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Econometrica,Vol.72,No. 4 (July,2004), 1023-1061FINANCIAL INTERMEDIARIES AND MARKETS

BY FRANKLIN ALLEN AND DOUGLAS GALEA complexfinancialsystem comprisesboth financialmarketsand financial nterme-diaries. We distinguish inancial ntermediariesaccording o whetherthey issue com-plete contingentcontractsor incompletecontracts.Intermediaries uch as banks thatissue incompletecontracts,e.g., demanddeposits, are subjectto runs,but this doesnot implya market failure.A sophisticated inancialsystem-a systemwith completemarketsfor aggregaterisk and limited marketparticipation-is incentive-efficient,fthe intermediaries ssue complete contingentcontracts,or else constrained-efficient,if they issue incompletecontracts.We argue that there maybe a role for regulatingliquidityprovision n an economyin which markets or aggregaterisks are incomplete.KEYWORDS:inancialintermediation,centralbanking, efficiency,financialcrises,completemarkets,general equilibrium.

1. MARKETS,INTERMEDIARIES,AND CRISESFOR A LONG TIME, it has been taken as axiomatic hat financialcrises are bestavoided. We confrontthis conventionalwisdomby showingthat,under certainconditions, a laisser-faire financialsystem achieves the incentive-efficientorconstrained-efficientallocation.1Furthermore,constrainedefficiencymay re-quirefinancialcrises in equilibrium.The assumptionsneeded to achieve theseefficiencyresults arerestrictive,but no more so than the assumptionsnormallyrequiredto ensure Pareto-efficiencyof Walrasianequilibrium.The importantpoint is that optimalityof avoidingcrises should not be taken as axiomatic.If regulationis requiredto minimizeor obviatethe costs of financialcrises, itshouldbe justifiedby a microeconomicwelfareanalysisbased on standardas-sumptions.Furthermore, he formof the interventionshould be derived frommicroeconomicprinciples.Financial nstitutionsand financialmarketsexist tofacilitate the efficientallocationof risks and resources.Any government nter-ventionwill have an impacton the normalfunctioningof the financialsystem.A policyof preventing inancialcrises will inevitablycreate distortions.One ofthe advantagesof a microeconomicanalysisof financialcrises is that it clarifiesthe costs and benefits of these distortions.

Policy analysesof bankingand securities markets tend to be based on veryspecificmodels.2In the absence of a general equilibrium ramework, t is hardto evaluate the robustness of the resultsand, ultimately,to answer the ques-tion: What precisely are the market failures associatedwith financial crises?1Wallace 1990) suggeststhat bank runsmight be efficient.Examplesof efficient bank runswere providedbyAlonso (1996) and Allen and Gale (1998).Here we provide generalsufficientconditionsfor the efficiencyof financialcrises.2See Bhattacharyaand Thakor(1993) for a survey.For examplesof more recent work thatstressesthe analysisof welfare,see Matutes and Vives(1996, 2000).

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E ALLEN AND D. GALEIn this paper, we take a step toward developing a general model to analyze mar-ket failures in the financial sector and study a complex, decentralized, financialsystem comprising both financial markets and financial intermediaries.3 Forthe most part, the seminal models of bank runs, such as Bryant (1980) andDiamond and Dybvig (1983), analyze the behavior of a single bank and consistof a contracting problem followed by a coordination problem.4 We combine re-cent developments in the theory of banking with further innovations to modela complex financial system. The model has several interesting features: it intro-duces markets into a general-equilibrium theory of institutions; it endogenizesthe cost of forced liquidation;5 it allows for a fairly general specification of theeconomic environment; it allows for interaction between liquidity and assetpricing;6and it allows us to analyze the regulation of the financial system usingthe standard tools of welfare economics.

In our model, intermediaries have two functions. Following Diamond andDybvig (1983), intermediaries are providers of insurance services. By poolingthe assets of individuals with uncertain preference for liquidity they can pro-vide a higher degree of liquidity for any given level of returns on the portfolio.Their second function is to provide risk sharing services by packaging existingclaims on behalf of investors who do not have access to markets. In this respectthe intermediary operates more like a mutual fund, but both functions are es-sential to the operation of an optimal intermediary. Financial intermediaries

3Inthispaperwe use the term financialmarkets narrowly o denote marketsfor securities.Otherauthorshave allowedfor markets nwhich mechanismsaretraded(e.g.,Bisin and Gottardi(2000)). We prefer to call this intermediation.Formally,the two activities are similar,but inpracticethe economicinstitutionsare quitedifferent.4Earlymodels of financialcrises were developedin the 1980'sby Bryant(1980)and Diamondand Dybvig(1983). Importantcontributionswere also made by Chari and Jagannathan 1988),Chari(1989), Champ,Smith,and Williamson(1996), Jacklin(1986), Jacklinand Bhattacharya(1988),Postlewaiteand Vives (1987), Wallace 1988, 1990),and others.Theoreticalresearchonspeculative currencyattacks,banking panics, the role of liquidityand contagionhave taken anumberof approaches.One is built on the foundationsprovidedby earlyresearchon bank runs(e.g., Hellwig (1994, 1998), Diamond (1997), Allen and Gale (1998, 1999, 2000a, 2000b),Peckand Shell (1999), Changand Velasco(2000, 2001), and Diamond and Rajan(2001)). Otherap-proaches nclude those based on macroeconomicmodelsof currencycrises that developedfromthe insightsof Krugman 1979), Obstfeld(1986), and Calvo (1988) (see, e.g., Corsetti, Pesenti,and Roubini(1999) for a recent contributionand Flood and Marion(1999) for a survey),gametheoretic models (see Morrisand Shin (1998),Morris(2000), andMorrisand Shin (2000)for anoverview),amplificationmechanisms e.g., Cole andKehoe(2000)and Chariand Kehoe (2000)),and the borrowingof foreign currencybyfirms(e.g.,Aghion,Bacchetta,andBanerjee(2000)).5Most of the literature,followingDiamond and Dybvig (1983), assumes the existence of atechnologyfor liquidatingprojects.Here we assume that a financiallydistressed nstitutionsellsassets to other institutions.This realistic feature of the model has important mplicationsforwelfareanalysis.Ex post, liquidationdoes not entail a deadweightcost becauseassets are merelytransferredrom one owner to another.Ex ante, liquidationcan result in inefficientrisksharing,butonlyif markets or hedgingthe risk are incomplete.6Inparticular,here is a role for cash-in-the-market ricing(Allen and Gale (1994)).

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FINANCIALNTERMEDIARIEShavemanyotherfunctions,of course,includingpayments, nformationgather-ing, lending,andunderwriting,but we ignorethese for the purposeof focusingon risksharingandmacroeconomicstability.We distinguishbetween intermediariesthat can offer complete contingentcontractsand intermediaries hat canonlyofferincomplete ontracts.An exam-ple of the latterwouldbe banksthat canonlyofferdepositcontracts.Withcom-plete contracts,the consequences of default can be anticipatedand includedin the contract,so without loss of generalitywe can assume default does notoccur.Withincompletecontracts,however,default can improvewelfareby in-creasingthe contingencyof the contract(see, e.g., Zame (1993)).Aggregate risk in our model takes the form of shocks to asset returnsandpreferences.Formally, here is a finite set of aggregatetates of nature that de-terminesasset returns and preferences.We contrasteconomieswith completemarkets, in which there is a complete set of Arrow securities, one for eachaggregatestate, from economies with incompletemarkets,in which the set ofArrow securities is less than the number of aggregatestates. This producesa 2 x 2 classificationof models, accordingto the completeness of contractsand markets:

Completemarkets IncompletemarketsComplete ontracts incentive-efficient notefficientIncompleteontracts constrained-efficientnot efficient

Financialcrises do occurin ourmodel,but arenot necessarilya source of mar-ket failure.A sophisticated inancialsystem providesoptimalliquidityand risksharing,where a financialsystem is sophisticated f markets for aggregaterisksare complete andmarketparticipation s incomplete.Efficiencydependson the completenessof marketsbut does not dependon whethercontractsarecomplete or incomplete.These resultsprovidea benchmark or evaluatinggovernmentinterventionandregulation.If a sophisticated inancialsystem eadsto an incentive-efficientor constrained-efficient llocation,whatprecisely s the role of the governmentor centralbank in interveningin the financialsystem?What can the govern-ment or centralbank do that privateinstitutionsand the market cannot do?Ourefficiencytheoremsassume that marketsfor aggregaterisk are complete.Missingmarketsmay providea role for government ntervention.In addition, the model providessome importantinsights into the workingof complexfinancialsystems.We include a series of examplesto illustrate thepropertiesof the model. In particular,we show that, in some cases, (a) exis-tence andoptimalityrequirethatequilibriabe mixed, hatis, identicalbankschoose very different risk strategies; (b) default and crises are optimalwhenmarketsare complete and contractsincomplete; (c) asset pricingin a crisis isdeterminedby the amount of liquidityin the market as well as by the assetreturns;and (d) risksharing s suboptimalwith incompletemarkets.

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E ALLEN AND D. GALEOur results are related to a small but important iterature that seeks to ex-tend the traditionalintermediationliterature in a more general equilibrium

direction. von Thadden (1999) also studies an integratedmodel of demanddeposits and anonymousmarkets. Martin (2000) addresses the question ofwhetherliquidityprovisionbythe central bankcan preventcrises without cre-atinga moralhazardproblem.Gromb and Vayanos(2001) studyasset pricingin a model of collateralizedarbitrage.The rest of the paperis organizedas follows. Section 2 describes the prim-itives of the model. Section 3 exploresthe welfarepropertiesof liquiditypro-vision and risk sharing in the context of an economy with a sophisticatedfinancialsystemin whichinstitutionstake the form of general intermediaries.Section 4 showsthat incompleteparticipation s critical for the optimalityre-sults achieved in the previoustwo sections. Section 5 extends this analysistoan economywith a sophisticatedfinancialsystemin which institutionsuse in-complete contracts. In Section 6, we consider an economy with incompletemarketsandcharacterize he conditions underwhich welfarecan be increasedby in some cases increasingand in some cases decreasingliquidity.Section 7containssome finalremarksand some of the proofs are gatheredtogether inthe Appendix.

2. THE BASICECONOMYThere are three dates t = 0, 1, 2 and a single good at each date. The good isused for consumptionandinvestment.The economyis subjectto two kindsof uncertainty.First, individualagentsare subjectto idiosyncraticpreference shocks,which affect their demandforliquidity(these will be describedlater). Second, the entire economyis subjectto aggregateshocksthat affectasset returnsandthe cross-sectionaldistributionof preferences. The aggregateshocks are representedby a finite number ofstates of nature,indexedby r7E H. At date 0, all agentshave a commonpriorprobabilitydensityv(r/) overthe states of nature.All uncertaintys resolved at

the beginningof date 1,when the state qris revealedandeach agentdiscovershis individualpreferenceshock.Each agent has an endowment of one unit of the good at date 0 and noendowment at dates 1 and 2. So, in order to provide consumptionat dates1 and2, theyneed to invest.There are two assets distinguishedby their returns and liquiditystructure.One is a short-termasset (the shortasset), and the other is a long-termasset(the long asset). The short asset is representedby a storage technology:oneunit invested in the short asset at date t = 0, 1 yields a return of one unit atdate t + 1. The long asset yields a return after two periods. One unit of thegood invested in the long asset at date 0 yields a randomreturn of R(r1) > 1units of the good at date 2 if state r is realized.Investors'preferencesare distinguishedex ante and ex post. At date 0 thereis a finite number n of types of investors, indexed by i = 1,..., n. We call i an

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FINANCIAL INTERMEDIARIESinvestor's ex ante type. An investor'sex ante type is common knowledgeandhence contractible.The measure of investors of type i is denoted by Aui 0.The total measureof investors s normalizedto one so that i ,ui= 1.While investorsof a givenex ante typeare identical at date 0, theyreceive aprivate,idiosyncratic,preferenceshock at the beginningof date 1. The date 1preferenceshock is denoted by OiE Oi,where Oi is a finite set. We call Oi heinvestor'sex post type. Because 0i is privateinformation,contractscannotbeexplicitlycontingenton 0i.Investorsonlyvalueconsumptionat dates 1 and 2. An investor'spreferencesarerepresentedbya von Neumann-Morgensternutilityfunction,ui(c1, c2; Oi),where ct denotes consumptionat date t = 1, 2. The utilityfunction ui(.; 0i) isassumedto be concave,increasing,andcontinuousfor everytype Oi.Diamondand Dybvig (1983) assumed that consumers were one of two ex post types,either early diers who valued consumptionat date 1 or late diers who val-ued consumptionat date 2. This is a special case of the preference shock 0i.The presentframeworkallows for much more general preferenceuncertainty.The probabilityof being an investor of type (i, Oi)conditionalon state r7is denoted by Ai(0i, r7) > 0. The probabilityof being an agent of type i is ui.Consistency hereforerequiresthat

E Ai(0i,r) =i, V /cE .Oi

By the usual lawof large numbers convention,the cross-sectionaldistribu-tion of typesis assumedto be the sameasthe probabilitydistributionA.WecanthereforeinterpretAi(0i,7r)as the numberof agentsof type (i, 0i) in stater.3. OPTIMALINTERMEDIATION

Intermediarieshave two broad functions in this model. First, because in-dividualinvestors do not have access to marketsfor sharingaggregaterisk,intermediariestrade existing claims on their behalf to produce a syntheticrisk-sharingcontract for the investors. In this respect, they are like mutualfunds. Secondly, as in the Diamond-Dybvig model, intermediariesprovideinvestorswith insuranceagainstthe preference shocks. One difference fromthe Diamond-Dybvigmodel is that intermediariescannotphysically iquidateprojectswhen they need liquidity.Instead,they sell assets on the capitalmar-ket at date 1.Because there is no physicalcost of liquidatingassets,liquidationis ex post efficient:the buyer's oss is the seller'sgainand vice versa.In this sectionwe assumethatfinancial nstitutionstake the formof generalintermediaries.Each intermediaryoffers a single contract and each ex antetype is attractedto a different intermediary.Contractsare contingenton theaggregatestates r7and individuals'reports of their ex post types, subject toincentivecompatibilityconstraints.

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F.ALLEN AND D. GALEOne can,of course, imaginea worldin which a single universal ntermedi-aryoffers contractsto all ex ante types of investors.A universal ntermediary

could act as a centralplannerand implementthe first-best allocation of risk.Therewouldbe no reason to resortto markets at all. Our world view is basedon the assumptionthat transactioncosts precludethis kind of centralizedso-lution and that decentralizedintermediariesare restrictedin the number ofdifferent contracts heycan offer.Thisassumptionprovidesa role for financialmarkets n which financial ntermediariescan share risk and obtainliquidity.At the same time, financialmarkets alone will not suffice to achieve opti-mal risksharing.Becauseindividual conomicagentshaveprivate nformation,markets or individualrisksareincomplete.The markets hatare availablewillnot achieve an incentive-efficientallocation of risk. Intermediaries,by con-trast,can offer individuals ncentive-compatible ontracts and improveon therisksharingprovidedby the market.In the Diamond-Dybvig(Diamond and Dybvig(1983)) model, all investorsare ex ante identical.Consequently,a single representativebank can providecomplete risk sharing and there is no need for markets to provide cross-sectionalrisksharingacross banks. Allen and Gale (1994) showed that differ-ences in risk andliquiditypreferencescanbe crucial n explainingasset prices.This is another reasonfor allowingfor ex ante heterogeneity.

3.1. MarketsAt date 0 investorsdeposit their endowments with an intermediary n ex-changefor a generalrisksharingcontract.The intermediarieshaveaccess to acomplete set of Arrowsecuritymarkets at date 0. For each aggregatestate r7there is a security raded at date 0 thatpromisesone unit of the good at date 1if state 17s observed andnothingotherwise.Let q(r/) denote the priceof oneunitof the Arrowsecuritycorresponding o state r1,hatis, the numberof unitsof the good at date 0 needed to buyone unit of the good in state r at date 1.All uncertainty s resolved at the beginningof date 1. Consequentlythereis no need to trade contingentsecurities at date 1. Instead,we assumethereis a spot market and a forward market for the good at date 1. The good atdate 1 is the numeraire so pi (r) = 1; the price of the good at date 2 for saleat date 1 is denoted by p2(n7), i.e., p2(n) is the number of units of the goodat date 1 needed to purchaseone unit of the good at date 2 in state r. Let

p(r/) = (pl (r), p2(r/)) = (1, P2(r/)) denote the vector of goods prices at date 1in state r7.Note that we do not assumethe existence of a technologyfor physically iq-uidatingprojects.Instead,we follow Allen and Gale (1998, 2000b)in assumingthat an institution n distress sells long-termassets to other institutions.Fromthe point of view of the economy as a whole, the long-termassets cannot beliquidated-someone has to hold them.

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FINANCIALINTERMEDIARIES3.2. IntermediationMechanisms

Investorsparticipatein markets indirectly, through intermediaries.An in-termediary s a risk-sharingnstitutionthat investsin the shortand long assetson behalf of investorsand providesthem with consumptionat dates 1 and 2.Intermediariesuse marketsto hedge the risks thatthey managefor investors.Each investor of type i gives his endowment (one unit of the good) to anintermediaryof type i at date 0. In exchange, he gets a bundle of goodsxi(Oi, 7T)ER2 at dates 1 and 2 in state r7if he reports the ex post type Oi.In effect, the functionxi = {xi(0i, r1)} s a directmechanismthat maps agents'reportsinto feasible consumptionallocations.7A feasible mechanism is incentive-compatible.The appropriatedefinitionof the incentive-compatibility onstraintmust take into account the fact thatagentscanuse the shortasset to store the good fromdate 1 to date 2. Supposethat the agentreceivesa consumptionbundlexi(0i, r7) rom the intermediary.By saving,he can obtainanyconsumptionbundlec ER2 suchthat2 2

(1) C1 , Xil(Oi, 7), c,t < Xi,t(i, q).t=l t=l

Let C(xi(0i,r7)) denote the set of consumptionbundles satisfying(1). Themaximumutilitythat can be obtainedfrom the consumptionbundle xi(Oi,17)bysavingis denoted by u (xi(0i, r7),Oi)and definedbyU*(xi(Oi, r7), Oi) = sup{ui(c, Oi): c E C(xi(Oi, 7T))}.

Then the incentiveconstraintwith savingcan be writtenas:(2) Ui(xi(i, ), Oi) > (Xi()i, T),Oi)EO, VE, e 7 E H.Notice thatby placingui(.) on the left-hand side of (2), we ensure that even atruth-telling agent will not want to save outside of the intermediary.Thereis no loss of generality in this restriction,since the intermediarycan tailorthe timing of consumption to the agent's needs. Let Xi denote the set ofincentive-compatiblemechanismswhen the depositorhas access to the stor-age technology.

7A direct mechanism s normallya function that assignsa uniqueoutcome to each profileoftypeschosenbythe investors.In a symmetricdirectmechanism, he outcome for a singleinvestordepends only on the individual'sreportand the distributionof reportsby other investors.In atruth-tellingequilibrium, he reportsof other investors are given by the distributionA(.,r7)soa symmetricdirectmechanism should properlybe written xi(Oi, A(-,71),71)but since A(., 7) isgiven as a function of 77thereis no loss of generality n suppressing he reference to A(., r7).

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E ALLEN AND D. GALE3.3. Equilibrium

Recall that each intermediary ssues a single contract and serves a singleex ante type of investor. We denote by i the representative ntermediary hattrades withthe ex ante investortype i. The representative ntermediary takesin deposits of ,ui units of the good at date 0 and investsin yi > 0 units of theshort asset and j,ui Yi> 0 unitsof the long asset. In exchangeit offersdeposi-tors an incentive-compatiblemechanismxi. Given the prevailingprices (p, q),a mechanismxi and investmentportfolio yi yield nonnegativeprofitsfor theintermediaryf it satisfies the following budgetconstraint:(3) ,q(-7) Ai(Oi, ri)p(Tr) Xi(Oi, 7)

Oi< Y q(t)P(l) ' Yi, /ti - yi)R(r1)).

(In equilibrium, ree entrywill drive the value of profitsto zero.) In state r7,the cost of goods givento investorswho report 0i is p(ri) ?xi(0i, 7q)and thereare A(0i,r]) such agents, so summingacrossex post types Oiwe get the totalcost of the mechanism in state r7as oiA(0i, 7r)p(r7) xi(0,, r7).Multiplying bythe cost of one unit of the good at date 1 in state r7andsummingover states r1givesthe total cost of the mechanism, n terms of units of the good at date 0, asthe left-hand side of (3). The right-handside is the total value of investmentsby the intermediary.In state r7the short asset yields yi units of the good atdate 1 and the long asset yields (Ati- yi)R(rq)units of the good at date 2 sothe total value of the portfoliois p(r7) (yi, (ui - yi)R(,7)). Multiplyingbythepriceof a unitof the good at date 1 in state r7and summingacrossstatesgivesthe total value of the investmentsby the intermediary,n termsof unitsof thegood at date 0.An intermediated llocationspecifiesan incentive-compatiblemechanismxiand a feasible portfolio yi for each representative intermediary i = 1,..., n.An intermediated allocation {(xi, yi)) is attainable f it satisfies the market-clearingconditionsat dates 1 and 2, that is,(4) i(0i, 7i)xil(0i, 77) Yi, v7r

i oi i

and(5) A(0i, 77)(Xil(0i, 7r)+ Xi2(0i, 77))i Vq.

= ~yi + (Hi-yi)R(r), V 7.

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FINANCIALINTERMEDIARIESCondition(4) says that the total consumptionat date 1 in each state 7rmustbe less than or equal to the supplyof the good (equal to the amount of theshortasset). Condition(4) is an inequalitybecause it ispossibleto transformanexcessof the short assetat date 1 intoconsumptionat date 2. Condition(5) saysthat the sumof consumptionover the two periods is equal to the total returnsfromthe two assets.Alternatively,we canreadthis as sayingthatconsumptionat date 2 is equalto the returnon the long assetplus whatever s left over fromdate 1.An intermediaryoffers a contractxi EXi that maximizesthe profitper con-tract,subjectto a participationconstraint

, A(0i, In)Ui(xi(0i, T?), Hi) > ui,r1 Qiwhere ui is the maximumexpected utility the ith type can obtain from anyother contract in the market.Hence, apureintermediatedquilibrium onsistsof a price system (p*, q*) and an attainable allocation{(x*,yi*)} uchthat, forevery intermediary , the choice of mechanism x* and portfolio y7 solves thedecisionproblem

maxE q*(r)p*(77) (Yi,(-i - yi)R(r1))- q*(r?) Ai(Oi, )p*(rq) xi(Oi,Tr) subjectto

r7 oiXi E Xi,

y E A(Oi, T)Ui(xi(0i, T), Oi) > A(0i, '])Uii(x* (Oi, ,Oi),17 01 17 Q] oi r] 0i

andequilibriumprofitsare zero:y q*(q) E Ai(0i, T7)p*(r). Xi(0i, Tr)

17 Oi

= q*(rl)p*(r). (Yi,(tli - yi)R(r)).

Competition and free entry force the intermediariesto undercut one an-other by offering more attractive contractsto the investors. In equilibrium,the contract chosen maximizes the welfare of the typical depositor subjecttoa zero-profitconstraint.(On the one hand, no intermediarywill offer a con-tract that earnsnegative profitsand,on the other, if the contractearnspositive

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E ALLEN AND D. GALE

profitsor fails to maximizethe investor'sexpectedutility,an entrant can stealcustomersand still make positiveprofits.)Hence, the intermediary'sdecisionproblemis equivalentto the following:(6) max A(Oi, i)ui(xi(Oi, 7}),Oi) subject to

7 OiXi E Xi,

q(r) y Ai(0i, xq)p(rl) xi(i, r)7

< q(ri)p({q). (yi, (zi - yi)R(/)).

In a pure equilibrium,we assume that all intermediariesserving type ichoose the same portfolio and contract.To ensure the existence of equilib-rium,we need to allow for the possibilitythat intermediariesof type i makedifferent choices. A mixedallocationis definedby a finite set of numbers{pj}and allocations {(xi, y')} such that pi > 0 and j pJ = 1. A mixed allocationrepresents he followingsituation.The population s divided nto groups.Eachgroupj is a representativesampleof size pJ,that is, the relativefrequencyofex ante typesin groupj is the same as in the populationat large.Each ex antetype i in group j is served by an intermediary hat offers a contractxJ andchooses a portfolioy[. In a sense, we can thinkof (xi, yi) as a pure allocationfor groupj, exceptthat there is no requirement orgroupj to be self-sufficient(there can be tradeamong groups)and each type i must be indifferentamongthe differentgroupsin equilibrium.A mixed allocation is attainable f the mean {(xi, yi)} = j pJ{(x, yI)} sat-isfies the market-clearing onditions(4) and (5). Note that {(xi, yi)} maynotbe an allocation:even if each (xi, yJ)belongs to Xi the mean (xi, yi) maynot,because the set Xi is not convex.A mixedintermediated quilibrium onsists of a price system (p, q) and amixed attainableallocation{(pi, xi, yj)} such that forevery ntermediary, andeverysubgroup , the choice (x', yJ)solves the decisionproblem (6).

ASSUMPTION (Nonempty Interior): For any ex ante type i = 1, ..., n, forany consumptionbundlexi EXi andpricesystemp e P, andfor any s > 0, either

P(n) (E A( i,)i(0i, )) =rl \ O,

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FINANCIALINTERMEDIARIESorthereexistsa bundlex' withina distancee of xi such that

p(q) (A(E i, )X'i(Oi, ]))< Ep( ) (e i, )xi(Oi,- ))r77 ii r OTHEOREM 1: Under hemaintainedassumptions,hereexistsa mixed nterme-diatedequilibrium,f Assumption1 is satisfied.For the proof see Existence of Equilibrium:An Addendum to 'Finan-cial Intermediaries and Markets ' at http://www.nyu.edu/econ/user/galed/papers.html.A pure equilibrium s a special case of a mixedequilibrium,but there is noguaranteethat pure equilibriaexist. In fact, we show in Section 5.1 that pureequilibria ail to exist in straightforwardases.For some purposes it is useful to interpreta mixed intermediatedequilib-riumas a pure intermediatedequilibriumwith a different set of ex ante types.Let (i, j) denote the new ex ante subtype consistingof investorsin the sub-groupj of ex ante type i and let -ijP pi-,i denote the measure of investors inthe new ex ante type (i, j). Notice thatwe have to define a distincteconomyforeverymixedintermediatedequilibrium.This is because theweights/ij dependon the endogenousvariablespi. In a mixedequilibrium,all agentsof given exante type i receive the same expected utility.Thus, the expectedutilityof sub-type (i, j) is the same as the expected utilityof type (i, j') for any given type i.Taking heweights {,uij}asgiven,we mightbe able to find otherpureequilibriaof the artificialeconomy, but they would not necessarilybe mixed equilibriaof the original economy,because they would not necessarilysatisfythis equi-librium condition. For most purposes, this is not an issue, so without loss ofgeneralitywe can restrict attention to pureequilibria.

3.4. EfficiencyUnder mild assumptions,we can show that the equilibriumallocation isincentive-efficient.An attainableallocation (x, y) = {(xi, yi)} is incentiveef-ficient if there does not exist an attainable mixed allocation {(pi,xi, yi)}such that

7 A(ei,q)ui(xJ (0i ), Oi) A(0i, Tr)ui(xi(0i, r), Oi)1 Oi rl Oi

forevery(i, j) with strict nequality or some (i, j). (ThisdefinitiondiffersfromParetoefficiencyonly to the extent that we restrict attention to the incentive-compatiblemechanismsxi EXi.)In order to prove the incentive-efficiencyof equilibrium,we need an addi-tionalregularitycondition:

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E ALLEN AND D. GALEASSUMPTION 2 (Local Nonsatiation): For any ex ante type i = 1, ..., n, forany mechanismx, E Xi, andfor any s > 0, thereexistsa mechanism x' E Xi

withina distance8 of xi suchthatA(0i, r)ui(x'(0, r)i) > LE,A(i, r7)u(xi(i, ri), O).

71 oi ~ oi

REMARK: Thisassumption sthe counterpartof the nonsatiabilityassump-tionsused in the classical heoremsof welfareeconomics.Itrequiresmore thanthe nonsatiabilityof each ex post type'sutilityfunction, ui(, 0i), however,be-cause the incentive constraintsmustbe satisfied also.Toillustratethe meaningof Assumption2, consider the followingexample.There are two ex post types,0i = 1, 2, with utilityfunctions ui(c, 1) and ui(c, 2). The utilityfunctionsaredefined as follows:i(C, 1) = C1 + C2,

ui(c, 2) = c, +c2 if c + c21,and for any utility level u > 1, we denote the indifference curve consistingof the locus of consumptionbundlesyielding u to an agent of type Oi= 2 byIi(u, 2) and define it byputting

Ii(u, 2)= (l, C2) = ((1-a) au: a


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FINANCIAL INTERMEDIARIESTHEOREM2: Under the maintainedassumptions, f (p, q, x, y) is an inter-mediatedequilibrium ndAssumption2 is satisfied, hen the allocation (x, y) is

incentive-efficient.For the proof see the Appendix.Theorem2 is in the spiritof Prescott and Townsend 1984a, 1984b),but thepresent model takes the decentralizationof the incentive-efficientallocationa step further. Markets are used in Prescott and Townsend(1984a, 1984b)toallocate mechanismsto agentsat the first date. After the firstdate all trade isintermediatedbythe mechanism.Here, markets are also used for sharingriskand for intertemporalsmoothingand intermediariesare activeparticipants n

markets at each date. In Section 4 we show that efficiencyrequires that in-dividualsdo not have access to financialmarkets. To this extent, we followPrescott-Townsend n assumingthat individuals' rade is intermediatedby themechanism.In Section 5, we considerincompletecontractsand the possibilityof default.In the event of default,individuals'rade is no longerintermediatedbythe mechanism.REMARK: An importantqualification o the incentive-efficiencyof the in-termediated equilibrium s the equilibrium election implicit in the defin-

ition of equilibrium. Following the standardprincipal-agentapproach(see,e.g., Grossmanand Hart (1983)), we allow the principal (the intermediary),to choose the actions of the agents (the investors), subject to an incentive-compatibilityconstraint.REMARK: The incentive-efficiency f equilibrium s in marked contrasttothe results in Bhattacharyaand Gale (1987). The difference is explained bythe informationalassumptions.In the model above, there is no asymmetryofinformation n the marketsfor Arrowsecurities. Once the state 77s observed,all aggregate uncertainty s resolved.The distributionof ex post types in eachintermediary s a function of r1and hence becomes common knowledgeoncer7 s revealed. TradingArrow securities at date 0 is sufficient to provideop-timal insuranceagainst all aggregateshocks at date 1. In BhattacharyaandGale (1987),bycontrast,an intermediary'sruedemandfor liquidity s privateinformationat date 1. Marketsfor Arrow securitiescannotprovideincentive-efficientinsuranceagainstprivateshocks.While symmetryof information n financialmarketsis a useful benchmark,one can easily imaginecircumstances n whichintermediarieshaveprivatein-formation, for example, the intermediaryknows the distributionof ex posttypes among its depositors, but outsiders do not. In that case, providingincentive-efficient insurance to the intermediarieswould require us to sup-plement markets for Arrowsecuritieswith an incentive-compatiblensurancemechanism,as in Bhattacharyaand Gale (1987).

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E ALLEN AND D. GALE3.5. Examples

In this section we consider a series of numericalexampleswith completemarketsand complete contracts with respect to aggregatestates. These pro-vide an efficient benchmark or subsequentexampleswithincompletemarketsand/orcontracts.We look first at an exampleof asset-returnshocks and thenlook at an examplewith liquidityshocks.ReturnShocks

EXAMPLE: We assume there is a single ex ante type of investor withDiamond-Dybvigpreferences.There are two ex post types, early consumers(0= 1) and late consumers(0 = 2). The utilityfunction is

Ulogc, 0-= 1,u(c, C2,))= 1 ,logc2, 0= 2.The probabilityof being an earlyconsumer is one half, independentlyof theaggregatestate r/. There are two equallylikely aggregatestates r-= 1, 2 andthe returnon the long asset at date 2 is contingenton the state:

R(1) = r; R(2) = 2.Different equilibriaaregenerated byvaryingr.Since investors are ex ante identical, the open interest in Arrow securitieswill be 0 in equilibrium.Since the investors are either earlyor late consumers,the optimalbundleswill be of the formx(0, 17)= {(c (7), 0), (0, c2(r7))}withearlyconsumerscon-suming cl(17) at date 1 and late consumersconsumingc2(r7)at date 2. Themarket-clearing onditionsare(7) .5cl()

1, equilibrium is unique and .5cl(r7) = y inboth states.A typicalequilibriumwith r = 1.5 is shownin TableI. The inter-mediaryputs .5 in the safe asset and .5 in the riskyasset. The earlyconsumersreceive 1 at date 1 irrespectiveof the state, and late consumers receive 1.5in state 1 and 2 in state 2. With these allocations it is clear that the incentiveconstraint does not bind. No late consumer would choose 1 at date 1 rather

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FINANCIAL INTERMEDIARIESTABLE I

EXAMPLESExample r y (q(1), p2(1)) (q(2), 2(2)) (c1(1), 2(1)) (c1(2), c2(2)) E[U]1A 1.5 .5 (.5, .67) (.5, .5) 5).5) (1,2) .2751B .5 .64 (.61,1) (.39, .89) (.82, .82) (1.28,1.44) .0541C .3 .79 (.59,1) (.41,1) (.85, .85) (1.21,1.21) .0162 A: .5 .5 (.5, .8) (.5, .8) (1,1.25) (1,1.25) .1122 N: 1.0 (0, .25) (0,4.75) 1.2503 1.6 .5 (.4, .94) (.6, .42) (1.25,1.33) (.83, 2) .2134A 1.5 .5 (.33-.75, .25-1) (.25-.67, .38-1) (1,1.5) (1, 2) .2754B .95 .49 (.55-.77, .70-.98) (.45-.33, .53-.73) (.97, .97) (.97,2.05) .1574C S: .5 .67 (.65, .90) (.35,1) (.95,.95) (.95,1.41) .0454C B: 0 (.77, .77) (1.41,1.41) .045

than 1.5 or 2 at date 2. The Arrow securitiesare priced at .5 for both states.The date 1 spot priceof consumption s higherin state 1 than state 2 becausedate 2 consumption s lower andmarginalutilityis higherin state 1.EXAMPLEB: For .4 < r < 1, the unique pure intermediatedequilibriumis such that the consumptionprofile satisfies .5c1(1) < y and .5c1(2)= y. Instate 1, the returnsto the long asset are so low that some of the short asset

mustbe savedto provide consumptionat date2;in state 2, all of the shortassetis consumedat date 1. In order for the intermediaries o be willingto hold theshortasset between date 1 and date 2 in state 1, the price mustbe equal to 1.Thus,consumption s equalizedacrosstime in state 1. The amount invested inthe short asset rises as the expected payoffon the long asset falls. TableI illus-trates the equilibriumvalues for the case where r = .5. The amount investedin the safe asset has movedup to .64, in state 1 p2(1) = 1, andconsumption sthe same at both dates.EXAMPLE1C: It remains to consider the case 0 < r < .4. As r continuestofall the intermediary nvests more in the short asset and less in the long asset.The total output and consumptionin state 2 are also falling and for r < .4the consumptionat date 1 is less than the holding of the short asset in bothstates: .5c1(r7)< y. Consumption s equalized across time in both states. Theequilibriumvalues corresponding o r = .3 are given in TableI. Here y = .78,p (1) = p2(1) = 1, and in both states consumptions are equated at each date.The next exampleintroduces a second ex ante typeconsistingof risk neutralinvestors.The purpose of this example is to illustratehow the Arrow secu-

rities allow cross-sectionalrisk sharingand to show their role in ensuringanincentive-efficientallocation.EXAMPLE2: There are two ex ante types of investors,the risk averse in-vestors,denotedby A, and the risk neutral investorsdenotedbyN. The period

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E ALLEN AND D. GALEutility functions are UA(c) = log(c) and UN(c) = c, respectively. The measureof each type is normalized to 1 and each type has a probability 5 of being anearlyor late consumer.Otherwisethingsare the same as in Example1.As in Example 1, the qualitativefeatures of the equilibriumdepend onwhether some of the short asset is carried over from date 1 to date 2. Onecase is sufficientto illustratehow the Arrowsecuritiesallow riskto be shared.For r > .4 the allocation is the same as in Example 1A, except that all un-certaintyis absorbedby the risk neutraltype. The equilibriumportfolios areYA= .5 andYN= O,for the representative ntermediariesof typeA andtypeN,respectively.TableI gives the equilibriumvalues for the case where r = .5. The market-clearing prices are q(1) = .5, q(2) = .5, P(1) = .8, p2(2) = .8. The portfolioreturns(receipts)of each intermediary ndconsumptionof eachtypearegivenas follows:

State1 State2(yA,R(r)(l-YA)) (.5,.25) (.5,1)(yN, R(7)(1 -YN)) (0,.5) (0,2)(CAl(T]), CA2(r)) (1,1.25) (1,1.25)(CNl(rl), CN2(r7)) (0, .25) (0,4.75)

Risksharingbetweenthe intermediaries orthe twogroups s achievedthroughtrading in the Arrow securitymarkets at date 0 and the spot consumptionmarkets at date 1. In state 1 the type A intermediarieshave .5 of the goodfrom their short asset holdings. They have promised each of their .5 earlyconsumers 1 unit of the good each so they can simply pay out what they re-ceive from the short asset to meet theirobligations.At date 2 they receive .25from their holdingsof the long asset. They have promisedtheir .5 late con-sumers 1.25 each. The deficit is .5 x 1.25- .25 = .375. To ensure that theyhave this, they need to use the r = 1 Arrow security and the spot marketat date 1. They purchase .3 units of the r = 1 Arrow securityat date 0 forq(l) x .3 = .5 x .3 = .15 and then use the .3 they receive at date 1 in state 1 topurchase .3/p2(1) = .3/.8 = .375 of the date 2 good in the date 1 spot market.At date 0 they finance the .15 needed to purchasethe r-= 1 Arrowsecuritybyselling .375 of the good at date 2 in state 2 and .3 of the r}= 2 Arrow secu-rity.Since they have 1 unit of output from their long asset at date 2, this willleave them with .625 of the good to payout to their .5 late consumerswho re-ceive 1.25 each. The typeN's arewillingto take the other side of these tradessince theyare risk neutral. It can be seen thatthis improvement n risksharingincreasesthe expected utilityof typeA's from .054in Example1Bto .112here.There are other intermediatedequilibriacorresponding o these parametervalues. In particular, he portfolio holdingsof the type-A intermediariesarenot uniquelydetermined.For example,the type-A intermediariescould holdless of the short term asset and more of the long term asset, as long as this is

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FINANCIALNTERMEDIARIESoffset by changesin the holdingsof the type-N intermediaries.As long as theaggregate portfolio remainsthe same, investors'consumptionand welfare isunchanged.LiquidityShocks

The uncertainty n Examples1 and2 is generated by shocks to asset returns.Similarresults can be obtainedby assumingaggregateuncertaintyabout thedemandfor liquidity.EXAMPLE: This exampleis similarto Example1. There are two maindif-ferences. First, it is assumed that there is no uncertaintyabout the returntothe long asset:

R(1) = R(2)=r.Second, there is aggregate uncertaintyabout the demand for liquidity.Thestates r = 1, 2 are equally likely but now the proportionsof early and lateconsumersdiffer across states:

Proportion State 1 State2Earlyconsumers .4 .6Late consumers .6 .4

By varyingr we can generate the same phenomena as in the case of returnshocks. Rather than go through all the different cases, we illustrate a pure in-termediatedequilibrium or the case r = 1.6. The equilibrium alues areshownin TableI. For this case the aggregate liquidityconstraintbinds in both statesbecause the returnon the long asset is high.At date 1 all the proceeds fromthe short asset are used for consumption.In state 1 aggregateliquidityneedsare low at date 1. Since consumptionis split evenly among early consumers,per capitaconsumption s high comparedto state 2 where aggregate liquidityneeds are high at date 1. At date 2 there are manylate consumers in state 1,so consumptionper capita is lower than in state 2 where there are relativelyfew late consumers.4. COMPLETEARTICIPATIONNDREDUNDANCY

Cone (1983) and Jacklin (1986) have pointed out that the beneficial effectsof banks in the Diamond-Dybvig (Diamond and Dybvig (1983)) model de-pend on the assumptionthat individualsare not allowedto trade assets at theintermediate date. Increasingaccess to financial marketsactually owers wel-fare. If investors are allowed to participate in asset markets, then the marketallocationweaklydominates the allocationimplementedbythe banks.

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E ALLEN AND D. GALEIn this section, we show that with complete markets and complete marketparticipation,banksare redundant n the sense that they cannot improveon

the risksharingachievedbymarketsalone. The abilityof intermediaries o pro-vide insuranceagainstshocks to liquiditypreference dependscruciallyon theassumption hat investorscannotparticipatedirectly n asset markets.Toshowthis,we first characterizean equilibriumwith marketsbut without intermedi-aries. Thenwe show that the introductionof intermediaries s redundant.The market data is the same as in Section 2. Investorsare allowedto tradein marketsfor Arrow securities at date 0 and can trade in the spot marketsfor the good at date 1. Let y, denote the portfolio chosen by a representativeinvestor of type i and let xi(0i, q7)denote the consumptionbundle chosen byan investor of type Oi n state r7.The investor'schoice of (xi, Yi)must solve thedecisionproblem:max YA(0i, ])ui(xi(0i, 1), Oi) subjectto

r7 &i

E q(7r) xi(0i,qr)< E q(7r)p(77) (yi,(1- yi)R(r7)), V i.17 17

The maximizationproblemfor the individual s similarto that of an intermedi-aryof type i exceptfor twothings.First,thereis no explicit ncentive constraintxi e Xi. Second, there is no insuranceprovidedagainstthe realization of 0i.Thisis reflectedin the fact that,instead of summing he budgetconstraintoverrl and 0i, it is summedover q only and must be satisfied for each Oi.The exante type i can redistributewealth across states r7 n any wayhe likes,but eachtype 0iwill get the same amount to spendin state r7.As a result,for each real-izationof 0i in a givenstate 77,he choice of xi(0O,77)must maximize he utilityfunctionui(xi(0i, r7),0i) subjectto a budgetconstraint hat dependson 7butnot on 0i. Thisensuresthat incentivecompatibilitys satisfied.An equilibriumfor this economy consists of a price system {p, q} and anattainable allocation {(xi, yi)} such that, for every type i, (xi, Yi) solves the in-dividual nvestor'sproblemabove.Comparing his definition of equilibriumwith the intermediary quilibrium,it is clear thatanequilibriumwithcompletemarketparticipation annotimple-ment the incentive-efficientequilibriumexcept in the specialcase where thereareno gainsfromsharingriskagainst liquiditypreferenceshocks.The more interesting question is whetherintroducing ntermediaries n thiscontextcan make investorsanybetter off. The answer,as we havesuggested, snegative.Tosee this,note first of all that in an equilibriumwithcomplete par-ticipation,all that investorscare about in each state is the marketvalue of thebundlethey receive fromthe intermediary. f an intermediaryoffers a mecha-nism xi, the investors of type 0i will reportthe ex post type 0i that maximizesp(r) -xi(0i, 77)in state 77.Using the Revelation Principle,there is no loss of

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FINANCIAL INTERMEDIARIES

generality n assumingthat the value of consumptionp(r) . xi(Oi, 71)= Wi(1)is independentof Oi.Butthis means that the intermediary an do nothingmorethan replicate the effect of the marketsfor Arrow securities. More precisely,the intermediarys offeringa securitythatpayswi(-q)in state 17and the inter-mediary'sbudgetconstraint

E q(rl)wi(7r)< , q(r1)p(r). (yi, (]li - yi)R(rl))r] T]

ensures that {wi(77)}can be reproducedas a portfolio of Arrow securities atthe same cost.THEOREM3: In an equilibriumwith completemarketparticipation, he in-troductionof intermediariess redundant n the sense that an equilibrium llo-cation in an economywithintermediariess payoff-equivalento (i.e.,yields thesame expectedutilitiesas) an equilibrium llocationof the correspondingcon-omywithout ntermediaries.This is, essentially,a Modigliani-Miller esult,sayingthat whatever an in-termediarycan do can be done (and undone) by individuals radingon theirown in the financialmarkets.Theorem 2 providesconditionsunderwhichanequilibriumwillbe incentive-efficient. However, the equilibriumdescribedin Theorem 2 does not neces-sarily Pareto-dominate the equilibriumdescribed in Theorem 3: with manyex ante types, it is possible that one type i will be worse off in the incentive-efficient equilibrium.In the special case where there is a single ex ante type,complete participationmakeseveryoneworse off. The results of Cone (1983)and Jacklin(1986) to which we referred earlier assume a single ex ante typeof investor.

5. INCOMPLETECONTRACTSAND DEFAULTIn the benchmarkmodel defined in Section 3, intermediariesuse general,incentive-compatible ontracts.Inreality,we do not observe suchcomplexcon-tracts.There aremanyreasonsforthis,whichare well documented n the liter-ature. These includetransactioncosts, asymmetricnformation,and the natureof the legal system.These kinds of assumptionscanjustifythe use of debt andmanyotherkindsof contractsthat intermediariesuse in practice.Since we areinterested in developing a general framework,we do not model any specificjustification or incompletecontracts. Insteadwe assumethat the contractin-

completeness can take any form and prove results that hold for any type ofincompletecontract.With complete contracts, intermediariescan always meet their commit-ments. Default cannot occur. Once contracts are constrained to be incom-plete, then it may be optimal for the intermediary o default in some states.

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E ALLENAND D. GALEIn the event of default, it is assumed that the intermediary'sassets, includ-ing the Arrowsecurities it holds, are liquidatedand the proceeds distributedamong the intermediary'snvestors.For markets to be complete, which is anassumptionwe maintainfor the moment, the Arrow securities that the bankissues mustbe default-free.Hence, we assumethat these securitiesare collat-eralized and their holdershave priority.Anythingthat is left after the Arrowsecurityholdershavebeen paidoff is paidout pro rata to the depositors.An intermediaryof type i chooses a formal contractxi E X' and an indi-cator function B:H -- {0, 1), where B(r) = 1 means that the intermediarydefaultson the formalcontract.There is also a defaultcontract, whichde-scribeswhat happenswhen the intermediarydefaults.The defaultcontractxi is defined by x(0i, r/) = (w1i(r), 0), where wi(r7) is the liquidated value ofassets in state 1r.The defaultcontract s not partof the formalcontract; t isdeterminedbythe institutionof bankruptcythe nexus of law,courts,creditors,and debtors)when the contracthas been breached.Formally, t is possible todescribeall this as a single mechanism,whichis what we do, but conceptuallythere is an importantdifference between what the formal contractsays andwhathappensif the formalcontractis breached.The institutionof bankruptcyis exogenous to the model, somethingthe intermediary akes as given whenwritingthe contract.

Given this bankruptcyprocedure,an intermediaryhas to choose two con-tractsfor individuals,a formalcontractxi EX/ and a default contractxi, anda decision rule B: H --- 0, 1 that indicateswhen the bankdefaults. Then theeffectivecontractXi s definedby(1 - B(r))xi(0i, r) + B(r/)xi(0i, r)

for every (0,, r7). Since everything s paidout at date 1 in the event of default,investorsuse the short asset to provideconsumptionat date 2. Then the util-ity of a contract xi generated in this way is given by u*(xi(Oi,qr),06)in thestate (0i, 17)andthe incentiveconstraint sU*(xi(Oi, T/), Oi) > u*(xi(Oi, 77), Oi), V Oi, Oi E {i.

Let Xi denote the set of incentive-compatible ontractsgeneratedin thisway.To demonstratehow defaultworks,considerthe followingsimpleillustration.

EXAMPLE:uppose that intermediariesare restrictedto offering noncon-tingentcontracts.Then xi EX' if andonlyifxi(0i, 1) = xi(Oi, 71')

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FINANCIAL INTERMEDIARIESfor anyrq,r4'EH. If we assumedfurther hatagentsareeitherearlyconsumers,who value consumptionat date 1, or late consumerswho valueconsumptionatdate 2, we couldwithout loss of generalitywritethe contract as

(cl,0O) if0i=0,xi(Oi, rf) --

(O,C2) if 0i 1.This contractis highlyinflexiblebecause it does not allow the intermediary oadjust he paymentacross states in responseto changingdemandsfor liquidityor changingasset returns. If default is not allowed,the intermediary s forcedto distortthe contract.Default allows the intermediaryo replacethis inflexiblecontractwith the following:

(cl, 0) if Oi= 0, Bi(77) = 0,Xi(Oi, ) (0, C2) if i = 1, Bi(r7) = 0,

(wi(r/), 0) if Bi(r) = 1.The possibility of incompletecontracts is represented by restricting thechoice of contractto a subset Xi c Xi. The theory developed in Section 3 is

essentially the same, but with Xi replaced by Xi. With this substitution,anintermediatedequilibriumbecomes an intermediatedquilibriumwithincom-plete contracts,an incentive-efficientallocation becomes a constrained-efficientequilibrium,andAssumption2 becomes Assumption2'. Then Theorem2 canbe interpretedas a theorem about constrained-efficiency f equilibriumwithincompletecontractsbut complete markets.THEOREM: Under he maintainedassumptions,f (p, q, x, y) is an interme-diatedequilibriumwith ncompletecontractsandAssumption2' is satisfied, henthe allocation(x, y) is constrained-efficient.Although this result is a direct corollaryof Theorem 2, it is substantivelythe most importantresult of the paper, for it shows that, in the presence ofcomplete markets,the incidence of default is optimal in a laisser-faireequi-librium.For example, consider the case where banks are restrictedto usingdemand deposits that promise a fixed payment at date 1. A bank may de-fault in some states because its assets areinsufficient o meet itsnoncontingentcommitments.Default allows the bank to make its risk-sharing ontract morecontingent, that is, more complete. But if several banks default in the samestate, this may cause a financialcrisis, as the simultaneous iquidationof sev-eral banks forces down asset prices,whichmayin turncause distress in otherbanks.Nonetheless, the theorem showsthat,givencompletemarkets or insur-ing against aggregateshocks,the occurrenceof these crises is optimalandnota marketfailure. There is no scope forwelfare-improvingovernment nterven-tion to preventfinancialcrises.

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E ALLEN AND D. GALEREMARK: The definition of constrainedefficiency is formallysimilar tothe definition of incentive-efficiency-we simply substituteXi for Xi in the

latter-but there are importantsubstantivedifferences. The definition of Ximplicitly ncorporatesthe bankruptcy ode and requiresconsumers to beautarkiconce the intermediarydefaults.Allowing trade after default wouldcomplicatethe theorybut shouldnot changethe basic result as longasmarketsfor aggregateriskarecomplete.REMARK: Alonso (1996) and Allen and Gale (1998) have arguedthat fi-nancial crises can sometimes be optimal,or at least Pareto-preferred o gov-ernment interventionto suppresscrises entirely.While these argumentshave

a similar flavor to the present results, they are substantiallydifferent. BothAlonso (1996)and Allen andGale (1998)relyon very specialexamplesto showthat laisser faire may be preferredto a policy of preventingfinancial crises.There is no general efficiencyresult. In fact,Allen and Gale (1998) show thateven within the context of their example, the efficiencyof financial crises isnot robust.More importantly, here is no role for Arrowsecurities n the examplespro-videdbyAlonso (1996)and Allen and Gale (1998).In theirmodels,a represen-tativebankchooses a portfolioand depositcontract to maximizethe expectedutilityof the representativeagent.Becauseof the representativeagentassump-tion, banks are autarkicand complete markets are irrelevant. So the resultspresentedhere are based on a differentargumentandapplyto a muchbroaderclass of economies.REMARK6: The role of Arrow securities deserves further attention in amodel with default.An Arrowsecurity s a promiseto deliver one unit of thegood in some state at date 1. It is assumed,crucially, hat Arrowsecuritiesarefully collateralized so there is no risk of default on these promises. Equiva-lently,traders n Arrowsecurities can anticipateeach intermediary's bilityto

repay borrowing hroughArrowsecuritiesin each state and Arrow securitiestake priorityover other claimson the intermediary'sassets. This means that,at date 0, the intermediarywill not be allowed to execute trades that cannotbe fulfilled at date 1 andthat the intermediary's ositionin Arrowsecurities issettled at the beginningof date 1 before anythingelse happens,in particular,before the default decisionis made and before investorsreceiveany payments.These assumptionsare restrictive but they are essential to the definition ofArrowsecuritymarkets.REMARK: Default is completelyvoluntary n the sense that the intermedi-arychooses at date 0 those states in which it wishes to default. Default is notforcedon the intermediarybythe budgetconstraint; n fact, in the presenceofcompletemarkets t is not clear what this would mean. Because Arrowsecuritymarketsarecomplete, there is a single budgetconstraintand the intermediary

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FINANCIAL INTERMEDIARIESchooses the amountof wealth allocated to each state. We do however assumethatthe intermediarys committedto hisdefault decisionatdate 0 andfurther,that he makesthe default decisionin the ex ante interest of the representativeinvestor. These assumptionsare restrictivebut interestinginsofar as they arerequiredfor efficiency.We can interpretthem as a descriptionof an optimalbankruptcynstitution.

5.1. ExamplesIn this section we demonstrate a number of results using examples.Theseresults include the following:(1) the existence (indeed necessity)of mixedequilibria;(2) the optimalityof default and crises;(3) cash-in-the-market ricingwhen there is a crisis;(4) suboptimalrisksharingwith incompletemarkets.There are manyexamplesof intermediariesthat use incomplete contractswith their depositors.As we have seen, as soon as incomplete contracts areintroduced the possibilityof default must be allowed.This in turn introducesthe possibilityof financial crises. In practice crises can be precipitated by awide arrayof intermediaries.Historically,however, t has been banks thathavecaused most crises. In order to illustratethe model with incompletecontracts

and default we considerexampleswhere the intermediariesare banks,that is,they are restrictedto issuingdeposit contracts.A deposit contractpromisesafixed paymentat date 1. If the bank is unable to meet its commitmentto itsdepositors, then it is liquidated and all its assets are sold. If it can meet itscommitment,depositorscan keep their funds in the bank untildate 2.EXAMPLE: To illustratethe propertiesof a bankingequilibrium,we revertto the parameters romExample1.EXAMPLEA: Forr > 1the firstbest canagainbe implementedasabankingequilibrium.The first-best allocationrequiresthat consumptionat date 1 bethe same in both states, so the optimal incentive-compatible ontract s in facta deposit contract.Moreover,bankruptcydoes not occur in equilibrium.Theallocationcorresponding o a pure bankingequilibrium s exactlythe same asthe allocationcorresponding o a pureintermediatedequilibrium,whichas wehave seen is the same as the firstbest. Forr = 1.5,forexample,the equilibriumvalueswillbe the same as for Example1A, as shownin TableI.The main difference between Examples 1A and 4A is that the prices sup-

porting the first best allocation are unique in Example 1A but are not inExample4A. In Example 1A the general intermediariescan freely varypay-offs across states and dates so only one set of prices can support the opti-mal allocation.Because the banks are restrictedto using deposit contractsinExample4A, consumptionat date 1 is independentof the state as longas there

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F. ALLEN AND D. GALEis no bankruptcy.Thispreventsthe banks from arbitragingbetween the statesat date 1. In effect, we have lost one equilibriumconditioncomparedwith theintermediatedequilibrium.Anyvectorof prices (q(l), q(2), p2(1), p2(2)) thatsupports he equilibriumallocationat date 2 is consistentwith a bankingequi-librium.The following no-arbitrageconditionsare satisfied:(9) q()p2(1) =.33,(10) q(2)p2(2) = .25,(11) q(l) + q(2) = 1,and the existenceof the short asset impliesthat

P2(l) < 1, p2(2) < 1.Combiningthese conditions, it follows that the range of possible prices forq(1) is

.33 < q(1) < .75.The otherpricesare determinedby (9)-(11).For r < 1 deposit contracts cannot be used to implement the first best orincentive-efficientallocation.Inparticular,a depositcontractthat is consistentwith the incentive-efficientconsumption n state 2 would imply bankruptcynstate 1 (see, e.g., Example1B).If allbankswere to offera contractthatallowedbankruptcyn state 1, theywould all be forced to liquidatetheir assets. If theyall do this,there is no bankon the othersideof the marketto buythe assetsandthepriceof futureconsumption,p2(1), falls to 0. This cannot be anequilibriumbecause it would be worthwhile or an individualbank to offer a contractthatdid not involvebankruptcy.Thisbank could then buyup all the longterm assetin state 1 at date 1 for p2(1) = 0 and make a large profit.

This argument ndicates thatfor r < 1 there cannot be a pure bankingequi-libriumwithbankruptcy.There are two otherpossibilities.One is that there isa pure equilibrium nwhichall banks remain solvent. The other is that there isa mixedequilibrium n which banks choose different contracts andportfolios.In particular,a groupof banks with measure p makes choices that allowit toremain solvent in state 1 while the remaining1 - p banksgo bankrupt.Thebanks that are ensured of solvencyare denoted type S while those that can gobankruptare denoted typeB. It turns out that both the pure equilibriumwithall banksremainingsolventandthe mixedequilibriumcan occurwhen r < 1.EXAMPLE B: For .90 < r < 1 there is a pure banking equilibrium in whichall the banks choose to remain solvent (p = 1). There is again a range ofprices that can supportthe allocation.Type-Sbanksstay solvent by loweringthe amount promised at date 1, loweringtheir investmentin the short asset

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F ALLEN AND D. GALEMany analysesof bankingassume that there is a technologyfor early liqui-dationof the long term asset. The resultingequilibriaare typicallysymmetric.

This example illustrates that this assumption materiallychanges the form ofthe equilibrium.With endogenous liquidationone group of banks must pro-vide liquidity o the market and this resultsin mixedequilibria.A comparisonof Example4CwithExample1 shows the importanceof non-contingentcontracts n causingfinancialcrises. In Example1 the abilityto usestate contingentcontractswith investors means that bankruptcynever occursand as a resultthere is never a crisis.However,with depositcontractscrises ofvaryingseverityalwaysoccurin the low outputstate.There are a numberof othertypesof equilibrium,dependingon whether theshortasset is held between date 1 and date2 and the numberof ex antegroups.The propertiesof these equilibriaare similarto those illustrated n Examples1 and2 andwill not be repeatedhere.In Example4 it was assumed that marketsare complete. However, it canstraightforwardlye seen that with incompletemarkets the allocationin eachcase would be the same. In Examples4A and 4B there is onlyone ex ante typein equilibriumandthey all do the same thingso the open interestin Arrow se-curities s zero and there is no role for risksharing.In the mixedequilibriumofExample4C there are two typesof banksprovidingdifferent allocations eventhoughex ante all investorsare identical.Complete marketsdo not help risksharing,however. The reason is that the contractforms restrictthe amountsthat can be paidout andthe trade-offsgiventhese mean that no improvementcan be made. The only difference in Example4C if markets are incompleteis the equilibriumholdings of assets of the two banks are now unique. Thetype-Sbanks hold .82 of the short asset and the type-B banks hold .59 of it.This fact that complete marketsdoes not make a difference here is clearlyaspecial case as the next example, which mixes Example2 and Example4C,demonstrates.

EXAMPLEA: The example is the same as Example 2 except that con-tracts are incomplete.There are risk aversetype-A investorsand riskneutraltype-N investors,r = .5, andtherearecompleteArrowsecuritymarkets.It canstraightforwardlye seen that theequilibriums exactly he same as thatshownin TableI for Example2. The type-A banks can use a depositcontractpromis-ing 1 at date 1to implementthe optimalallocation.Similarly he type-N bankscan use a depositcontractpromising0 at date 1.EXAMPLEB: This example is the same as Example5A except now thereare no Arrow securitiesso markets are incomplete. Here it can be seen thatthe equilibrium s essentially the same as in Example4C. The banks for thetype-A investorsbehaveexactlyasin Example4C. The banksfor the type-N in-vestors do not find it worthwhile o enter the markets hatdo exist.They simplyput all their customers'endowment in the long asset and pay out 0 at date 1and whatever s producedat date 2.

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FINANCIALINTERMEDIARIESA comparisonof Examples5A and 5B demonstratesthe crucial role thatcomplete markets can play.With complete marketsthe financialsystempro-

duces the first best allocation.However,with incompletemarketsthe alloca-tion is muchworse. Moreoverthe incompletenessof markets leads to defaultand crises in equilibrium.In conclusion, the examples in this section have demonstratedthe resultsmentionedinitially.First,Example4C has shownthat a mixedequilibriumcanexist. When there is only one type of bankthey cannot all go bankruptat thesametime. If theydid therewould be nobodyto buytheirassets andthe assets'price would fall to 0. This cannot be an equilibrium hough since it would beworthwhile for a bankto staysolvent and buy up all the assets at the 0 price.As a result the equilibriummust be mixed.Second,Example4C showsthatdefault canbe optimalfor the banks.Theo-rem 4 showsthat the crises that resultfrombanks'defaultingare also sociallyoptimal.In fact crises are necessaryfor the constrained-efficientallocation tobe achieved.Third,the logic demonstrating he existence of mixed equilibriashowshowcash-in-the-marketpricingoccurs. In makingtheir decision on how much ofthe short asset to hold, the solvent banks take into account the fact that ifthere is a crisistheycanbuyup the long assetcheaplyandtradethis off againstthe cost of not investing n the long asset at date 0 androllingoverthe liquidityin state 2.Fourth,Example5 demonstrates the importantrole of complete markets.Withincompletemarketsthe equilibrium s completelydifferentfrom the onewith complete markets. Not only is risk sharingmuch better with completemarketsbut also there are no crises.

6. INCOMPLETEMARKETSAND LIQUIDITY REGULATIONThe absence of markets for insuring ndividual iquidityshocksdoes not by

itself lead to marketfailure. As we showed in Section 3, intermediarieswithcomplete contracts can achieve an incentive-efficient llocation of risk undercertain conditions. We showed in Section 5, under the same conditions, thatintermediarieswith incompletecontractscan achieve a constrained-efficientl-location of risk. Twoassumptionsare crucialfor these results:marketsfor ag-gregate uncertaintyare complete (there is an Arrowsecurityfor each state r1)and marketparticipation s limited. Marketsfor aggregateriskallowinterme-diaries to shareriskamongdifferentex ante typesof investor andlimitedpar-ticipation allows intermediariesto offer insuranceagainst the realization ofex post types.In order to generate a market failure and provide some role for govern-ment intervention, an additional friction must be introduced. Here we as-sume that it comes in the form of incompletemarketsor aggregateisk.Thereare two sources of aggregate risk in this model. One is the return to the

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F ALLEN AND D. GALErisky asset R(r/); the other is the distributionof investors' liquidity prefer-ences A(0i, qr).Althoughthere is a continuumof investors,correlated iquidityshocksgive rise to aggregatefluctuations n the demand for liquidity.Marketincompletenessmaybe expected to varyaccordingto the source ofthe risk. On the one hand,asset returnscanbe hedged usingmarketsfor stockand interestrate options and futures.Forexample,financialmarketsallowin-termediaries o takehedging positionson futureassetpricesand interestrates.Additionalhedges can be synthesizedusing dynamic radingstrategies.On theother hand, liquidityshocks seem harderto hedge. Asset prices and interestrates are functionsof the aggregate istributionof liquidityshocksin the econ-omy. Invertingthis relationship,we can use asset prices and interest rates toinfer the averagedemandfor liquidity,but not the liquidityshockexperiencedby a particular ntermediary.Hence, tradingoptionsand futureswill not allowintermediaries o provideinsuranceagainsttheir ownliquidityshocks. So mar-kets for hedging liquidityshocks are likely to be more incomplete than themarketsfor hedgingasset returns.Weanalyzea polarcase below: we assumethat there existcompletemarketsfor hedging asset returnshocks,but no marketsfor hedging liquidityshocks.Sincecompletemarketsfor asset-return hockspermitefficientsharingof thatrisk,we mightas well assumethat asset returnsare nonstochastic.Inwhat fol-lows,we adoptthe simplifyingassumption hatthe returnto the long asset is aconstantr > 1 and the only aggregateuncertaintycomes fromliquidityshocks.We continue to assume that each intermediaryserves a single type of in-vestor. Let xi denote the mechanism and yi the portfoliochosen bythe repre-sentativeintermediaryof type i. The problemfaced by the intermediary s tochoose (xi, yi) to maximizethe expectedutilityof the representativedepositorsubjectto multiple,state-contingentbudgetconstraints:

max E A(0i, 7r)ui(xi(0i, r1), 0,) subject to7 Oi

Xi E Xi,

E A(0i, r)p(r) .Xi(Oi,7) _P() p(Yi,,Li - y,)R(7r)), Vl7.oi

The incompletenessof markets is reflected in the fact that there is a separatebudget constraintfor each aggregatestate of nature r7,rather than a singlebudgetconstraint hat integrates surplusesand deficitsacross all states.With incompletemarkets,the existence of welfare-improvingnterventionsis well known in manycontexts (cf. Geanakoplosand Polemarchakis 1996)).Rather than pursuinggeneral inefficiencyresults,we investigatea more spe-cific policy intervention:Is there too much or too little liquidityin a laisser-faire equilibrium?The interest of the results followsnot from the existenceofa policy that increaseswelfare, but rather from the precise characterization

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FINANCIALINTERMEDIARIESof the optimal (small) intervention o regulate liquidity.Our intuitionsuggeststhatmarketprovisionof liquiditywill be too lowwhenmarketsareincomplete.However,even in an equilibriumwitha lot of structure,ouranalysisshows thatwhether there is too much or too little liquiditydependscruciallyon the degreeof relativerisk aversion.We identify liquidity egulationwith manipulationof portfoliochoices at thefirst date. For example,a lower bound on the amountof the short asset heldby intermediariescan be interpreted as a reserve requirement.We examineregulationin the context of an economywithDiamond-Dybvigpreferences.As usual, there are n ex ante types of investors i = 1, ..., n. The ex ante typesare symmetricandthe size of each type is normalizedto /a,= 1. There are twoexpost types,calledearlyconsumersandlate consumers.Earlyconsumersonlyvalue consumptionat date 1, while late consumersonlyvalue consumptionatdate 2. The ex ante utilityfunction of type i is definedby

Ui(Cl, C2, Oi) = (1 - Oi)U(c1) + OiU(c2),where 0i E {0, 1}. The period utility function U: R+-R is twice continu-ously differentiable and satisfies the usual neoclassicalproperties, U'(c) > 0,U (c) < 0, andlimc,, U'(c) = oo.Each intermediarys characterizedat date 1 by the proportionsof earlyandlate consumersamongits customers.Callthese proportions he intermediary'stype. Then aggregateriskis characterizedbythe distributionof typesof inter-mediaryat date 1. We assume that the cross-sectionaldistributionof types isconstant and that there is no variation n the total demand for liquidity.In thissense there is no aggregateuncertainty.However,individual ntermediariesre-ceive differentliquidityshocks:some intermediarieswillhavehighdemand forliquidityand some will have low demand for liquidity.The assetmarket s usedto reallocate liquidityfrom intermediarieswith a surplusof liquidityto inter-mediarieswith a deficit of liquidity.It is onlythe cross-sectionaldistributionofliquidityshocks that does not vary.To make this precise, we need additionalnotation. We identifythe state r7with the n-tuple (T71, .., rn^), where 77i s thefractionof investors of type i who are early consumersin state rq.The frac-tion of earlyconsumerstakesa finite numberof values,denotedby0 < ark< 1,where k = 1, ..., K. We assume that the marginal distribution of r1i s the samefor every type i and that the cross-sectionaldistribution s the same for everystate 7r.Formally, et Ak denote the ex ante probability hat the proportionofearlyconsumersis ok. Let Hik = {r E H: 7i = ok}. Then Ak= -EHik Ai(0, r7),foreveryk andeverytype i. Expost, Akequalsthe fractionof ex antetypeswitha proportiono-kof earlyconsumers.Then Ak = n-'#{i: rli = ok}, for every kandeverystate 7/.The returnon the long asset is nonstochastic,

R(7r)=r>1, Vr/.

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E ALLENAND D. GALEThe only aggregate uncertainty n the model relates to the distributionof liq-uidityshocks across the differentex ante types i.

In what follows,we focus on symmetricequilibrium, n which the prices atdates 1 and 2 are independentof the realized state rj and the choices of theintermediariesat date 0 are independentof the ex ante type i. Let the goodat date 1 be the numeraire and let p denote the price of the good at date 2in terms of the good at date 1. At date 0 the representative ntermediaryoftype i chooses a portfolioyi = y and a consumptionmechanismxi(r7)= x(rli)to solve the followingproblem:(12) maxL Ak{okU(X1((k)) + (1 - k)U(x2(k))} subjectok

O'kxl(Ok) + (1 - Ok)px2(o-k) < y + pr(l - y), Vk,Xl(o-k) < X2(rk), Vk.

The incentive constraint x1(o-k) < x2(0rk) is never binding. To see this, dropthe incentive constraint and solve the relaxed problem. The special early-consumer, late-consumerstructureof preferences and the assumptionr > 1implythat the incentiveconstraint s automatically atisfiedby the solutiontothe relaxedproblem.A symmetric equilibriumconsists of an array (p,x,y) such that (x,y)solves the problem (12) for the equilibriumprice p and the market-clearingconditions

KE Ak kX(ok) < y,k=l

KAk{okXl(k)- + (1- o-k)x2(rk)} = y + r(1 -y),

k=lare satisfied.To study the effect of regulation of intermediaries,we take as given thechoice of portfolio y at date 0 and consider the determinationof the con-sumptionmechanismx and the market-clearingprice p. Call (p, x, y) a regu-latedequilibriumf x solves the problem (12) for the givenvalues of p and y,and (x, y) satisfiesthe market-clearing onditions(15).Suppose that (p, x, y) is a regulated equilibriumand p > p* = 1/r. Thenthe short asset is dominatedby the long asset between dates 0 and 1 and noone will wantto hold the shortasset. If the regulatorrequires ntermediaries ohold a minimumamountof the shortasset,then each intermediarywill wanttohold the minimum. In that case, (x, y) solves the maximizationproblem (12)subject to the additional constrainty > y for an appropriatelychosen value

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FINANCIALINTERMEDIARIESof y. Similarly, f p < p*, then the long asset is dominated and (x, y) solvesthemaximizationproblemsubjectto an additionalconstrainty < y. Thus,any reg-ulated equilibriumcan be interpretedas an equilibrium n whichthe interme-diaries choose the mechanismx andthe portfolioy to maximizethe expectedutilityof the depositorssubject o a constraint hatrequires hemto holda min-imum amount of the short asset in the case p > p* or of the long asset in thecase p < p*. In this sense, the regulatorcan implementthe regulated equilib-rium (p, x, y) by imposingan appropriateconstrainty on the intermediaries'portfoliochoice.By manipulating he intermediaries'portfolio choice, the regulatoris ableto manipulatethe equilibriumprice p. The welfare effect of this change inprice depends on the degree of risk aversion. We can state a positive resultfor regulatorypolicy as follows: it is possible to increasewelfare by imposinga lower bound on intermediaries'holdingsof the short asset if the degree ofrelativerisk aversion s greaterthanone. More precisely, et W(p) denote theexpectedutilityof the typical depositorin the regulatedequilibrium p, x, y).

THEOREM: If (p*,x*,y*) is a symmetricequilibriumand -U (c)c/U'(c) > 1, thenfor some y > y* there is a regulatedequilibrium p, x, y) inwhich intermediaries re required o hold at least y unitsof theshort asset andW(p) > W(p*).

The proof of this result is in the Appendix.The key step in the argument sprovidedbythe followinglemma,which shows how the degree of risk aversiondeterminesthe welfareeffect of a changein the regulatedequilibriumprice.LEMMA: Forany p > p*sufficiently lose to p*, W(p) > W(p*) if

U (c)c(13) - > 1.Foranyp < p*sufficiently lose to p*, W(p) > W(p*) if

U (c)c


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E ALLEN AND D. GALEpresent value of total consumptioncC1+ (1 - o)pc2. In order to satisfythebudgetconstraint,

oc1 + (1 - a)pc2 = y + pr(1 - y),both cl and c2mustfall (efficient risk-sharing mpliesthat c1 and c2alwaysriseand fall together). Then (1 - a)c2, the total consumptionat date 2, falls andthe budgetconstraint mpliesthat aoc, the total demand for liquidityat date 1,rises. So per capita consumptionat dates 1 and 2 and demand for liquiditygoin opposite directions. From this it follows that,when aois high, o-cl > y and(1 - -)c2 < r(l - y), so an increasein liquidity(increasein p), has a positiveincome effect and raisesconsumptionand,when ro s low,anincreasein liquid-ityhasthe oppositeeffect. Thus,an increasein liquidity owersconsumption ngood states (whereper capita consumption s high) and raises it in bad states(where per capita consumptionis low). This transferfrom high-consumptionstates to low-consumptionstates raises welfare. Whenrelative risk aversion islow, the argumentworksin the oppositedirection.The welfare effect dependson the correlationbetween equilibriumconsumptionandthe liquidityshocks.The preceding analysis assumes intermediariesoffer complete contracts.A parallel analysiswith incomplete (deposit) contracts reaches similarcon-clusions(Allen and Gale (2000c)).

7. CONCLUDING REMARKSWehavearguedthat,contrary o the conventionalwisdom, t is not axiomaticthat financialcrises are bad from a welfarepoint of view. In fact,we havepre-sented conditions under which they are essential for constrainedefficiency.This should give economists pause for thought. Although the conditions forefficiencyare restrictive,they are no more so than the conditionsusuallyre-quiredfor efficiencyof general equilibrium.We should also note that, while

the focus of this paper is on a rudimentarymodel of banks and financialin-termediaries, he basic ideas have a widerapplication.Wherever ncompletelycontingentcontracts,suchasdebt contracts,areused, thereis the possibilityofdefault.In the absence of completemarkets or the aggregaterisks nvolved, i-nancialcrisesin the form of widespreaddefaultmayconstitutea marketfailurerequiringsome form of interventionbythe financialregulator.The model we have used is specialin severalrespects.Forexample,we haveassumedthat investors are condemned to autarkyafter a default.A more re-alisticmodel would allow intermediaries o enter and offer deposit contractsto investors whose intermediarieshave defaulted at date 1. We have seen, inSection4, that incentive-efficientrisksharingrequireslimitedparticipation nmarkets.In particular,we cannot allow investors to deal with more than onebank at a time. If investorscould deal with a second bank ex post, this couldundermine he incentive constraints n the originalcontractand resultin lower

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FINANCIALINTERMEDIARIESwelfare. However, once an intermediaryhas defaulted, there is no reason toprevent investorsfrom acceptingcontracts with the surviving ntermediariesand thus avoidingthe inefficiencythat arises from being excludedfrom assetmarkets.This would be more complicated,butwould not fundamentallyalterthe results,as long asentryoccursafter the original ntermediaryhas defaultedandliquidated ts assets.Althoughwe have tried to lay the groundwork or a welfare analysisof fi-nancialcrises,we haveonlyscratched he surface as far as the studyof optimalregulationof the financialsystem s concerned.Inparticular,he roleof centralbanking requiresa much richer model of the monetary systemand monetarypolicy.We have also avoidedanalyzingaggregateuncertainty n the context ofincompletemarketsandthe possiblerole thisprovidesfor the centralbank as alenderof last resort. We have not exploredthe interactionof the financialsec-tor and the real sector,which is where the impactof financialcrises is largelyfelt (Bernankeand Gertler(1989)).These are all clearly subjectsfor future work.

Dept. of Finance, WhartonSchool, University f Pennsylvania,Philadelphia,PA 19104, U.S.A.;[email protected]. of Economics, New YorkUniversity, 69 MercerStreet,New York,NY 10003, U.S.A.;[email protected], 001; inal revision eceived anuary, 004.

APPENDIX: PROOFSA.1. Proofof Theorem

The argument ollows the lines of the familiarproof of the first fundamental heoremof wel-fareeconomics.Let (x, y, p, q) be anequilibriumandsuppose,contraryo whatwe want to show,that there is an attainablemixedallocation{(pj,xi, yi)} that makessome ex ante typesbetter offandnone worse off. If type (i, j) is (strictly)betteroff under the newallocation,then clearlyE q(7) E A(Oi,q)p(n) . xij(i, r) > E q(7)P(7). (y, Li - yi)R(ri)),71 6i 1)

or (x', y) would have been chosen. The nextthingto show is that for everyex ante type (i, j)Eq() A(0i, 7 )P(T7) . xij(i, , ) > q(71)P(7) ' (yij, (i- yij)R(fr)).

1 Oi '1

If not, then by Assumption 2 it is possible to find a feasible mechanismxi such that (x'j,Yi)satisfiesthe budgetconstraintand makestype i (strictly)betteroff than (xi, yi), contradictinghedefinitionof equilibrium.Thensumming he budgetconstraints or all types (i, j) we haveE p E q(r) EA(0i,)p(r) *xi(Oi, T)i,j 7' Oi

> E pjE q(r)p() - (yi, (~i- yi)R(rt)),i,j 'n

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F ALLEN AND D. GALEor(14) E q(7)p(7) . EE A(i, 7)pxj(0i, 77)> E(r)p(7). E pJ(i, (Ai-Yi)R(7) )

?l i,j Oi rq ijThe market-clearingonditions mplythat,for each state r1, ither

E E A(Oi,1)Epixix(Oi, 71)- E pij(y, ('i,- y/)Rg(7))i j i j

if all the good is consumed at date 1, or pI (1) = P2) andi'J ))E A(Oi, )pj(xi,(ii, 7)+x2(i)) = Epi + i-y)R(7))i,j i i,

if some of the good is storeduntil date 2. In eithercase,p(T1) 'E E A(Oi, 7})Pjxj(Oi, 1) = P()p() Epi(, (i - yi)R(7r))-i'j i i,j

Multiplyinghisequationby q(rl) andsummingover 71yieldsE q(7r)p(r7) E. A(0i 77)p , 77)

7) i,j Oi

= q(r?)p(r) ). pi(y,i (i - y)R(77)),71 aoj

in contradictionof the inequality 14). Thiscompletesthe proof.A.2. Proofof Lemma 6

PROPOSITION: There xistsa uniquesymmetricquilibriump*, x*,y*).PROOF: Uniqueness.The Inadaconditions mply hat the optimalmechanismwillprovidepos-itiveconsumptionat both dates, so both assetsare held in equilibrium.Both assets will be heldbetween date 0 and date 1 only if the returns on the two assets are equal. If pr > 1, then thereturnon the long asset is greaterthan the returnon the short asset between dates 0 and 1 andno one will hold the latter; f pr < 1, then the returnon the short asset is greaterthan the return

on the long asset between dates 0 and 1 and no one will hold the long asset. Then equilibriumrequires hatp = 1/r at date 1 (Allen and Gale (1994)).At the price p = 1/r, the short asset is dominatedbetween dates 1 and 2 so no one will holdit. It follows that all consumptionat date 1 is providedbythe shortasset and all consumptionatdate 2 is providedbythe long asset.Then the market-clearingonditions areKL,ku kXI(7k ) = ny,

k=lE Ak I - rk )X2(ok) = nr(l - y).k=l

At the price p = 1/r, intermediariesare indifferentbetween the two assets at date 0. Thus,thequantitiesof the assetsheld in equilibriumare determinedbythe market-clearingonditions.Finally,note that at the price p = 1/r, the right-handside of the budgetconstraint n (12)is equal to 1 for all k, so the choice of x is independentof y. Strictconcavityimplies that

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FINANCIALINTERMEDIARIES 1057there is at most one solution to the maximizationproblem.Hence, the equilibriumvalues areuniquelydetermined.Existence.The existenceof a symmetricequilibrium ollows directlyfrom the existence of asolution of the intermediary'sdecisionproblem(12). A solutionexistsbecause the choice set iscompactand the objectivefunction is continuous. The objectivefunction is increasing,so thebudgetconstraintholds as an equationfor each k. Summing he budgetconstraintover k we get

Ak{o'kx(o-k) + (1- o-k)PX2(oJk)}=n,k

where p = 1/r. Clearly,there exists a value of 0 < y < 1 such that the market-clearing ondi-tions (15) are satisfied.Then (p, x, y) constitutesa symmetricequilibrium. Q.E.D.Tostudythe effect of regulationof intermediaries,we take as giventhe choice of portfolioy atdate0 and considerthe determinationof the consumptionmechanismx andthe market-clearing

pricep. Call (p, x, y) a regulatedequilibriumf x solves the problem (12) for the givenvalues ofp andy, and (x, y) satisfiesthe market-clearingonditions(15).PROPOSITION: For any p sufficientlyclose to p*, there existsa uniqueregulatedequilib-rium(p, x, y).PROOF:Uniqueness.At any price p close to p*, the short asset is dominated between dates1 and 2 so no one will hold it. It follows that all consumptionat date 1 is providedby the shortasset and all consumptionat date 2 is providedby the long asset. Then the market-clearingon-ditions are

K K: Ako-kXl(rk) =ny, Ak l - O'k)x2(Ok)= nr(l -y).k=l k=l

The budgetconstraintsareokxl(ak) + (1 - ak)pPX2(Jk) = y + pr( - y),

for each k. For the given (p, y), strictconcavity mpliesthat there is at most one solution to themaximizationproblem.Hence, the equilibrium aluesareuniquelydetermined. f there existtwo,distinct,regulated equilibria(p, x, y) and (p, x', y'), say, then from the budgetconstraintsandthe fact that consumptionat each date is a normalgood, consumptionmust be uniformlyhigherat each date in one regulatedequilibrium.But this is clearly mpossible romthe market-clearingconditions.Hence, there is at most one regulatedequilibrium.Existence.Similar o the proofof Proposition7. Q.E.D.

A.2.1. Proofof the LemmaLet X(ak, p, y) = (Xl(ock, p, y), X2(o-k, p, y)) denote the optimal mechanism for each valueof (p, y). An optimalmechanism s one that solves theproblem:

maxurkU(xl(Crk)) + (1 -k)U(x2(oJk)) subject too'kxl(ok) + (1 - Uk)x2(ok)/r < y + pr(1 - y)

for each k. The necessaryand sufficientconditions for this are(16) U'(X (ok; p, y)) = rU'(X2(k; p, y))and(17) okXl ('k; p, y) + (1 - k)pX2(k; p, y) = y + pr(l - y),for everyk.


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F. ALLEN AND D. GALEThe functionX(o-k,p, y) is well definedand continuouslydifferentiable or every (p, y) in asufficiently mallneighborhoodof (p*, y*). This follows from the implicit unction theorem andthe regularityof the system (16)-(17) at (p*, y*):

eU (X (ok; p* ,y)) -rU (X2(k; p*, y*)) ]det --0.L k (1- ok)P*The market-clearingondition can be writtenas

KAk okXI (k; p,y) = ,k=l

since the other market-clearingondition is automatically atisfiedby Walras' aw.To show theexistenceof a regulated equilibrium or each p sufficientlyclose to p* we can use the implicitfunctiontheoremagain,notingthat,(? o.kXork 'Xl (k; p*,y)yy ( AkOkX (ok;p,Y)-Y3=E AkSk-1

- -1,k=l k=l

because y + p*r(1 - y) = 1 is independent of y.We next calculate the changein expectedutilitycausedby a smallchangein p at p*:d Ak{kU(X (ok; p, y)) + (1 - ok)U(X2(ok; p, ))}

= Ak 0o-kU'(X((ok; p, Y))-- (k; p, y)

+ (1- 0k)U'(X2(rk; P, Y))d (7k; p, Y)89pdxl ?X2= YAkU(X2(ok; p,y)) r(r ((7k; p, y) + (1 k) - (k; P,y)k 89P 8P

= Ak U'(X2(-k p, y)){r(1- y) - (1- ok)X2(k; P, y)}.kNow supposethat the degreeof relativerisk aversion s less than one (the othercase canbe dealtwith in exactly he sameway).Then (16) impliesthat

rx (ork)


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FINANCIALINTERMEDIARIESwhere the last equationfollows from the market-clearingondition,

E Ak(l - ok)X2(ok; p, y) = r(l - y).k Q.E.D.REFERENCES

AGHION, .,P.BACCHETTA,NDA. BANERJEE2000): CurrencyCrises andMonetaryPolicy nan Economywith CreditConstraints, WorkingPaper,UniversityCollege,London.ALLEN, , ANDD. GALE 1994): LiquidityPreference,MarketParticipationand Asset PriceVolatility, AmericanEconomicReview,84, 933-955.(1998): OptimalFinancialCrises, ournalof Finance,53, 1245-1284.(1999):Comparing inancialSystems.Cambridge:MIT Press.(2000a): FinancialContagion, ournalof PoliticalEconomy,108,1-33.(2000b): OptimalCurrencyCrises, Carnegie-RochesteronferenceSeries on PublicPolicy,53, 177-230.(2000c): FinancialIntermediariesand Markets Addendum:PrudentialRegulationofBanks, http://www.econ.nyu.edu/user/galed/papers.ALONSO,. (1996): OnAvoidingBankRuns, Journalof MonetaryEconomics,37, 73-87.BERNANKE,.,ANDM. GERTLER1989): AgencyCosts,Net Worth,and BusinessFluctuations,AmericanEconomicReview,79, 14-31.

BHATTACHARYA,., ANDD. GALE(1987): Preference Shocks, Liquidity and Central Bank Pol-icy, nNewApproache

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