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Essays on banking Wong, Kit P. 1993

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ESSAYS ON BANKINGbyKIT PONG WONGB. SSc., The Chinese University of Hong Kong, 1987M. A., The University of Western Ontario, 1989A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESTHE FACULTY OF COMMERCE AND BUSINESS ADMINISTRATIONWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJuly 1993© Kit Pong Wong, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature) Department ofThe University of British ColumbiaVancouver, CanadaDate Tv, („DE-6 (2/88)ABSTRACTThis dissertation contains three essays which look at the role of price competition inbanking. The method of investigation is a theoretical one. The first two essays examine therelative efficiency of relationship banking and price banking. The third essay discusses thedetermination of bank interest margin.Conventional wisdom suggests that increased interbank competition should improve so-cial welfare and thus price banking should dominate relationship banking. Essay one showsthat the opposite result may occur when the product market is imperfect and the lendinginstruments are loan commitments. Under relationship banking both banks and borrowershave bargaining power. The borrowers have substantial bargaining power when the costs ofstwitching banks are small. In this case, it pays the banks to charge interest rates belowthe competitive rates in order to keep their customers. The interest losses are compensatedfor by higher commitment fees paid upfront by the borrowers. Since interest costs are lowerunder relationship banking than under price banking, borrowers produce more and outputprice declines. Social welfare thus unambiguously increases. Essay two goes on to examinethe relative efficiency of relationship banking and price banking under the asset substitu-tion problem. The bank-customer relationship is assumed to provide a credible commitmentfor a borrower to refrain from transacting with other banks. The outcome under relation-ship banking is second-best since underinvestment results in solving the asset substitutionproblem. The multilateral credit transactions permitted by price banking impose negativeexternalities to existing loans by inducing the borrower to substitute riskier project. Moreunderinvestment is needed to resolve the dual incentive problem and equilibrium results inreduced welfare for borrowers. Essay three tackles the determination of bank interest mar-gins using a simple production-based model of risk-neutral banks which face (i) loan defaultrisk, (ii) interest rate risk, (iii) capital regulation, and (iv) deposit insurance. The optimalbank interest margin is shown to be increasing with the variability of the short-term moneymarket rate, but decreasing with either a stiffer capital requirement or an increase in theflat-rate deposit insurance premium.iiTABLE OF CONTENTSAbstract^ iiTable of Contents^ iiiList of FiguresAcknowledgement^ viChapter 1 Introduction and Overview^ 11.1 Introduction^ 11.2 Historical Background^ :31.3 Literature Review 51.4 Overview^ 9References 12Chapter 2 Banking Competition and Efficiency underProduct Market Imperfection^ 172.1 Introduction^ 172.2 The Basic Model 212.3 Analysis of Production-Search-Production Subgames^ 232.3.1 Subgame (reject, reject)^ 252.3.2 Subgame (accept, reject) 252.3.3 Subgame (accept, accept) 272.4 Equilibrium for the Full Game^ 272.4.1 Full Game 1: Both Firms Have Relationships with Bank 1 in Stage 1^272.4.2 Full Game 2: Firm i Has a Relationship with Bank i in Stage 1^292.5 Banking Competition and Welfare^ 302.6 Conclusion^ :31Appendix 33References^ 38iiiChapter 3 Debt, Asset Substitution, and Further Borrowing^433.1 Introduction^ 433.2 Model and the First-Best Solution^ 463.2.1 The Model^ 463.2.2 The First-Best Solution^ 483.3 Asset Substitution under Relationship Banking: The Second-Best Equilibrium^493.4 Asset Substitution under Price Banking: The Third-Best Equilibrium^523.4.1 Incentives for Further Borrowing^ 533.4.2 The Third-Best Solution^ 543.5 Discussion and Extensions 583.5.1 Renegotiation^ 583.5.2 Effort Incentive Problem^ 593.5.3 Managerial Compensation 603.6 Conclusion^ 60Appendix 62References^ 69Chapter 4 On the Determinants of Bank Interest Marginunder Capital Regulation and Deposit Insurance^ 744.1 Introduction^ 744.2 The Model 774.3 The Solution^ 794.4 Comparative Statics^ 824.5 Conclusions^ 86References 87ivLIST OF FIGURESChapter 2Figure 1 Extensive Form of Production-Search-Production Subgames^41Figure 2 The Normal Form of Production-Search-Production Subgames^42Chapter 3Figure 1 The Second-Best Equilibrium^ 71Figure 2 Incentives for Further Borrowing Given /sB < 1H^72Figure 3 The Third-Best Equilibrium^ 73ACKNOWLEDGEMENTI wish to express my most sincere thanks to the members of my dissertation committee,Professor Josef Zechner and Professor Paul Fisher, for their assistance and, especially, tomy dissertation supervisor, Professor Ron Giammarino, for his encouragement and helpfulguidance. I would also like to thank Rob Heinkel, Burton Hollifield, Helena Mullins, RamanUppal and seminar participants at the University of British Columbia for helpful comments.My special thanks also go to David Downie and Max Maksimovic who made personal effortto assist my work.Financial support from the University of British Columbia and the Social Sciences andHumanities Council of Canada is gratefully acknowledged.viChapter 1Introduction and Overview1.1 IntroductionThis dissertation contains three essays which look at the role of price competition inbanking. The importance of this issue is highlighted by the fact that financial marketdevelopments and financial deregulation have induced a move from less price competitive"relationship banking" to more price competition.' As Keeley (1990) points out, increasesin competition seem to be related to decline in bank charter values, which in turn mayhave caused banks to increase default risk through increases in asset risk and reductions incapita1. 2 The failure of the Continental Illinois Bank' in the United States (1984) and thecollapse of the Canadian Commercial Bank and Northland Bank' in Canada (1985) maybe viewed as the most significant examples of the competition/financial stability tradeoffthat accompanies the transition to the new price-driven system all three failed banks hadheavily purchased loans and deposits from others who had no ongoing relationship with thebanks prior to default. The weakening of relationship links between banks and customers issuggested by Davis (1992) to be an important factor of the financial fragility and instability'This transition is well documented by Crum and Meerschwam (1986) and Meerschwam (1989, 1991).2 See also Chan, Greenbaum and Thakor (1992).3 See Swary (1986) and Saunders (1987).4See Estey (1986).1of recent years.In spite of the significance of the above issues, very few studies—either empirical ortheoretical have been focused on them. Shockley and Thakor (1993) state: "[The] smallbut growing literature on the theoretical and empirical significance of relationship banking... should shed further light on the role of banks in an era of intense debate about the relativeimportance of banks versus capital markets in the capital allocation process (p. 32)." Inparticular, little is known about what relationship banking means for welfare. The purposeof this dissertation is to add to the understanding of this welfare aspect.Although relationship banking is discussed in the literature, it is not specifically defined.While related to a lack of competition, relationship banking is not explicitly linked to anyparticular market imperfection. In this study, the term "relationship banking" will referto model specific imperfections. In chapter 2 the bank-customer relationship arises fromthe cost of educating more than one bank. Chapter 3 assumes that the bank-customerrelationship takes the form of an exogenous mechanism to commit not to borrow further. Inchapter 4, the bank is assumed to be a monopolist in the loan market.The deregulation of banking and demise of local financial boundaries have squeezed bankinterest margins (the difference between average lending rates and average borrowing rates)to an extent which "did not properly reflect the risks that were being taken on." 5 As pointedout by Blanden (1993): "The problem for the banks in every industrialized country is thatthe search for other sources of income has dominated their thinking (p. 20)." He concludes:"The banks have been diverted from their fundamental role in life [of borrowing and lendingmoney]. They need to get back to basics, even if that means becoming even more unpopularwith customers (p. 20)." 6 Thus, this dissertation will also tackle the determination of bankinterest margins in order to add to our understanding of the sources of change in bankinterest margins.'This is a comment made by Robin Leigh-Pemberton, Governor of the Bank of England, in a speech toThe Chartered Institute of Bankers in Scotland, as reported by Blanden (1993), p. 19.6 Words inside the brackets are added by the author.2The remaining part of this chapter is organized as follows. The next section presentsa brief historical account of the changes in the banking systems within and between majordeveloped countries. Section 1.3 provides a concise literature review on relationship banking.Section 1.4 offers an overview of this dissertation.1.2 Historical Background'Until the mid-1970s, in the major developed countries, such as the United States, Japan,and the United Kingdom, the financial regulatory structure brought forth similar forms ofbanking systems. Crum and Meerschwam (1986) refer to these systems as "relationshipbanking." Under these highly regulated banking regimes, the freedom to transact financiallyat mutually beneficial prices was seriously suppressed. For example, in the United States, asystem of geographic and product market segmentation, imposed through branching restric-tions (McFadden Act) and product restrictions (Glass-Steagall Act), was bolstered by variousinterest rate restrictions (e.g., Regulation Q). In the United Kingdom, interest rates werenot allowed to reflect market conditions but were determined by a bank cartel composedof the dominant clearing banks. In Japan, a strict interest rate law (Temporary InterestRate Adjustment Law, 1947) was imposed to link various rates throughout the economy.Therefore, non-price competition in the form of long-term relationships between banks andcustomers evolved as a primary means of competition. Through such relationships, lenderswere able to accumulate information and build up confidence in borrowers; while borrow-ers, through continued good repayment, warranted the right to receive future credit. Theseongoing relationships, fostered by regulation, permitted both parties to credibly commit tobehaving responsibly and to forbear temporary problems.' Relationship banking, therefore,"represented a stable, effective financial system, and one that developed as a common re-sponse to ... [different] national cultures and strategies (Crum and Meerschwam (1986) pp.7For a detailed documentation, see Meerschwam (1991).'Relationship banking is in fact a form of implicit contract upon which the nature of the agreement toprovide credit (by the lender) and to remain a customer (by the borrower) cannot be formally contracted.3262-263)."During the 1970s, in which the Bretton Woods system of fixed exchange rates collapsedand massive international capital flows began to emerge, interest rates became more volatile.This situation provided investors with the incentive to shop for the best financial productprices and banks with the incentive to be innovative—both for profits and market share.New, price-oriented financial instruments were rapidly developed to take advantage of theseemerging market opportunities. This led to a move away from the well established bank-customer relationships and, as a result, the price-regulated domestic systems began to decay.Many forces encouraged the observed changes from local relationships towards price-driven transactions. Below, some major types of deregulation are highlighted (see Davis(1992) p. 26):• abolition of interest-rate controls, or cartels that fixed rates;• abolition of exchange controls;• removal of regulations restricting establishment of foreign institutions;• development and improvement of money, bond, and equity markets;• removal of regulations segmenting financial markets;• deregulation of fees and commissions in financial services;• tightening of prudential supervision.Crum and Meerschwam (1986) refer to the new systems, which emphasized impersonal price-oriented financial transactions, as "price banking."41.3 Literature Review9The importance of the bank-customer relationship to bank lending policy was first as-serted by Hodgman (1961, 1963). Then, the bank-customer nexus was tied to the studiesof credit rationing in a variety of contexts (see Kane and Malkiel (1965), Wood (1975), andBlackwell and Santomero (1982)). The basic ingredient of this literature is the presence ofexternalities to bank lending which yield additional benefits to the bank from either sub-sequent loans or cross-selling other services to the borrower. In a full information world,the loan pricing should reflect the externalities. Blackwell and Santomero (1982), therefore,suggest that "models based on the customer relation are inadequate to explain the specialstatus of the prime customer in a competitive world (p. 129)." In fact, on the assumptionthat borrowers with low credit risk have a more elastic demand for credit, they show thatthese borrowers will be the least costly to ration, whatever the nature of the bank-customerrelationships.Greenbaum, Kanatas and Venezia (1989) point out that "the existing literature providesno formal explanation for ... the benefit of forming a bank-customer relationship in a com-petitive environment (p. 222)." Rather than taking the relationships as given, they providea rationale by constructing a stylized search mode1. 1° The incumbent bank is assumed toenjoy an advantage of information reusability (see Chan, Greenbaum and Thakor (1986))over the competing banks about the borrower's probability of repayment as a consequenceof previous lending." In addition, the borrower is assumed to incur exogeneous search costsin shopping for more favourable loan terms.' Thus, the incumbent bank obtains some9 A broader survey of banking theory is provided by Bhattacharya and Thakor (1991).10A preliminary outline of this idea is suggested by Greenbaum and Venezia (1985) in a context of pricingloan commitments. Easterwood and Morgan (1991) also establish a search model of a financial market(generalized to include costly contracting). However, their model is a certainty model and focuses on theborrower's optimal choice between depository and brokerage financial intermediaries.Specifically, Greenbaum, Kanatas and Venezia assume symmetric information between the incumbentbank and the borrower, but asymmetric information between the incumbent bank and the competing banks.12 This captures Wood's (1975) suggestion that there are fixed costs to borrowers of changing their bankingconnection.5monopoly power owing to its private information and to the borrower's search costs. Theyshow that the longer the relationship between the borrower and the bank, the higher theloan rate that the bank will charge" and the shorter the expected remaining duration of therelationship. They also show that the price charged by the incumbent bank will tend to behigher than the average price offered by competing banks, since the latter must price lowerin order to lure the borrower, establish a relationship, and reap the future monopoly profits.They conclude: "From the bank's viewpoint, the client relationship is a wasting asset witha strictly finite duration (p. 232).""Similar to Greenbaum, Kanatas and Venezia (1989), Sharpe (1990) casts the bank-customer relationships as a consequence of the asymmetric evolution of information. Theincumbent bank is assumed to have superior interim information since it can observe theex post return on the borrower's investment.' This creates ex post monopoly power forthe incumbent bank, who will then be tempted to opportunistically increase its loan inter-est rate to successful borrowers about whom it knows more than the competing banks. Exante interbank competition results in banks bidding away these anticipated ex post expectedprofits via lower initial loan interest rates. With diminishing marginal returns of investmentin any given project, the final outcome is second-best. Because the size of the loan taken byborrowers depends on the interest rate, too much capital is loaned out when less is knownabout the quality of borrowers and too little capital is allocated to the successful borrow-ers. As a result, low quality borrowers employ a greater proportion of the capital loanedout, relative to the first-best symmetric information case. Bank-customer relationships ariseendogenously in Sharpe's model not because the bank treats borrowers preferentially, butbecause high quality borrowers are, in a sense, "informationally captured" the problem ofadverse selection hinders the competing banks from drawing off the incumbent bank's good13This result seems to be consistent with Hester's (1979) empirical finding that new loans to borrowerswho were profitable customers to banks in the past bear higher interest rates than loans to other borrowers."For some empirical findings on borrowers' shopping behaviour, see Haines, Riding and Thomas (1991).'Specifically, Sharpe assumes that borrowers do not know their own types (either high quality or lowquality). All agents (the borrower and all banks) are initially symmetrically informed.6customers without a concomitant attraction of the worse ones.The papers mentioned so far have not addressed the important welfare question about therelative efficiency of relationship banking and price banking.' This question has recentlydrawn a great deal of academic attention. Conventional wisdom suggests that increasedinterbank competition should improve social welfare. This assertion is formally verifiedby Besanko and Thakor (1992) in a context of spatial oligopolistic banking competition.They show that increased competition caused by a relaxation of entry barriers into bankingimproves the welfare of depositors and borrowers at the expense of banks' shareholders.Besanko and Thakor (forthcoming) comment that their earlier model precludes the poten-tial interesting interactions between the bank's portfolio choice and the banking structure.Due to the static nature of that model, relationship banking issues are also ignored. Infact, relationship banking adds value to the bank charter since relationship-specific informa-tional advantages generate rents.' The concern with protecting these rents mitigates thedistortionary effects of the fixed-rate deposit insurance system 18 and thus improves banksoundness.' Besanko and Thakor are particularly interested in the manner in which dereg-ulated entry into banking impinges on borrowers' welfare. They show that there are twoconflicting effects that increased interbank competition has on a borrower's welfare. The di-rect effect is that his borrowing cost is lowered due to more fierce competition among banks,which benefits him. However, this increase in the borrower's surplus reduces the value of thebank-customer relationship to the bank, which in turn dampens the pivotal counterveilingforce to the bank's propensity to exploit the deposit insurance put option by appropriatelyincreasing risk. The higher bank insolvency probability resulting from riskier loans increasesthe chance of disruption of the bank-customer relationship and the associated destruction of16 Although Greenbaum, Kanatas and Venezia (1989) and Sharpe (1990) have shown that the outcomeunder relationship banking is second-best, they do not contrast it with the one under price banking.17See also Sharpe (1990) and Rajan (1992) for how informational rents can arise from bank-customerrelationships.'See Merton (1977) for the moral hazard problem inherent in the risk-insensitive deposit insurance system.'This argument is similar to Keeley (1990).7valuable information. The cost of this for some borrowers may exceed the benefit of lowerloan interest rates. Thus, the conventional wisdom that increased interbank competition im-proves borrowers' welfare is not quite right when the impact of market structure on banks'portfolio choices is accounted for.Petersen and Rajan (1993) provide another argument for relationship banking. Theyshow (both theoretically and empirically) that borrowers are more likely to get finance frombanks in a concentrated market than in a competitive one. This is because the individualrationality constraints in concentrated markets do not bind period by period as they dofor competitive markets so that surplus can be re-allocated between time periods. Mayer(1988) and Hellwig (1991) also make the observation that bank may rescue firms that arein financial difficulty if they anticipate being able to participate in the returns from suchrescues.Rajan (1992) points out that there are both advantages and disadvantages of relationshipbanking over price banking. Under relationship banking, banks have incentives to monitorborrowers and control their investment decisions. In the process of doing so, banks acquirethe ability to appropriate borrowers' rents because of information monopoly. This adverselyaffects borrowers' incentives to exert effort. Thus, Rajan concludes that relationship bank-ing and price banking represent two starkly different control-rent trade-offs such that anyunidimensional comparisons are misleading.Bhattacharya (1993) examines the relative efficiency of relationship banking and pricebanking for financing R&D-intensive investments by firms competing in production markets.The R&D investments may yield technological knowledge of potential benefit to society, aswell as to competing firms. The generation of R&D knowledge requires unobservable effortwith non-pecuniary costs on the part of borrowers. Such knowledge is not only private toborrowers, but also proprietary in the sense that disclosure of it to product market com-petitors would reduce its private value. Under relationship banking such knowledge is not8revealed to product market competitors; under price banking it may lead to such knowledgebeing shared across product market competitors. Bhattacharya is able to show that if theprivate costs are small, price banking will be weakly dominating. On the other hand, whenthese costs are sufficiently high, relationship banking induces greater incentives for borrowersto do R&D and this may or may not coincide with the level of R&D that is in the socialinterest.The papers by Maksimovic (1990) and Bizer and DeMarzo (1992) also relate to thisdissertation. Maksimovic (1990) shows that a bank loan commitment, by precommittingthe firm to aggressive product market behaviour, increases the value of a firm in a Cournotoligopoly. While it is individually rational for each firm to acquire a loan commitment,however, when all firms in the industry enter into such arrangement, they are made worseoff. The strategic value of a loan commitment is a cornerstone of the model explained inchapter 2. Bizer and DeMarzo (1992) study the negative externalities arising from additionalborrowing in the presence of moral hazard. While each new bank does not pay for theseexternalities, prior banks recognize the potential victimization that they may suffer and thusreact accordingly. This contrasts with a one-bank environment in which all effects on earlierloans are internalized by the sole lender. Bizer and DeMarzo show that borrowers are worseoff than they would be if they could credibly commit to refraining from further borrowing.1.4 OverviewChapters 2 and 3 of this dissertation try to further explore the efficiency implications ofmarket structure in banking. In chapter 2, I argue that relationship banking may improvesocial welfare which is measured as the sum of producers' and consumers' surplus 2° whenthe product market is imperfect and the lending instruments are loan commitments. In mymodel, there are no asymmetric information or agency problems. The seemingly counter-"This should be contrasted with Besanko and Thakor's (forthcoming) interest in borrowers' welfare only.They have not actually shown that relationship banking may benefit the whole society.9intuitive result is driven by the second-best nature of the economy where both the capitaland product markets are imperfect. The adverse effects imposed by one imperfect marketupon the economy may offset (at least partially) those caused by the other and, becauseof the bank-customer relationships, banks are able to exploit their borrowers. However,borrowers are free to switch to other banks and thus this provides a force to discipline thebehaviour of the incumbent bank. Of course, the effectiveness of this force depends on howcostly it is for borrowers to switch banks. Borrowers' search costs are modelled as costs ofdelayed production.' I show that if borrowers are tough (i.e., the search costs are small),the incumbent bank has to charge an interest rate below the competitive one in order tokeep its customers. The interest losses are compensated for by higher commitment fees paidupfront by the borrowers. Since interest costs are lower under relationship banking thanunder price banking, borrowers produce more and output price declines. Social welfare thusunambiguously increases.Chapter 3 goes on to examine the relative efficiency of relationship banking and pricebanking under the situation where risk-neutral borrowers can choose among projects ofvarying degree of riskiness which is unobservable to banks (i.e., under the asset substitutionproblem). I assume that under relationship banking each borrower can only deal with onebank. I show that the outcome of this bilateral credit transaction is second-best sinceunderinvestment results in solving the asset substitution problem. In the absence of sucha bank-customer relationship (so that borrowers cannot credibly commit to refraining fromfuture borrowing), I show that the outcome is third-best. Within the multilateral credittransactions, borrowers can borrow as many times as they want, each time going to a newbank. The crux of the problem is that additional loans impose negative externalities toexisting loans by inducing borrowers to substitute riskier projects.' Since new banks donot pay for these externalities, prior banks recognize the potential victimization that they21 In explaining firm's dynamic financing and investment decisions, Berkovitch and Narayanan (1993)assume that switching banks involve dissipative costs which are proportional to the net present value of thefirm's investment project. My model shares some of this view but endogenizes the switching costs.22 1n the existing literature, the negative externaltiies described are not formally modelled.1 0may suffer and thus react accordingly. More underinvestment is needed to reslove the dualincentive problem and equilibrium results in reduced welfare for borrowers.'The final chapter of this dissertation tackles the determination of bank interest marginsin response to the concerns raised by Blanden (1993) and others as described in section 1.1.I develop a simple production-based model of risk-neutral banks which face (i) loan defaultrisk, (ii) interest rate risk, (iii) capital regulation, and (iv) deposit insurance. The modelis richer in structure than the existing ones in explaining the behaviour of bank interestmargins.' I am able to decompose the optimal bank interest margin into two components:the spread due to default risk and the spread due to banking regulation. I show that theoptimal bank interest margin increases with the variability of the short-term money marketrate, but decreases with either a stiffer capital requirement or an increase in the flat-ratedeposit insurance premium. The chapter concludes with empirical implications unique tothis literature.23This reinforces the finding by Bizer and DeMarzo (1992) who consider a similar situation in whichrisk-averse borrowers can choose the effort required to avoid bankruptcy.24There are only a few theoretical models for this issue (Ho and Saunders (1981), McShane and Sharpe(1985), Allen (1988), Zarruk (1988) and Zarruk and Madura (1992)). In contrast to my model, all theexisting ones focus on risk-averse banks and pay little attention to banking regulation.11References[1] Allen, Linda, 1988, "The Determinants of Bank Interest Margins: A Note," Journal ofFinancial and Quantitative Analysis, Vol. 23, pp. 231-235.[2] Berkovitch, Elazar and M. P. Narayanan, 1993, "Timing of Investment and FinancingDecisions in Imperfectly Competitive Financial Markets," Journal of Business, Vol. 66,pp. 219-248.[3] Besanko, David and Anjan V. Thakor, 1992, "Banking Deregulation: Allocational Con-sequences of Relaxing Entry Barriers," Journal of Banking and Finance, Vol. 16, pp.909-932.[4] Besanko, David and Anjan V. Thakor, 1992, "Relationship Banking, Deposit Insur-ance, and Bank Portfolio Choice," Dicussion Paper #511, Graduate School of Business,Indiana University, forthcoming in Xavier Vives and Colin Mayer, eds., Financial In-termediation in the Construction of Europe, Cambridge: Cambridge University Press.[5] Bhattacharya, Sudipto, 1993, "Financial Intermediation with Proprietary Information,"Working Paper 205.93, Institut d'Analisi Econômica, CSIC, Universitat AutOnoma deBarcelona.[6] Bhattacharya, Sudipto and Anjan V. 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Sharpe, 1985, "A Time Series/Cross Section Analysis of theDeterminants of Australian Trading Bank Loan/Deposit Interest Margins: 1962-1981,"Journal of Banking and Finance, Vol. 9, pp. 115-136.14[29] Meerschwam, David M., 1989, "International Capital Imbalances: The Demise of Lo-cal Financial Boundaries," in Richard O'Brien and Tapan Datta, eds., InternationalEconomics and Financial Markets, Oxford: Oxford University Press, pp. 289-307.[30] Meerschwam, David M., 1991, Breaking Financial Boundaries: Global Capital, NationalDeregulation, and Financial Services Firms, Boston: Harvard Business School Press.[31] Merton, Robert C., 1977, "An Analytic Derivation of the Cost of Deposit InsuranceLoans Guarantees," Journal of Banking and Finance, Vol. 1, pp. 3-11.[32] Petersen, Mitchell A. and Raghuram G. Rajan, 1993, "The Effect of Credit Market Com-petition on Firm-Creditor Relationships," Working Paper, Graduate School of Business,University of Chicago .[33] Rajan, Raghuram G., 1992, "Insiders and Outsiders: The Choice Between Informedand Arm's-Length Debt," Journal of Finance, Vol. 47, pp. 1367-1400.[34] Saunders, Anthony, 1987, "The Inter-Bank Market, Contagion Effects and InternationalFinancial Crises," in Richard Portes and Alexander K. Swoboda, eds., Threats to In-ternational Financial Stability, Cambridge: Cambridge University Press, pp. 196-232[35] Sharpe, Steven A., 1990, "Asymmetric Information, Bank Lending, and Implicit Con-tracts: A Stylized Model of Customer Relationships," Journal of Finance, Vol. 45, pp.1069-1087.[36] Shockley, Richard L. and Anjan V. Thakor, 1993, "Information Content of Commit-ments to Lend in the Future: Theory and Evidence on the Gains from RelationshipBanking," Discussion Paper #523, Graduate School of Business, Indiana University.[37] Swary, Itzhak, 1986, "The Stock Market Reaction to Regulatory Action in the Conti-nental Illinois Crisis," Journal of Business, Vol. 59, pp. 457-473.15[38] Wood, John Arnold, 1975, Commercial Bank Loan and Investment Behavior, New York:.John Wiley k Sons.[39] Zarruk, Emilio R., 1988, "Bank Spread with Uncertain Deposit Level and Risk Aver-sion," Journal of Banking and Finance, Vol. 13, pp. 797-810.[40] Zarruk, Emilio R. and Jeff Madura, 1992, "Optimal Bank Interest Margin under CapitalRegulation and Deposit Insurance," Journal of Financial and Quantitative Analysis,Vol. 27, pp. 143-149.16Chapter 2Banking Competition and Efficiency underProduct Market Imperfection2.1 IntroductionThe purpose of this chapter is to examine the relative efficiency of relationship bankingand price banking. By the term relationship banking I refer to a system of repeated bilateralcredit transactions between each borrower and a particular bank. Price banking, on the otherhand, is characterized by many banks bidding competitively for each transaction undertakenby a borrower. As documented by Crum and Meerschwam (1986) and Meerschwam (1989,1991), there is a transition from relationship banking to price banking in developed countriessuch as the United States, the United Kingdom and Japan. This gives rise to the question ofthe relative efficiency of these two systems. Conventional wisdom suggests that a competitiveprice mechanism should achieve an optimal allocation of all available funds.' However,Meerschwam (1991) argues that imperfections such as asymmetric information and agencyproblems divert the achievement of the first best outcome, and a relationship between aborrower and a bank may be an efficient institutional attempt to deal with these information1 Besanko and Thakor (1992) verify this assertion using a spatial model of oligopolistic banking.17problems.'In this chapter I demonstrate that, even in the absence of any information problems,relationship banking may yield a higher level of social welfare than price banking. The keyfactor that drives this result is product market imperfection. In a second-best world whereboth the capital market and the product market are imperfect, it is possible that the adverseeffects imposed by one imperfect market upon the economy may offset (at least partially)those caused by the other.'To illustrate the above "two wrongs make a right" argument, I construct a two-periodCournot duopoly model with a stylized relationship-based banking system. Outputs areproduced and sold to consumers in each period by firms that have no capital and thereforemust borrow from banks. Each firm has a lock-in relationship with a bank at the beginningof the first period 4 and prefers to start its production in that period so that it can capture alarger market share from its rivals. The firm's desire to produce earlier gives the incumbentbank a chance to exploit it. However, if the firm thinks that it is overly exploited, it canspend some time to educate other banks. After a new relationship is established, the firmcan ask the two banks to bid for its loan. Assuming the banks cannot collude, the firm isable to obtain better terms for the loan.' The firm's search cost is endogenized to the extentthat the cost of switching banks comes from the loss of market share due to the delay ofproduction. It does not pay for the firm to build up a relationship with other banks whenthe benefit from receiving better terms of the loan cannot fully cover the search cost. Theability to switch to other banks provides the firm with an outside option that defines itsbargaining position (i.e., its reservation profit) when it bargains with the incumbent bank.2 See Sharpe (1990) for a formal model that shows how such a relationship arises endogeneously as aconsequence of asymmetric information.3 Brander and Lewis (1986) and Maksimovic (1986) are the first seminal studies that emphasize howimportant the linkages between the product market and the capital market are on the equilibrium outcomes.'Petersen and Rajan (1992) present empirical evidence that small firms tend to concentrate their bor-rowing from a single bank, although this tendency declines as firm size increases.5 A similar stylized imperfectly competitive financial market is used by Berkovitch and Narayanan (1993).They assume that switching banks involve dissipative costs which are proportional to the net present valueof the borrower firm's investment project. My model goes further to endogenize these costs.18The argument, however, also requires that the bank's actions are able to affect the firm'sproduct market behaviour. This is added to the model by letting banks offer terms for aloan commitment to firms. A loan commitment obligates a bank to grant loans to a firmup to a maximum amount called the commitment size at a predetermined interest rate. Inaddition to the interest paid when the commitment is used, the contract itself entails a costcalled the commitment fee which the firm pays upfront to the bank for acquiring the standbycredit option.' Maksimovic (1990) argues that a loan commitment increases the value of aborrower firm from a Cournot oligopoly by enhancing the firm's strategic power.' While itis individually rational for each firm to acquire a loan commitment, he shows that all firmsin the industry taken together are made worse off by the existence of loan commitments.The equilibrium for the above model is contrasted with the one for an otherwise equivalentmodel but with a transactions-based competitive banking system. The main findings aresomewhat surprising: output price is lower (and thus consumers are better off) and socialwelfare (the sum of producers' surplus and consumers' surplus) is higher under relationshipbanking than under price banking. To understand these seemingly counterintuitive results,first note that banks are not passive investors under relationship banking. Suppose that asingle bank is dealing with competing borrowers. The incumbent bank can exert substantialinfluence over the firm's production decision. This influence in turn indirectly affects thevalue of the firm's outside option. For example, suppose that an incumbent bank offers twocompeting firms low interest rates. If the firm rejects the offer while its rival accepts, its rivalwill flood the product market in the first period because of the low interest costs (i.e., lowmarginal production costs): the value of the firm's outside option is small in this case. Sincethe bank can extract all the surplus above what the firm can get from its outside option by6 A typical loan commitment also contains restrictive covenants such as collateral requirements and limiteddividends. These are designed either to monitor the behavior of the borrower or to enhance the safty of thecontract. As documented by Duca and VanHoose (1990), bank loan commitments comprise about 80% ofall commercial lending in the United States.7There are many theoretical papers in the banking literature showing the importance of loan commit-ments. See Campbell (1978), Thakor, Hong and Greenbaum (1981), Thakor (1982), Deshmukh, Greenbaumand Kanatas (1983), Kanatas (1987), Boot, Thakor and Udell (1987, 1991), and Berkovitch and Greenbaum(1991), to name just a few.19means of the commitment fee, it pays the bank to charge a low interest rate to keep the valueof the firm's outside option small. While doing this implies a smaller producers' surplus,as long as the portion shared by the bank is larger, the bank is willing to pick this as theequilibrium outcome. As a result, relationship banking is more pro-competitive.This chapter is related to the growing literature on relationship banking which is re-viewed in chapter 1. My work is distinguished from this area of research on the basis ofits special focus on the product market competition. The exceptions in the literature whichalso emphasize the product market side are the papers by Bhattacharya (1993) and Poitevin(1989). While Bhattacharya (1993) looks at the R&D race in the product market, I examinea more traditional Cournot model of competition. The rationale of the results is also verydifferent. Bhattacharya shows that relationship banking may be efficient because it inducesgreater incentives for borrowers to do R&D. In contrast, the driving force in my model isthe allocation of bargaining power between borrowers and banks.My work is also related to the paper by Poitevin (1989). He shows that the extent ofcompetition in downstream industries may depend on the choice of banks. It is now well-known that in a Cournot oligopoly, debt is pro-competitive, since it gives incentives to theborrowing firm to undertake an aggressive output strategy.' Poitevin shows that membersof the industry may be able to achieve a partial collusion in the output market by borrowingfrom the same bank. A common bank can better control the incentive effects induced bydebt and thus limit the extent of competition in the output market. I extend Poitevin's workby considering interbank competition and the important implications that arise from it.The remaining part of this chapter is organized as follows. In the next section, I presenta formal description of the model in the context of a linear Cournot duopoly and a stylizedrelationship-based banking system. Sections 2.3 and 2.4 characterize the equilibrium loancommitments and the product market outcome. In section 2.5, I contrast the above equilib-8See Brander and Lewis (1986) for the arguments under standard debt contracts and Maksimovic (1990)under loan commitments.20rium with the one in an otherwise equivalent economy with a transactions-based competitivebanking system and show that the former banking regime is in fact more pro-competitive.Section 2.6 concludes.2.2 The Basic Model9Consider an industry comprised of two identical firms (indexed by i = 1, 2) producinga homogeneous good.' Each firm has two production periods and is endowed with a point-input point-output technology that transforms one unit of capital into one unit of output.After production in each period, the output is sold immediately to consumers. The industry'sinverse demand function is given by p = a —b(qi + q2 ), where p is the price of the homogeneousgood, qi is firm i's two-period cumulative output, and a and b are positive constants.Each firm has zero capital and must therefore rely on a banking sector (consisting ofat least two banks) to grant loans in the form of loan commitments. A loan commitment,(r, f), is a pair that specifies a predetermined interest rate, r, at which the bank is obligatedto guarantee future availability of credit to the firm, and a commitment fee, f , paid upfrontby the firm to acquire this privilege of having standby credit." In other words, a loancommitment is viewed as a two-part tariff, which is one of the most basic and commonpricing schemes in economics.' For simplicity, each bank faces a perfectly elastic supply ofdeposits at the zero two-period interest rate. There is no discounting in the economy.'The model described below can be viewed as a two-period extension of Maksimovic's (1990) model."Since firms in the industry are asymmetric in the sense that they have different production costs, forthe tractability reason I have to restrict my attention to the duopoly case."Since it is not an important issue here, the question of where the firm obtains the funds for paying thecommitment fee is suppressed. Little generality is lost by assuming that the firm's initial wealth endowmentaccommodates the commitment fee but is insufficient to permit self-financing for production."Another distinct feature of the loan commitment contract is the quantity setting clause. This commit-ment obligation places an upper limit (called commitment size) on the firm's borrowing. Within this limit,the firm may set future loan size unilaterally at the agreed upon rate. From a survey by Ham and Melnik(1987), less than 20% of the firms in their sample ever reached the maximum limit of the firms' commitmentsizes. Hence, for simplicity, I assume away the possibility that the firm faces credit rationing, in the sensethat it cannot borrow as much as it wants at the prespecified interest rate. This assumption is innocuousin my model because the commitment size is isomorphic to the production capacity of the firm. It is alwaysoptimal for the firm not to be capacity-constrained as this will reduce its competitiveness over its rival.21At the beginning of the first period, each firm has a lock-in relationship with a bankand the firm requests a loan commitment from this bank to fund its two-period production.The bank then makes a take-it-or-leave-it offer to the firm. If the firm accepts the offer,it receives capital to start production right away. If the firm is not satisfied by the bank'soffer, it is free to visit other banks and educate them. After a second relationship is builtup, the firm asks both banks to bid for its loan. Of course, doing so is time-consuming andI assume that it inevitably leads to a delay of production for the entire first period. Hence,the firm's search cost is to some extent endogenously determined: Getting the line of creditearlier provides the firm with a first-mover advantage on the one hand, but may involveless favourable terms of the loan on the other. The endogenous component of the searchcosts, not found in existing search models in the literature, allows me to consider welfareimplications and leads directly to the surprising conclusions.Formally, the set up is a two-period four-stage complete information game. At the be-ginning of the first period, each firm solicits a loan for production from the incumbent bank.Both firms may happen to have relationships with the same bank or different banks andI treat these two possibilities as two different full games, denoted as full game 1 for theone-bank case and full game 2 for the two-bank case. Without loss of generality, I assumethat both firms have relationships with bank 1 in full game 1 and firm i has a relationshipwith bank i in full game 2.Stage 1. In full game 1, bank 1 announces a pair of loan commitments [(r1,(r2 , 12)], one for each firm. These are take-it-or-leave-it offers and firms can observe theirrival's offer.' In full game 2, banks 1 and 2 simultaneously announce take-it-or-leave-it offers(r 1 , fl ) and (r 2 , 12 ) to firms 1 and 2, respectively. Firms cannot observe their rival's offer.For the rest of both full games, all actions are the same. Firms either accept or reject the13If firms cannot observe their rival's offer, then they will fear "third-party" opportunism by the bank:Once a firm has paid its commitment fee the bank is no longer concerned with protecting that firm's profitand is therefore tempted to reduce the interest rate to the other firm in exchange for a higher commitmentfee. To rule out this incentive problem, I simply assume that offers by the bank is publicly observable. See,for example, Hart and Tirole (1990) and McAfee and Schwartz (1990).22offers. Accepting involves paying the commitment fee upfront to the bank. Each firm thenlearns its rival's loan commitment and acceptance decision.Stage 2. If both firms accepted the offers, they simultaneously choose their first-periodoutputs, qll and qn . If firm i accepted the offer while firm j rejected, firm i chooses its first-period output (p i , knowing that = 0. If both firms rejected the offers, then q ll = q21 = 0.These outputs are then publicly known and sold to consumers at the end of the first period.Stage 3.^At the beginning of the second period, any firm i which rejected the offer instage 1 now gathers banks to bid for its loan commitment,^fi). Again, accepting involvesfirm i paying the commitment fee, f2 , upfront to the bank which wins the bid. The loancommitments are then publicly known.Stage 4.^Both firms choose simultaneously their second-period outputs, q12 and q22,which are then sold to consumers at the end of the second period.The equilibrium concept employed is Selten's (1975) subgame perfect Nash equilibrium(SPNE) and I restrict attention to pure strategy equilibria only. A set of pure strategiesfor a game is a SPNE if it is a Nash equilibrium for the entire game and its revelant actionrules are a Nash equilibrium for every proper subgame. This is the appropriate equilibriumconcept for the given complete information game. For the following analysis, a variablewith an asterisk means a SPNE strategy for the subgame in question. I also adopt thetie-breaking rule that a firm will choose a loan commitment offered in the first period if thefirm is indifferent between this loan commitment and another loan commitment offered inthe second period."2.3 Analysis of Production-Search-Production SubgamesIn this section, I characterize the SPNE for subgames starting from stages 2 to 4 (hence-14This assumption is justified when there are transaction costs such as transportation costs and biddingarrangement costs, no matter how small they are.23forth called production-search-production subgames). Given a pair of loan commitments[(r1 , (r2, f2)] offered to firms in stage 1, there are four possible histories associated withproduction-search-production subgames. These are denoted as follows: (accept, accept) de-notes the history in which both firms accepted the offers; similarly, (reject, reject) denotesthe history in which both firms rejected the offers; finally, (accept, reject) and (reject, ac-cept) denote the histories in which one firm accepted its offer while the other rejected. Ishall analyze each of these in turn, specifying how equilibrium play unfolds for each possiblehistory. Since the two subgames (accept, reject) and (reject, accept) have parallel analy-ses, we shall address only the subgame (accept, reject) in detail. The extensive form ofproduction-search-production subgames is depicted in figure 1.(Figure 1 about here)The calculation of the SPNE for a production-search-production subgame proceeds bysolving backward. Given complete information and zero discounting, rational consumersaccurately anticipate all equilibrium actions taken by firms and the prices paid by them arethe same in both periods and equal to the market clearing price: a — b(q1 q2 ).In the final stage, each firm knows its rival's interest cost and first-period output. As thefirst-period profit from sales to consumers has alreadly been realized, firm i's second-periodprofit maximization problem given (qii, qji, qj2, ri, rj) ismax [a — b(qii qi2 q + qj2) - (1 + 1, 2.q,2 >0It is straightforward to solve the second-period Cournot-Nash equilibrium outputs, q1 2 andq2*2 , which are given by,1 ,qi*2^T., ri) =3b ta — 1 — b(qii^(Lid — 2ri + rj], i = 1, 2. (2.1)Now, go back one stage and suppose that firm j rejected the offer in stage 1. Then, qji = 0and r i and q21 are known. In stage 3, banks compete for the loan commitment (1- j , 4). For24any (r i , 7"•.; ), they can correctly anticipate the second-period equilibrium outputs to be givenby (2.1). Through Bertrand competition, they charge = —4i (fjc2 so that they break even.Again Bertrand competition drives them to set an interest rate that maximizes firm j'sex-commitment fee profit, i.e., it solves the following profit maximization problem given(qa, ri):max [a — b(q:2 + qii + q:;2 ) — (1 + f. i )](/;2 —7 •3max [a — b(q:2 qii q;2 ) —7,2since^= —7•i (6. Using (2.1), it is easy to show that the optimal interest rate is given by13 = 3 (g i rx ) = --4(a — 1 —^+ ri)•In stage 2, looking ahead to the second-period equilibrium outputs, firm i is solving thefollowing two-period profit maximization problem given (qii , ri , rj):max [a — b(qi i^q:2 + q3. 1 +^) — (1 + ri)](qii^i = 1, 2.32q,i>0 (2.3)2.3.1 Subgame (reject, reject)Since both firms rejected the offers in this subgame,q' 1 = q2*1 = 0. Solving (2.2) simul-taneously for both firms, the equilibrium interest rates are = ( — 1)/5, and theequilibrium ex-commitment fee profit for each firm is 2(a — 1) 2 /25b.2.3.2 Subgame (accept, reject)Suppose that firm i accepted its offer (ri , fi ) but firm j rejected. Then (6 = 0 and firmj knows r i and qil before it visits other banks. Since my goal is to characterize the SPNEfor the full game, I have to consider the SPNE for this subgame following all possible pairsof loan commitments offered in stage 1.(2.2)25The set of all possible loan commitment contracts offered in stage 1 can be partitioned intothree mutually exclusive subsets according to the firms' optimal response in each subgame. 15The first subset is the one in which firm i produces nothing in each period in equilibrium.Notice that firm i can always earn 2(a — 1) 2 /25b if it rejects the offer. Hence, this subgamecannot be on the equilibrium path. This implies that the consideration of this subset isirrelevalent to the characterization of the full equilibrium.The second subset is the one in which firm i produces positive output but firm j produceszero output in equilibrium. In the appendix, I show that no pair of loan commitments inthis subset is subgame perfect. This result follows from the fact that the interest rate mustbe set low enough to induce firm i to adopt a very aggressive output strategy so as to drivefirm j out of the industry. If the bank did so, it could not fully cover its interest loss fromthe commitment fee. As a result, I can restrict the attention to the third subset in whichboth firms produce positive outputs in equilibrium.Since the equilibrium is interior, substituting (2.1) and (2.2) into (2.3) yields firm i's two-period profit maximization problem given r i . By simple calculation, the optimal first-periodoutput of firm i is given by1q71 =^(ri ) = —3b(a — 1 — 3r1 ).Hence, firm i's cum-commitment fee profit given r i is given by7,(ri ) = 121 b (a — 1 — 3r i ) 2 ,and firm j's ex-commitment fee profit given r i is given by1(ri) = 1817^b (a^1 + 3ri)2.'These subsets are mutually exclusive because the optimal response of each firm is unique following anypair of loan commitments offered in stage 1. To save notation, I am deliberately informal here.262.3.3 Subgame (accept, accept)From the analysis in the above subsection, it is no loss of generality to assume an interiorequilibrium for this subgame. Substituting (2.1) into (2.3), the first-period Cournot-Nashequilibrium outputs are given by1q71. = galls, ri) ^5b(a — 1 — 3ri^2r.7), i = 1, 2.Hence, the cum-commitment fee profit given (ri , rj) for each firm in this subgame isrj) = 225b (a — 1 — 3ri^27-i ) 2, i = 1, 2.2.4 Equilibrium for the Full GameNow, proceed to analyze the stage 1 decision. Since there are two different full gamesdepending on whether firms have relationships with the same bank or not at the beginningof the first period, I shall analyze each of them in turn. In the next subsection, we considerfull game 1 first and delay the analysis of full game 2.2.4.1 Full Game 1: Both Firms Have Relationships with Bank 1 in Stage 1Suppose that bank 1 offers the loan commitment (r 1 , fi ) to firm 1 and (r2 , f2 ) to firm2 at the beginning of the first period. I summarize each firm's ex-commitment fee profitcalculated in section 2.3 by the normal form game depicted in figure 2.(Figure 2 about here)Proposition 2.1.^In full game 1, any pairs of SPNE loan commitments solve: (P1)2max!^E^+ q:2) + fi]ri ,r2 J1 >0,2?_° j=.1,43(2.5)(2.4)27s. t.^ri)^- 7r,i(rJ) > 0 ,^ (2.6)7T2(Ti) -^— 25b(a — 1) 2 > 0, (2.7)71-;(ri, r) —^— 7r3f(ri ) > 0,^ (2.8)where q71 and (172 are defined by (2.4) and (2.1) respectively.Proof See the appendix. ^Conditions (2.6) and (2.7) are the necessary and sufficient conditions for accepting tobe the dominant strategy of firm i. Given that firm i accepts the offer, condition (2.8) isthe necessary and sufficient condition for firm j to accept its offer. Hence, proposition 2.1simply says that bank l's expected profit, (2.5), is maximized subject to the acceptance ofthe offers by both firms.Proposition 2.2.^There exists a unique SPNE for full game 1 which is asymmetricwith firm i's offer being worse than firm j's offer. Firm i produces less than firm j and theindustry output is 0.905187(a — 1)/b.Proof See the appendix. ^Given that firms i and j are identical, the fact that they are treated asymmetrically issurprising. The intuition supporting this result is as follows. Since the offers presented to thefirms are publically observable, the bank can, through terms of the loan, credibly commit oneof the firm to act more aggressively, thereby reducing the value of the other firm's outsideoption. By giving firm j a lower rate, the bank is able to extract a larger fraction of theremaining surplus from firm i and increases total profits by doing so. The industry ouput isbigger than the traditional Cournot output 2(a — 1)/3b but smaller than the Pareto efficientoutput (a — 1)/b. 16' 6The Pareto efficient output is one in which the output price is equal to the marginal interest cost.282.4.2 Full Game 2: Firm i Has a Relationship with Bank i in Stage 1Consider full game 2 in which bank i offers the contract (r i , fi ) to firm i without observingbank j's offer, (rj , fi ), to firm j (and vice versa) at the beginning of the first period. Banki must offer a contract that firm i will definitely accept or otherwise it will earn zero profit.Thus, using similar argument as above, bank is profit maximization problem given (r , li )is: (P2)max ri^q72)^s.t . (2.6) and (2.7),ri, fi >0where (CA and q72 are defined by (2.4) and (2.1) respectively.Proposition 2.3.^There exists a unique. SPNE for full game 2 which is symmetric.The industry output is 0.9052(a — 1)/b.Proof. See the appendix.^^The following two propositions follow immediately from propositions 2.2 and 2.3.Proposition 2.4.^The equilibrium industry output in full game 1 is lower than thatin full game 2.Proposition 2.5.^If firms are risk-neutral and there. is an equal chance for them tobe treated preferentially by the bank with which they both have relationships, they will strictlyprefer to have relationships with the same, bank.These two results are similar to the findings of Poitevin (1989). Proposition 2.4 says thatif both firms borrow from the same bank, the product market becomes more collusive. Thisoutcome is achieved through a better coordination of the firms' production activities whenthe bank can partially internalize the external effect of one firm's marginal interest cost on its29rival's output. Proposition 2.5 says that the expected ex-commitment fee profits are higherwhen firms borrow from the same bank, despite the fact that the bank has stronger powerto exploit them in this case. This is the case because the producers' surplus is larger due toa more collusive industry output and both the firms and the bank are able to benefit fromsharing a bigger pie.2.5 Banking Competition and WelfareConsider an otherwise equivalent economy with a transactions-based banking system.Everything is the same as in the previous section except that banks compete directly inloan prices. Through competition, banks will charge fi =^+ (2 ) so that they breakeven, where^q:2 is the equilibrium total output of firm i anticipated by banks givenri ). Again, competition drives them to set an interest rate that maximizes firm i'sex-commitment fee profit, i.e., it solves the following profit maximization problem given r j :max 711(r i , rj)^q:2),riwhere^and q:2 are defined by (2.4) and (2.1), respectively. It is straightforward to calculatethe equilibrium interest rates to be 7 -1 = 7-2* = —(a — 1)/14 and the equilibrium industryoutput to be 6(a — 1)/7b. It follows immediately that the equilibrium industry output inthis case is lower than that under the imperfectly competitive banking regime. Hence, themajor results are summarized as follows:Proposition 2.6.^The output price is lower (and thus consumers are better off) andsocial welfare is higher under relationship banking than under price-driven banking.The intuition of these seemingly inconsistent results is as follows. With perfectly com-petitive banks, firms have all the bargaining power and banks are passive. As a result,the main concern in the bargaining process is to maximize firms' profits. However, under30relationship banking, both banks and firms have some bargaining power. The incumbentbank has bargaining power because it is a first mover and the firm has bargaining powerbecause it posesses an outside option by switching to other banks. Also note that the bankcan indirectly influence the value of firm's outside option. Now suppose that the bank setsa low interest rate. If the firm rejects the offer while its rival accepts, its rival will floodthe output market in the first period because its interest cost is low. Thus, setting a lowinterest rate reduces the firm's bargaining strength through the reduction of the value of itsoutside option. Since the bank can use the commitment fee to extract the surplus abovewhat the firm can get from its outside option, it pays for the bank to charge a low interestrate. Even though this implies a smaller producers' surplus, as long as the portion sharedby the bank is larger, the bank is willing to pick this as the equilibrium outcome. Thus, inequilibrium, interest rates are lower than those with perfectly competitive banks. This inturn causes firms to be more aggressive in production. Consequently, relationship bankingis more pro-competitive.2.6 ConclusionIn developed countries such as the United States, the United Kingdom and Japan, thereis a transition from relationship banking to price-driven banking. This gives rise to somedebate about the relative efficiency of these two banking systems. In this chapter, I developeda stylized model of relationship banking under product market imperfection. I find that evenin the absence of any information problems, a relationship-based banking system may yielda higher social welfare than a transactions-based competitive system. Interactions betweenthe imperfect product and capital markets may offset each market's adverse effects on societyand thus improve overall efficiency.3132AppendixSubgame (accept, reject).^Consider the subgame (accept, reject) following a pair ofloan commitments in the second subset in which firm i becomes the monopoly producer ofthe industry in equilibrium. Notice first that firm j chooses q;2 = 0 if and only if qii >(a — 1 + r,)/b. The proof of this observation is as follows.To prove the sufficiency part, suppose that firm i anticipates that firm j will cease pro-duction. Then, the monopoly second-period output of firm i is1 ,gig =^ri) = —2b (a — 1 — bgil — ri )•Using (A2.1), the output price, given that firm j produces nothing is1a — b(qii q72 ) = —2(a + 1 — bqii + 7-) <1,(A2.1)where the inequality follows from the restriction on q ii . Since the output price is below firmj's interest plus commitment fee (i.e., 1) even if it does not produce, firm j will not produceand firm i's conjecture is correct. To prove the necessity part, suppose that q;2 = 0. Then,it must be the case that the output price is below firm j's true interest cost. Using (A2.1),I get the restriction on qa .Therefore, using (A2.1), firm i's two-period profit maximization problem given r i and itsmonopoly position is1max^—, (a — 1 — bqa — ri )(a — 1 + — r i ).qi>+;(a-1-1-7-,) 4o(A2.2)The derivative of this objective function is —4 1 /2 < 0 for all q, i > 0. Hence, the optimalsolution for this problem is=^(a — 1 + r i )/b if r, > 1 — a,q:1 0 if r, < 1 — a.(A2.3)Substituting (A2.3) into (A2.2), firm i's cum-commitment fee monopoly profit given r, is{—r i (a — 1)/b^if r, > 1 — a,— 1 (a — 1 — ri ) 2 /4b if r, < 1 — a. (A2.4):33Since firm i can always earn the ex-commitment fee profit of 2(a — 1) 2 /25b if it rejectsthe offer, bank 1 must allow it to retain at least this profit level. Thus, the maximumcommitment fee given r i that bank 1 can charge is2fi( ) = 7rNri)^25b (a — 1) 2.Using (A2.1) to (A2.5), the bank's profit as a function of r i given that firm i is a monopolyisf —2(a — 1) 2 /25b^ if r i > 1 — a,1 —(a — 1 — r i )(1 — a — r,)/4b — 2(a — 1) 2 /25b if rt < - a ,which is always negative.^^Proof of Proposition 2.1.^To prove this proposition, I first prove the following lemma.Lemma. In the subgame (accept, reject), the maximum profit of bank 1 is (a— 1) 2 /300h.Proof^Using an argument that is similar to the proof used above, the maximumcommitment fee given ri that bank 1 can charge isfi (ri ) =^— 225b (a — 1) 2 .^ (A2.6)Then, using (2.5) to (2.8) and (A2.6), bank l's profit maximization problem is1^2max 1— r, (a — 1 — 3r, )2b(a — 1 — 3ri )2 — 25b (a — 1) 2 .rt 2The first-order condition yields r: = 0. Thus, in this case, the maximum profit of bank 1 is(a — 1) 2 /300b.^^Now, I am ready to prove proposition 1. If both firms reject the offers, bank 1 earnszero profit. If bank 1 sets the pair of loan commitments such that only one firm accepts theoffer, then by the above lemma bank 1 can at most earn (a — 1) 2 /300b. Now, suppose thatbank 1 sets r 1 = r2 = 0 and fi = f2 = (a — 1) 2 /300b. It is easy to show that this pair of(A2.5)34symmetric loan commitments satisfies constraints (2.6) to (2.8) and gives bank 1 the profitof (a — 1) 2 /150b. Hence, in any SPNE for the full game, the bank must induce both firmsto accept the offers.^^Proof of Proposition 2.2.^Denote Q(r i , rj , fi , fj ) and Ak (ri , rj , fi , fj ) for k = 1, 2 and3 be the objective function and constraints (2.6) to (2.8) of (P1) respectively. It is easyto check that Q is strictly concave while Ak is strictly quasi-concave for all k. Then, byTheorems 1 and 2 in Arrow and Enthoven (1961) (see also Takayama (1985) pp. 114-115),the Kuhn-Tucker conditions are the necessary and sufficient conditions for a global maximumof the given problem. The Kuhn-Tucker conditions are3Q(r7,7',07'i^3^30 00 1 ) • P, I) + ^Ak(7 r;^f7)^(A2.8)(A2.9)Ai(r7,r;^f7) = 0,^ (A2.10)A2(71, r;,^f7) = 0, (A2.11)A3 (71, r .7 ,^,fj)^0, (A2.12)> 0, f; >0, AZ > 0 for all k,^ (A2.13)where A i , A2 and A3 are the Lagrangean multipliers and (71,7-;,^f7, , A 2*, A 3*) is a solutionto the Kuhn-Tucker conditions.Subtracting (A2.11) from (A2.10), I get2771 2 — [18(a — 1) — 864r'jr: — [19(a — 1) 2 — 12(a — 1)r 3*. — 1627..7 2 ] = 0.^(A2.14)Note that I can write7-** =^k(a — 1)3^(A2.15)for some constant k. Since k is allowed to depend on a, this representation of 7.; is alwayspossible. In fact, I shall show later that k is indeed a constant independent of a. Substitutingk=1^A*  —AkW^=0,15; ori^z 3^3 (A2.7)3dr •^33 k=1+ A; = 1, A; = 1,35ris =186 — 4293k + V3600 — 1344600k + 29907225k 22214^(a^1).(A2.15) into (A2.14), I can solve the quadratic equation (A2.14) to get the two real rootsfor r:, 7- 7b and 7S, as functions of k:1188k — 6 — V ^(a80064 — 64800k + 729000k 2r:b =^ 1),^(A2.16)21061188k — 6 + V80064 — 64800k + 729000k 2r = (a1).2106^ (A2.17)Substituting (A2.9) into (A2.7), I can solve AI as a function of r: and 7-;. Similarly,substituting (A2.9) into (A2.8) yields Al' as another function of r7 and 7-;. These two functionsmust be the same and thus I get28087-7 2 + [243(a — 1) + 891r;]7-7 + [7(a — 1) 2 — 57(a — 1)7-; — 2592TV] = 0.^(A2.18)Substituting (A2.15) into (A2.18), I can solve the quadratic equation (A2.18) to get the tworeal roots for ri as functions of k:186 — 4293k — 0600 — 1344600k + 29907225k 22214(A2.20)Since (A2.16) and (A2.19) must be the same, using numerical methods, I get k = 0.0583which is independent of a. Substituting the corresponding value of r: into (A2.11) yieldsf,* = —0.04(a — 1) 2 /b, which violates (A2.13). Hence, I know that the solution must bethe one that solves (A2.17) and (A2.20). Using numerical methods, I get k = —0.003which is independent of a. The uniqueness of the global maximum solution follows fromthe uniqueness of the solution for the Kuhn-Tucker conditions. The SPNE outcome for fullgame 1 isr: = —0.129983928(a — 1), = —0.132984454(a — 1),= 0.080997163(a — 1) 2 /b, f; = 0.083107548(a — 1) 2 /b,q:1 = qi2 = 0.224796575(a — 1)/b, q;1 = q;2 = 0.227797101(a — 1)/b,= 0.020069836(a — 1) 2 /b, = 0.02067549(a — 1) 2 /b.^^(a — 1),^(A2.19)36Proof of Proposition 2.3.^By analogous arguments as used in the proof of proposition2.2, the global maximum solution for (P2) given (7.;, f7) is one at which constraints (2.6) and(2.7) are binding, i.e., it satisfies (A2.14). In equilibrium, (A2.14) must hold simultaneouslyfor i = 1 and 2 at (r1,7-2). Subtracting (A2.14) for i = 2 by (A2.14) for i 1 yields(ri* — r 2*)[9(r i*^7- 2*) + 2(a — 1)] = 0. I have either 7- 1* = r2* or7.7 + 7- 2* = —92(a^1 )-Suppose that (A2.21) holds. Substituting (A2.21) into (A2.14) yields2025r: 2 + 450(a — 1)r: + 41(a — 1) 2 = 0.However, there are no real roots for (A2.22) because450 2 (a — 1) 2 — 4(2025)(41)(a — 1) 2 = —[360(a — 1)1 2 .(A2.21)(A2.22)Since the relevant domain of the objective function is compact, by the Weierstrass Theorem(see Takayama (1985) p. 33), there must exist a real maximum solution. By the Arrow andEnthoven Theorem, the maximum solution must satisfy the Kuhn-Tucker conditions. Hence,(A2.21) cannot hold.Since r i* = r2 , I have k = 0 from (A2.15). Thus, by (A2.16) and (A2.17), I get 71 1, =0.137205995(a — 1) and 71, = —0.13150799(a — 1). Using similar arguments as in the proofof proposition 2.2, 71 is not the solution. The uniqueness of the equilibrium follows fromthe uniqueness of the solution for the Kuhn-Tucker conditions. The SPNE outcome for fullgame 2 is—0.13150799(a — 1), = 0.082058091(a — 1) 2 /b,q1 = (172 = 0.226301598(a — 1)/b, 711 = 0.020366734(a — 1) 2 /b,for i = 1, 2.^^37References[1] Arrow, Kenneth J. and A. C. Enthoven, 1961, "Quasi-Concave Programming," EC0110-metrica, Vol. 29, pp. 779-800.[2] Berkovitch, Elazar and Stuart I. Greenbaum, 1991, "The Loan Commitment as anOptimal Financing Contract," Journal of Financial and Quantitative Analysis, Vol. 26,pp. 83-96.[3] Berkovitch, Elazar and M. P. Narayanan, 1993, "Timing of Investment and FinancingDecisions in Imperfectly Competitive Financial Markets," Journal of Business, Vol. 66,pp. 219-248.[4] Besanko, David and Anjan V. Thakor, 1992, "Banking Deregulation: Allocational Con-sequences of Relaxing Entry Barriers," Journal of Banking and Finance, Vol. 16, pp.909-932.[5] Bhattacharya, Sudipto, 1993, "Financial Intermediation with Proprietary Information,"Working Paper 205.93, Institut d'Analisi EconOmica, CSIC, Universitat Autenoma deBarcelona.[6] Boot, Arnoud W. A., Anjan V. Thakor and Gregory F. Udell, 1987, "Competition,Risk Neutrality and Loan Commitments," Journal of Banking and Finance, Vol. 11,pp. 449-471.[7] Boot, Arnoud W. A., Anjan V. Thakor and Gregory F. Udell, 1991, "Credible Com-mitments, Contract Enforcement Problems and Banks: Intermediation as CredibilityAssurance," Journal of Banking and Finance, Vol. 15, pp. 605-632.[8] Brander, James A. and Tracy R. Lewis, 1986, "Oligopoly and Financial Structure: TheLimited Liability Effect," American Economic Review, Vol. 76, pp. 956-970.38[9] Campbell, Tim, 1978, "A Model of the Market for Lines of Credit," Journal of Finance,Vol. 33, pp. 231-244.[10] Crum, M. Colyer and David M. Meerschwam, 1986, "From Relationship to Price Bank-ing: The Loss of Regulatory Control," in Thomas K. McCraw, ed., America VersusJapan, Boston: Harvard Business School Press, pp. 261-297.[11] Deshmukh, Sudhakar D., Stuart I. Greenbaum and George Kanatas, 1983, "LendingPolicies of Financial Intermediaries Facing Credit and Funding Risk," Journal of Fi-nance, Vol. 38, pp. 873-886.[12] Duca, John and David D. VanHoose, 1990, "Loan Commitments and Optimal MonetaryPolicy," Journal of Money, Credit and Banking, Vol. 22, pp. 178-194.[13] Ham, John C. and Arie Melnik, 1987, "Loan Demand: An Empirical Analysis UsingMicro Data," Review of Economics and Statistics, Vol. 69, pp. 704-709.[14] Hart, Oliver and Jean Tirole, 1990, "Vertical Integation and Market Foreclosure,"Brookings Papers on Economic Activity: Microeconomics, pp. 205-286.[15] Kanatas, George, 1987, "Commercial Paper, Bank Reserve Requirements, and the In-formation Role of Loan Commitments," Journal of Banking and Finance, Vol. 11, pp.425-448.[16] Maksimovic, Vojislav, 1986, Optimal Financial Structure in a Stochastic Oligopoly, Un-published Ph.D. Dissertation, Harvard University.[17] Maksimovic, Vojislav, 1990, "Product Market Imperfections and Loan Commitments,"Journal of Finance, Vol. 45, pp. 1641-1653.[18] McAfee, R. Preston and Marius Schwartz, 1990, "Two-Part Tariffs to Competing Firms:Destructive Recontracting, Nondiscrimination, and Exclusivity," Mimeo, Department ofEconomics, University of Texas at Austin.39[19] Meerschwam, David M., 1989, "International Capital Imbalances: The Demise of Lo-cal Financial Boundaries," in Richard O'Brien and Tapan Datta, eds., InternationalEconomics and Financial Markets, Oxford: Oxford University Press, pp. 289-307.[20] Meerschwam, David M., 1991, Breaking Financial Boundaries: Global Capital, NationalDeregulation, and Financial Services Firms, Boston: Harvard Business School Press.[21] Petersen, Mitchell A. and Raghuram G. Rajan, 1992, "The Benefits of Firm-CreditorRelationships: Evidence form Small Business Data," Working Paper, Graduate Schoolof Business, University of Chicageo.[22] Poitevin, Michel, 1989, "Collusion and the Banking Structure of a Duopoly," CanadianJournal of Economics, Vol. 22, pp. 263-277.[23] Selten, Reinhard, 1975, "Re-Examination of the Perfectness Concept for EquilibriumPoints in Extensive Games," International Journal of Game Theory, Vol. 4, pp. 25-53.[24] Sharpe, Steven A., 1990, "Asymmetric Information, Bank Lending and Implicit Con-tracts: A Stylized Model of Customer Relationships," Journal of Finance, Vol. 45, pp.1069-1087.[25] Takayama, Akira, 1985, Mathematical Economics, 2nd edn., Cambridge: CambridgeUniversity Press.[26] Thakor, Anjan V., 1982, "Toward a Theory of Bank Loan Commitments," Journal ofBanking and Finance, Vol. 6, pp. 55-84.[27] Thakor, Anjan V., Hai Hong and Stuart I. Greenbaum, 1981, "Bank Loan Commitmentsand Interest Rate Volatility," Journal of Banking and Finance, Vol. 5, pp. 497-510.40accept^ rejectaccept^reject^accept^rejectFigure 1.^Extensive Form of Production-Search-Production Subgames41firm 2accept^ rejectreject7r1(r i , f2) — fi — fi7r4(r i , r2 ) — 12 712(7-1)rt(r2 ) 2(a — 1) 2 /25b72(r2) — f2 2(a — 1)2/25bacceptfirm 1Figure 2.^The Normal Form of Production-Search-Production Subgames.The ex-commitment fee profits of firm 1 are in the upper left corners and those of firm 2 arein the lower right corners.42Chapter 3Debt, Asset Substitution, andFurther Borrowing3.1 IntroductionThe previous chapter demonstrated that relationship banking may be more efficient thanprice banking when product markets are imperfect. The setting considered was one ofsymmetric information. This chapter goes on to examine this issue under moral hazard.Moral hazard arises when borrowers, after receiving their loans, take ex post unobservableactions that jeopardizes their loan repayments!When a borrower is allowed to engage in multilateral credit transactions with differentbanks, there exist negative externalities that are notably ignored by the contracting lit-erature. These negative externalities stem from the public good nature of the borrower'sunobservable action: the effect of varying the loan amount provided by one bank affects theborrower's action choice, and hence the profits of the other banks which transacted withhim. Bizer and DeMarzo (1992) are the first to incorporate such externalities into the designof optimal loan contracts. They consider the effort-incentive problem under a two-periodconsumption smoothing model. The borrower borrows in the first period and repays his debt'See, for example, Jensen and Meckling (1976) and Myers (1977).43from his random second period income whose distribution is affected by his unobservablework effort. Unlike the traditional approach, the borrower is allowed to borrow sequentiallyfrom more than one bank and debt is fully prioritized so that prior loans retain seniorityover new ones. However, future loan requests by the borrower are unobservable to thosebanks which have already transacted with the borrower. If the borrower is heavily leveragedso he risks bankruptcy in the bad state, increases in indebtedness will decrease his workeffort which in turn lowers the probability of repayment of earlier loans. Since new banksdo not pay for the negative externalities that they impose on prior lenders, their loan termswill not reflect the resulting devaluation of existing debt. This contrasts with a one-bankenvironment in which all effects on prior credit are internalized by the sole creditor. Bizerand DeMarzo show that the borrower is worse off than he would be if he could crediblycommit to refraining from further borrowing.While Bizer and DeMarzo's model is appropriate for the study of consumer loans, theeffort-incentive problem may be less important for corporate loans than the risk-shiftingincentive. Jensen and Meckling (1976) argue that the agency cost of corporate debt resultsmainly from the risk-incentive problem (hereafter referred as the asset substitution problem),which occurs when the managers of a firm can choose among projects of varying riskiness.'As pointed out by Black and Scholes (1973), stockholders of the levered firm can be viewedas holders of a European call option on the firm with an exercise price equal to the facevalue of the debt. The value of this call option can be increased if stockholders can inducemanagers to shift into high risk projects in their investment policies. This type of incentiveproblem is the focus of this chapter. 3My model is similar in spirit of Bizer and DeMarzo's in that I retain the assumptions of2 The possible impact of unobservable further borrowing under the asset substitution problem is alsomentioned by Bizer and DeMarzo, although it is not possible to convert their model to address this issue.In contrast, my model can be easily modified to discuss the effort incentive problem. See section 3.5.30ther formal models discussing the asset substitution problem include Gavish and Kalay (1983), Green(1984), and Green and Talmor (1986).44full prioritization of debt and unobservability of future borrowing. 4 The main issue of thischapter, however, is the asset substitution problem (vis-a-vis the effort incentive problem inBizer and DeMarzo): the borrower can choose among projects of varying riskiness and hisproject choice is ex post unobservable to banks. I assume that under relationship bankingthe borrower can credibly commit not to borrow further and I show that underinvestmentarises when the asset substitution problem is present. This generates a second-best outcome.I also assume that borrowers cannot credibly commit to refrain from future borrowing underprice banking and I show that more underinvestment is needed to overcome the given dualincentive problem. This gives rise to a third-best outcome.The incentive problem induced by further borrowing can in principle be eliminated bydebt covenants restricting future debt issues. However, such covenants are not widely used inpractice (see Asquith and Wizman (1990)). One reason for this may be the underinvestmentproblem identified by Myers (1977). If new investment of a firm can only be financed by newequity issues or by reduced dividends, then with risky debt outstanding part of the gains fromthe investment goes to debtholders, rather than shareholders. Covenants prohibiting futureborrowing would induce perverse investment incentives so that not all positive net presentvalue projects would be undertaken. 5 Another way to overcome this incentive problem isthrough the use of contingent claims such as callable debts or warrants (see Green (1984)).Bizer and DeMarzo (1992) argue that, as a firm's investment opportunity set might evolveover time, pricing a fully state-contingent claim is extremely difficult.The remainder of this chapter is organized as follows. Section 3.2 describes the ba-sic model and the first-best equilibrium. Section 3.3 characterizes the equilibrium underrelationship banking, along with a comparative statics analysis. Section 3.4 develops theequilibrium under price banking and also obtains some comparative statics results. Section:3.5 discusses the robustness of the analysis and suggests possible extensions. Section 3.6'See Bizer and DeMarzo for arguments and empirical evidence that justify these two assumptions.'See also the discussions by Smith and Warner (1979) and Bizer and DeMarzo (1992).45concludes. All proofs are given in the appendix.3.2 Model and the First-Best Solution3.2.1 The ModelConsider a risk-neutral entrepreneur (hereafter called the borrower) who has monopolyaccess to two mutually exclusive, nontransferable, divisible investment projects, indexed by0 E {L, H}. The borrower has no initial wealth and must solicit one or more loans from arisk-neutral competitive banking sector in order to fund either project. The banking sectorconsists of at least two banks. Deposit insurance is (de facto) complete and thus each bank'sdeposit funding cost is the gross riskless interest rate i > I.In this subsection I adopt a one-shot perfect information game with two stages: thebanking stage and the investment stage. In the banking stage, the borrower may requestloans repeatedly, each time going to a new bank. In other words, there may be multiplerounds in this stage. At each round, the borrower visits a new bank and offers a loan contract.The bank either accepts or rejects the loan request based on the borrower's current credithistory.' The borrower's current credit history is a sequence of outcomes (loan requests andacceptance of banks) of each round of banking prior to the visit of the bank. ? The bank,however, cannot observe future borrowings by the borrower. The loan contract, (r, /), is apair that specifies a loan amount, I > 0, provided by the bank if it accepts the contract, anda repayment amount, r > I, that the borrower must repay to the bank after the project'scash flow is realized. The project's cash flow is ex post observable and verifiable. Debt is fullyprioritized so that each loan is repaid only after all earlier commitments are satisfied. Thisassumption, together with the observability of the borrower's previous borrowings, ensures6 The results are robust to alternative institutional arrangements, since there are no search or time costsfor the borrower and banks are competitive.71f banks cannot observe the borrower's current credit history, there will be a market failure similar tothe one described in Akerlof (1970). Banks always infer that the borrower will only undertake the high-riskproject.46that loan contract terms (based solely on the project's realized cash flow) are enforceable.The borrower stays in the banking stage until he is satisfied with his loan portfolio, then heenters into the investment stage by choosing his optimal project given his loan portfolio. Hisproject choice is ex post unobservable to banks.In the investment stage, the borrower may invest any I dollars in either project but notboth.' In return, project H yields a random terminal cash flow of RH(I) with probabilitypH, or zero with probability 1 — pH . 9 Project H is referred to as the high-risk project.Project L yields a random terminal cash flow of RL (I) kRH(I) with probability pL, orzero with probability 1 — pL , where 0 < pH < pi, <1 and pH /pL < k. 19 Project L is referredto as the low-risk project. The restriction on the scaling factor, k, is a sufficient conditionfor the low-risk project to be first-best, since it ensures that the low-risk project alwaysgenerates a higher expected value than the high-risk project does." The return functionRH : 34 is strictly increasing and concave with RH (0) = 0, limy„, RH (I) < oo andR/H (/) = oo.Suppose that the borrower obtains a loan portfolio, { (r t , it )}nt_ i , where t denotes thepriority of the loan contract, and undertakes project 0. Then his expected terminal payoff isPe[R9 (E It) —t=1^t=1n^nFor the loan contract^IT), the borrower's total debt commitment prior to it is ETt_-11 Pt-Since debt is fully prioritized, a bank with this contract receives[( np o min^, R9^It ^,---1— E r t]rr^— if,.t=1^t--.1The equilibrium concept employed is Selten's (1975) subgame perfect Nash equilibrium.'Since the borrower is an expected terminal payoffs maximizer, he has no incentive to invest less thanwhat he has borrowed in the banking stage.'The assumption of zero cash flow in the bad state is for the sake of simplicity. Relaxing this assumptionwill not affect the results qualitatively.'The assumption of RL(I) being proportional to RH(I) gives nice graphical illustrations of the results. Infact, the following less restrictive assumptions are necessary: (i) pL RL (I) > pHRH(I), (ii) RL(I) < RH(I),(iii) RiL (/) < R i i (/), for all I larger than a sufficiently small number.11 See proposition 3.1.47A strategy for the borrower is a combination of the loan portfolio choice and the projectchoice. A strategy for a bank is a set of loan proposals that the bank is willing to accept fromthe borrower. A subgame perfect Nash equilibrium (hereafter called equilibrium) requiresthat the equilibrium strategy of each player maximizes his expected terminal payoff giventhe strategies of the other players, contingent upon all possible current credit histories of theborrower. An allocation, [(r, I), 0], is a pair consisting of an initial loan contract, (r, I), anda project, 0, chosen by the borrower. Any equilibrium can then be specified by an allocation.3.2.2 The First-Best SolutionIn the first-best case, the borrower's choice of project is ex post observable to banks andis contractible. The borrower chooses the project that yields the highest expected value netof the investment compounded at the riskless rate i. That is, he solvesmax^Po Re(/) — i/.OE{L, H}, 1>0Denote IL and IH be the respective solutions of the following first-order conditions:PLOVH(h) = i,^ (3.1)andpHR/H (/H) =^ (3.2)By the assumptions on RH, IL and IH are the unique interior maximum solutions. Clearly,the low-risk project is optimal whenpLkRH(h) — ZIL > pH RH (IH) -^ (3.3)It is easy to show that (3.3) holds if k > pH IpL . Thus, the following proposition is immediate(all proofs are given in the appendix).Proposition 3.1.^The first-best equilibrium is the allocation RiI"' pL, IFBwhere I' solves (3.1).483.3 Asset Substitution under Relationship Banking:The Second-Best EquilibriumUnder relationship banking, the borrower's choice of project is ex post unobservable tobanks, but the borrower can credibly commit to borrow from one bank only. The credibilityof the commitment may arise from the fact that there are high costs to prevent the borrowerfrom switching banks. The asset substitution problem is said to be present if the borrowerchooses the high-risk project when he receives the first-best loan contract (jJFB/pL , iFB).The condition for this to be true isPH [RH(IFB) J. FB I > pL[kRH(IFB) iIFB1j,PLwhich is equivalent toPH^pH^i IFBk < —PL + (1 (3.4)PLL ) 13 L RH (IFB) *Since pLkRH(IFB) > aFF3, it follows immediately that (3.4) implies k < 1. Throughout thepaper, condition (3.4) is assumed.The second-best equilibrium is obtained by solving the following program: (P1)max po.[Re.(I)— r]1>o, r>Is.t. po .r — iI > 0,^ (3.6)0* E arg max pe[Re (I) — r].^(3.7)BE{L, H}In words, (P1) says that the borrower's expected terminal payoff, (3.5), is maximized subjectto the bank at least breaking even, (3.6), and the borrower choosing an incentive-compatibleproject 0*, (3.7).Proposition 3.2.^The second-best equilibrium (under asset substitution and relation-ship banking) is the allocation [(iIsB1pL IsB\ L], where IsB solves(Mk — pH)RH(IsB) (1 _PH)iisBPL(3.8)(3.5)49if the following condition holds:PLkRH(IsB )— iISB > pHRH(IH)^(3.9)where IH solves (3.2). Otherwise, the second-best equilibrium is the allocation [(iI H I pH , IH ), H].Condition (3.9) states that the borrower's expected payoff when he receives the loan,(iisn/pL 7-sscontract^) (and he chooses the low-risk project) is no less than that whenhe receives the loan contract (iIH IpH , IH) (and he chooses the high-risk project). It canbe interpreted as requiring the scaling factor k to be large enough. 12 Since it is not aninteresting issue if the low-risk project is not the second-best project, hereafter condition(3.9) is assumed.(Figure 1 about here)The second-best equilibrium is illustrated in figure 1. The borrower's expected payoffincreases to the northwest, and the bank's expected profit increases to the southeast. DenoteU 9 and II 9 respectively as the borrower's indifference curve (iso-expected-payoff curve) andthe bank's break-even line given that the borrower undertakes project 0 E {L, H}. It is easyto show that the borrower's indifference curves are strictly increasing and convex. Followingfrom the fact that k < 1, UL (having slope 1/kR'H (/)) is steeper than UH (having slope1/R'H (/)). The I.C. curve depicts the locus of the loan contracts at which the incentivecompatibility constraint, (3.7), binds, that is,PL[kRH(/) — = pH[RH(I) — r].It is easy to see that the I.C. curve is strictly increasing and convex with sloped/dr PH ^1(PLk PH)R 1H(I)^kR/H(I)'I.C. curve(3.10)12 It is easy to show that condition (3.9) implies k> pH/PL.50which is steeper than the borrower's indifference curves. Loan contracts on or above the I.C.curve induce the borrower to choose the low-risk project. However, for these contracts tobe admissible, the bank must be willing to participate. Hence, they must be on or belowH L . Thus, the shaded region is the set of feasible loan contracts which induce the borrowerto choose the low-risk project." Contract F = (iIFB IpL , IFB) is the first-best contractwhich, by condition (3.4) (the presence of the asset substitution problem), lies below theI.C. curve. Thus, this contract cannot be implemented. Contract E = (iI sB1pL,maximizes the borrower's expected payoff in the shaded region. On the other hand, theborrower's maximum expected payoff given that he chooses the high-risk project and thebank breaks even is corresponded by UH passing through contract G = (iIH/pH, IH)•Since this indifference curve lies below contract E (this follows from condition (3.9) whichguarantees the optimality of the low-risk project), inducing the borrower to undertake thelow-risk project by contract E is indeed optimal.Proposition 3.3. The second-best loan amount and repayment amount are less thantheir respective first-best levels.In the second-best equilibrium, the borrower can be motivated to choose his first-bestproject by undercutting the loan amount. The reason why this can resolve the asset substi-tution problem is as follows. The differential between the cash flows of the high-risk projectand the low-risk project in the successful state is (1 — k)RH(I), which is strictly increasing inthe investment amount, I. By reducing the loan amount, the high-risk project becomes lesssuperior in the successful state and, at the same time, the probability of success is lower ifthe borrower undertakes this project. Hence, provided that the loan amount is small enough,the low-risk project becomes the optimal project for the borrower.13 Note that since the I.C. curve has a slope of zero as I approaches 0+, the shaded region is alwaysnon-empty.51Proposition 3.4.^The second-best loan amount and repayment amount are increasingwith (i) an increase in the scaling factor, k, (ii) a decrease in the gross riskless interest rate,(iii) an increase in the probability of success of the low-risk project, p L , or (iv) a decreasein the probability of success of the high-risk project, p H .This proposition provides some comparative statics properties of the second-best equi-librium. The intuition underlying (i) is that an increase in the scaling factor increases theborrower's net payoff from the low-risk project in the successful state for a given repaymentamount. Thus, the bank can raise this amount at least a little bit without causing the bor-rower to switch to the high-risk project. By the same token, the loan size can be increased.To see why (ii) obtains, note that a decrease in the riskless rate lowers the bank's cost offunds. The bank can now demand a smaller repayment amount while still breaking even.This in turn reduces the borrower's incentive to substitute projects. As the borrower issuffering from underinvestment, it is efficient for the bank to increase the loan size at thesame time. The intuition behind (iii) is similar to that of (i). An increase in the probabilityof success of the low-risk project raises the net expected payoff from the low-risk project.Hence, both the loan amount and the repayment amount can be increased. Finally, (iv)obtains because a decrease in the probability of success of the high-risk project dampensthe asset substitution problem and hence less underinvestment is needed. Larger loan sizeimplies a larger repayment obligation.3.4 Asset Substitution under Price Banking: The Third-Best EquilibriumI assume that price banking allows the borrower to obtain additional loans sequentially,each time from a new bank, after an initial loan contract has been signed.3.4.1 Incentives for Further Borrowing52Suppose that the borrower has received the second-best loan contract as his initial loan.Figure 2 illustrates why the borrower may prefer to find a new bank from whom to seek anadditional loan. Given the additional loan, the borrower switches to the high-risk project.Nonetheless, the borrower can offer an additional loan contract that gives the bank a nonneg-ative profit even if the bank assumes that the borrower will undertake the high-risk projectand the borrower finds this additional loan desirable.(Figure 2 about here)Figure 2 is drawn such that the second-best loan size, / sB , is smaller than the optimalinvestment amount, IH , given that the borrower undertakes the high-risk project." If theborrower does not borrow further, his expected payoff is corresponded by his indifferencecurves passing through the second-best loan contract, E, regardless of whether he choosesthe low-risk or high-risk project. Suppose now that the borrower borrows further by offeringan additional loan contract (say contract H) in the lower shaded region. Obviously, thisadditional loan will be acceptable by any new banks irrespective of their beliefs about theborrower's project choice. The borrower's loan portfolio (the combined loan contract, saycontract E+ H) will end up in the upper shaded region.' The borrower will be better off tooffer this additional contract and switch to the high-risk project. Given that the borrowerchooses the high-risk project, the initial bank which supplies contract E will then sustain aloss. Thus, the second-best outcome is not attainable in this case if the borrower can engagein unobservable future borrowing.The intuition behind the above observation is that the additional loan imposes a negativeexternality to the initial loan. Given that / sB < Ili, the borrower will have an incentive to14If this condition does not hold, there will be no incentive for the borrower to borrow further. Seeproposition 3.6.15These two shaded regions are simply one-to-one transformations of each other by shifting point E to theorigin. It is easy to show that U H , which passes through contract E, has a slope less than pHli at point E,which is the slope of R H (as well as 11 //' ). This follows from the fact that / sB < /H and from (3.2). Hence,the shaded regions are non-empty.53switch to the high-risk project as long as he receives more funds. The asset substitution thenreduces the probability of repayment of the initial loan and causes a loss to the initial bankwhich supplies it. It is this externality that hinders the implementability of the second-bestoutcome.4.2 The Third-Best SolutionBanks, however, are not naive. They will take the borrower's vulnerable behaviourinto considerations. As a result, only those initial loan requests under which the borrowercan credibly promise to refrain from future borrowing are acceptable to banks. Thus, thefollowing proposition follows.Proposition 3.5.^When the borrower can engage in unobservable future borrowing,the equilibrium is characterized by solving the following program: (P2)max pet[119*(I) —^ (3.11)1 >0, r>/s.t. po*r — iI > 0,^ (3.12)0* E arg max pe [Ro (I) — r],^ (3.13)0E{L, HI[(0, 0), 0*] E arg^max^pe[Ro(I + AI) — (r + AO], (3.14)(Lr, AI)ET, OE{L, 11}(where T = {(Ar, AI) e V+ : pH Ar — iAI > 0}), there is no further borrowing in equilib-rium.(P2) looks similar to (P1), the program solving the second-best outcome, except the im-position of constraint (3.14). Loosely speaking, (3.14) is the no further borrowing constraintwhich guarantees that the borrower will not offer any additional loans. In words, it says thatthere is no additional loan request from the borrower that is desirable to him and, at thesame time, acceptable to new banks no matter what the borrower's project choice. Clearly,54(3.14) provides a necessary condition for no further borrowing. However, it may not be clearthat (3.14) is also sufficient to rule out all self-enforcing additional loans since it only con-siders those additional contracts which make nonnegative profits for new banks irrespectiveof the borrower's project choice. Given the structure of the problem, (3.14) happens to besufficient as well. The reason is as follows: Note that the second-best project is the low-riskproject. If the borrower finds it desirable to make an additional loan, it must be the casethat he wants to switch to the high-risk project given his new loan portfolio. New banksanticipate the borrower's motive and price the additional loan request accordingly as if theborrower chooses the high-risk project. Thus, the solution of (P2) has the property that theborrower has no incentive to make any self-enforcing future borrowing.Proposition 3.6.^The second-best loan contract ( iisE pL, IsB\) is the optimal initialloan contract when the borrower can engage in unobservable future borrowing if, and only if,ISB >This proposition says that the second-best outcome is implementable when / sB > IH ,even though the borrower is free to make unobservable additional loans. The underlyingintuition is that for any investment level greater than IH, the borrower will be worse offif he increases the investment amount and switches to the high-risk project. Hence, giveniss > IH, receiving / sB as the initial loan ensures that the borrower will have no incentiveto borrow further.Proposition 3.7.^/sB < IH if, and only if,k < PH (,^PHA PL^PL pL(1 + A)'where A is the Lagrange multiplier defined in the appendix as (A3.1).(3.15)This proposition says that the second-best outcome is not attainable when the scaling55factor, k, is sufficiently small so that a substantial underinvestment is needed to resolve theincentive problem in the second-best case. In order to study the third-best outcome whenthe second-best outcome is not feasible, condition (3.15) is assumed from now on.Proposition 3.8.^If there exists an I E [0, IH ) satisfying^pLkRH (I) — (2 — P—H )iI > pH RH (IH ) — iIH ,^(3.16)pLthen the third-best equilibrium (under asset substitution and further borrowing) is the allo-\cation [(iITB^ITB) L], where ITB solves (3.16) with equality. Otherwise, the third-bestequilibrium is the allocation [(iIHIN' , IH ), H].(Figure 3 about here)Graphically, the third-best outcome is illustrated in figure 3. The NFB curve depicts thelocus of the loan contracts at which the no further borrowing constraint (3.14) binds. Supposethat 0* = L. Take 0 = H for the right hand side of (3.14). Then, the optimal additional loan(Ar, Al) E T that maximizes it given an initial loan (r, I) can be computed. It is easy toshow that the optimal additional loan is (0, 0) when I > IH and is (i(IH — I)/pH , IH — I)when I < 1H. Hence, for I > IH, the NFB curve coincides with the I.C. curve, and forI < IH, the NFB curve is given by^PL[kRH(I) — = PH [RH(/H) — — i(IH — I).^(3.17)In words, (3.17) says that at any point on this portion of the NFB curve (say at contractE'), the borrower's expected payoff given that he undertakes the low-risk project with thiscontract (say corresponded by UL ) is the same as the one corresponded by the indifferencecurve assuming high-risk project (say U *H) that is tangent at IH to the half-line (say I-v„)extended from the point parrallel to 1-1 1/ (i.e. UL* and UH* cross at the I.C. curve). It is easyto see that this portion of the NFB curve is strictly increasing and convex with sloped/dr PL PH NFB curve pL kRVI)—(3.18)56which is flatter than the I.C. curve, since RIH (I) > ilpH for I < IH . From (3.17), this portionof the NFB curve lies above the I.C. curve. Hence, any loan contracts on or above the NFBcurve must not give the borrower the incentive to borrow further and switch to the high-riskproject. For these contracts to be admissible they must be on or below II L so the bank iswilling to participate. Condition (3.16) simply says that the NFB curve intersects H L atsome point with I < IH. This implies that the shaded region is non-empty and gives the setof feasible initial loans. Contract E' = ( i 'TB p ITB ) maximizes the borrower's expectedpayoff in the shaded region. As / TB solves (3.16) with equality, the borrower obtains anexpected payoff which is bigger than pHRH(IH) — iIH , the borrower's maximum expectedpayoff given that he chooses the high-risk project and banks price the initial loan contractcorrectly. Hence, the low-risk project is indeed optimal given condition (3.16). However, ifcondition (3.16) does not hold, the shaded region is empty and thus there is no feasible initialloan that can induce the borrower to choose the low-risk project. In this case, undertakingthe high-risk project is the only credible outcome.Proposition 3.9.^The third-best loan amount and repayment amount are less thantheir respective second-best levels.In the third-best equilibrium, the borrower is induced to undertake the low-risk projectby a further cut in the initial loan size. The intuition underlying this result is that thefreedom of seeking additional loans by the borrower makes the high-risk project even moreattractive. As a result, to resolve the given dual incentive problem, a larger underinvestmentis needed to ensure the low-risk project to be optimal.57Proposition 3.10,^The third-best loan amount and repayment amount are increasingwith (i) an increase in the scaling factor, k, (ii) a decrease in the gross riskless interest rate,(iii) an increase in the probability of success of the low-risk project, p L , or (iv) a decreasein the probability of success of the high-risk project, pH .The comparative static properties of the third-best equilibrium are the same as those ofthe second-best equilibrium. The intuition described before applies here.3.5 Discussion and Extensions3.5.1 RenegotiationIn the third-best equilibrium, there is an additional efficiency loss. An immediate questionarises as whether the reported equilibrium is robust to renegotiation between the borrowerand the initial bank. Suppose that the borrower is allowed to seek new loans from the initialbank after he started his project but before the project's cash flow is realized (the projectonce started is irreversible). The bank can either accept or reject the new loan. My claimis that the third-best loan contract is also an optimal initial loan under renegotiation. Thatis, given the third-best contract, the borrower optimally chooses the low-risk project. In theregotiation stage, the borrower requests the additional loan that yields a combinded loancontract equal to the second-best one. The reason why renegotiation can improve efficiencyis as follows. If the borrower offers a new loan that is below the I.C. curve, the bank will pricethe loan as if the borrower has chosen the high-risk project. From figure 3, the maximumexpected payoff that the borrower can get is corresponded by UH if the high-risk project ischosen. This is definitely less than his second-best expected payoff. The borrower anticipatesthis outcome and thus optimally undertakes the low-risk project given the third-best contractas his initial loan.'l6See Wong (1992) for a similar observation when debt contracts include collateral, and Matthews (1991)583.5.2 Effort-Incentive ProblemA trivial extension of the model is to consider the effort incentive problem. Supposethat the borrower has monopoly access to a nontransferable, divisible project that requiresboth capital and the borrower's work effort to operate. The borrower can choose a workeffort e E {e, T}, where 0 < e < T < 1. For an investment amount I and a work efforte, the project yields a random terminal cash flow of R e (I) with probability e, or zero withprobability 1 — e. 17 The borrower's disutility of effort in monetary terms is V(e), withcc > V(T) > V(e) > 0. The return function R e n+ is strictly increasing andconcave, with Re(I) < oo and lim i_o+ R',(/) = oo, for all e. Moreover, the returnfunctions satisfy (i) Re(I) > R e (/), (ii) eRe(0) — V(T) > eRe (0) — V(e) > 0, and (iii)-ê.RW) < elre (I), for all I > 0. Condition (i) says that work effort is productive. Conditions(i) and (ii) guarantee that high work effort always generates a higher expected payoff than lowwork effort does. Condition (iii) says that capital and work effort are substitutes. This givesrise an incentive for the borrower to increase his borrowing and shirk. All the qualitativeresults reported before remain valid in this modified problem. The borrower will borrow lessin the third-best case. This contrasts with the findings by Bizer and DeMarzo (1992) whoshow that the converse may be true. The key reason for this difference is that the borrowerseeks loans in order to smooth his intertemporal consumption in Bizer and DeMarzo's model.Thus, the only way to correct the dual incentive problem is by charging a high interest rateto internalize the potential impact of unobservable further borrowing. However, resolvingthis problem by interst rates alone is inefficient, as the borrower may borrow more as wellto ease part of the inefficiency.under a general principal-agent model.'Since the probability of success increases with effort, normalizing the effort level to be equal to thisprobability is without loss of generality.593.5.3 Managerial CompensationAnother possible extension is to use the given framework to study the conflict of in-terests among managers, stockholders and bondholders. If managers' objective is differentfrom stockholders', stockholders should design an optimal compensation scheme for man-agers before the banking stage begins. Obviously it is inefficient for stockholders to write acompensation contract that can perfectly align managers' interests with theirs, as this willsimply produce the third-best outcome discussed above. Thus, the optimal compensationscheme will be departed from those characterized in the principal-agent literature in orderto alleviate the efficiency loss. In my framework, there exists a trivial managerial com-pensation contract that can achieve the first-best outcome: Stockholders offer managers afixed salary when the firm is solvent and zero otherwise. Then, it is always in the man-agers' best interest to choose the low-risk project, as it has a higher probability of success.As long as side-payments are not allowed, this contract credibly commits the firm to thechoice of the low-risk project and thus removes the agency cost of debt. Of course, witha more complicated situation, such an efficient managerial compensation contract may notexist. Nevertheless, the agency problem can still be mitigated by an appropriate choice ofmanagerial compensation. For further discussion, see Brander and Poitevin (1992).3.6 ConclusionIn this chapter, I study loan contract design problems under both relationship bankingand price banking. The opportunity to bank sequentially under price banking is shown toaffect the equilibrium outcome by introducing a time-consistency requirement. Borrowerswould be better off if they could credibly commit to refrain from additional loan requestsbut their vulnerable behaviour cracks the commitment. Borrowers end up receiving fewerfunds for investment and this generates an efficiency loss. The crux of the problem is thatnew loans impose negative externalities to existing loans by inducing borrowers to substitute60riskier projects. Since new banks do not pay for these externalities, prior banks recognizethe potential victimization that they may suffer and thus react accordingly. Equilibriumresults in reduced welfare for borrowers.In an empirical study of firms involved in leveraged buyouts, Asquith and Wizman (1990)report that corporate bonds containing clauses restricting total debt tend to maintain oreven gain value in leveraged buyouts. In contrast, corporate bonds with weaker covenantsenforcing priority alone suffered a decline in value from the substantial increase in juniordebt associated with leveraged buyouts. This empirical evidence seems to be consistent withthe finding that future borrowing imposes negative externalities on existing debt.The negative externalities addressed here are not unique to banking markets. Kahn andMookherjee (1991) suggest that "the externalities from side-trades are likely to pose difficul-ties whenever hidden information or hidden actions are present."' A deeper understandingon this issue is no doubt needed in order to improve the efficiency of optimal contract design.'See Kahn and Mookherjee (1991) for other possible applications.61AppendixProof of Proposition 3.1. Since IL is the maximum solution of pL kRH(I)—iI, it must betrue that pLkRx(IL)—ilL> PLkRH(-111)—iIH, which in turn is greater than pH RH(IH)- ZIHby k >^Proof of Proposition 3.2.^Suppose that the low-risk project is optimal (i.e. 0* = L).Then, the Lagrangian for (P1) isPL[kRH(I) —^P(PLr — in+ A[(pLk — pH)RH(I) — (PL — PO],where it is the Lagrange multiplier for (3.6) and A is the multiplier for (3.7). By the assump-tions on RH , the first-order conditions are necessary and sufficient for a global maximum.The first-order condition with respect to r yields= 1 + A (1 — P-- ^>0,PLsince A > 0. Therefore, (3.6) is binding. The first-order condition with respect to I yieldspLkR'H (/BB ) — i^A =^ (A3.1)-^(pLk - Hp )RIH (ISB) 1using (3.6) and substituting the value of ft. The denominator of (A3.1) is positive if, andonly if,- PH (PLk — PH)ffH(I SB)^(A3.2)By (3.10), the left hand side of (A3.2) is equal to the slope of the I.C. curve. In words, (A3.2)means that the I.C. curve cuts H L from below, which is always true. By (3.4), the first-bestloan contract lies below the I.C. curve. Hence, using (3.4), (A3.2) and a simple geometricargument, I have / sB < /FB . But then by (3.1), the numerator of (A3.1) is positive sinceit can be rewritten as pLk[R'H (IsB ) — R'H (IFB )]. Thus, A > 0 and (3.7) is binding. Solving(3.6) and (3.7) yields (3.8).62a r SB^a SBNow, suppose that the high-risk project is optimal (i.e. 0* = H), then there is no incentiveproblem. Hence, the optimal loan contract is (iIH /pH , IH ), the one characterized in section3.2.2. Which project is optimal then hinges on condition (3.9). ^Proof of Proposition 3.3.^By (3.4) and (3.8), I haveRH(/sB ) > RH (IFB)SB^IFB •Since RH is strictly concave, (A3.3) implies /sB < /FE3 .(A3.3)Proof of Proposition 3.4.^Denote D = (1 — pH IpL)i — (pL k — PH)R'H (I sB ) and rsBi/ sB/pL . By (A3.2), D > 0.Differentiating (3.8) with respect to k and rearranging terms, I geta ISB pLRH(ISB)Differentiating rSB with respect to k yields> 0.Ok^akDifferentiating (3.8) with respect to i yieldsars^(1 — pH /pL ) /ss < 0.SiDifferentiatingarsB^IsB^ai sBaiL +^ai PL(pLk—pH)reH (IsB) <0.PL DDifferentiating (3.8) with respect to pi, yieldsai sB pi, kRH(i si3 ) _ (pHhoii sB^ =api, pLD> piiRH (IH )— ail +(1— pidpL)iisB >0,pLDOk^D^> 0.rSB with respect to i yields63where the first inequality follows from (3.9). Differentiating 7- 813 with respect to pi, yieldsarSB^iiss^aisB-^2^ +^,aPL^PL PL uPLi[PLkRH(I sB ) — iIsB + (pL k — pH )MH (IsB )Is E3 1=  > 0.ADDifferentiating (3.8) with respect to pH yieldsaisB^iisB/pL RH(ISB)apHpL(1 — k)RH (I sB )< 0,(PL — PH)Dwhere the first inequality follows from (3.8). Differentiating 7,5B with respect to pH yieldsarsB^aisBOPH pL OPHProof of Proposition 3.5. First, I show that constraint (3.14) gives a necessary conditionfor no further borrowing. Suppose that (3.14) does not hold. Then, there exists an additionalloan contract (A7-, Al) E T such thatPe[Re(I + Al) — (r Ar)] > po•[Re•(/) — 7],for some 0 E {L, H}. That is, the additional loan contract makes the borrower better off ifhe offers it and switches to project 0. At the same time, new banks are willing to enter intoany loan contracts in T irrespective of the borrower's project choice. This implies that theallocation [(r, I), 01 is not immune to this additional loan.Next, an obvious sufficient condition for no further borrowing is: (SC) There does notexist an additional loan contract (Ar, Al) such that^B E arg max pe[Ro(/ + Al) — (r AO],^(A3.4)^+ Al) — (r Or)] > pe.[Ro.(/) — 7],^(A3.5)andNAT- — iAI > 0.^ (A3.6)< 0. ^64In words, there is no additional loan request from the borrower that the borrower finds itdesirable, (A3.5), and new banks will accept, (A3.6), knowing that the additional loan willgive them nonnegative profits as long as the borrower chooses his project optimally givenhis new loan portfolio, (A3.4). If I can show that maximizing (3.11) subject to (3.12), (3.13)and (SC) is equivalent to (P2), we are done.For any additional contract (sr, AI) that induces the borrower to choose the high-riskproject and is acceptable to new banks, by (3.14), the borrower will not be better off byoffering it. For any additional contract (sr, DI) that induces the borrower to choose thelow-risk project and is acceptable to new banks, maximizing (3.11) subject to (3.12) and(3.13) guarantees that (A3.5) cannot hold at the optimal initial loan contract. Hence, thesolution of (P2) turns out to have the property that the borrower has no incentive to makeany self-enforcing future borrowing. ^Proof of Proposition 3.6.^By proposition 3.2, the second-best contract (i/sB/pL, isB)solves (P2) without imposing constraint (3.14). If I can show that this contract satisfies(3.14) if, and only if, /' > IH , I am done.Suppose that the low-risk project is optimal (i.e. 0* = L). For any I E [0, IH), (3.14)can be written asPL[kRH(/) — > px[RH(/H) — — i(IH — I).^(A3.7)For any I E [IH, co), (3.14) becomesPL[kRH(/) — r] > pH [RH (I) — r].), i sB\Hence, the second-best contract (iISB/pL  ^satisfies (3.14) if, and only if, / sB > IH . Thelow-risk project is indeed optimal by proposition 3.2.^^Proof of Proposition 3.7.^From (A3.1), I have/rH (/sB ) = z[PL(1 + A) — pHA] PL[PLk(1+ A) — pH A] •(A3.8)65isBSince RH is strictly concave, /sB < IH if, and only if, R'H (^) > RH (/H ), which in turn isequivalent toi[pL(1 + A) —PHA] >PL_PLk( 1 + A) — PHA]^PH'^(A3.9)by (A3.8) and (3.2). Substracting the left hand side of (A3.9) from its right hand side yields— PH) — PL(PLk — PH)( 1 + A)] PLPH[PLk( 1 + A) — pH A]which is positive if, and only if, (3.15) holds.^^Proof of Proposition 3.8.^Given that condition (3.16) holds. Suppose that the low-riskproject is optimal (i.e. 0* = L). I shall verify this conjecture later.By propositions 3.6 and 3.7, the third-best loan size must be less than IH, or otherwise itwill be equal to /sB , which violates condition (3.15). Hence, I can restrict / E [0, fromwhich (3.14) can be written as (A3.7). Since I < IH, (A3.7) implies (3.13). The Lagrangianfor (P2) isPL[kRH(/)^(PL7. —^+ CIPLkRH(I) — PHRH(IH) — (pL — PH)r^— I)],where and ( are the Lagrange multipliers for (3.12) and (A3.7) respectively. By the assump-tions on RH, the first-order conditions are necessary and sufficient for a global maximum.The first-order condition with respect to r yields(2 — pH/PL)i — pL kR/H (ITB) ,using (3.12) and substituting the value of^By (3.2), pLkR1H (ITB ) — i = pLkIrH (ITB)PHR'H(IH) > 0 since k > pH /pL and IT B < IH. It follows that C > 0 if, and only if,pL — PH > pLPLk1VH(ITB ) — i^i •(A3.10)= 1 +^— —PH ) >0,PLsince ( > 0. Therefore, (3.12) is binding. The first-order condition with respect to I yieldspLkR1H (/TB ) — i=66By (3.18), the left hand side of (A3.10) is equal to the slope of the NFB curve. In words,(A3.10) means that the NFB curve cuts H L from below, which is always true when the NFBcurve and H L have non-empty intersection. Hence, given condition (3.16), (A3.7) is binding.Solving (3.12) and (A3.8) yields /TB .Finally, it remains to verify the initial conjecture. Since /TB solves (3.16) with equality,I know pL kRH (ITB ) — zITB > pHRH(IH) —) iIH • The former is the expected payoff for theborrower if he undertakes the low-risk project while the latter is that for the high-risk project.Hence, I can conclude that the low-risk project is indeed optimal. ^Proof of Proposition 3.9. Since I H maximizes pH RH (I) — iI, it must be true that> pHRH(1-13) —pHRH(/H) — i/HUsing (A3.11) and (3.16), I havePH ) TTB.(pLk PH)RH(ITB ) > ( - -By (3.9) and (A3.12), we knowRH(ITB) RH(ISB)ITB > ISBSince RH is strictly concave, (A3.13) implies /TB < isB.^^(A3.11)(A:3.12)(A3.13)Proof of Proposition 3.10. Denote D' = (2—pH IPL)i—mkR/H(ITB) and rTB = iITB/pL.By (A3.10), D' > 0.Differentiating (3.16) with respect to k, I getalrB pi, RH ( ITB> 0.ak^D'Differentiating rTB with respect to k yields> 0.ak^akarTB^aiTB67apH D'pL k RH (Fs) —^— 'TB)pH D' < 0,Differentiating (3.16) with respect to i yieldsa/TB ^(2 —pH/pL)/TB + IH ^ai D'where I use (3.2) to simplify the numerator. Differentiating 7 TB with respect to i yields^arTB^/TB^i a/TBDi^PL^aipLkki,(/TB)/TB+ii„PL D'Differentiating (3.16) with respect to pi, yieldsITB^pLk RH (ITB ) — (pH I pL )iITBpdYpHRH(IH)-ibir + 2 ( 1 - PHIpl)iITBpL D'> 0,where the second equality follows from (3.16). Differentiating PTB with respect to pi, yieldspL k^— (pH poirrs}pi D'Differentiating (3.16) with respect to pH yieldsarB^RH (110 — i IT B I pL < 0,<0.arTB^i /TB^i afTB2apL^PL^pi, ami{pHRH(IH)- 2IH +>0.where the numerator of the first equality is simplified by using (3.2) and the second equalityfollows from (3.16). Differentiating TM with respect to pH yieldsarTB^i a/TBapH pL apH < 0. ^68References[1] Akerlof, George A., 1970, "The Market for Lemons: Quality Uncertainty and the MarketMechanism," Quarterly Journal of Economics, Vol. 84, pp. 488-500.[2] Asquith, Paul and Thierry A. Wizman, 1990, "Event Risk, Bond Covenants, and the Re-turn to Existing Bondholders in Corporate Buyouts," Mimeo, Harvard Business School,Harvard University.[3] Bizer, David S. and Peter M. DeMarzo, 1992, "Sequential Banking," Journal of PoliticalEconomy, Vol. 100, pp. 41-61.[4] Brander, James A. and Michel Poitevin, 1992, "Managerial Compensation and theAgency Costs of Debt Finance," Managerial and Decision Economics, Vol. 13, pp. 55-64.[5] Gavish, Bezalel and Avner Kalay, 1983, "On the Asset Substitution Problem," Journalof Financial and Quantitative Analysis, Vol. 18, pp. 21-30.[6] Black, Fischer and Myron Scholes, 1973, "The Pricing of Options and Corporate Lia-bilities," Journal of Political Economy, Vol. 81, pp. 637-659.[7] Green, Richard C., 1984, "Investment Incentives, Debt, and Warrants," Journal ofFinancial Economics, Vol. 13, pp. 115-136.[8] Green, Richard C. and Eli Talmor, 1986, "Asset Substitution and the Agency Costs ofDebt," Journal of Banking and Finance, Vol. 10, pp. 391-399.[9] Jensen, Michael C. and William H. Meckling, 1976, "Theory of the Firm: ManagerialBehavior, Agency Costs and Ownership Structure," Journal of Financial Economics,Vol. 3, pp. 305-360.69[10] Kahn, Charles M. and Dilip Mookherjee, 1991, "Efficiency of Markets under MoralHazard with Side-trading," Faculty Working Paper No. 91-0157, College of Commerceand Business Administration, University of Illinois at Urbana-Champaign.[11] Matthews, Steven A., 1991, "Renegotiation of Sales Contracts under Moral Hazard,"Discussion Paper No. 950, Department of Economics, Northwestern University.[12] Myers, Stewart C., 1977, "Determinants of Corporate Borrowing," Journal of FinancialEconomics, Vol. 5, pp. 147-175.[13] Selten, Reinhard, 1975, "Reexamination of the Perfectness Concept of EquilibriumPoints in Extensive Games," International Journal of Game Theory, Vol. 4, pp. 25-55.[14] Smith, Clifford W., Jr. and Jerold B. Warner, 1979, "On Financial Contracting: AnAnalysis of Bond Covenants," Journal of Financial Economics, Vol. 7, pp. 117-161.[15] Wong, Kit Pong, 1992, "Debt, Collateral, and Renegotiation under Moral Hazard,"Economics Letters, Vol. 40, pp. 465-471.70I•I.C. curve1FBINiSBr 0^ irSB^iiFB^UlfPL PL PHFigure 1. The Second-Best Equilibrium. U9 and 119 denote respectively the bor-rower's indifference curve and the bank's break-even line given that the borrower undertakesproject 9 E {L, H}. The shaded region is the set of incentive compatible and individuallyrational contracts which induce the borrower to choose the low-risk project. Contract Emaximizes the borrower's expected payoff in the shaded region and lies above LI H. Hence,in the second-best equilibrium. the borrower chooses the low-risk project with contract E.71IIllrSB0 i(Im - ISB) irSB^irSB + jaw - ISB)PL^PHFigure 2. Incentives for Further Borrowing Given ISB < ill. CH (r,; ) and IIHdenote respectively the borrower's indifference curve and the bank's break-even line giventhat the borrower undertakes the high-risk project. rtH corresponds to the borrower's second-best expected payoff. 1TH is a line parallel to IT H. Given that contract E (the second-bestloan contract) is the initial loan, any additional loan (say contract H) in the lower shadedregion will give the borrower a new loan portfolio (say contract E + H) in the upper shadedregion. This additional loan is acceptable by the bank irrespective of the borrower's projectchoice, and the borrower is better off to offer it and switch to the high-risk project. Hence,the second-best outcome is not implementable in this case.PH PL72IIHISB/TB0^1I TB uS13^irTB jug ITB)PL^PL^PL^PHFigure 3. The Third-Best Equilibrium. 79 and He denote respectively the bor-rower's indifference curve and the bank's break-even line given that the borrower undertakesproject 9 E {L, H}. The shaded region is the set of incentive compatible and individuallyrational contracts which give the borrower no incentive to borrow further and choose thehigh-risk project. Contract E' maximizes the borrower's expected payoff in the shaded re-gion. The borrower's third-best expected payoff can be corresponded by either 72 or U H.Since ITH lies above rr, in the third-best equilibrium the borrower chooses the low-riskproject with contract E' as the optimal initial loan.73Chapter 4On the Determinants of Bank Interest Marginunder Capital Regulation andDeposit Insurance4.1 IntroductionUnlike the previous two chapters which examine the relative efficiency of relationshipbanking and price banking, this chapter is devoted to the study of the determinants of bankinterest margin (the difference between loan and deposit rates).The move away from relationship banking towards price banking is believed to create afrenzy of competition among banks to attract funds and to pass them onto borrowers—bothdomestically and internationally. It is argued that bank interest margins are squeezed to anextent which ignores the possibility of the world economic downturn such that banks wouldneed to prepare against the threat of loan losses. Robin Leigh-Pemberton, Governor of theBank of England, has recently commented on this problem: 1It can be argued, particularly with hindsight, that banks expanded their balance'See the report by Blanden (1993).74sheets too rapidly, notably by lending to risky businesses (including propertycompanies) at margins that did not properly reflect the risks that were beingtaken on. ... It is certainly true that there was a very pronounced reduction inbank margins during the 1980s. Moreover, although there have been many claimsto the contrary, margins have in fact not risen very much in the early 1990s.And, given their new freedom and a growing economy, it was not surprisingthat banks sought to increase the volume of their lending. ... There are severalquestions banks must ask themselves. Do they really pay attention to the lessonsof history for example the property crisis of the early 1970s? Did they reallymonitor the credit criteria which had served them well in the past? And werethe incentives given to loan officers really appropriate, or did they encouragenew business at the expense of sound business? ... There are lessons for bankers.Close attention to the control and pricing of risk is a theme that I have broughtto your attention before now, but I make no apology for doing so again. It isat the heart of the banker's professional life, and no amount of competition ormarketing strategy should ever divert us from it.What went wrong? Blanden (1993) argues that banks forgot the fundamental role ofborrowing and lending money. Furthermore, he feels that banks spent too much effortsearching for other sources of income and that they need to get back to basics. But itis not clear whether changes in margins are due to increased competition, lax regulatorysupervision, or greater economic risk. It is, therefore, the purpose of this chapter to studythe determinants of bank interest margins.To date, few theoretical models have been developed to analyze the determinants ofoptimal bank interest margins. Ho and Saunders (1981) utilize a framework similar to thebid-ask spread model of Stoll (1978). In their model, a bank is viewed as a risk-aversedealer who pays for deposits at a bid price and lends funds at an ask price. This modelingstrategy is further extended by McShane and Sharpe (1985) and Allen (1988). Lerner (1981),75the discussant of Ho and Saunders' paper, argues that by "considering banking to be only atrading activity, the insights that arise from reconizing that a production function exists maybe lost. (p. 601)" In addition, it is hard to address regulatory issues in this setting. Sincebanks are among the most heavily regulated firms in the economy, an important componentof the banking environment is eliminated.Recently, Zarruk (1989) and Zarruk and Madura (1992) go some distance in filling thisgap by adopting a production-based model of risk-averse banks under uncertainty as analternative approach. Their model is similar to traditional banking models. However, theysuffer from two shortcomings: First, bank risk aversion seems to be an awkward assumption.As argued by Santomero (1984) and Flannery (1989), given perfect capital markets theappropriate objective for a bank is to maximize its market value. Second, in their modelsbanks are not subject to limited liability. This excludes the moral hazard problem arisingfrom the mispriced deposit insurance system from their analysis. Thus, the deposit insurancesystem is nothing but a supplementary administrative cost to deposits in their framework.The purpose of this chapter is to present a production-based model of risk-neutral banksthat are subject to prevailing capital regulation and deposit insurance. Banks are protectedby limited liability and deposit insurance is interpreted as a put option (see Merton (1977)).Uncertainty in the model arises from random loan defaults and a stochastic short-termmoney market rates. The model is used to characterize interest margins and comparativestatics provide alternative explanations of a number of empirical observations concerning thebehavior of a bank's interest margin. The results indicate that a bank's interest margin in-creases with the short-term money market rate variability (in the sense of a mean-preservingspread). Empirical evidence consistent with this is documented by Ho and Saunders (1981)and McShane and Sharpe (1985). The results also show that a bank's interest margin is adecreasing function of its capital-to-deposits ratio and the flat-rate deposit insurance pre-mium. Indirect empirical support of the former result is provided by Furlong (1992), whoshows that bank loan growth rates and capital-to-assets ratio are positively related.76The remaining part of this chapter is organized as follows. Section 4.2 presents the basicstructure of the model. The solution of the model is characterized in section 4.3, and iscontrasted with the one in which deposit insurance is fairly priced. Section 4.4 develops thecomparative static properties of the model. The final section summarizes and concludes.4.2 The ModelConsider an insured risk-neutral bank which operates for one period. At the beginningof the period, the bank issues two types of liabilities: deposits, D, and equity capital, K.Regulators provide full deposit insurance and charge the bank a flat-rate premium, p > 0, perdollar of deposits. The deposit supply is perfectly elastic at the one-plus deposit rate RD . 2The amount of equity capital, K, is assumed to be fixed and is not part of the bank's decision,instead it is tied by regulation to satisfy the following capital requirement constraint:K > iD,^ (4.1)where t > 0 is the required minimum capital-to-deposits ratio.'Using the proceeds from its liabilities, the bank can acquire two kinds of assets: riskyloans, L, and short-term money market assets, C. It has monopoly access to a segment of therisky loan market and is a loan rate setter. 4 Loan demand is given by a downward-slopingfunction, L(RL), where RL is the one-plus loan rate and L'(RL) < 0. The initial balancesheet identity of the bank is given by'L(RL) C = D(1 — p) + K.^ (4.2)Uncertainty in the short-term money market rate is modeled through the use of an2 The qualitative results are robust to an upward-sloping deposit supply function.'The constant minimum capital-to-deposits ratio is adopted to simplify the exposition. None of thequalitative results will change if the ratio is an increasing function of the amount of risky loans held by thebank (i.e. risk-based), as modeled by Zarruk and Madura (1992).'Loan rate-setting behavior by banks is well documented by Slovin and Sushka (1983) and Hancock(1986).'The inclusion of legal reserve requirements does not alter the qualitative results.77V 1 f0additive random variable, E , having a known cumulative distribution function, F(c), over[c, with mean zero and variance a2 . Succinctly, the one-plus short-term money marketrate is given by R 7E, where R > RD is the expected one-plus rate and y > 0 is a shiftparameter affecting interest rate uncertainty. Since investment in short-term money marketassets should be relatively safe, throughout the paper I assume the following condition:R(1 — p > RD. (4.3)Condition (4.3) simply implies that should the bank invest all its equity capital, K, anddeposits (with the capital requirement constraint binding), K/it, less the deposit insurancepremium, pK/i, into short-term money market assets, it is expected to remain solvent.'At the end of the period, loans are repaid and the bank receives an uncertain gross loanrevenue, OG(RL ) = aRL L(RL ), where 0 is a random variable distributed over [0, 1] andis independent of "J. 7 Thus, for a given level of risky loans, L(RL ), the bank will receiveat most its total contractual loan repayments, G(RL ), and it may receive less, dependingon the uncertain state of nature, U. The bank also receives an uncertain payment from itsinvestment in short-term money market assets, (R -yE)C. For the sake of tractability, Iassume that 0 is uniformly distributed with unit density. It should be noted that most ofthe qualitative results also hold under more general distribution functions.Equity is a residual claim against the bank's assets. If the bank's end-of-period rev-enues exceed its deposit obligations, the remainder is distributed as liquidating dividends toshareholders. Otherwise, the bank defaults, and the regulators take control of the remainingassets and pay all deposit obligations. The expected value of the bank's equity is given by 8max [GG(RL) (R 7c)C — RDD, 0] dO dF(e)1= —2 G( RL ) R[D K — L(RL)] PRD — RDDl(4.4)8 In practice, IC > p and thus this assumption is satisfied.'This formulation of uncertain loan defaults is taken from Taggart and Greenbaum (1977).8The inclusion of administrative costs of loans and deposits does not affect the qualitative results.78f 0 max [RED — OG(RL) —^76)[D(1 — p) K — L(RL)1, 0] dO dF(c),where I have used the fact that max [X — Y, 0] = X —Y + max [Y — X, 0] and the balancesheet identity, (4.2). The sum of the first two terms of (4.4) is the total expected value of thebank. The third term is the deposit insurance premium paid to the regulators. The fourthterm is the payoff to depositors. The fifth term is equal to the expected payout to depositorsby the regulators when the bank's assets fall short of the claims of depositors. This termcorresponds to Merton's (1977) put option value of deposit insurance.The objective and decisions of the bank are as follows. At the beginning of the period,the bank chooses the one-plus loan rate, RL , and the amount of deposits, D, (before anyuncertainty is resolved) to maximize its equity claim, (4.4), subject to the capital requirementconstraint, (4.1).4.3 The SolutionLet r represent the Lagrange function, and A > 0 denote the Lagrange multiplier asso-ciated with the capital requirement constraint, (4.1). The Kuhn-Tucker conditions areRL^(c)Or = -1-2-q(KL) — RIAR*L ) — cr [OG'(R*L ) — (R +^)1 de dF(€) = 0, (,)[(R -yc)(1 — p) — RD ] dO dF(€) — A* = 0,OD Orwhere (RI, D*, X') is the optimal interior solution, and O'(€) determines the maximum loanrevenue given E below which the bank is insolvent, i.e.,1 91‘(c) G(Rfl IRDD* (R + 7c)[D*(1 — p) K —In order to study the impact of the capital regulation, throughout the paper the capitalrequirement constraint is assumed to be binding. Thus, we have A* > 0 and D* = KIK. TheOA = K — K,D* > 0,OrA*— = 0OA79first-order conditions then reduce toVRL = 1 ( fin -^7 IOM(Kj) —^[OG'(ffj) — (R -yf)L 1 (RI,)] dO dF(E) = 0, (4.5)andI7(0 [(R -y €) (1 — p) — RD ] dO dF(e) > 0,0where01(e) = G(RL)K{ RD^(R 'Y E )^( 1 — p) + K — L(RL)1}.The first-order condition (4.5) implies that the bank sets the loan rate up to the point wherethe expected marginal appreciation of the bank's value exactly offsets the expected marginaldepreciation of the deposit insurance put option value.To gain more insight into the above solution, it is contrasted with the one in whichdeposit insurance is actuarially fair. In this case, the deposit insurance premium (the thirdterm of (4.4)) is always set equal to the value of the deposit insurance subsidy (the fifth termof (4.4)). Thus, the expected value of the bank's equity is simply equal to the total expectedbank value net of the payoff to depositors:1V = —2 G(RL ) R[D K — L(RL )]— RD D.The bank's objective is to maximize its equity claim, (4.8), subject to the capital requirementconstraint, (4.1). The first-order condition with respect to RL is given by1—2G' (Rr ) — ROR*L*) = 0 , (4.9)where RL is the optimal loan rate with fairly priced deposit insurance. Comparing (4.5)and (4.9) yields1—2 G' (R7,) — RIAR*L ) > —2 C(RL**) — RIARr) = 0.By the second-order condition of the latter optimization problem, it follows immediately thatRL < RL**. This implies that the bank will grant more risky loans when deposit insurance ismispriced. The difference between RL and RL, AWL = RL - RL , represents a subsidy from(4. 6)(4.7)( 4 . 8 )80deposit insurance. It is noteworthy mentioning that given actuarially fair deposit insurancethe bank's optimal loan rate depends only on the loan demand function and the expectedshort-term money market rate.The insolvency risk of the bank is given byP(RL)TI1(6)dO dF(E).Differentiating P(RL) with respect to RL yieldsP' ( RL ) maRLi(c) dF(c)le ORL )RR + ,70(1 p h.,) RD}^RL0  dF(c)KG(RL) 2K^(RL) [RoIKG(R.02^p + K) RD] R1L T 91(0 dF(c)< 0.The insolvency risk is a decreasing function of the loan rate. As a result, the presence of themispriced deposit insurance induces the bank to increase its portfolio risk.Now, I proceed to decompose the optimal bank interst margin. Suppose that all loansare safe (i.e., 0 = 1 with probability one) and deposit insurance is fairly priced. In this case,the expected value of the bank's equity value isV = G(RL) + R[D + K — L(RL)] — RDD.^(4.10)The bank's objective is to maximize its equity claim, (4.10), subject to the capital require-ment constraint, (4.1). The first-order condition yieldsOR*L**) — RIAR*L**) = 0,where RL  is the optimal loan rate with fairly priced deposit insurance and without loandefault risk. Since Gi(R*L**) < 0, it must be the case that1^ 1(-1' (R17) — RIAR*L**) > ( WE') — RIARr) = —2 OR*L ') — ROR2:k) = 0.81Thus, it follows from the second-order condition that R L*** < R. The difference betweenfrz, and kr , AR1 = kz, - kr , represents a premium for loan default risk. As a result,the optimal bank interest margin can be decomposed as— RD =- Rr * - RD - ARIL +In words, the optimal bank interest margin is equal to the interest spread under fairly priceddeposit insurance and no loan default risk minus a component due to mispriced depositinsurance and plus a component due to loan default.4.4 Comparative StaticsHaving examined the solution to the bank's optimization problem, I am ready to analysethe comparative statics of the bank's optimal interest margin. The following observationswill prove to be useful in the comparative statics analysis.Rewritting (4.5) asf r(o1^VRL = I^[OG/ (R*L) - (R + -yE)// (Ha dO dF(c) = 0,implies thatG'(1:11,` ) < 0.Denote M(c) = OT(c)C(R*L ) — (R 7c).L' (RI). Then, substituting (4.7) and rearrangingterms yieldsM(c) = L(R*L) (R + 7E) KG'(R*L) RR + 7€)(1 — P K) — RD].^(4.11)^RL^K,G(111,)Note that the effect of a change in any parameter, x, of the model on the bank's optimalloan rate (and thus the bank's interest margin) is given byd^I7RLx dx^VRLRL82By the second-order condition, VRL RL < 0 and the sign of dR*L /dx is the same as that ofVRL ,,. The comparative static analysis can thus be carried out by considering the sign ofFirst consider the effect of a change in the capital-to-deposits ratio, K. Partially differ-entiating (4.7) with respect to , yieldsOK^K2G(R*L)[(R -ye)(1 — p) — RD ].From (4.5), (4.11) and (4.12), we have— Jr M(6) aoT(E) dF(e)OKVRL(4.12)K 2 G'( fin { [Ro^RD] [Rop to RD] 4. ,720.2(1 - p)(1 - p K)}tc3G(R*L)2*2 {R[R(1 — p) — RD ] + o-2 (1 — p)}RLIf R(1 — p) — RD > 0, it is clear that VRL ,. < 0 and thus dR*L /dh-, < 0. On the other hand, ifR(1 — p) — RD < 0, it is not difficult to show that VR L , < 0 when720.2 ^R(1 — p) — RD R.1 — pSubstituting (4.7) into the left hand side of (4.6) and rearranging terms yields[R(1 — p) — RD] [l — 7 07(c) dF(e)]^C* ^2 2G(Rt) 7 °- (1 — 1))1 C* G(Rn[R(1 p) RD ][1?*1, L R*L) RD + RC 1 + G(Iti) 72 a2 (1 — p). (4.14)If R(1 — p) — RD < 0, then for (4.14) to be positive it must be the case thaty 2 0 2 > R(1 — p) — RD I RI, L(R si — RD K ic 1 — p^L^c*which implies (4.13). Thus, we can stateProposition 4.1.^An increase in the capital- to -deposits ratio decreases the bank'soptimal interest margin.(4.13)83K ( R2 + 7 2 2K, R*L2To understand the intuition behind this result, note that, ceteris paribus, an increasein K, lowers the critical bankruptcy state, 0 1*(€), only for sufficiently bad realizations of E,and raises the critical bankruptcy state otherwise. This implies that the deposit insurancesubsidy is more valuable on the margin to the bank facing a stiffer capital requirementThus, the distortion inducing the bank to take on inordinate risky loans is fortified. Giventhat the loan market is imperfect, the bank reduces the size of its interest margin to attractmore loans. 9Beginning in 1990, the stiffening of capital regulation is argued to have curtailed banklending, and, thereby, contributed to a credit crunch. Furlong (1992) does find empiricalevidence that suggests the stiffening of capital regulation in the 1990s, but contrary to theaforementioned view, his analysis shows a positive relationship between bank loan growthrates and capital-to-assets ratio. Thus, proposition 4.1 provides a theoretical rationale for hisobservation and suggests that the so-called credit crunch in the 1990s is not due to changesin capital regulation.Now consider the effect of changes in the flat-rate deposit insurance premium, p. Partiallydifferentiating (4.7) with respect to p yieldsa°i(c) ^K  (R 7c).ap^KG(R*L)From (4.5), (4.11) and (4.15), I haveao ac(E) — m(e) p  dF(c)K 2 G'(Rn {R[R(i p K) RD} + 720-2(1^K)}tc, 2 G(R) 2< 0.VRL P(4.15)9 Zarruk and Madura (1992) obtain the same result in their model. However, their derivation is notcorrect because they ignore the term L(RL ) on the upper limit of their intergal when they take the first-order condition. By Leibniz's rule, this term will affect the first derivative. Mullins and Pyle (forthcoming)derive similar result using simulation.84Proposition 4.2.^An increase in the flat-rate deposit insurance premium decreasesthe bank's optimal interest margin.The intuition underlying this result follows a similar argument as in the case of a change inK. Increases in the cost of deposit insurance increase the critical bankruptcy state, 0 1*(0, forall realizations of "e, other things being equal. This encourages the bank to shift investmentsto risky loans, and thus the optimal loan rate drops.Next, I examine the impact of mean-preserving spread changes in the short-term moneymarket rate uncertainty, 7. Partially differentiating (4.7) with respect to -y yieldsFrom (4.5), (4.11) and (4.16), I haveDCVO^EC*0-y^GIRL)(4.16)fl^ T^a01VRL = I T jevo — Cl/ 1(Rn dO dF(e) — f M(e)  a,),(E) dF(e)f-ya 2 C* IL(R*L )^KG'(R*L) _^. _ , ,, I(1 p + K) L (RL)=G(RnL R*L,^ KG(R*L)> 0.Proposition 4.3. A mean-preserving spread increase in the short-term money marketrate uncertainty increases the bank's optimal interest margin.Intuitively, increased uncertainty about the short-term money market rate induces thebank to shift investments to short-term money market assets. Doing so allows the bankto further exploit the mispriced deposit insurance. Thus, the optimal loan rate goes upto reduce loan demand. Ho and Saunders (1981) and McShane and Sharpe (1985) presentstrong empirical evidence that there is a positive relationship between bank interest marginsand interest rate risk. Thus, proposition 4.3 provides a theoretical rationale for this empiricalobservation.854.5 ConclusionsIn this chapter I combine the option view of bank value maximization with the deter-minants of optimal bank interest margins. The results offer alternative explanations of theobserved behavior of bank interest margins, based on the fact that deposit insurance is nottruely risk-adjusted (i.e. actuarially fair). Specifically, the results rationalize the observa-tions that the bank interest margin increases with short-term money market rate variability.Further, the results have implications for bank regulation: Propositions 4.1 and 4.2 showthat either a stiffer capital requirement or a higher deposit insurance premium reduces thebank interest margin. This directly weakens the ability of the bank to sustain loan lossesand thus the soundness of the bank. In the restructuring of the deposit insurance systemand capital regulation, these effects must be taken into consideration when making a policyprescription.86References[1] Allen, Linda, 1988, "The Determinants of Bank Interest Margins: A Note," Journal ofFinancial and Quantitative Analysis, Vol. 23, pp. 231-235.[2] Blanden, Michael, 1993, "Bank Lending: Even More Risky," The Banker, Vol. 143,February pp. 18-20.[3] Flannery, Mark, 1989, "Capital Regulation and Insured Banks' Choice of IndividualLoan Default Risks," Journal of Monetary Economics, Vol. 24, pp. 235-258.[4] Furlong, Frederick T., 1988, "Changes in Bank Risk-Taking," Economic Review, FederalReserve Bank of San Francisco, pp. 45-56.[5] Hancock, Diana, 1986, "A Model of the Financial Firm with Imperfect Asset and De-posit Elasticities," Journal of Banking and Finance, Vol. 10, pp. 37-54.[6] Ho, Thomas S. Y. and Anthony Saunders, 1981, "The Determinants of Bank Inter-est Margins: Theory and Empirical Evidence," Journal of Financial and QuantitativeAnalysis, Vol. 16, pp. 581-600.[7] Lerner, Eugene M., 1981, "Discussion: The Determinants of Bank Interest Margins:Theory and Empirical Evidence," Journal of Financial and Quantitative Analysis, Vol.16, pp. 601-602.[8] Merton, Robert C., 1977, "An Analytical Derivation of the Cost of Deposit Insuranceand Loan Guarantees," Journal of Banking and Finance, Vol. 1, pp. 3-11.[9] McShane, R. W. and I. G. Sharpe, 1985, "A Time Series/Cross Section Analysis of theDeterminants of Australian Trading Bank Loan/Deposit Interest Margins: 1962-1981,"Journal of Banking and Finance, Vol. 9, pp. 115-136.87[10] Mullins, Helena M. and David H. Pyle, 1993, "Liquidation Costs and Risk-Based BankCapital," Finance Working Paper No. 225, Walter A. Haas School of Business, Univer-sity of California at Berkeley, forthcoming in Journal of Banking and Finance.[11] Santomero, Anthony M., 1984, "Modelling the Banking Firm," Journal of Money,Credit, and Banking, Vol. 16, pp. 576-712.[12] Slovin, Myron B. and Marie Elizabeth Sushka, 1983, "A Model of the Commercial LoanRate," Journal of Finance, Vol. 38, pp. 1583-1596.[13] Stoll, Han R., 1978, "The Supply of Dealer Services in Security Markets," Journal ofFinance, Vol. 33, pp. 1133-1151.[14] Taggart, Robert A. Jr. and Stuart I. Greenbaum, 1978, "Bank Capital and PublicRegulation," Journal of Money, Credit, and Banking, Vol. 10, pp. 158-169.[15] Zarruk, Emilio R., 1988, "Bank Spread with Uncertain Deposit Level and Risk Aver-sion," Journal of Banking and Finance, Vol. 13, pp. 797-810.[16] Zarruk, Emilio R. and Jeff Madura, 1992, "Optimal Bank Interest Margin under CapitalRegulation and Deposit Insurance," Journal of Financial and Quantitative Analysis,Vol. 27, pp. 143-149.88

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