European Economic Review 52 (2008) 2854

Received 18 March 2005; accepted 15 January 2007

Available online 3 April 2007

Keywords: Credit market; Bond market; Risk aversion; Adverse selection

ARTICLE IN PRESS

www.elsevier.com/locate/eer

Corresponding author at: Swiss Banking Institute, University of Zurich, Plattenstr. 32, 8032 Zurich,Switzerland. Tel.: +41 1 6343708; fax: +411 6344970.0014-2921/$ - see front matter r 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.euroecorev.2007.01.009

E-mail address: agerber@isb.unizh.ch.1See the Flow of Funds Account of the Federal Reserve System, March 9, 2006, Table F.102.1. Introduction

Corporations have to rely on external nancing whenever internal cash ows are toosmall to nance new projects. The largest fraction of external nancing by U.S.nonnancial corporations comes from debt, while net equity issues are mostly negative.1

The starting point of our paper is a striking observation concerning the corporate debtstructure in the presence of private (bank) and public debt markets (bonds). There isAbstract

We consider a closed economy where a risk neutral bank competes with a competitive bond

market. Firms can nance a risky project either by a bank credit or by issuing a bond which is

directly sold to risk averse investors who also hold safe deposits at the bank. We show that the bank

tends to allocate more capital to lower quality projects but there are some interesting qualications.

If the asymmetric information concerns only the success probability, then we observe adverse

selection while if it concerns only the expected return, bad types are driven out of the market.

r 2007 Elsevier B.V. All rights reserved.

JEL classification: D82; G21Direct versus intermediated nance:An old question and a new answer

Anke Gerber

Swiss Banking Institute, Plattenstr. 32, CH-8032 Zurich, Switzerland

signicant empirical evidence of the fact that bad risks are predominantly nanced bybanks while good risks obtain most of their funds from public debt markets. An early

ARTICLE IN PRESSA. Gerber / European Economic Review 52 (2008) 2854 29study by James (1987) shows that rms announcing new bank loans and privately placeddebt have a higher default risk and are on average smaller than those announcing publiclyplaced straight debt offerings. In an analysis of corporate nancing in Japan, Hoshi et al.(1993) nd that high net worth rms are more likely to use public than bank debt. Johnson(1997) presents evidence of the fact that the proportion of bank debt is negatively relatedto rm size and age and positively related to leverage and to earnings growth volatility. Allthese characteristics can serve as proxies for credit risk: Small and young rms as well ashighly leveraged rms or rms with high earnings growth volatility can generally beconsidered more risky than rms with the opposite characteristics. Similar results for moregeneral private debt are reported in Krishnaswami et al. (1999). Finally, the tremendousgrowth of the market for credit derivatives in the last decade can also be viewed as evidencefor the risk accumulation by banks.2

The theoretical literature that provides explanations for the observed debt structure ofrms is large and too extensive to be reviewed in detail. Hence, we only give a briefoverview over the proposed explanations and contrast them with the contribution of thispaper. The literature mainly focusses on three aspects that can explain the choice betweenbank loans and public debt, namely information costs, monitoring and renegotiation. Theinformation cost aspect was stressed by Fama (1985) who pointed out that the issue ofpublic debt requires the provision of information for a large group of potential debtholders (via bond ratings or audits) while there is only one contracting party in case of abank loan. Fama (1985) concluded that the information cost for public debt nancing ishigher for smaller than for larger rms, so that small rms prefer bank loans while largerms prefer public debt. This prediction is consistent with the empirical evidence in James(1987) and Johnson (1997). In the sequel several authors have pointed out that a bank loaninvolves close monitoring of the nanced project which is not the case for publicly tradeddebt. Several papers take this monitoring activity to be the dening characteristic of abank. Since monitoring is costly bank loans are more expensive than public debt and henceonly those rms rely on bank credit which require monitoring. These can be rms thathave not built up a reputation of repaying their debt as in Diamond (1991) or rms that arehighly leveraged so that the market does not expect them to behave diligently if they arenot monitored as in Holmstrom and Tirole (1997). These predictions are consistent withthe empirical observation that rms announcing new bank loans have a higher default risk(James, 1987) and have a higher leverage (Johnson, 1997) than rms announcing the issueof public debt. Finally, it is much easier to renegotiate a bank loan than publicly tradeddebt since the holders of public debt are typically widely dispersed while a bank loan onlyinvolves one contracting partner. Debt renegotiation can prevent inefcient liquidationand hence rms with a high probability of nancial distress prefer bank loans over bonds(see Chemmanur and Fulghieri, 1994; Bolton and Freixas, 2000). This prediction isconsistent with the observation that bank debt is increasing in the earnings growthvolatility (see Johnson, 1997). Also, as Rajan (1992) argues, debt renegotiation is costlysince banks gain bargaining power over the rms prots once projects have begun.

2Banks and other nancial institutions are the main participants in the credit derivatives market, which

according to the Credit Derivatives Report 2003/2004 of the British Bankers Association grew from $180 billionin 1997 to about $5,000 billion in 2004 and is expected to exceed $8,000 billion in 2006.

He concludes that rms with high quality ( high success probability) projects will preferpublic debt while those with low quality projects prefer bank loans, which is consistentwith the empirical evidence in James (1987).

ARTICLE IN PRESSA. Gerber / European Economic Review 52 (2008) 285430The contribution of our paper is to offer an alternative and to some extent much simpleranswer to the old question, why banks tend to allocate more capital to riskier or moregeneral lower quality projects. Our emphasis is on the role of banks in diversifying risks.While the previous literature has focussed on a risk neutral world, where there is aninnitely elastic supply of funds at an exogenously given interest rate, we take the view thatrisk aversion is a predominant phenomenon, which has to be taken into account in any fullmodel of bank and capital market lending. We will show that the investors risk aversionalone together with the banks ability to diversify risk can explain the debt allocationbetween banks and bond markets.In order to derive the implications of a banks risk diversication activity we disregard

the monitoring role of banks and choose a setting with asymmetric information prior tocontracting (rms have a given project type prior to contracting). In our model amonopolistic bank faces competition from a bond market which limits the rent extractionof the bank. Hence we combine a principal-agent situation with a competitive market.3

Risk neutral rms seek nance for projects and can obtain credit from the bank or issuebonds. There are two types of rms which differ in the quality of their project. Only theproportions of the two types of rms are common knowledge while the type of a single rmis its private information. Following Hellmann and Stiglitz (2000) we measure projectquality along two dimensions, namely the expected return and the success probability of aproject. Risk averse investors can invest their capital in safe bank deposits or in riskybonds. Investment in bonds is risky because by assumption investors cannot diversify theirrisk at the bond market. This assumption can be justied by the fact that building a welldiversied fund of bonds is too costly for individual investors. By contrast the bank candiversify its risk by giving loans to a pool of rms. The bank sets the credit volume and theinterest rates on deposits and credit such as to maximize expected prots. In doing so itanticipates the equilibrium on the bond market. The rms in our model have no nancialresources and there is no equity market. Hence, collateral cannot serve as a screeningdevice and the bank can only screen the rms by setting the interest rate so high that onerm type does not invest at all. In this case we say that the economy is in a screeningequilibrium, while a pooling equilibrium refers to the case where both types of rmsinvest.Different from most models on direct versus intermediated nance4 we nd that

generically there is a mix of nancing source in equilibrium, i.e. rms obtain nance fromthe bank as well as by issuing bonds. This prediction is conrmed by several empiricalstudies (see, for example, Houston and James, 1996; Johnson, 1997). Moreover, our modelcan explain the stylized fact that bad risks receive more nance from the bank relative tothe bond market than good risks. The reason is that in the presence of a riskless depositcontract, which is offered by the bank, the investors demand for bonds increases with theexpected return and decreases with the default risk of the bonds. In equilibrium the credit

3There is some empirical evidence of monopoly power at the level of individual banks (see Cosimano and

McDonald, 1998). However, it turns out that the results in our paper are not driven by the monopoly power of the

bank. Similar qualitative results can be obtained for an oligopolistic banking sector (see Section 3.1).

4Notable exceptions are the papers by Besanko and Kanatas (1993) and Rajan (1992).

volume is equal to the capital required by the rms in excess over what they obtain onthe bond market. Hence, the credit volume is decreasing in the expected return and

ARTICLE IN PRESSA. Gerber / European Economic Review 52 (2008) 2854 31increasing in the default risk. The latter result is consistent with the observation that theproportion of bank debt is increasing in credit risk (see, for example, James, 1987;Johnson, 1997).5

In addition we obtain interesting results concerning the nature of the equilibrium(pooling or screening). If rms only differ in the success probability of their projects ourmodel predicts that there will be a pooling equilibrium, whenever the proportion of highrisk projects in the economy is small, and a screening equilibrium, whenever the proportionis large. In the latter case the low risks are driven out of the market. This is the classicadverse selection effect. If, on the contrary, rms only differ in the expected return of theirprojects, then there is pooling for small proportions of the good project (high expectedreturn) while there is screening if its proportion in the economy is large. If there isscreening we observe a positive selection effect, meaning that the low return project isdriven out of the market. These results are very intuitive: If the proportion of rms that areable to pay a high interest rate on credit gets sufciently large, it is optimal for the bank toviolate the participation constraint for the rms with a low willingness to pay, i.e. we havea screening equilibrium. Since the rms willingness to pay is increasing in the expectedreturn of the project and decreasing in the success probability we obtain the selectioneffects described above. These predictions have not been obtained by the previousliterature and to our knowledge there is no empirical study that has compared thecharacteristics of projects that have been nanced by bank debt with those that wererefused a credit. Subject to the availability of detailed data on bank lending this wouldclearly be an interesting research topic.Our paper is organized as follows. In Section 2 we set up the model and derive rst

results concerning admissible contracts for the bank. The optimal contract for the bank isdetermined in Section 3, where we also discuss some extensions of the model. We concludethe paper in Section 4. All proofs are in the appendix.

2. The model

There are three types of agents in our economy, investors, rms and a monopolisticbank. There are two periods t 0; 1, and there is one good (capital) in each period. Firmsseek nance for a risky project either at a bank or on the bond market and we explicitlyallow for a mix of nancing source. Each project has a xed scale and requires aninvestment of one unit in t 0. The returns of the projects are realized in t 1. Investorsare the only agents who possess funds in our economy. Thus, if the bank wants to give acredit to a rm, it rst has to raise funds from the investors. The precise characteristics ofthe agents are as follows.Investors: There is a continuum of identical investors with mass equal to 1. Each investor

has an endowment of one unit (of capital) in t 0 and nothing in t 1. All consumptiontakes place at t 1.6 Investors are risk-averse expected utility maximizers and theirpreferences for consumption in t 1 are represented by the von NeumannMorgenstern

5We are not aware of any empirical analysis that studies the inuence of the project return on the nancing

source.

6That is, we assume that consumption in t 0 is already completed.

utility function ux lnx for x40.7 If there is no bank and no bond market, anyinvestor consumes her period zero endowment in t 1, i.e. we assume that there is astorage technology and that capital is nonperishable.8

Firms: There is a continuum of rms that are owned and run by managers who aim tomaximize rms expected prots at t 1. The mass of rms is 1 and hence equals the massof investors. We can think of the rm as a start-up. The manager has a project idea but nonancial resources to realize the xed scale project, which requires an investment of oneunit in t 0. The manager will only engage in the project if the expected prot covers hisopportunity cost K40. This participation constraint may, for example, reect hisopportunities to work as an employee for some other rm instead of starting his ownbusiness.

ARTICLE IN PRESSA. Gerber / European Economic Review 52 (2008) 285432We distinguish rms by the quality of their projects which we measure along twodimensions, namely expected return and risk as reected by the projects successprobability (cf. Hellmann and Stiglitz, 2000). Hence, the type of a rm is given by y m;s with m 2 R; s41, where 1=s is the success probability and m is the expected returnof the project. The return of a project with type y m;s then is Qy:sm with probabilitypy:1=s and 0 with probability 1 py. The distribution of project returns across rmswith the same type y is assumed to satisfy a law of large numbers, i.e. there is no aggregaterisk.9 Moreover, we assume that the distribution of returns is independent across projectsof different types.There is asymmetric information concerning the type of a rm which is only known to

the manager. Since the rm has no nancial resources, collateral cannot serve as ascreening device between different projects as in Bester (1985). Managers are not subject tomoral hazard, though, i.e. they cannot choose the quality of their project. Project returnsare not publicly observable unless a rm does not repay its debt and is declared bankrupt.Thus, ex post verication of types is only possible in case of bankruptcy. We will comeback to this point when we discuss credit contracts.We will assume that there are two types of rms in the economy, y1 m1;s1 and

y2 m2;s2, with corresponding proportions l40 and 1 l40 that are commonknowledge among all agents in the economy. Moreover, we assume that

mi K41 for i 1; 2,

i.e. that it is strictly efcient to carry out both projects. In the following let pi pyi 1=si and Qi Qyi simi for i 1; 2.

7The results of this paper are not specic to logarithmic utility functions. The same implications can be derived

for the class of utility functions with constant relative risk aversion (CRRA) r with 0orp1 and r sufcientlyclose to 1. The essential property that we need to obtain the results of the paper is that the investment in the bond

is increasing in the return it delivers in the no-default state. As is well known, CRRA utility functions with

0orp1 have this property.8This assumption could easily be relaxed to allow for a positive depreciation rate without any qualitative change

of our results.9Observe that there is no law of large numbers for a continuum of independent and identically distributed

random variables (see Judd, 1985; Feldman and Gilles, 1985). On the other hand, as Feldman and Gilles (1985)

have shown, our distributional assumptions are innocuous. There is a continuum of random variables, which are

identically distributed according to the distribution we have specied and which satisfy a law of large numbers.We merely have to abandon independence, which is not needed for our results.

Bank: The bank has no nancial resources in either period and has to obtain funds frominvestors in order to lend money to a rm. The bank is risk-neutral and aims to maximizeits expected period 1 prot from lending to rms and borrowing from investors.There are three assets that can be traded in our economy, a deposit contract, a credit

ARTICLE IN PRESSA. Gerber / European Economic Review 52 (2008) 2854 33contract and a bond. There is restricted participation in these markets and there are shortselling constraints and buying oors which are different for the different agents in oureconomy. In our simple model some of these constraints cannot be obtained endogenouslybut they are exogenously justied.Deposit contract: A deposit contract is characterized by a safe return rDX0 which is

independent of the state of the world that is realized in t 1, i.e. independent of the failureof any rms project. Hence, rD 1 is the interest rate on deposits. Only the bank and theinvestors are allowed to trade in the deposit contract and the bank is restricted to go short,while the investors are restricted to go long in this contract. We will see later that thedeposit contract is indeed a safe asset, i.e. that the bank, by choosing interest rates andcredit volumes appropriately, returns rD almost surely.Credit contract (private debt): A credit contract delivers the return rCX0 in t 1 if the

debtor has enough funds to fulll his obligations. Hence, rC 1 is the interest rate oncredit. If the debtor does not repay the credit he has to declare himself bankrupt at no costand the bankrupts funds, if any, go to the creditor. Thus, there is limited liability on thepart of the debtor. Under such a contractual arrangement the creditor has no incentive todeclare himself bankrupt if he has enough funds to repay his debt. Hence, if rC does notexceed the projects return in the good state, a credit contract will always deliver rC or 0depending on whether the project was successful or not. Observe that, as we alreadymentioned above, ex post verication of returns is not possible since there is no way toforce a rm into bankruptcy unless it does not repay its credit. Hence, a standard debtcontract (cf. Gale and Hellwig, 1985) is the only feasible contractual arrangement underlimited liability and there is no way to screen the rms using this instrument withoutdriving one rm out of the market.We restrict the bank to go long and the rms to go short in the credit contract while

investors are not allowed to trade in this contract.Bond contract (public debt): A bond contract is identical to a credit contract except for

different restrictions on market participation. Firms are restricted to go short and investorsare restricted to go long in the bond contract. For ease of presentation we do not allow thebank to trade in bonds. This assumption is innocuous since we would obtain the sameresults if the bank were active on the bond market.10 The risky return of the bond contractin t 1 is denoted by rBX0, i.e. rB 1 is the interest an investor receives for one unit ofbond if the project is successful.The bank chooses the interest rate on deposits and on credit as well as the credit volume

C, where 0pCp1. In our model the bank does not set the deposit volume it is willing toaccept, which again can be justied exogenously.11 All other agents act as price takers, i.e.investors take rD and rB as given and choose the optimal portfolio of deposits and bonds

10It is straightforward to show that for any admissible contract (cf. Denitions 2.1 and 2.2) where the bank

trades in bonds, there exists an admissible contract without trading activity of the bank on the bond market such

that the banks prot is the same. Hence, without loss of generality, we can restrict to admissible contracts, where

the bank does not trade in bonds.

11We have in mind standard savings deposits here.

and rms take as given rC and rB and the credit volume C and choose the optimal nancingstrategies for their projects. We will now analyze the optimization problems of the differentagents.Optimization problem of a firm: Let the rm be of type y m;s. Given

C; 0pCp1; rCX0; rBX0, the rm chooses whether to participate or not and whether toaccept the banks offer or else obtain all nance for the project on the bond market. If

ARTICLE IN PRESSA. Gerber / European Economic Review 52 (2008) 285434CX0 is the amount of credit the rm borrows from the bank and if BX0 is the amount ofbonds it sells on the bond market, then its expected period 1 prots are

Py pymaxf0;Qy CrC BrBg if C BX1;

0 else:

(

The rm will only participate if the expected prots cover its opportunity cost, i.e. ifPyXK . The prot maximizing choice of the rm is independent of its type and is given by

CyC; rC ; rB C if rBXrC ;

0 else;

(ByC; rC ; rB 1 CyC; rC ; rB

and the rm participates if and only if

pyQy CyC; rC ; rBrC ByC; rC ; rBrBXK() m pyCyC; rC ; rBrC ByC; rC ; rBrBXK :12

Since each type of rm is interested in minimizing the repayment in the good state,CyC; rC ; rBrC ByC; rC ; rBrB, and since this repayment is independent of the type, thebank cannot separate types by offering two different credit contracts C1; rC1 and C2; rC2 so that each contract is chosen by exactly one type. Indeed, if the bank would offer twodifferent credit contracts C1; rC1 and C2; rC2 , then either both types would choose thesame offer, or both would obtain all nance on the bond market, or at least one type wouldnot participate. Hence, the bank cannot do better than by offering exactly one contractand the only way it can separate the rms is by offering a credit contract that is acceptableto one type only.Optimization problem of an investor: Each investor can invest in deposits, yielding the

riskless return rD, and in a bond, yielding the return rB unless the rm that issued the bondis bankrupt. If rDo1, an investor will not invest into deposits but rather store her capitaluntil period 1.13 Hence, the bank will never offer rDo1 and we will only consider the caserDX1 in the following. The investor (knowing the rms participation constraint) correctlyanticipates that bankruptcy can only occur in case of a failure of the project.14 We assumethat investors cannot diversify the risk at the bond market, i.e. there is no poolingsecurity as in Bisin and Gottardi (1999) and investors cannot build a pooling bond bythemselves, which is justied if pooling on a small scale is too costly. Hence, risk pooling is

12To be precise, the maximizer of the expected prot function is not unique if prots are never positive for the

given C; rC ; rB. However, in this case, the rm will not participate so there is no harm in selecting a particularmaximizer then. Also, we assume that the rm accepts the banks offer in case of indifference, i.e. whenever

rB rC . Introducing another tie breaking rule would not change our result qualitatively as long as each rmaccepts the banks offer with a positive probability if rC rB.

13Recall that we assumed the depreciation rate to be zero.14Recall from the optimization problem of a rm that it will only participate if expected prots exceed K40,hence in equilibrium there is no bankruptcy in case of a success of the project.

sible. However, the analysis becomes moreinsights. Thus, for ease of presentation wek on the bond market at all. This implies, inbond, then chance determines whether an

Given her belief b about the proportion of type y1 rms in the pool of rms offering a

ARTICLE IN PRESSbond contract, and taking as given rDX1 and rBX0 each investor solves

max b1 p1uDrD p1uDrD BrB 1 b1 p2uDrD p2uDrD BrB

s:t: D;BX0 and D Bp1,() max 1 pbuDrD pbuDrD BrB

s:t: D;BX0 and D Bp1,where pb bp1 1 bp2. Since ux lnx is strictly monotone the investoroptimally chooses B 1D. From the rst order condition, which is necessary andsufcient for a maximum since u is strictly concave, we obtain

Db; rB; rD 1 pbrB

rB rD if pbrBXrD;

1 else:

8>>:

A. Gerber / European Economic Review 52 (2008) 2854 39

If ~rC4s2m2 K, then the solution to (S) is given by ~rC ; ~rD with ~rD 1. In this case the

ARTICLE IN PRESSA. Gerber / European Economic Review 52 (2008) 285440profit maximizing screening contract is ~C; ~rC ; ~rD with

~C 1l

1 p1~rC

~rC 1 1 l

o1

and it yields strictly positive profits for the bank.

If ~rCps2m2 K, then there exists no profit maximizing screening contract.Again the banks prots are decreasing in rD so that it is optimal for the bank to set

~rD 1. However, contrary to the case of a pooling contract, prots are not necessarilyincreasing in rC over the whole domain. Hence, the bank may maximize prots at anintermediate interest rate on credit and the type y1 rm gets a positive rent. This will bethe case if l is sufciently small. Then, the increase in prot due to the increase in rC istoo small to compensate for the decrease in the credit volume and hence the decrease inprot that is caused by the reduction in the investors demand for deposits. If at theintermediate interest rate the type y2 rm would enter the market as well, there existsno prot maximizing screening contract since the bank would like to charge an interestrate factor rC arbitrary close to s2m2 K in this case. Finally, we observe that, when-ever there exists a prot maximizing screening contract, then there is trade on the bondmarket.We now come back to the problem of multiplicity of equilibria on the bond market.

Obviously, the prot maximizing screening contract ~C; ~rC ; ~rD can never be an admissiblepooling contract: Any admissible pooling contract has to satisfy C40. If ~rB is the interestrate factor on bonds supporting ~C; ~rC ; ~rD as a screening contract, then any r^B supportingthis contract as a pooling contract would have to satisfy r^B4~rC ~rB since the rms acceptC^40 only if r^BX~rC . However, if r^B4~rC ~rB, then the participation constraint of the typey2 rm is violated.On the contrary, Example 2.1 shows that the prot maximizing poolingcontract can be an admissible screening contract.16 If the bank proposes this contract itmay turn out that the resulting equilibrium on the bond market leads to screening andgives the bank a lower prot than the best pooling contract. In this case it is difcult topredict the banks behavior. Under a pessimistic attitude the bank will never propose apooling contract if this may result in a screening equilibrium giving the bank a lower protthan the pooling equilibrium. On the contrary, if the bank is optimistic, it will act as ifthere were no multiplicity of equilibria believing that the pooling equilibrium will arise.Since it is not clear whether one should assume an optimistic or pessimistic attitude on thepart of the bank we look for conditions on the parameters of our model under which thereexists a unique equilibrium on the bond market. The following lemma provides suchconditions.

Lemma 3.2. Let y1 m1;s1 and y2 m2;s2 be given with s1m1 K4s2m2 K. LetC^; r^C ; r^D be a profit maximizing pooling contract such that GPC^; r^C ; r^DX0. If

s1 s2 or s1p2s2m2 K

m2 K 1, (1)

then C^; r^C ; r^D is not an admissible screening contract.16Observe that the contract in Example 2.1 is indeed the prot maximizing pooling contract for the givenparameters.

noncon

ARTICLE IN PRESSGPC; rC ; rDX0, then there exists l^40 such that the corresponding profit maximizing poolingcontract C^; r^C ; r^D yields nonnegative profits and is also an admissible screening contract.In the following we will assume that one of the conditions in (1) is fullled so that there

is a unique equilibrium on the bond market. Let y1 m1;s1; y2 m2;s2 be given withs1m1 K4s2m2 K and let a1 m1 K and a2 m2 K . From Theorems 3.1 and 3.2we recall that

GPl 1 pl a2

a2 p2plp2

a2 1

is the prot from the best pooling contract whenever this prot is nonnegative and

GSl p1 ~rC 1 1 p1

~rC

~rC 1 1 l

is the prot from the best screening contract whenever ~rC4s2a2, where

~rC s1a1; lXp1;

min 1 1 p1p1p1 l

p ; s1a1( )

else:

8>>>:

Moreover, from the proof of Theorem 3.2 we know that GSl gives an upper bound for the

prot from a screening contract even if there exists no prot maximizing screening contract,

i.e. if ~rCps2a2. Comparing GSl with G

Pl then gives the main result of our paper:

Theorem 3.3 (Optimal contract). Let y1 m1;s1 and y2 m2;s2 be given withs1m1 K4s2m2 K.

(i) If s1 s2, then there exists 0olo1 such that the banks profits are maximized at theprofit maximizing pooling contract for lpl and at the profit maximizing screeningcontract for lXl.

(ii) If s1as2 and s1p2s2m2 K=m2 K 1, then there exists 0olo1 such that thebanks profits are maximized at the profit maximizing pooling contract for lpl and at ascreening contract for lXl. If m1 K41 is sufficiently small, then for all lXl thereexists a profit maximizing screening contract.

The profit maximizing pooling and screening contracts are the contracts defined inIf fThes1as2 and s142s2m2 K

m2 K 1.

or some l40 there exists a profit maximizing pooling contract C; rC ; rD withLemquilibria for all l for which the prot maximizing pooling contract gives the bank anegative prot. This is shown by the next lemma (see also Example 2.1 where bothditions in (1) are violated).

ma 3.3. Let y1 m1;s1 and y2 m2; s2 be given with s1m1 K4s2m2 K and letThe conditions in (1) are not only sufcient but also necessary to rule out the multiplicityof e

A. Gerber / European Economic Review 52 (2008) 2854 41orems 3.1 and 3.2, respectively.

The theorem shows that the bank optimally chooses a pooling contract whenever theproportion of type y1 rms is small while it optimally chooses a screening contractwhenever their proportion is large. Under a screening contract type y2 rms drop out oftheassuhighto c

ARTICLE IN PRESSA. Gerber / European Economic Review 52 (2008) 285442keep interest rates low and contract with all rms. Clearly, the details that lead to thisresult are more involved as can be seen from the proof of Theorem 3.3 but the mainmechanism is the one described before.As we have seen it is always the rm with the lower willingness to pay that is driven out

of the market under a screening contract. Since a rms willingness to pay is given bysimi K there is a trade-off between the success probability and the expected return thatdetermines which rm is driven out of the market. Hence, it is not clear whether the betteror lower quality project obtains nance, i.e. whether we have a positive or an adverseselection effect. Let us consider two interesting extreme cases.The case m14m2; s1 s2: In this case the return distribution of type y1s project

dominates the one of type y2s project in the sense of rst order stochastic dominance.Since the repayment probability is the same for both projects but type y1s projectdelivers a higher expected return, rms with the high return project are willing to pay ahigher interest rate on credit than rms with the low return project. Hence, there is apositive selection effect: We observe screening if the proportion of good projects (highexpected return) is large, in which case the rms with low return projects drop out ofthe market.The case m1 m2;s14s2: In this case the return of type y1s project is a mean preserving

spread of the return of type y2s project, i.e. the return distribution of type y2 dominates theone of type y1 in the sense of second order stochastic dominance. Since both projects havethe same expected return but type y1 has a lower success and hence repayment probabilitythan type y2, rms with the high risk project (type y1) are willing to pay a higher interestrate on credit than rms with the low risk project (type y2). Hence, there is an adverseselection effect: If the proportion of high risk projects is large, there is screening and thelow risks drop out of the market.In addition to the selection effects described above our results show that lower quality

projects obtain more capital from the bank relative to the bond market than higher qualityprojects. From Theorem 3.3 and the characteristics of the optimal screening and poolingcontracts we obtain the following comparative statics results.17

(i) In the region where pooling is the optimal choice for the bank (lol), the creditvolume is increasing in the risk of type y1s project and it is decreasing in the expectedreturn of type y2s project.18 Moreover, the credit volume is increasing in theproportion of the project with the higher risk, i.e. if s14s2, then the credit volume isincreasing in l, and if s1os2, then the credit volume is increasing in 1 l.

17We have to restrict to local statements since we do not know how the credit volume in the optimal pooling

contract compares to the one in the optimal screening contract at the threshold l, where we have a change ofregimes from pooling to screening.the18market and only type y1 rms obtain nance. This result is very intuitive. From ourmption that s1m1 K4s2m2 K it follows that type y1 rms are willing to pay aer interest than type y2 rms. Hence, if the proportion of type y1 rms is large, it paysontract with these rms only by asking for an interest rate on credit that is too high fortype y2 rms. On the contrary, if the proportion of type y1 rms is small it is better toObserve that the credit volume is independent of the expected return of project 1.

banobttheindcom

ARTICLE IN PRESSCequ

theks (see Cosimano and McDonald, 1998) it is important to note that the resultsained in this paper are not driven by the monopoly power of the bank. A full analysis ofcompetitive case clearly goes beyond the scope of this paper. Nevertheless, we want toicate why our results are quite robust with respect to changes in the degree ofpetition in the banking sector.onsider the case where there is competition between two banks. Obviously, inilibrium both banks must make zero prots. Moreover, since banks compete for funds,(ii) In the region where screening is the optimal choice for the bank (lXl) and wherem1 K is sufciently small the credit volume is increasing in the risk of the nancedproject and decreasing in its expected return. Also, the credit volume is increasing inthe proportion of the nanced project. Hence, if the nanced project is the riskier one,then the bank allocates more capital to this project if its proportion in the economyincreases.

Hence, in a nutshell, the higher the risk of a project or the lower its expected return, themore capital it obtains from the bank relative to the bond market. The intuition for thisresult is very simple: The lower the expected return of a project, the lower the interest rateon credit and bonds a rm is willing to pay. In the presence of a riskless deposit contract alow interest rate on bonds decreases the investors demand for bonds and hence increasesthe credit volume by the market clearing condition. Similarly, a higher default risk onbonds decreases the investors demand for bonds and therefore increases the credit volume.We have seen that the banks preference for lower quality projects does not only concernthe credit volume but also the type of contract (pooling or screening) that is chosen. Eventhough there can be a positive selection effect, where the low return project does not obtainnance, this positive selection is easily turned into an adverse selection if the risk of the lowreturn project becomes sufciently large. To see this observe that if m1om2 and s1 issufciently large compared to s2, then type y

2 is driven out of the market. Thus, overallour results conrm the stylized fact that bad risks obtain more nance from banks relativeto bond markets than good types. Our ndings are natural in a context where a risk neutralbank has to raise funds from risk averse investors whose demand for the safe depositincreases with the riskiness of the bond and decreases with its return. This effect is notcaptured in partial equilibrium models where it is usually assumed that banks face aninnitely elastic supply of funds by investors at some exogenously given interest rate. Inour model the supply of funds varies with the types of rms seeking nance on the bondand credit market and the bank takes this into account when setting the credit volume andthe interest rates. Hence, our approach offers a new explanation for the stylized factmentioned above and this is the risk aversion of investors.Observe that, by denition, in our model there is no credit rationing in equilibrium (cf.

Stiglitz and Weiss, 1981). If at the given interest rates for a bank credit and for bonds arm is willing to obtain nance its demand is satised. Even if the bank does not offer tonance the whole project, the remaining funds can be raised on the competitive bondmarket, where interest rates adjust such that demand equals supply.

3.1. Competitive banking sector

Although there is some empirical evidence of monopoly power at the level of individual

A. Gerber / European Economic Review 52 (2008) 2854 43equilibrium interest rate on deposits will be maximal subject to the participation

constraints of the rms and the zero prot condition of the banks. Hence, in a pooling

ARTICLE IN PRESSA. Gerber / European Economic Review 52 (2008) 285444equilibrium the maximal interest rate factor that banks can pay is given by rD pls2m2 K since rC s2m2 K is the maximal interest rate factor on credit thattype y2 rms are willing to pay. One can show that banks can pay a higher interest rate ondeposits under pooling than under screening if l is small. For l large screening allows for ahigher interest rate than pooling. Hence, we obtain the same selection effects as in themonopoly case: For l small there will be a pooling equilibrium and for l large there will bea screening equilibrium.Concerning changes in the credit volume with respect to changes in risk and return of

the nanced projects we rst observe that there is no trade on the bond market if theeconomy is in a pooling equilibrium. This follows from the fact that investors invest alltheir capital at the bank if rD plrB.19 Hence, in a pooling equilibrium rms obtain alltheir nance from the bank and the credit volume is invariant with respect to changes inrisk and return. In case of a screening equilibrium, however, there is trade on the bondmarket. For the same reason as in the monopoly case the credit volume is decreasing in theexpected return and increasing in the proportion of the nanced project. Moreover, for abroad range of parameters the credit volume is increasing in the risk of the nancedprojects.

3.2. Screening of projects

In our model rms have no capital, so that the bank can only use interest rates to screenthe projects. As we have seen, in this case screening necessarily drives one rm type out ofthe market. In the following we briey discuss the effect of introducing internal rm capitalwhich allows for collateral to be used as an additional contractual instrument for screeningprojects. We will provide an informal and intuitive argument for why we expect that themain insight of our benchmark model remains true in this extended model, namely thatlower quality projects receive more nance from the bank relative to the bond market thanhigher quality projects.Consider a situation, where rms have some capital A40 which is not sufcient to self-

nance the project, i.e. Ao1. For simplicity suppose that rms differ only with respect tothe success probability of their projects, i.e. there is a high risk and a low risk type withsuccess probability 1=sH and 1=sL, respectively, where sH4sL. In addition to setting thecredit volume and interest rate on credit the bank can now ask for a collateral S 2 0;A tobe paid in case the nanced project is not successful. Clearly, with a positive collateral forbank credit the interest rate on bonds has to be larger than the interest rate on credit.Otherwise rms would obtain all nance from the bond market. Also, the high risk typerm suffers more from an increase in collateral than the low risk type rm and hence, forany given credit volume, the high risk rm requires a larger decrease in the interest rate oncredit than the low risk rm for any marginal increase in collateral. Suppose now that thebank offers the same contract S;C; rC to all rms. Then it can improve by offering anadditional contract SH;C; rCH with SHoS and rCH4rC which will be preferred to theoriginal contract by the high risk type only. In order to satisfy the market clearingcondition on the bond market, the bank has to decrease the credit volume it offers to the

19As in the monopolistic bank case one can show that in equilibrium the interest rate on bonds equals theinterest rate on credit, i.e. rB rC .

ARTICLE IN PRESSlow risk rm, i.e. CLoC. To see this observe that a decrease in collateral decreases thesupply of bonds. One way to restore market clearing then is to decrease the credit volumefor the low risk type which increases the supply of bonds. Given these observations weconjecture that the bank maximizes prots by offering a menu of separating contractsSH;CH; rCH; SL;CL; rCL with SHoSL;CH4CL and rCH4rCL such that the high risktype chooses the contract with low collateral, large credit volume and large interest rate,while the low risk type prefers the contract with high collateral, small credit volume andlow interest rate. Moreover, we conjecture that the total credit volume is decreasing in theexpected return of the projects (which is assumed to be the same for both rm types) forthe same reason as in our benchmark model: An increase in the expected return allows forhigher interest rates on credit which will drive up the interest rate on bonds and henceincreases the demand for bonds. Market clearing then requires that the credit volume mustdecrease.Summarizing, as in our benchmark model without internal rm capital we expect to see

more bank nance relative to the bond market for high risk and low return projects.A rigorous analysis which requires more subtle arguments than the ones we have providedabove is left for future research.

4. Conclusion

We have presented a simple model of a closed economy where a bank competes with abond market on both sides of its balance sheet: Consumers can invest their capital in riskybonds and in safe deposits and rms can issue bonds and obtain a bank credit. Therefore,when setting the credit volume and the interest rates, the bank has to take into account theresulting equilibrium on the bond market. In particular, the bank does not face aninnitely elastic supply of funds at an exogenously given interest rate as it is commonlyassumed in the literature.Different from most models on direct versus intermediated nance in the literature our

results predict a mix of nancing source, i.e. neither the bank nor the bond market has itsown clientele. This is conrmed by several empirical studies showing that a large fractionof rms has a mix of bank and public debt outstanding (Houston and James, 1996;Johnson, 1997), i.e. bank lending remains an important source of nance even if rms haveaccess to public debt markets. Moreover, we have seen that the credit volume is increasingin the default risk and decreasing in the expected return of the nanced projects. Thisprovides a new explanation for the fact that banks allocate more capital to lower qualityprojects. Unlike earlier work, which has focused on monitoring and relationship banking,our results are driven by the risk aversion of investors and the banks ability to diversifyrisk. We do not claim that the monitoring or relationship aspect of banking is notimportant in explaining the allocation of capital across risky projects. Instead, we want topoint out that there is an additional, much simpler explanation, which does not appeal tovery sophisticated bank services like monitoring or contract renegotiation and which seemsto have been overlooked so far. Risk aversion is a predominant phenomenon and it shouldcome at no surprise that it has signicant inuence on the allocation of capital and risk inan economy.Our model also has interesting implications for the case where the proportion of the high

risk and/or high expected return project gets large. In this case we have seen that there will

A. Gerber / European Economic Review 52 (2008) 2854 45be a screening equilibrium with adverse selection, whenever rms only differ in the default

riskexp

proHoandto t

ARTICLE IN PRESSA. Gerber / European Economic Review 52 (2008) 285446Acknowledgments

I am grateful to Thorsten Hens, Jorg Naeve, Georg Noldeke, Bodo Vogt and, inparticular, to two referees for valuable comments. I also thank seminar participants atBonn, Hohenheim, Zurich, and to participants at the EEA meeting in Lausanne and theannual meeting of the association Verein fur Socialpolitik in Innsbruck.

Appendix A. Proofs

Proof of Lemma 2.1. Let C; rC ; rD be an admissible contract with C40 and let rBXrC bethe interest rate factor on the bond market supporting C; rC ; rD as a pooling or screeningcontract. Suppose by way of contradiction that r^BorC clears the bond market as well. IfC; rC ; rD is an admissible pooling contract it follows that given r^B both types of rms willreject the banks offer and demand one unit of capital on the bond market. Hence, for thebond market to clear each investor has to invest all his capital on the bond market whichhe will never do given his preferences as we have seen before.It remains to consider the case where C; rC ; rD is an admissible screening contract.

W.l.o.g. let it be the type y1 rm that participates at this contract. If r^BorC then eitherboth types of rms will participate and reject the banks offer leading to the samecontradiction as above. Or else only the type y1 rm participates. But then the bondmarket cannot clear at the interest rate factor r^B since it cleared at rBXrC : The type y1 rmdemands more capital on the bond market at r^B while the investors will invest less capitalthan at rB. &

Proof of Lemma 2.2.

(i) Let C; rC ; rD be an admissible pooling contract and let b be the corresponding beliefof the investors and rB the interest rate factor on bonds supporting C; rC ; rD. Thenb l and C40 immediately follows from the market clearing condition and the factthat Db; rB; rD40. This implies rBXrC from the optimization problem of the rms. If,in addition, GPC; rC ; rD CplrC rDX0, then it follows that plrCXrD sinceC40.

(ii) Let C; rC ; rD be an admissible screening contract and let b be the corresponding beliefof the investors and rB the interest rate factor on bonds supporting C; rC ; rD. W.l.o.g.let b 1. From the market clearing condition on the bond market (condition (iv) ofperf20t would be desirable to distinguish empirically our explanation for the role of banks inject nancing from other explanations that have been provided in the literature.wever, hard evidence on the relative importance of monitoring, relationship bankingrisk diversication may be difcult if not impossible to obtain, so the best we can do isest the predictions of the theoretical models. In this respect, as we have seen, our modelorms very well.insiI, and a screening equilibrium with positive selection, whenever rms only differ in theected return of their projects. Putting this prediction to an empirical test could be veryghtful since other theoretical models have not produced similar results.20Such a test would require detailed data about bank lending, which may not be easily available.

sinc

C^4and

inve

ARTICLE IN PRESSDl; r^B; rD 1 pl r^B

r^B rD41 plrB

rB rD C

e 1 plrB=rB rD is decreasing in rB. Let C^ Dl; r^B; rD and r^D rD. Then,C and we will show that C^; r^C ; r^D is an admissible pooling contract supported by r^Bthat GPC^; r^C ; r^D4GPC; rC ; rD. To this end let b l be the correct belief of the

iDenition 2.2) it follows that

Db; rB; rD 1 l1 C.Hence, GSC; rC ; rD lCp1rC rD 1 l1 rDX0 immediately implies thatp1r

CXrD if C40 since rDX1. rBXrC follows as above. Since pb p1 this proves ourlemma. &

Proof of Lemma 2.3. Let rB and r^B be two interest factors on bonds supporting theadmissible contract C; rC ; rD and let b be the corresponding belief of the investors.Observe that independently of rB it is always the same rm that is driven out of the market.Let GPC; rC ; rDX0, respectively GSC; rC ; rDX0. Consider rst the case where C40.Then, by Lemma 2.2 it follows that rBXrC ; r^BXrC ; pbrBXrD and pbr^BXrD: SinceDb; rB; rD Db; r^B; rD by the market clearing condition (iv) in Denitions 2.1 and 2.2, itthen follows that

Db; rB; rD 1 pb rB

rB rD 1 pbr^B

r^B rD Db; r^B; rD.

Since rDX1 this implies rB r^B.It remains to consider the case where C 0. By Lemma 2.2 this can only be the case if

C; rC ; rD is a screening contract. By the market clearing condition (iv) in Denition 2.2 itfollows that Db; rB; rD Db; r^B; rD 1 bl 1 b1 lo1. Hence, as above weconclude that rB r^B. &

Proof of Lemma 3.1. Let C; rC ; rD be an admissible pooling (screening) contract and let bbe the corresponding belief of the investors and rB the interest rate factor on the bond

market supporting C; rC ; rD as a pooling (screening) contract. By Lemma 2.2 we knowthat C40; rBXrC and pbrCXrD since GPC; rC ; rDX0, respectively GSC; rC ; rD40.(Observe that GSC; rC ; rD40 implies that C40.) Hence Db; rB; rD 1 pb rB

rBrDfollows from the optimization problem of the investors. Now suppose rB4rC . Then Co1since C 1 would imply Db; rB; rD 1 and hence rD pbrB4pbrCXrD which isimpossible. We dene

r^C CrC 1 CrB.Then, obviously, rB4r^C4rC . Let r^B r^C . We will now consider separately the cases of apooling and a screening contract.Pooling: If C; rC ; rD is an admissible pooling contract, it follows that b l by

condition (iii) of a pooling contract. Then, by denition, plr^B4plrCXrD which implies

A. Gerber / European Economic Review 52 (2008) 2854 47stors. Since r^B r^C we get Cy C^; r^C ; r^B C^ for i 1; 2, so that condition (i) of an

ARTICLE IN PRESSadmissible pooling contract is satised. Moreover, for i 1; 2,mi piC^r^C 1 C^r^B mi pir^C

mi piCrC 1 CrBXK ,by denition of r^C so that the participation constrained (ii) is satised. Conditions (iii) and(iv) are fullled by denition. Finally,

GPC^; r^C ; r^D C^plr^C r^D4CplrC rD GPC; rC ; rD.Screening: We only consider the case where b 1 is the belief of the investors

corresponding to the admissible screening contract C; rC ; rD. This implies pb p1.From p1r^

B4p1rCXrD it follows that D1; r^B; rD 1 p1r^B=r^B rD41 p1rB=rB

rD D1; rB; rD by the same monotonicity argument as above. Also, from condition (iv)of an admissible screening contract it follows that D1; rB; rD 1 l1 C41 l sothat D1; r^B; rD41 l. We dene

C^ D1; r^B; rD 1 l

l

and will argue that C^; r^C ; r^D with r^D rD is an admissible screening contract that strictlyincreases the banks prot over the contract C; rC ; rD. To this end let b 1 be the correctbelief of the investors. Since r^B r^C we get Cy1 C^; r^C ; r^B C^, so that condition (i) of anadmissible screening contract is satised. Moreover,

m1 p1C^r^C 1 C^r^B m1 p1r^C m1 p1CrC 1 CrBXK

4m2 p2CrC 1 CrB m2 p2r^C

m2 p2C^r^C 1 C^r^Bby denition of r^C so that condition (ii) is satised. Conditions (iii) and (iv) are fullled bydenition. Finally,

GSC; rC ; rD lp1CrC 1 lD1; rB; rDrD D1; rB; rD 1 lp1rC 1 lD1; rB; rDrD D1; rB; rDp1rC rD 1 l1 p1rCoD1; r^B; r^Dp1rC r^D 1 l1 p1rCoD1; r^B; r^Dp1r^C r^D 1 l1 p1r^C GSC^; r^C ; r^D,

where the second equality follows from the market clearing condition on the bond market

and the last inequality follows from the fact that r^C4rC and D1; r^B; r^D41 l. &Proof of Theorem 3.1. The necessity and sufciency of the condition plmini1;2 fsimi

A. Gerber / European Economic Review 52 (2008) 285448KgX1 immediately follows from the constraints of the optimization problem (P). Now let

ARTICLE IN PRESSthis condition be satised. Then we compute

qqrC

FPrC ; rD 1 plrC rD2 plrC2 2plrCrD rD2

which is easily seen to be positive for all rCX0; rDX1, since plo1. AlsoqqrD

FPrC ; rD 1 pl2 rC2

rC rD2

which is negative for all rCX0; rDX1 since plo1. Hence, the solution to (P) is given byr^C mini1;2 fsimi Kg; r^D 1. Since FPr^C ; r^D is the maximum prot the bank canachieve with a pooling contract, the optimal pooling contract is given by C^; r^C ; r^D withC^ 1 plr^C=r^C 1. Obviously, C^o1 if and only if plr^C41. &

Proof of Theorem 3.2. Let p1rCXrDX1. Then

qqrD

FSrC ; rD 1 p12rC2

rC rD2o0

which implies that FSrC ; rD is maximized for ~rD 1. Hence, it remains to solve thefollowing optimization problem:

maxrC

FSrC ; 1 p1rC 1 1 p1rC

rC 1 1 l

s:t: p1rCX1 and

s1m1 KXrC4s2m2 K. 2We compute

qqrC

FSrC ; 1 1 p1rC 12 p1rC2 2p1rC 1 1 lp1 (3)

and q2=qrC2FSrC ; 1o0 since rC41. Hence, FSrC ; 1 is strictly concave in rC for rCwith p1r

CX1. From (3) it follows that q=qrCFSrC ; 140 for all rC if lXp1. In this casethe solution to (2) is given by ~rC s1m1 K.If lop1, then for rC with p1rCX1, q=qrCFSrC ; 140 if and only if rCo1

1 p1=p1p1 l

p. Hence, the solution to (2) is given by

~rC min 1 1 p1p1p1 l

p ; s1m1 K( )

whenever ~rC4s2m2 K. In this case let

~C 1l

1 p1~rC

~rC 1 1 l

.

Since p1 ~rC41 it follows that

qFS~rC ; 1 p ~rC 140.

A. Gerber / European Economic Review 52 (2008) 2854 49ql 1

ARTICLE IN PRESSMoreover, liml!0 FS~rC ; 1 0. Together this implies FS~rC ; 140 for all l40. Hence, ~C; ~rC ; ~rD is the prot maximizing screening contract and it yields strictly positive protsfor the bank.If ~rCps2m2 K there exists no solution to (2) and hence no prot maximizing

screening contract. &

Proof of Lemma 3.2. Let C; rC ; rD be an admissible pooling contract with plrCXrD. LetrB be the interest rate factor supporting C; rC ; rD as a pooling contract and let ~rB be theinterest rate factor supporting C; rC ; rD as a screening contract. Then ~rB4rB and

C 1 pl rB

rB rD 1

l1 p1

~rB

~rB rD 1 l

by the market clearing conditions in the denition of an admissible pooling and screeningcontract. This implies

lCo 1 p11 plC 1 l,

() 1 loC 1 p1 l1 pl1 pl . (4)

Now let C^; r^C ; r^D be a prot maximizing pooling contract with GPC^; r^C ; r^DX0. Inorder to show that C^; r^C ; r^D is not an admissible screening contract it sufces to show thatC^ violates (4). Let a2 m2 K . Since plr^CXr^D (nonnegative prots of the bank) we canapply the argument above. Consider rst the case where s1 s2. Then p1 p2 pl forall l and (4) transforms to

1 loC^ 1 p11 l1 p1

which is impossible since C^p1.Now consider the case where s1as2 and s1p2s2a2=a2 1. By Theorem 3.1 we know

that C^ 1 pla2=a2 p2. Using this expression (4) is equivalent to

a2p1 p2 l2 lp21 a2a2p1 p2

p2 a2p1a2p1 p2

40. (5)

If p14p2, then (5) is violated for all l withp2 a2p1a2p1 p2

plp1.

Since plr^CXr^D 1 one immediately veries that (5) is indeed violated for the given l, i.e.C^; r^C ; r^D is not an admissible screening contract.If p1op2, then (5) is violated for all lp1. Hence, as in the previous case we have proved

that C^; r^C ; r^D is not an admissible screening contract. &Proof of Lemma 3.3. Dene a1 m1 K and a2 m2 K . Under the assumptions in thestatement of the lemma let C^; r^C ; r^D be a prot maximizing pooling contract for somel^40 such that GPC^; r^C ; r^DX0. Let rB be the interest rate factor on the bond market

A. Gerber / European Economic Review 52 (2008) 285450supporting C^; r^C ; r^D as a pooling contract. From Theorem 3.1 we know that

ARTICLE IN PRESSrB r^C a2=p2; r^D 1, and C^ 1 pl^a2=a2 p2. Then, C^; r^C ; r^D is also anadmissible screening contract if and only if there exists r^B4rB such that

C^ 1l^

1 p1r^B

r^B rD 1 l^

(6)

and C^r^C 1 C^r^Bps1a1. (7)From Eq. (6) it follows that r^B4rB if and only if

l^C^o 1 p11 pl^

C^ 1 l^ (8)

() a2p1 p2 l^2 l^p21 a2a2p1 p2

p2 a2p1a2p1 p2

40. (9)

Also, from Eq. (6) it follows that

r^B l^C^ 1 l^l^C^ p1 l^

. (10)

Case 1: p14p2. Let C; rC ; rD be a prot maximizing pooling contract for some l40such that GPC; rC ; rDX0. Since p14p2 it follows that pl is increasing in l. Hence, for alll^Xl there exists a prot maximizing pooling contract giving the bank a nonnegativeprot. One veries that the inequality in (9) is satised for all l^ under the conditions of thelemma and given that p14p2. Moreover, from (10) and C^ ! 1 p1a2=a2 p2 forl^! 1, it follows that r^B ! a2=p2 for l^! 1. Hence, r^Boa1=p1 if l^ is close to 1 since byassumption a2=p2oa1=p1.Choose l^ such that the corresponding r^B as dened in (10)satises r^Boa1=p1 and let C^; r^C ; r^D be the prot maximizing pooling contract for l^. Ourarguments show that (6) and (7) are satised so that C^; r^C ; r^D is also an admissiblescreening contract.

Case 2: p1op2. Let C; rC ; rD be a prot maximizing pooling contract for some l40such that GPC; rC ; rDX0. It is straightforward to see that the inequality in (9) is satisedfor all l^ such that l1ol^o1, where

l1 a2p1 p2a2p2 p1

.

Consider rst the case where l1X0. Then pl^r^C pl^a2=p2X1 for all l^, whereC^; r^C ; r^D is the corresponding prot maximizing pooling contract. Hence, as above wecan choose l^ close to 1 such that r^Boa1=p1 and the corresponding prot maximizingpooling contract gives nonnegative prots and is an admissible screening contract.

Consider now the case where l1o0. Then the inequality in (9) is fullled for all l^. SinceplrC pla2=p2X1, pl is decreasing in l (because p1op2) and p1a2=p2o1 (becausel1o0), it follows that there exists l^Xl such that pl^a2=p2 1. Hence, for the

A. Gerber / European Economic Review 52 (2008) 2854 51corresponding prot maximizing pooling contract C^; r^C ; r^D it is true that C^ 1 and

ARTICLE IN PRESStherefore

C^r^C 1 C^r^B r^C a2=p2oa1=p1.Thus, again (6) and (7) are satised, i.e. C^; r^C ; r^D is an admissible screening contract. &Proof of Theorem 3.3. Let y1 m1;s1 and y2 m2; s2 be given with s1m1 K4s2m2 K. Dene a1 m1 K and a2 m2 K . Then

GPl 1 pl a2

a2 p2plp2

a2 1

is the prot at the best pooling contract whenever this prot is nonnegative and

GSl p1 ~rC 1 1 p1

~rC

~rC 1 1 l

is the prot from the best screening contract whenever ~rC4s2a2, where

~rC s1a1; lXp1;

min 1 1 p1p1p1 l

p ;s1a1( )

else:

8>>>: (11)

Moreover, from the proof of Theorem 3.2 we know that GSl gives an upper bound for

the prot from a screening contract also in the case where ~rCps2a2.Since

plp2

a2Xminfp1; p2g

p2a241

for all l whenever s1 s2 or s1as2 and s1p2s2 a2a21, it follows that GPl40 is bounded

away from 0 for all l40. Hence, from liml!0 GSl 0 we conclude that G

SloG

Pl if

l is sufciently small. Moreover,

liml!1

GSl 1 p1

a1a1 p1

a1 1 and

liml!1

GPl 1 p1

a2a2 p2

p1p2a2 1

.

Hence, liml!1 GSl4liml!1 G

Pl if and only if

a1a1 p1

a1 14a2

a2 p2p1p2a2 1

. (12)

The right-hand side of this inequality is easily seen to be increasing in a2. Since byassumption s1a14s2a2 it follows that a2oa1p2=p1 and therefore (12) is satised.We further observe that G

Pl is a concave function in l while G

Sl is an increasing

convex function in l since

d

dlGSl p1 ~rC 1

A. Gerber / European Economic Review 52 (2008) 285452is positive (~rC4s1) and nondecreasing in l because ~rC is nondecreasing in l.

(ii)

T

s1a

GP

sinc

ARTICLE IN PRESSBesanko, D., Kanatas, G., 1993. Credit market equilibrium with bank monitoring and moral hazard. The ReviewRefoGPlvG

Sl if and only if lbl.

Thus, the banks prots are maximized at a pooling contract for lpl and at ascreening contract for lXl. It remains to show that there exists a prot maximizingscreening contract for lXl whenever a1 is sufciently small.

here exists a prot maximizing screening contract for lXp1 since in this case ~rC

14s2a2 for ~rC as dened in (11). Hence, to prove our claim it is sufcient to show that

p14GSp1. Provided that a1 is sufciently small this immediately follows from

GSp1 p11 p1

a1 1a1 p1

,

GPp1 1 pp1

a2a2 p2

pp1p2

a2 1

e a1 1=a1 p1 is increasing in a1. This concludes the proof. &

erencesa2 pGPl0 1 p a2

a2 pa2 1.

Hence, GPl04G

Sl0, i.e. l4l0.

Let s1as2 and s1p2s2a2=a2 1. As we have noted above, GSl is convex in l.

Since s1as2 it follows that GPl is strictly concave in l. Hence, from

liml!1 GSl4liml!1 G

Pl and liml!0 G

Sloliml!0 G

Pl it follows that there

exists a unique l; 0olo1, such thatConsider the following cases.

(i) Let s1 s2 s and p 1=s. Then GPl is constant while G

Sl is strictly increasing

in l. Hence from our observations above it follows that there exists a uniquel; 0olo1, such that

GPlvG

Sl if and only if lbl.

It remains to show that there exists a prot maximizing screening contract for l4l.From the denition of ~rC in (11) it follows that ~rC ! sosa2 for l! 0. Hence, thereexists l0 such that ~rC sa2 and there exists a prot maximizing screening contract ifand only if l4l0. We will show that l4l0 for the intersection point l between G

Pl

and GSl.

GSl0 a2 1 1 p

a2 1 l0

,

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Direct versus intermediated finance: An old question and a new answerIntroductionThe modelThe optimal contract for the bankCompetitive banking sectorScreening of projects

ConclusionAcknowledgmentsProofsReferences