal of Banking & Finance 31 (2007) 81101 93to liquidate the long-term asset when they do not have enough liquidity. The analysis canbe summarized as follows.

Let qi be the fraction of the value of deposits in the representative bank in region i att = 1, i.e., each depositor with a claim worth $1 is actually able to withdraw 0 6 qi 6 1 dol-lars. Since the proceeds of the deposits are assumed to be split proportionally, qi must bedetermined simultaneously for all regions by the two sides of the market. The fraction qi isdetermined in equilibrium equating the total demand with the total supply of liquidity. Aslong as the total supply is greater (or equal to) the total demand of liquidity then qi = 1. Iftotal demand is greater than total supply of liquidity, we have qi < 1, and the region isbankrupt in t = 1. In the latter case, qi represents the liquidation value of the banks assets.

Consider, for example, region A. Depositors of region A have claims worth 1 towardsthe bank, while the bank in region D has claims worth z. Let bcA1 be the amount that con-sumers of region A desire to withdraw from the bank at time 1, and bzA the amount thatbank D desires to withdraw. Thus, bcA1 bzA is the total demand for liquidity. The total sup-ply is given by region As asset (short and long) plus its holding in region B that is,y + rx + zqB. Therefore, if bcA1 bzA 6 y rx zqB then bank of region A is able to meetthe demand for liquidity, and qA = 1. Otherwise qA yrxzqBbcA

1bzA , and the upper bound of

the liquidation value of region A under bankruptcy is given by qA 6 qA yrxzbcA1bzA , namely

when region B is not bankrupt (i.e., qB = 1). Notice that, as observed above, (qA,qB,qC,qD)have to be determined simultaneously. Furthermore, the demands bci1 and bzi are also deter-mined endogenously.

When the state of the world is either S1 or S2 then bci1 bzi y in each region, so thatqi = 1 for all i. However, when S occurs the global demand for liquidity exceeds the totalamount invested in the short-term asset. This implies that some quantity of the long-termasset has to be liquidated.6 Notice, however, that the demand for liquidity in t = 1 mayincrease, since late consumers may opt for early withdrawal if they fear that early liquida-tion of the long-term asset will reduce the payo in t = 2 The maximum amount of thelong-term asset that can be liquidated at t = 1 without triggering a bank run by late con-sumers is x 1 x c10R . In fact, the representative bank has to give at least c1(0) to lateconsumers in t = 2 (otherwise, they would withdraw in t = 1). Therefore, a bank with xearly consumers keeps at least 1 x c10R units of the long asset. The maximum amount

5 Therefore at t = 1, the representative bank may be in any one of the following situations: solvent (if it can meetthe demand with the short asset plus the deposits held in the other region); insolvent (if it has to liquidate some ofthe long asset); or bankrupt (if it cannot meet the demand after liquidating all assets).6 Deposits from the interbank market are of no use in the unexpected state. In fact, once region A withdraws its

interbank deposits from region B to pay its early consumers, region B is forced to withdraw its amount ofinterbank deposit from region C and so on. Since no representative bank wants to liquidate the long asset if it canbe avoided, all banks simultaneously withdraw their deposits in banks in the other regions in t = 1. These mutual

withdrawals oset each other, so each region is forced to be self-sucient.

of consumption that can be obtained by liquidating the long asset without causing a run is

then bx r x 1 x c10Rh i

: In state S, the representative bank in region A is able to

94 F. Castiglionesi / Journal of Banking & Finance 31 (2007) 81101We have seen in the previous section that an unexpected aggregate liquidity shortage int = 1 can cause the collapse of the entire economic system. We show in this section that theCB can prevent nancial fragility, allowing the economy to reach the rst-best allocationalso with an aggregate shortage of liquidity. The CB, by xing reserve requirements on theamount of deposits collected by the commercial banks, can ensure that the inecient liq-uidation of the long-term asset is avoided. Without the costly liquidation of the long asset,the possibility of bankruptcy and, consequently, of contagion is completely avoided. TheCB ensures that a possible nancial crisis in one region does not spill over other regions.

When the unexpected state S appears in t = 1, the representative bank in region A willneed ec1(q) units of consumption by liquidating the long asset. Without other forms ofintervention, region A can get extra-liquidity by liquidating the long-term asset only if

7 Other interventions, such as suspension of convertibility or liquidity provision by means of ex-post loansmeet demands from early consumers, without help from other regions, if, and only if,ec1(0) 6 b(c + e). If this condition is violated, then region A is bankrupt and the possibilityof contagion arises.

Proposition 4. Consider the incomplete market structure of the interbank deposits market

and perturb it with a zero-probability state S. Suppose that each representative bank choosesfirst-best investment portfolio and offers first-best consumption allocation through deposit

contracts. If the following two conditions, regarding region As bankruptcy and spillover to

region D respectively, are satisfied: (a) ec1(0) > b(c + e); (b) z1 qA > bc, then, in anycontinuation equilibrium, the representative banks in all regions are bankrupt at date 1 in

state S.

Proof. See Allen and Gale (2000). h

Contagion occurs under more restrictive conditions if the structure of the interbankdeposit market is complete. In this case the cross-holding is z = (xH c)/2, which is smal-ler than the amount requested in incomplete markets [z = (xH c)].

A possible remedy to avoid contagion could be that each region in t = 0 commits itself(directly with the other regions, or through the CB) to provide the liquidity according tothe state of nature that occurs. This mechanism lacks of ex-post credibility. In fact, instates S1 and S2, it is not optimal for the regions to deviate from the commitment (other-wise they would have to liquidate their long-term asset, which is more costly than liquidat-ing the cross-holding deposits). However, it is optimal to deviate in state S. In this case, thethree regions that are not hit by the shock have an incentive to deviate in order not to liq-uidate the long-term asset (they would have to liquidate some of it in order to avoid Asbankruptcy).7 Consequently, the necessary liquidity in order to be able to intervene also instate S has to be guaranteed in t = 0.

5.2. Financial fragility with the Central Bankmarket, do not work in preventing region As bankruptcy and contagion (see Allen and Gale, 2000).

ec1(q) 6 b(c + e). If this is the case, late consumers in region A are worse o because thepremature liquidation of the long asset prevents to pay the promised consumption c2(q)in t = 2. If ec1(q) > b(c + e), contagion could arise according to Proposition 4.

c4

F. Castiglionesi / Journal of Banking & Finance 31 (2007) 81101 95so that by shifting reserves across banks appropriately, the CB can prevent liquidation ofthe long-term asset in each region. When q = y*, also the investment in the long-term assetis equal to the rst-best since x(q) = 1 q = 1 y* = x*. h

This result implies that the CB intervention avoids either the worsening of the conditionof the late consumers (in case region A is insolvent) or the possibility of contagion (in caseregion A is bankrupt). The amount c1 given to the early consumers is less than what theywould expect, namely c1q yc . This is equivalent to say that the CB declares the insol-vency of the banking system in t = 1 and pays early consumers c1. In case of a liquiditycrisis, the amount of reserves collected in t = 0 are sucient to reach eciency and thebanking system does not need extra-liquidity, which could have been obtained by liquidat-ing the long asset or by injecting at money into the system. Using all the reserves to payearly consumers, the CB reach rst-best consumption allocation when an aggregate short-age of liquidity occurs (as analyzed in the planner problem).8

8 Note that the declaration of insolvency does not cause any turmoil. In fact, since the CBs interventiondelivers rst-best consumption, late consumers do not run the banks because the incentive-ecient constraintholds, and early consumers are satised with the amount they withdraw because they perfectly realize what is inHowever, the problem can be avoided altogether if the CB can reduce the value of thedeposits of early withdrawal. In this way, the rst-best allocation can be reached. This isdone as follows. Suppose that q is suciently high, so that the deposit contract oered bythe banks gives the right to withdraw an amount qc in t = 1, and the banks do not invest inthe short-term asset (this will actually be the case when q is determined in order to achievethe rst-best). When state S occurs, the total demand for liquidity exceeds the quantity ofreserves since 4c e qc > 4q. In this case, the CB simply states that the amount that canbe withdrawn at t = 1 is equal to qce4

, and shifts part of the reserves to the region hit by theliquidity shock. We have the following result.

Proposition 5. If qCB = y*, the decentralized allocation d(q) coincides with the first-bestallocation d*. In the continuation equilibrium in state S in t = 1, the CB allows region A toface the liquidity shock avoiding the premature liquidation of the long-term asset, and

consequently the contagion outcome.

Proof. If q = y* > cc1(0) the contract offered by the banks promises a payment c1q qcin t = 1. The rst-best outcome is achieved when states S1 or S2 realize. It is thereforeenough to check that the rst-best outcome is achieved also in state S. In that case therst-best consumption allocation is c1 yce4 and c2

R1y1ce4

: Now suppose that when S

occurs the bank reduces the value of promised payment at time 1 to y

ce4. In this case the

total demand for deposit withdrawal by early consumers is equal to the total quantityof reserves

4c e ye 4y;their best interest once the unexpected shock appears.

When there is a liquidity shock, able to determine an aggregate liquidity shortage,then the CB need to use two instruments: reserve requirements, and the declarationof the insolvency of the banking system. The rst tool assures the ecient level of

96 F. Castiglionesi / Journal of Banking & Finance 31 (2007) 81101investment in the liquid asset y, which would not be chosen by commercial banks; thesecond tool assures that in state S the level of consumption in t = 1 can be properlyreduced.9

However, if we accept the idea that crises have to be dealt with a fail safe principle(namely, to minimize damages when a crisis occur), then reserve requirements xed bythe CB guarantee the right amount of liquidity in order to avoid contagion eciently.Otherwise, if nancial crises have to be avoided no matter what, which in the presentmodel it means to allow early consumers to withdraw the promised c1(q), then no levelof reserves is sucient to avoid the risk of contagion. This because commercial banks oera level of consumption in t = 1 that is not feasible in the unexpected state.

Declaring the insolvency of the banking system in t = 1 can be considered an applica-tion of the fail safe principle, and xing qCB is then necessary to avoid contagion in anecient way. In order to clarify this statement, assume that the CB xes reserve require-ments 0 6 q < cc1(0), then the consumption promised by the deposit contract in t = 1 isc1(0). Assume also that the CB has the possibility to declare the insolvency of the system.This kind of intervention, not only does not reach rst-best in the expected states of nature(since c1(0) < c1), but also in the unexpected state. In fact, to avoid the crisis and the risk ofcontagion the CB will allow to withdraw in t = 1 only

cc10c e

4

< c10:

This amount is less than the ecient amount c1, since

cc10c e

4

< c1 cc1c e

4

< c10 < c1:

Without xing the optimal qCB, any ex-post intervention that prevents contagion wouldbe inecient and somehow arbitrary. The CB could even decide to liquidate part of thelong asset to penalize also late consumers.

Finally, it is important to notice that the role of the CB in preventing nancial crisisand contagion does not depend on the structure of the interbank deposit market. Evenwith a complete market structure, when there is an aggregate excess demand for liquidity,region A has to be self-sucient. Consequently, the CB may improve not only upon theincomplete interbank deposit market structure (which is quite intuitive), but also uponthe complete one. The reason for this result is that commercial banks and depositors signdeposit contracts that are not contingent on all the states of nature. The CB becomes away to complete the deposit contracts, independently of the structure of the interbankdeposit market.

9 Declaration of insolvency is strictly related to the notion of incentive-eciency. Indeed, if constrained-eciency is taken into account, then declaration of insolvency is not needed in order to avoid contagion (see

Section 6).

6. Constrained eciency

F. Castiglionesi / Journal of Banking & Finance 31 (2007) 81101 97In Section 3, the rst-best allocation has been characterized by a planners problem,where the planner can condition the consumption allocation on the liquidity shock.Accordingly, the optimal level of consumption in the expected states (c1,c2) is dierentwith respect the optimal level of consumption when the unexpected liquidity shock strikes(c1;c2). The planners investment in the short asset turns out to be higher than in thedecentralized equilibrium (y* > y = cc1(0)) to insure against the liquidity shock. To repli-cate the rst-best allocation, it is necessary that the CB induces commercial banks toincrease their investment in the short asset (which is achieved by mean of reserve require-ments), and that, when the shock appears, commercial banks must condition their con-sumption payments on the liquidity shock (which is obtained by declaring theinsolvency of the banking system). The last intervention, instead of an application ofthe fail safe principle, could be interpreted as suspension of convertibility. Assumingaway non-contingent contracts could be considered a way to eliminate the problem ofcontagion.10

In this section, we restrict also the planner to use non-contingent (demand deposits)contracts. In this way we consider the notion of constrained-eciency, rather than incen-tive-eciency, as the appropriate benchmark. The planner in t = 1 has to assure to earlyconsumers the amount c1 no matter what is the liquidity demand (i.e., c1 c1). In order toavoid inecient waste of consumption, deposits not withdrawn in t = 1 are reinvested inthe storage liquid asset y and given to late consumers in t = 2. The level of consumption int = 2 then depends on how many early consumers withdraw in t = 1. However, at least theamount c2 has to be guaranteed to late consumers. The planners problem is

maxfx;y;c1;c2g

p1 p2cuc1 1 cuc2 p3 ce4uc1 1 c e

4uc2

h isubject to: x y 6 1;

cc1 6 y;

c e4

c1 6 y;

1 cc2 6 Rx y cc1;1 c e

4

c2 6 Rx y c e

4

c1;

c1 6 c2 6 c2:The optimal amount invested in the short asset is clearly yC c e

4c1, since the non-

contingent contract now forces the planner to hold liquidity in excess of what is requiredto pay the promised consumption to the early consumers in normal times (i.e., cc1).Accordingly, the optimal long-term investment is xC = 1 yC. The excess of liquidity isused to provide the same level of consumption to the additional early consumers if theaggregate liquidity shock strikes. If the shock does not realize, late consumers getc2 11c RxC e4 c1. Otherwise, the optimal level of consumption in t = 2 is c2 Rx

C

1ce4.

We indicate the constrained rst-best allocation as dC fyC; xC; c1; c2;c2g. It is easy toverify that for e small enough the allocation dC satises incentive-eciency, that is c1 6 c2.10 For a general analysis on this issue, see Allen and Gale (2004).

Indeed, a liquidity shock too high would reduce too much the long-term investment xC

and, consequently, the consumption for late consumers. Finally, note that for e = 0 wehave c2 c2 and the constraint c2 6 c2 is satised. Moreover, given that dxCde c14 , we havedc2de c11R41c < 0 and dc2de R1c141ce42. Then, for c1 6 1 it is

dc2de P 0 and the constraint c2 6 c2 is

satised for all parameters values. Otherwise, for c1 > 1, it isdc2de < 0 and then we need the

further condition dc2de >dc2de , which implies

e4< 1 c R1c11cc11R

h i1=2.

As already seen in Section 4, the imposition of reserve requirements in the decentralizedeconomy does not cause commercial banks to hold the excess of liquidity since they do notanticipate the liquidity shock. Accordingly, they increase their promises to early consum-ers in line with their increased holdings of the short asset, so that they still have inadequateaggregate supplies of liquidity when the unanticipated liquidity shock hits. With contin-gent contract, the declaration of insolvency introduced the right amount of contingencyto reach eciency and avoid contagion. Then with non-contingent contract, a dierentreserve policy is required to protect against contagion. We dene reserves as the excessof the short asset over the normal commitments in t = 1, that is q y cc1(0), wherec1(0) is the promised consumption to early consumers in the decentralized economy. Thencommercial banks provide liquidity according to the normal commitments in t = 1, andonly consumption in t = 2 depends on the reserve xed by the CB. In order to not wasteconsumption, deposits not withdrawn in t = 1 are rolled over for last period consumption.The decentralized problem is

maxfx;y;c10;c2qg

cuc10 1 cuc2q

subject to: xq 6 1 y q;0 6 y cc10 6 q;1 cc2q 6 Rxq q:

Independently of the level of reserve requirements xed by the CB, commercial banks in-vest an amount cc1(0) in the short asset y, which is less then the optimal amountyC c e

4

c1 since c1(0) < c1. Indeed, like in the incentive-eciency analysis, the con-

sumption c1(0) does not take into account the possibility of the liquidity shock, whilethe consumption c1 is computed considering such event. This implies that optimal con-sumption for early consumers should be higher than what characterized by the decentral-ized economy.

In order to implement the constrained rst-best investment in the short liquid asset, theCB need to impose a reserve such that y + q = yC. This implies that q c e

4

c1 cc10.

Accordingly, also the investment in the long asset is optimal since x = 1 y q =1 yC = xC, so the optimal investment portfolio is achieved.

Let us analyze what happen in the expected states characterized by absence of liquidityshock. The promised consumption c1(0) to the early consumers is obviously met. In thiscase, in fact, the total demand from early consumers is cc1(0) and the supply of liquidity

is c e4

c1. Late consumers will get a level of consumption c2q Rx

Ce4c1cc1c101c . Note

that early consumers withdraw less than what is optimal, while late consumers withdrawmore than the optimal amount (that is, c2(q) > c2).

If the unexpected state occurs, early withdrawals are c e4

c10 and the total liquidity

98 F. Castiglionesi / Journal of Banking & Finance 31 (2007) 81101is y q c e4c1, which is enough to assure the committed level of consumption to all

Since the incentive-ecient allocation analyzed in Section 3 guarantees an higher

F. Castiglionesi / Journal of Banking & Finance 31 (2007) 81101 99expected utility than the constrained-ecient allocation analyzed in this section, the CBshould x reserve requirements letting banks to choose investment and consumption allo-cations and, in case of an unexpected shock, declare the insolvency of the banking system.However, even if it is not possible to have such desirable ex-post exibility, xing reserverequirements is still an eective instrument to avoid contagion.

7. Conclusion

The contribution of this paper is mainly normative, namely to analyze the role of a CBin preventing nancial contagion. The CB nds its rationale on the bad distribution ofliquidity among dierent regions and on its dierent a-priori on the possible occurrenceof an aggregate shortage of liquidity. The CB, imposing reserve requirements on theamount of deposits collected by commercial banks, is able to avoid contagion and to reachrst-best allocation when contingent contracts are considered. If the analysis is restrictedto non-contingent contracts, the CB still avoids contagion without, however, reachingrst-best consumption allocation.

Both the Diamond and Dybvig (1983) model and the one analyzed in this paper studythe optimal intervention of the CB in the presence of aggregate uncertainty. The distinc-tion between the two models is that, while in the former the intervention of the CB takesthe form of deposit insurance that is enough to avoid the bad equilibrium (i.e., the bankruns), in the latter the intervention of the CB needs to be more explicit through the impo-sition of reserve requirements.

The existence of contagion in Allen and Gale (2000) is related to the incomplete struc-ture of the interbank deposit market. We show that contagion is not rooted in theinterbank deposit market, but in the impossibility for agents to sign contracts contingentearly consumers (again because c1(0) < c1). Late consumers would get the amount

c2q RxC ce4 c1c10

1ce4. The banking system, with the intervention of the CB, avoids con-

tagion since in t = 1 there is enough liquidity to face also the unexpected early consumers.However, also in the unexpected state, consumption of early consumers is ineciently lowand late consumers consumption is higher than the level of constrained eciency (that is,c2q > c2).

With the more restricted notion of constrained-eciency, reserve requirements guaran-tee the sucient liquidity to avoid contagion and the CB does not need to declare theinsolvency of the banking system (reducing the promised consumption in t = 1). However,the consumption levels dictated by constrained-eciency cannot be implemented. The rea-son is that, with non-contingent contract, reserve requirements have to be xed on top ofthe normal commitment in t = 1, which implies that commercial banks do not internalizethe cost of the reserve and do not raise the inecient (low) amount of consumption prom-ised to the early consumers. This makes it impossible to achieve constrained-eciency,with early consumers penalized in favor of late consumers, but it still allows to invest inthe ecient portfolio (yC,xC) and to avoid crisis and contagion. Note that, even if theCB cannot deliver the ecient consumption levels, no turmoil will occur since early con-sumers withdraw their promised amount c1(0) and incentive compatibility constraint forlate consumers always holds (since c10 < c1 6 c2 6 c2 < c2q).on unforeseen contingencies. Consequently, policy proposals that aim to improve the

100 F. Castiglionesi / Journal of Banking & Finance 31 (2007) 81101working of the nancial markets turn out to be necessary, but not sucient, measures inorder to deal successfully with nancial contagion. The presence of unforeseen contingen-cies is a form of market incompleteness. The CB then can be seen as a way to complete themarkets.

The present work nds a rationale for the presence of reserve requirements that is dif-ferent from previous contributions. First, the CB intervention in our framework does notrely on creation of liquidity. In order to prevent nancial contagion, the CB optimally xesthe liquidity that the banking system needs in the intermediate period. This implies thatholding reserves per se could be crucial, and not only a tool for stabilizing monetary pol-icy. Second, in our model banks do not face asymmetric information problems, then reg-ulatory intervention by means of reserve requirements does not derive from the fact thatbanks activity is characterized by costly state verication. Third, the CB intervention doesnot put any burden on taxpayer for bail out the distressed region.

The presence of reserve requirements has important implication for the welfare proper-ties of the model. Using all the reserve requirements in case of an aggregate liquidity shockimplies that insolvency (or bankruptcy) involves no ex-post ineciency. From a theoret-ical point of view, also the existence of markets on which assets can be liquidated couldensure ex-post eciency (resale prices would transfer value to the buyer but would notconstitute a deadweight loss). In our model, reserve requirements can be seen as a substi-tute for the missing markets on which is possible to liquidate the long asset. Without thepresence of these markets, eciency is reached by means of the imposition of reserverequirements. The implication is that in a well developed market economy, where marketsfor asset liquidation exist and are well functioning, the presence of reserve requirementsbecomes less crucial. Our model predicts that the observed reduction of reserve require-ments in well developed economies should not expose them to contagion. On the otherhand, in economies where nancial market are less developed, reducing reserve require-ments would expose their systems to systemic risk.

From an historical point of view, nancial contagion cases have occurred mainly in theUS in the National Banking System period (18631913), and after the institution ofthe Federal Reserve, which imposed reserve requirements, the frequency of nancial con-tagion is steadily decreased, as the model predicts. In the last decade, contagion phenom-ena aected mostly developing countries: South American countries after the Mexicoscrisis in 1994, or the SouthEast Asian countries after the crisis of Thailand in 1998. Thiscontagion eects usually followed processes of nancial liberalization, which impliedreduction on reserve requirements. If capital markets do not allow for liquidation oflong-term assets, then the model predicts that these economies are more exposed tocontagion.

Some caution is needed in interpreting our results. In particular, a very important con-cern in the debate on the opportunity to bail out distressed banks is the moral hazardissue. Banks that know that there is the possibility to be bailed out by a CB in case of theirbankruptcy, have an incentive not to make the best eort in monitoring the investmentopportunities or can invest their resources in risky assets. In this study, the problem isavoided altogether since the representative banks do not consider the possibility of anaggregate liquidity shortage. This implies that banks do not expect to be bankrupt, andconsequently to be bailed out by the CB. This does not give any room for moral hazardproblem. A future research topic is to investigate the CB best policy given the trade-o

between reducing contagion and avoiding moral hazard problem.

Acknowledgements

This paper is a slightly revised version of the rst chapter of my PhD dissertation atUniversidad Carlos III de Madrid. A special thank goes to Sandro Brusco who providedinvaluable feedback on each version of the paper. I am also indebted to Fabio Feriozzi,Charles Goodharth, Guido Lorenzoni, Manuel Santos, an anonymous referee, seminarparticipants at Universidad Carlos III, Universitat Auto`noma de Barcelona, XXVIIISimposio de Analisis Economico, and XII Foro de Finanzas for useful comments. Theusual disclaimer applies.

F. Castiglionesi / Journal of Banking & Finance 31 (2007) 81101 101References

Allen, F., Gale, D., 2000. Financial contagion. Journal of Political Economy 108, 133.Allen, F., Gale, D., 2004. Financial intermediaries and markets. Econometrica 72, 10231061.Bhattacharya, S., Gale, D., 1987. Preference shocks, liquidity and central bank policy. In: Barnett, W.A.,

Singleton, K.J. (Eds.), New Approaches to Monetary Economy. Cambridge University Press.Brunner, K., Meltzer, A.H., 1990. Money supply. In: Friedman, B.M., Hahn, F.H. (Eds.), Handbook of

Monetary Economy, 1. North-Holland, Amsterdam, pp. 357396.Chen, Y., 1999. Banking panics: The role of the rst-come, rst-served rule and information externalities. Journal

of Political Economy 107, 946968.Dasgupta, A., 2004. Financial contagion through capital connections: A model of the origin and spread of bank

panics. Journal of the European Economic Association 2, 10481084.Diamond, D., 1984. Financial intermediation and delegated monitoring. Review of Economic Studies 51, 393

414.Diamond, D., Dybvig, P., 1983. Bunk runs, deposit insurance and liquidity. Journal of Political Economy 91,

401419.Di Giorgio, G., 1999. Financial development and reserves requirements. Journal of Banking and Finance 23,

10311041.Freixas, X., Parigi, B., Rochet, J.C., 2000. Systemic risk, interbank relations and liquidity provision by the central

bank. Journal of Money, Credit and Banking 32, 611638.Jacklin, C., Bhattacharya, S., 1988. Distinguishing panics and information-based bunk runs: welfare and policy

implication. Journal of Political Economy 96, 568592.Lorenzoni, G., 2001. Interest Rate Stabilization and Monetary Control: A Reconciliation, Princeton University.Rochet, J.C., Tirole, J., 1996. Interbank lending and systemic risk. Journal of Money, Credit and Banking 28,

733762.Sargent, T.J., Wallace, N., 1982. The real bills doctrine vs. the quantity theory: A reconsideration. Journal of

Political Economy 90, 12121236.Williamson, S., 1986. Costly monitoring, nancial intermediation and equilibrium credit rationing. Journal of

Monetary Economics 18, 159179.

Financial contagion and the role of the central bankIntroductionThe modelOptimal risk-sharingDecentralized economyFinancial fragilityFinancial fragility without the CBFinancial fragility with the Central Bank

Constrained efficiencyConclusionAcknowledgementsReferences