Working Paper Series. Government guarantees and financial stability. No 2032 / February 2017

Transcription

1 Working Paper Series Franklin Allen, Elena Carletti, Itay Goldstein, Agnese Leonello Government garantees and financial stability No 2032 / Febrary 2017 Disclaimer: This paper shold not be reported as representing the views of the Eropean Central Bank (ECB). The views expressed are those of the athors and do not necessarily reflect those of the ECB.

2 Abstract Banks are intrinsically fragile becase of their role as liqidity providers. This reslts in nderprovision of liqidity. We analyze the effect of government garantees on the interconnection between banks liqidity creation and likelihood of rns in a model of global games, where banks and depositors behavior are endogenos and affected by the amont and form of garantee. The main insight of or analysis is that garantees are welfare improving becase they indce banks to improve liqidity provision althogh in a way that sometimes increases the likelihood of rns or creates distortions in banks behavior. Keywords: panic rns, fndamental rns, government garantees, bank moral hazard JEL classifications: G21, G28 ECB Working Paper 2032, Febrary

3 Non-Technical Smmary Government garantees to financial instittions are common all over the world. They come in different forms, ranging from standard deposit insrance schemes to the promise of an ex-post bailot in the case of a bank s failre. The recent financial crisis was characterized by a massive se of government garantees, which lead to a renewed interest and debate abot government intervention in the financial sector. While pblic intervention proved to be effective in restoring confidence and preserving financial stability dring the crisis, it also had significant negative conseqences in terms of sovereigns fiscal positions, and, in trn, banks and firms health and cost of fnding. In the crrent academic and policy debate, government garantees are considered to be an effective tool to prevent the occrrence of panic crises and mitigate their negative effects. However, their provision can distort banks risk-taking incentives and indce them to take excessive risk. Becase of this moral hazard problem, the provision of garantees can lead to the perverse otcome of increasing the overall instability in the banking sector (Demirgc-Knt and Detraigiache, 1998) and magnify the costs for the government providing them. Are garantees effective in preventing banking crises? What are the implications they have for banks role as Iiqidity providers and their risk-taking decisions? How do garantees affect the interaction between liqidity provision and risk-taking? This paper tackles these qestions and provides new insights abot the desirability of government garantees, their effectiveness in preventing rns, as well as the type and severity of the associated moral hazard problem. In the paper, we develop a theoretical framework where banks raise fnds from risk-averse depositors in the form of demandable deposit contracts and, ths provide liqidity and risk-sharing to them by allowing depositors to access risky bt profitable long-term investment opportnities, while still being able to obtain liqidity when needed. As a reslt, in or framework, banks are exposed to two sorces of risk. One the one hand, they face insolvency risk, in that they can fail as a conseqence of a bad realization of bank s investment projects (fndamental crises). On the other hand, they are exposed to liqidity risk, as they can fail as a conseqence of large prematre withdrawals by depositors, driven by the fear that others wold withdraw and, ths deplete banks resorces (panic crises). In or framework, the probability of both crises is endogenos and depends on the banks risk choice, as well as on the (type and size of) government garantees. This allows analysing how the bank s risk choice and the garantees interact with the probability of fndamental and panic crises, as well as, with each other. In or model, the effect of the garantees on the probability of a crisis is twofold. On the one hand, garantees have a positive direct effect, since, by increasing depositors repayments, they redce their incentive to withdraw early and ths, banks exposre to liqidity risk. On the other hand, they affect banks risk-taking decisions and, ths have a negative indirect effect on the probability of a banking crisis. To flly exploit these effects, we consider two garantee schemes. The first one is only meant to prevent the occrrence of panic-driven crises and so depositors are garanteed to receive at least a ECB Working Paper 2032, Febrary

4 minimm repayment if the project of the bank is solvent irrespective of what other depositors do. The second one, which resembles a standard deposit insrance scheme, garantees depositors to receive a minimm repayment whenever the bank is not able to repay them the promised repayment, ths affecting both the probability of fndamental- and panic-driven crises. The main insight of or analysis is that garantees are welfare improving becase they indce banks to improve liqidity provision althogh in a way that sometimes increases the likelihood of rns or creates distortions in banks' behavior. This reslt hinges on the fact that some risk taking by banks is desirable, as it is an inherent featre of the liqidity transformation fnction that banks perform in the economy, and it might be sppressed by the concern for financial fragility. By relaxing these concerns to some extent, garantees allow banks to provide greater liqidity transformation and so are desirable. This effect, which is captred by or model thanks to the endogenization of banks' and depositors' behavior, is missing from the crrent debate and academic literatre and generates a more nanced assessment of government garantees. Another interesting reslt of or analysis regards the direction of the distortions in banks behavior (i.e., moral hazard) indced by the garantees. As typical in models where private agents, enjoy the benefits of a pblic form of insrance withot internalizing its costs, also in or model the introdction of garantees creates a wedge between the deposit rate chosen by banks and the one that the government wold like to choose. However, nlike conventional wisdom, we show that government garantees do not always lead to more risk-taking by banks, and, in trn, to an excessively high exposre to rns. Sometimes their introdction leads to the exactly opposite effect: banks choose to be less exposed to rns than what it wold be socially optimal. The important detail is whether the government ends p paying depositors more in the case the bank ends p failing in the longer term for fndamental reasons or in case there is a rn and the bank faces a shortage of liqidity. If the former holds, then the cost of a rn from the point of view of banks is higher than from the point of the government and the banks choose to limit their exposre to rn. The reslt that banks choose to be less exposed to rns than it wold be desirable resembles the idea of prompt corrective actions. Liqidating banks early rather than letting them operate longer and intervene when banks' resorces are completely deployed may be desirable when it allows to minimize the costs associated with pblic intervention. ECB Working Paper 2032, Febrary

5 1 Introdction Government garantees to financial instittions are common all over the world. They come in different forms, sch as deposit insrance provided to depositors who pt their money in commercial banks, or implicit garantees for a bailot provided ex post pon the bank s failre. The recent financial crisis has led to renewed interest and debate abot the role of government garantees and their desirability. On the one hand, government garantees are thoght to have a positive role in preventing panic among investors, and hence help stabilize the financial system. On the other hand, the common belief is that they might create adverse incentives for banks to engage in excessive risk taking, which might lead to an increase in financial fragility. 1 In light of this trade-off, evalating the overall effects of government garantees to banks is challenging, as it reqires a framework in which the behavior of banks and their investors interacts with the amont and form of garantees. Sch a model is known to be notoriosly rich and hard to solve. It needs to endogenize the probability of rns and how it is affected by banks risk choices and government garantees. It also needs to endogenize banks risk choices and how they vary with the garantee, taking into accont investors expected rn behavior. We make technical progress in this paper by ptting all these ingredients together in a tractable framework. Or framework generates some srprising reslts on the effects of government garantees. Most notably, we show that the increase in bank risk taking following the introdction of government garantees may sometimes be a desirable conseqence of their introdction. Hence, the downside of garantees that is often broght p in the policy debate might be exaggerated. To the extent that banks perform a welfare-enhancing role with their liqidity transformation activities and that sch liqidity transformation is inherently risky, they might provide too little liqidity in the face of rn risk. As they alleviate rn risk, government garantees may in trn indce banks to increase the scope of these activities, ths improving welfare. This effect, which is captred by or model thanks to the endogenization of banks and depositors behavior, is missing from the crrent debate and academic literatre and generates a more nanced assessment of government garantees. Or starting point is the seminal Diamond and Dybvig (1983) economy, which has served researchers for years in stdying rns and financial fragility. In this framework, banks offer deposit contracts to investors, who might face early liqidity needs, and by that provide liqidity transformation enabling risk sharing 1 See, e.g., Calomiris (1990), Demirgc-Knt and Detragiache (1998), Gropp, Grendl and Gettler (2014), and Acharya and Mora (2015). ECB Working Paper 2032, Febrary

6 among depositors. While banks may improve investors welfare de to the risk sharing they provide, the deposit contracts also expose banks to the risk of a bank rn, where many depositors panic and withdraw early ot of the self-flfilling belief that other depositors will do so and the bank will fail. In the original framework, Diamond and Dybvig (1983) propose a deposit insrance scheme that eliminates rns altogether and restores fll effi ciency. In their model, banking crises happen only de to a coordination failre. Then, by ensring that depositors will receive the promised payment independently of the other depositors withdrawal decisions, deposit insrance prevents the bank-rn eqilibrim withot entailing any disbrsement for the government and so the first best allocation is achieved. The literatre that followed Diamond and Dybvig (1983) recognized that the effects of deposit insrance are more complicated and that there might be tradeoffs involved with increasing the amont of coverage. When rns are not prely driven by panics bt sometimes occr as a reslt of deteriorating fndamentals of the banks assets (see evidence in Gorton (1988), Calomiris and Gorton (1991) and Calomiris and Mason (2003)), deposit insrance may not flly prevent rns. This implies that actal costs of paying for failed banks will be incrred, and these might be increased by the fact that banks elevate their risk taking when they know they are insred. Despite enriching Diamond and Dybvig analysis with more realistic featres, this literatre cannot flly evalate the effects of government garantees as, by and large, it does not endogenize the probability of rns and, ths, cannot captre how this is affected by the risk taking decisions of banks. 2 This is what we do in this paper. To condct or analysis, we bild on the model developed in Goldstein and Pazner (2005), in which depositors withdrawal decisions are niqely determined sing the global-game methodology, and so the probability of a rn and how it is affected by the banking contract and by government policy can be determined. Goldstein and Pazner (2005) stdy the interaction between the demand deposit contract and the probability of a rn. In their model the rn probability depends on the banking contract, and the bank decides on the banking contract taking into accont its effect on the probability of a rn. We add a government to this model to stdy how the government garantee policy interacts with the banking contract and the probability of a rn. In or model, there are two periods. Banks raise fnds from risk-averse consmers in the form of deposits and invest them in risky projects whose retrn depends on the fndamentals of the economy. Depositors derive tility from consming both a private and a pblic good. At the interim date, each depositor learns 2 We provide a literatre review in the next section. ECB Working Paper 2032, Febrary

7 whether he needs to consme early or not and receives an imperfect signal regarding the fndamentals of the bank. Impatient depositors withdraw at that point and patient ones decide when to withdraw based on the information received. In deciding whether to rn or not, depositors compare the payoff they wold get from going to the bank prematrely and waiting ntil matrity. These payoffs depend on the fndamentals and the expectation abot the proportion of depositors rnning. In this setting, the eqilibrim otcome is that rns occr when the fndamentals are below a niqe threshold. Within the range where they occr, they can be classified into panic-based rns or fndamentalbased rns. The former type of rn is one that is generated by the self-flfilling belief of depositors that other depositors will rn. The latter type of rn happens at the lower part of the rn region, where the signal on the fndamentals is low enogh to make rnning a dominant strategy for depositors. Overall, the probability of the occrrence of a rn (and of both types of rns) is niqely determined and depends on the deposit contract offered to depositors. As in Diamond and Dybvig (1983), there is perfect competition with free entry in the banking sector, and so banks offer a contract that maximizes depositors expected tility. Unlike in Diamond and Dybvig (1983), however, they recognize the implications that the contract has for the possibility of a rn and take them into accont when deciding on the contract. As in Goldstein and Pazner (2005), we first show that the decentralized soltion, i.e., withot government intervention, is ineffi cient. There are two sorces of ineffi ciency. First, in eqilibrim, banks choose to offer deposit contracts that lead to ineffi cient fndamental-based and panic-based rns. While they internalize the cost of the rns, the benefit from risk sharing is large enogh to lead banks to offer contracts that entail some ineffi cient rns. Second, since they internalize the probability of ineffi cient rns, banks redce the amont of liqidity they offer to depositors demanding early withdrawal. Hence, in eqilibrim, the amont of risk sharing that is offered to depositors is lower than what depositors wold have liked if there was no concern of a rn. Then, we enrich the model by adding the government, which attempts to alleviate these ineffi ciencies by garanteeing depositors to receive a minimm repayment throgh the transfer of resorces from the pblic good to the banking sector. We start by considering a simple scheme of garantees that is the analog to the one in Diamond and Dybvig (1983), in that it is only meant to prevent the occrrence of panic rns de to coordination failres. To this end, the scheme foresees that depositors are garanteed to receive a minimm payment if the bank s project is sccessfl irrespective of what the other depositors do. By eliminating the negative externality that a rn imposes, this scheme prevents the occrrence of panic rns with a mere ECB Working Paper 2032, Febrary

8 annoncement effect and ths it does not entail any actal disbrsement in eqilibrim for the government. Hence, it does not lead to distortions in the bank s choice of the deposit contract. The contract chosen by the bank internalizes all the eqilibrim effects and so is identical to the one that the government wold have chosen. However, nlike in Diamond and Dybvig (1983), fndamental rns still occr in or framework, as depositors are not protected against the risk that the assets of their bank fail to prodce the reqired retrn. An important reslt is that when this garantee scheme is in place, banks increase the amont they offer to depositors in case of early withdrawal, and so create more liqidity. This leads to an increase in the probability of fndamental-based rns. This is one form of what researchers may refer to as an increase in risk taking following the introdction of garantees (e.g., Calomiris, (1990)). However, in or model, this scheme always promotes welfare relative to the decentralized soltion. The increase in welfare reslts from addressing both ineffi ciencies in the decentralized case. The garantee scheme eliminates panic-based rns and encorages banks to perform more liqidity transformation. Intitively, banks provide contracts that maximize depositors expected welfare nder the constraints. With this garantee scheme in place, the implications of increasing the short-term rate for the probability of a rn are less severe, and so banks choose to increase it more, redcing the extent of the ineffi ciency of the decentralized soltion de to ineffi cient risk sharing mentioned above. The apparent increase in risk taking is in fact a desirable otcome. Interestingly, we can show that this garantee scheme sometimes leads to an increase in the overall probability of a rn, that is, the increase in the probability of fndamental-based rns is large enogh to more than offset the elimination of panic-based rns. This is consistent with evidence presented by Demirgc-Knt and Detragiache (1998) that crises might become more likely in the presence of deposit insrance. However, welfare is still higher nder this insrance scheme than in the decentralized soltion. The fact that banks increase the amont they offer for early withdrawals and might increase the likelihood of rns overall does not imply they are acting against depositors interests. Hence, the model demonstrates the need for cation in interpreting often-mentioned empirical reslts. The above garantee scheme still exhibits ineffi ciency. The facts that depositors are not protected against the failre of the banks projects and that ineffi cient fndamental-based rns might be triggered as a reslt limit the effi ciency increase coming from this garantee scheme. 3 This scheme might also be hard to implement in the real world as in principle the payment is only triggered in case of panic, which is perhaps 3 Fndamental rns can also be ineffi cient: even thogh it is a dominant strategy to rn, a rn might still be collectively ineffi cient. ECB Working Paper 2032, Febrary

9 not easily verifiable. Hence, we also consider a second garantee scheme, in which depositors receive a minimm garanteed payment, irrespective of what the others do and irrespective of the bank s available resorces. That is, they get some protection against panic rns and fndamental failres either in the form of fndamental-based rns or bank insolvency. This scheme resembles the real-world deposit insrance. We show that, for a given short-term rate set by the banking contract, this garantee scheme redces the probability of both panic-based and fndamental-based crises. Bt, crises still occr, leading to actal disbrsements, and so leading to non-trivial costs of increasing the level of garantees. Hence, the government is limited here in how mch it helps the banking system. As with the previos scheme, we show that the introdction of garantees addressing both panic and fndamental rns leads banks to increase the short-term payment they offer to depositors, which acts to improve welfare becase of the increase in risk sharing. However, de to the fact that disbrsement actally happens in eqilibrim, there is a wedge between the optimal short-term rate (that the government wold like to set) and the one that banks set in their contracts. Banks internalize the effect of the short-term rate on the probability of a rn among their depositors, bt they do not internalize the effect it has on the amonts that the government needs to spend and so on the level of pblic good. This is becase overall government spending and the amont left for the pblic good are determined by the decisions of all banks combined and are very slightly affected by the decision of each particlar bank. This is where the intition of moral hazard often featred in the pblic debate according to which banks incentives are distorted by garantees starts to show p in or model. Interestingly, however, while it is commonly thoght that banks set short-term deposit rates too high in response to garantees, or framework shows that the distortion can go in both directions. The important detail determining the direction of the distortion is whether the government ends p paying to depositors more in case there is no rn and the bank ends p failing in the longer term for fndamental reasons or in case there is a rn and the bank faces a shortage of liqidity. If the former holds, then the cost of a rn from the point of view of banks is higher than from the point of view of the government and the banks set too low of a deposit rate (generating the opposite of the common wisdom). If the latter holds, then the cost of a rn from the point of view of banks is lower than from the point of view of the government and the banks set too high of a deposit rate (generating the common wisdom). In any case, however, the government can choose the amont of garantee sch that welfare will always increase, despite the distortion. In smmary, a carefl analysis of the effects of government garantees shows that they have an important ECB Working Paper 2032, Febrary

10 role helping the financial system to provide risk sharing to investors while mitigating the problems associated with coordination failres and ineffi cient liqidations. The common criticism against garantees that they encorage excessive risk taking neglects to consider that some risk taking by banks de to liqidity transformation is desirable and might be sppressed by the concern for financial fragility. Garantees, in trn, relax these concerns to some extent allowing banks to provide greater liqidity transformation, which is welfare improving. Of corse, or paper does not cover all possible aspects of government garantees; for example, we do not model the choice of assets by banks, bt rather all risk taking in or model is captred on the liability side. It is possible that extending the model frther will ncover darker sides of government garantees. Also, we make several simplifying assmptions on the form of the banking contract and government garantees, which keep the analysis tractable, bt might prevent additional implications from being revealed. Still, or framework, to the best of or knowledge, is the first one that allows stdying the endogenos probability of rns and the endogenos risk choice by banks and how they interact with each other and with the government s garantees policy. The paper proceeds as follows. Section 2 contains a literatre review. Section 3 describes the model withot government intervention. Section 4 derives the decentralized eqilibrim. Section 5 analyzes the garantee schemes. It first characterizes a scheme against panic rns and then one protecting depositors against both panic rns and fndamental failres. Section 6 ses a parametric example to illstrate the properties of the model. Section 7 contains discssion and conclsion. All proofs are contained in the appendix. 2 Related literatre The analysis of or paper provides a step towards nderstanding the interconnection between garantees, fragility and bank s behavior. Or starting point is the literatre on deposit insrance originated with the seminal paper by Diamond and Dybvig (1983). That framework featres mltiple eqilibria: one where banks provide optimal risk sharing and no rn occrs; another one where a panic rn occrs de to the coordination failre among depositors. The introdction of deposit insrance works as an eqilibrim selection device. The garantee of receiving the promised repayment removes depositors incentives to rn. As a conseqence, only the good eqilibrim srvives and the maximm amont of risk-sharing is achieved. Deposit insrance neither entails any cost nor it affects banks behavior. 4 4 Similar environments where rns are driven by agents expectations and pblic intervention is desirable to eliminate the panic eqilibrim are analyzed in sbseqent papers inclding, recently, Cooper and Kempf (2015). ECB Working Paper 2032, Febrary

11 Sbseqent contribtions (see Allen, Carletti, Goldstein and Leonello (2015) for a srvey) have instead looked at the effect of deposit insrance on banks and investors behavior in frameworks where banks invest in risky projects and rns are de to the expectation of bad bank fndamentals. For example, Cooper and Ross (2002) extend Diamond and Dybvig (1983) by allowing banks to invest in a risky technology and depositors to monitor banks investment strategies. They show that, when deposit insrance is sffi ciently generos, banks find it optimal to invest in excessively risky projects and depositors have no or little incentives to monitor banks. As a conseqence, althogh it prevents rns, a complete deposit insrance scheme fails to achieve the first-best allocation becase of the greater bank risk taking. Importantly, this literatre considers rns as snspot phenomena triggered by some exogenos shift in depositors expectations and independent of bank s behavior. Hence, it does not endogenize the probability of a rn and its interaction with banks choices and government garantees. Or model combines the two above described approaches to crises in that it featres both fndamental and panic based rns. However, differently from previos stdies, in or model the probability of either type of rns is flly endogenos and affected by banks risk choice, as well as by the presence of garantees. Or pblic intervention resembles standard deposit insrance schemes and differs from bailots, which represent a form of ex post intervention aimed at mitigating the negative conseqences of a crisis rather than preventing it. Despite these differences, or analysis shares some featres with the literatre on bailots (see, among others, Farhi and Tirole (2012), Nosal and Ordonez (2016), Keister (2016) and Keister and Narasiman, (2016)), in that also these contribtions analyze how the (anticipation of) bailots may adversely affect banks risk taking incentives, and ltimately the desirability of pblic intervention. Among these contribtions, the closest papers to ors are Keister (2016) and Keister and Narasiman (2016). Both contribtions extend Diamond and Dybvig (1983) and consider the effect of pblic intervention on depositors withdrawal decisions and banks behavior. The anticipation of a bailot introdces a trade-off: on the one hand, it indces banks to engage in more liqidity creation, ths increasing depositors incentives to rn; on the other hand, it improves investors payoffs, ths redcing their incentives to rn if they expect others to do the same. Whether the bailot improves welfare and leads to more or less fragility depends on which of these two effects dominates. In both frameworks, the occrrence of rns depends on the realization of a snspot variable, whose probability is exogenos and not affected by the anticipation of bailots, and there is always a no rn eqilibrim irrespective of the bailot policy chosen by the government. An advantage of or paper is that ECB Working Paper 2032, Febrary

12 the probability of rns is flly endogenos in or model, and so we are able to better characterize the interconnection between fragility, pblic garantees and bank behavior. Or reslts that garantees enable banks to perform more welfare-improving liqidity transformation, which is tre even if the probability of crisis increases, and the characterization of the direction of distortions cased by garantees are not present in the other papers. The disadvantage of or model is perhaps the assmptions on the form of the banking contract and government garantees, which are taken as given, bt this enables s to flly endogenize investors rns decisions, banks risk choices, and the interaction between them and with the garantees regime. This is something that the previos literatre was not able to do. The ability to endogenize the probability of panics- and fndamental-driven rns relies on the se of global games. This approach, which was first stdied in the seminal paper by Carlsson and van Damme (1993), allows deriving niqe eqilibria in contexts where agents have private information on some random variable. In the first application, Morris and Shin (1998) se this featre to stdy the occrrence of crrency crises. Since then, global games have been sed in finance to analyze, among others, isses of contagion (Goldstein and Pazner (2004)), the role of large traders on the occrrence of crrency crises (Corsetti, Dasgpta, Morris and Shin (2004)), twin crises (Goldstein (2005)), central bank lending (Rochet and Vives (2004)) and the fragility of demand deposit contracts (Goldstein and Pazner (2005)). 5 Generally, in the global games literatre the proof of the niqeness of the eqilibrim bilds on the existence of global strategic complementarities between agents actions, in that an agent s incentive to take a specific action increases with the nmber of other agents taking the same action. This is the case in all papers mentioned above besides Goldstein and Pazner (2005). In their framework, a depositor s incentive to rn does not monotonically increase with the proportion of other depositors rnning. As noted in the introdction, or paper extends Goldstein and Pazner (2005) by stdying how the provision of garantees (their design and size) affects fragility in the banking sector and interacts with the bank s choice of the deposit contract. Or framework bilds on theirs and ths shares the same technical challenge of characterizing the existence of a niqe eqilibrim in a context in which there are no global strategic complementarities. As they allow to endogenize the probability of crises and derive niqe eqilibria in contexts characterized by strategic complementarities, global games techniqes have also been increasingly sed in recent years to analyze relevant policy qestions concerning financial reglation and pblic intervention (e.g., Bebchk and Goldstein (2011), Choi (2014), Vives (2014) and Eisenbach (2016)). In this respect, they represent a powerfl 5 See also Morris and Shin (2003) for a srvey on the theory and application of global games. ECB Working Paper 2032, Febrary

13 tool for policy analysis. As emerged in or analysis, having a niqe eqilibrim and being able to disentangle the varios effects of a specific policy is key to evalate its desirability, effectiveness and costs. 3 The basic model The basic model is based on Goldstein and Pazner (2005), agmented to inclde a government for the prpose of stdying garantee policies. There are three dates (t = 0, 1, 2), a continm 0, 1] of banks and a continm 0, 1] of consmers in every bank. There is perfect competition among banks, so that they make no profit. Banks raise one nit of fnds from consmers in exchange for a deposit contract as specified below, and invest in a risky project. For each nit invested at date 0, the project retrns 1 if liqidated at date 1 and a stochastic retrn R at date 2 given by R = { R > 1 w.p. p(θ) 0 w.p. 1 p(θ). The variable θ, which represents the state of the economy, is niformly distribted over 0, 1]. We assme that p(θ) = θ and E θ p(θ)]r > 1, which implies that the expected long term retrn of the project is sperior to the short term retrn. Each consmer is endowed with one nit at date 0 and nothing thereafter. At date 0, each consmer deposits his endowment at the bank. The bank promises a fixed payment c 1 > 1 to depositors withdrawing at date 1. Alternatively, depositors can choose to wait ntil date 2 and receive a risky payoff c 2, as specified below. Consmers are ex ante identical bt can be of two types ex post: each of them has a probability λ of being an early consmer (impatient) and consming at date 1, and a probability 1 λ of being a late consmer (patient) and consming at either date (we sally refer to them as early depositors and late depositors, respectively). Consmers privately learn their type at date 1. The government has an endowment g, which, for the moment, it can only se to provide pblic goods to consmers in addition to the deposit payments they obtain from banks. Consmers preferences are then given by U(c, g) = (c) + v(g), where (c) represents the tility from the consmption of the payments obtained from banks and v(g) is the tility from the consmption of the pblic good provided by the government. In what follows, we will refer ECB Working Paper 2032, Febrary

14 to (c) and v(g) also as private and pblic tility, respectively. 6 The fnction U(c, g) satisfies (c) > 0, v (g) > 0, (c) < 0, v (g) < 0, (0) = v(0) = 0 and relative risk aversion coeffi cient, c (c)/ (c), greater than one for any c 1. The state of the economy θ is realized at the beginning of date 1, bt is pblicly revealed only at date 2. After θ is realized at date 1, each consmer receives a private signal x i of the form x i = θ + ε i, (1) where ε i are small error terms that are independently and niformly distribted over ε, +ε]. After the signal is realized, consmers decide whether to withdraw at date 1 or wait ntil date 2. The bank satisfies consmers withdrawal demands at date 1 by liqidating the long term asset. If the liqidation proceeds are not enogh to repay the promised c 1 to the withdrawing depositors, each of them receives a pro-rata share of the liqidation proceeds. 7 Since the banking sector is perfectly competitive, banks choose the deposit contract (c 1, c 2 ) at date 0 that maximizes depositors expected tility. As standard in the financial crisis literatre (e.g. Diamond and Dybvig (1983) and nmeros papers thereafter), the deposit contract involves a non-contingent date 1 payment c 1 and a date 2 payment c 2 eqal to the retrn of the non-liqidated nits of the bank s project divided by the nmber of remaining late depositors. The payment c 1 mst be lower than the amont 1 λc1 1 λ R that each late depositor receives at date 2 when only the λ early depositors withdraw early and the project scceeds. Otherwise, the deposit contract is never incentive compatible and late consmers always have an incentive to withdraw early and ths generate a rn. The timing of the model is as follows. At date 0, each bank chooses the promised payment c 1. At date 1, after realizing their type and receiving the private signal abot the state of the fndamentals θ, depositors decide whether to withdraw early or wait ntil date 2. At date 2, the bank s project realizes and waiting late depositors receive a pro-rata share of the realized retrns of the project nits remaining at the bank. The model is solved backwards. 6 Consmers receive the same amont of pblic good irrespective of their type. As with the good provided by the bank, early consmers enjoy the pblic good at date 1 while late consmers enjoy it at either date. Given there is no disconting, the timing of the provision does not matter for the late types. 7 The assmption that depositors repayments follow a pro-rata share rle rather than a seqential service constraint as in Goldstein and Pazner (2005) simplifies the analysis withot affecting the qalitative reslts. ECB Working Paper 2032, Febrary

15 4 The decentralized eqilibrim We first analyze the model withot government garantees and refer to the reslt as the decentralized eqilibrim. We start by analyzing depositors withdrawal decisions at date 1 for a given fixed payment c 1. We then move to date 0 and analyze the bank s choice of c 1. Early consmers always withdraw at date 1 to satisfy their consmption needs. By contrast, late consmers compare the expected payoffs from going to the bank at date 1 or 2 and withdraw at the date when they expect to obtain the highest tility. Their expected payoffs depend both on the realization of the fndamentals θ as well as on the proportion n of depositors withdrawing early. Since the signal x i provides information on both θ and other depositors actions, late consmers base their withdrawal decision only on the signal they receive. When the signal is high, a late consmer attribtes a high posterior probability to the event that the bank s project yields the positive retrn R at date 2. Also, pon observing a high signal, he infers that the others have also received a high signal. This lowers his belief abot the likelihood of a rn and ths his own incentive to withdraw at date 1. Conversely, when the signal is low, a late consmer has a high incentive to withdraw early as he attribtes a high likelihood to the possibility that the project s date 2 retrn will be zero and that the other depositors rn. This sggests that late consmers withdraw at date 1 when the signal is low enogh, and wait ntil date 2 if, by contrast, the signal is sffi ciently high. To show this formally, we first assme that there are two regions of extremely bad or extremely good fndamentals, where each late consmer s action is based on the realization of the fndamentals irrespective of his beliefs abot the others behavior. The existence of these two extreme regions, no matter how small they are, garantees the niqeness of the eqilibrim in depositors withdrawal decisions. We then analyze the intermediate region where beliefs abot the others behavior play an important role in the determination of the eqilibrim. We start with the lower region. Lower Dominance Region. When the fndamentals are very bad (θ is very low), the expected tility from waiting ntil date 2, θ ( 1 λc1 1 λ R ), is lower than that from withdrawing at date 1, (c 1 ), even if only the early depositors were to withdraw (n = λ). If, given his signal, a late depositor is sre that this is the case, rnning is a dominant strategy. We then denote by θ(c 1 ) the vale of θ that solves ( ) 1 λc1 (c 1 ) = θ 1 λ R, (2) ECB Working Paper 2032, Febrary

16 that is θ(c 1 ) = (c 1 ) ( 1 λc1 1 λ R ). (3) We refer to the interval 0, θ(c 1 )) as the lower dominance region, where rns are only driven by bad fndamentals. For the lower dominance region to exist for any c 1 1, there mst be feasible vales of θ for which all late depositors receive signals that assre them to be in this region. Since the noise contained in the signal x i is at most ε, each late depositor withdraws at date 1 if he observes x i < θ(c 1 ) ε. It follows that all depositors receive signals that assre them that θ is in the lower dominance region when θ < θ(c 1 ) 2ε. Given that θ is increasing in c 1, the condition for the lower dominance region to exist is satisfied for any c 1 1 if θ(1) > 2ε. Upper Dominance Region. The pper dominance region of θ corresponds to the range ( θ, 1 ] in which fndamentals are so good that no late depositors withdraw at date 1. As in Goldstein and Pazner (2005), we constrct this region by assming that in the range (θ, 1] the project is safe, i.e., p(θ) = 1, and yields the same retrn R > 1 at dates 1 and 2. Given c 1 < 1 λc1 1 λ R R, this ensres that the bank does not need to liqidate more nits than the n depositors withdrawing at date 1. Then, when a late depositor observes a signal sch that he believes that the fndamentals θ are in the pper dominance region, he is certain to receive his payment 1 λc1 1 λ R at date 2, irrespective of his beliefs on other depositors actions, and ths he has no incentives to rn. Similarly to before, the pper dominance region exists if there are feasible vales of θ for which all late depositors receive signals that assre them to be in this range. This is the case if θ < 1 2ε. The Intermediate Region The two dominance regions are jst extreme ranges of fndamentals in which late depositors have a dominant strategy that depends only on the fndamentals θ. When the signal indicates that θ is in the intermediate range θ(c 1 ), θ ], a depositor s decision to withdraw early depends on the realization of θ as well as on his beliefs regarding other late depositors actions. To determine late depositors withdrawal decisions in this region, we calclate their tility differential between withdrawing at date 2 and at date 1 as given by ( ) θ 1 nc1 1 n R (c 1 ) if λ n n v(θ, n) = (4) 0 ( 1 n ) if n n 1, ECB Working Paper 2032, Febrary

17 where n represents the proportion of depositors withdrawing at date 1 and n=1/c 1 (5) is the vale of n at which the bank exhasts its resorces if it pays c 1 1 to all withdrawing depositors. The expression for v(θ, n) states that as long as the bank does not exhast its resorces at date 1, i.e., for n n, late depositors waiting ntil date 2 obtain the residal 1 nc1 1 n R with probability θ while those withdrawing early obtain c 1. By contrast, for n n the bank liqidates its entire project at date 1. Each late depositor receives nothing if he waits ntil date 2 and the pro-rata share 1/n when withdrawing early. Insert Figre 1 As Figre 1 illstrates, the fnction v(θ, n) decreases in n for n n and increases with it afterwards. This implies that a late depositor s incentive to withdraw early is highest when n = n rather than when n = 1, as it is sally the case in standard global games where the eqilibrim bilds on the property of global strategic complementarity (e.g., Morris and Shin (1998)). However, since v(θ, n) decreases in n whenever it is positive and crosses zero only once for n n remaining always below zero afterwards, the model exhibits the property of one-sided strategic complementarity as in Goldstein and Pazner (2005). This implies that there still exists a niqe eqilibrim in which a late depositor rns if and only if his signal is below the threshold x (c 1 ). At this signal vale, a late depositor is indifferent between withdrawing at date 1 and waiting ntil date 2. Formally, x (c 1 ) is sch that, conditional on this signal, the expected tility n λ (c 1)dn + n ( 1 n )dn from withdrawing at date 1 is eqal to the expected tility ( ) n λ θ 1 λc 1 1 λ R dn + (0)dn from waiting n ntil date 2. Here, the marginal depositor expects the proportion n of rnning depositors to be niformly distribted between λ and 1. This is a reslt of the fact that impatient depositors always rn and patient depositors receive signals with niformly distribted noise and in eqilibrim they rn below x (c 1 ) and do not rn above it. The following reslt holds. Proposition 1 The model has a niqe eqilibrim in which late depositors rn if they observe a signal below the threshold x (c 1 ) and do not rn above. At the limit, as ε 0, x (c 1 ) simplifies to θ (c 1 ) = (c 1) 1 λc 1 ] + c 1 n= n ( 1 n ) c 1 n ( 1 nc1 1 n R ). (6) ECB Working Paper 2032, Febrary

18 The proposition states that even in the intermediate region a late depositor s action depends niqely on the signal he receives as this provides information both on the fndamentals θ and the other depositors actions. For θ in the interval θ(c 1 ), θ (c 1 )) there is strategic complementarity in consmers withdrawal decisions: If c 1 > 1, the bank has to liqidate more than one nit for each withdrawing depositor. This implies that late depositors date 2 payoff is decreasing in the proportion n of early withdrawing depositors and so their incentives to rn increases with n. In the limit case when ε 0, all late depositors behave alike as they receive approximately the same signal and take the same action. This implies that only complete rns, where all late depositors withdraw at date 1, occr. In what follows, we will focs on this limit case. Insert Figre 2 Proposition 1 implies that a rn occrs for any θ < θ (c 1 ), bt for different reasons as also illstrated in Figre 2. For θ in the interval 0, θ(c 1 )) rns are fndamental-based: Late depositors withdraw early becase they expect the fndamentals to be bad so that rnning is a dominant strategy. For θ in the interval θ(c 1 ), θ (c 1 )) rns are panic-based: Late depositors withdraw becase they expect the others to do the same. The two types of rns differ significantly in terms of effi ciency. Panic rns are always ineffi cient as they reslt from a coordination failre among depositors. By contrast, fndamental rns can be effi cient if they lead to the early liqidation of nprofitable investments. For each nit that the bank liqidates at date 1 to repay the withdrawing depositors, the retrn R is foregone with probability θ. Liqidating the project is then ineffi cient for any θ > θ(1) since the tility (1) that a depositor obtains from the liqidated nit is lower than the expected tility θ (R) he wold obtain from the same nit if invested ntil date 2. If c 1 > 1, then θ(1) < θ(c 1 ) < θ (c 1 ). Ths, fndamental rns are effi cient in the range 0, θ(1)) bt ineffi cient in the range θ(1), θ(c 1 )). The likelihood of both types of rn as given by the thresholds θ(c 1 ) and θ (c 1 ) is affected by the promised payment c 1 offered by the bank to early withdrawers. We have the following reslt. ( ) Corollary 1 Both thresholds θ(c 1 ) and θ (c 1 ) are increasing in c 1 i.e., θ(c1) > 0 and θ (c 1) > 0 θ (c 1) > θ(c1). with The corollary sggests that both rn thresholds increase with the deposit rates offered by banks, althogh their sensitivity is different. The reason is that the higher c 1, the lower is the payoff c 2 accred by the late depositors at date 2 and ths the stronger is the incentive for each late depositor to withdraw early. The ECB Working Paper 2032, Febrary

19 panic threshold θ (c 1 ) is more sensitive to changes in c 1 than the fndamental threshold θ(c 1 ) becase in the case of panic rns an increase in c 1 also changes the beliefs that each depositor has on the others behavior and on the damage that their withdrawals will case to the remaining investors retrns. This reinforces each late depositor s incentive to rn, ths making θ (c 1 ) more sensitive to changes in c 1 than θ(c 1 ). Now that we have characterized depositors withdrawal decisions at date 1, we trn to date 0 and compte the optimal deposit contract c 1. Each bank chooses c 1 at date 0 to maximize the expected tility of a representative depositor, which is given by θ (c 1) 0 ( )] 1 λc1 (1) dθ + λ(c 1 ) + (1 λ)θ θ (c 1) 1 λ R dθ + 0 v(g)dθ. (7) The first term represents depositors expected tility from depositing at the bank for θ < θ (c 1 ) when, given that a rn occrs, the bank liqidates its entire project and each depositor obtains 1 instead of the promised payment c 1. The second term is depositors expected tility for θ θ (c 1 ) when the bank contines ntil date 2. The λ early consmers withdraw early and obtain c 1, while the (1 λ) late depositors wait and receive the payment 1 λc1 1 λ R with probability θ. The last term is the tility that depositors receive from the consmption of the pblic good. Since the entire government s endowment g is sed to provide the pblic good, depositors tility v(g) is naffected by the occrrence of rns. We have the following reslt. Proposition 2 The optimal deposit contract c D 1 ( 1 λ (c 1 ) θr λc1 θ (c 1) θ (c 1 ) > 1 in the decentralized soltion solves )] 1 λ R λ(c 1 ) + (1 λ)θ (c 1 ) dθ + ( ) ] 1 λc1 1 λ R (1) = 0 (8) In choosing the promised payment to early depositors the bank trades off the marginal benefit of a higher c 1 with its marginal cost. The former, as represented by the first term in (8), is the better risk sharing obtained from the transfer of consmption from late to early depositors. The latter, which is captred by ( ) ] the second term in (8), is the loss in expected tility λ(c 1 ) + (1 λ)θ (c 1 ) 1 λc1 1 λ R (1) de to the increased probability of rns, as measred by θ (c 1). At the optimm, the bank chooses c D 1 > 1 even if this entails panic rns. The reason is that when c 1 = 1, the difference between early and late depositors expected payment is maximal. A slight increase of c 1 provides a large benefit in terms of risk sharing given that depositors have a relative risk aversion coeffi cient greater than 1. Frthermore, at c 1 = 1 the loss in terms of expected tility in the case of a rn approaches zero. 8 Ths, increasing c 1 slightly above 1 entails ( ) ] 8 When c 1 = 1, the term λ(c 1 ) + (1 λ)θ (c 1 ) 1 λc1 1 λ R (1) simplifies to (1 λ) θ (1) (R) (1)] = 0 with ECB Working Paper 2032, Febrary

20 a second-order cost and a first-order benefit and so it is always optimal. The optimal c D 1 is chosen so that rns occr only for θ < θ (c 1 ) < θ. If this was not the case and rns occrred for any θ, consmers wold obtain a tility (1), which wold be lower than the tility they reach with the optimal c D 1. The bank s choice of c D 1 > 1 entails a trade-off in or model. On the one hand, c 1 represents the amont of risk sharing banks offer to depositors. On the other hand, given that the probability of rns is endogenos and is linked to the parameters of the deposit contract, c 1 represents a form of risk as it determines banks exposre to rns. Hence, the higher payment c 1 the greater is banks liqidity creation bt also their fragility. Since banks anticipate the effect of a higher c 1 on their fragility, they redce deposit rates ths scaling down liqidity creation. In eqilibrim, banks offer too little risk sharing relative to the level they wold choose if rns were not considered. In the latter case, each bank wold in fact choose c 1 to maximize 0 ( )] 1 λc1 λ(c 1 ) + (1 λ)θ 1 λ R dθ + 0 v(g)dθ. The soltion to this problem, which we denote c NR 1, is obtained from the following eqation: ( )] 1 λ (c 1 ) θr λc1 1 λ R dθ = 0. (9) 0 The soltion c NR 1 maximizes the gains from risk sharing and it coincides with the promised repayment to depositors in Diamond and Dybvig (1983), with the difference that the bank s project is now risky and ths the tility of late consmers at date 2 incldes the probability θ that the project scceeds. Comparing (8) and (9), it can be easily seen that c NR 1 > c D 1. The intition is simple: c NR 1 maximizes the gain from risk sharing while ignoring the possibility of bank rns. As sch, it does not consider the effect of c 1 on the probability of rns and the associated costs. To sm p, the decentralized eqilibrim is characterized by too little risk sharing and the possibility of ineffi cient rns for θ(1), θ (c 1 )). Both ineffi ciencies depend on the bank s choice of c 1. A higher c 1 improves liqidity creation bt also enlarges the range θ(1), θ (c 1 )) where rns destroy good investments. The qestion is whether and how the decentralized soltion can be improved pon with the help of pblic intervention. 5 Introdcing government garantees In this section, we analyze whether the introdction of government garantees can improve pon the decentralized allocation. The starting point to nderstand the role of pblic intervention in or model is Diamond θ (1) = ] (1). (R) ECB Working Paper 2032, Febrary