Relationship Lending, Transaction Costs, andLikelihood of Bank Runs

Jan MutlGraduate Assistant

University of Maryland at College ParkCollege Park, MD 20742mutl@econ.bsos.umd.edu

February 7, 2002

Abstract

I study properties of an optimal banking contract in an environmentsubject to a possibility of bank runs. I show that increases in the expectedreturn on the bank’s portfolio reduce the likelihood of a bank run andthat an increase in the variance of the portfolio has the opposite e¤ect.I then analyze such banking contract in a model of relationship banking.I conclude that higher transaction costs (that led to the emergence ofrelationships among banks and borrowers) increase the likelihood of abank run. I interpret this as a theoretical explanation of cross-countryvariation in the number of bank runs observed.

1 IntroductionThere are various theoretical explanations for the occurrences of …nancial tur-moil in both the emerging markets as well as the developed economies (suchas the EMU crises in early 90s). The models use various channels to show thepossibility of a crisis. The seminal paper by Krugman (1972) assumes that thegovernment pursues inconsistent policies (such as …xed exchange rate togetherwith unsustainable expansionary policies) that eventually lead to an attack onthe …xed peg. Later models introduce possibility of self-ful…lling runs due tofor example con‡icting objectives of the government (Obstfeld 1994 and 1995),herding behavior of international investors (Calvo 1995), informational asymme-tries (Calvo and Mendoza 1998), or due to implicit deposit insurance providedby the government (Dooley 1997). Other papers use the Diamond and Dyb-vig (1983) model in an international context to show that currency crises couldessentially be banking panics (Goldfajn and Valdez 1995, Chang and Velasco1998, 1999, 2000). Finally, the ‡uctuations in the value of domestic collateralcan be also shown to produce crises Caballero and Krishnamurthy 2000).

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However, most of the literature has ignored the e¤ect of industrial structurein either generating …nancial crises or in making it more likely. In this paper Iwill examine how the organizational structure of …nancial intermediation a¤ectsthe probability of currency crisis in the context of liquidity crisis model. Mymodel brings together the industrial organization literature with the liquiditycrises models in international …nance.For simplicity I construct a closed economy version of the bank run model.

I do not distinguish between domestic and foreign depositors and in doing soI ignore the interplay between domestic depositor panics and foreign creditorpanics (as in Chang and Velasco 1999, 2000). My model serves the purposeof pointing out the e¤ects of transaction costs on the probability of a liquiditycrisis and hence provides a natural explanation why developing countries aremore likely to experience a crisis.I start with the observation that banks and borrowers form long term re-

lationships. The empirical evidence suggests that such relationship has a pos-itive value to both borrowers (it enables them to obtain lower interest rate)and lenders (long term relationship enables them to have valuable informationabout the borrowers).1 Clearly then, …rms and banks are willing to invest intobuilding such relationships with the expectations that they will be able to splitthe resulting surplus. Another way to describe the empirical evidence is to notethat the value of transaction (lending/borrowing) is higher if the two partieshave a long-term relationship.In this paper, I build a simple model of such relationship lending. I am

not interested in explaining why relationship lending arises, neither in the ex-act mechanism that allows the lender and the borrower to overcome the infor-mational asymmetry.2 Instead, I want to derive the properties of an optimalbanking contract and study how the transaction costs (of building the long-termrelationships) a¤ect the stability of the banking system.In order to simplify my analysis, I take the extreme assumption that lending

outside of the long-term relationship does not generate any positive surplus.As a result, lending is only possible among parties that have already investedinto a relationship. Such setup leads to a two-stage game: in the …rst stage

1This fact has been, among others, documented by Berger and Udell (1994) who …ndevidence that longer relationship with a bank allow small and medium sized …rms to obtaina lower interest rate. Similarly, Petersen and Rajan (1993, 1994) have documented that…rms with longer relationships with banks have to rely less on trade credit arrangements.Another strand of literature has examined whether banks produce valuable information aboutborrowers (e.g. James 1987, Lummer and McConnel 1989, Hoshi et al. 1990, James and Weir1990). These studies found that existence of a relationship with a bank increases …rm’s value.

2Theoretical models of relationship lending concentrated on the informational advantageof banks in long-tem relationships with their borrowers (Diamond 1984, 1991, Ramakrishnanand Thakor 1984 and Boyd and Prescott 1986). Di¤erent models then predicted who bearsthe cost of creating the relationship. Boot and Thakor (1995) and Petersen and Rajan (1993)predict that borrowers pay the cost by incurring higher interest rates or higher collateralrequirements at the beginning of the relationship. Greenbaum et al. (1989), Sharpe (1990)and Wilson (1993) predict that the interest rate increases over the length of the relationshipso that the cost is borne by the banks that charge lower interest rate at the beginning of therelationship.

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relationships are formed without exact knowledge of …nancing needs and returns;in the second stage this uncertainty is resolved and lending is realized. I solvefor the sub-game perfect Bayesian equilibrium of the game and show that thereturn on bank’s portfolio is negatively linked to the costs of forming the long-term relationships.To evaluate the e¤ect the necessity of relationship lending has on the sta-

bility of the banking system, I extend the Diamond and Dybvig (1983) modelalong the lines of Morris and Shin (2000) and Goldstein and Pauzner (2000). Iderive the endogenous probability of a bank run that an optimizing bank willchoose and show that it increases with return on bank’s assets and decreaseswith its variance. Linking this result with the costs of forming the borrower-lender relationships, I conclude that higher transaction costs lead to increasedprobability of bank runs.The remainder of the paper is organized as follows. Next section will present

the bank run model and derive the single Nash equilibrium of the game the bankdepositors are facing. Given the depositors’ behavior I will then determine thechoice of a bank that maximizes its depositors’ welfare and show how it reacts tochanges in the probability distribution of the return on bank’s assets. Section IIIwill concentrate on the determinants of the return on the assets. I will solve forthe sub-game perfect Bayesian equilibrium in the relationship formation gamethat borrowers and lenders are playing. The solution will link the propertiesof the return realized by the bank with the costs of forming the long-termborrower-lender relationships. Section IV will conclude.

2 Bank RunsMy model is a variant of Diamond and Dybvig (1983).3 I modify the model andintroduce a small amount of noise into the information structure of the agentsin the economy. The noise reduces the indeterminacy of equilibria and allowsthe probability of the bank run to be a well-de…ned function of the economy’sfundamentals. This lets me consider the solution a bank optimizing depositors’welfare would choose. My main contribution is summarized in the propositiondescribing the properties of the solution.

2.1 The Economy

There are three time periods 0, 1 and 2 and there is one perfectly storable good.The economy is populated by a continuum [0,1] of agents, each endowed withone unit of the good in period 0. Agents have access to a random productiontechnology: they can invest any amount of their endowment in period 0 and geta gross return of L if the investment is liquidated in period 1 or gross return of R

3There has been a lot of work on bank runs following the original article by Diamond andDybvig. @@…nd references@@ The role of noise in the information structure in ruling outmultiplicity of equilibria was suggested by Morris and Shin (2000) and Goldstein and Pauzner(2000).

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in period 2. I assume that L is a non-stochastic commonly known constant and issmaller or equal to 1 (so that there are costs to liquidating the investment early),while R s (R;¾2R) is a random variable with commonly known distribution andwith mean R higher than 1. Both returns are same for all agents and I assumethat the realization of R is revealed in period 2 but each agent receives a privatesignal Xi about the realization of R where

Xi = R+ "i (1)

with "i being iid(0; ¾2"); the distribution of "’s is again assumed to be commonknowledge.The agents have the same ex ante utility function

U(c1; c2) = ¸ ¢ u(c1) + (1¡ ¸) ¢ u(c2) (2)

where c1 and c2 denote consumption at period 1 and 2 respectively and ¸ is adummy variable taking on values 0 or 1. I make the standard assumptions thatu0(:) > 0 and u00(:) < 0. The agents do not know their value of ¸ in period0 and it is only revealed in period 1. ¸ is each agent’s private information.However, there is no aggregate uncertainty and the economy-wide proportion ofrealizations of 1 is µ. This form of utility function captures the fact that someagents might have early liquidity needs that are not known ex ante. Since thegood is perfectly storable, the agents that do not have early liquidity needs areindi¤erent about the timing of their returns. In the rest of the paper, I will callagents with realization of ¸ = 0 ’patient’ agents.

2.1.1 Autarky Solution

If agents cannot pool their resources, they maximize their expected utility sub-ject to the technology constraint. Speci…cally, they have to decide the proportionIA of their endowment they will invest into the risky and illiquid technology. Ifan agent’s realization of ¸ is 1, she only values consumption in period 1 and willliquidate all her investment early receiving IA ¢L. On the other hand, an agentwith realization ¸ = 0 will let her investment mature (as long as the expectedreturn R is higher than L) and will receive a return of IA ¢R. Both agents willalso have the (1¡ IA) part of their endowment they did not invest.Formally, each agent solves:

max Ef¸u(c1) + (1¡ ¸)u(c2)g (3)

subject to

c1 = (1¡ IA) + L ¢ IA (4)

c2 = (1¡ IA) +R ¢ IA if E(RjXi) > L (5)

c2 = (1¡ IA) + L ¢ IA otherwise (6)

0 · IA · 1 (7)

0 · c1; c2 (8)

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The solution to the above problem is often assumed to be a corner solutionwith IA being zero or one depending on the exact speci…cation of the parametersof the model.4

2.1.2 Social Planner’s Solution

Social planner can pool agents’ resources and maximizes the expected utilityof a representative agent. In this section I solve the social planner’s problemunder the assumption that the social planner can distinguish the di¤erent typesof agents (¸ is not private information). I show that the social planner’s solutioncannot be implemented via demand deposits if ¸ is private information. Thenext section will solve the social planner’s problem in the case of ¸ being aprivate information.The social planner chooses the proportion ISP of endowments to be invested

into the risky technology. Since she can distinguish types, she can allocatedi¤erent consumption levels to the two types of agents. Furthermore, as thereis no aggregate uncertainty about the proportion of agents with early liquidityneeds, she can choose her investment so that there will be no early liquidation.Formally, the social planner solves:

max Efµu(c1) + (1¡ µ)u(c2)g (9)

subject to

c1 =1¡ ISPµ

(10)

c2 =R ¢ ISP1¡ µ if E(RjXSP ) > L (11)

c2 =L ¢ ISP1¡ µ otherwise (12)

0 · ISP · 1 (13)

0 · c1; c2 (14)

where XSP is the social planner’s signal about the realization of R, I assume itis of the same form as the agents’ signals. I denote the optimal choices of thesocial planner with superscript SP, e.g. (cSP1 ; cSP2 ; ISP ).

Impossibility of demand deposit implementation Since the planner isable to distinguish types, she can maximize each agent’s utility as if her type wasknown and o¤er her an optimal level of consumption given her preferences andinvestment possibility. Let us consider the case of a social planner that tries toimplement the optimal consumption allocation (cSP1 ; cSP2 ) by o¤ering a demanddeposit paying a gross rate of return cSP1 in period 1 and cSP2 in period 2. If

4Diamond and Dybvig (1983) as well as Goldstein and Pauzner (2000) and Morris and Shin(2000) assume L = 1 and hence in their model IA = 1. Chang and Velasco (1998,1999,2000)assume ... xx. Goldfajn and Valdez (xx) ...

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only agents with early liquidity needs withdraw their deposit in period 1, thenthe social planner has exactly the required resources to satisfy their withdrawalswithout liquidating any investment early. Conversely, only without liquidatingany investment early, the period 2 consumption of patient agents is equal tocSP2 and is strictly preferred to cSP1 . If more than a critical proportion of agents

¸c = 1¡ISP (1¡L)cSP1

< 1 withdraw their deposits in period 1,5 all investment has tobe liquidated in period 1 and the period 2 return is zero with certainty. Henceif an agent expects more than ¸c proportion of withdrawals, she would alsowithdraw.The patient agents have a strategic decision to make. The above argument

seem to suggest that there are two Nash equilibria in the game - all patientagents withdraw their deposits (a bank run equilibrium) or all patient agentsdo not withdraw their deposits (a ’good’ equilibrium implementing the socialplanner’s allocation). These are indeed equilibria in the game without privateinformation about R, e.g. when R is …xed and is common knowledge (Diamondand Dybvig, 1983) or when R is revealed in period 1 (Chang and Velasco, 2000or Goldfajn and Valdes, 1995).However, the ’good’ equilibrium is eliminated in a game with private infor-

mation about R. Suppose that there is an agent who expects only µ withdrawalsin period 1 (only agent with early liquidity needs withdraw). Suppose that thisagent receives a very ’bad’ signal about R and expects the period 2 returns cSP2(which are a function of R) to be lower than period 1 returns cSP1 . This agentwould than rationally withdraw her deposit in period 1. Hence, agents whosesignal Xi is below a critical threshold level Xc will always withdraw.6

Consider an agent whose signal is just above such threshold (Xc+¢). Sincethe signals are independently and symmetrically distributed around the truevalue of R, she would expect that half of the patient agents received a signalbelow the threshold and would always withdraw. Hence she would also withdrawif the period two returns conditional on the proportion of withdrawals beingµ+ 1

2 ¢ (1¡ µ) is smaller than the period 1 return.Lemma 1 The expected period 2 return of the demand deposit is a decreasingfunction of the proportion of withdrawals in period 1, provided that the volumeof withdrawals requires liquidating some of the investment early and that thesocial planner would not be able to satisfy all withdrawals in period 1 had all theagents try to withdraw in period 1.

Proof. See the appendix.When implementing the social planner’s solution, there some of the invest-

ment has to be liquidated if at least one patient agent withdraws her depositin period 1. Hence the above lemma applies and the agent that receives a sig-nal Xc + ¢ would withdraw (for small enough ¢). Therefore the withdrawal

5The total resources the social planner has after liquidating all of the investment is (1 ¡ISP ) + ISP ¢ L and this has to be equal to the resources needed to pay out cSP1 to critical

proportion of depositors ¸c which is ¸c ¢ cSP1 . Hence ¸c = 1¡ISP ¢(1¡L)cSP1

.6 If the E(RjXi) < c1 than the agent i always withdraws.

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threshold for Xi is at least Xc +¢. Since the expected period 2 is decreasingin the proportion of withdrawals, the above reasoning also applies to an agentreceiving a signal Xc + 2¢, Xc + 3¢, etc.On the other hand since the social planner is not solvent in period 1,7 no level

of expected returns E(RjXi) would induce the agents to leave their deposits tomature until period 2 if they expect all other agents to withdraw in period 1.As a result, there is only one dominant strategy for the patient agents: alwayswithdraw their deposit in period 1 (a bank run equilibrium). The argumentsketched above if formally summarized in the following proposition.

Proposition 2 If L < 1 and the social planner o¤ers a demand deposit withgross returns cSP1 ; cSP2 in periods 1 and 2 respectively, while investing ISP intothe risky asset, the depositors would always withdraw their deposits in period 1regardless of their information.

Proof. See the appendix.

2.1.3 Social Planner Constrained by Demand Deposits

Even if the social planner is constrained by demand deposits, she could make the…rst period return conditional on the number of withdrawals and still be ableto implement the previous optimal solution even without the knowledge of theprivate information about ¸. The social planner could, for example announcethe total period 1 resources available for withdrawal in period 1 (equal to cSP1 µ)and let agents decide whether to withdraw or not. Such mechanism inducesonly patient agents to withdraw in period 1 and implements the social plannerallocation (cSP1 ; cSP2 ).However, such conditioning of the …rst period returns is not consistent with

a sequential service constraint.8 In this section I take up the interesting problemof a social planner that is constrained to demand deposits and faces a sequentialservice constraint as well.I will solve the problem in two stages. First, taking the action of the social

planner as given (the returns on the deposits and the proportion of endowmentinvested into the risky technology), I will determine the response of agents inthe economy and derive the probability of a run (withdrawal of all deposits inperiod 1). Second, I solve for the optimal action of the social planner takinginto account the reaction of depositors (represented by the probability of a runas a function of social planner’s choices).

Probability of a Run I take the action of the social planner as given (c1, c2and proportion I are given) and solve for the equilibrium of the game that thepatient agents face. I show that the game has a unique solution - a switching

7Solvency in period 1 means that the social planner has enough resources (after liquidatingthe investment) to satisfy withdrawals of all the agents in period 1.

8The gross period 1 return in that case is not known until the last agent makes a requestfor withdrawal. Under a sequential service constraint, the return has to be known when the…rst agent makes a withdrawal request.

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strategy for patient agents in which they withdraw if E(RjXi) < Rc where Rcis a critical value of the return R. The mechanism that makes this a uniquesolution is the same as mechanism that made the bank run equilibrium the onlyequilibrium of the game in the previous section.I show that if the social planner is not solvent in period 1, then there is

unique bank run equilibrium.9 If the social planner is solvent in period 1, thereare two cuto¤ points in the distribution of the signal Xi; X and X. If thesignal an agent receives is higher than X, she will not withdraw her deposit inperiod 1, regardless of what she believes about other agents’ actions and beliefs.Similarly, if the signal an agent receives is lower than X, she will withdraw herdeposit in period 1, regardless of what she believes about other agents’ actionsand beliefs. Following the intuition outlined in the previous section, these cuto¤points are moved until they coincide.

Proposition 3 If c1 > 1¡ I ¢ (1¡ L) then the only equilibrium is a bank run.For c1 < 1¡ I ¢ (1¡ L), there exists

Rc =c1 ¡ 1¡ I

Iif µ +

1

2(1¡ µ) < 1¡ I

c1(15)

= c1 ¡ 1 + (1¡ L) ¢ I otherwise

such that if E(RjXi) < Rc the agent withdraws her deposit in period 1 and doesnot withdraw otherwise.

Proof. See the appendix.

Figure 1 plots the proportion of withdrawals as a function of R.Figure 1

Provided that ¾2" is small enough, for any realization of R lower than Rc,there will be a bank run (withdrawal of all deposits in period 1). The probabilityof run is then the probability of R < Rc which is

P = F (Rc) (16)

where F (:) is a cumulative distribution function of the random variable R.

Optimal Solution Social planner constrained by the demand deposit andsequential service technology has to take into account the possibility of a bankrun. I assume that in the event of a run, all the investment is liquidated inperiod 1 and the resources split evenly among the depositors.10

9The solvency constraint can be relaxed in a model where there exists uncertainty aboutthe liquidation value of the investment. In such model, there would exist the possibility thatsome agents could be misled by favorable signals about both L and R and would not withdrawtheir investment even when the social planner is insolvent in period 1.10Technically, under a sequential service constraint, only a fraction of depositors receive

the promised rate c1 and the rest receives no return. If agents are risk neutral, then ourformulation is equivalent, however, even with risk aversion, the qualitative features of thesolution are not a¤ected.

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Formally, the social planner solves:

max Ef(1¡ P ) ¢ [µu(c1) + (1¡ µ)u(c2)] + P ¢ u(cr)g (17)

subject to the de…nitions of Rc and P in equations (15) to (17), the de…nitionof cr - the consumption level under a bank run

cr = 1¡ I ¢ (1¡ L) (18)

and the constraints

c1 · 1¡ ISPµ

(19)

c2 =1¡ µc1 +R ¢ ISP

1¡ µ if E(RjXSP ) > L (20)

c2 =1¡ µc1 + L ¢ ISP

1¡ µ otherwise (21)

0 · ISP · 1 (22)

0 · c1; c2 (23)

The social planner has the option of keeping ’unproductive’ cash that isneither invested nor kept for the expected withdrawals of patient agents butserves as a deterent of a run - it lowers the equilibrium level of the likelihood ofthe run. In the equilibrium, the bank chooses an investment level and o¤ers aschedule of returns on deposits. These choices imply a choice of a probabilityof run P . The characteristics of the optimal solution is summarized in thefollowing proposition.

Proposition 4 A bank constrained by a demand deposit technology and a se-quential service will choose a positive probability of a run P > 0 and its optimalchoice of P will be positively related to E(R) = R and negatively to var(R) = ¾2R,i.e.

@P

@R

¯̄̄̄at the optimum

< 0 anddP

d¾2R

¯̄̄̄at the optimum

> 0 (24)

Proof. See the appendix.The relationship between the return on bank’s assets and the probability

of a bank run is very intuitive. Degradation of bank’s assets (a decrease inthe expected return or increase in variance) lead to a higher likelihood of abank run. However, models of bank runs that have multiple equilibria do notpredict such connection. The empirical literature on bank runs has found someevidence of an element of predictability of bank runs, leading to the conclusionthat fundamentals as opposed to sunspots play a role in determining whether arun will occur in a particular bank (Gorton 1988, D’Amato, Grubisic and Powell1997, Peria and Schmukler 1998 or Rojas-Suarez 1998).

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3 Lending NetworksPrevious section has build a model of banks’ liabilities that explained the exis-tence of banks by their ability to pool resources and transform maturity. In thissection, I will turn to the asset side of banks and present a model that deter-mines the return on bank’s assets R, which was so far treated as given randomvariable. The properties of R were derived by aggregation of the returns onindividual projects to which the agents in the economy had access. The agentshad enough resources to carry out all the projects in the economy, however,the possibility of facing an early liquidity need prevented them from doing so.Hence, in this section I will model a situation where there are not enough re-sources to …nance all available projects in the economy and concentrate on themechanism by which scarce …nancing is allocated to abundant projects.11

I keep the basic structure of the economy described in the previous section -the agents have access to a risky technology that requires a unit of …nancing andearns a gross return of Ri. I reinterpret the signals that agents receive aboutthe aggregate return to be the actual realization of returns on their individualprojects. The timing of the model is as follows: agents …rst establish links with…nancial intermediaries and in the second stage they bargain to get fundingfrom the intermediaries with which they have a connection. The connectionwith the …nancial intermediary allows the agents to overcome the asymmetricinformation problem and hence adds value to the transaction between a borrowerand lender. I assume that without such relationship, lending is not possible - seethe introduction for motivation and references to empirical literature supportingthis. The two key assumptions are:

1. Borrowers and lenders have to have a (costly) relationship to be able tonegotiate a loan.

2. The relationships with …nancial intermediaries have to be formed beforethe exact valuation of the projects to be …nanced is known.

My model is a model of relationship lending. The existing theoretical paperson relationship lending concentrated on exclusive bank-borrower relationships. Iwant to build a model that allows …rms to obtain …nancing from several sources(e.g. bank or trade credit, bond or stock issue, etc.) and/or form relationshipswith several di¤erent banks.The existence of a lender/borrower relationship helps the parties to overcome

asymmetric information between them. Hence lenders are able to verify infor-mation about borrowers and this in turn implies that they have a bargainingadvantage. It is then natural to assume that lenders are willing, and do bear thecost of establishing the relationship. I will show that under these assumptions,the networks that emerge as equilibria in my model are constrained e¢cient in11By one of the earliest de…nitions, economics is indeed a discipline concerned with allocation

of scarce resources to alternative uses (xx).

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the sense that they maximize the overall expected payo¤.12 If both borrowersand lenders had to share the relationship formation costs (which I also refer toas link costs), the network structures that emerge as equilibria of the game arenot guaranteed to be e¢cient.13 I do not concentrate on the dynamic problem ofhow such lending networks are formed14 but instead consider the properties ofthe equilibria and examine how the link costs a¤ect the …nancial system relativeto a zero link cost world.The situation corresponds to buyer-seller game that has been studied in the

IO literature (e.g. Kranton and Minehart, 2000). There is a …xed number of…nancial intermediaries (sellers) and each is endowed with an exogenous amountof funds.15 There is a larger number of potential borrowers (buyers) that seekfunding of their projects. The returns on the projects (corresponding to thevalue of the traded goods - loans) are uncertain ex ante. The intermediaries canonly lend to borrowers that they linked by incurring the …xed cost per link.The game has two stages. Second stage is after the links are formed. The

borrowers discover the value (return) of the project they are seeking …nancingfor and this determines their demand for …nancing as a function of the o¤eredinterest rate. The lenders linked to these borrowers are able to verify the returnsand compete with other linked lenders for the borrowers with best returns. Firststage of the game is the link formation stage. Based on the knowledge of thesubsequent stages of the game, the borrowers and lenders form links. I usethe results in Kranton and Minehart (2000) and show that under my speci…cassumptions on the market structure and market allocation mechanism, thelinks form a constrained e¢cient network. I then derive the properties of thereturn on the bank’s portfolio and present several simple simulated examplesfor illustration.

3.1 Network Formation Game

Let me …rst introduce some notation. There is a …nite set of lenders L and a…nite set of potential borrowers B. For simplicity, I assume that each seller isendowed with a unit of liquidity and that each borrower would like to borrowexactly one unit of liquidity.16 There are many more borrowers than lenders,NL = jLj << jBj = NB. Each borrower bj²B has a valuation vj > 0 for the12 If relationships are not formed e¢ciently (maximizing overall welfare), link costs create

larger distirtions and this further butresses my argument that link/transaction costs decreasereturn on bank’s assets thus increasing vulnerability to runs.13 In general, borrowers and lenders bargain over the surplus generated by the link. In

my model, lenders obtain the entire surplus and are assumed to pay for the link formation,hence the networks form e¢ciently. If the surpuls was allocated in a di¤erent way, I couldassume corresponding sharing of the cost of the links and still preserve the e¢cient formationof networks.14 See the related works of Jackson and Watts (2000) on dynamic network formation.15 I assume that liquidity cannot be transferred among intermediaries. This is an extreme

assumption that is suppose to capture the fact that inter-bank markets are subject moralhazards problems and do not function perfectly.16Banks usually have a capacity to satisfy more than one borrower’s needs. The results

below would not be substantially di¤erent in this case.

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loan, given by the realization of the return on her individual project. A borrowercan borrow only if she is linked to a lender. I denote the link pattern as NL£NBmatrix G with elements gij²f0; 1g. If lender li is linked with borrower bj thengij = 1 and gij = 0 otherwise. The set of lenders linked to a particular borrowerbj is denoted as L(bj) := fli²Ljgij = 1g.An allocation of loans is an NL £ NB matrix A with elements aij²f0; 1g

where aij = 1 indicates that borrower bj obtains a loan from a lender li. Anallocation is feasible if gij ¸ aij for all i; j (only linked borrowers can obtain aloan) and for all lenders li²L the sum

Pbj²B

aij · 1 (lenders make at most oneloan each).

3.1.1 Lending in a Network

First, I take the network structure as given and study the expected payo¤sto borrowers and lenders. In order to do so, I have to specify an allocationmechanism that determines which borrowers obtain the loans, as well as a payo¤(price formation) mechanism that speci…es how the parties divide the surplusgenerated by their transaction.

De…nition 5 Consider a price vector p = (p1; ::; pNL) which allocates a priceto each lender. For a network G and a vector of valuations v = (v1; ::; vNB), theprice vector and allocation (p;A) is competitive i¤:

1. A is feasible.

2. If borrower bj borrows from lender li then vj ¸ pi ¸ 0 and pi = minfpkjlk²L(bj)g.3. If borrower bj does not obtain a loan in A then vj · minfpkjlk²L(bj)g.4. If lender li does not lend to any borrower then pi = 0.

Condition 2 and 3 require that there is no excess demand, condition 4 re-quires that there is no excess supply at prices p. Kranton and Minehart (1999)prove that the same set of borrowers (buyers) obtains the loans in all com-petitive allocations.17 However, the price associated with the above conceptof competitive equilibria is not unique. In particular, the authors show thatthe competitive prices form a nonempty convex lattice. The exact payo¤ isundetermined by the requirement of competitiveness of the equilibria becausethe borrowers and lenders have to bargain over the surplus generated by theexistence of a link between them. Since lenders have gained information aboutborrowers linked to them, I assume that lenders are able to extract the entiresurplus.17Proposition 3, part (a). The proof requires that borrowers valuations are generic, i.e. no

two borrowers have exactly the same valuation. Since valuation are assumed to be realizationof a non-degenerate random variable with continuus support, the likelihood of two borrowershaving the same valuation is zero.

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A speci…c mechanism leading to a competitive equilibrium and giving themaximum payo¤ to lenders for a given network structure is the following de-scending bid auction. The lenders simultaneously hold descending-bid auctionswith the same going price of a loan across lenders. Lenders start by o¤ering highinterest rates to their borrowers. If at any time there exists a unique clearableset of borrowers and lenders, they transact at the going price and drop out fromthe bidding. The price of loans declines for the remaining borrowers until allfeasible loans are made and there are no borrowers with links to lenders thatdid not make a loan.

De…nition 6 Let L µ L and B µ B be a subset of lenders and borrowers. LetG0 be the sub-network of the network G that these borrowers and lenders consti-tute. The pair (B;L) is called clearable if and only if there exists a competitiveallocation on the network G0 with the current going price as its competitive price.

The above de…nition of clearable set is not very constructive. Let p be thegoing price, a lender li and a borrower bj constitute a clearable set if theyare linked and if vj · p. However there might be a larger set of lenders andborrowers that can clear at the going price. Consider the following network withvaluations v2 > v1 > v3.

Figure xx Illustration of a clearable setb1 b2 b3

l1 l2

When the going price for a loan is higher than the highest valuation v2no borrowers are interested in the loan. As the price drops to v2 borrowerb2 is interested in a loan but she still has two options for getting the loanand there exist two clearable sets (fb2g; fl1g) and (fb2g; fl2g). Hence the pricedrops further until it reaches v1. At that point, there is a unique clearableset (fb1; b2g; fl1; l2g) and borrower b1 obtains the loan from lender l1, whileborrower b2 obtains the loan from lender l2. The price of both loans is v1.The descending price auction is a well speci…ed game with competitive allo-

cation as the unique equilibrium.

Lemma 7 For a given network G and a vector of valuations v the only equi-librium of the descending price auction is a competitive allocation (A; p). Fur-thermore there does not exist any other competitive allocation (A; p0) with somep0i > pi.

Proof. See the appendix.Although the lenders make their loans to borrowers that are directly linked

to them, the competition from indirectly linked borrowers and lenders does playa role in determining the price that a lender achieves. The following de…nitionof opportunity path characterizes the notion of borrower’s outside option.

13

De…nition 8 Given a network G and an allocation A, an opportunity pathconnecting a borrower bj1 with a borrower bjn is a sequence of lenders I =(i1; ::; in¡1) and borrowers J = (j1; ::; jn) such that 8k²f1; ::n ¡ 1g : gikjk = 1,aikjk = 0, gikjk+1 = 1 and aikjk+1 = 1. A trivial opportunity path connecting aborrower with herself is I = ; and J = ;.Using the notation li¡ bj for a lender and a borrower that are linked but do

not engage in a transaction in A, and bj ! li for a borrower j that obtains aloan from a lender i, we can write an opportunity path as

bj1 ¡ li1 ! bj2 ¡ li2 ! :::! bjn¡1 ¡ lin¡1 ! bjn

If a borrower bj1 is connected to another borrower bjn by an opportunitypath, the valuation of borrower bjn sets an upper bound on the price that theany lender can charge her in a competitive allocation. Suppose that for a givencompetitive allocation, borrower bj1 obtains the loan from a lender lx. Since theborrower obtained the loan from lx and not from li1 , it has to be the case thatpi1 ¸ px (by condition 2 in the de…nition of a competitive allocation). Similarly,borrower bj2 obtained her loan from li1 and not from li2 , and hence pi2 ¸ pi1and so on. Furthermore, since borrower bjn received a loan from lender lin¡1 ,by condition 3 in the de…nition of competitive allocation, her valuation of theloan has to be higher or equal to the price charged for the loan, vjn ¸ pin¡1 .Therefore, we have that vjn ¸ pin¡1 ¸ pin¡2 ¸ ::: ¸ pi2 ¸ pi1 .Therefore, the maximum price that a lender can extract from a borrower

bj is given by the lowest valuation of any borrower linked borrower bj by anopportunity path. Since the descending price auction chooses the price that ismost favorable to lenders, in the equilibrium each lender receives a maximumprice she can extract from borrowers she has connected. The following lemmaformalizes the above argument.

Lemma 9 Given a network G and valuation v, the price p obtained as an equi-librium outcome of the simultaneous descending-bid auction allocation mecha-nism is

p = (maxjfg1j ¢ v(bj)g; ::;max

jfgNLj ¢ v(bj)g)

where v(bj) is the lowest valuation of any borrower linked to borrower bj by anopportunity path.

Proof. See the appendix, the argument in the text above and proposition5 in Kranton and Minehart (1999).

3.1.2 Network Formation

The competitive allocation for a given network G and valuation v is unique upto the price vector. The price is unique given that competition for loans ina network is characterized by the descending price auction. Given the aboveLemma, I can denote the expected payo¤ of each lender for a given network as

Vi(G) = Efpig = Efmaxj[gij ¢ v(bi)]g (25)

14

where the expectation is taken over all possible realizations of the vector ofvaluations v. Given these expected payo¤s, lenders then noncooperatively andsimultaneously form their links and incur a …xed cost c > 0 per link. Strategyof a lender li is a vector gi = (gi1; :::; giNB), where gij = 1 when lender li formsa link with borrower bj . The strategies of the lenders form rows of matrix Gdescribing the network structure.In an equilibrium, lender’s strategy has to maximize her expected pro…ts

which are ¦i(G) = Vi(G) ¡ c ¢PNb

j=1 gij. Lenders cannot have any incentiveto rearrange the pattern of their links given the strategies of other lenders.Therefore, an equilibrium network structure G¤ has to satisfy

8i = 1; ::; NL : g¤i = argmaxgi Vi(G)¡ c ¢NbXj=1

gij (26)

There are e¢ciency gains if the number of links in a network is increased.More links provide more ‡exibility in allocation of loans and borrowers withhigher valuations are more likely to get loans. Consider the extreme cases -if the network consists of exclusive one-to-one relationships between lendersand borrowers (each lender has a link to exactly one borrower and the rest ofborrowers do not have access to any lending facilities),18 then a …xed set ofborrowers always obtains loans, regardless of the realization of the valuationsfor the loans. On the other hand, if there is perfect competition for loans (thenetwork is complete - all borrowers are linked to all lenders), borrowers withhighest valuation for the loans get …nancing and aggregate welfare is higher(without taking into account the costs of building the links).

Figure xx.Exclusive Network vs. Complete Networkb1 b2 b3 b4 b1 b2 b3 b4

l1 l2 l3 l1 l2 l3

In the next section I will show that when one takes into account the linkbuilding costs, the optimal network structure is between these two extremes andthat it is the structure that emerges as the equilibrium of the network formationgame.

3.2 Equilibria of the Game

In this section I want to show two main properties of equilibria of the networkformation game. First, the networks that emerge are constrained e¢cient in the18 Such structure might seem very extreme and unrealistic. However, it represents for ex-

ample the situation where upstream and downstream …rms are integrated. In terms of the…nancing structure, this would correspond to situation where …nancial intermediation andproduction of goods are integrated in one group (e.g. korean chaebols) or where …rms areconstrained to …nance investment from their own cash-‡ow.

15

sense that they maximize the expected overall welfare. Secondly, I will studyhow link building costs a¤ect the pro…ts lenders can expect. In particular, I willshow that the lowest expected pro…t decreases with link building costs. Thelender that will do the worst for a given link building cost (and a network struc-ture) will on average do better if the link costs decreases (and the equilibriumnetwork structure changes).The payo¤ to lenders was characterized by Lemma 9. I …rst show that such

payo¤ is equivalent to the social value of the links each lender has. The socialvalue of a particular link is de…ned as the change in the expected sum of thevaluations for loans that are granted in a network with and without that link.Figure xx Example of Social Payo¤b1 b2 b3

l1 l2

Consider a simple example of the network in …gure xx. The social value ofthe link between lender l1 and borrower b1 is v1. In the network without the linkborrower b1 never obtains the loan and hence the change in social surplus is herexpected valuation for the loan. What would be the change in social surplus if alink between lender l1 and borrower b2 was added? In the original network, theexpected valuations of the borrowers obtaining the loans are v1 +maxfv2; v3g.With the link added, the two borrowers with highest valuations get the loansand the sum of their valuations is (v1+v2+v3)¡minfv1; v2; v3g. The di¤erenceis not trivial expression, however, one can show the it is equal to the change inthe payo¤ of the borrower b1.19

Lemma 10 The payo¤ to lenders is exactly equivalent to the social surplusgenerated by their links.

Proof. See the appendix.

Lenders pay for the creation of the links and on the other hand receive thesocial value of their links. An e¢cient equilibrium network structure maximizesoverall social value and hence maximizes each lender’s contribution to the socialsurplus. As a result, no lenders can deviate and by rearranging her links increaseher contribution to social surplus (which is equivalent to her expected payo¤).Hence we can establish the following proposition.

Proposition 11 The network structure that maximizes overall welfare is anequilibrium of the network formation game.

Proof. See the text above.

Given this result, I can assert that networks can be formed e¢ciently. Ana-logically to Kranton and Minehart (xx) I can show that for if the link building19The payo¤ to lender l1 or the social surplus are di¤erent in the two networks only if

v2 > v3 > v1 or v3 > v2 > v1. In the …rst case the change in social surplus is v3 ¡ v1, in thesecond case it is v2¡ v1. In both cases it is equal to the change in the price lender l1 charges.

16

costs are low enough, only e¢cient network structures are equilibria of the game.The e¢cient network structures can be characterized as ’least link allocativelycomplete’ (LAC). Furthermore, in all LAC networks, each lender has the samenumber of links and, therefore, lenders’ payo¤s are strictly decreasing with linkbuilding costs over this range.20

The situation is somewhat more complicated in networks that are not alloca-tively complete. When link building costs decrease, there might be an incentivefor lenders to add link and improve the allocative e¢ciency of the network.Adding a link increases a lender’s payo¤ because it increases her contributionto social surplus by more than the cost of the link. It might, however, decreasethe payo¤ to some other lender(s).My model of bank runs has made the connection between the expected return

on bank’s assets to the likelihood of run against that bank. If we interpret thedi¤erent level of link building costs and the network structures as situationin two di¤erent countries, the likelihood of an occurrence of a bank run in acountry is determined by the expected return on assets of the worst bank inthat country. Hence I need to show a connection between the lowest expectedreturn to a lender and the link building costs. It turns out the one can showthe following.

Proposition 12 Take the two network structures that emerge as equilibria fortwo given levels of link building costs. Take the lenders that have lowest expectedpro…ts in such networks. The pro…t of such lender in the network with lowerlink building costs is at least as large as the pro…t of such lender in the networkwith higher link building costs.

Proof. See the appendix.

Summary The level of transaction/link costs (together with the parametersof the distribution of project returns) determines the shape of the network thatemerges as an (e¢cient) equilibrium of the game. If link costs are zero or lowenough, the network that emerges is ’allocatively complete’ (ACN), meaningthat under all possible realization of project values, the projects with highestvalues will receive …nancing. If transaction costs are prohibitively large, thenetwork structure that would emerge (provided the costs are not high enough toshut the market altogether) is that of exclusive one-to-one lending relationships.Lenders do not have any choice in their lending decision and have to lend to…rms that have linked to them. Their only choice left is not to lend at all whenthe project is not perceived as pro…table given the costs of funds to the lender- in this case the lender is left with liquid resources that are not generating anyreturn.The incompleteness of the network changes the distribution of R - the econ-

omy is less e¢cient in allocating resources to the best possible uses. Under anACN, since the lender extracted the entire surplus from its borrower, the ex-pected R is the average of the NL highest valuations of the NB borrowers. If the20 See propositions 3,4 and 5 in Kranton and Minehart (2000).

17

network is not AC, some borrowers with the higher valuations cannot borrow.Let us take the exclusive relationships as the polar case. The expected returnR is then given by average of the valuations of the borrowers that are linkedto a lender (and there are NL of these), i.e. average of NB draws from f(Ri).The di¤erence can be substantial - the table below shows some simulations fora speci…c f(Ri) and di¤erent NB and NL.Table 1. Simulated Expected Payo¤s

NL = 1 NL = 3NB ACN Exclusive ACN Exclusive1 0.00 0.00 na na4 1.01 0.00 0.99 -0.035 1.16 0.01 1.67 0.00The above is average of 10,000 draws based on Ri s N(0; 1).The results of the simulations show that the expected highest realization

of standard normal distribution out of 4 trials is about 1.01, compared to theexpectations of a …xed realization from the same distribution which is zero.The probability of getting a draw of 1.01 from a standard normal distributionis 12.3%. The results reported in the table above are in fact a …xed statisticsassociated with distribution functions. xx - introduce the Xi:j notation.

4 ConclusionLess developed countries have higher transaction costs and less developed …nan-cial markets and, hence there will be disjoint lending clubs or unique lendingrelationships. In developed countries, there might be multiple lending channelsfor borrowers. My analysis showed that the nature of the lending network thatemerges in the economy is determined in large part by the level of transactioncosts (in our narrow sense). The network structure, in turn, determines thestatistical properties of the return on banks’ assets. The results from my bankrun model then imply that changes in the properties of the distribution functionof bank’s assets (mean, variance) have consequences on the likelihood of a runon that bank (positive and negative respectively). Hence my argument is thathigher transaction costs lead to ’less allocatively complete’ lending networkswhich in turn lowers the expected returns on banks’ assets and leads to higherlikelihood of bank runs. The same reasoning would apply to any distortion thatin‡uences the allocative e¢ciency of the bank: lower allocative e¢ciency implieslower expected returns and that means higher likelihood of a run.My model has policy implications for crisis prevention. In the short run, the

implications are the same as those of the liquidity crisis models I built upon.However, in the long run, the crises prevention policies have to concentrateon improving the allocative e¢ciency of the economy, e.g. through loweringtransaction or contracting costs (costs of forming the links).It must be stressed that in my model I have concentrated on a single form of

market distortion and the main result was that such distortion makes bank runsmore likely to happen. The policy implication is that in the long run, one must

18

remove such distortion in order to achieve …nancial stability. However, if thereare other market distortions that are more damaging to the economy’s perfor-mance, these have to be addressed …rst. For example, if the largest distortionis the moral hazard in the banking system due to implicit government depositinsurance policy (as in Dooley 1997 model), that distortion is more likely toplay a role in pushing the economy into a currency/banking crises and has tobe addressed …rst.

5 Appendix AProof. Lemma 1Denote the proportion of withdrawals in period 1 by º. To satisfy requested

withdrawals in period 1, the bank liquidates its investment until it has enoughresources to pay out º ¢ c1. The bank has 1¡ I cash on hand and hence if it isnot enough to pay the withdrawals (1¡ I < c1 ¢º), it has to liquidate º¢c1¡(1¡I)

L

of its total resources and has I ¡ º¢c1¡(1¡I)L of its resources left as investment

in the risky asset. Therefore the return for the 1¡ º patient agents is

c2 =R ¢ (I ¡ º¢c1¡(1¡I)

L )

1¡ º (27)

The derivative of c2 with respect to º is:

@c2@º

=¡R ¢ c1L ¢ (1¡ º) +R ¢ (I ¡ º¢c1¡(1¡I)

L )

(1¡ º)2 =R ¢ (¡c1 + I ¢ L+ 1¡ I)

L ¢ (1¡ º)2 < 0

(28)by assumption in the lemma.

Proof. Proposition 2Follows from the text.

Proof. Proposition 3The proof that the switching strategy is the only equilibrium follows directly

the argument sketched in the text. The switching thresholds is derived notingthat an agent that received a signal such that her updated belief about thereturn is Rc, has to be indi¤erent between withdrawing her deposit in period1 or 2. That is, the expected return on the deposit in period 2 conditional onR = Rc has to be equal the return on the deposit in period 1:

E(c2jR = Rc) = r1 (29)

Noting that if the return is equal to its switching threshold, half of the patientagents withdraw their deposit, we have from the proof of Lemma 1 for ¸c =µ+ 1

2(1¡ µ) < 1¡Ic1

(i.e. when the bank had to liquidate some of its investmentwhen ¸c agents withdraw at period 1):

Rc ¢ (I ¡ ¸c¢c1¡(1¡I)L )

1¡ ¸c = r1

19

and for ¸c ¸ 1¡Ic1:

1¡ µ ¢ c1 +Rc ¢ I = r1 (31)

Solving these equations for Rc gives:

Rc =c1 ¡ 1¡ I

Iif µ +

1

2(1¡ µ) < 1¡ I

c1(32)

= c1 ¡ 1 + (1¡ L) ¢ I otherwise

Proof of Proposition 4 Note: the proof here is for the special case of L = 1which reduces the number of …rst order conditions. The general case of L < 1cannot be made with additional regularity conditions on the utility functions.A note on the L > 1 case is available on request from the author. The exist-ing literature has simpli…ed the problem as well: Diamond and Dybvig, Morrisand Shin and Goldstein and Pauzner only consider the case of L = 1, and al-though Chang and Velasco have L < 1, they postulate a speci…c utility functionand (lack of) uncertainty structure so that the optimal solution implies thatthe social planner invests all of her resources into the risky technology (whicheliminates one FOC condition).To prove proposition 4, we will need the following lemma.

Lemma 13 At the optimum choice of the bank:@P@c1

> 0@2P@c1

> 0@P@R< 0

@2P@c1@R

< 0.

Proof. First, note that @Rc

@c1= 1¡¸c

(1¡¸c)2 > 0 and @2Rc

@(c1)2= 2¢(1¡¸c)¢¸c

(1¡¸c)3 > 0.Second, I assume that the distribution F (:) is unimodal. i.e. there are no humpsin the density function.

(i) @P@c1

= @F@R

¯̄R=Rc ¢ @R

c

@c1= f(Rc) ¢ @Rc@c1

> 0 because f(:) is a probabilitydistribution function and hence f(Rc) > 0 since Rc is in the support of R.

(ii) @2P@c1

= @F@R

¯̄R=Rc ¢ @2Rc

@(c1)2+ @2F

@R2

¯̄̄R=Rc

¢³@Rc

@c1

´2> 0 since @F

@R

¯̄R=Rc =

f(Rc) > 0 and @2F@R2

¯̄̄R=Rc

> 0 because at Rc we are to the left of expected

value of R and the distribution is unimodal.

(iii) @P@R= @

@R[F (Rc)] < 0 because increase in mean decreases the value of the

CDF at all points for unimodal distributions.

(iv) @2P@c1@R

= @2

@R2 [F (Rc)] ¢ @Rc@c1

> 0 because at Rc we are to the left of expected

value of R and since distribution is unimodal then @2

@R2 [F (Rc)] > 0.

20

Equation (18) contains a random component u(c2) = u(R¢(1¡c1¢µ)1¡µ ). Weapproximate this random expression by its Taylor expansion around the meanof R:

u(c2):= u(c2) + (R¡R) ¢ u0(c2)¡ 1

2(R¡R)2 ¢ u00(c2) (33)

where c2 =R¢(1¡c1¢µ)

1¡µ . The …rst order condition for the bank maximizing equa-tion (18) with respect to c1 is then approximated as:

0 =@U

@c1= (1¡ P ) ¢ µ ¢ [u0(c1)¡EfR ¢ u0(c2)g] (34)

+@P

@c1¢ [u(1)¡ µ ¢ u(c1)¡ (1¡ µ) ¢ Efu(c2)g]

:= (1¡ P ) ¢ µ ¢

·u0(c1)¡R ¢ u0(c2) +R ¢ u000(c2) ¢ 1

2¾2R]

¸+@P

@c1¢ [u(1)¡ µ ¢ u(c1)¡ (1¡ µ) ¢ Efu(c2)g]

Lemma 14 At the optimal choice of the bank:@2U(@c1)2

< 0@2U@c1@R

< 0

@2U@c1@(¾2R)

> 0

Proof.

1. Proof.

(a) The partial derivative of the FOC (xx) with respect to c1 is:

@2U

(@c1)2:= (1¡ P ) ¢ µ ¢ (35)

¢[u00(c1) +R ¢ u00(c2) ¢ R ¢ µ1¡ µ ¡

1

2¾2R ¢R ¢ uiv(c2) ¢

R ¢ µ1¡ µ ]

+@2P

@c1¢ [u(1)¡ µ ¢ u(c1)¡ (1¡ µ) ¢Efu(c2)g]

¡2 ¢ @P@c1

¢ µ ¢·u0(c1)¡R ¢ [u0(c2)¡ u00(c2) ¢ 1

2¾2R]

¸= (1¡ P ) ¢ µ ¢A+ @

2P

@c1¢B ¡ @P

@c1¢ µ ¢ C < 0

at the optimum choice of c1 by a second order condition of the bank’smaximization problem. To verify this I will prove that A < 0, B < 0and C > 0 which together with the signs of the partial derivatives of

21

P implies that the second order condition holds. B is the expecteddi¤erence of the utility with and without a bank run and hence isnegative (utility is lower with a bank run). To determine the sign ofC we note that the FOC is:

0 = (1¡ P ) ¢ µ ¢C + @P

@c1¢B =) C = ¡

@P@c1

¢B(1¡ P ) ¢ µ (36)

and hence C has the opposite sign than B. The sign of A can bedetermined by noting that:

A = u00(c1) +R2 ¢ µ1¡ µ ¢

·u00(c2)¡ 1

2¾2R ¢ uiv(c2)

¸(37)

:= u00(c1) +

R2 ¢ µ1¡ µ ¢Efu

00(c2)g

and the fact that u"(:) < 0 by assumption on the utility function.

(b) The partial derivative of the FOC (xx) with respect to R is:

@2U

@c1@R= (1¡ P ) ¢ µ ¢ (38)

¢[¡u0(c2)¡R ¢ u00(c2) ¢ 1¡ c1 ¢ µ1¡ µ +

+u000(c2) ¢ 12¾2R +R ¢ uiv(c2) ¢

1

2¾2R ¢

1¡ c1 ¢ µ1¡ µ ]

¡@P@R

¢ µ ¢·u0(c1)¡R ¢ [u0(c2)¡ u00(c2) ¢ 1

2¾2R]

¸¡ @P@c1

¢ (1¡ c1 ¢ µ)

+@2P

@c1@R¢ [u(1)¡ µ ¢ u(c1)¡ (1¡ µ) ¢ Efu(c2)g]

= (1¡ P ) ¢ µ ¢D¡ @P@R

¢ µ ¢C ¡ @P

@c1¢ E + @2P

@c1@R¢B

I have already shown that at the optimum choice of c1, C > 0 andB < 0. We will show that D < 0 and E > 0 which implies that@2U@c1@R

< 0: We can rewrite D as:

D = ¡u0(c2)¡R ¢ u00(c2) ¢ 1¡ c1 ¢ µ1¡ µ + (39a)

+u000(c2) ¢ 12¾2R +R ¢ uiv(c2) ¢

1

2¾2R ¢

1¡ c1 ¢ µ1¡ µ

= ¡u0(c2) + 12¾2R ¢ u000(c2)¡R ¢

1¡ c1 ¢ µ1¡ µ ¢ [u00(c2)¡ 1

2¾2R ¢ uiv(c2)]

:= ¡E fu0(c2) + c2 ¢ u00(c2)g

22

This expression is not negative for all possible speci…cations of util-ity functions - we have to assume that the curvature of the utilityfunction is not excessive. If u(.) was a CRRA utility, we would haveto assume that the risk aversion is less than xx.

(c) The partial derivative of the FOC (xx) with respect to R is:

@2U

@c1@(¾2R)= (1¡P )¢µ ¢ 1

2¢u000(c2)¢R¡(1¡µ)¢ @P

@c1¢ 12¢u00(c2) > 0 (40)

Prove this - the second term is positive and the …rst could be zero(quadratic utility) or we have to assume that u000 is positive (this holdfor example for CRRA utility).

Totally di¤erentiating the FOC (xx) with dc1 6= 0 and dR 6= 0 we obtain:

dc1

dR= ¡

@2U@c1@R

@2U(@c1)2

(41)

and given the signs of the partial derivatives at the FOC, we can conclude thatit is negative at the optimal solution of the bank.Total di¤erentiating the FOC (xx) with dc1 6= 0 and d(¾2R) 6= 0 we obtain:

dc1d(¾2R)

= ¡@2U

@c1@(¾2R)

@2U(@c1)2

(42)

and given the signs of the partial derivatives at the FOC, we can conclude thatit is positive at the optimal solution of the bank. Therefore

@P

@R

¯̄̄̄at optimum

=@P

@c1¢ @c1@R

< 0 and@P

@(¾2R)

¯̄̄̄at optimum

=@P

@c1¢ @c1@(¾2R)

> 0

6 Appendix BProof. Lemma 7The price formation mechanism (the descending price auction) is constructed

so that Lemma 7 holds. Kranton and Minehart (1999) show in proposition 4 thepossible range of competitive price with pmax being the maximum price lenderscan achieve.

Proof. Lemma 9

23

See the characterization of pmax in proposition 5, Kranton and Minehart(1999).

Proof. Lemma 10(Lenders receive the social value of their links)Pick lender l1 and remove all her links. Observe the change in the social

surplus. Take any valuation v. Suppose that with v, borrower b1 would havereceived the loan from lender l1 at a price p1. Consider two cases:

1. p1 = v1: Borrower b1 does not have any opportunity path and hence isnot linked to any other lender. Therefore, the change in social surplus isv1 which is also lender’s l1 payo¤.

2. p1 < v1: There has to be an opportunity path from borrower b1 to bor-rower bn such that vn = p1. This opportunity path represents borrowerb1’s best outside option. Thus when lender l1 disappears, borrower b1displaces borrower bn (see the picture). Borrower bn does not have anoutside option (because that would extend borrower b1’s oppotunity path)and hence does not obtain the loan in the new network. The change insocial surplus is then vn and it is also equivalent to lender l1’s payo¤.

Figure xx. Borrower Replacement along an Opportunity Pathb1 b2 b3

l1 l2

The above holds for arbitrary valuation v and this implies the Lemma.

Proof. Proposition 11Proved in the text.

Proof. Proposition 12(The worst of lender is better o¤ when link costs decrease)Sketch: I conjecture that lower link costs increase the incentives for adding

links. I then prove the proposition in two steps. First I show that when theworst o¤ lender adds a new link and nobody else does, she is better o¤. Second,I show that no lender has an incentive to add a link to a borrower that the worsto¤ lender has linked and, as a result, the worst o¤ lender is not any worse o¤when links are added.The …rst part is trivial. The worst o¤ lender would only add the link if it

made her better o¤.Second part: denote the worst o¤ lender as lw and take arbitrary other two

lenders l1 and l2. I will show that when that lender l1 considers adding a link,she would be better o¤ by linking a borrower connected to lender l2 then aborrower connected to lw. The situation is depicted in …gure xx.

Figure xx.

24

Bw bw B1 b2 B2

lw l1 l2

Borrower bw represents a particular borrower linked to lender l1 and the setBw is a set of the other borrowers linked to lw. Similar notation applies to b2and B2. The set B1 is a set of all borrowers linked to lender l1. Since the lenderlw is the worst o¤ lender, the oportunity starting with bw is expected to havelower valuation of the terminating borrower than an opportunity path startingwith borrower b2. As a result, the price lender can expect to be able to chargeborrower bw, if she links him, is lower than the price she can expect to be ableto charge to borrower b2. Hence she has an incentive to link b2 rather then bw

25

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