Incentive-Based Lending Capacity, Competition and Regulation in Banking

Chiesa, Gabriella Working Paper Incentive-Based Lending Capacity, Competition and Regulation in Banking Quaderni - Working Paper DSE, No. 397 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Chiesa, Gabriella (2001) : Incentive-Based Lending Capacity, Competition and Regulation in Banking, Quaderni - Working Paper DSE, No. 397, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4904 This Version is available at: https://hdl.handle.net/10419/159238 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ Incentive-BasedLendingCapacity,Competition and RegulationinBanking¤ GabriellaChiesay DepartmentofEconomics UniversityofBologna December11,2000 Abstract Thispaperstudiesmoralhazardin bankingduetodelegatedmonitoringinanenvironmentofaggregateriskand examinesitsimplicationsforcreditmarketequilibriumand regulation, inamodelwhere banksareprice competitorsforloansand deposits.Itprovidesarationaleforanincentive-basedlendingcapacitypositively linkedtothe bank’scapitaland pro…tmargin,foranoligopolisticmarketstructurewhereverbankshavemarketpower,and forcapitalrequirements. Social-welfare-maximizingcapitalrequirementsareloweredinrecessions,arehigherthemorefragmentedthebankingsector,and areincreasedwhenanti-competitivemeasuresareremoved.Inequilibrium banksearnexcessivepro…tsand creditmayberationed. JEL-Classi…cation:D82,G28,L13 Keywords:bank-moralhazard,capitalrequirements,competition ¤IamgratefultoJohn Mooreforhishelpsince the earlystageofthisproject.For theirhelpfulcomments,IthankSudiptoBhattacharya,VincenzoDenicolo’, BrunoParigi, and especiallyananonymousreferee and AnjanThakor(the editor).Financialsupport byCNRand MURST40%isacknowledged. yA¢liation:DipartimentodiScienze Economiche,PiazzaScaravilli 2,40126 Bologna. Tel. +051-2098154;fax:+051-2098040;e-mail: [email protected] 1 1Introduction Since theremovalofregulatoryrestrictionson branching,depositrates, scopeof business and so oninthelateseventiesand early eightiesinorder tofacilitateincreasedinterbankcompetition,therehasbeena generalmove towardsre-regulationof banking onaprudentialbasis,primarily capital requirements.Interestingly, thisgeneraltrend hasbeeninterruptedonly duringrecessionswhenregulatorshavebeenmorelenient,justifyingthis approachonthegroundsthatcapitalrequirements should beindexedto thebusiness cycle(see Tirole1994).Thereisasuggestionthatperhaps suchleniencymayhaveundesirable consequences,basedonrecentempirical evidence thatsuggeststhat…nancialmarketliberalizationmaycontribute tobankcrises(see Demirguc-Kuntand Detrajache1998). Thesedevelopmentsraisethefollowingimportantquestions.Should prudentialregulation berelatedtotheintensityof bankingcompetition (marketstructure)?Shoulditbelinkedtothebusinesscycle,withregulators actingmoreleniently duringrecessions?And,morefundamentally, what determinestheintensityof bankingcompetition,and shouldthebanking systemberegulatedatall? Three strandsoftheliteraturetrytoclarifythesepolicy issuesand to provideargumentsinsupportof policymakers’attitude. Onestrand focuses onwhethercapitalrequirementsareane¤ectivetoolforlimitingtherisk onanassetportfoliowhosereturnistakenasexogenously giveninapartial equilibriumframework(Sharpe1978; KarakenandWallace1978; Kohenand Santomero 1980;Bhattacharya 1982;Furlong and Keeley1989; Keeleyand Furlong 1990;Rochet1992).Thebasicinsightisthatcapitalrequirements on banksattenuatemoralhazard,reducingbanks’incentivetotake excessive risk.1Asecond strand analyzesthemacroeconomicimplicationsofcapital 1Capitalrequirementsmaybe costly,though.Thishasbeenshown byBesanko and Kanatas(1996)forabankwhosetotalassetsaregivenand capitalrequirementsdictate the…nancingmixbetween debt–deposits– and outside equity.The costsofcapitalrequirementsaretheagencycostsofoutside equityalaJensenand Meckling(1976).Moreover, 2 requirements,showingthat theymayreinforce macroeconomic‡uctuations (see Blumand Hellwig 1995;Thakor1996).Implicitly, thisprovidesan argumentforlinkingcapitalrequirementstothebusiness cycle.However, thesemodelsdonotexplainwhycapitalrequirementsarethereinthe…rst place,orwhat the costsofliftingthem mightbe.Athirdstrand notesthat bankrentsactasamitigatingfactoragainstrisk-takingbymakingitmore costly forabanktofail (Bhattacharya 1982;Chan,Greenbaumand Thakor 1992;Hellman,Murdockand Stiglitz2000).2This suggeststhatcapital requirementsand pro…tscan beviewedas substitutesincontrollingmoral hazardin banking. Theimplicitassumption underlying all three strandsisthatfully diversi…edassetportfoliosarenotfeasible.Indeed,accordingto a fourthstrand, delegated-monitoringtheory(Diamond 1984;Ramakrishnanand Thakor 1984;Diamond 1991;Hellwig 1991;Bhattacharya and Thakor1993),the constructionofafully diversi…ed portfoliowhosereturniscertaineliminates moralhazardentirely and withit,all need for regulation(see Diamond 1984; DewatripontandTirole1994).Toreintroducemoralhazard,Holmstromand Tirole(1997)excludediversi…cation byallowingforaggregaterisk–project returnsare correlated.Thiscreatesanincentiveforbankstotakebadrisk and providesaroleforbankcapital inattenuatingmoralhazard. Fromthesefourstrandsofliteraturewelearnthatsystematicriskmay leadtomoralhazardand thatcapitalrequirementsand pro…tsplayarolein controllingit,althoughtheformermaybeharmfully pro-cyclical. Butwedo notlearnwhythebankingsystemshould beregulatedatall asopposedto beingdisciplined bymarketforces,orhowregulationshould bedesignedin termsofitsrelationshiptointerbankcompetitionand thebusiness cycle.To address theseunansweredquestions,thispaperdevelopsamodel inwhich theintensityofcompetition,theroleofcapitalrequirements,and bank pro…tsareallendogenouslydetermined.Moreover,thee¤ectsofthebusiness cycle,of banking-sectorconcentrationand ofstructuralregulation(entry capital iscostlybecauseoftheliquiditycostsofequity versusdemand deposits(Gorton and Winton1995). 2Keeley(1990)…ndsadirectrelationship betweentheUSreformsthatincreasedcompetitionand theincreaseinthenumberofbankfailuresinthe1980s. 3 barriersand marketliberalization)onall thesevariablesareanalyzed. Abankactsasadelegatedmonitorinanenvironmentofaggregaterisk; thatis,projectreturnsare correlatedasinHolmstromand Tirole(1997).3 Banksborrowfrominvestorsand lend to…rms,monitoringtheminequilibrium,at termsthatresultfrominterbankprice competition.4Theyacton behalfofshareholders(insiders)whose equityholdingsconstitutethebank’s capital; additional inside capital istoo costly toraise(see Smith1986 fora surveyofevidence). Theessentialingredientofthemodelismoralhazardinbanking: insiders maygain by…nancingprivately pro…tableprojectsthathavenegativesocial value.Indeed,bankmonitoringcostscan beinterpretedastheopportunity coststoinsidersof notengagingininsiderlending,orcolludingwith borrowerstoundertakehigh-risk, high-return projectsor,moregenerally, those bringingprivatebene…ts.Theresultsareasfollows.Inequilibrium,banks engageinmonitoring:theamountoflendingthatabankundertakesis subject to a ceiling—thebank’sincentive-basedlendingcapacity—thatis positively relatedtoitscapitaland itsendogenouspro…tmargin(thespread betweenlending and borrowingrates).Acorollary isthatevenif bankscompetein pricesand thereisnoproductdi¤erentiation,bankcompetitionis imperfect:competition forloansisBertrand-Edgeworthwithcapacityconstraints,the constraintsresultingfromincentiveproblems.Theincentivebasedlendingcapacityexceedsthebank’scapital, i.e.thereis scopefor outside…nancing and hence for…nancial intermediation.However, inthe absence ofregulation(bymarketdiscipline) thereisano-intermediation equilibrium,unless investorsactincoordinated fashionandbanks’pro…t marginsarepublicinformation.Bycontrast,underthesameinformational 3FollowingHellwig(1991),Bhattacharya and Thakor(1993),Dewatripontand Tirole (1994)and Freixasand Rochet (1997),weinterpret”monitoring” broadly.Itcanconsist of: i)screeningprojectsinanadverse-selectionenvironment (asinRamakrishnanand Thakor1984,Broecker1990);orii)preventing opportunisticbehaviourbytheborrower duringtherealizationoftheproject (asinDiamond 1991,Holmstromand Tirole1997). 4Wethusexplicitlymodelmonitoringinstitutionsasintermediariesthatcompetein pricesforloansand deposits.Thisisincontrast toHolmstromand Tirole(1997),where monitoringinstitutionsareprice takers, inthat therateofreturnoncapital isdetermined bymarketclearing. 4 constraints,optimalcapitalrequirementspermitanintermediationequilibriuminwhich banksraiseoutside…nancing and lend inexcess oftheir capital. Thus,capitalrequirementsareasolution notonly tothelackofcoordinationamongdispersedinvestorsbutalsototheunobservabilityofsuch importantfeaturesascontractualtermswith borrowers,and hence pro…t margins.Capitalrequirementsworkby limitingbanks’scaleof business. Theyrestrict thebank’slendingnot toexceeditsincentive-basedlending capacity, inordertoretainthebank’sincentivetomonitor.Rational, albeit unprotected, investorsarethenwillingtofund banks.The equilibriumvaluesofoptimalcapitalrequirementsand banks’pro…tmarginsareinversely related.Social-welfare-maximizingcapitalrequirementsarerelatedtothe business cycle, viathelinkbetweenrecession, insolvenciesand bankcapital, and regulatorsactmoreleniently duringrecessions.Capitalrequirements areraisedwhenentrybarriersareremovedand arelowerthemore concentratedisthebankingsector.Inotherwords,theydepend ontheintensity of bankingcompetition.Finally, the equilibriumisaconstrainedoptimum; itdi¤ersfromthe…rst-bestinthatbanksearnexcessivepro…tsand credit mayberationed. Ouranalysistherefore complementsWinton(1995)and Yanelle(1997),whorejectperfectcompetitioninDiamond’s(1984)environmentofcostly observabilityof projectreturns,whereportfoliodiversi…cationreducesmonitoring(auditing)costsbutintroducesincreasingreturns toscale.5Winton(1995)allowsfora…nite economywherediversi…cation islimited.Heshowsthatbankcapitalizationreplacesorcomplementsdiversi…cationinreducingdefaultprobabilityand thereforelowering auditing costs,and that the equilibriumbankingsectormayhaveseveralbanks.In thispaper,projectreturnsareobservableatnocost,but thereisbankmoral hazard.Capitalizationand pro…tsjointly determinethebank’sriskchoice, and, inequilibrium,severalbanksareactive,butcapitalregulationmaybe essentialtotheviabilityofintermediation. Therestofthepaperisorganizedasfollows.Section2presentsthe model. Section3derivesthebank’sincentive-basedlendingcapacityand 5Indeed,with uncorrelated projectreturns,thelargerthenumberofloans,themore diversi…edthe creditportfolio,thelowertheprobabilityofdefaultand thelowerauditing costs;forafullydiversi…ed bank,the costsarenil. 5 characterizesthedistortionsthatmotivate capitalrequirements.Section 4derives social-welfare-maximizingbank-capitalregulationand the creditmarketequilibrium(borrowing and lendingratesand lendingvolumes)as afunctionofstructuralparameters(aggregatebankcapital, aggregateloan demand,numberof banks).Section5delineatesthe empirical implications ofthemodel, and Section6concludes. 2 TheModel Weconsideraone-periodcreditmarketconsistingoftotalmeasureMofnonatomistic entrepreneursor…rms,totalmeasureIof non-atomisticinvestors, and nbanks(indexed byi=1,2..n),withn¸2.Agentsarerisk-neutral, havelimitedliability, and maximize theirend-of-periodexpectedwealth. Eachentrepreneurcan undertakeaninvestmentproject thatrequiresone unitofresources.Eachinvestorisendowedwithoneunitofresources,which caneitherbestoredat thegross returnRdordepositedinabank. Investors arelargeinnumber(I>M).Abanktakesdepositsfrominvestorsandlends toentrepreneurs.Itactson behalfofits shareholders, insiders,whose equity holdingsconstitutethebank’sendowmentofinside capital. Thiscapitalcan be eitherlent toentrepreneursorstoredat thegross rateRd.Banksare endowedwithaggregate capitalA>0thatisdistributedsymmetrically across banks.Thus,A nisanindexofconcentrationofthebankingsector. 2.1ProjectTechnologyand Monitoring Bank lendingisintheformof project…nancing.Aprojectrequiresoneunit ofresourcesat thebeginning oftheperiodand deliversarandomreturnat the end.Therealizationofthisreturn dependsonthemacro-staterealizationat the end of periodµ2©µ;µª,whereµoccurswith probabilityp,and ontheproject type.A good project (typeg)deliversareturnofxboth inthegoodstateµand inthebadstateµ;abad project (typeb)delivers areturnofxinµand ofzeroinµ.Returnsareobservableand veri…able. Whetherabank-funded projectisoftypegdependsonthebank’schoice ofactionat thebeginning of perioda2fm;?g,wheremindicates”monitor”,and ?”notmonitor”:amonitored projectistypeg, i.e. itsucceeds 6 whateverthemacro-staterealization;an unmonitored projectistypeb, i.e. itsucceedsonly inthefavorablemacro-staterealizationµ: Monitoringmayconsistintheprovisionofservicestailoredtothe…rm, oraconstraintonthe entrepreneur’schoice of project throughappropriate debtcovenants,whoseful…llmentisthenmonitored.Thiswould bethe caseifatypebprojecto¤ered privatebene…tslarge enoughtoleadthe entrepreneurtopreferthebad project.Inthiscasenon-monitoring, i.e.?, impliesthat the entrepreneurwill undertakethetypebproject.6 BankmonitoringcostsF>0perproject.Thisisanon-pecuniarye¤ort cost tothebank. Fmayalsobeinterpretedastheopportunitycostof eschewinginsiderlending,forgoingthepotentialprivatebene…tsofatypeb projectincollusionwiththeborrower. AssumptionA1 (an unmonitored projecthasnegativenetpresentvalue:) px<Rd(1) AssumptionA2 (amonitored projecthaspositivenetpresentvalue:) x>Rd+F(2) AssumptionA3 Thebank’schoice ofactionisunobservable. Ourassumptionsimply that:(i)bankmonitoringincreasesloanvalue and isunobservableto outsidersand costly tothebank; (ii) theaverage loanreturn,conditionalupon non-monitoring, isuncertain(itsrealization dependsonthemacro-staterealization).The…rstassumptionisanecessary condition foraproblemof delegationorbankmoralhazardvis-a-visoutsiders(investors);thesecond assumptionensuresthat thisproblemcannot besolvedthrough diversi…cation. 6Monitoringmayalsoconsistintesting ofanentrepreneur’screditworthiness (ata cost) inanadverse-selectionenvironmentwherea given percentageofthe entrepreneur population havetypegprojectsand therest typebprojects,and thetestresultisasuccess orafailuredepending on project type.Inthiscase,Fisthe costofperformingatest, divided bytheprobabilityofanentrepreneur’sbeingendowedwithatypegproject. 7 Inourframework, abank’soutside equityand debtareperfectly equivalent.Therelevantdi¤erence isbetweenoutsideand inside…nancing,where thelatterisbounded byA n. Weassumethatoutside…nancingisintheform of deposits. 2.2TheCreditGame Thegameproceedsasfollows. Atdate0,thesocial-welfare-maximizingregulatorpublicly announces banks’capitalrequirements.Thisistheregulationstage.Atdate1,each banki=1;2::::no¤ersa gross rateRiatwhichitiswillingtolend and a gross rateRdiatwhichitiswillingtoborrow.Thisisthebankcompetition stage.Finally, atdate2,bankschoosewhethertomonitortheirloan portfolios.7Thisisthemonitoringstage.8Figure1summarizesthis sequence of events. FIGURE1 ABOUTHERE Theoutcomeofthebankcompetitionstagedetermines:i’slendingrate, Ri;thedepositrate,Rdi;thevolumeoflending,Li;and deposits,Di,where Lisatis…esi’s‡owof fundsconstraint,Li·Di+A n. Asusual, thegameis solved backwards. Weprovidebelow,asabenchmark, the…rstbestsolutionobtainedintheabsence ofanydelegation problem, i.e. if banks’choicesofactionwereobservableand contractibleor, equivalently, if bankshad unlimitedliability. 2.3FirstBest LetR¤denotethelendingratethatmakesitpossibletorecoup, inexpected value,theresourcesinvestedinamonitored project: 7Arguably,abankcouldmonitoronlypartofitsloans.However,suchastrategyis strictlydominatedeitherbymonitoring all loansornone.Thiswill be clari…edin Section 3(see footnote9). 8Since thebank’sactionisunobservable, itstimingisirrelevant;thusa gameinwhich thebankmakesitsmonitoringchoice (contingentonitslending and borrowingratesand lendingvolume)at theoutsetisperfectlyequivalent totheonede…ned here. 8 increasewhenaggregatebankcapitaland/orthenumberof banksincreases relativetotheaggregatedemand forlending and bankingcompetitionis moreintense;(iii) the equilibriumisaconstrainedoptimum.Itdi¤ersfrom the…rst-bestinthatbanksearnexcessivepro…tsand creditmayberationed. Weshall nowderivetheseresultsin formalterms. Weassumethatcontractualtermsbetweenthebankand itsborrowers(namely, thelending ratesRi)are eitherunobservableortoo costly to observe.Thusabank’s capitalrequirementcannotbe conditionedonitspro…tmargin;sotheregulatorsetsan unconditionalcapitalrequirementc,and each banki=1;:::n cannotlend inexcess ofA n c, i.e.: Li·A n c;i=1;2:::::n: Thisallowsusnotonly toderiveanexplicitcredit-market-equilibriumsolution,butalsotoextractclear-cutpredictionson unconditionalcapital requirementsthatare consistentwithwhatisactually observed. An unmonitored projecthasnegativenetpresentvalue,and amonitored one,positive.Tomaximize socialwelfare,therefore,theprojectsthatare undertakenmustbemonitored.ByProposition3, itfollowsthattheoptimal capitalrequirement,denoted byc¤,necessarily satis…esbankmonitoring incentive constraints: c¤¸c(Ri;Rd);i=1;2::::n; wherec(Ri;Rd)isi0sminimum(incentive-based)capitalrequirementas given by(9);i0slendingrateRiistheoutcomeoflendingcompetition atdate1,givenstructuralparametersand the capitalrequirementsetat date0. Underoursimplifying assumptionofrectangulardemand forloans,maximizingsocialwelfareistantamount tomaximizingthemeasureof projects undertakenor,equivalently, aggregatelending,subject tobankmonitoring incentive constraints: max c n X i=1Li(12) s:t: 15 n P i=1Li=min(M;A c) c¸c(Ri;Rd);i=1;2::::n whereA cistheoverall amountoflendingthatbankscan undertakegiventhe capitalrequirementc,and Misthetotalmeasureof projects,orequivalently theaggregatedemand forloans.However,wewishour resultsto apply, at leastqualitatively, tothemoregeneralcaseof downward-sloping aggregate demand forlending.12 Wethusassumethat theregulator’sobjectiveisto maximize the expectedsurplusofreal investmentactivitythataccruesto …rms,orequivalently tominimizebankrents subject tobankmonitoring incentive constraints.Theregulator’sproblemthenamountstochoosingc so astomaximize theoverall lendingthatbankscan undertake,subject to bankmonitoringincentive constraints: max cA c(13) s:t: c¸c(Ri;Rd);i=1;2::::n(14) Notethat theoptimalcthatsolvesproblem(13)isasolutiontotheproblem ofsocial-welfaremaximization(12);thiscisthelowestwithintheinterval ofvaluesthatmaximize socialwelfare.Solvingproblem(13)amountsto minimizingthe capitalrequirementcsubject tothe constraint thateach banki=1;:::n…ndsitincentive-compatibletomonitor(condition(14)). Indeed,thelowerthe capitalrequirement,thegreatertheamountoflending thatbankscan undertake,themoreintensethe competitioninlending,the lowerthelendingratesand hence thehigheristhe expectedsurplusof real investmentactivitythataccruesto…rms,providedthatcsatis…esthe incentive constraint(14):Butifcviolatedinequality(14),thenregulation would fail toconstrainlendingnot toexceedthebank’sincentive-based capacity: inequilibriumdepositorswould notfund banks(byProposition3 and itsCorollary3). 12Thiswould bethe caseifundertakingaprojectrequiredentrepreneuriale¤ortand its (non-pecuniary)costdi¤eredacross entrepreneursaccordingtosomedistributionfunction overthetotalmeasureof…rms–entrepreneurs. 16 4.1Optimal Capital Requirementand CreditMarket Equilibrium Wesolveforanequilibriumofcompetitioninlending and fortheoptimalc. Thisisthelowestcthatsatis…esinequality(14),giventhatdepositors,who learncfromtheregulator’spublicannouncementatdate0,correctly infer that theincentive constraint(14)is satis…edand thatbankswill monitor. Giventhisrationalinference,depositorsarewillingtofund banks,andbanks remuneratedepositsat therateRd(byProposition3). Structuralparameters,A;n;andM,andthecapitalrequirementcjointly determinethelendingcompetitionregime,which dependsonwhichofthe followingmutually exclusive conditionsis satis…ed: nA n c·M(15) (n¡1)A n c¸M(16) M n¡1>A n c>M n(17) Ifinequality(15)holds,thentheoverall amountoflendingthatbanks can undertakedoesnotexceedaggregatedemand,and the equilibriumstrategyofabank isto o¤erthemonopoly ratex.Bycontrast, ifinequality(16) holds,thenabank’scompetitorscancoverthewholemarket,competitionin lendingisunrestricted,and inequilibriumthe expected pro…tsof banksare driventozero.If neithercondition(15)nor(16)holds,thatisifA;n;M, and csatisfy inequality(17),thenthereisnoequilibriumin purestrategies, forthesamereasonsasinthestandardBertrand-Edgeworthmodel(Tirole 1988,Chapt.5,and theliteraturethere cited).13 Theoptimalcapitalrequirementsatis…estheincentive constraint(14); and i’slendingrate,Ri,satis…esinequality(4):ThatisR¤·Ri·x;8i 13Notethat,fora givenn;and,A M,ascdecreasesthelendingcompetitionregimeshifts from monopoly–forcthatsatis…esinequality(15)–totheBertrand-Edgeworthregime –forcthatsatis…esinequality(17)– and fromthistotheperfectlycompetitive(zeropro…t) regime–forcthatsatis…esinequality(16):Thusminimizingthe capitalrequirement subject tomonitoring-incentive constraint(14),doesindeedamount tominimizingbank rents,subject tobanks’…ndingitincentive-compatibletomonitor. 17 and banks’expected pro…tsarenon-negative.Thus,theoptimalcapital requirement,c¤,necessarily satis…es: c¸c¤¸c(18) c´c(R¤;Rd)=1+F (1¡p)Rd¡R¤ Rd c´c(x;Rd)=1+F (1¡p)Rd¡x Rd where:1>c>c(byassumptionsA1-A2). Supposethat theminimumvalueofthe capitalrequirement,c,satis…es inequality(15),whichistrueifand only if: A M·c(19) then forc=cbankscompeteaccordingtothe(collusive)regimede…ned by condition(15). Lemma1Ifcondition(19)holds, i.e. if aggregate bank capitalis su¢- cientlyscarce,then: c¤=c´c(x;Rd)(20) andcreditisrationedif inequality(19)is strict. Proof.See Appendix. Ifaggregatebankcapital is scarce (i.e., ifcondition(19)is satis…ed), thentheoptimalcapitalrequirementc¤attainsitsminimumc.Thetotal amountoflendingthatbankscan undertakedoesnotexceedaggregatedemand (becauseA,Mand c=c¤satisfy inequality(15)),banksimplicitly colludeonthemonopolylendingrateandundertakethemaximumincentivecompatiblevolumeoflending: XLi=nLc(x;Rd;A n)(21) Ifinequality(19)is strict,thisfallsbelowaggregatedemand,M. Supposethat themaximumcapitalrequirement,c;satis…esinequality (16),whichistrueifand only if: A M¸cµn n¡1 ¶(22) 18 then forc=c;bankscompeteaccordingtothe(perfect) competitionregime de…ned bycondition(16). Lemma2Ifcondition(22)holds, i.e. if aggregate bank capitalis su¢- cientlyabundant,then: c¤=c´c(R¤;Rd)(23) andthe…rst-bestoptimumisattained. Proof.See Appendix. Ifaggregatebankcapital is su¢ciently abundant (i.e., ifcondition(22) is satis…ed),thentheoptimalcapitalrequirementc¤attainsitsmaximumc, and A;M;nand c=c¤satisfy inequality(16),sothatabank’scompetitors cancoverthewholemarket.Competitioninlendingis sointensethatrates aredriventothe zero-pro…tvalueR¤,and since nobank’slendingexceeds itsincentive-basedcapacity, all banksmonitor.The equilibriumattainedis the…rst-bestoptimum. Itisworthremarkingthatasnincreases,condition(22)weakens;that is,thelargerthenumberofcompetingbanks,themorelikely anequilibrium wherebankslend at the zero-pro…trateand the capitalrequirementattains itspeak valuec. If neithercondition(19)nor(22)holds, i.e. if: c<A M<cµn n¡1 ¶;(24) thenatc¤neithercondition(15)nor(16)holds. Wheninequality(24)holds, namely forintermediatevaluesofaggregatebankcapital, the competition regimeisthatde…ned bycondition(17): inequilibriumbankswill usemixed lending-ratestrategies.Lemma 3 belowderivesc¤forA;nand Mthat satisfy inequality(24).Thekeytothelemmaisasfollows. (i)Assume condition(17)is satis…ed.Then,asweprovebelow(Proofof Lemma 3),each banki=1;:::nrandomizesitslendingrateaccordingto an atomless distribution function,¹;withsupport£R;R¤,whereRdenotes thelowerbound,and Rtheupperbound.Let the capitalrequirement,c; satisfy: c¸c(R;Rd):(25) 19 Then,becausethelendingraterealizationofeach banki=1;:::nsatis…es R·Ri·R,theincentive constraint(14)ismet.Therefore,a¤ i=m; i=1;:::n,and all banksmonitor. All ratesinthesupportof¹yieldthesame(maximal)payo¤,denoted by¼.Thedistribution¹;R,Rand ¼arethesolutionto: [Ri¡(Rd+F)] (" M¡(n¡1)A n c # (¹(Ri))n¡1+A n c h1¡(¹(Ri))n¡1i) =¼ (26) R=argmax Ri·x[Ri¡(Rd+F)] " M¡(n¡1)A n c # (27) ¹(R)=0(28) ¹¡R¢=1(29) Theleft-hand sideof functionalequation(26)givesi’sexpected pro…t atany lendingrateRi2£R;R¤,giventhat:a)itscompetitorsrandomize theirlendingratesaccordingtodistribution¹withsupport£R;R¤;b)it monitorslending,thatisa¤ i=m(which holdsby(25));and remunerates depositorsatRd.Indeed,the expression[Ri¡(Rd+F)]isi’spro…tperunit oflendingconditionalupona¤ i=mand itsunitcostof fundingbeingRd(by (5)):The expressionincurly bracketsgivesi’sexpectedlendingvolume: ifi isthebankwiththehighestlendingrate,thenitfacestheresidualdemand forlending,M¡(n¡1)A n c,and the capitalrequirementisnotbinding;that is, itlendsM¡(n¡1)A n c(by(17));thisoccurswith probability(¹(Ri))n¡1. Inanyotherevent,the capitalrequirementisbinding;thatis,i’slending volumeisA n c(by(17)).Theright-hand sideofequation(27)isthemonopoly rateontheresidualdemand left to a bankwhenall therivalsundercutits rate.Becauseabankthatsetsitsratetotheupperbound ofthesupport isundercutbyall therivals,thehighestrate everchargedinequilibriumis Rgiven by(27):Clearly R=x,themonopoly rategiventherectangular demand curve. Solvingfunctionalequation(26)for¹(Ri)leadsto: ¹(Ri)= 8 > > < > > : [Ri¡(Rd+F)]A n c¡¼ µnA n c¡M¶[Ri¡(Rd+F)] 9 > > = > > ; 1 n¡1 (30) 20 ¹(R)=0() [R¡(Rd+F)]A n c=¼(31) ¹¡R¢=1() ¼= [x¡(Rd+F)] " M¡(n¡1)A n c # (32) whereR´x(by(27)):Wethen havethat theupperbound ofthesupport Risthemonopoly ratex;thelowerbound Risthesolutionto(31),and the expected pro…tofeach banki=1;:::n;is¼asgiven by(32);by(17), thisis strictly positive.Notethatdistribution¹,Rand ¼areuniquely determined fora givenc.Moreover,bothRand ¼arestrictly increasing inc;thatis,thehigherthe capitalrequirement,thehigherarebankrents. Foranygivenc;equations(30)¡(32)and R=xcharacterize theBertrandEdgeworthequilibriumforarectangulardemand curve(forthemoregeneral caseofadownward-slopingdemand curvesee Tirole1988,Chapt.5,and the referencesthere cited).Thisistheuniquesymmetric equilibriumfora given c(by(30)¡(32)and (27)): (ii)Giventhatinthe competitionregimede…ned bycondition(17),each banki=1;:::nrandomizesitslendingrateaccordingtodistribution¹with support[R;x],theoptimalcis: c¤=c(R;Rd)´1+F (1¡p)Rd¡R Rd:(33) Thatis,c¤istheminimumvaluethatsatis…esthemonitoringincentive constraint(14).Forc=c¤asgiven by(33);thebankwiththelowest lendingrateisexactly indi¤erentbetweenmonitoring and notmonitoring. (iii)Wheninequality(24)holdsand c=c¤asgiven by(33),then condition(17)isful…lled. Lemma3Ifcondition(24)holds, i.e. if aggregate bank capitalisneither scarce norabundant,then: c¤=c(R;Rd)´1+F (1¡p)Rd¡R Rd where,Risthesolutionto(31)givenc=c¤´c(R;Rd): Proof.See Appendix. 21 ForA;nand Mthatsatisfy inequality(24),theoptimalcapitalrequirementc¤exceedstheminimumvaluec,attainedwhen bankscolludeonthe monopoly rate(whencondition(19)holds),and fallsbelowthemaximum, c,attainedwhencompetition drivespro…tstozero(whencondition(22) holds).Furthermore,thelendingcompetitionregimeisthatde…ned bycondition(17): inthesymmetric equilibriumeach banki=1;:::n;randomizes accordingtodistribution¹withsupport[R;x],monitorslending,and earns strictly-positivepro…t¼asgiven by(32). Figure2illustratesthe comparativestaticsofc¤,withrespect toA;M;n thatsatisfy inequality(24). FIGURE2 ABOUTHERE ForcurveC,c=c(R;Rd);thatis,a¤ i=m,i=1;:::n(banksmonitor)on andabovecurveC.ForcurveB,[R¡(Rd+F)]A n c= [x¡(Rd+F)]·M¡(n¡1)A n c ¸; thatis,abank’sexpected pro…t (conditionalupona¤ i=m)at thelower bound Requalsthepayo¤atanyrateinthesupport,whichis¼asgiven by (32):Theintersectionofthese curvesgivesc¤:AsA Mornincreases,curve BshiftsupwardstoB’. Accordingly, c¤increasesasA Mornincreases.The reasonis simple:aseitherA Mornincreases,theamountoflendingbya bankwhenitisundercutby itscompetitors shrinks,and whenitsetsits rateitplacesmoreweighton undercuttingconsiderations.Thismeansthat pro…tmargins,and hence marginalreturnstomonitoring,decrease.Raising the capitalrequirementrestoresmonitoringincentives.The comparisonof optimalcapitalrequirementsderived fromLemmas1-3con…rmsthat this resultholdsforanyA;Mand n. Proposition4The equilibriumvaluesofbanks’pro…tmarginsand optimalcapitalrequirementsareinverselyrelated.Social-welfare-maximizing capitalrequirementsincreasewhenaggregate bank capitaland/orthenumberofbanksincreasesrelative totheaggregatedemandforlending.The (constrained)optimumentails…nancialintermediationand deviatesfrom the…rstbestwheneveraggregate bank capitalisnotsu¢cientlyabundant (i.e., whencondition(22)failsto hold).Inthiscase,banksearnexcessive pro…tsandcreditisrationedif inequality(19)is strict. 22 Proposition4summarizestheresultsinLemmas1-3.Astheratio of aggregatebankcapitaltotheaggregatelendingdemand,A M, increases,the economymovesfromanequilibriuminwhich banksmakemonopoly profitsand the capitalrequirementattainsitsminimumc(forA Mthatsatis…es (19),Lemma 1) to oneinwhich pro…tsarestrictly positivebutbelowthe monopoly leveland the capitalrequirementisinanintermediaterange, thatisc<c¤<c(forA Mthatsatis…es(24),Lemma 3),and …nally to an equilibriuminwhich pro…tsare zero and the capitalrequirementattains itsmaximumc(forA Mthatsatis…es(22),Lemma 2).Anincreaseinthe numberof banksn,produces similare¤ects.Ifaggregatebankcapital is notsoscarce astosatisfycondition(19),thenasnincreasespro…tsfall and the capitalrequirementrises(byLemma 3).Furthermore,asnincreases, condition(22)weakens;thereforethehigherthenumberof banks,themore likely anequilibriuminwhich bankslend at the zeropro…trateand the capitalrequirementattainsitsmaximumc.Theseresults stemfromthefact thatA M,nand cjointly determinetheintensityofcompetitionand hence pro…tmarginsand monitoringincentives.Foranygivenc,asA Mand/orn increases,thebank’smarketpowerdiminishesbecauseitscompetitorscan servea greaterportionofaggregatedemand,bankscompetemoreaggressively, and theirpro…tmargins shrink. Themarginalreturntomonitoring thusdecreasesasA Mand/ornincreases.Raisingthe capitalrequirement restoresmonitoringincentives. Wetherefore concludethatunlikemarketdiscipline(Corollary3),optimalcapitalrequirementsmake…nancial intermediation possible. Optimal capitalrequirementsdiminishat the end ofarecession,since bankshaveless capitalasaresultofcyclically-inducedinsolvencies;theyarehigherwhen thebankingsystemismorefragmented,and increasewhenentrybarriers areremovedand competition becomes…ercer. 5 ThePredictionsoftheModel This sectionsummarizesthepredictionsofthemodelwithregardtothe endogenousvariables: lendingrates,returnoncapital, optimalcapitalrequirementand theprobabilityofa…rm’sbeingdeniedcredit. 23 Depending on parametervalues(A M;n;Rd;x;F)therearethree possible regimes:themonopoly, credit-rationingregime(condition19);theBertrandEdgeworthmixed-strategyregime(condition24);and the zero-pro…tperfectly competitiveregime(condition22).ThisisdepictedinFigure3. FIGURE3 HERE where c´c(R¤;Rd)=1+F (1¡p)Rd¡R¤ Rd R¤´Rd+F c´c(x;Rd)=1+F (1¡p)Rd¡x Rd Exogenousparametervaluesdeterminewhichregimethe endogenous variablesbelongto and theirvalueswithinthatregime.Figure4depicts thebank lendingrate,Ri;asafunctionofA Mforanygiven(n;Rd;x;F).14 Similarly, Figure5depictsbankpro…tperunitofcapital, ¼i A n;Figure6depictstheoptimalcapitalrequirement,c¤:Figure7depictstheprobabilityof a…rm’sbeingdeniedcredit,Pr=max·1¡A c¤ M;0¸;whereA c¤istheoverall lendingvolumegivenaggregate capitalAand theoptimalcapitalrequirementc¤:Thisprobability is strictly positiveand decreasinginA Mwhenever condition(19)holdsand aggregate capital is scarce. FIGURES4,5,6,7 ABOUTHERE Anincreaseinn,numberofcompetingbanks,shiftsn n¡1ctotheleft. 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Econ.Stud.64,215-39. 33 34 Figure 1 Sequence of Events Date 0 Regulation stage: Bank regulator announces capital requirements Date 1 Bank competition stage: - Bank i = 1,2...n sets its lending and borrowing rates - Borrowing and lending takes place Date 2 Monitoring stage: Bank i = 1,2...n chooses whether to monitor lending Date 3 Project returns are realized, borrowing and lending contracts are executed 35 c*' c* c R CB ’ B Figure 2 c* under condition (24) 36 c n n 1 − Monopoly Regime BertrandEdgeworth Regime Perfectlycompetitive Regime M A c Figure 3 Equilibrium Regimes 37 Ri x R* M A c n n 1 − c n n 1 ' ' − n n >' c Figure 4 Lending Rate 38 n A i π ( ) FRxd + − 0 c n n 1 ' ' − c n n 1 − M A n n > ' c Figure 5 Bank Profit per Unit of Capital 39 c * c n n 1 − c c nn > ' c c n n 1' ' − Figure 6 Optimal Capital Requirement M A 40 M A Pr 'c c 0 1 d d RRc< ' :' Figure 7 Credit Rationing