Deposit insurance and subsidized recapitalizations

Journal of Banking & Finance 35 (2011) 3400–3416

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Deposit insurance and subsidized recapitalizations Alan D. Morrison a,⇑, Lucy White b a b

Saïd Business School, Park End Street, Oxford OX1 1HP, UK Harvard Business School, Baker Library 345, Soldiers Field Road, Boston, MA 02163, USA

a r t i c l e

i n f o

Article history: Received 9 August 2010 Accepted 26 May 2011 Available online 1 June 2011 JEL classification: G21 G28 Keywords: Deposit insurance Bank regulation Adverse selection Moral hazard

a b s t r a c t The 2007–2009 financial crisis saw a vast expansion in deposit insurance guarantees around the world and yet our understanding of the design and consequences of deposit insurance schemes is in its infancy. We provide a new rationale for the provision of deposit insurance. In our model the banking sector exhibits both adverse selection and moral hazard, which implies that the social benefits of bank monitoring must for incentive reasons be shared between depositors and banks. Consequently, socially too few deposits are made in equilibrium. Deposit insurance – or, equivalently, bank recapitalization – corrects this market failure. We find that deposit insurance should be funded not by banks or depositors but out of general taxation. The optimal level of deposit insurance varies inversely with the quality of the banking system. Hence, when the soundness of the financial sector is uncertain, governments should consider supporting deposit insurance schemes and undertaking subsidized recapitalizations. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction Deposit insurance schemes are of increasing importance around the world. The number of such schemes has almost tripled in the last 20 years (Demirgüç-Kunt and Sobaci, 2001), and deposit insurance schemes now form part of the IMF’s best practice recommendations to developing countries. Also, in response to the financial crisis, many countries have increased the generosity of their deposit insurance schemes recently. For example, in October 2008, the level of depositor protection provided to US depositors was increased from $100,000 to $250,000, and the level provided to British depositors increased from 100% of the first £2000 lost and 90% of the next £31,000 to 100% of the first £50,000 lost. Many other countries simply ended up providing blanket guarantees to all depositors.1 Deposit insurance in the United States is provided by the Federal Deposit Insurance Corporation from a bank-financed fund which, as of September 2008, stood at $45 billion2; British ⇑ Corresponding author. Tel.: +44 1865 288 921; fax: +44 1865 288 805. E-mail addresses: [email protected] (A.D. Morrison), [email protected] (L. White). 1 Feyen and Vittas (2009) survey changes to deposit insurance schemes in the wake of the crisis. Austria, Denmark, Germany, Greece, Hong Kong, Iceland, Ireland, Malaysia, New Zealand and Singapore all provided blanket guarantees; amongst countries that did not, the average deposit coverage as a multiple of GDP per capita increased from 1.5 to 5. Details of deposit insurance schemes in 46 countries appear in the Appendix to Financial Stability Board (2009). 2 See http://www.fdic.gov/news/news/press/2008/pr08084.html. 0378-4266/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankfin.2011.05.017

depositors are protected by the Financial Services Compensation Scheme, which originally paid for bail-outs via an ex post tax on financial institutions, but which will, as a result of the 2009 Banking Act, move to a pre-funded system like that in the United States. Despite an apparently high level of consensus amongst regulators and policy-makers that deposit insurance schemes are desirable, and that they should be pre-funded along US lines, the evidence on their value is at best mixed.3 Moreover, many financial economists are critical of the design of real-world deposit insurance schemes. For example, Kane (1989) argues that the levies paid by banks for deposit insurance are too small, and fail adequately to account for the riskiness of their portfolios. The 2007–2009 financial crisis highlighted the implicit subsidy provided by deposit insurance: many governments were forced to provide explicit guarantees for bank deposits and bank debt that significantly exceeded those offered by formal deposit insurance schemes. Politicians are now attempting to claw back some of these subsidies through focussed taxes. Doing so may ameliorate moral hazard problems in the banking sector, but there is a risk that academic arguments will be displaced by the political backlash. This paper provides a more nuanced perspective on these questions. We construct a formal model of a banking sector that is characterized by both moral hazard and adverse selection. We are able to draw conclusions regarding the optimal level of deposit insurance provision, the right way to pay for deposit insurance, and 3

See Demirgüç-Kunt and Kane (2002) for a recent survey.

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the relationship between banking sector quality and the extent of deposit insurance provision. Deposit insurance is valuable in our model because it serves to maintain the banking sector at a socially desirable size, and so to stimulate bank investment. As a consequence, we find that only if deposit insurance is funded through general taxation does it have any beneficial effect. Subsidized deposit insurance is valuable in our model provided that it is accompanied by sufficiently tight capital requirements, and provided that it is not so generous that bank investors are completely indifferent as to whether banks monitor their investments or not. Most of the previous theoretical literature on deposit insurance takes its existence as given, and assesses its effects. This literature starts from the observation that deposit insurance is essentially a put option held by the bank’s shareholders on its assets (see Merton, 1977, 1978; Pennacchi, 1987; Hwang et al., 2009). When the deposit insurance put option’s price does not fully reflect the riskiness of the bank’s assets, bank equity holders have incentives to increase bank risk to socially excessive levels (see Kim and Santomero, 1988; Nagarajan and Sealey, 1995; Penati and Protopapadakis, 1988). More recent work has shown that in the presence of such moral hazard by or adverse selection of banks, fairly priced deposit insurance may be either impossible (Chan et al., 1992) or, possible but undesirable (Freixas and Rochet, 1998).4 We depart from most of the previous literature by presenting a model in which we consider the design of deposit insurance schemes. We model an economy in which there is a moral hazard problem between depositors and bankers. Banks in our model are socially valuable because they can use a costly monitoring technology to transform depositor funds into more productive investments. However, since monitoring is not verifiable, banks must be rewarded for deploying their costly monitoring technology. Hence, in common with most of the existing models that emphasize the problems created by deposit insurance schemes, we analyze the effects of moral hazard in the banking sector.5 In contrast to the bulk of the existing literature, however, we also allow for adverse selection in the banking sector. We assume that some, unsound, bankers are simply unable to monitor investments; we refer to banks that can monitor as sound. Unsound bankers have the same investment technology as depositors, so their presence makes depositors reluctant to deposit in banks. Depositing in unsound banks is no less productively efficient than not depositing at all, but, because unsound bankers are indistinguishable from sound bankers, all bankers must be compensated at the level required to motivate monitoring by sound bankers. As a result, depositors lose out relative to their outside option when they deposit with unsound bankers. The danger that they will do so causes depositors socially to undervalue bank investment; as a result, from a social point of view, there is insufficient depositing. The greater the perceived proportion of unsound bankers in the economy, the less confidence depositors have in the quality of their banks, and hence the greater the degree of underdepositing. In this set-up there is a role for deposit insurance, which makes bank investment more attractive for depositors. We show that when the adverse selection problem is sufficiently severe, deposit insurance can improve social welfare. The deposit insurance fund does not have better information than depositors, but it internalizes all of the expected social benefits of bank depositing, and the deposit insurance scheme is designed accordingly. We require the deposit insurance fund to balance its books in expectation, but

4

For a useful survey of the literature, see Freixas and Rochet (1997). In our model, moral hazard takes the form that banks fail to monitor their investments. Our treatment follows Holmström and Tirole (1997). However, we could equally well allow moral hazard to involve NPV-reducing risk-enhancing investments, as is often done in the literature. 5

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allow it to be funded by payments from bankers or depositors, or through general taxation. In contrast to the existing literature, in our model we find that for deposit insurance to have a beneficial effect, it must be funded through general taxation. It is the tax levied on agents outside the banking system that provides a net subsidy to the system and thus encourages depositing.6 This result runs contrary to the conventional wisdom and to majority of the existing literature, which assumes that deposit insurance ought to be funded by bankers and then examines the pricing to individual banks.7 It is, however, entirely consistent with the policy-makers’ recent response to the financial crisis, which has been to provide much wider guarantees to the banking sector, arguably at taxpayers’ expense.8 In contrast, in our model, increasing the safety net by increasing bank contributions to a deposit insurance scheme is completely welfare-neutral. To see why, note that requiring banks to contribute part of their capital to a deposit insurance fund reduces the banker’s stake in any investment of a given size, thus reducing his incentives (or alternatively, reducing the size of the project undertaken). But, on the positive side, providing more generous deposit insurance reduces the interest rate that bankers have to pay to entice depositors to deposit, and so increases the share of any given return from a successful project that a banker can extract, so improving incentives.9 These two effects exactly cancel each other out in a world where all agents are risk-neutral.10 Our analysis yields some interesting comparative statics. Because deposit insurance is a necessary response to a lack of confidence in the quality of the banking sector, deposit insurance must be made more generous when confidence drops. Hence, our analysis goes some way towards explaining the puzzling empirical finding that deposit insurance schemes are on the one hand consistently associated with a higher probability of banking crisis (see Section 7.2) and yet are at the same time being widely adopted across the world and promoted as part of good practice by institutions such as the IMF.

6 In a different setting to ours, Andries and Billon (2010) consider the effect of state-funded deposit insurance upon the transmission of monetary policy. They find that a greater degree of state funding serves to reduce the impact of monetary tightening upon the supply of loans. 7 But see The Economist (‘‘Still Money in that Franchise’’, April 30, 1994): Give them half a chance and bankers in both Europe and America will complain ad nauseum about the iniquitous obligations that fall upon them as a result of possessing a bank charter. Ask them why they do not give their charter up and you will receive a funny look. [. . .] Ignore all this. The truth of the matter is that however much banks moan about the restrictions and obligations imposed by governments and regulators, they know that this is the price they pay in return for something hugely valuable. That something is a public subsidy, and the subsidy takes the form of deposit insurance. 8 In Section 8.2 of our paper we show that when all agents are risk neutral, subsidized deposit insurance and subsidized recapitalizations are equivalent policies. In line with our conclusions, Diamond and Rajan (2009) argue that, if banking sector subsidies are to be effective, they must use funds taken from outside the banking sector. In their model, underinvestment is a consequence of high real interest rates in the wake of a liquidity shock that causes withdrawals from the banking sector. They show that if governments attempt to fix the problem with a bank subsidy paid for by a tax on depositors, then the depositors will undo the subsidy by withdrawing more from their bank accounts; the only effective subsidies are therefore those paid from using taxes upon households that do not hold bank deposits. 9 In our model this arises because with lower deposit rates, bankers receive a higher return on monitoring when they are successful. In a more general multi-period model, the effect could also manifest itself through an increase in charter value. In this sense, a deposit insurance subsidy can be seen as an alternative to allowing banks to extract rents through excessive market power. See Bhattacharya (1982) and Keeley (1990). 10 For a model where the risk aversion of depositors provides a rationale for deposit insurance, see Park (1996). For a model where forcing banks to subscribe to deposit insurance schemes reduces differentiation between more and less risky banks, thus increasing bank competition, see Matutes and Vives (1996). De Giuli et al. (2009) present a model in which mutual insurance by bankers generates a free-rider problem that results in excessive risk-taking by banks. In contrast, the deposit insurance scheme in our model is designed to alleviate the bankers’ moral hazard problem.

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Our model sheds some light upon events in the banking sector following the onset of the financial crisis. Losses in the opaque and poorly understood market for sub-prime mortgage securitizations left market participants unsure whether banks that had previously received a clean regulatory bill of health were indeed sound. As a result, creditors withdrew their investments in the banks.11 Precisely as in our model, the consequence was a reduction in the capacity of banks to extend loans: this was socially damaging but, as in our analysis, its effects were not internalized by bank creditors. More generous deposit insurance was required to entice depositors back to the banks, and so to counter credit-rationing problems. Recapitalization and other forms of explicit and implicit guarantees were provided to uninsured bank creditors.12 In most of the prior literature, deposit insurance is exogenously set at such a high level than depositors no longer care about the risks that their bankers take. As a result, deposit insurance exacerbates moral hazard problems. In contrastto this previous work, we have an endogenous rationale for deposit insurance, and hence can analyze the optimal deposit insurance contract. We find that it is never optimal to set deposit insurance at levels so generous that depositors cease to care about bank risk. In fact, as noted above, properly designed deposit insurance schemes reduce the tendency towards moral hazard. The reason for this is that deposit insurance reduces the interest rate that depositors require to induce depositing, and so increases the return that a successful banker earns, raising the banker’s effort incentives. Thus, our model has implications for the optimal level of deposit insurance. As remarked above, contrary to what one might expect, we show that the deposit insurance subsidy should be greater, the lower the expected quality of the banking system. Note further that if depositors cannot easily observe the level of deposits in each bank, the regulator should supplement deposit insurance schemes with a capital adequacy policy in order to prevent banks becoming too large and gambling with their funds (failing to monitor their investments). Interestingly, however, as deposit insurance protection is increased from zero to the optimal level, capital requirements can be correspondingly eased. This is simply because, as highlighted above, subsidized deposit insurance alleviates the moral hazard problem.13

11 One of the earliest casualties of the current financial crisis was the British bank, Northern Rock. Between the 14th and the 17th September 2007, Northern Rock experienced the first depositor run on a British bank since the 1866 failure of Overend Gurney. An announcement by the British authorities at 5.00 pm on the 17th September that all deposits in Northern Rock would henceforth be guaranteed by the government stopped the run. Shin (2009) argues that Northern Rock was not a classic Diamond and Dybvig-style depositor run, but rather that Northern Rock failed because its wholesale funding dried up, and the depositor run began only when the authorities announced emergency liquidity support. However, to the extent that problems in the wholesale market arguably stemmed from a loss of faith in received assumptions about the valuation of securitizations, and in particular in the assetbacked commercial paper upon which Northern Rock relied, our analysis can be applied. Wholesale depositors in the United Kingdom benefited during the financial crisis from subsidies to UK banks, for example through a widening of the population of eligible collateral at the Bank of England’s Discount Window Facility and via the Credit Guarantee Scheme and the Bank Recapitalization Fund. This support substituted for wholesale confidence in UK bank assets, and so prevented disinvestment from occurring. 12 For simplicity, our model treats all bank creditors as identical and equally unable to assess individual bank quality; in reality unsophisticated investors make small deposits and are insured whereas larger, more sophisticated investors lend at higher rates and at least prior to the Crisis, they appeared to enjoy fewer guarantees. After the onset of the Crisis, however, as Shin argues, the latter appear to have behaved like classic Diamond–Dybvig depositors, voting with their feet rather than monitoring and receiving explicit and implicit state guarantees. 13 In line with this observation, Fonseca and Francisco (2010) demonstrate using data for 1337 banks in 70 countries that capital buffers are higher in countries with less generous deposit insurance schemes. For a recent interesting contribution with contrasting results, see Hellman et al. (2000). They impose full (100%) deposit insurance protection for depositors and show that this results in moral hazard by banks. They argue that capital requirements and deposit rate ceilings should be imposed, but do not tackle the question of the optimal level and design of deposit insurance schemes.

Our work supplements explanations for deposit insurance based upon the Diamond and Dybvig (1983) story of bank runs. In their model, banks take liquid demand deposits and investment them in long-term, illiquid assets. Demand deposits are desirable because they provide depositors with insurance against non-verifiable liquidity shocks, as they can withdraw their funds early should they need them. However, demand deposits also open the door to undesirable Nash equilibria in which every depositor withdraws his or her funds early, and so precipitates a socially costly liquidation of long-term assets. Deposit insurance makes late withdrawal as profitable as early withdrawal for depositors with no short-term consumption needs, and hence stops welfare-reducing ‘‘bank runs’’ from occurring. Hence, in contrast to our emphasis upon sustaining bank lending at a socially optimal level, deposit insurance in the Diamond and Dybvig world is a response to liquidity shocks. Moreover, while the Diamond and Dybvig model provides an important and influential explanation for the utility of deposit insurance, it does not, as it stands, provide much guidance as to the form that deposit insurance should take nor how it should be funded. In contrast, deposit insurance payouts are an equilibrium phenomenon in our model, and we are able to provide detailed comment upon the design of deposit insurance schemes. The paper proceeds as follows. In the next section we set out a model of an unregulated economy in which bankers have a role in channeling funds from the household sector into productive investments, but in which adverse selection of and moral hazard by banks acts as an impediment to socially desirable depositing. In Sections 3 and 4 we show how an appropriately designed deposit insurance scheme can address this problem and we investigate the appropriate way to fund such a scheme. In Section 5 we show that our results are unaffected if our analysis is extended to multiple time periods. In Section 6 we examine various schemes by which the regulator might attempt to resolve adverse selection problems by separating sound from unsound bankers, obviating the need for deposit insurance. In Section 7 we discuss the empirical implications of our theory. Section 8 examines some other extensions of our model and Section 9 concludes.

2. The model We consider an economy consisting of N + l agents, each of whom is endowed with $1 to invest. Consumption occurs after one period and every agent has access to a storage technology, or bond, which for each dollar invested yields R > 0 with probability pL and which otherwise returns 0. N of the agents, to whom we refer as households, have simply to select an investment for their funds: they can either invest in the bond, or they can place their funds with one of the remaining l agents, or banks. Banks are able to collect and invest household funds and use them to finance investment projects. There are two types of banks; a banker’s type is his private information. Unsound banks have no technology that is unavailable to households, and so they simply invest their deposits in bonds. Sound banks have access to projects which require the use of a costly monitoring technology, but which enjoy a higher probability of success (i.e., returning R) equal to pH = pL + D > pL. The cost of monitoring such a project of size $k is $Ck, so that there are constant returns to scale in monitoring.14 The model is a simplified version of Morrison and White (2005); the monitoring technology follows Holmström and Tirole (1997). The literal interpretation of the costs of monitoring in this set up is that they are either private effort costs that the banker must 14 For evidence that monitoring borrowers is an important function of banks, see the classic papers by Fama (1985), James (1987) and, more recently, Ashcraft (2005).

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incur in order to generate higher expected returns, or the costs of foregoing the private benefits that the bank could enjoy if it did not work hard to make sound investments, but instead contented itself with lower expected returns. However, this interpretation is not important for the qualitative results – with small changes to the algebra we could allow monitoring costs to be financial and subtracted from the amount available for investment. What we require is that it is not verifiable whether the monitoring costs C are spent on genuine monitoring, or consumed in perquisites. For simplicity we also assume that all of the projects in which a bank invests are perfectly correlated, so that no diversification benefits are derived from running a large bank.15 The bond earns an expected return of RpL. Deposited funds therefore earn an expected return of RpL when managed by an unsound banker and, net of monitoring costs, earn RpH–C when managed by a sound banker who chooses to monitor. We assume that monitoring is efficient:

RD > C:

ðA1Þ

Note that, although the expected return on deposited funds depends on the type of banker with whom they are deposited, it is at least as high as the return on bonds. It follows that the social optimum is for all households to deposit their funds with banks.16 The relationship between depositors and bankers is governed by a deposit contract, which stipulates the fee Q that the depositors will pay to the bankers in the event that their project succeeds. When a bank of size k succeeds, the banker therefore receives an income R from his own $1 endowment, and Q from each of the $ (k  1) that he manages on behalf of depositors. The cost of monitoring is $C per dollar invested; since monitored investments succeed with probability pH, the sound banker’s payoff from monitoring is therefore

ðR þ Q ðk  1ÞÞpH  Ck; and the expected return to sound bankers who do not monitor and to unsound bankers is

ðR þ Q ðk  1ÞÞpL : Monitoring is therefore incentive compatible for sound bankers if and only if:

Q P Q MIC ðkÞ 

Ck  RD : ðk  1ÞD

ðMICÞ

The deposit rate that the depositors receive if their bank succeeds is R  Q. We wish to examine the relationship between the quality of the banking sector and optimal deposit insurance policy. 15 This assumption is not essential to our results, but modeling in this way allows us to abstract from any economies of scale in banking, which would complicate our analysis. We require that bank returns be risky and that bankers cannot completely diversify by being sufficiently large, á la Diamond (1984). For a rationale of why this will occur in the presence of moral hazard by banks, see Hellwig (2001), who shows using the Diamond model that, unless banks can contract on how they distribute their investments, as they become larger they will nevertheless find it optimal to concentrate their funds on a single investment rather than diversifying, for the usual reason that risk-shifting will benefit equity-holders at the expense of creditors. See also (Tirole (2006, Section 4.2.3, p. 161)) for reasons why diversification is limited in practice. 16 We set out our welfare criterion in Section 3 below. We have normalized the return on unsound banks’ project to be the same as that on the bond. The idea is that unsound banks do not add any value to the projects in which they invest, yet they extract fees for intermediating the investment. Our results are robust to assuming that unsound banks’ investments are worse than the bond provided that bank investments are on average more productive than households’ investments. We have also implicitly assumed that the social return on bank deposits is equal to the financial return R on investments. As will become clear in the following, if there is a positive externality associated with bank projects (e.g., the entrepreneurs – outside the model – to whom banks lend – obtain some consumer surplus on their loans) this would only strengthen our result that bank services are under-provided in equilibrium and should be subsidized. For an interesting analysis of the case where banks are unable to capture all the surplus generated by their monitoring, see Allen et al. (2009).

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To this end, let g 2 [0, 1] be the probability that a randomly selected bank is sound.17 The expected return to a bank depositor is then

ðR  Q ÞðpL þ g DÞ;

ð1Þ

provided condition (MIC) is satisfied. Agents will be prepared to deposit provided this expression dominates the expected income RpL from the bond, which is equivalent to the following:

Q < Q IRNoReg ðgÞ 

Rg D : pL þ g D

ðIRUnReg Þ

Banks of size k are feasible if there exists a deposit contract that satisfies conditions (MIC) and (IRUnReg): this is the case precisely when condition (2) is satisfied:

Q IRNoReg ðgÞ  Q MIC ðkÞ ¼

RDðpL þ gkDÞ  CkðpL þ g DÞ > 0: ðk  1ÞDðpL þ g DÞ

ð2Þ

It is immediate from condition (2) that there exists a maximum feasible bank size kB < 1 precisely when condition (A2) is satisfied:

CðpL þ g DÞ  Rg D2 > 0:

ðA2Þ

Condition (A2) has the following intuitive explanation. Suppose that banks had no capital. In that case, monitoring would be incentive compatible provided Q P DC ; since Q is paid only in success states, the minimum expected cost to depositors of inducing monitoring in the absence of bank capital would therefore be C D ðpL þ g DÞ. Condition (A2) states that this cost exceeds the marginal benefit RgD of monitoring, and hence that, absent banker capital, it is impossible simultaneously to satisfy the monitoring incentive compatibility constraint and the depositors’ participation constraint. Hence depositors will invest in banks only if the banker has some capital at stake: this serves to relax the monitoring incentive compatibility constraint, and hence leaves more funds to cover the depositor’s individual rationality constraint. The need for bank capital places an upper bound on the feasible bank size. The interesting case clearly arises when Eq. (A2) is satisfied, so that banks require capital to operate, and there is an upper bound on their size. The upper bound arises at the intersection of the constraints (MIC) and (IRUnReg), which is at bank size kB:

kB ¼

RpL D : CðpL þ g DÞ  Rg D2

In the absence of regulation, kB is the maximum size of a bank: if banks were larger, depositors would be unwilling to pay fees sufficient to prevent moral hazard by sound banks. The total volume of funds deposited in the unregulated banking sector is therefore lkB: we can think of k1B as the capital requirement for unregulated banks.18 We assume that 17 For simplicity we treat g as exogenous. Morrison and White (2005) show how g can be endogenized in a general equilibrium setting, when it is an increasing function of the quality of regulation. We also assume that each bank has the same probability g of being sound. At the cost of additional notation, the model could be extended to allow depositors to perceive a bank specific probability of gi of each bank i being sound. In this extension, our main results would be unchanged, but capital requirements and deposit insurance subsidies should then be tailored to each individual bank according to its perceived probability of soundness. In particular, and perhaps counter-intuitively, weaker banks should receive larger deposit insurance subsidies, which is the way that most deposit insurance schemes work in practice. 18 There are two interpretations of how exactly the enforcement of capital requirements occurs. These are essentially equivalent from a modeling point of view, though very different in spirit. It can be assumed that banks state their anticipated deposit-to-capital ratio; depositors observe the level of deposits banks collect and before banks invest they have an opportunity to withdraw without penalty if banks exceed their promised ratios. This can be termed market monitoring of capital adequacy. Alternatively if one supposes that depositors do not observe the level of deposits but that the regulator does, one can assume that there is regulatory monitoring of capital requirements. The regulator should immediately close any bank which accepts more deposits than would be compatible with monitoring by sound banks.

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lkB < N þ l:

ðA3Þ

Since N + l is the total size of the economy’s endowment, assumption (A3) states that the volume of funds under bank management in the pure market equilibrium is less than the total available: in other words, that the level of deposits is sub-optimal. The under-depositing problem is illustrated in Fig. 1. Sound bankers will decide to monitor only when they are compensated for doing so, which is when (k, Q) lies above (MIC). Households will elect to deposit only when the deposit rate is sufficiently high: in other words, below the (IRUnReg) constraint. Bankers earn the highest income from their activities at the point where the two constraints cross and they will therefore offer this contract to the household sector. When the intersection point kB is below 1 þ Nl as illustrated, assumption (A3) holds and there is underdepositing. To understand the root of the under-depositing problem, we can re-write assumption (A3) as follows:

g < g 

CðN þ lÞpL  RpL lD : ðRD  CÞðN þ lÞD

ðA30 Þ

This expression highlights the role of adverse selection in our model. Recall from assumption (A2) that there is a conflict between the banker’s incentive constraint and the depositors’ participation constraint: in the absence of capital, when bankers are guaranteed a sufficiently high return to induce them to monitor, depositors will not think it worth their while to invest in banks. Hence bankers need to commit some of their own capital so as to relax their incentive compatibility constraint, and hence to attract depositors. In this set-up, the effect of adverse selection is to make the depositors’ participation constraint (IRUnReg) harder to satisfy, because with probability (1  g) they will place their funds with an unsound banker who will take the monitoring fee Q, but who will not perform any monitoring: from the regulator’s perspective, this payment is a welfare-neutral wealth transfer, but the danger that it will occur increases the return that depositors demand from successful bankers. As a result, adverse selection serves to increase the amount of capital that the banker has to deploy per unit of depositor funds: when it is sufficiently severe, banks will be socially too small. This completes our description of the economy. Our model deliberately avoids introducing various features of banking models such as depositor risk aversion and self-fulfilling panics which have been used to justify deposit insurance in the past in order to show that deposit insurance may be desirable even if, as may be the case in many developed economies, pure sunspot panics are thought to be unlikely. Our model captures two important aspects of the banking literature which we believe are common to virtually all modern economies. Firstly, banks have an allocational role: they take funds from unproductive agents and place them where they can be more profitably used. Secondly, banks have monitoring and project management skills which tend to augment the return which entrepreneurs achieve on their projects. 3. The impact of deposit insurance on the banking sector We now introduce a benevolent regulator with a utilitarian welfare function. That is, the regulator aims to maximize the unweighted sum of banker and household utility.19 Since all agents are assumed to be risk-neutral, and production is divided between them, this amounts to maximizing production levels in the economy. Further, since unsound banks and household investments have the same productivity, the regulator aims to maximize the volume of funds managed by sound agents. As we have seen above, in the 19 It is easy to see that if the regulator were to maximize the weighted sum of the utilities then the qualitative conclusions would be similar.

Fig. 1. Constraints in the unregulated economy.

unregulated economy, it may be that not all funds are deposited with the banking sector (and hence not all are managed by sound agents), so that there is room for the regulator to improve productivity. The regulator has the same information about the banking sector as the depositors and the only tool at its disposal is a deposit insurance scheme, which must be paid for out of taxes levied on the bankers and households who comprise the economy.20 The timing of the game which the regulator plays with the depositors is illustrated in Fig. 2. First, the regulator announces lump sum taxes sN, sB and sD 2 [0, 1] which will apply to nondepositors, bankers and depositors respectively. Each banker then offers a contract (k, Q) to depositors. As in Section 2, 1k is the bank’s capital adequacy requirement. With a tax rate sB upon bankers this implies that each bank will have size k(1  sB). The fee Q which bankers receive per dollar for managing depositor funds is defined in Section 2. After bank contracts have been announced, agents decide how to allocate their funds. The regulator then collects tax and invests the tax revenue either in the banking sector or in the bond (in which case it will return in expectation RpL). After investment returns are realized and distributed, the proceeds from the deposit insurance fund will be distributed to depositors in failed institutions in proportion to the amount of their deposit. We assume that the deposit insurance fund must balance its budget: it cannot pay out more than it has collected in tax. In this simple one-period model there will be no benefit to the fund in holding over surpluses, so it will always pay out all the taxes it has collected. 3.1. Deposit rationing When there is no taxation, assumption (A3) implies that there will be ‘‘rationing’’ of deposits. This occurs because with a bank CkB RD size kB and a deposit rate ðk , the depositor individual rationality B 1ÞD constraint is just satisfied, so that all (non-bank) agents would be happy to deposit their endowment with a bank, but the banking sector will not be big enough to absorb all of their funds. The banking sector cannot expand to absorb the excess funds without either reducing capital requirements (and so violating the monitoring incentive compatibility constraint (MIC)) or reducing deposit rates (so violating depositors’ individual rationality constraint (IRUnReg)). We assume that depositors are rationed in such a way that each bank takes deposits up to the point where the monitoring incentive compatibility constraint and the individual rationality constraint are just binding. 20 In practice banking regulators often have other tools at their disposal, including the ability to set capital requirements and to close banks which violate these. We will discuss the reasons why capital requirements alone cannot be used to screen banks in this setting in Section 6.1 below. We explain in footnote 18 why capital regulations may nevertheless be useful if depositors cannot observe bank size. In Morrison and White (2005) we set out a general equilibrium model where capital requirements may be used for screening purposes and the regulator can audit banks ex post.

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Fig. 2. Time line for the regulatory game.

Deposits could still be rationed when the regulator levies taxes. We assume that if rationing occurs, all agents who attempt to deposit succeed in placing a fraction q of their pre-tax endowment with a bank.21 Notice that as the total revenue T from taxation is the sum of taxes from banks, depositors and non-depositors, it depends on the extent to which deposits are rationed:

T  lsB þ NqsD þ Nð1  qÞsN :

ð3Þ

The expression for q depends upon whether the revenue T is invested in the banking sector or in the (inefficient) bond. In both cases the size of the banking sector will be kl(1  sB), of which l(1  sB) will be bank capital. If T is invested in the storage technology, all of the remaining (k  1)l(1  sB) in the banking sector will be depositor funds. This lð1sB Þ corresponds to a pre-tax figure of ðk1Þ and there are a total of ð1sD Þ N potential depositors in the economy, each of whom is endowed with a single pre-tax dollar. If all the non-bank agents wish to deposit each will therefore succeed in depositing the following pre-tax quantity:

qst 

ðk  1Þlð1  sB Þ ; Nð1  sD Þ

ð4Þ

whenever this expression is less than 1, and 1 otherwise. If T is invested in the banking sector, of the (k  1)l(1  sB) dollars in the banking sector which are not banker endowment, max{(k  1)l(1  sB)  T,0} will be depositor funds. In practice the regulator has nothing to gain from collecting a tax revenue greater than the banking system can absorb, so we can assume that T 6 (k  1)l(1  sB). It follows that if all non-bank agents were to deposit, rationing would be given by:

qbk 

ðk  1Þlð1  sB Þ  T : Nð1  sD Þ

ð5Þ

Fig. 3. Constraints when the deposit insurance fund is invested in the banking sector.

likely to invest in a failed institution and all invest the same amount, they each expect an equal ex post payment from the fund of NT ðR  Q ÞðpL þ g DÞ. Note that the extent of rationing of deposits qbk does not appear in this expression, because we have assumed that the deposit insurance fund must balance its budget. As in the unregulated economy, the maximum possible bank size with deposit insurance occurs at the intersection of the monitoring incentive compatibility constraint (MIC) and the depositors’ participation constraint. The (MIC) constraint is unaffected by deposit insurance.22 The IR constraint is affected by the deposit insurance scheme and becomes:



 T þ qbk ð1  sD Þ ðR  Q ÞðpL þ g DÞ þ ð1  qbk Þð1  sN ÞRpL N P RpL ð1  sN Þ;

3.2. Deposit insurance fund invested in banks

ð7Þ

or, rearranging, In this section we examine the optimal level of deposit insurance when the regulator invests the deposit insurance fund in the banking technology. The regulator is unable to distinguish between sound and unsound bankers but, like the depositors, the regulator knows that investments in the banking sector are on average more productive than other investments. In this case we can substitute Eq. (3) into Eq. (5) and rearrange to obtain the extent of deposit rationing as a function of the taxes imposed by the regulator to fund the deposit insurance scheme:

qbk ¼

l½kð1  sB Þ  1 sN :  Nð1  sN Þ 1  sN

ð6Þ

Since the gross return on bank deposits is (R  Q)(pL + gD) (see Eq. (1)), the expected total return from the deposit insurance scheme in this case is T(R  Q)(pL + gD). Since all agents are ex ante equally 21 One could equivalently assume that a fraction q of agents who wish to deposit succeed in depositing all of their funds, and the remainder are forced to invest in bonds.

Q 6 R  RpL

lðkð1  sB Þ  1Þ  NsN : ðpL þ g DÞ½lðkð1  sB Þ  1Þ þ lsB 

ðIRBankInv Þ

This expression is decreasing in k and has limiting value p RgþgDD, which L is equal to (IRUnReg). The situation is illustrated in Fig. 3, from which the following result is immediate. Proposition 1. Let kB and kbk be the respective intersections of (MIC) with (IRUnReg) and (IRBankInv). Then kB < kbk. When a deposit insurance fund is created using ex ante taxation and invested in the banking sector, bank capital requirements may be loosened from k1B to k1 . bk

22 The monitoring requirement becomes (1  s B ){(R + (k  1)Q)p H  Ck} P (1  sB){(R + (k  1)Q )pL } from which the taxation terms cancel. Since much of the literature has emphasized that deposit insurance causes moral hazard by banks, this might appear paradoxical. The result follows because in our model, the return to depositors depends upon the bank’s monitoring effort even with deposit insuranceso that banks will be charged ex ante for slacking. This is not true when deposit insurance renders the depositor return insensitive to banker behavior.

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The intuition for Proposition 1 is straightforward. Tax is levied ex ante to create a deposit insurance fund which is invested in the banking sector. The returns on the deposit insurance fund are in expectation divided equally amongst depositors and so create an additional incentive for households to deposit their endowment. All else equal, households are therefore prepared to accept a lower deposit rate, shifting up (IRBankInv) relative to (IRUnReg). Bankers can thus earn a correspondingly higher return on their monitoring activities for a given bank size. This reduces their temptation to moral hazard and so reduces the amount of capital required to induce them to monitor a dollar of depositor funds. Note that we have not yet proved that deposit insurance will increase bank size, however. The ex ante taxes which bankers pay reduce their capital base, as well as increasing the leverage of their banks: bank size is k(1  sB). Social welfare W bk is proportional to the volume lk(1  sB) of funds managed by banks. For simplicity, and without loss of generality, we set the constant of proportionality equal to 1. Proposition 2 provides an expression for W bk and gives its maximal value. Proposition 2. When the proceeds of the ex ante taxation scheme (sN, sB, sD) are invested in the banking sector, the following social welfare level is achieved:

W bk ¼

ðl þ NsN ÞRpL D CðpL þ g DÞ  Rg D2

ð8Þ

:

This has maximal value

W bk ¼ l þ N; which is attained when sN assumes the following value:



sN ¼ 1 þ

l CðpL þ g DÞ  Rg D2 N

RpL D



l N

ð9Þ

:

Proof. Setting the (IRBankInv) constraint equal to the (MIC) constraint yields the following equation for bank capital:

kbk ð1  sB Þ

l N

½CðpL þ g DÞ  Rg D2  ¼

l N

 þ sN RpL D2 :

ð10Þ

Social welfare W bk is equal to the volume of funds managed by banks, lkbk(1  sB). Rearranging Eq. (10) yields Eq. (8). Welfare is increasing in sN.23 Our discussion so far has assumed that qbk 6 1: welfare is therefore maximized when sN is at its maximum value subject to this constraint. Substituting from Eq. (10) for k(1  sB) into the bank rationing Eq. (5) we obtain Eq. (9), as required. h Proposition 2 has several interesting consequences. Firstly, note that neither sB nor sD appears in the social welfare function (Eq. (8)). In other words, taxing bankers and depositors is welfare-neutral. When depositors are risk-neutral, taxing them ex ante and using the proceeds to make ex post payouts has no effect upon their incentives. Deposit insurance paid for by depositors does nothing to improve welfare in this model. Moreover, the negative effects of bank taxation upon bankers’ capital levels are precisely offset by the positive effects which follow from the relaxation of capital requirements highlighted in Proposition 1. Thus the common prescription that deposit insurance schemes ought to be paid for by banks does not hold in this model. A deposit insurance scheme funded only by banks would not improve welfare because of the adverse effects that it would have upon bank capital.24 23

Note that, in the light of (A2), the denominator of expression (8) is positive. 24 Although bank capital is fixed in our model this insight would be unaffected if bank capital could be raised externally by issuing equity or subordinated debt. Bank taxation would then affect external capital in the same way as it does banker capital in our model and hence would have the same incentive consequences.

The only tax rate that has an effect upon welfare is the rate sN that is levied upon non-depositors. Taxation of non-depositors increases the expected returns from depositing, and it makes depositing more attractive. It thus slackens the depositors’ participation constraint. As we discuss above (Proposition 1), this loosens optimal capital requirements by allowing bankers to extract higher fees from depositors. It has no effect upon bank capital and so increases bank size and hence welfare. At the optimum taxation level sN the first best level of welfare is achieved, in which all agents deposit their endowments in banks, which then invest in projects and monitor them. In equilibrium there is no deposit rationing and no taxes are gathered from non-depositors since all households opt to deposit. One possible optimal taxation arrangement which achieves this has sB = sD = 0 and sN ¼ sN . In this case the taxes will be collected and the deposit insurance fund will be empty. The deposit insurance scheme then functions more as a ‘‘stick’’ than as a ‘‘carrot’’: it threatens agents with penal taxation if they do not allow banks to reallocate their funds to socially more productive uses. The socially optimal allocation can also be achieved by flat rate taxation of sN upon all agents: banks and households. This alternative is likely to be politically more palatable. Although all agents are taxed at the same rate, only the taxes on non-depositors have any real effect. The deposit insurance scheme in this case collects general taxation and then re-allocates it only to those who deposit in the banking system. Since those who invest elsewhere will not benefit from the scheme, it creates a strong incentive to invest in the banking system. This does not seem a bad approximation to the way that (government funded) deposit insurance schemes have operated in practice. Such schemes will produce social benefits if (as in our model) the average productivity of the funds invested by the banking sector is higher than the average productivity of the funds which it attracts from other uses (i.e., bond investment). This is likely in practice if deposit insurance mostly attracts funds which would otherwise be held unproductively by agents ‘‘under the mattress’’, rather than funds that would otherwise be invested in projects with a higher yield than that earned by banks. Given that most real-world deposit insurance schemes are designed to reassure the small investor (most schemes have caps on the amount which is insured), this assumption does not seem too unrealistic.25

3.3. Deposit insurance fund invested in the storage technology In the previous section we assumed that the deposit insurance fund was invested in the banking sector. Intuitively, one might expect this to yield higher welfare for society than investing the fund in the relatively unproductive storage technology, or bond. However, this policy might not always be very realistic in practice.26 For example, in a world where investors are risk-averse, one might obtain diversification benefits from investing the deposit insurance fund less productively outside the banking sector. Therefore in this section, we analyze the case where the deposit insurance fund is invested in the unproductive storage technology. We demonstrate that in fact the social first best can still be attained, provided the regulator’s taxation policy is unconstrained. When the regulator is constrained to use a flat rate taxation scheme the first best cannot be achieved, although the deposit insurance scheme is still welfareincreasing. Proposition 3 summarizes our results. 25 This intuition would not be affected by the introduction of alternative, highyielding projects for non-depositors. In this case sN would be optimally set so as to discourage unproductive bond investment, whilst not discouraging households from investing in high yielding projects. 26 For example, the FDIC is required by statute to invest its funds in US Treasury obligations.

A.D. Morrison, L. White / Journal of Banking & Finance 35 (2011) 3400–3416

Proposition 3. When the proceeds of the ex ante taxation scheme (sN, sB, sD) are invested in the storage technology, social welfare W st is achieved, provided the rationing fraction qst of Eq. (4) satisfies the constraint qst 6 1:

ðl þ NsN ÞRpL D W st ¼ : CðpL þ g DÞ  Rg D2

ð11Þ

The maximal welfare level W st is

W st ¼ lð1  sB Þ þ Nð1  sD Þ:

ð12Þ

Proof. The (MIC) constraint is unchanged from Section 3.2, but the depositors’ IR constraint now takes the following form:

TRpL þ qst ð1  sD ÞðR  Q ÞðpL þ g DÞ þ ð1  qst Þð1  sN Þ N P RpL ð1  sN Þ:

ðIRStoreInv Þ

ðk1Þlð1sB Þ Nð1sD Þ

Assume for now that 6 1, so that qst is given by Eq. (4). Rearranging and substituting in Eq. (IRStoreInv) for qst yields the following expression for bankers’ fees Q: N

Q 6R

sN þ sB

g Dðk  1Þ þ l 1sB pL ðk  1ÞðpL þ g DÞ

ð13Þ

:

Solving Eq. (13) simultaneously with the monitoring incentive compatibility constraint (MIC) yields an expression for k(1  sB) as in Section 3.2. Multiplying this expression by l immediately yields Eq. (11) for welfare when qst 6 1. h Note that, since taxes are invested in the unproductive storage technology, it will never pay the regulator to raise more taxes than are required to remove deposit rationing. In other words, the limiting value for W st will occur when

ðk1Þlð1sB Þ Nð1sD Þ

¼ 1. This expression

can be solved for kl(1  sB) to yield Eq. (12). lð1sB Þ Note that W st ¼ W bk . In other words, provided ðk1Þ 6 1, Nð1sD Þ the welfare effects of deposit insurance are the same, irrespective of the way in which the deposit insurance fund is invested. The conclusions of Section 3.2 above are unchanged. At the maximal welfare level W st , there is no rationing and all after tax household funds are invested in the banking sector. As in Section 3.2, the taxation scheme increases welfare by discouraging households from unproductive bond investment. When sB = sD = 0, W st ¼ W bk ¼ N þ l and the social first best is achieved by levying taxes upon non-depositors at sN , as in Eq. (9). However, taxes on bankers and depositors are no longer welfare-neutral, but welfare-decreasing. This is because any tax raised from these sources is diverted from the banking sector to be invested into the unproductive storage technology, which is inefficient. Thus it is positively damaging to ask bankers or depositors to contribute to the deposit insurance fund in this case: ideally they should not be taxed. Notwithstanding this observation, we are able to prove that there should still be a deposit insurance scheme: Lemma 1. Deposit insurance funded via a flat rate ex ante tax which is invested in the storage technology is welfare increasing. The optimal tax rate is s⁄:

s



ðN þ lÞ½CðpL þ g DÞ  Rg D2   lRpL D ¼ > 0: ðN þ lÞ½CðpL þ g DÞ  Rg D2  þ NRpL D

taxationÞ

¼ ð1  sÞðl þ NÞ:

To see that s⁄ > 0 and hence that flat rate taxation is welfare increasing, note that assumption (A3) implies that (RgD2  Cg D ) (N + l) < C(N + l)pL  RlD, from which it follows that

ðN þ lÞ½CðpL þ g DÞ  Rg D2  > RlD  CðN þ lÞpL þ CðN þ lÞpL > RlpL D: 

ð16Þ

4. Ex post taxation So far, we have assumed that deposit insurance schemes are funded through ex ante taxation. In this section we investigate schemes which are paid for through ex post taxation of agents. Perhaps surprisingly, we derive the same welfare levels as those of Section 3.2, so that it is unimportant for welfare whether taxes are levied ex post or ex ante. The game which we study in this section differs from that illustrated in Fig. 2 in the following way. Although the regulator announces the taxation schedule (sN, sB, sD) at the start of the game, taxes are collected after projects end and are then distributed as deposit insurance payments. As in Section 3, we assume that the regulator’s budget must balance so that all taxes are distributed to depositors in failed institutions. With taxation levied in this way the banker’s ex ante capital level is $1 and, excluding monitoring costs and taxes, his return from running a successful bank is therefore [R + Q(k  1)]. This quantity is taxed at sB so the monitoring incentive compatibility condition in this case is the following:

QP

Ck  Rð1  sB ÞD : ðk  1Þð1  sB ÞD

ðMICep Þ

Without ex ante taxes, the rationing Eqs. (4) and (5) when all households wish to deposit both reduce to

lðk  1Þ

qep ¼

N

:

Since all depositors are equally likely to invest in both successful and unsuccessful banks, the expected return per dollar invested in the banking system is

rep 

ð1  sD ÞðR  Q Þðk  1ÞlðpL þ g DÞ T ep þ : lðk  1Þ lðk  1Þ

The first term in this expression is the total after tax return on bank deposits divided by the dollar amount l(k  1) invested in the banking system. Tep is the expected amount raised by the ex post taxation scheme:

T ep ¼ lðpL þ g DÞ½sB ðR þ Qðk  1ÞÞ þ sD ðR  Q Þðk  1Þ þ ½N  lðk  1ÞsN RpL :

ð17Þ

The second term in the expression for rep is Tep divided by amount invested in the banking system. The depositors’ participation constraint is as follows:

qrep þ ð1  qÞRpL ð1  sN Þ P RpL ð1  sN Þ: This reduces after some manipulation to the following:

ð14Þ

Proof. With a flat tax rate of s, the maximal welfare (Eq. (12)) is

W stðgeneral

3407

ð15Þ

Setting this equal to Eq. (11) allows us to solve for the optimal flat rate tax s⁄, which is given by Eq. (14).

Q 6R

l½ðk  1Þg D þ sB ðpL þ g DÞ þ sN NpL : lð1  sB Þðk  1ÞðpL þ g DÞ

ðIRep Þ

Eqs. (MICep) and (IRep) can be used to determine social welfare with this taxation scheme. Proposition 4 summarizes our findings: Proposition 4. Social welfare W ep with ex post taxation is equal to the welfare level W bk attained when deposit insurance is funded through ex ante taxes which are invested in the banking sector.

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Proof. Per bank social welfare in an equilibrium with symmetric banks is given by the amount invested in the banking system. Without ex ante taxes, this is equal to k. Simultaneous solution of Eqs. (MICep) and (IRep) yields k ¼ W bk , as required. h This result is somewhat surprising: the received wisdom on ex post taxation is that it is bad for bankers’ incentives as it takes away from successful bankers some of the fruits of their labors to give these away to the unlucky depositors in failed institutions. Thus it reduces the effort which sound bankers will devote to monitoring. In fact, as Proposition 4 demonstrates, this is not the whole story: this intuition holds only for a given bank size. If bankers do not have to pay taxes ex ante, their starting capital will be larger and this will improve their incentive to monitor in a way which exactly offsets the reduction due to taxation ex post. Another way to think about this is that taxation which occurs ex post may be regarded as simply the retrieval of ex ante taxes which have been deposited in the banking system. When ex post taxation is levied at levels determined ex ante, it has the same welfare effects as the ex ante taxes investigated in Section 3.2. 5. Deposit insurance with inter-generational transfers In practice, deposit insurance schemes typically guarantee a specific payment to depositors if their bank fails. In the model which we have so far considered the deposit insurance fund’s budget balance constraint precludes such a scheme, since (even with a stochastic ex post taxation rate) it would be impossible for the fund to meet its obligations in the unlikely event that every bank failed. In this section we allow for inter-generational transfers via the deposit insurance fund. This allows us to examine guaranteed payout deposit insurance funds. For simplicity (and without loss of generality when inter-temporal transfers are allowed), we consider ex post taxation schemes. It transpires that the welfare conclusions of Section 4 are unaffected. In this section we make assumptions in addition to those of previous sections. First, we assume that the deposit insurance fund, which is infinitely lived, is able to borrow and to lend unlimited quantities from an external capital market. Second, we assume that the interest rate on the associated borrowing and lending is 0%. This is equivalent to assuming that, in real terms, the economy, and hence taxation revenues, grows at the same rate as the interest rate to which the fund is subject. Third, we assume that banking licences are reassigned in every period, so that in each period the depositors use the same value for the quality parameter g.27 The possibility of inter-temporal wealth transfers via the deposit insurance fund will not affect the incentives of bankers and so the monitoring incentive compatibility constraint is given by Eq. (MICep). To understand the depositors’ participation constraint, note that with inter-generational transfers funded at an interest rate of 0%, we only require the budget to balance in expectation. In other words,

T ep ¼ lðk  1Þð1  ðpL þ g DÞÞP;

ð18Þ

where Tep is the total revenue from taxation, given by Eq. (17), and P is the payment which depositors in failed institutions receive. The depositors’ IR constraint is then

q

ð1  sD ÞðR  Q Þðk  1ÞlðpL þ g DÞ þ qð1  ðpL þ g DÞÞP lðk  1Þ

Substituting for P from the budget balance constraint (18) we obtain (IRep). The constraints of this section are therefore the same as those of Section 4. We have therefore proved the following result: Proposition 5. Social welfare W ig with fixed deposit insurance payouts and inter generational wealth transfers allowed is equal to the welfare level W bk ¼ W ep attained when the deposit insurance fund must balance its budget in each period Of course, Proposition 5 requires agents to be risk-neutral. If inter-temporal transfers can smooth expected payouts from the deposit insurance fund then they will increase the welfare of risk-averse agents. 6. Sorting bank types In the foregoing we assumed that welfare was maximized when all funds were invested in the banking sector, since all banks do at least as well as the bond. Yet since some banks are better than others at investing deposits, one might wonder whether the regulator can do better by trying to select out the unsound banks and channel more funds to the good banks. Certainly if the regulator can employ any policy such as periodic inspections or audits to deter unsound banks and reduce the severity of the adverse selection problem by raising the value of g, then, all other things being equal, this will improve welfare. So far, we have implicitly assumed that the regulator is already employing such complementary policies to the full extent of its ability, so that the parameter g can be considered a function of the regulator’s ability (for further discussion, see Section 7.2 below). In this section we ask whether, given the tools available, one might expect either the regulator or the good bankers collectively to attempt to find a separating mechanism to resolve the adverse selection problem. We examine the use of capital requirements imposed upon bankers by the regulator; cross-subsidization schemes operated by the regulator to bribe unsound bankers to leave the market; and bankers’ clubs (‘‘co-insurance’’) set up by bankers in an attempt to screen out poor bankers. We show that our basic model is robust to these extensions: banks cannot be sorted when their types are pure private information. We conclude this section with an analysis of two substantial extensions of our model where sorting of banks is possible. In the first of these, projects have a positive return in failure states. Bankers can therefore offer contracts under which they give up this return if they fail, and hence are only rewarded after project success. We show that, provided bank size is small enough, it is possible to separate good from bad bankers in this set-up. We extend this model further by allowing the regulator to perform an imperfect interim audit of banks, and to close those that are found to be of low quality. This ability increases the maximum bank size that can exist in a separating equilibrium. Deposit insurance is not useful in the separating equilibrium. When the audit is sufficiently reliable, the separating equilibrium dominates the equilibrium with deposit insurance. However, we demonstrate that, for a non-trivial range of parameter values, the pooling equilibrium with deposit insurance is socially preferred to the separating equilibrium. 6.1. Capital requirements

þ ð1  qÞRpL ð1  sN Þ P RpL ð1  sN Þ: 27 Without this assumption, depositors would become increasingly certain that long-lived banks were sound. In such a situation, new banks would be squeezed out of the market, or older banks would earn equilibrium reputational rents. Either scenario would introduce complications which are outside the scope of this paper.

The regulator could impose severe capital requirements on the banking industry in an attempt to select out the bad bankers. Note however that the unsound bankers’ outside option is to run their own project. This is equivalent to running a bank with a capital requirement of 100%. The regulator must set a lower capital

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requirement to avoid autarky for the sound bankers and such a requirement will always be accepted by the unsound bankers. The reason is that they can extract fees Q on any deposits which they manage, and for unsound bankers, managing deposits is costless. Thus capital requirements alone cannot resolve the adverse selection problem in this model, because unsound bankers will always mimic sound bankers.28 The argument here is similar to the reason Chan et al. (1992) find that it is impossible to induce depository institutions to reveal their risk levels by choosing a revealing capital requirement and deposit insurance premium pair. However, their result holds only for unsubsidized schemes. In our analysis we allow for a net subsidy to the deposit insurance system. 6.2. Cross-subsidization schemes We now consider schemes which may involve some kind of subsidy to unsound bankers to induce them not to participate in the banking sector. Consider in particular a scheme under which the regulator selects l volunteers from the pool of potential bankers, imposes an ex ante tax of sB upon those who elect to run banks and then, instead of using the proceeds to fund a deposit insurance scheme, pays them to those agents from this pool who elect not to bank.29 These agents can use this sum to augment their initial endowment for investment, but they cannot collect deposits. We call such a scheme ‘‘cross subsidization’’. Cross subsidization schemes will only be effective if the non-deposit-taking option is more attractive to the unsound bankers than to the sound ones. This is difficult to achieve since monitoring is efficient, so the sound bankers generate greater profits from subsidies than unsound ones. Moreover, if the monitoring incentive compatibility constraint binds, sound and unsound bankers will earn the same profits from running a bank, so that separation will be impossible. Below we demonstrate that this intuition extends even to the case where capital requirements are tightened so that the MIC is slack: if a sound agent is willing to run a bank, an unsound agent will also wish to do so. The single-crossing condition runs in the wrong direction. Intuitively, the only way to support an equilibrium where only sound agents run banks is if the number (1  g)l of unsound bankers receiving subsidies is sufficiently low, so that if one agent elects to stop taking deposits he will so significantly reduce the per agent subsidy for not banking that he prefers not to do so. This is obviously impossible if the number of banks is large enough. Lemma 2 provides a sufficient condition for cross subsidization schemes to be ineffective. Lemma 2. Cross subsidization schemes cannot separate sound from unsound bankers when condition (19) is satisfied.

ð1  gÞl >

ðRD  CÞpH : CpL

ð19Þ

28 Morrison and White (2005) demonstrate in a general equilibrium model that the regulator can use capital requirements to resolve the adverse selection problem when banker entry is endogenous and the regulator is able to screen potential bankers. Unsound bankers then have depositing as an outside option. But they show that only regulators with poor screening technology will resolve the adverse selection problem: better regulators prefer not to separate good from bad bankers as this would involve shrinking the banking sector more than is desirable. Thus our assumption that screening using capital requirements does not occur is robust to the use of a more general setting for appropriate parameter ranges. In Section 6.4 below we examine another setting where it is feasible but not necessarily desirable to employ capital requirements for screening. 29 Without loss of generality we consider the case where all the proceeds from this tax are paid out to the non-participating banks, since this allows the largest possible ‘‘bribe’’ and thus the greatest chance of separation.

Proof. A cross subsidization scheme that resolves the adverse selection problem must render deposit taking incentive compatible for sound bankers. Suppose that all sound bankers elect to take deposits. If one sound banker deviates then the total tax subsidy per non-deposit-taking banker is

g l1 ð1gÞlþ1

sB . The deposit taking IC

constraint for sound bankers is therefore given by Eq. (20):  ½R þ ðk  1ÞQ pH  Ckgð1  sB Þ P 1 þ

 gl  1 sB ðRpH  CÞ: ð1  gÞl þ 1

ð20Þ

A successful cross subsidization scheme must in addition render non-deposit-taking incentive compatible for unsound bankers:

 ½R þ ðk  1ÞQ pL ð1  sB Þ 6 1 þ

 g sB RpL : 1g

ð21Þ

Rearranging these equations yields the following constraint upon sB:

ðk  1ÞðQpH  CÞ g l1 ½R þ ðk  1ÞQ pH  Ck þ ð1gÞ lþ1 ðRpH  CÞ

P

P sB

ðk  1ÞQpL : g ½R þ ðk  1ÞQ pL þ 1g RpL

ð22Þ

Cross multiplying the outside terms in this expression and rearranging yields the following necessary condition for a suitable tax rate sB to exist:

CR½ð1  gÞl þ 1 þ CQ lð1  gÞ þ QRpH P 0:

ð23Þ

Note that, if separation occurs, the depositors’ IR constraint reduces to Q 6 RDp/pH. So a sufficient condition for a separating equilibrium not to exist is for condition (23) to be violated when Q = RDp/pH. In other words, for condition (24) to be satisfied:

CpH ½ð1  gÞl þ 1 þ C lð1  gÞD þ RpH D < 0:

ð24Þ

Rearranging this expression yields Eq. (19). h In fact, even if an equilibrium exists in which sound agents run banks and unsound agents do not, it may be only one of many: Lemma 3. In a neighborhood of the MIC constraint, if a crosssubsidization equilibrium exists in which each of the gl banks is sound, then provided depositors’ IR constraints are satisfied, any number of sound banks is sustainable through a cross-subsidization equilibrium.

Proof. In the Appendix. h Lemma 3 indicates that, even when a good equilibrium can be supported by cross-subsidization, there is no particular reason to suppose that this is the equilibrium which will arise. The results of this section suggest that attempting to separate out bank types by combining capital requirements and cross-subsidization schemes is not likely to form the basis of effective policy. Given the adverse selection problem, the regulator cannot separate bank types and rather than subsidizing non-entrants is best off providing a deposit insurance subsidy to all banks which choose to enter, as we have assumed in deriving the results of previous sections. For similar reasons we conjecture that it cannot be optimal to charge a licence fee to those wishing to run a bank. 6.3. Voluntary deposit insurance schemes and coinsurance Coinsurance schemes between banks have operated in the past with varying degrees of success (see for example Calomiris, 1990; Calomiris and Schweikart, 1991; Kumbhakar and Wheelock, 1995). Under such schemes bankers form clubs to insure their members,

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with strict entry requirements designed to ensure quality of membership. Such clubs are ineffective in a model such as ours where agents are asymmetrically informed. Since unsound bankers are less likely than sound ones to be required to pay out under such schemes, they have stronger incentives to enter them than sound bankers.30 Screening of members through the imposition of capital requirements cannot succeed for the reasons outlined in Section 6.1. Similarly, willingness to enter a voluntary deposit insurance scheme cannot itself serve as a separation device. (We assume that a voluntary scheme does not receive any net subsidy from outside the scheme; we considered schemes involving cross-subsidization in the previous subsection.) In our model, the general public would never wish to deposit in an unsound bank if they knew its type, so unsound banks will simply mimic the decision of sound banks in entering a voluntary scheme or not. Since this is a signaling game, whether any bank joins a voluntary deposit scheme depends on the general public’s off-the-equilibrium-path beliefs about the type which is most likely to join. An equilibrium where all join would be sustained by beliefs that any bank outside the scheme is an unsound bank; an equilibrium where no one joins would be supported by beliefs that joiners are unsound types. In our model of risk-neutral agents, voluntary deposit insurance schemes, since they cannot function as screening devices, do not serve any useful purpose. Having banks pay a fee to enter the scheme which is then paid out to depositors of the failing institutions in the scheme is equivalent to taxing bankers to pay for deposit insurance in our earlier model, and is thus welfare-neutral. More generally, one might expect coinsurance schemes to arise when bankers are better able to learn about one another than is the regulator (see Rochet and Tirole (1996) for a model where banks monitor one another). Our model indicates a possible reason why private schemes have in the last half century or so been replaced by schemes funded through public taxation. We have argued that deposit insurance is most effective when it is not funded by bankers: that deposit insurance should instead be funded through general taxation. One would therefore expect a state-operated deposit insurance scheme to crowd out less efficient private schemes. 6.4. Separating equilibria with positive failure payouts and interim screening We now consider two extensions of our model. First, we assume that banks investments’ yield positive returns even in the ‘‘low’’ or ‘‘failure’’ state. Second, we allow for positive failure state payoffs and at the same time force banks to undergo an audit of their investments at an interim date. The purpose of these extensions is to demonstrate circumstances in which it is possible to use contracts on capital requirements and payments to bankers (deposit rates) in such a way as to separate sound from unsound bankers (which was impossible in the basic model). We show that when the adverse selection problem is resolved in this way, deposit insurance is not useful. But we also show that from a welfare point of view, it may be preferable to allow sound and unsound bankers to pool and to use deposit insurance to expand the banking system, as in the basic model that we have studied up until now. Because our purpose is to provide a benchmark with which the value of deposit insurance can be compared, we concentrate throughout on contracts which will implement the separating equilibrium with the largest possible banks.31 30 For evidence that this was indeed the case for the deposit insurance scheme which operated in Kansas 1909–1929, see Kumbhakar and Wheelock (1995). 31 In the tradition of contract theory, we also suppress the dependence of equilibria on out-of-equilibrium beliefs. One interpretation is that the banking regulator is the uninformed principal specifying the set of contracts which banks are allowed to offer.

6.4.1. Positive failure payoffs Consider a modification of our basic model in which projects pay r > 0 after failure, and r + R > r after success. Bankers can now receive a non-zero payment after failure, so we denote by q and Q the payment per dollar of depositor funds that bankers receive after project failure and success, respectively.32 Sound bankers earn the following expected return monitoring when they run a bank of size k:

r þ RpH þ ðk  1Þðq þ QpH Þ  Ck;

ð25Þ

and their outside option is r + RpH  C, so that their participation constraint is given by Eq. (26):

q P C  QpH :

ð26Þ

Unsound bankers and sound bankers who elect not to monitor receive the following expected income from a bank of size k:

r þ RpL þ ðk  1Þðq þ QpL Þ:

ð27Þ

Hence unsound bankers will wish to run banks as long as Eq. (28) is satisfied:

q P QpL :

ð28Þ

Banks have limited liability, but bankers’ capital generates a return r in the low state, so banks can receive negative fees in the low state only to the extent that they can pay these fees from this wealth. Therefore the limited liability constraint in the low state is given by Eq. (29):

r þ ðk  1Þq P 0:

ð29Þ

It is immediate from expressions (25) and (27) that the monitoring incentive compatibility constraint for sound bankers is still given by expression (MIC). In order to separate sound banks from unsound ones, we need to ensure the contract between bankers and depositors specifies downstate payments q that violate the unsound bankers’ IR constraint (28), and satisfy the sound bankers’ IR constraint (26) and the limited liability constraint (29). In other words we must have r QpL > q P C  QpH and QpL > q P  k1 . There exists a q that satisfies these conditions precisely when conditions (30) and (31) are satisfied:

Q>

C ; D

r : Q< pL ðk  1Þ

ð30Þ ð31Þ

Eqs. (30) and (31) are illustrated in Fig. 4: both are satisfied in the shaded region. Note that the monitoring constraint (MIC) approaches DC as Q increases, so that monitoring is incentive compatible whenever it is possible to separate sound from unsound bankers. Our first result is that, although there is a positive failure state payoff, the equilibrium contract leaves no income to bankers after failure: Lemma 4. In any separating equilibrium, the banker’s low state limited liability constraint (29) binds. Hence all of the income from a failing project is paid to depositors.

32 In allowing payments contingent only on success or failure, we are implicitly ruling out contracts where unsound bankers are cross-subsidized by sound ones and receive payments to induce them not to accept deposits. Whilst theoretically possible, such contracts are unattractive in practice because they would tend to induce a lot of entry from agents purporting to be unsound bankers.

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6.4.2. Interim regulatory auditing We now extend the model of Section 6.4.1 by allowing for interim regulatory auditing. Specifically, we force banks to have their investment projects audited at an interim date after they have made their investments, but before these investments have yielded any returns. For each bank, the audit reveals the expected returns of the banker’s investments with probability k, and otherwise it reveals nothing. k can thus be considered to measure the quality of banks’ disclosures at the interim date.33 We assume that if an audit identifies a non-monitored project then that project will be immediately liquidated for its expected value, r + RpL. The simplest interpretation is that the regulator recognizes that the bank is insolvent and steps into close the bank. We assume that in this case, the regulator can distribute the returns from liquidated projects as it sees fit: in particular, the regulator can pay all of the proceeds of liquidation to depositors.34 The possibility that an interim audit will lead to the liquidation and confiscation of investments reduces the expected return to unsound bankers, and to sound bankers that do not to monitor: the return to both under this regulatory regime is given by expression (35): Fig. 4. Separating region with positive low state payoff.

ð1  kÞðr þ RpL þ ðk  1Þðq þ QpL ÞÞ:

Proof. The separating equilibria are precisely those that lie within the shaded region of Fig. 4. Within this region, social welfare and sound banker profitability are both increasing in k, since larger banks result in more monitoring, and more information rent for bankers. Hence any separating equilibrium will lie on the locus r of Eq. (31). Along this line, QpL ¼ q ¼  k1 , so that the limited liability constraint (29) binds, as required. h Lemma 4 states that, when our model is extended to allow for positive failure state payoffs, the constrained optimal contract between the banker and the depositors is a standard debt contract: the depositor is promised a payment of R  Q and when the bank is unable to make this payment, the depositor receives all of the banker’s assets (including the returns on the banker’s own invested capital). The banker prefers this contract ex ante because it allows him to run the largest possible bank while still committing to monitor, and hence to extract the highest possible information rent from his monitoring ability. We now consider the depositors’ participation constraint. In a separating equilibrium, depositors know that they are placing their funds with a sound bank and will be willing to do so provided that the return r  q + (R  Q)pH from doing so exceeds their outside option, r + RpL. This is the case precisely when condition (32) is satisfied:

Q6

RD  q : pH

ð32Þ

Recall from the proof of Lemma 4 that separating equilibria lie on the locus of Eq. (31) so that, in any separating equilibrium, QpL = q. Hence Eq. (32) reduces to QpH 6 RD + QpL, or

Q 6 R:

ð33Þ

The region where Eqs. (30), (31), and (33) are satisfied is indicated in Fig. 4 with cross-hatching. Since the regulator is concerned only to maximize the volume of funds under skilled management, the maximum possible welfare in the separating equilibrium occurs at the intersection of Eqs. (30) and (31): that is, where 

k ¼ ksep  1 þ

rD : CpL

ð34Þ

We postpone our discussion of the welfare properties of the separating equilibrium to Section 6.4.3.

ð35Þ

Monitoring is incentive compatible under interim audits when expression (25) exceeds expression (35); that is, when condition (36) is satisfied:

Q P Q MIC ðk; kÞ 

Ck  RDk  kðr þ qðk  1ÞÞ ; ðk  1ÞDk

ð36Þ

where

Dk  D þ kpL :

ð37Þ

Note that condition (36) reduces to condition (MIC) when k = 0. Moreover,

Q MIC ðk; 0Þ  Q MIC ðk; kÞ ¼ k

Dðr þ ðk  1ÞqÞ þ CkpL > 0; ðk  1ÞDDk

ð38Þ

so that interim auditing relaxes the monitoring incentive constraint as one would expect. The expected return to unsound bankers who choose to run a bank when there is interim auditing (Eq. (35)) exceeds their outside option r + RpL precisely when the participation constraint (39) is satisfied:

qP

kðr þ RpL Þ  QpL : ð1  kÞðk  1Þ

ð39Þ

Interim audits affect only unmonitored investments. Hence the return to depositors from investing in a bank is unaffected in separating equilibria where monitoring is incentive compatible, and hence the depositor IR constraint is again given by Eq. (32). Similarly, because the return to sound bankers who monitor is unaffected by the introduction of the auditing technology, their participation constraint is once again given by expression (26), and the limited liability constraint in the low state is again given by Eq. (29). As in Section 6.4.1, two constraints on Q and k are necessary for separating equilibria to exist. First, condition (39) can be violated 33 Cordella and Yeyati (2002) also study a model of deposit insurance and disclosure in banking. However, they focus on the two limiting cases of full risk-based deposit insurance and uninsured but fully informed depositors. They show that the two extreme regimes yield equal levels of risk taking in equilibrium. They do not consider how the optimal level of deposit insurance varies with the extent of disclosure, as we do here. 34 An alternative interpretation is that depositors see that the bank is insolvent and run on the bank, forcing the bank to liquidate its investments and pay out all the proceeds to depositors.

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Proof. Recall from Section 3.2 and Fig. 3 that deposit insurance has no effect on the monitoring constraint: its only effect is to relax the depositors’ participation constraint. Hence deposit insurance can raise welfare only when this constraint binds. But we have already demonstrated that, in the separating equilibrium with the largest banks, the depositors’ IR constraint is slack. h We now turn to a comparison of welfare levels in the regulator’s preferred separating equilibrium with welfare the best pooling equilibrium with deposit insurance, as studied in the rest of the paper. It transpires that the separating equilibrium dominates the pooling equilibrium with deposit insurance when the quality k of the regulator’s screening technology is high enough: The regulator prefers separating equilibria to pooling equilibria precisely when condition (42) is satisfied:

k P kSc 

Fig. 5. Separating region with interim regulator audit.

while the limited liability constraint (29) is satisfied precisely when condition (40) is satisfied:

Q6

  1 kðr þ RpL Þ r ; þ pL ð1  kÞðk  1Þ k  1

ð40Þ

second, condition (39) can be violated while the sound banker IR constraint (26) is satisfied precisely when condition (41) is satisfied:

QP

  1 kðr þ RpL Þ : C ð1  kÞðk  1Þ D

ð41Þ

Constraints (40) and (41) are illustrated in Fig. 5; separating equilibria are possible within the shaded region of this figure. The maximum possible bank size in a separating equilibrium with auditing is at the intersection (kScReg, QScReg) of the two constraints. The following result, whose proof appears in the Appendix, demonstrates that banks of this size are feasible: Lemma 5. The depositors’ participation constraint (32) and the banker’s monitoring constraint (MIC) are both satisfied for banks at the intersection of constraints (40) and (41). Hence the maximum bank size in a separating equilibrium is

kScReg ¼ 1 þ

r Dk þ RkpL pH : Cð1  kÞpL 

Note that when k = 0, kScReg is equal ksep (see Eq. (34)). Lemma (5) demonstrates that, with positive low-state project returns and regulatory audits, it is possible through a judicious selection of k and Q (and appropriate out-of-equilibrium beliefs) to separate sound from unsound bankers without using deposit insurance, as in the rest of the paper. It does not however follow that there is no need for deposit insurance: even when screening is possible, the resultant banks may be so small that it is better not to screen. The following section determines conditions under which screening is the socially best policy. 6.4.3. Welfare value of screening We now welfare-rank equilibria with screening, and equilibria without. As in the rest of the paper, the appropriate measure of welfare is the volume of depositor funds that is placed under the management of sound bankers. We start by observing that deposit insurance is relevant only in the latter case: Deposit insurance cannot raise welfare in separating equilibria.

CðN  2lÞpL  rlD : ðCðN  2lÞ þ lðr þ RpH ÞÞpL

ð42Þ

When there is no screening, every depositor invests, and a total of Nl is invested in sound institutions; when there is screening, a total of k  1 dollars is invested in each of the lg sound banks, so that the total volume of depositor funds under sound banker management is lg(k  1). Hence it is better to screen only when Nl < lg(kScReg  1); this requirement reduces to Eq. (42), as required. When r = 0, 0 < k < 1; for sufficiently positive r, kSc is negative, so that separating equilibria are always preferred. Finally in this section, we examine the effect of better auditing upon the level of deposit insurance required to support the banking system. Clearly, no deposit insurance is required in separating equilibria. In pooling equilibria some deposit insurance is required. We assume, in line with Section 3.2, that deposit insurance is financed via a tax sN levied upon non-depositors, and that taxation revenues are invested in the banking sector. Since the role of deposit insurance is to relax the depositors’ participation constraint, we assume that sB = sD = 0. With these assumptions, each agent deposits a total q ¼ ðk1ÞNlT in the banking sector, where T = N(1  q)sN is the total tax take (see Eq. (6)). Each depositor will receive (R  Q) if its bank is not liquidated after audit, and it subsequently succeeds; this happens with probability

pSuccAud  kgpH þ ð1  kÞðpL þ g DÞ ¼ gpH þ ð1  gÞð1  kÞpL ; with complementary probability, the depositor invests in a bank that fails, and hence generates an income r  q. Bankers maximize their depositor base by setting q = r/(k  1) so as most efficiently to satisfy their monitoring incentive constraint, so depositors’ failure state income is rk/(k  1). Finally, bankers that are liquidated after an audit yield a return r + RpL per dollar invested, and this occurs with probability k(1  g). Funds that are not invested in the banking sector are self-managed, so the depositor’s IR constraint in pooling equilibria is given by Eq. (43):



 T þ q ððR  Q ÞpSuccAud þ rkð1  pSuccAud Þ=ðk  1Þ N

ð43Þ

þ ðkð1  gÞðr þ RpL ÞÞ P RpL ð1  sN Þ: Substituting for q and T in this expression and rearranging yields Eq. (44):

Q 6 Q IRaudit ðk; s; kÞ 

Num ; ðk  1ÞlðgpH þ ð1  gÞð1  kÞpL Þ

where

Num ¼ rlðk þ ð1  gÞðk  1Þk  kðgpH þ ð1  gÞð1  kÞpL ÞÞ þ Rðgðk  1ÞlD þ NsL pL Þ:

ð44Þ

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The maximum bank size in an equilibrium with interim auditing is at the (k, Q) values where QIRaudit(k, s, k) and QMIC(k, k) intersect. QIRaudit is increasing in both sN and k, so that taxation and auditing are substitute ways of relaxing the depositor IR constraint; at the same time, QMIC is decreasing in k, so that better auditing hardens the bank’s monitoring incentive compatibility constraint. The following proposition is therefore immediate: The equilibrium level of taxation required to support the deposit insurance scheme in pooling equilibria is decreasing in the quality k of the regulator’s auditing technology.

7. Empirical implications 7.1. Deposit insurance and banking sector quality Examination of Propositions 2 and 3 and Lemma 1 shows that independently of how deposit insurance funds are invested and when taxes are levied, the optimal level of deposit insurance will be decreasing in the quality of the banking system: Corollary 1. The tax rate s and hence the optimal size of the deposit insurance fund is decreasing in banking sector quality g. Before we can comment on the empirical validity of corollary 1, we must consider how to treat countries without an explicit deposit insurance scheme. In their study of deposit insurance around the world, DemirgucKunt et al. (2005) simply treat all countries which do not have an explicit deposit insurance scheme as having an implicit deposit insurance scheme. In other words, countries with no explicit deposit insurance scheme are not regarded as lacking deposit insurance but rather as having implicit deposit insurance with an unspecified but potentially high degree of coverage. This is because the problem in practice with implicit deposit insurance is that since its level is not explicitly limited ex ante it can be difficult to impose limits on its levels ex post in a crisis situation. Gropp and Vesala (2004) provide evidence which supports this idea within the EU. They show that the establishment of an explicit deposit insurance scheme in a country reduces bank risk-taking, and their interpretation of their result is that the introduction of explicit deposit insurance is associated with a de facto reduction in the scope of the safety net. Hence, countries without an explicit deposit insurance scheme may be implicitly committed to excessively large state guarantees of banks. Our theory suggests it would be better for such countries to establish a scheme with explicit limits on coverage, because too much deposit insurance allows banks to violate their MIC constraints, eliminating the value-added of the banking sector and the rationale for deposit insurance. We can also consider the positive implications of our theory. Suppose that at least those governments that have considered the issue of deposit insurance thoroughly enough to have adopted an explicit scheme have tried to adopt one with desirable features. Corollary 1 above then suggests that better-regulated banking sectors should provide less deposit insurance. We test this prediction with the data in Demirguc-Kunt et al. (2005), using GDP per capita as a crude proxy for quality of regulation. Consistent with our model, we find that high-income countries have schemes with relatively lower coverage than low-income countries.35

35 Taking logs, a regression of 2003 deposit insurance coverage limits as a percentage of GDP per capita on 2002 GDP per capita yields a coefficient of 0.4 (significant at the 1% level). Repeating the exercise using the ratio of insurance coverage to deposits per capita gives a similar result: a coefficient of 0.5 on GDP per capita (also significant at the 1% level).

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7.2. Deposit insurance and banking crises The result in corollary 1 that the extent of deposit insurance is inversely correlated with the quality of the banking sector may also help us reconcile the apparent contradiction between the increasing adoption of deposit insurance schemes on the one hand with recent findings in the empirical literature that deposit insurance schemes seem to be associated with reduced banking sector stability on the other (see for example Demirgüç-Kunt and Detragiache, 1998; Demirgüç-Kunt and Kane, 2002; Barth et al., 2004). For example, Barth et al. (2004) report univariate regressions between a dummy variable indicating whether a banking crisis has taken place or not and their ‘‘moral hazard index’’. Their ‘‘moral hazard index’’ is taken from Demirgüç-Kunt and Detragiache (2002), who used a principal components analysis to capture the presence of different features of deposit insurance schemes such as whether there is any co-insurance, whether the scheme covers inter-bank and foreign currency deposits, whether membership is compulsory or voluntary, how the scheme is managed and how high is explicit coverage. The index is designed so that higher values suggest that the design of the scheme is more likely to encourage moral hazard by banks. Therefore, the empirical literature suggests that (for instance) greater deposit insurance coverage is associated with a higher probability of a banking crisis. The problem with these empirical analyses is that they are unable to fully control for the quality of the banking sector. Our model suggests that this lack of control variables should lead one to take cautious view of the positive association between deposit insurance and crisis incidence for the following reason. A banking system with a lower fraction of sound banks g is more likely to experience widespread failures (which in most empirical models would be classified as a banking crisis). But it is precisely in such a system that depositors are most reluctant to invest their funds in banks, and therefore optimal policy calls for larger deposit insurance subsidies.36 So one would expect to see a positive association between the generosity of deposit insurance and the occurrence of banking crises in the data, without this necessarily implying that one is causing the other. Rather, a weak banking system, a lack of regulatory resources for screening and monitoring banks and enforcing capital regulations, or a weak regulatory environment, may each cause banking sector instability and at the same time make a generous deposit insurance scheme optimal.37 Indeed, empirically, it is quite common for deposit insurance schemes to be introduced during or in response to times of stress for the banking system, so that the reverse causality problem is not purely a theoretical issue. For example, deposit insurance was first introduced in the US following the banking crisis of the Great Depression. Similarly, the UK government responded to recent financial sector fragility by announcing in the autumn of 2007 an explicit 100% government guarantee on all retail deposits at Northern Rock, and, as discussed in the Introduction, it subsequently extended the UK deposit insurance scheme.38

36 Of course, our model assumes that regulators can be relied upon to act in the public interest. Demirgüç-Kunt and Kane (2002) argue that generous deposit insurance schemes may be problematic where corruption is rife and the rule of law is erratic, so that legal reform should accompany deposit insurance reform. 37 To be more explicit, let the average quality of banks g(a), be a function of the regulator’s ability a to prevent unsound banks from getting licences. Then if countries with worse regulators (low a) have both worse quality banks and more banking crises, but deposit insurance is more likely to be adopted where it is most beneficial, then our model would predict a positive correlation between banking crises and deposit insurance, although deposit insurance would not be causing crises. 38 See ‘‘Darling revises compensation rules for savers’’, Scheherazade Daneshkhu, p. 4, Financial Times, 2 October Daneshkhu (2007) and ‘‘Lose the safety net and banks might find balance’’, Paul Amery, p. 11, Financial Times, 23 July 2008.

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7.3. Deposit insurance and banking sector size

8.2. Recapitalizations

According to our model, the reason for introducing deposit insurance is to expand the size of the banking sector, which is otherwise inefficiently small. So, controlling for banking sector quality, introducing a deposit insurance scheme (provided it is somewhat subsidized and not funded entirely by depositors and bankers, which as we have seen, has no net effect) should lead to a larger banking system. Barth et al. (2004) report univariate regressions between ‘‘bank development’’ and their ‘‘moral hazard index’’ (described above).39 As mentioned above, Barth et al. (2004) unfortunately do not use any controls for banking quality, but they nonetheless find a positive though insignificant relationship between deposit insurance and bank development, suggesting that the presence of deposit insurance may indeed expand the size of the banking sector. Barth et al. (2004) also find that the stringency of capital regulation (the ‘‘capital regulatory index’’) is positively associated with the ‘‘moral hazard index’’ (see their Table 2). Again, this is also consistent with our theory, since the expected quality of banks is omitted from the relation. When the quality of the banking system is poor, capital regulation must be tighter to prevent moral hazard by banks – so the banking system would have to be smaller. The regulator should optimally try to offset this effect by putting in place a more generous deposit insurance scheme. However, if the offsetting is incomplete then in the data we will see a correlation between generous deposit insurance and tighter capital requirements – both driven by a weaker banking system.

An alternative to using the money raised though the deposit insurance fund to pay out depositors would be to use it to augment banks’ capital through an ex ante subsidy. One could consider such a policy to be a ‘recapitalization’ of banks. Given the irrelevance of the timing of the deposit insurance subsidy to banks (ex post versus ex ante) already proved, it should be clear that such a policy can be welfare-enhancing and substitute for the imposition of deposit insurance. General taxation could be used to collect funds to augment banks’ capital and allow them to expand to absorb remaining funds without violating their monitoring incentive compatibility constraints. (Note that for the usual reasons, recapitalizations using funds drawn from the banking sector itself will be ineffective, at least in our model where all banks are symmetrically placed.) In our stripped-down model of risk-neutral depositors, recapitalizations and deposit insurance serve exactly the same purpose, and so the choice between them must be one of political expediency. It would be interesting to see in a richer model which factors should lead a government to choose recapitalization over deposit insurance.40 For example, if depositors are risk-averse, rather than risk-neutral as we have assumed here, then it seems that it would be preferable to use any net subsidy to the banking system to fund deposit insurance rather than recapitalizations, because this provides depositors with better insurance against poor bank outcomes ex post. By the same token, with risk-averse depositors, taxation of depositors to fund deposit insurance will now be welfare-enhancing rather than welfare-neutral, as the provision of such insurance on an actuarially fair basis will now encourage depositors to deposit, whereas previously it had no impact.

7.4. Deposit insurance and the quality of bank disclosures According to Proposition 7 above, as the quality of bank disclosures to depositors or regulators improves, the need for deposit insurance declines. So, other things (in particular, the quality of the banking sector) being equal, we should see a negative relationship between deposit insurance coverage and disclosure quality. Also, the better is disclosure, the larger the banking system for a given level of deposit insurance. As far as we are aware, these propositions have yet to be tested by the empirical literature.

8. Extensions, robustness checks and discussion 8.1. Systemic risk This discussion of the possibility of banking crises leads us to the issue of systemic risk. In our analysis of using ex post taxation to fund deposit insurance (Section 4), we have implicitly assumed that, although projects are perfectly correlated within banks, they are independent across banks. This set up might be considered appropriate for a regional banking network, where banks invest in the industries present in their regions, and industries differ across regions, but it is clearly very stylized. The assumption is useful in simplifying the analysis of ex post funding of deposit insurance because, if there are sufficiently many independent ‘‘regions’’, we can be sure that there will be sufficient funds to raise the appropriate level of deposit insurance ex post. If, however, the outcomes of bank investments are correlated (e.g., all banks invest in the same industries and assets) then this may not be the case. Thus, when systemic risk is present we conjecture that it may be preferable to fund deposit insurance through ex ante rather than ex post payments. 39 Bank development is measured as claims on he private sector by deposit money banks as a percentage of GDP, averaged over 1997–1999.

8.3. Deposit insurance, banking competition and capital requirements In our model, the presence of binding capital requirements means that there is no competition between banks for deposits.41 A bank cannot attract too many deposits relative to its own start up capital because if it were to do so, it would violate its monitoring incentive compatibility constraint. It could not credibly commit to monitoring and earning a high expected return, and so would be shunned by depositors and/or closed or forced to reduce its borrowing by the regulator. Because banks cannot expand deposits beyond this threshold, if there is not enough banking capital l relative to potential depositors N, deposits will be rationed – depositors have more funds available to deposit than banks are able to accept.42 In this environment, which is precisely the environment where deposit insurance is useful in our model, there is no reason for banks to compete for deposits.43 The effect of subsidized deposit insurance is then benign. It would be interesting to extend the model to examine to what extent this would hold true if banks did compete for deposits. Interestingly, Keeley (1990) argues that an important factor behind the US Savings and Loan debacle of the 1980s was an increase in competition between banks and thrifts. He argues that, as in our model, the deposit insurance scheme worked very well during the decades when competition was limited, and shows empirically that 40 Sweden, for example, had no explicit system of deposit insurance, but rather recapitalized its banks after the banking crisis of the early 1990s. Following EU guidelines, it then introduced a system of explicit deposit insurance in 1996 (see e.g. ‘‘Cleaning up’’, The Economist, June 28, 1997). 41 Acharya (2003) also has this feature. Essentially, the flip-side of banks’ capital requirements binding is that depositors would like to deposit more than banks are able to accept, so there is no reason for banks to compete for deposits. 42 The deposit rate does not fall because this would violate depositors’ individual rationality constraint. 43 As mentioned in Section 2 above, if Nl were large enough that there were no deposit rationing, deposit insurance would have no impact in our model.

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a proxy for increased competition is associated with a higher probability of bank failure. Hellman et al. (2000) argue similarly that financial liberalization was a major culprit in the banking crises of the 1990s. Their model assumes that depositors are 100% insured, and so do not care about the fact that banks offering higher returns are also more risky.44 They show in this environment that capital requirements are insufficient to prevent excessive risk-taking without also implementing deposit rate ceilings to limit banking competition. We conjecture that in their environment it may be socially preferable simply to reduce the level of deposit insurance protection from a super-optimal level.

9. Concluding remarks We present a model in which bankers play a socially useful role in monitoring investments to improve their productivity, but where these banking services are subject to both adverse selection and moral hazard. We show that, in this set-up, deposit insurance is welfare enhancing to the extent that it constitutes a net subsidy to the banking system. This result runs contrary to the received wisdom that deposit insurance should be financed through the contributions of banks, and that the objective of the deposit insurance fund should be to minimize losses to the taxpayer,45 but can provide some theoretical foundation for the large subsidies that governments around the world have provided to banks in the recent financial crisis.46 The reason behind our result is that in the presence of moral hazard, banks must earn rents if they are to perform monitoring. Depositors do not value the rents that accrue when they deposit, so, as a consequence, the banking system will be socially too small unless the regulator can provide some additional incentive to deposit. Taxing bankers and then paying this back to depositors in the case of ex post failure – as has traditionally been advocated – is a welfare-neutral policy because it has the adverse effect of reducing the banks’ capital base, making moral hazard more likely. Similarly, a tax upon depositors acts to reduce their investable funds and to deter them from investing. When the proceeds of the ex ante taxation are invested productively, we demonstrate that each of these negative effects exactly balances the positive incentive effects of deposit insurance. Taxation of those households who choose to remain outside the banking system (nondepositors) has neither of these adverse side-effects however, and hence increases welfare. Funding the deposit insurance scheme through general taxation is therefore welfare-enhancing since its net impact is on taxpayers who are neither depositors nor bankers. We derive our results in a stripped-down model, in which the need for deposit insurance arises not from depositor risk aversion or the possibility of banking panics, but purely from the existence of informational problems – adverse selection and moral hazard – within the banking sector. Thus, in contrast to much of the previous literature, we are able to derive comparative statics for the optimal deposit insurance policy. We show that, as the level of public confidence in the banking sector falls, so that depositors are more inclined to withdraw their funds from the banking sector, the regulator should counter this tendency with a more generous deposit insurance scheme. This was what happened in Japan between 1996 and 2003 Cargill (2002), and, starting in 2007, across

the world Feyen and Vittas (2009). Worse regulators should provide more generous deposit insurance, a result which may go some way towards an understanding of why on the one hand deposit insurance seems to be associated with banking crises (see Demirgüç-Kunt and Detragiache, 1998; Demirgüç-Kunt and Detragiache, 2002), yet on the other such schemes are being widely adopted across the world and increasingly recommended as part of good practice (see for example Garcia (2001) and references therein). Acknowledgements We are grateful to Liam Brunt, Mark Flannery, Ed Fraser, Xavier Freixas, Charles Kahn, Hassan Naqvi, Colin Mayer, Oren Sussman, and Jean Tirole for helpful comments and suggestions, and also to seminar participants in Oxford, the September 2005 FDIC’s Bank Research Conference and the 2006 Financial Intermediation Research Society Conference. Appendix A. Proofs Proof of Lemma 3. Note that a cross-subsidization equilibrium exists with gl banks of which any number are sound provided depositors wish to deposit, monitoring is incentive compatible, and the following conditions are satisfied:

 1þ

 gl  1 sB ðRpH  CÞ ð1  gÞl þ 1

6 f½R þ ðk  1ÞQ pH  Ckgð1  sB Þ   g sB ðRpH  CÞ; 6 1þ 1g   gl  1 sB RpL 6 ½R þ ðk  1ÞQpL ð1  sB Þ 1þ ð1  gÞl þ 1   g sB RpL : 6 1þ 1g

ð45Þ

ð46Þ

The first inequality in Eq. (45) and the second in (46) state that a sound bankers and unsound non-bankers do not choose to change their roles. To prove the lemma, we need to demonstrate that they imply the second inequality in Eq. (45) and the first in (46): these state respectively that sound non-bankers and unsound bankers do not choose to deviate, either. The sound banker non-deviation condition reduces to

ðk  1ÞðQpH  CÞð1  sB Þ P ðRpH  CÞsB

l ð1  gÞl þ 1

;

ð47Þ

the unsound banker non-deviation condition is

ðk  1ÞQpL 1  sB P RpL sB

l ð1  g l þ 1Þ

:

Given the former condition, a sufficient condition for the latter is

ðk  1Þð1  sB ÞðQ Dp  CÞ 6 sB ðRDp  CÞ

l ð1  gÞl þ 1

:

ð48Þ

Similarly, a sufficient condition given non-deviation by unsound non-bankers for non-deviation by sound non-bankers is

ðk  1Þð1  sB ÞðQ Dp  CÞ 6

sB 1g

ðRDp  CÞ:

ð49Þ

44

For evidence that depositors do in fact care about the level of risk taken by banks, and that the market discipline they exert is reduced by the introduction of deposit insurance, see Demirgüç-Kunt and Huizinga (2004). 45 See e.g. Mishkin’s (2001, p. 301) description of the goals of the Federal Deposit Insurance Corporation Improvement Act. 46 The Financial Stability Board (2009) survey financial sector support in the wake of the crisis.

We know from the MIC constraint that

ðk  1Þð1  sB ÞðQ Dp  CÞ P ðRDp  CÞð1  sB Þ: So in a close enough region to the MIC line (where this expression holds with equality), conditions (48) and (49) are satisfied.

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Proof of Lemma 5. The intersection point of constraints (40) and (41) has the following coordinates:

r Dk þ RkpL pH ; Cð1  kÞpL Cðr þ kpL Þ ¼ : r Dk þ RkpL pH

kScReg ¼ 1 þ Q ScReg

Because the intersection point lies on the locus of Eq. (40), we r must have q ¼ k1 , so that the depositor IR constraint at this point is given by Eq. (33). To demonstrate that this constraint is satisfied, it suffices to observe that

R  Q ScReg ¼

rðRD  CÞ þ RkpL ðr þ RpH  CÞ > 0: rDk þ RkpL pH

To check that monitoring is incentive compatible at (kScReg, r QScReg) this is the case, we substitute q ¼ k1 in the monitoring constraint (36) to obtain the following requirement:

Q 6 Q MIC:H ðkÞ 

Ck  RDk  2rk : ðk  1ÞDk

Monitoring is incentive compatible for sound bankers with bank parameters (kScReg, QScReg) provided QScReg exceeds the following value:

Q MIC:H ðkScReg Þ ¼

C ðrDk þ RkpL pH Þ Dk ðrDk þ RkÞpL pH C ð1  kÞðRDk  C þ 2rkÞpL :  Dk ðrDk þ RkÞpL pH

Straightforward manipulations give us the following:

Q ScReg  Q MIC:H ðkScReg Þ ¼

Cð1  kÞpL ðRD  C þ 2rkÞ > 0; Dk ðr Dk þ RkpL pH Þ

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