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Essays in banking Downie, David Craig 1995

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ESSAYS IN BANKING By David Craig Downie B.Comm., The University of Alberta,, 1986 M.Sc, The University of British Columbia, 1988 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES FACULTY OF COMMERCE AND BUSINESS ADMINISTRATION We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 1995 © David Craig Downie, 1995 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature) Department of Commerce and Business Administration The University of British Columbia Vancouver, Canada Date April 26, 1995 DE-6 (2/88) Abstract This dissertation examines two issues in the theory of banking: the role and efficiency of a monopoly bank in a spatial economy and, the design of a deposit insurance contract. Chapters 2 and 3 of the thesis present the development and analysis of a simple production economy with two types of agents. Lenders have an endowment of one unit of a good that may be consumed or invested in a firm. Firms have access to a project but lack the capital necessary to operate it and thus are forced to borrow: firms' projects are identically independently distributed cross-sectionally. A simple information asymmetry prevents efficient contracting by lenders and firms and results in deadweight default costs being incurred. One way these deadweight costs could be avoided is to establish a "delegated monitor"—a bank—who collects deposits from the lenders and makes loans to firms. This may result in an efficiency gain since the firms' projects are Ltd. so, as the bank makes more loans, the probability that it defaults will be lower than the probability that an individual firm defaults. This diversification reduces the probability that the bank will fail and the probability that default costs are incurred. However, I assume that these costs are related to distance. This restricts the bank's ability to diversify and may induce costly strategic behavior on the part of the bank. The bank may also lend 'locally' in that it may attract deposits in a region yet not make loans to firms near those depositors. The social welfare implications of this bank are examined in Chapter 3. The results show that the socially optimal outcome is one that restricts the firms' ability to compete with the bank in the debt market and that credit rationing may also be efficient. Chapter 4 examines a model where a deposit insurance scheme is designed by a regulator whose objective is to maximize social welfare. There is a single bank in the economy which ii can be one of two types: the true type is unknown to the regulator. The results show that the regulator's efficacy is improved when regular insurance premia are combined with a premia that are refunded to solvent banks—akin to a deposit insurance fund. i i i Table of Contents Abstract ii Acknowledgement vii 1 Introduction and Overview 1 1.1 Introduction 1 1.2 Overview 2 2 Financial Intermediation and Efficiency: The Monopoly Bank Case 10 2.1 Introduction 10 2.2 The Model 13 2.3 Direct Lending 16 2.3.1 Contracting 16 2.3.2 Direct Lending Equilibrium 18 2.4 Monopoly Bank 22 2.4.1 The Bank's Profit Function 24 2.4.2 Discussion 33 2.5 Numerical Examples 34 2.5.1 No Direct Lending but Banking 34 2.5.2 Direct Lending and Banking are Feasible 36 2.6 Conclusions and Future Research 39 2.7 Appendix 40 iv 3 Welfare Properties of a Banking Monopoly 48 3.1 Introduction 48 3.2 The Economic Setting 50 3.2.1 Bank Lending 51 3.2.2 Direct Lending 54 3.2.3 Social Welfare 55 3.3 Results 55 3.3.1 Discussion 59 3.4 Numerical Example 59 3.5 Conclusions and Future Research 64 4 The Role of Deposit Insurance in the Regulation of Banks 65 4.1 Introduction 65 4.2 The Model 67 4.2.1 The Bank 68 4.2.2 Capital Structure and Bank Value 71 4.2.3 The Regulator 73 4.3 The Regulator's Problem 75 4.3.1 Full Information 75 4.3.2 First-Best 77 4.3.3 Second Best 78 4.3.4 Results 79 4.3.5 Numerical Example 82 4.4 Conclusion 84 4.5 Appendix 84 v Bibliography 95 vi Acknowledgement There are a number of individuals who contributed to the completion of this thesis. My co-chairs, Ron Giammarino and Burton Hollifield, provided boundless encouragement and constructive comments at all stages of the dissertation's development. JimBrander's assistance as the third member of the committee was also very helpful. On a personal note, I would like to thank Ron Giammarino for his help and support throughout my years at the University of British Columbia both as an informal advisor during my early years and later as my thesis co-chair. Outside of school, my friends provided support as well as numerous distractions which I indulged in perhaps too often. To Craig, Dafna, John, Kevin, Mamie, Sati, and Tracy: a huge thanks! I was also extremely fortunate to have been adopted as an honourary Haden 'bro' by Mark, Kent, Bruce, and Paul each of whom, in their unique way, gave me strength and emotional support when I needed it. Paul deserves special mention for being a patient room-mate, a great friend, and a general partner-in-crime. Finally, my family was patient and understanding throughout my undergraduate and grad-uate education. My parents especially showed me that the basic economic problem of scarcity does not apply to their allocation of love and support and for this, I dedicate the dissertation to them. vii Chapter 1 Introduction and Overview 1.1 Introduction During the past decade the banking sector has come under increased scrutiny from the popular press, regulators, policy makers, and academics. Much of this attention has come as a result of the failure of a number of financial institutions in Canada and the United States. These failures have come at a large social cost: conservative estimates have placed the cost accruing to failures of commercial banks and savings and loan organizations in the United States at US$240 billion (1991 dollars)1 while the largest insurer of bank deposits in Canada, the Canada Deposit Insurance Corporation, has been forced to borrow C$3.4 billion from the federal government to cover payouts to depositors and the rehabilitation of distressed banks and trust companies that it insured.2 To understand why banks fail and why there are accompanying social costs requires a theory of why these institutions arise endogenously in an economy as well as an understanding of their social benefits. This goal of this dissertation is to contribute to the theory underlying these issues by examining the economic role of a bank in a spatial economy, the social impact that this bank may have, and finally, how a deposit insurance fund may affect the efficacy of a regulator who insures bank deposits. 'Kenneth E. Scott and Barry R. Weingast, "Banking Reform: Economic Propellants, Political Impediments," in Reforming Financial Institutions and Markets in the United States, ed. George G. Kaufman (Boston: Kluwer Academic Publishers, 1994), 20. 2 Canada Deposit Insurance Corporation, CDIC Annual Report (Ottawa: CDIC, 1994), 2. 1 Chapter 1. Introduction and Overview 2 1.2 Overview There are many theories of why intermediation arises in a modern economy. Among these is the notion that intermediaries can provide monitoring services more economically than individual savers alone. In this story, a simple information asymmetry prevents lenders and borrowers from achieving efficiency in their contracting. For example, suppose that lenders have capital but only have access to a riskless storage technology and firms have valuable projects but lack the required capital to operate it. Furthermore, the payoff on this project is private information that can only be observed at a cost by the lender and individual lenders are unable to share their information. Under these conditions the Nash loan contract involves costly verification by lenders and, in equilibrium, this monitoring will only occur when the borrower announces that it is in default: this is the standard debt contract (SDC) described by Townsend (1979) and Gale and Hellwig (1985). If borrowers must contract with more than one lender this leads to duplicated monitoring since each of the lenders monitors when it is optimal to do so. When verification costs are sufficiently large, this may lead to a collapse of the private credit market since it may prevent lenders and firms from writing contracts. Now consider a delegated monitor (a bank) which writes SDC's with lenders and with firms. The bank pools deposits and makes loans, reducing the number of borrowing contracts for firms who deal with the bank to one. Thus, if monitoring of firms is required, the bank alone has to incur the cost of verifying these firms' returns while the lender is left with the task of monitoring the bank. However, in the case of a default by the bank, all of its depositors would be forced to monitor, again resulting in duplicated monitoring. Efficiency gains result, however, if the bank makes many loans. Unless the payoff on these loans are perfectly correlated with one another, the bank's probability of failure will decrease as it makes more loans. If the marginal cost of monitoring is bounded (see Krasa and VUlamil (1991)), as the number of loans made by the bank approaches infinity, the probability that the bank defaults may approach zero. Chapter 1. Introduction and Overview 3 The social benefit of the bank arises from two sources: first, the ability of the bank to perform delegated monitoring may avoid the social loss that results from forfeit of valuable firm projects when the private debt market fails. That is, if the cost of monitoring is too high, firms may be unable to offer attractive enough contracts to lenders to fund their projects. Lenders would revert to autarky: using the riskless storage technology and consuming their own endowments a period later. If the bank can use its ability to diversify to offer attractive "enough" contracts to lenders and to firms, it may be able to overcome this autarky in the private debt market and firms can produce their valuable projects. Second, if the private debt market is feasible, the bank may increase social welfare by reducing aggregate expected deadweight monitoring costs. This is the Diamond (1984)/WUliamson (1986) theory of financial intermediation: in each of these studies, the bank arises from this simple information asymmetry and, given the constant monitoring costs assumed, yields a single bank that is infinitely large and riskless. The simple structure of the intermediation industry also yields a straightforward division of the rents from the savings in monitoring: lenders are offered a contract that has the same expected return as the riskless storage technology (the same return earned by the bank) and the firms earn all of the surplus through lower loan rates. While this theory is plausible and somewhat intuitive, it yields results that are not entirely believable. The efficiency gains just discussed are maximized through very large intermediaries. Yet, empirically there are a large number of financial intermediaries that are of a variety of sizes and the existence of financial intermediaries has not eliminated the need for other forms of credit. There are a number of modifications to the Diamond/Williamson model that will yield a finite-sized intermediary and/or a more complicated industry structure. Some examples include increasing marginal monitoring costs (Krasa and Villamil (1991)), undiversifiable macroeconomic risk (Krasa and Villamil (1992)) and different game specifications (Winton (1992), Daltung (1991), and Yanelle (1989)). In Chapters 2 and 3 of this dissertation, I add to Chapter 1. Introduction and Overview 4 these by considering the role distance related monitoring costs have in a Diamond/Williamson economy. I assume that monitoring costs are proportional to the distance between lenders and bor-rowers. The interpretation of distance can be more general than simple physical distance. For example, it could be thought of as cultural differences or industry specialization: lenders who specialize in loans to the oil industry may have difficulties evaluating loans made to an electronics firm. In an international setting, we might think that Canadian firms would find it more difficult to acquire financing in Mexico because of the physical distance and also because the corporate and social cultures differ between the two countries. Even within a single country such as Canada or the United States, specializing in commercial loans to blue chip corporations may be 'distant' from loans to small inter-city businesses. The higher cost could be reflective of liquidating or taking over a firm outside of the lender's area of expertise and/or engaging lawyers and receivers in other legal jurisdictions. In Chapter 2,1 examine a monopolistic banking sector where the existence and location of the bank are exogenously specified. The bank is operated by a lender and is allowed to move first and offer contracts to lenders and firms. After that, firms who are not offered loans (or refuse the bank's offer) may make offers to lenders who did not deposit with the bank. That is, the bank is forced to compete with the bond market: its initial offers must be such that it is able to attract enough depositors to make enough loans to operate. This underscores the bank's dilemma: it is forced to compete on prices for inputs (deposits) and for outputs (loans). Moreover, the more loans it makes, the cheaper the deposits become (net of monitoring costs) and, given that there are a finite number of lenders and firms, if the bank is aggressive in the loan and/or deposit market, it can also affect the price of its inputs and outputs. I am able to show that the bank may indeed be able to overcome autarky: if direct lending without the bank is not feasible, the introduction of a bank may allow some firms to produce. The converse is also true: when direct lending is feasible, the monopoly bank may not be able Chapter 1. Introduction and Overview 5 to operate. The reason for this is that the competition from the bond market may mean that the bank has to pay deposit rates that are too high and charge loan rates that are too low to support its operation. That is, even though direct lending through the bond market results in duplicated deadweight monitoring costs, the bank cannot exploit depositors and firms enough to take advantage of this situation and the potential social benefit of the bank goes unrealized. When both direct lending through the bond market and the bank co-exist, the bank's need to be more competitive squeezes its financial margin and increases its probability of default. This offsets some of the efficiency gained through the decreased expected monitoring costs. A unique feature of my model is that the bank has some flexibility in loan size. In the Diamond/Williamson model of the bank, the investment size of firms was fixed exogenously. Here, I assume that the firm needs a rmnimum amount of capital to operate the project (K) but the project is constant returns to scale after that. That is, for every investment of K and above, a successful project returns a constant amount, R > 1, per unit of capital invested. The bank, then, chooses both the number of loans and their size. When loans have a variable size, once the bank is "sufficiently" diversified, it may stop making loans but may keep attracting depositors. That is, the bank may choose to attract depositors around firms that do not receive funding. This is a form of credit rationing: firms in an area (or industry) may see their source of funds siphoned off by the bank, but not be offered loans. This has allocational consequences since it means that an economy that is dominated by a large bank will be more specialized than one where the firms sought funds only through the bond market. In the case where distance is physical, the economy may have production clustered around the bank and (some) firms in the extremities of the economy may be non-producing. Diversification is not the bank's only motive for making loans, however. Since the market for the bank's inputs (deposits) and outputs (loans) are linked, the bank may have a strategic reason to make loans. For example, suppose that direct lending is feasible and lenders are Chapter 1. Introduction and Overview 6 able to extract most of the rents from the firms—this will make their reservation returns for dealing with the bank very high. If the bank takes away the firms a direct source of investment for lenders by making loans to all of the firms, it can leave lenders with only their riskless technology or bank deposits as a means of storing their endowment. The bank's offer in this case need only match the expected utility offered by the riskless technology. This sort of behaviour can be socially costly though, since the bank will, in general, lend to more firms than is optimal and will be forced to incur the expected monitoring costs associated with these loans. This is illustrated by a numerical example in Chapter 2. The surplus from the bank's operation is shared to some extent by all three types of agents: lenders and firms get some rents because the bank has to bribe them both to stay out of the bond market and the bank gets some rents through its ability to act strategically in the lending and borrowing markets. The trade-off between the lower expected deadweight monitoring costs provided by the bank and the possibility of regional credit rationing raises interesting social welfare issues. With this in mind, in Chapter 3 I describe a simple social planner's problem where the planner is now assumed to run the bank described in Chapter 2 in a manner that maximizes a utilitarian social welfare function. Under this specification, the following results are obtained: first, maximizing social welfare may well entail regional credit rationing and, second, the planner will want to restrict the ability of firms to obtain direct financing from the bond market. The reason why regional credit rationing may be socially desirable is that the planner views loans the same way as the (private) owner of the bank—simply as a mechanism for reducing the probability that the bank fails. The planner will equate the marginal decrease in expected monitoring costs with the marginal monitoring cost of the last firm it attracts. Since convergence in the law of large numbers is exponential (Krasa and Villamil (1991)) the number of loans required to make the bank nearly riskless will be quite small. The reason that the planner would like to restrict competition is that by having to compete, Chapter 1. Introduction and Overview 7 the bank is forced to lower its rate on loans and increase its deposit rates. This means for the same number of loans made, more of these loans are going to have to pay off (since the loan rate is lower), and, since the deposit rate is higher, for a given number of loans made, more of these loans must pay off. This increases the probability that the bank will fail and consequently increases the deadweight monitoring costs that society will have to bear. Ideally, the planner would like to operate the bank unhindered by direct lending and then allow the left-over firms to offer contracts to any depositors that are not served by the bank. This is an important result since it points out that the monopoly bank, shown to be the most efficient industrial organization in the Diamond/Williamson model of financial intermediation, may not be the most efficient when distance related costs are considered. The focus of the dissertation changes somewhat in Chapter 4 where I address the problem of the design of a deposit insurance contract. In Chapters 2 and 3, the bank operated without the benefit of deposit insurance: the bank was "forced" to diversify its loan portfolio in order to be profitable. In Chapter 4,1 again focus on the interaction between a monopoly bank and a social planner. Since Merton (1977) recognized that deposit insurance can be thought of as a put option sold by the insurer to the bank, there has been a great deal of work done on the pricing of this insurance. One of the major implications of this body of work is that since the insurance sold is typically risk insensitive (i.e., all banks, regardless of their true risk, pay the same price for deposit insurance), it would be in the bank's best interest to maximize the value of this put. This has led to two different streams of research: one focussing on "deposit insurance as a put option" and the other, on the design of a regulatory system that has, as its primary focus, the maximization of social welfare. Most of the work in the area of pricing the put option focusses on the fair pricing of deposit insurance. Since banks differ greatly from one another in terms of their risk, a flat-rate deposit insurance premium, it is argued, induces a wealth transfer from the equityholders of safe banks Chapter 1. Introduction and Overview 8 to those of risky banks3 and that, implicitly, this has a social cost. This research, it should be pointed out, typically concentrates on symmetric information models of banking. That is, the bank and the insurer are assumed to have access to the same information set. For this type of model to be implimentable, the insurer would have to know the value and distributional properties of the bank's assets at (almost) every instant. More recently, authors such as Chan, Greenbaum, and Thakor (1992) and Giammarino, Lewis, and Sappington (1993) have focussed on the social welfare considerations of deposit insurance. These two studies in particular focus on the design of a regulatory scheme, rather than the pricing of the insurance alone. In the Giammarino, et. al. study, the optimal scheme impliments a deposit insurance system in which the insurance is not fairly priced. While both of these streams focus on the pricing of deposit insurance, they both ignore an aspect of deposit insurance that practitioners and regulators seem particularly concerned about: the deposit insurance fund. Most, if not all, deposit insurance systems maintain a pool of funds that the banks pay into and (explictly or implictly) have a claim to. The goal of Chapter 4 is to examine what role this fund has on the optimal regulation of banks. In the model developed, deposits issued by the monopoly bank provide a social benefit and are required to be completely insured by a benevolent bank regulator. The bank can be one of two types: risky or safe and the bank's type (which determines its probability of default) is unknown to the regulator when she makes the offer of a deposit insurance contract to the bank. Ideally, the regulator would like to design the contract so that each type of bank makes the first-best level of investments into loans and chooses its socially optimal capital structure. However, because of the information asymmetry, the insurer cannot observe the true risk of the bank. This gives the risky type the incentive to select the deposit insurance contract meant for its less risky counterpart. 3See, for example, Marcus and Snaked (1984) for estimates of the "correct" premia for FDIC provided insurance and Giammarino, Schwartz, and Zechner (1989) for estimates for Canadian chartered banks. Chapter 1. Introduction and Overview 9 Deposit insurance in this model is financed in two ways: regular, non-refundable premia and a fund contribution. The regular deposit insurance premium is paid up front and is not refunded to the bank at the end of the period. The fund contribution is made at the start of the period as well but, unlike the premium, if the bank is solvent at the end of the period, the bank's contribution is refunded. This is not unlike a scheme where the insurer overcharges for deposit insurance and refunds some amount to solvent institutions or where the insurer sells debentures to the bank which revert to the insurer if the bank fails. The regulator is allowed to mis-price the deposit insurance in order to enhance social welfare. Her objective function is to jointly maximize the value of the bank (which will completely capture the social value of deposits) less the social cost of providing deposit insurance coverage. The bank, after accepting the insurance contract from the regulator, chooses its optimal capital structure. At the end of the period, the revenue of the bank is revealed and, if the bank fails, the regulator seizes its licence and pays off the depositors. The results show that when default is costly for the bank (if it fails, it loses a valuable bank charter), the deposit insurance fund can help mitigate the bank's incentive to take mispriced insurance contracts. In a first-best world the fund is shown to have no value: the regulator would fairly price deposit insurance using non-refundable premia only. When the true risk of the bank is not observable, the fund does have a value: by requiring the bank to invest in the fund, the regulator, in part, is able to tie the cost of the deposit insurance to the bank's (unobservable) risk. The Chapter concludes with a numerical example. Chapter 2 Financial Intermediation and Efficiency: The Monopoly Bank Case 2.1 Introduction The value of well functioning financial intermediaries to a modern economy is without doubt. Less is known, however, about the exact origin of these benefits. One common assertion is that these agents reduce "transaction costs" within the economy—the most basic of these being the elimination of the need for a "double coincidence of wants" between traders (i.e., a reduction of search costs1) and the production and/or processing of information (Campbell and Kracaw (1980)). Others would argue that financial intermediaries pool risky assets with less than perfect correlation with one another and issue claims on this diversified portfolio that are less risky than the individual assets that make up the pool. Social welfare is enhanced by a reduction in aggregate risk (if agents are risk-averse) or by a reduction in costs associated with informational asymmetries. A specific example of the latter view, sometimes referred to as the 'delegated monitoring' literature, provides the basis for this study. The delegated monitoring theory of financial intermediation proposed by Diamond (1984) and later by Williamson (1986), is driven by the ability of "large" economic agents to reduce or eliminate deadweight costs imposed by information asymmetries. These papers consider two classes of agents: lenders, who have wealth but only access to ariskless storage technology; and firms who have a valuable project, but lack the required funds needed to operate the technology. The payoff on these projects are identically, independently distributed (i.i.d.) and the realized return is private information to the borrowers. Since lenders cannot costlessly observe the 'See, for example, Rubinstein and Wolinsky (1987). 10 Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 11 payoff on the project, they must purchase a monitoring technology (or equivalently expend effort) to confirm that the borrower is truthfully stating the project's return. Since all lenders are forced to do this monitoring, there are duplicated deadweight costs incurred when firms obtain direct financing. This implies that delegated monitors (banks) may be able to reduce these costs by gathering deposits and making loans to firms. As the bank gets large, diversification in the loan portfolio makes the probability that the bank defaults closer and closer to zero. When there are a (countable) infinity of firms, a monopoly bank becomes asymptotically riskless. By (implicitly) appealing to the contestability hypothesis2 Diamond concludes that the monopoly bank is the unique outcome if intermediation is feasible and the savings from the duplicated monitoring accrue to firms by way of lower interest rates on their loans. A by-product of these models is that the efficiency gain is monotone in the size of the bank and thus the intermediary will be an infinite-sized monopolist. There is substantial empirical evidence to suggest that this may not be the case: while regulation, taxes and culture differ throughout the world, no developed economy has a single intermediary. This would seem to suggest that "something" interferes with the efficiency monotonicity. In this Chapter, I explore the role that distance3 related costs have on the allocation of credit and the structure of the banking system. I preserve the structure of previous models but explicitly consider a finite number of agents and model a spatial component to the economy. As with the earlier literature, agents are prevented from first-best contracting by a simple information asymmetry; namely that the payoff of the project is unobservable. This gives the project's owner the incentive to lie about the true state of nature and consume the part (or all) of the project's payoff. To overcome 2See Tirole (1988) p. 308, for example. 3Distance should be thought of in a very broad sense: it could be interpreted as cultural differences or information about specific markets that is more costly for lenders to acquire. As an example, a certain bank may have a comparative advantage evaluating loans made to high-technology firms and other banks who lack this advantage may find it more costly to lend in this industry. Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 12 this moral hazard problem, the uninformed agents must travel to verify the outcome and this becomes more costly the further away the borrower is. This cost will be reflected in the terms of the contract that lenders will demand from borrowers. As a benchmark, I first consider a case where intermediation is not allowed and firms offer contracts directly to lenders. With high enough travel costs, the only equilibrium will be autarky: lenders consume their endowment and firms cannot produce. In the cases where firms can offer attractive enough contracts to lenders and are able to produce, the equilibrium is characterized by "excess" monitoring costs since, when a firm's output is verified, all of its lenders incur the cost. This suggests a role for an intermediary that reduces these costs through diversification. Next, I consider a lender who is allowed to form a monopoly bank which is chartered to have the exclusive and unrestricted right to operate as a bank in this economy. I am able to show that the introduction of a bank does not ensure that production will take place, and interestingly, in the case where production was feasible without banks, it is not always the case that the bank will earn a sufficient enough profit to operate. This is in contrast to the results of Diamond and Williamson where the intermediary was always able to operate when direct lending was feasible. When the monopoly bank does operate, it may distort the allocation of income and production in the economy since it may choose to fund only those firms close to it but may attract lenders from other areas—a form of regional credit rationing. A numerical example is presented to illustrate this result The bank in this economy will be under-diversified for two reasons: first, the marginal cost of monitoring is not constant and this may induce the bank to choose not to offer loans to all firms. Secondly, the bank has to compete with the firms themselves for lenders (and also for loans). This restricts its ability to diversify and affects the probability of default by squeezing the bank's financial margin—the difference between the expected return on its loans and its deposit rate. Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 13 Another paper which directly addresses inter-bank competition in a spatial context is Be-sanko and Thakor (1992). In their model, agents are also located around a circle and must travel to the bank in order to deposit and borrow. Their focus is on the allocational effects that banks have and hence they abstract from the reasons for why banks exist. They show that increasing bank competition (by relaxing barriers to entry) increases the welfare of firms and depositors but decreases the welfare of bank shareholders. This is consistent with my findings that increased competition from the bond market increases the utility of most lenders and some firms but decreases the utility of the bank's owner. The plan of this Chapter is as follows: section 2.2 describes the model considered and sets up the direct lending (no intermediation) game presented in section 2.3. A bank is introduced in section 2.4 and its effect on the number of firms funded and the expected returns of lenders is examined. In Section 2.5,1 present a numerical example that illustrates some of the Chapter's results. The final section concludes the Chapter and discusses possible extensions to this research. 2.2 The Model Consider a production economy where economic activity takes place on the perimeter of a circle of unit circumference. The economy has one period and two dates (t0 and U) and two types of agents: firms and lenders. There is a single good in this economy that can be used as a production input at tQ or consumed at t\. there is no consumption at t0. Firms: There are m < oo risk-neutral entrepreneurs ("firms") in this economy (distributed uniformly about the circle) each of whom is endowed with an identical project that has a risky payoff at t\. The project requires a niinimum investment of K > 1 units of capital (made at io) in order to operate but, provided that the minimum scale has been reached, production is constant returns to scale past K. Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 14 The payoff on the project has a two-point distribution: with probability 0 < p < 1 the project returns a constant R > 1 per unit of capital invested and with probability 1 — p, the project returns zero. The returns of firms' projects are independently and identically distributed (i.i.d.) cross-sectionally. Call the project's return y: then for an investment of K > K units of capital K R with probability p y=\ (2.1) 0 with probability 1 — p Moreover, the realization of y at *i is private information—no other agent is able to costlessly observe this value. The total investment, however, is observable. Firms have no endowment of the goods necessary to produce the project so without an outside source of funds, they would earn zero from the project. Denote the utility of firm / as IT/, which, if it can raise K > K units of capital at to by promising to repay a gross amount of D at ti can be written as Uf=pKR-pD (2.2) (note that this takes the firm's limited liability into account by recognizing that it would only make a payment to lenders when the project pays off). The firm's objective then, is to pick K and D to maximize (2.2) subject to 11/ > 0. Lenders: There are n > m, n < oo risk-neutral lenders (also uniformly distributed about circle) each endowed with a single unit of the consumption good and an unlimited amount of effort at time zero. At t0 they can make two investment choices: they can either enter into a contract with a firm to lend their endowment for the period, or they can store the good for the period in a riskless storage technology that will return /? > 1 units of the good at t\. The choice of lending to a firm will be complicated by the fact that lenders are unable to observe the true output from the firm's project and therefore must rely on a report issued by the firm at t\. This leaves open the possibility that the firm, regardless of the true outcome Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 15 of the project, will claim that its return is zero. Without a means to verify the true payoff, no lender would engage in a lending contract with a firm, leading to a collapse of the credit market. However, I assume that the lenders are able to purchase a technology at tx that will allow them to verify the true output. Further, I assume that the cost of this technology has a fixed component, ao > 0, and a variable component, ax > 0, that depends on the distance, dif, between the monitoring lender, i, and the firm, / . Then the total cost paid by the lender to monitor the firm's output will be C(d) = a>o + etidij For analytic tractability, I assume that these costs are paid out of the lenders' endowment of effort rather than the consumption good4 and that, in order to avoid problems defining initial distance, dif is defined to be the number of lenders between lender i and the firm, / . So, for example, if there are three lenders between i and firm / , i's monitoring cost is a0 + 3c*i. Then, conditional on the contracts written (discussed below), the lender's objective will be to maximize t\ consumption, c, less any effort, e, expended, V(c, e) = c — e subject to the constraint that the lender, whatever his investment choice, earns at least his reservation return in expected utility terms, V (c, e) > /?. Other Assumptions: For the project to be operated, lenders and firms will have to write contracts that specify the transfer made from firms to lenders. The feasible contract set will depend on the scarcity of credit and the values of the exogenous parameters above. To simplify this set somewhat, the following additional assumptions are made. 1. Since there is one unit of capital per lender, there will be n total units of capital in the economy and since there are m firms that need this capital, I assume that there is enough 4This will allow for situations where utility could be negative but not consumption. Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 16 capital to fund each project to at least its minimum scale. That is, m 2. The expected return per unit of capital is strictly greater than the return on the riskless storage technology: pR>(3 3. Lenders cannot share information they learn from monitoring and cannot make side-payments with one-another. 2.3 Direct Lending Since the firms have no capital of their own, they must write contracts with lenders that allow firms to borrow from the lenders. I assume that firms make the contract offer and negotiate with each lender separately. This means that the firms will have to design a contract to offer to lenders that takes into account the asymmetry of information at t\. These contracts must provide lenders with at least their reservation return of /3 and must completely specify the payments for all contingencies. 2.3.1 Contracting We restrict attention to the following contracts for a representative firm, / , writing contracts with K > K individual lenders. Each contract specifies: (i) that the lender transfers one unit of the consumption good to the firm at t0; («") that the firm makes a (not necessarily truthful) announcement of the project's outcome at t\ by selecting s G S = {R, 0}; (Hi) the set of announcements that will cause the lenders to monitor; and (iv) the transfers from the firm to the lender in cases where monitoring occurs or does not occur. Specifically, I assume that if the firm is monitored by K lenders, these lenders have the right to seize an equal share of any Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 17 output that they find upon monitoring. Thus, if the firm realizes an output of y > 0 and claims that the project returned y = 0, if K lenders monitor this report, they will each receive y/K. The contract design problem can then be cast as a standard "sender-receiver" game and we will look for equilibria using the Nash solution concept. Since, by assumption, the total investment is costlessly observable and the possible payoffs to the project are known, the only credible announcements that the firm could make is that the project returned y = KR or y = 0. Call the set of messages that the firm can issue S = {R, 0}5 so that, conditional on the true outcome of the project, it makes an announcement s e S. For example, if the project returns y = KR, the firm's action could be to announce that the project returned zero, s = 0, or that the project returned KR, s = R. Since the lending contract must specify the set of announcements that force lenders to monitor, their set of pure strategies has four elements: to monitor for any s e S; to not monitor for any s G S; to monitor if s = 0, but not if s = R; and to monitor if s = R but not if s = 0. The following is a standard debt contract: 1. Firms are always truthful in their announcements: they announce s = R if the project returns y = KR and they announce s = 0 if y = 0. 2. Lenders do not monitor if the firm announces s = R and receive a constant amount, JD, < R, (where V denotes some lender i) from the firm. If the firm announces s = 0, the lenders monitor and receive y/K. Comment: This is the standard debt contract (SDC) of Gale and Hellwig (1985) and others: the lender is promised a constant payment, regardless of the true outcome of the project and, if the announcement by the firm is in the monitoring set, the firm is essentially liquidated and the proceeds distributed to the lenders. There are some problems with contracts of these 5Note that this is without loss of generality since total investment, K, is observable. Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 18 sorts, however, since the lenders' strategy requires a precommittment to monitor. Since the equilibrium is one in which the firm tells the truth and lenders only monitor when the firm claims that the project returned nothing, lenders, if they believe that the firm is telling the truth, might be tempted to shirk on their monitoring. Monitoring is costly, and if they monitor the firm and find that the firm was telling the truth, their effort is, in a sense, wasted. Hence, if the lender believed that the firm was always telling the truth, they might be tempted not to monitor when the firm announced that the project returned zero. However, doing this will destroy the equilibrium and optimality of the contract It should also be pointed out that the monitoring strategies are deterministic in nature: if the firm sends s = 0 as a message, the lenders strategy tells them that they monitor with probability 1. Mookherjee and Png (1989) point out that in more general models these contracts are dominated by stochastic monitoring schemes where the lenders would monitor all reports with some positive probability strictly less than one.6 This will, in general, break the monotonicity of the contracts. The empirical evidence in Boyd and Smith (1994), however, shows "that the gains due to stochastic monitoring are small, at least when reasonable exogenous parameter values are used" (Boyd and Smith (1994), p. 560.). 2.3.2 Direct Lending Equilibrium An equilibrium exists in which the firms offer SDC to lenders: the firm will transfer A to a lender i when it is solvent—this occurs with probability p—and the lender will monitor the firm when it announces that it is insolvent which occurs with probability 1 — p. Thus, the expected utility of lender i, d{f lenders from the firm, / , offering the contract, will be V(c,e) = pDi + il-pftO-C&f)] 6There are differences in the economic environment that they model most notably that the borrower in their model is risk-averse (as opposed to the risk-neutral firm in this model) and that they also consider the problem of inducing and rewarding effort on the borrower's part. Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 19 = pDi - (1 - p) (aQ + ai dif) and, to be acceptable to lender i, it must be the case that this expected utility is at least as high as his reservation return. Since firms make offers to lenders based on the face value of the debt contract and since firms can always 'scale up' production past K, it will be the case that the firms compete in a Bertrand manner for lenders. That is, since any firm is free to make an offer to any lender, then, to be successful, the firm must make an offer that just beats the offer made by the competing firm. Hence, a lender's reservation return may exceed /?. To determine this return, consider a lender i and two adjacent firms / and / ' . Since there are n lenders and m firms, the distance between the two firms will be n/m and if lender i is dif lenders away from firm / , he will be (n/m) — dif lenders away from / ' . The highest promised payment that a firm would ever make is a face value of R (the marginal value of a unit of investment) and suppose that / is closer to i than is / ' . Then i's expected utility from /"s maximum offer will be pR - (1 - p) (a0 + ax I — - dif\) so, if these two firms compete for this lender, i will be able to extract an expected return equal to /"s maximum offer out of / . This means that this lenders' reservation return, £,, can be written & = max P,pR - (1 - p) a0 + ax ( — - dif) (2.3) I will refer to the set of lenders whose reservation return is greater than /? as the competitive region. Now we can write a representative firm's problem as one where it chooses the number of lenders to attract (K) and the face values to offer lenders, (£),), K max nfL = p(KR) -p^^i (2.4) {Di},K . = 1 Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 20 subject to: pDi - (1 - p)(a0 + axdi) > & V* (2.5) K > K (2.6) nfL > 0 (2.7) Constraint (2.5) is the lenders participation constraint. It requires that the expected utility of the firm's offer of a face value of A must be at least equal to lender i's reservation return. (2.6) is the requirement that the inputs to the production technology must be at least K and, finally, (2.7) recognizes that the firm, having limited liability, must be guaranteed at least an expected profit of zero. The characterization of the equilibrium is straightforward and is summarized in Proposition 1 Suppose R 4/3+(l-p){4ao + ai(K-2)] ~ Ap then direct lending is feasible. Proof: Note that to prove feasibility of direct lending, we need only concern ourselves with cases where & = 0 since & > ft only if direct lending is feasible. Next, at an optimum, (2.5) will hold with equality. We can sum over all i to get an expression for the total obligations of the firm, D=i:£i{P+(l-p)[ao + o<idif]} P By symmetry7, and by the assumption that distance was measured as the number of lenders 7Note that even if n/m is an odd number (meaning that the number of lenders per firm is odd) we can optimize as if the firm always attracts an even number of lenders. This is a result of the Bertrand game for lenders: since the lender for whom the cost of going to the firm on the left or right is the same, this lender can extract all of the firm's marginal profit from the dollar it lends. For this reason, the firm will be indifferent to attracting the odd lenders since they add zero marginal profit, and, by assumption (n/m > K) they would already have enough to fund the project Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 21 between the firm and the lenders, we can write the last term in the numerator of D as K K ]T) [a0 + audi/] = Ka0 + ax ^ d{ «=i «=i (K-2)/2 = Ka0 + 2ai J2 i t = i 4Ka0 + aiK(K - 2) Then the firm's objective function, (2.4), can be written as maxnfL = p(KR) -Kfi-K(l-p) 4ao + ati(K-2) (2.9) subject to: K> K n fL>o To show the condition, substitute K = K into (2.9) and divide through by K. This gives '4ao + ai(K-2)' nfL=pR-P-(l-p) The above must be greater than or equal to zero at K to satisfy the firm's incentive compatibility constraint. Setting it equal to zero and solving for R gives the result. • Note that the direct lending equilibrium (if it exists) is characterized by a duplication of expected bankruptcy costs since, if the firm defaults, all of its lenders incur the utility loss from their monitoring. The symmetry in the model means that if one firm produces in the direct lending game, all m will produce and if these firms are funded by K investors each, the expected default costs will be m(l -p)K[4a0 + ax{K-2)] 4 This section points out two costs incurred by an economy without banks: in the case where direct lending is not viable, the economy is forced to forfeit valuable projects. When direct lending is feasible, the deadweight expected default costs decrease social welfare. Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 22 2.4 Monopoly Bank Given the duplication in monitoring in the direct lending case, it would seem that there would be scope for a single agent to collect lenders' funds and make loans to firms. In this section, we consider the case where a lender is designated the delegated monitor and has the exclusive right to act as a bank in this economy. It performs this task by writing lending contracts with firms and borrowing contracts with lenders. If a firm receiving a bank loan defaults, only the "delegated monitor" or bank alone has to verify the firm's output instead of all of the lenders, as in the direct lending case presented above. This, however, does not address the problem of who monitors the bank: when the bank defaults, all of its depositors would have to monitor it. But, the firms' projects (and hence the loans of the bank) are i.i.d. which means that the more loans the bank makes the lower its probability of default becomes and the less likely the need for all of the depositors to monitor becomes. This is the crux of the Diamond/Williamson model of a bank: as the bank gets large in the sense that the law of large numbers can be applied, the probability that it will default goes to zero. In this economy, however, there is the additional complication that monitoring costs depend on distance. The bank's growth will be hindered by the fact that loans further away will be more costly to monitor and this will affect its ability to diversify. Here, I make some additional assumptions: 1. The bank moves first and offers contracts to all lenders and to some set of firms. These contracts are costlessly observable to all agents and have the following properties: (a) Loans: A loan contract will be written at t0 by the bank and a firm the bank wishes to lend to which will specify: (/) the amount transferred from the bank to the firm at to; (ii) that the firm will make a (not necessarily truthful) announcement of the Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 23 project's outcome at t\\ (Hi) the set of announcements that the bank will monitor at t\; and (iv) the transfers from the firm to the bank in cases where monitoring does, or does not occur. These contracts are further constrained in the following way: • The bank is restricted to offer the same loan size, K, to all firms they choose to give credit. Note that this implies that the loan size will be _ $ of Deposits Raised + 1 Number of Loans Made ' • All firms receiving credit are offered the same face value, r/, per dollar of loan; • The bank offers contracts to firms symmetrically and sequentially about the circle. Given these restrictions, the Nash equilibrium will be one where: (i) firms make truthful announcements about their project's outcome; (ii) the bank does not monitor if the firm repays the loan in full and always monitors announcements that the firm is in default seizing any of the output it finds upon monitoring. (b) Deposits: A deposit contract is offered by the bank to all lenders in the economy and will specify: (0 the amount transferred from the lender to the bank at t0; (ii) that the bank will make a (not necessarily truthful) announcement of its value at t\\ (Hi) the set of announcements that the lenders will monitor at t\\ and (iv) the transfers from the bank to the lenders in cases where monitoring does, or does not occur. Since the bank will have a number of loans, the set of announcements it can make will have more than two possible elements. Call rj. the (common) face value of the debt contract offered to lenders. Without loss of generality, the bank's message, s, will be stated as the realized value of the loans per depositor, and will thus belong to the set S = [0, r{[. Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 24 Let a; be the bank's loan revenue per dollar of deposits. We can now define the SDC for deposits as: i. Lenders precommit to monitor any report s e [0, rd). ii. If x > rd, the bank reports s = rj and transfers rd to every lender holding a deposit contract with the bank. iii. If x < r&, the bank reports x and is monitored. Lenders seize any residual value in the bank. 2. Firms that do not receive loan offers from the bank are free to offer contracts to lenders who do not purchase deposits from the bank; and 3. Lenders and firms have rational expectations and will not make side-payments amongst one-another. The monopoly bank's problem is to choose the number of firms to fund, iV>, the face value of the individual deposit contracts, rj, and the gross interest rates (1+ the interest rate) on the loans made to firms, r;. I maintain the same distance convention so the distance to a firm is measured as the number of firms between the bank and the firm being monitored times the number of lenders per firm. For example, if there are 3 firms between the bank and the fourth firm, it costs QQ + 3(n/m)ai to monitor firm four. 2.4.1 The Bank's Profit Function The bank gathers deposits from lenders and makes loans to firms. Its comparative advantage is that it can diversify away some of its default risk and reduce the face value of the deposit contract without violating the lenders' participation constraint. The assumptions above make it simple to characterize the bank's probability of solvency. Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 25 It will be convenient to make some normalizations: first let y/ be the bank's income from making a loan of size K and rate r/ to firm / . Given the payoff on the project, this implies that Vf = Kri with probability p 0 with probability 1 — p Then the total income earned by the bank when it makes Np loans will be yr = Y^yf The bank's expected total revenue, E[yT], will be NF E[yT] = £ i % ] = NF{pKr,) = p{ND + l)ri Since the bank is going to sell Nr> deposit contracts, the income per depositor will be x = VT/ND. Now we can state the properties of the distribution ofx: E[X] = ~J^pri (2.10) al = Var k^Vf -i vD / = -^NFp(l-p)K2rj ~ NFN2D P[1 p)Tl (2.11) Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 26 In general, x will be distributed binomially, however, given the computational difficulty of the binomial distribution, we appeal to the Central Limit Theorem and use the normal approximation to the binomial8. Then, for a given deposit rate, r<*, the probability that the bank is solvent will be \ ox ox ) Let z be the standardized score, rd - E[x] z = —-Ox then, if <£(•) is the c.d.f. of the standard normal Pr(a; > rd) = 1 - O(z) Some properties of the bank's probability of solvency are outlined in the following lemma Lemma 1 Ceterus paribus, the probability that the bank is solvent, 1 — O(z), is: 1. Increasing in r/; 2. Decreasing in rd; 3. Increasing in Np (provided that E[x] > ri); 4. Decreasing in ND. Proof: See the appendix. Some comments about the properties of the bank's probability of solvency are in order. 1 and 2 together imply that a lower financial margin makes the bank riskier and while this is hardly surprising, it has important implications later on. 3 is the intuitive result that by adding i.i.d. loans, the bank lowers the risk of its loan portfolio. 4 is a "capital" effect: since the bank 8This can be justified through the De Moivre-Laplace central limit theorem (see Spanos (1986), pages 63-64). For the exact characterization of this distribution, see appendix 2.7. Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 27 is a lender, it also lends its endowment and this provides a cushion against losses and, as the bank brings in more lenders, this benefit is dampened. Let dib be the distance from lender i to the bank, djb be the distance from firm / to the bank and dif be the distance between lender i and the nearest firm, / . It will also be useful to define N to be the number of lenders "left over" per unfunded firm in the after-market, that is m — NF to attract firms, the bank has to offer loan terms such that the last firm it wishes to attract is indifferent between borrowing from the bank and offering direct lending contracts to the lenders who do not deposit with the bank. A firm offered a contract {r/, K} by the bank will have an expected value of, Ilf(rl,K)=pK(R-rl) If the firm is not offered a loan contract or rejects the bank's offer, it will have to try to get funds in the after-market of which the probability that this is successful and the cost of the financing will be functions of N. For simplicity, we denote this expected value by Tif(N) > 0. Then a firm will accept the bank's offer if, and only if, Uf(rhK)>Uj(N) Since 11/ = pK(R—r/), the bank can attempt to satisfy this constraint by adjusting the interest rate, r\, or the loan size, K. Next, by the symmetry assumption, the expected cost of monitoring firms by the bank is given by NF (NF-2)/2 J2(l-p)[a0+(n/m)aidfb] = JV>(1 -p)a0 + 2(1 -p)ai(n/m) £ 3 (1 - p)NF[4a0 + {n/mfa {NF - 2)] Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 28 To attract depositors, the bank is going to have to set a deposit rate such that the lenders it wishes to attract will not reject its offer in order to take firms' direct lending contracts in the after-market. If the (potential) depositor anticipates that the nearest unfunded firm (dty lenders away from the potential depositor) will offer D,; < R in the after-market then this offer will yield expected utility of pDi - (1 - p)[a0 + aidif] This depositor's reservation return will then be £ = max {P,pDi - (1 - p)[ao + aidif]] (2.12) If a lender accepts the bank's offer, with probability 1 — O(z) it will earn rd and with probability Q>(z) it will monitor the bank and seize its residual value. The expected value of each depositor's claim in the case of the bank's failure will be9 J rrd ' x<f>(x)dx 0 where fa) = _J_e-(*-W/2^ <rxV2ir is the p.d.f. for x. A lender i will accept the bank's offer, if, and only if, (1 - ®{z))rd - ®(z)[aQ + axdib - x~] > & (2.13) If we compare (2.12) and (2.13) we can see how the bank is able to attract depositors: most depositors are closer to the nearest firm than they are to the bank and this makes the monitoring cost higher for the bank, on the other hand, the bank's ability to diversify makes the probability of having to monitor much lower. The net result is that the expected monitoring cost may be lower for a depositor dealing with a bank rather than with a firm directly. 9This assumes that each creditor has an equal claim in bankruptcy and, while consistent with the manner in which the contracts are written, may not be optimal. Winton (1995) shows that when there is more than one creditor in models such as these, optimal contracts will have different classes of debt Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 29 Comments: The bank's offer of rd, together with O(-) and monitoring costs will determine the number of lenders that are attracted by the bank. One difficulty presented by this is that the number of lenders is non-monotone and not everywhere continuous in the offered deposit rate.10 The nonmonotonicity can be illustrated as follows: suppose that a given deposit rate, fd, attracts No depositors. Now increase rd towards R. For small changes around fd, the bank's default risk will not increase that much and the number of lenders attracted will increase. After some point however (as rd —y R) the default probability of the bank will approach 1 and the expected utility of the offer will start to decrease: the number of lenders attracted will fall. To illustrate the lack of continuity, suppose the bank initially offers a deposit rate of rd = 1. At rd — 1, no lenders are attracted but as rd increases there will be a rate where the number of depositors attracted "jumps" to two. For these reasons, differentiation of the bank's objective function with respect to its choice variables will not yield global optima. The bank's objective function can now be written as '4a0 + (n/m)ai(NF - 2)' max V*» = pn(ND + l)-{l-p)NF {rd,{r,,NF}} subject to: -ND(1-Q(z))rd (2.14) (1 - 0(z))rd - ®{z)[ao + axdih - x~] > £ (2.15) Uf(rhK)>Uf (2.16) WMB > 0 (2.17) ND<n-l, NF<m, ND* * > K I\F The first term in (2.14) is the expected return from making the NF loans, the second term is the expected monitoring costs for loans made by the bank to the NF firms and the final term 10Yanelle (1989) refers to this as the "sustainability of good offers". Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 30 is the expected deposit cost. The program's constraints are that lenders and borrowers earn at least their reservation returns (2.15,2.16) and that the bank, being a lender, earns at least its reservation return of/?, (2.17). In contrast to models where the loan size cannot be increased by the bank, the probability of default becomes a function, not only of the number of firms funded, but also of the number of depositors attracted. In general, the only benefit from funding firms comes from diversification; the optimal number of firms funded will equate the marginal cost (the expected bankruptcy costs) with the marginal benefit (lower aggregate deposit cost). Attracting an additional unit of deposits has a first-order effect on profits through the spread between loan and deposit rates, prj — r<i(l — $>(z)), but it also has a second order effect through O, which also affects r^. An important benefit of the Diamond/Williamson intermediary is that in the event that a direct lending equilibrium is not feasible, the bank may be able to overcome this autarky through its economies of scale. If, on the other hand, direct lending was indeed feasible, then the bank would always be feasible and would provide an unambiguous increase in social welfare. This is not always the case here. Since the bank has to compete with firms for lender's funds, it may not be possible for it to write contracts with firms and lenders such that it is viable. The two possible outcomes of introducing a bank into this economy are outlined in: Proposition 2 (i) A monopoly bank may be feasible when direct lending is not; (ii) A monopoly bank may not be feasible even though direct lending is. Proof: See the appendix. Part (i) is the standard result that the bank's ability to diversify may allow the economy to overcome autarky. By gathering deposits and making "enough" loans to diversify, the bank is able to eliminate a large portion of the expected monitoring costs of depositors. In this case, it earns the lion's share of the rents, (ii) is the more interesting result: in this case, the monopoly bank cannot earn enough to operate even though, if left to themselves, the firms could obtain Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 31 funding and the economy would have to bear the excessive expected monitoring costs. What is happening here is that the competition between the bank and firms for lenders drives up lenders' reservation returns and the need for the bank to offer some rents to firms and lenders drives down loan rates and increases deposit rates. This increases the bank's probability of default and also increases the deposit rates needed to attract lenders. The net result is that the bank may not be able to earn the reservation return necessary to operate. This points out that the bank's incentive to make loans also may have a strategic element to it. For example, it could be the case that travel costs are so low that the lending market is close to competitive and lenders are able to extract most of the project's surplus from firms in the direct lending game. Firms would earn only a small profit by writing direct lending contracts and could be "bought off by the bank offering the same small profit on a loan contract. If the bank lends to all of the firms in the economy it will be able to eliminate almost all of the lenders' rent-extraction capability and capture most of the diversification rents. This strategy is costly though: by issuing loans to all of the firms, the bank has to deviate from the strategy of picking JV> to iriinimize its probability of default. An example illustrating this is presented in Section 2.5. The ability of the bank to increase or decrease the size of loans to firms also provides some interesting incentives for the bank. Krasa and Villamil (1991) point out that convergence in the law of large numbers is exponential, thus, the probability that the bank defaults goes to zero exponentially. This has implications for the number of firms the bank will offer loan contracts. Given the production technology, once the bank has "enough" loans to diversify its loan portfolio, it has no incentive to incur the extra expected monitoring cost from funding additional firms. However, since it can use more deposits to increase the size of loans, attracting more lenders may be profitable. This can lead to "regional" credit rationing; a situation where the bank raises funds from one region, but does not make loans there. Formally, this is outlined in, Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 32 Proposition 3 The monopoly bank will ration regional credit; that is, raise deposits "near' some firms it chooses not to fund: (i) when direct lending is not feasible if AT ^ 4a0[(n/m) - 1] + K(n/m)ai J\F > 7—7—r (n/m)ai and, (u) when direct lending is feasible if (l-p)a1(NF-2) 4 fa -P- ( A & ) fotfUv) + *W(«a + "1*-*-*-)) - ^ r ] > 1 Proof: See the appendix. The bank also has an incentive to use its size to act strategically in the deposit market when direct lending is feasible. By attracting depositors close to firms (but not those firms themselves), they can reduce both the marginal lenders' reservation return and the firm's reservation profit. This advantage comes at the expense of the bank's financial margin: as the bank becomes more aggressive in the deposit or loan market its margin is squeezed. This relationship is outlined in Corollary 1 Regional credit rationing and the bank's financial margin are negatively related. Proof: A bank's financial margin is commonly defined as the loan rate less the cost of capital. The numerator of (2.34) in the Appendix is ™ - P ~ (A^TT) kW*'/)+ *Wfa> + **•» - *~» - (1 ~N» + \F pri is the expect loan return per dollar and P + G V ^ T T ) (5(^' di'f) + ° w ( a ° + axdi'b ~ x~]) + (1 ND + I** is the cost of that dollar (including the cost of capital for the bank's endowment). Thus this denominator is proportional to the financial margin. Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 33 As this margin gets smaller, the right-hand side of (2.34) will get larger and credit rationing will increase. • Corollary 2 Ceteris paribus, as the fixed cost of monitoring, ao, increases, regional credit rationing is reduced, and, as the variable cost of monitoring, a\, increases regional credit rationing increases. Proof: See the appendix. This has important implications for a social planner who may worry about regional economic development. If there is some social benefit to having more firms operate, the monopoly banking system may not necessarily be more desirable than direct lending even with its associated duplication of bankruptcy costs. 2.4.2 Discussion The monopoly banking structure is an interesting exercise since the models of Diamond and Williamson imply that this is where the analysis should stop. However, in the economy I consider this is not necessarily the case. The benefits of having an intermediary in this model affects only one of two costs: the expected cost incurred by depositors monitoring the bank; the cost of monitoring firms when they default is still present. In the absence of verification costs the standard optimality of the monopoly bank would carry through. Here, though, these costs limit the ability of the bank to perfectly diversify and hence, the optimal number of loans made will trade off the costs of monitoring loans with their diversification benefit. The result is that a small, local bank may or may not increase social welfare, and if direct lending is feasible, could distort the distribution of investment in the economy or increase aggregate default risk by engaging in strategic behavior in the loan or deposit market. Unlike the Diamond/Williamson results, when the bank is feasible, the rents do not always accrue to one source. When both direct lending and the bank co-exist, the bank will earn some Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 34 rents as will depositors and some firms. When direct lending on its own is not feasible and the bank is, the bank and some of the lenders will earn some surplus. 2.5 Numerical Examples In order to illustrate some results from the paper, we consider some numerical examples. For the following examples, assume these values for the exogenous parameters: • R = 2.25; • p = 0.7; • n = 1,000, m = 100; • K = S; For the specific cases, the monitoring costs, ao and <*i, will be specified to illustrate some of the conditions noted in the paper. 2.5.1 No Direct Lending but Banking This example will illustrate how the bank may be able to overcome autarky in the direct lending market. Here, assume that the fixed cost of monitoring is a0 — 2.00, and the variable cost is OLX = 0.025. From Proposition 1, the necessary condition for direct lending to be feasible, (2.8), is D . 4P+{l-p)[4ao + ai(K-2)] K > Ap 4+(l-0.7)[4(2)+0.025(6)] 2'2 5 " 4(0.7) 2.25 t 2.302 Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 35 Which means that firms cannot raise funds on their own by selling debt contracts directly to lenders. The fact that direct lending is not feasible means that the bank can offer loan contracts that have a face value of R and a deposit contract that offers that last lender it wishes to attract, i*, an expected return of /?. The bank's problem is to pick {rj, Np} to maximize: '4ao + (n/m)al(NF-2)' VMB = PR(ND + 1) - (1 -p)NF subject to ND (l - <&(*)) n (1 - ®(z))rd - <J>(z)(a0 + otidi.h - x~) = j3 Results: Numerical calculations11 show that the following banking equilibrium exists: the bank makes a total of 34 loans (out of 100 firms) and takes deposits from all of the lenders in the economy: credit is rationed. The deposit rate that attracts the last two lenders is 1.012: all of the lenders up to the last two will earn some rents due to the fact that they are compensated for the expected monitoring costs of the last two lenders—in this case the total of these rents is 5.40. The bank earns an expected profit of 525.27, much greater than its reservation profit of (5 = 1. The probability that the bank will fail, <£(z), is 0.0007. Discussion: This example shows that even when the monitoring costs are very high (close to R), the fact that the bank is able to diversify these away means that the deposits it issues are nearly riskless. The lack of competition from the private debt market also means that the bank is 'free' to exploit lenders and firms to their reservation returns and this negates the strategic element to loan granting. With only the diversification motive for making loans the bank selects the number of firms that equates the marginal benefit (diversification) to the marginal monitoring cost of firms. 11 Performed in Maple. Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 36 2.5.2 Direct Lending and Banking are Feasible Direct Lending: Consider next fixed monitoring costs of ao = 0.025 and variable costs of «i = 0.0075. Checking condition (2.8) in Proposition 1, shows: 4/3 + (l-p)[4a0 + ai(K-2)] R > 2.25 > Ap 4 + (1 - 0.7)[4(0.025) + 0.0075(6)] 4(0.7) 2.25 > 1.444 so direct lending is indeed feasible. In fact, because monitoring costs are quite low, it is likely that the firms will have to compete in a Bertrand manner for some or all of the lenders. To check for this we have to look at condition (2.3), & = max /3,pR-(l-p) Qo + a x ^ - ^ ] ] Since n/m =10, the lender closest to firm (/) will be 10 — 1 = 9 lenders away from the next closest firm (/') and the maximum offer that / ' can make to this lender will be pR-(l-p) a0 + ai( dA\ 0.7(2.25) - 0.3(0.025 + 0.0075(9)] = 1.57425 > (5 which means that firms compete for all lenders. Since the competitive region extends over the entire set of lenders, & = pR - (1 - p ) \a0 + «i (— - dif) for each lender i—firms will be able to attract n/(2m) = 5 lenders on each side. Using (2.3) for lenders i = 1,... ,5 6 = .7(2.25) - .3[0.025 + 0.0075(10 - 1)] = 1.54725 Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 37 6 = 1.54900 6 = 1.55175 & = 1.55400 6 = 1.55625 Symmetry dictates that these will be the reservation returns for the 5 lenders on the other side of/. Using these reservation returns, we can calculate the face values of the contracts that / offers (again only for lenders on one side): & + (l -p)[ao + aidi] Di = P and for i = 1, . . . , 5 f, 1.54725+ .3[0.Q25 + 0] = 2.2211 D2 = 2.2275 D3 = 2.2339 D4 = 2.2404 D5 = 2.2468 .7 which means that the total debt obligations of the / (and hence all of the firms) will be D = ij^Dj 3=1 = 22.3394 Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 38 and since ^ = 5 x 2 = 1 0 , the firms' values are Tlf = pKR-pD = 0.7(22.5-22.3394) = 0.1124 Intermediation: In this case the bank knows the following: all firms produce and make an expected profit of 0.1124 if there is no bank. Lenders also receive most of the rents. Therefore, if the bank leaves any firms unfunded, they will be very aggressive competitors since the distance related costs are quite small. If the bank makes an offer to all of the firms that gives them an expected profit of 0.1124 then they'll cease to compete for lenders and the bank can offer the last lender a deposit rate that nets ft in expected value terms. For example, suppose that the bank offers r; = 2.23394 to NF = 100 firms and a deposit contract with a face value of rj = 1.001. Then the loan size will be 10 and each firm will earn an expected profit of 0.1124—exactly the same as they expected to earn when they financed in the bond market when there was no bank. They will opt for the bank's offer. Lenders, realizing that there will be no after-market, will take any offer from the bank which provides them with an expected utility equal to at least /? = 1, thus ND = 999.12 Here X¥MB = 508.38 and the probability that the bank defaults is 0.1825 x 10~7. To verify that the last depositor attracted (499 lenders away from the bank) will have an expected utility equal to p = 1, this is calculated below:13 V = (l-<t>(z))rd-®(z)(a0 + <*idi.b-x-) 12This turns out to be the only equilibrium for the following reason: if the bank doesn't give all of the firms loans, even a single firm 90 lenders away from the last one the bank needs to attract to make its loans will drive up the reservation return of that lender to a point where the bank cannot operate. Of course, if we allowed the bank to make discriminating deposit offers, this problem would be mitigated. 13To simplify these calculations, I approximated x~ by r<j/2. To calculate this exactly, one could use Monte Carlo techniques, for example. However, since the probability that the bank fails is very small, little precision is lost by this approximation. Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 39 = 1.001(0.99999998175) - 0.00000001825(0.025 + 0.0075(499) - 0.5005) « 0=1 his reservation return. Discussion: This example illustrates that the threat of competition may impose large costs on an economy such as this. In this case, the low travel costs made the bond market very competitive: for the bank to operate it had to eliminate that market all together. 2.6 Conclusions and Future Research The purpose of this Chapter was to examine the role of a delegated monitor in an economy with distance related costs. These costs affect the viability of the private debt market and, in some circumstances, can provide scope for a financial intermediary. The bank provides monitoring of firms on behalf of its depositors and, through diversification, is able to reduce the probability that these depositors will have to monitor the bank. The findings point out that when monitoring costs depend on distance, the division of rents between the bank, lenders, and firms becomes a function of the degree of competition between the bank and the private debt market. By driving up the reservation returns of lenders and borrowers, this competition may prevent the bank from operating despite the efficiency gains from delegated monitoring. If the bank can operate, competition may force the bank to act strategically in the lending market and this may cause the bank to deviate from choosing the number of loans which equates the marginal cost of lending with the marginal benefit. Distance related costs are also shown to affect the regional allocation of credit. If the bank is well diversified it may find it optimal to raise deposits in one area and yet not make loans there. This regional credit rationing is also related to the degree of competition in the lending and borrowing markets. Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 40 These results point out the importance of competition to stability in the banking sector. This naturally begs the question of what effect competition from other banks, as well as the bond market, would have on this economy. Since banks are likely to be better competitors for one another than is the bond market, this is likely to affect most, if not all, of the results presented here. 2.7 Appendix The Bank's Probability of Default The probability that the bank fails, O, can be derived through the binomial distribution: each successful loan returns [(No + 1)/Np]ri to the bank, thus, if a loans are successful the bank's revenue will be, The bank will fail if the above expression is less than its total deposit obligations, Nprj. Hence, the probability of default can be viewed as the probability that less than a loans are successful out of Np made. Solving for a from (2.18) we find NFNDrd a~ r,(l+ND) The minimum number of loans that have to be successful is the smallest integer greater than or equal to a, [a] = ac. The probability that the bank is insolvent is <zc-l f Np\ ° = E i?(l-p){NF-j) (2.19) j=o \ j J Proof of Lemma 1 The change in the bank's probability of solvency, 1 — 0>{z), with respect to changes in z is —&(z) and, from the properties of the normal density <f/(z) > 0. Since —<&(z) < 0, the sign Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 41 of the derivative will depend on the negative of the sign of the derivative of z with respect to each of the variables, everything else held constant. So for each of the cases: 1. rr. and do. dz _-*%?*.-(u-E[x))% dri ot dE[x] dri = P ND + l N, D dax dri ND + l ND p(l -p) NF The numerator of (2.20) can be written •ND + lf lp(l-p) -P ND Nf ri - (rd - E[x]) ND + 1 which can be simplified to -E[x] Nt D p{\-P) dividing by gives so NF ND + l ~ (r* ~ E[x]) ND ND + l N, D ND IP(I-P) NF -r-i<o 9(1-*{*)) dri > 0 2. rd: dz I n — = — >o ord ox so drd < 0 W-P) Np NF (2.20) (2.21) Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 3. NF: ' dNF dz (rd-E[x])&* which, after dNF some simplification can dz °l be written {rd~ -E[x))ND 9NF 2{ND + l)r, y/NFp(l-p) which will be negative as long as rj — E[x] < 0—if this is the case, then ON, > 0 4. ND: and dND ol dE[x] pri dND =~Nl dND N2D then after some simplification, (2.23) can be written y/W(N° + l>1 > 0 so 9(1 ~ *(*)) dND D Proposition 2 : < 0 (2 (2 (X) A monopoly bank may be feasible when direct lending is not. Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 43 Proof: Assume that none of the resource constraints in the bank's optimization program bind at an optimum. Then, if direct lending is infeasible, 'Aao + a^K-2)' pR-(l-p) <P (2.24) The monopoly bank's expected profit from making NF loans of size K must be at least equal to P for the bank to form so (2.17) can be written as pn(ND + l)-(l-p)NF '4a0 + (n/m)ai(JV>-2) -{\-<b{z))NDrd>p (2.25) Denote by i* the lender just indifferent between accepting or rejecting the bank's offer of rj. Now, since direct lending is not feasible, we know that i*'s expected return will be /?. That is, (1 - Q(z))rd - ®{z)[ao + M,-.6 - x~] = (3 and we know that the firm's reservation profit will be zero. This means that the bank can exploit the firm the maximum possible amount by setting n = R. We can use these to write (2.25) as PR(ND + 1) - (1 -p)NF r ^D + ( " / " 0 « i ( ^ - 2 ) --ND\p + $(z){a0 + aldi.b-x-)] > (3 but, since O(z) (a0 + airf,»& — x~) is a positive constant, the above can be written '4a0 + (n /m)a 1 (A^ F -2) ' pR(ND + l)-(l-p)NF or 4a0 + (n/m)ai(NF - 2) -NDp>p > P(ND + 1) pR(ND + l)-(l-p)NF dividing through by ND + 1 and recognizing that K = (ND + \)/NF gives '4a0 + (n/m)ai(NF-2) pR-(l-p) AK >P (2.26) Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 44 Subtracting (2.24) from (2.26) and simplifying, we find that for the monopoly bank to be feasible (when direct lending is not) requires 4a0(K - 1) + ax[K{K - 2) - (NF - 2)] > 0 (ii) A monopoly bank may not be feasible even though direct lending is. (2.27) Proof: We have to show that there exist parameters such that the expected profit from banking is less than j3 and the return from direct lending is greater than /?. To do this we proceed in exactly the same manner except that now both the lender's reservation returns and the firm's reservation profit will be functions of the size of the after-market, N. Without loss of generality we can write the lender z'*'s reservation return as &• =P + 9(N,di.f) (2.28) where # : R + x R + + - » R + i s a positive function, N = (n — iV£>)/(m - NF) and d,-.y is the distance to the nearest unfunded firm to lender i*. The loan rate, ru can be written in a similar manner, rt = R- h(N) (2.29) where / i : R + - > R + i s a measure of the firms' bargaining strength.14 For ease of exposition, we'll consider the case where direct lending is strictly profitable. This means that, PR-(l-p) and if the bank is not feasible 4a0 + a1( Jr?-2) >P (2.30) pr,(ND + l)-(l-p)NF 4a0 + (n/m)ai(NF-2) - (1 - <&{z))NDrd < p (2.31) 14 All that the proof requires is that we be able to decompose the lender's reservation rate into the minumim they would ever accept (/?) plus "something" and, for firms, the highest rate they would ever accept (R) less "something". Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 45 Substituting (2.28) and (2.29) into (2.31) and simplifying yields ~4a0 + (n/m)ai(NF - 2) P « - ( 1 ^ K ND[g(N, dj.f) + <E(z)(a0 + M,>6 - *")] ND + l -ph{N) (2.32) Then subtracting (2.33) from (2.30) and simplifying shows that (1 - p) f [4a0 + {n/m)ax (JV> - 2)] - K[a0 + at (K - 2)]' 4 1 K + ph(N) ND[g(N,dj.f) + ®(z)(a0 + axdi*b - x~)} ND + \ which is positive over a non-empty set of parameters. Proof of Proposition 3 The monopoly bank will ration regional credit; that is, raise deposits "near" some firms it chooses not to fund: (0 When direct lending is not feasible if: 4a0[(n/m) - 1] + K(n/m)ai I\F > 7~T~\ {n/m)ai Proof: There will be regional credit rationing if K > n/m (since there are n/m lenders per firm and K = (ND + \)/NF). Recall equation (2.27): this inequality must be satisfied for the monopoly bank to overcome autarky, K > 4 Q ° + (n/m)ai(NF ~ 2) 4a0 + « i (A"-2) and we can rewrite this as K 4a0 + (n/m)ai(NF-2) (n/m) (n/m)[4aQ + a^K - 2)] Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 46 Then, since there is regional credit rationing if the right-hand side of the above is greater than 1, we can solve for the number of loans that induces credit rationing, Np, where c _ 4ao[(n/m) — 1] + K(n/m)a\ {n/m)a\ N° = (2.33) in this case, if the bank makes Np > Np loans, it will ration credit regionally. • (K) When direct lending is feasible, if: (l-p)a1(NF-2) 4 h - P - G f e ) (9(N,di.f) + *(*)(<*, + on*.* - x - ) ) - G=ga£ > 1 Proof: Assume again that the last lender attracted by the bank, i*, has a reservation return, f,-., that can be written where d^f is the distance from i* to the nearest unfunded firm. Then, following the procedure in the proof of part (ii) of Proposition 2, we can write (l-&(z))rd-&(z)(4ao + aidi.h-x) = &. (1-Q(z))rd = p + g(N,di.f) + 0(z)(ao + a1di.b-x-) We can combine this with the expression for the bank's expected profit '4a0 + (n/m)ai(NF-2) vp MB 4oto + (n/m)ai{NF-2) = pri(ND + 1) - (1 -p)NF = prl(ND + l)-(l-p)NF -ND\p + g(N, di.f) + <D(z)(a0 + M t- . 6 - x~)] We know that a feasible monopoly bank must have ND(l-0(z))rd ,MB *¥M" >fi Chapter 2. Financial Intermediation and Efficiency: The Monopoly Bank Case 47 So P < pr,(ND + l)-(l-p)NF 4a0 + (n/m)a1(NF-2) -ND[p + g(N, di.f) + ®(z)(a0 + axd^b - x~)] '4a0 + (n/m)ai(NF-2) P(ND + 1) < pn(ND + 1) - (1 -p)NF -ND[g(N,di*f) + ®(z)(a0 + aid,-.6 - x)] Dividing through by No + 1 and simplifying gives K > (l-p)a1(NF-2) (n/m) - 4 [pr, - /? - (jgo.) (g(N,*.,) + O(*)(a0 + a ^ - x~)) - < ^ g ^ ] (2.34) Then, if the right hand side of (2.34) is greater than 1, the monopoly bank will ration credit. • Proof of Corollary 2 Diffentiating (2.33) with respect to «o and ct\ yields: dN$ _ 4[(n/m) - 1] dao (n/m)a.\ and dN$ _ -4a0[(n/m) - 1] da\ (n/m)2 oil > 0 < 0 Chapter 3 Welfare Properties of a Banking Monopoly 3.1 Introduction It is well known that the incentives of a monopolist do not normally align themselves with those of society in general. However, Diamond (1984) presents an economy where the bank and society do share common goals. In his model, as the size of the bank increases, its probability of default approaches zero and this means that the expected value of deadweight monitoring costs from depositors monitoring the bank also approaches zero. This has led to the interpretation of the delegated monitoring process as a natural monopoly. At first glance the Diamond equilibrium has some surprising properties: lenders and the bank earn no rents; all of the rents from delegated monitoring accrue to the firms. Essentially, this result obtains because of the threat of competition from the bond market or from potential entrants into the banking sector. These threats cause the bank to price its loans and deposits such that it earns zero profit in equilibrium. From the theory of contestable markets (see Tirole (1988) pp. 307-311, for example), this can lead to the outcome being socially efficient. While Diamond does not specifically address the issue of social welfare, it is clear that when intermediation is feasible in his model, the monopoly bank minimizes the information cost associated with lending (the average cost function of the bank is monotone decreasing in the number of loans) and thus maximizes social value. Unfortunately, as Yanelle (1989) points out, this result is not robust to different specifications in the lending game. In the previous Chapter we saw that the monopolist bank can potentially offer some improvements over an 48 Chapter 3. Welfare Properties of a Banking Monopoly 49 economy where firms use the bond market as an exclusive source of funds. This Chapter asks the question of whether we are always better off in aggregate by having a bank when monitoring costs depend on distance. That is, does the Diamond result that the bank is at least welfare neutral hold in the economy presented in Chapter 2? The answer, not surprisingly, is no. In this Chapter I discuss the nature of banking when a social planner has control over the nature of contracting in the economy including the operation of the bank and the degree of competition in the market for lending. In order to study the social value of the bank, I adopt a simple utilitarian social welfare function where the planner maximizes the sum of firm profits, lender returns and the profit of the bank. Consistent with Chapter 2, the planner is subject to the same information asymmetries as the other agents as well as the same contracting restrictions imposed on the bank in that setting. The results confirm that a single bank in this model does not necessarily generate the socially optimal outcome. If direct lending and the bank co-exist, the bank is forced to compete with firms for lenders and, as we saw in Chapter 2, this competition can lead the bank to engage in strategic lending. That is, when direct lending is feasible, the bank knows that its offers to lenders and to firms must take this threat into account and may lead to higher deposit rates and lower loan rates. This in-turn decreases the probability that the bank will be solvent and thus increases the probability of socially costly monitoring by the bank's depositors. The planner can avoid this problem by placing restrictions on the firms' and lenders' rent extraction abilities. By restricting the firms' ability to compete with the bank, the absence of the threat of competition will allow the bank to exploit the marginal lender and funded firms to their reservation returns which in-turn, increases the probability that the bank is solvent. Another result presented in Chapter 2 is that the privately run bank may ration credit regionally. That is, the bank may take deposits from one region and yet may not make loans there. This incentive arises from the fact that the bank creates value by diversifying. Holding the strategic incentive for making loans constant, the bank will equate the marginal cost of Chapter 3. Welfare Properties of a Banking Monopoly 50 monitoring firms with the marginal benefit from reduced default costs. Since the bank has flexibility in its choice of loan size, once it is well-diversified it can take deposits from around firms that are not offered loans and use those funds to make larger loans locally. This property also manifests itself in the planners solution: if the planner-run bank is well diversified in equilibrium, then the bank will almost always ration credit regionally. The plan of this Chapter is as follows: section 3.2 outlines the economic environment and develops the social welfare function. Section 3.3 presents the major findings of the Chapter and section 3.4 extends an example from Chapter 2 to illustrate these results. The final section concludes the Chapter and discusses possible extensions to this research. 3.2 The Economic Setting The economic environment is identical to that considered in Chapter 2: lenders and firms have the same preferences, endowments, and technologies as before, and assumptions \-A on page 15 are maintained. The planner's bank, like the privately run monopoly bank of Chapter 2, is given first mover advantage in the lending and borrowing markets. Subsequent to the bank's offer, unfunded firms can offer direct lending contracts to lenders who did not invest their endowment with the bank. Since the information structure is identical to that of Chapter 2, the contracting problem between the planner's bank and the lenders and firms is also the same. This means that both the lending and borrowing contracts will be the standard debt contracts presented in the previous Chapter. A policy instrument that is available to the planner is competition policy: I assume that the planner can dictate the degree of competition that the bank faces. Chapter 3. Welfare Properties of a Banking Monopoly 51 3.2.1 Bank Lending The lending problem here is identical to that considered in Chapter 2: the planner chooses the number of firms that it wishes to lend to, N§.p, the per-dollar face value of the loan, rfp, and the deposit rate offered to lenders, r | p : the number of depositors attracted, N^p, will be implicitly determined by the deposit rate. Both firms and lenders will make conjectures about the private debt market when deciding on whether to accept these offers from the bank. Some differences to note here are that, unlike the lender-run bank considered above, the planner has no unit of capital to lend. This means that the loan size will be ~ Nfp which makes the expected revenue of the planner's bank Nsp E[ySTP] = ±PKSPrfp SP~ (ND | rSP = WP = NsDPprfP The expected revenue per depositor will then be E[XSF] _ S j j p = p/p with a standard deviation of CrSP = \ P(1~P)rSP mp l and means that the number of depositors attracted by the planner-run bank, N^p, will not affect the bank's mean return per depositor nor standard deviation of that return and thus has no effect on the probability of default. With this one exception, all of the other results of Lemma 1 in Chapter 2 will carry through. Chapter 3. Welfare Properties of a Banking Monopoly 52 The probability that the bank fails can again be approximated by the normal distribution. Define zSP as the standardized score „SP _ r§p - E[xsp] then, if <!>(•) is the c.d.f. of the standard normal, the probability that the bank is solvent will be Pr(xSP>r$p) = l-<!>(zsp) If the bank defaults, its residual value per depositor, xSP~, can be written XSP- = j ^ xSP(f) ^ S P ) dxS „SP where j (^) = 1 e-^'-^'B'/^sP is the p.d.f. of the normal distribution. Given the above, we can write the expected profit of the planner's bank as follows: ^SP = NsPprfP _ ( 1 _p)NsP Rao + (nMMNsr - 2) -NsDp(l - ^{zsp))rs/ (3.35) Reservation Returns The bank's offer must satisfy individual rationality constraints for lenders and for firms. For a lender, i, d^ lenders away from the bank to accept the bank's offer, he must earn at least his reservation return, & > /?, so Vfp = (1 - ®(zSP))r$p - Q>(zsp)[aQ + axdib - xSP~] > ft (3.36) As before, £,• will be determined by anticipated competition from the private debt market1 and will depend on the distance i is from an unfunded firm. The total utility for lenders who take 'See equation (2.12) in Chapter 2. Chapter 3. Welfare Properties of a Banking Monopoly 53 deposits from the bank will be NSP ySP = Y>ySP i = l = NSDP(1 - *(zSP))rSdP • - <!>(zSP) Nsr N%pa0 + ax J2 dib - N% SP„SP X t = l (3.37) Firms who are offered a loan contract {KSP,rfp} from the bank will weigh this offer against the profit they anticipate they could earn by selling bonds in the private debt market: denote the reservation profit for firm / to be Tlf > 0. The firm's participation constraint can then be written Uf(Ksp, rfp) = pKSP(R - rfp) > Ylf (3.38) The total expected profits of firms who are offered (and accept) a loan contract from the bank will be nsp = E n ^ V f ) = NSFPP(^f)(R-rn = PNsDp(R-rfp) (3.39) Welfare From Banking The welfare generated by the banking sector will be the sum of the bank's profit, VF5P, total depositor utility, VSP, and total firm profits, n 5 p . Denoting this sum by Ssp and simplifying gives, ?SP _ xpSP + ySP + JJSP = pN¥R-(l-p)N§. SP 4a0 + (n/m)a1(JV|p-2) -0(2*0 NSP SP„SP-N^ao + ^^dib-N^x «=i (3.40) Chapter 3. Welfare Properties of a Banking Monopoly 54 3.2.2 Direct Lending Firms that are not offered (or who rejected) loan contracts from the bank may make offers to lenders who chose not to take a deposit with the bank. Denote by NpL the number of these firms who are successful in their acquisition of financing through the bond market and by N£L, the number of lenders who purchase this debt. Let Dk be the face value of the debt offered to lender k = 1, . . . , N^p, dkj lenders away from the borrowing firm / . This lender will earn an expected utility equal to VkDL = pDk - (1 -p)(a0 + axdkf) Summing over all of these lenders, gives a total expected utility of NDL VDL = Y,]pDk-{\-p){aQ + axdkf)} "BL = pJ2Dk-(l-p) NSL NELa0 + axJ2 dkf k=\ (3.41) Aggregate profits for firms who succeed in securing financing directly from lenders will be NPL nDL = En / *BL = NEL(pR)-pJ2Dk J t = l (3.42) Welfare From Direct Lending The welfare generated by direct lending between unfunded firms and non-depositor lenders will be the sum of the lenders' utility, VDL, and total firm profits, JJDL. Denoting this total by SDL and simplifying gives, SDL _ yDL+UDL = NEL{pR)-(l-P) NgL NgLaQ + a, £ dkf k=l (3.43) Chapter 3. Welfare Properties of a Banking Monopoly 55 3.2.3 Social Welfare Total social welfare, S, will be the sum of the welfare from banking, SSP, and the welfare from direct lending, SDL. That is S = gSP + SDL TSP , ATDL\ = PR(N*  + N°L) - 0(zSF) Nsp N^ao + at^du-N^x SP„SP-i-\ -(i -P)NS/1"4"0"1" (nMMNlp-2l -(l-p) NSL NELa0 + ax J ) dkf fc=i (3.44) Discussion Since the planner can restrict competition, her problem is to choose the number of firms to lend to, NpP, the per-doUar face value of these loans, rfp, and the face value of the deposit contract, rfp, subject to satisfying lenders' and firms' participation constraints. As before, I assume that these face values are common across all lenders and to all of the firms offered loans: that is, no discriminating contracts are allowed. This prevents the planner from fully exploiting her power but maintains consistency with the restrictions placed on the privately run bank in Chapter 2. The planner is allowed to control firms' entry into the private debt market, however, which offsets this disadvantage to a large degree. 3.3 Results The maximization of social welfare has two components: the first is maximizing the allocation of lenders' endowments to the firms' production technology; and, second is the minimization of expected deadweight monitoring costs. From (3.44) we can see that while direct lending is consistent with the first component, it may not be consistent with the second component. That is, utilizing the direct lending market is socially costly since, when the bank operates, Chapter 3. Welfare Properties of a Banking Monopoly 56 competition from the bond market may require the bank to pay rents to the firms and lenders who accept loan and deposit contracts. These transfers increase the probability that the bank will fail and increase expected dead-weight monitoring costs, reducing social welfare in the process. This leads to the following proposition Proposition 4 If direct lending is feasible the social optimum involves: (i) no rents paid to firms funded by the bank; and (ii) restrictions on the firms' ability to compete with banks. Proof: The proof involves showing that the two policy decisions made above will decrease the bank's default risk. (/) Hold r | p fixed and recall that the bank's default risk will fall as zSP = ,1^3 decreases. Next note that firms' participation constraint pKSP(R-rfp)>Tlf means that the loan rate can be written as = R = R „SP - n ' PKSP N§ptif pNsDp when this constraint binds. Thus, the mean return per dollar of deposits, is decreasing in ft/, so for rfp fixed, the numerator of (3.45) is increasing in fi/. The standard deviation of xsp can be similarly written NfpUf arsp = P(l-P) \ N§.p R- PNSDP Chapter 3. Welfare Properties of a Banking Monopoly 57 which is decreasing in 11/. Since the numerator is increasing and the denominator is decreasing in flf, then zSP must be increasing in fi/ and this will lead to an increase in default risk, Q>(zSP). This means that the optimal policy for the planner is to pay no rents to the firms since employed capital will not change and default risk will fall, implying that social welfare is increasing. («") Suppose there is competition from firms for lenders. Then £,., the reservation return of the marginal lender, will be and £. = max ]B,pR- (1 - p ) \a0 + ai( d{A SP _ &• + ®(zsp)(a0 + arfi.b - xSP~) >P rr = I - ®(zsp) is increasing in f,-.. Since default risk is increasing in r$p, the planner should make £,-* = /?, the minimum value it can be. • Since the planner can eliminate the need for the bank to lend strategically, she is free to select the number of firms offered loan contracts to that which minimizes the probability that the bank will fail. As we saw in Chapter 2, this may lead a bank to ration credit regionally—the bank may borrow from regions where it does not make loans. This result also obtains in the planner's problem. Here the planner knows that deposits can be channeled into the production technology most efficiently via the bank and that once the bank is well diversified, there is no value to a larger number of loans: she will simply choose to make larger loans. This is stated in the following corollary, Corollary 3 Suppose the bank is well diversified in equilibrium, then the planner's solution may involve credit rationing. Chapter 3. Welfare Properties of a Banking Monopoly 58 Proof: First note that if the bank is well diversified, then 0(zsp) -» e (where e is a small finite number) and, from the properties of the normal distribution we know that lim ®{zSP) = 0 and lim &(zSP) = 0 (3.46) zsp-+—oo which implies that changes in zsp have vanishingly small changes in default risk when we're near the tail of the distribution. Next, we know from the discussion above that the 'best' solution is one where the planner only trades off expected monitoring costs on loans with lower probabilities of bank default (that is, she makes no strategic loan or deposit offers). Then, at the margin for the bank - M ^ I ^ L M - (l-p)[2a&+(n/m)a1(JV#''-2)] { }dNsFp 2 where or M-NSDP NsDpaQ + axy£dii,-NsDpx" rSP-t = l T ) R _ rSP ° ' ^ O / ^ P ^ ^ = & -* a° + Wm)atN$P ~ 2)1 Ry/N$pp(l-p) Then from (3.46), changes in NpP will have vanishing effects on O(-) but will increase expected monitoring costs by (1 -p)[2a0 + (n/m)a1(NF'p - 2)] > 0 This means that once the bank is diversified 'enough', the planner will stop making loans. However, since O(-) is small and if there is no competition for lenders, attracting more depositors has a small (but finite) effect on costs but will increase expected social welfare by a factor of pR for each additional depositor attracted. This will lead the planner to ration credit. • Chapter 3. Welfare Properties of a Banking Monopoly 59 3.3.1 Discussion The social planner's problem provides some interesting results. In this economy the incentives of the benevolent planner and a profit seeking monopolist are exactly aligned. Both want to decrease aggregate default costs and this is done by exploiting lenders and firms to their minimum reservation returns. Moreover, the planner finds it optimal to distort the allocation of credit by lending close to its location, again for the reason that this is consistent with its goal of minimizing default costs while maximizing the amount of capital channeled into the production technology. The objectives of the planner-operated and the lender-operated banks will differ with respect to direct lending. Since the privately operated bank does not have the ability to restrict the competition it faces from the bond market, it may attempt to eliminate this market all together. The planner, on the other hand, through the use of her policy instrument will allow firms to attract lenders that the bank cannot offer efficient deposit contracts to. This is captured in the SDL term in the social welfare function. 3.4 Numerical Example In order to illustrate some results from this Chapter, we'll consider a numerical example based on the second example in section 2.5 of Chapter 2. Recall that we assumed the following values for the exogenous parameters: • R = 2.25; • p = 0.7; • n= 1,000, ro= 100; • K = 8; Chapter 3. Welfare Properties of a Banking Monopoly 60 • 0 = U • aQ = 0.025; • <*j = 0.0075. Direct Lending: Direct lending was feasible in this economy and firms found it profitable to make offers to all of the lenders in their 'areas'. Since there are (n/m) = 10 lenders per firm, there will be five lenders on each side of every firm. As in Chapter 2, we can calculate the reservation returns for the lenders on one side of a representative firm, / : 6 = 1.54725 & = 1.54900 £$ = 1-55175 & = 1.55400 & = 1.55625 where the & is the reservation return for the lender closest to the firm and £5 is the reservation return for lender furthest from the firm. Symmetry dictates that these will be the reservation returns for the 5 lenders on the other side of / . Using these reservation returns, we calculated the face values of the contracts that / offers and saw that the total debt obligations of firm / (and hence all of the firms) were D = 2J2DJ i=\ = 22.3394 and since -ftT = 5 x 2 = 1 0 , the firms' expected profits were shown to be 11/ = pKR-pD Chapter 3. Welfare Properties of a Banking Monopoly 61 = 0.7(22.5-22.3394) = 0.1124 Social welfare here will be the sum of the firms' expected profits plus the expected utility of lenders. In this example the surplus will be (n/2m) m S = 2(n/m) £ 6 + E n / i=i / = i 5 = 200J2& + 100(0.1124) t=i = 1,551.65 + 11.24 = 1,562.89 Intermediation By a Private Bank: When the bank was operated by a lender it acted strategically in the lending market—offering all of the firms loan contracts—and, in the process eliminated the private debt market as a competitor. Each of the firms received a loan of size K = 10 with a face value per-dollar of loan of r/ = 2.23394: this provided all of the firms with the same expected profit as they earned under direct lending. All of the depositors accepted a contract with a face value of r<j = 1.001. The bank's probability of default, <f>(z), was calculated to be 0.1825 x 10~7 and the expected profit of the bank was X¥MB = 506.60. The last depositor attracted (499 lenders away from the bank) will have an expected utility equal to2 V = (l-<b(z))rd-®(z)(a0 + aidi.b-x-) = 1.001(0.99999998175) - 0.00000001825(0.025 + 0.0075(499) - 0.5005) « P=l 2To simplify these calculations, I approximated x~ by rj/2. To calculate this exactly, one could use Monte Carlo techniques, for example. However, since the probability that the bank fails is very small, little precision is lost by this approximation. Chapter 3. Welfare Properties of a Banking Monopoly 62 his reservation return. Social welfare will be 999 100 s = y ^ + Efi + S n / «=i / = i = 506.60+1,000+11.24 = 1,517.84 Which shows that social welfare when the private bank operates is actually lower than when the only source of funds was the private debt market in spite of the fact that the bank's default risk is very close to zero. The reason for this loss of efficiency is that the bank has to be aggressive in the loan and deposit market to compete with the bond market. The bank realizes that if it doesn't offer loan contracts to all of the firms then even a single unfunded firm could drive lenders' reservation returns to a point where the bank is unable to diversify sufficiently to operate. This aggression leads them to "over lend"—they offer too many loans and the excess expected monitoring cost incurred on these extra loans is a significant drag on the bank's value. In a sense, the bank's inability to ration credit is socially costly in this example. Intermediation By the Planner-Operated Bank: From Proposition 4, we know that the planner's best policy is to restrict the ability of the bond market to compete with the bank. That is, the planner would like the bank to be able to act as a monopolist in the loan and deposit market. Suppose in this example that she eliminates the bond market all together and allows the bank to operate as a true monopolist Solving this planner's problem under this condition yields the following equilibrium: N§p = 34 firms (out of a total of 100) are offered loan contracts with a loan size of 29.41 and a per-dollar of loan face value of rfp = 2.25 (providing firms with an expected profit of zero) and lenders are offered and accept a deposit contract with a face value of r$p = 1.004. Almost all of the rents are captured by the bank: its value is *¥sp = 565.63. The probability that the Chapter 3. Welfare Properties of a Banking Monopoly 63 bank defaults is <I>(z5P) = 0.0006.3 Social welfare in this case will be S = SSP + SDL _ xj/SP + ySP + JJSP + yDL + UDL = 565.63 + 1,002.24 + 0.00 + 0.00 + 0.00 = 1,567.86 This is higher than welfare under direct lending alone as well as that achieved with a privately run bank. Discussion: This example illustrates that the threat of competition may impose large costs on an economy such as this. In this example, low travel costs made the bond market very competitive: for the privately run bank to operate it had to eliminate that market all together. This strategic move against the bond market is socially wasteful since, while the bank eliminated some of the expected monitoring costs, it did so by lending to more firms than is optimal. The net result was that the bank was feasible but actually reduced social welfare from that achieved under direct lending. In the planner's problem, we see that she would like the bank to operate as a monopoly and exploit lenders and firms as much as possible. This is due to the fact that the planner's objectives and a monopoly bank's objectives are perfectly aligned: both would like to drive default costs to the point where the marginal default cost is equal to the marginal expected monitoring cost imposed by lending to firms. The welfare improvement results from the planner's ability to exploit firms and lenders to their reservation returns through the use of her policy instrument. 3 As a check, the marginal lender's expected utility is 1.004(0.99940) - 0.0006(0.025 + 0.0075(499) - 0.502) « 1 Chapter 3. Welfare Properties of a Banking Monopoly 64 The privately operated bank, on the other hand, is forced to utilize a costly strategy of capturing the entire borrowing market and forces this bank to lend to more firms than is optimal. The result is that the planner rations credit in equilibrium and, by doing this, is able to make a welfare improvement over direct lending. 3.5 Conclusions and Future Research This Chapter presented a welfare analysis of the bank discussed in Chapter 2. Unlike the results presented by Diamond (1984), the introduction of a bank may have a detrimental effect on social welfare. When faced with competition from the bond market, the bank may engage in strategic lending that is unrelated to its desire for diversification. This strategic lending is socially costly since it forces the bank to incur the expected monitoring costs associated with these loans. The results point to a need to restrict competition for the bank. It was also shown that the credit rationing result from Chapter 2 is also efficient. That is, regardless of whether the planner or a lender operates the bank, it would like to ration credit regionally. Natural extensions to this research would be to explore whether other types of industrial structures produce higher levels of social welfare than the one discussed here or possibly, the design of a tax and transfer scheme to implement the social optimum. Further work may also wish to examine different social welfare functions—perhaps one that places a social value on the number or location of loans made in the economy. Such a specification would certainly provide different conclusions. Chapter 4 The Role of Deposit Insurance in the Regulation of Banks 4.1 Introduction During the past ten years, the extent of the Savings and Loan crisis in the United States has become apparent and even the notoriously stable Canadian banking system has seen a number of costly failures.1 A contributing factor to this problem is that in both Canada and the United States, deposit insurance is provided to banks by government or quasi-government agencies at a fixed premium rate. That is, banks are always charged the same premium despite differences in their insurance risk. One implication of this is that banks may have the incentive to increase their risk and have the government bear the financial consequences. This has led most industry observers to conclude that reform of the deposit insurance system is essential for a return to stability in the banking sector. The academic and professional literature has been almost exclusively focused on the issue of the pricing of the insurance contract. This, however, neglects an important aspect of several deposit insurance systems, namely deposit insurance funds. In this Chapter, I examine what role these funds have in the optimal regulation of banks. Deposit insurance is typically funded by two different sources: premia (not unlike "normal" insurance) which are not refundable; and a contribution used to maintain a pool of funds in case of loss—a deposit insurance fund—of which a portion was usually refunded.2 What 'As an example, between 1975 and 1983 an average of three insured commercial banks or trust companies failed per year in the United States. Between 1984 and 1992, this average jumped to 145 per year (source FDIC: Historical Statistics on Banking, 6). In Canada, between 1967 and 1982 the Canada Deposit Insurance Corporation made payments to depositors or incurred rehabilitation costs on a total of 4 banks. During the 1984-1994 period, this number increased to 30 (source: CDIC Annual Report, 1994,25). 2 In the United States, the FDIC's premium assessment was well above the historical loss figure for many years 65 Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 66 distinguishes the "pure" premium portion from the fund portion of the premium is that while the banks remain solvent, they have a claim to the assets of the fund. Since the value of the fund to a particular bank depends on its default risk, this may be an important tool for regulators hoping to control the excessive risk-taking in the banking sector. The academic literature has largely ignored the issue of the fund and its size and has chosen instead to concentrate on the pricing of premia. Evidence of the pervasiveness of this can be found in a recent issue of the Journal of Banking and Finance3 which is entirely devoted to the topic of deposit insurance reform. In total, fourteen papers are presented and of those, roughly half discuss the well-documented problem of the pricing of deposit insurance. With one exception, this set of papers approach the problem from the perspective of "deposit insurance as a put option", a concept first proposed by Merton (1977): in fact, of the fourteen papers, ten cite Merton and none discuss the role of the fund. This is in contrast to bankers and regulators who do seem to be concerned with the size and composition of the fund. In February 1991, the FDIC outlined an eleven point plan on the reform of the deposit insurance and the recapitalization of the Bank Insurance Fund (BIF). On the issue of recapitalization, the plan called for legislation permitting the FDIC to borrow from "new sources such as the Fed and the banking industry"4 and for premium increases to re-build the fund. This was tacitly endorsed by the president of the American Banker's Association who stated Another premium increase is a bitter pill... [b]ut if it's needed to ensure that the Bank Insurance Fund is solvent, it's in everyone's best interest for banks to swallow that pill.5 and they maintained a policy of refunding a portion of the premium to solvent banks. In Canada, federally funded deposit insurance does not have this feature. Some provincial insurance agencies do, however. 3Volume 15, Number 4/5, September 1991, pp. 733-1040. 4 ABA Banking Journal. April 1991, p. 40. 5ibid Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 67 In this Chapter, I adopt a model very similar in spirit to those currently used in the literature for modeling an informational asymmetry between the bank and regulator to analyze the effects of the fund. The work closely follows Chan, Greenbaum, and Thakor (1993) and Giammarino, Lewis, and Sappington (1993) who view the problem from the insurer's perspective: banks have differing risk which is not observable to the insurer. When the insurer offers deposit insurance to the bank, the riskier banks have an incentive to mimic the safer banks and purchase a mispriced contract. Here though, I modify the instrument set available to the regulator: instead of offering a combination of deposits and premia like Chan, et. al. or a package of loan size, quality, and reserves as in Giammarino, et. al., the insurer can offer the combination of premia and fund size, and loans. After accepting the regulator's offer, the bank chooses a capital structure composed of new equity and deposits to finance its loans. The plan of the Chapter is as follows: in section 4.2,1 outline the economic environment under which these agents operate and the technology they can employ. In section 4.3,1 define and develop the proposed insurance scheme, derive the optimal premium schedule and level of loans that the regulator will offer each type and provide a numerical example showing the results. Section 4.4 concludes the Chapter. 4.2 The Model In order to study the relationship between the structure of the deposit insurance contract and the regulation of banks, I adopt a stylized model of the banking sector. Here, there exists a single bank and a regulator who also serves as the deposit insurer. Both types of agents operate in a one period setting: at time zero, the bank learns its type then the regulator moves and offers a contract comprised of a level of loans and a deposit insurance contract. The bank selects its preferred contract from the menu offered by the regulator and raises the funds it needs to do its lending by selling equity and fully insured deposits. At time one, payoffs are revealed to all Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 68 parties and consumption takes place. All agents are risk neutral. The regulator is charged with the task of insuring deposits and regulating the bank. As deposit insurer, the regulator must provide complete deposit insurance to banks and she is free to set the form of the deposit insurance premium and the lending of the bank. Using these regulatory tools, she maximizes a simple measure of social welfare. While the regulator knows a great deal about the bank, she cannot observe its default risk. 4.2.1 The Bank The bank learns of its type, 6, at tQ but this is unknown to the regulator. The parameter 6 is the probability that the bank's loans payoff: that is with probability 6, the loans pay off in full and with probability 1 — 6 they pay off zero. There are two types of banks: risky, R, and safe, S, and their solvency probabilities are 0 < 6R < 6s < 1. With probability /z > 0 a bank is of type R and with probability 1 — p. it is of type S. Loans The bank has a risky loan production technology whose payoff at t\ is a strictly concave function, F(-), of the loan size invested at to, L. Assume that F(0) = 0 and that F(-) obeys the Inada conditions: l i m ^ o F'(L) -* oo and lim^oo F'(L) = 0. Thus a type 6j bank's gross expected return from making L(6j) in loans will be 6jF(L(6j)). However, access to this technology does not depend on the issuance of deposits: that is, the loan may be funded entirely with an equity issue, should the bank choose to do so. Bank Charters As a regulated industry, entry into the banking sector may carry with it the ability to earn rents. The present value of these rents can be thought of as a 'charter' value, or the value of a license6 to operate a bank. 6Up until 1980, the formation of a chartered bank in Canada required an act of Parliament and, even today requires the approval of the federal Cabinet and the acquisition of a license from the Minister of Finance (Binhammer, 1982). Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 69 Rents from banking can accrue from market power on the asset side—by being able to purchase loans at below their market value—or on the deposit side—by being able to sell deposits at below their market value. This charter has value to the bank only while it is solvent: if the bank fails, the charter reverts back to the regulator. Keeley (1990) presents a model where power in one, or both markets can generate a value to a bank charter and derives a measure of market power (Tobin's V ) that is positively related to the value of the charter. In his model, banks will have an incentive to hold more capital (relative to assets) in order to 'insulate* themselves from default and the resulting loss of their valuable charter. In an empirical test, he finds that there is evidence to suggest that banks in the United States enjoyed greater market power prior to the extensive deregulation in the banking industry in the 1970's and 1980's and that during this period banks held more capital and fewer of them failed—evidence that banks may have been protecting valuable charters during the pre-deregulation period. Keeley further hypothesizes that a decrease in market power following deregulation eroded the value of bank charters which may have contributed to a larger number of failures in the post-deregulation era. In Canada the evidence for charter value is somewhat mixed. Binhammer (1982) cites a study by the Economic Council of Canada which "suggests that due to a lack of sufficient competition a significant degree of market power existed in the Canadian banking industry"7 during the late 1960's and early 1970's. A more recent study by Amoako-Adu and Smith (1994) finds that the 1980's deregulation in Canada was, for the most part, wealth neutral for shareholders of banks and near banks.8 Like Keeley, I assume that bank charters are valuable and that the regulator, if forced to pay 7Binhammer (1982), p.106. 8It is also important to note that the Canadian reforms affected almost all aspects of the financial services industry. Many of the banks used relaxed barriers to entry to move into other areas of financial services, most notably, securities underwriting. Thus, it could have been that the value of their bank charters fell and yet the decrease in the banks' value was offset by the value to entry in these other sectors, leaving shareholder wealth relatively unaffected. Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 70 out the depositors of the bank, will seize the charter. Since the return to the loan technology is independent of the manner in which it is financed, charter rents are assumed to come solely from the issuance of deposits.9 Deposit Insurance Banks who issue deposits are required to purchase insurance from the regulator. This insurance provides a complete guarantee to the purchasers of deposit contracts: any losses imposed on them by a bank default will be fully covered by the regulator. The deposit insurance premium (expressed on a per dollar of deposits level) can be comprised of two components: anon-refundable portion, ir, and a portion that may be reclaimed by the bank (provided it is solvent) at time one, denoted by / . The total required transfer from the bank to the regulator for insurance on D deposits at time zero is then D x (7r + / ) . Deposits There is a perfectly elastic supply of deposits in the economy which, since they are completely insured, can be issued at the riskless rate of interest by the bank: this riskless rate of interest is exogenously set equal to zero. Once the bank has accepted their insurance terms from the regulator, it optimally selects its deposit level. New Equity The bank can also finance its loans and deposit insurance premium and fund contribution by issuing new equity, E. Since I focus on a separating equilibrium, this equity is assumed to be correctly priced in equilibrium and I assume that the cost of equity, re > 1, is constant. The difference between the required return on equity, re, and the riskless rate of 1 captures an assumed preference for deposits over equity, perhaps generated by the liquidity benefit of bank deposits, as an example. 9KeeIey also finds that, while there is evidence that U.S. banks enjoyed market power in the deposit market, the measure of their power in the loan market was positive but not significant Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 71 4.2.2 Capital Structure and Bank Value The bank is required to issue equity, deposits, or both to finance its investment in the loan technology and to pay for deposit insurance. Since the return on the loan is independent of the capital structure choice made by the bank, the bank can choose to finance the project with equity alone. Denote the optimal loan level chosen by a type 0 'bank' if it finances with equity only as LAE{9) and assume that OF{LAE{0)) - reLAE{e) > 0 V0 € {6R,0S} (4.47) where LAE{0) solves 0F'(LAE(0)) = re V0e{9R,es} which means that the value of the 'bank' when it is financed entirely with equity is positive for both types. Also, since by assumption, OR < Os, the strict concavity of F{-) means that LAB(0R) < LAE(0s): the safe type invests more than the risky type with all-equity financing. If it chooses to finance with equity and deposits, the sources and uses of funds for the bank can then be written as, D + E = L + D[TT + f] or E = L - D[l - 7T - / ] (4.48) At t0, the expected total value of bank's equity will be 0[F(L) -D + fD] so if the original owners of the bank sell a fraction, a, of this in the form of new equity, the value of the old shareholders' claim will be (1 - a){0[F(L) - D(l - /)]} (4.49) Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 72 Since the new equity is fairly priced in equilibrium reE = a{6[F(L)-D(l-f)]} Substituting from the above into (4.48) and then into (4.49) yields the initial value of the bank (just after it selects L and D and pays fD and TTD to the regulator). 6[F{L) - D{\ - /)] - re[L -D{\-ir- /)] Charter Value: The present value of the bank's future rents from the charter will be a function of the deposit rents that it is able to earn in the future. Let this value be represented by a function B{D) where B{0) = 0, B'(D) > 0, and B"(D) > 0. This structure assumes that the value of the charter is an increasing function of the value of the deposits issued today which would reflect the value of establishing or maintaining economies to scale in the deposit market such as a branching system or the value of rents from maintaining the payments system. Then, since the bank loses the charter when it fails, the expected cost of failure will be (1 — 6)B{D). This makes the expected value of the bank at to, W(8) = 6[F(L) - D{\ - /)] - re[L - D{\ - TT - /)] - (1 - 9)B{D) (4.50) As we will see below, the regulator will offer the bank a deposit insurance premium, IT, and fund contribution, / , to finance deposit insurance. Let {/, f} be the deposit insurance contract selected by the bank. Then for any loan size, L, a type 6 bank would pick D to maximize - ( 1 -f)D + reD(l -7t-f)-(l- 6)B{D) Taking first-order conditions shows that the optimal deposit level, D*(9,f,n), solves B-(^,/,t))^'"-f72;'"1"/) <«» The second-order condition for a maximum - (1 - 0)B"(D*(0, / , TT)) < 0 (4.52) Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 73 is satisfied by the assumption that B{-) is strictly convex. The deposit level selected by the firm then, reflects the trade-off between capturing liquidity rents, paying deposit insurance and the bankruptcy costs (the loss of the valuable charter) imposed by deposits. The optimal level equates the net marginal value of deposits with their marginal cost. 4.2.3 The Regulator By virtue of its ability to issue deposits in place of more costly equity, the bank can generate socially valuable financing gains through its choice of capital structure. These deposits are fully guaranteed by the regulator whose problem is to balance the social gains (captured by the bank) with the social cost of this guarantee. If the regulator could observe each bank's type, the pricing of the insurance contract would be straightforward. However, when information is asymmetric, the design and pricing of the contract must take into account the incentives of the bank to misrepresent its type. In this section, I examine the design of this contract in the economy described above. Beginning with the seminal paper by Merton (1977), there has been a pronounced tendency for academics to focus on finding deposit insurance contracts that are priced in an actuarially fair manner. That is, each 'type' of bank pays a premium for deposit insurance equal to its expected loss. A more recent body of research points out that this may not be socially optimal. When information is not symmetric, the contracts offered by the regulator have to take the adverse incentives of the bank into account and this may entail forcing the bank to distort their investment level and/or choice of capital structure away from their optimal values. If we further restrict these contracts to be fairly priced, the social cost of these distortions will increase. Chan, Greenbaum, and Thakor (1992) point out that insurance/capital structure contracts that are "incentive compatible" (meaning that, in equilibrium, they are taken by the type they Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 74 were designed for) will be impossible to implement if the banking sector is competitive, or if the insurance is required to be fairly priced. The intuition for their result is that if deposits and equity have the same cost in equilibrium and incentive compatible deposit insurance contracts involve an information rent to one type, or a set of types, then the insurer is better off forcing that type (or set of types) to finance with equity only. Giammarino, Lewis, and Sappington (1993) present a model where deposits are socially valuable and where the regulator is allowed to finance expected shortfalls in the deposit insur-ance system through costly government financing. In this way their model captures the social tradeoffs that a social planner confronts when designing optimal regulatory policies. Their results show that incentive compatible deposit insurance is possible and that, in equilibrium, the planner trades off loan quality to induce incentive compatibility. While both of these papers approach the pricing problem from the perspective of a social planner, they abstract from the role that the deposit insurance fund may have in the regulatory scheme. Here, like Giammarino, et. al., I allow the regulator to finance the insurance mispricing through the use of public funds but I also allow her to utilize a deposit insurance fund. The regulator collects D x (n + f) at t0. At t\, if the bank fails the regulator keeps the fund contribution, / , and the premium, it. If the bank is solvent at t\, the regulator still keeps 7r, but refunds / . Since the bank is solvent with probability 6 and fails with probability 1—6, the expected value of the funds available to the regulator (per dollar of deposits) at ti will be = 07T+(l-0)(7r + / ) = 7 T + ( l - 0 ) / and the expected loss (per dollar of deposits) will be ( 1 - 0 ) Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 75 together, this means that the shortfall (surplus) that must be financed will be Z = (i-e)-f(l-0)-ir = ( l _ * ) ( l - / ) - * • As is common in these types of regulatory models (see Laffont and Tirole (1993), for example), I assume that the use of public funds is distortionary with a shadow cost of A, a constant.10 Hence, for a type 9 bank issuing D in deposits, the social cost of a deposit insurance contract of {/, 7r} will be ( 1 + A ) I > [ ( 1 - * ) ( 1 - / ) - * ] 4.3 The Regulator's Problem 4.3.1 Full Information Under full information, the regulator would solve the following problem for each type 9 e max SF< = m-Q+wv,m,*)w-w-w))-*] {L(0),f(6),Tr} where 4/(^) is the value of a type 6 bank. This is also subject to the bank's value being at least zero, Y(0) > 0 (4.53) Proposition 5 The optimal full information allocations (denoted by the superscript FI) for each type 9 e {9R, 9S}, are as follows: 10These costs may be generated by ".. . distortionary taxes on income, capital, or consumption..." [Laffont and Tirole (1993)]. Note as well that this shortfall could be negative: the total insurance premia collected could exceed the expected loss per dollar of deposits. If this is the case we could think of the expected surplus (1 + \)Z as the value of a transfer from one sector (banking) to another sector where government intervention would entail costly public borrowing or taxation. Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 76 1. The optimal loan size is the same as that chosen under all-equity financing. LFI(6) solves 6RF'(LFI(eR)) = re (4.54) esF'(LFI(9s)) = re (4.55) and, hence LFI{0R) < LFI(9S). 2. Deposit insurance fund contributions for each type will be: fFI(0) = 1 - ( - ^ - ) D*(9,fFr(9),nFI)B"(D*(9,fFI(9),7rFI)) - ^ (4.56) Proof: See the appendix. Discussion: If the regulator could observe the bank's type, the proposition tells us that the set of contracts she would offer would have bank choose its best loan portfolio and finance it with the amount deposits that exactly equated the marginal value of these deposits, (re — 1), to their marginal cost, and equity. It is also interesting to note that 2. above implies that (1 - 6){\ - fFI(9)) - *FI = ( ( 1 _ ^ ( 1 + A ) ) D*(9,fFI(9),nFI)B"(D*(9,fFI(9),nFI)) so if deposits are issued, the bank will be undercharged relative to the actuarially expected loss, i.e., TTF/ + (1 - 9)fFI(9) <(l-9) This means that the planner is willing to trade off the social cost of borrowing, A, against the social value of deposits, re — 1. That is, under full information, the planner would be willing to socially subsidize the issuance of deposits by the bank. Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 77 4.3.2 First-Best It will be useful to examine the first-best (FB) allocation for means of comparison later on.11 Here, we consider the cost of government intervention, A, to be equal to zero and use the results from full information above. Proposition 6 The first-best allocations (denoted by a superscript FB) for each type 6 e {QR,6S} we as follows: 1. Loan sizes are equal to their full information values: LFB{9) = LFI{6) 2. Deposit insurance is fairly priced and involves nFB = 1 3. Deposits chosen by the bank solve B'(D*(e,fFB(9UFB))={^ Proof: Since A does not affect the bank's choice of L in a full information setting, eRF'(LFB(eR)) = re 9SF'(LFB(9S)) = re From part 2. of Proposition 5, setting A = 0 gives TTFB fFB(6) = l-JZTe Wet f* ,**} (4.57) which means that the regulator fairly prices deposit insurance in a first-best world. Substituting this into ^(0) and simplifying, gives ¥(0) = 9F{LFB(0)) - reLFB(9) + {re - l)D9(l - fFB(9)) - (1 - 9)B(D) 11 Where first-best here means the allocation the regulator would enforce under symmetric information. Since there are default costs (the loss of a valuable charter), this should really be considered a constrained first-best or second-best allocation. -9andJFB{9) = 0. - 1 ) * -9 Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 78 which means that the deposit level, D*(9, f(9),n), chosen by the bank will solve then, to maximize socially valuable deposits outstanding, the planner should set fFB(9) = 0 and TTFB = 1 - 9. D 4.3.3 Second Best The regulator's sole objective in this economy is to maximize the expected values of the bank while providing complete deposit insurance.12 Here the regulator's problem is presented as a standard principal-agent program: the bank is offered a menu of contracts from which to select. I assume that the regulatory instruments are the size of loans, L, and the contribution to the deposit insurance fund, / . The regulator also chooses the flat rate premium, n which does not depend on the bank's type. This menu can then be written as {L(9), f(9), -K}. Let {L(9j\9i), f(9j\9i),7r} denote a contract designed for a type i bank and selected by a type j bank. Then the value of a type i bank taking this contract will be W^Oi) = 9i[F{L{9j))-D\9iJ{9j)^){\- f{9j))] -r«[L(^-) - D*(9i,f(9j),ir)(l-n-f(9j))} - (1 - 9i)B(D*(9i, f(9j),7r)) and for simplicity denote bank z's value under the separating contract By appealing to the Revelation Principle (Myerson, 1982), we can restrict our attention to those contracts that are incentive compatible and satisfy the bank's participation constraints. Under the separating contract, {L(9i\9i),f(9i\9i),7r} = {L(0,),/(#,),7r}, the regulator's expected 12The regulator does not have to separately accountfor the liquidity value of deposits since these are completely captured by the bank. Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 79 cost of providing deposit insurance will be Z = »D*(9R,f(0R),7r)[(l-eR)(l-f(eR))-n] +(1 - ii)D*{e8,ttOs),*W - **)(1 - f(0s)) - n] The regulator's problem, [RP], is max S = v*{0R) + (1 - fi)W(es) - (1 + X)Z [RP] (4.58) {{£(*)•/(*)}.*} subject to: VVR) > V(0s\eR) (4.59) n<?s) > Wnlds) (4.60) ^(0R) > 0 (4.61) V(6s) > 0 (4.62) Constraints (4.59,4.60) are the incentive compatibility constraints: these ensure that the bank selects the contract the regulator intends it to take. (4.61,4.62) provide the owners of the bank with limited liability. 4.3.4 Results The solution to the regulator's problem must balance the social value of deposits with their social costs. Incentive compatibility requires that equilibrium contracts make one type's contract (at least weakly) unattractive to the other type. This requirement comes at a cost: if an equilibrium with positive deposits exists, it may be distorted away from the full information values presented above. However, if the cost of implementing incentive compatible contracts that induce the bank to issue a positive amount of deposits is too high, the regulator may choose contracts that force both types to not issue any deposits. The following proposition shows that a solution like this always satisfies the constraints of [RP], Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 80 Proposition 7 A non-banking solution (denoted by the superscripts NBE) to [RP] always exists. This solution has the following properties: 1. LNBE(9) = LFB(9), for each type 9 e {eR,0s}; 2. fNBE{6) = 1, for each type 9 e {9R,9S}; 3. irNBE = 0; 4. D*(9,fNBE(9),nNBE)=0. Proof: Since the bank is financed entirely with equity, we know by assumption that it is at least weakly profitable, regardless of type. This means that (4.61) and (4.62) are satisfied. Also, since the loan sizes are first-best in this case 9SF(LFB(9S)) - reLFB(9s) > 9RF(LFB(9R)) - reLFB(9R) and 0RF(LFB(9R)) - reLFB(9R) > 9SF(LFB(9S)) - reLFB{9s) means that (4.59) and (4.60) are also satisfied. • In this case regulator chooses not to allow either bank to issue deposits in equilibrium by offering them a deposit insurance contract that induces them to choose D* = 0. But this solution is not necessarily unique: since there is slack in the incentive constraints for both types, it may be the case that the regulator could offer a contract that would induce one, or both types to issue deposits and, results in strictly higher welfare without violating the incentive compatibility constraints or the participation constraints. Then, depending on the parameters of the model: 9; re; A; //; and the functions B(-) and F(-), there may exist a banking equilibrium (BE), whose properties are outlined in the following proposition: Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 81 Proposition 8 At a banking equilibrium, (denoted by the superscript BE) the banks are offered the following allocations: 1. The risky bank invests in the socially optimal size of loans and the safe type over or under invests relative to its social optimum. That is, LBE{9S) ± LFB{6S) > LBE{6R) = LFB(0R) 2. The optimal fund contribution for the risky type is greater than that for the safe type. That is, fBE{oR) > fBE{es) 3. The deposits issued by the safe type are larger than those issued by the risky type: D*(esjBE(9s),7rBE) > D*(eR,fBE(eR),7rBE) Proof See the appendix. Discussion Here we see how the information asymmetry affects the regulator. Ideally, she would like to induce the first-best level of deposits and loans however, the fact that she cannot ex-ante distinguish between the types means that the risky type may have an incentive to take the safe type's contract. To circumvent this problem, she offers the safe type a 'distorted' contract—one that is away from his first-best contract. Both types are distorted away from their first-best deposit size by the fund allocation. Here the solution reflects the risky bank's incentive to take a mispriced insurance contract and hence the optimal allocation is designed to make the risky type's deposit choice when he takes the safe type's contract so large that the gains from mispriced insurance are totally offset by an increase in the expected value of a lost charter. This sorting benefit is also augmented by the fact that the fund is a claim issued by the bank that pays off if, and only if, the bank is solvent. By forcing Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 82 the risky bank to make a larger contribution to the fund, this type is forced to internalize at least part of the cost of providing the deposit insurance. The result is that the liquidity benefit of bank deposits may not be fully captured in equilib-rium and, as the following corollary points out, if re is too large, it may force the regulator to offer contracts that induce a zero deposit level. Corollary 4 Social welfare is non-monotone in the cost of equity. Proof See the Appendix. This result while seeming surprising at first, simply reflects the increased sorting costs as re increases. As the liquidity rents rise, each type has a greater incentive to issue deposits and, since the regulator has to keep the risky type from taking the safe type's contract, the level of deposits required for sorting increases with re. After a point, these sorting costs outweigh the value of the liquidity rents and the regulator opts for a non-banking equilibrium. 4.3.5 Numerical Example Here we consider a simple example to illustrate the tradeoffs the regulator makes at equilibrium and to show that there does exist a banking equilibrium as suggested in the previous section. Assume the following values: 1. Default risk: 6S = 0.8, 6R = 0.6; 2. Cost of equity: re = 1.1; 3. Shadow cost of public funds: A = 0.4; 4. Probability that the type is risky: p = 0.5; and assume that the production function for loans is of the Cobb-Douglas variety: F{L) - 4VI Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 83 and that the value of the bank charter is given by B(D) = l-D2 Then, given its type, 9, and the terms of the deposit insurance contract, {irBE, fBE(9)}, each bank will select a D* such that y m . / f l ^ - « « - r<(l-*BE-fBE(0))-9(l-fBE(9)) B'{D*{9,fBh{9)^Bt')) = i-e re(i _mBE _ fh D*(9jBE(e),*BE) = BE,*, BE, _ r*(l-KBE-fBE(9))-9(l-fBE(9)) 1-9 These assumptions yield the following equilibrium:13 1. LBE{9S) = 2.436, LBE(9R) = 1.190; 2. fBE(9s) = -0.814, fBE(9R) = -0.060, nBE = 0.419; 3. D*(9s,fBE(9s),nBE) = 0.417, D*(9RJBE(9R),7TBE) = 0.173. As a point of comparison, this banking equilibrium yields expected social welfare of 1.839 versus 1.818 for the non-banking equilibrium. To illustrate Corollary 4, if we increase the return on equity from 1.10 to 1.25 or reduce it to 1.00, the banking equilibrium is destroyed (and we get a non-banking equilibrium) and if we decrease it to 1.05, social welfare increases to 1.910. The first-best levels are: 1. LFB(9S) = 2.116, LFB(9R) = 1.190; 2. D*(0S,f(9s) = 0,TTFB = \-9R)= 0.400,D*(9R,f(9R) = 0,TTFB = 1-9R)= 0.150; 13Solved numerically in Maple. Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 84 4.4 Conclusion This Chapter presents a simple case for the optimality of deposit insurance funds to a regulator. Here it serves as both a means eliciting the true risk characteristics of a bank (its role as a "sorting" device) and as a means of providing insurance. It is shown that the regulator can provide "type" dependent deposit insurance but that it is costly in the sense that types are distorted away from their first-best production and financing decisions. The results also point to the fact that capital structure is a mechanism for regulation but that it is not a perfect substitute for the role that the fund plays. This is particularly germane in light of the recent multilateral efforts towards bank regulation. Also, although it isn't addressed here, one can conjecture that this scheme may have more power over 'moral hazard' problems than other premia-alone based systems. The reason for this is that the fund has a cost that depends on the bank's true type—hence choosing a riskier loan portfolio, for example, would increase the bank's cost of deposit insurance unlike 'regular' deposit insurance, where the cost is paid up front and is independent of post contracting actions. 4.5 Appendix Full Information Allocations We can write the Lagrangian for this problem as CFI = (1 + n) {e[F{L{6) - (1 - !{e))D{f{6U)} - r<[L{6) - D(f(6),*))(l - f(6) - *) - ( 1 -O)B(D(f(0),*))} - (1 + \)D(f(e),TT))[(1 - 6)(l - /(*)) - ,r] where rj is the Lagrange multiplier on the nonegativity constraint. Taking first-order conditions (denoting optimal values with FI superscripts) gives: | ^ : (l + r]FI){eF'(LFI(9))-r*} = 0 (4.63) Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 85 d£FI • {\ + rlFI)\eD*{9,fI{euFI)-e{\-fFI{6))dD*{djFI{e)^FI) ,fFI(t df(0) dC FI d-K dC FI drj reD*(9, fFI(9),O) + re(l - JFI{9) - TT F / )^ ' ( g . / " (*)» *" ) ( 4 . 6 4 ) +(l+\)D*(9,fFI(9),nFI)(l-9) -(1 + A)[(l -0)(1- fFI(9)) - ^ ] d D ^ Q ^ F I ) = 0 (4.65) (l+r1FI){-reD*(9,fFI(9),7rFr) +r^-fF^9)-^)dD^fy^ OTT +(l+\)D*(9,fFI(9),nFJ) -(1 + A)[(l -9){\- fFI{9)) - vFifD'V'f "&)>*") = o (4.66) OTX vFI > o, VFI{6[F(L(9)-(l-f(9))D(f(9U)] -r<[L(9)-D(f(9),7r))(l-f(9)-7r) -(l-9)B(D(f(9),7r))]=0 (4.67) It will be convenient to state the following lemma, Lemma 2 The participation constraint on the bank's value never binds: that is, 7]FI = 0. Proof: Suppose to the contrary that the bank's participation constraint is binding: rf1 > 0. From the first-order conditions on L{9) (l + r1FI){9F'(LFI(9))-re} = 0 which has a unique solution 9F'{LFI{6)) = re Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 86 the same loan size as the bank would choose under all equity financing, LAE, and, by assumption, 9F(LAE(9)) - reLAE(9) > 0 This means that for the bank's participation constraint to be binding it must be the case that - ( 1 - fFI(9))D*(9JFI(9),irFI)} +reD*(9,fFI(9),7rFI)(l - fFI(9) - TTF/) - (1 - 9)B(D*(9, fFI(9),nFI)) < 0 for all D*(9, fFI(9),iTFI). But the bank would never choose a deposit level that contributed a strictly negative value—it would just choose to finance with all equity which yields a strictly positive value. This contradicts the supposition and proves that rjFI = 0. • Equations (4.65) and (4.66) can, respectively, be re-written as dCFI df(9) 9D*(9, fFI(9), TTF/) - reD*(9, fFI(9), 7tFI) + [re(l - fFI(9) - nFI) - 9(1 - fFI{9)) +(1 + A)(l - 9)D*(9,fFI(9),irFI) - (1 + A)[(l - 9)(1 - fFI{9)) - nFI] dD*(9,fFI(9UFI) df(9) and DC FI 8-K -r'D'{e,!F,(B),-n") + [r'(l - fF'(0) - * " ) - 9(1 - / " (»)) -(i - t)ffMt.f"WS'))] ap ,^7>-*FJ> +(1+A)£>'(S,/"(»),*") - ( 1 + A ) [ ( 1 - ^ ( 1 - / » W ) - ^ ^ « . ^ W . ' " ) Next, from the bank's choice of D (implicitly defined in (4.51)), re(l - fFI{9) - nFI) - 9(1 - fFI(9)) - (1 - 9)B'(D*(9, fFI(9),nFI)) = 0 (4.68) Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 87 and using this property and the implicit function rule on the above we get dD*(9,fFI(9),irFI) -(re-9) df(9) (l-6)B"(D*(9,fFi(9),nFI)) dD*(9,fFI(9),nFI) _ -r* dn ~ (l-e)B"(D*(9,fFI(6),7rFI) so these two first-order conditions further reduce to dCFB (4.69) (4.70) df(9) = 0 dCFB On 9D*(9, fFI(9), 7tFI) - reD*(9, fFI{9), TTFI) +(l + \)D*(9,fFI(9),nFI)(l-9) (r'-9)(l+\)[(l-9)(l-fFI(9))-nF'} (1 - 9)B"(D*(9,fFI(0),irFI) -reD*(e,fFI(9),7rFI) + (l + \)D*(0,fFI(9),KFI) r%l + \)[(l-9)(l-fFr(9))-nF'] = {l-9)B"(D*(9,fFI(9),7rFI) Solving each of these for reD*(9, fFI(9), 7rFI), equating them and simplifying yields, fFI(9) = 1 - (-±-) D*(9,fFI(9UFI)B"(D*(9,fFI(9),nFI)) - **' \ + \) K" x n ' v K,J v " " 1-9 Proof of Proposition 8 I solve a 'relaxed' problem where I ignore the incentive compatibility constraint on the safe type, (4.60), and the individual rationality constraints on both types, (4.61,4.62) and then check that the solution to the relaxed problem does not violate any of these. For notational compactness, denote the deposit choices under the separating contract for R and S respectively as, DBRE = D*{9R,f{9Rl*) and DBE = D*(9Slf(9s),7r) Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 88 and, if the risky type selects the safe type's contract, DBfs = D*(9R,f(9s),7t) The Lagrangian for the relaxed problem can be written as: C = fW(9R) + (1 - n)V{9s) - (1 + \)Z + 7[*¥(9R) - V{0R\es)] (4.71) First-order conditions Denote the optimal values of the choice variables with BE super-scripts, and note that I use (4.68),(4.69), and (4.70) above to simplify the first-order conditions given below: dC dLR dC dLs dC dn = f,[9RF'(LBE(9R)) - re] + <yBE[9RF'(LBE(9R)) - re] = 0 BE\ •plfrBEi = (1 - lx)[9sF'{LBE{9s)) - re] - ^BE[9RF'(LBE(9S)) - re] = 0 £ = (1 + A - r')MDBE + (l-„)DBE] + ^BEre(DBfs-DBE) (4.72) (4.73) +(1 + A ) ^ + (!-//> (1 - 9R)B'(DBE) + (B"(DBE) - r*)9R(l - fBE(0R)) (l-9R)B"(DBE) (1 - 9S)B'(DBE) + (B"(DBE) - re)9s(l - fBE(9s)) (l-9s)B"(D§E) = 0 (4.74) dC BE df(9R) = -(» + rh)(re-9R)DBE-fi(l + \) x = 0 DF^-OR) (r* _ 0R)(r* _ i ) ( i _ fBEyj _ 7TBE) _ B'(DBE){\ - 9Ry (1-9R)B"(DBE) (4.75) dC df(9s) = - ( 1 - fi)(re - 9S)DBE + 7 * V " 0s)Difs - (1 - /*)(! + A) x = 0 DBsE{\-9s)- (r* - 9s)(r< - 1)(1 - fBE(9s) - nBE) - B'{DBE)(\ - 9S) (l-9s)B"(DBE) (4.76) dC dj = 9R[F(LBE(9R)) - F{LBE{9S))] - re[LBE(9R) - LBE{9S)) - M U " fBE(0R))DBE - (1 - }BE{9s))DBfs] Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 89 +re[DBE(l - **E - fBE(9R)) - DRls(l - *BE - fBE(9s))] +(l-9R)[B(DBE)-B(DBE)] = 0 (4. Banking Equilibrium Conjecture the following solution to the first-order conditions of the relaxed regulator's problem above: 1. DBE > 0 and DBE > 0; 2. LBE(eR) solves eRF'(LBE(9R)) = re; 3. 7B S > 0. First look at 2., this implies that R invests in the first-best loan size and, coupled with 3. allows us to state the following Lemma: Lemma 3 At the proposed equilibrium, LBE(6R) — LFB(6R) and j B E > 0 implies that LBE{9S) ^ LFB{6S) > LBE(6R) = LFB(eR). Proof : LBE(6R) = LFB(0R) implies that (4.72) is satisfied and if ^BE > 0 then there are three relevant cases to consider for (4.73) to be satisfied: 1. 6SF'{LBE{6S)) = re and eRF'{LBE{6s)) = re. But this requires that 6S = 6R: a contradiction. 2. esF'(LBE(es)) > re and 8RF'(LBE(6S)) > re. 3. esF'{LBE{9s)) < re and 9RF'(LBE(9S)) < re. Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 90 Neither case 2. nor case 3. above can be ruled out in the proposed equilibrium. • This shows that equations (4.72,4.73) are satisfied at the conjectured solution and completes the proof of the first part of the proposition. Next we need to show that if the bank is offered deposit insurance terms that induce the banking equilibrium, that (1) these contracts satisfy the regulator's problem, [RP], and (2) that fsE < fnE- The proof is broken into several steps. 1. Existence of a solution to the first-order conditions in the relaxed form of [RP] will depend on the exogenous parameters in the model. By examination, we can see that none of the first-order conditions above are non-zero for admissible parameters at the conjectured solution. To show that a solution does indeed exist, I provide a numerical example in the main body of the text. 2. To show that fBE(9R) > fBE(9s) is straightforward. As pointed out above, at the conjectured allocation, DBE > 0 and DRE > 0. The first-order condition on 7 is: 6R[F{LBE{eR)) - F(LBE(9S))] - re[LBE{9R) - LBE{9S)] -9R[(1 ~ fBE{eR))DBE - (1 - fBE(9s))DBE) +r<[DBE(l - nBE - fBE(0R)) - DBfs(l - vBE - fBE(9s))] +(l-eR)[B(DBE)-B(DBE)] = 0 From Lemma 3 we know that LBE(9R) = LFB(9R) which means that 9RF(LBE(9R)) - r<LBE(9R) > 9RF(LBE(9S)) - r'LBE(9s) Hence for the first-order condition on 7 to be satisfied, it must be the case that -eR[{\ - fBE(9R))DBE - (1 - fBE(0s))DBfs] + r*DBE(l - nBE - fBE(9R)) -r*DBH(l - KBE - fBE(9s)) + (1 - eR)[B(DBE) - B(DBE)} < 0 (4.78) Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 91 Define the function H (D) as H(D) = -0(1 - / ) + reD{\ - 7T - / ) - (1 - 9)B{D) and note that this is simply what determines the bank's optimal choice of D given the insurance terms, / and TT. This function is strictly concave and positive when D is positive. Then (4.78) can be written as H(DBE) < H(DBE) which means that DRFS > DRE and, from the strict convexity of B(-), B'{DBE) > B'{DBRE) (4.79) From the bank's optimal choice of D, (4.79) implies that r e ( 1 _ nBE _ fBE{ds)) _ 6R{1 _ fBE{es)) i-eR r e ( 1 _ ^BB _ fBE{6R)) _ 9R{1 _ fBE{6R)) 1-0R which can be simplified to (re-eR)(fBE(0R)-fBE(es))>o and, since re > 0R, means that fBE(9R) > fBE(9s). 3. Finally, it is necessary to show that the solution to the relaxed problem does not violate the participation constraints nor the incentive compatibility constraint on the safe type. (a) Suppose that the banking equilibrium {LBE(9s),LBE(9R)JBE(eR),fBE(es),nBE} Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 92 solves the regulator's problem but the participation constraints bind on either type. Then, since by assumption an all equity bank is profitable, 0F(LAE(9)) - reLAE{9) > 0 it must be the case that the regulator offered contract terms such that -0(1 - / ) + reD{\ - 7T - / ) - (1 - 9)B(D) < 0 for all D. But if this is the case, the bank would just choose D = 0, which we know does not violate their participation constraints. • (b) Next suppose that the banking equilibrium solves the regulator's problem and the incentive compatibility constraint on S is binding. The deposits issued by the safe type when taking the risky type's insurance contract, DB,R will be the solution to Rl(nBE, _ re(l-fBE(9R)-irBE)-9s(l-fEE(9R)) B (DS]R) — and, by the same reasoning as above, this will mean that -Dfi j | < DBE and that H(DBER) < H(DBE) Then if the incentive compatibility constraint of S binds esF(LBE(9s))-reLBE(9s) + H(DBE) < 9SF(LBE(9R)) - reLBE(9R) +H(DBER) 9sF(LBE(9s))-reLBE(9s) < 9SF(LBE(9R)) - r'LBE(9R) +H(DB{ER) - H(DBE) =>9sF(LBE(9s))-reLBE(0s) < 9SF(LBE(9R)) - reLBB(9R) Chapter 4. The Role of Deposit Insurance in the Regulation of Banks 93 Which implies that the safe bank will be strictly better off receiving a loan size of LBE(0R) instead of LBE{6S). But since LBE(6R) is the (unique) solution to F'{LBE{9R)) = f VR and from the first-order condition on L{6s), (1 - ,S)[9sF'{LBE{9s)) - re] - iBE[eRF'{LBE{es)) - re] = 0 if LBE(8s) = LBE(8R) then for the above equation to be satisfied, it must be the case that F>(LBE(8R)) = £ which cannot be true since F'(LFB(9S)) = fs and, LFB(9S) ^ LBE{9R) = LFB{9R). This contradicts the claim that the incentive constraint on S binds in a banking equilibrium. • Proof of Corollary 4 Differentiate £ with respect to re and apply the envelope condition. This will show that +1BE[DBE(1 - nBE - fBE(9R)) - DB?S(1 - nBE - fBE(0s))} which can be re-written as 0* + 1BE)DBE{\ - nBE - fBE(9R)) + (1-*BE- fBE{6s)[{\ - n)DBsE - -fED^s] (4.80) First note that when re = 1, the regulator will offer an insurance contract that induces a zero level of deposits for both types. Thus, the derivative of £ with respect to re, evaluated at re = 1 Chapter 4. The Role of Deposit Insurance in the Regulation of Banks will be zero. 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