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Bank Reputation, Bank Commitment,

and the Effects of Competition in

Credit Markets

I. Serdar Din¸c

University of Michigan

This article discusses the effects of credit market competition on a bank’s incentive to

keep its commitment to lend to a borrower when the borrower’s credit quality deterio-

rates. It is shown that, unlike in the borrower’s commitment problem to keep borrowing

from the same bank in “good” times, the increased competition may strengthen a bank’s

incentive to keep its commitment. Banks offer loans with commitment to the highest

quality borrowers but, when faced with competition from bond markets, they also give

these loans to lower quality borrowers. An increase in the number of banks has a non-

monotonic effect; new banks reinforce a bank’s incentive only if there are small number

of banks.

The inability to contract across all contingencies may result in inefﬁcien-

cies in bank lending. The borrower may not undertake efﬁcient investments

if future reﬁnancing is difﬁcult to obtain in the case of temporary distress.

The lender may not provide funds when the borrower is in distress if the

surplus cannot be shared in the long run.

1

This problem can be mitigated,

however, if the lender and the borrower interact repeatedly. For example,

a bank’s concern to maintain a “good” reputation can induce the bank to

keep its commitment to a costly action [see, e.g., Sharpe (1990), Boot,

Greenbaum, and Thakor (1993), Aoki (1994), Chemmanur and Fulghieri

(1994)]. Indeed, the use of bank reputation as an enforcement mechanism

seems to be widespread.

2

This article is based on a chapter in my Ph.D. dissertation. I am grateful to my committee members—Masahiko

Aoki (main advisor), Douglas Bernheim, Avner Greif, and Paul Pﬂeiderer—for their invaluable guidance. I also

thank Marco Da Rin, Thomas Hellmann, Kevin Murdock, two anonymous referees, the editor, and seminar

participants at numerous institutions for many helpful comments, and Kathryn Clark and Alexandra Haugh

for editorial help. Address correspondence to I. Serdar Din¸c, Department of Finance, University of Michigan

Business School, 701 Tappan St., Ann Arbor, MI 48109-1234, or e-mail: dincs@umich.edu.

1

Although this is an old problem in economics, Mayer (1988) appears to be the ﬁrst one to discuss it in the

context of ﬁnance. See also Hellwig (1991).

2

Duca and Vanhoose (1990) observe that 80% of commercial loans in the United States are made via loan

commitments that the bank has little or no legal obligation to honor. Boot, Greenbaum, and Thakor (1993)

give examples of bank off-the-balance-sheet activities in which bank reputation plays an important role. Aoki

(1994) discusses the importance of bank reputation in inducing a main bank to rescue its distressed borrowers

in Japan.

The Review of Financial Studies Fall 2000 Vol. 13, No. 3, pp. 781–812

© 2000 The Society for Financial Studies

The Review of Financial Studies/v13n32000

Although the importance of reputation in banking is well studied, our

understanding about the effects of credit market competition on a bank repu-

tation mechanism is limited because much of the existing theory assumes the

bank’s return to be independent from the competition it faces. This leaves

many important questions unanswered: Is a reputation mechanism sustainable

with increased competition? Does it matter if the competition comes mainly

from security markets instead of other banks? To what type of borrower does

a bank offer to lend with commitment? How do the characteristics of these

borrowers change with increased competition?

This article examines how credit market competition changes the effec-

tiveness of bank reputation in enforcing a bank’s commitment. It provides a

theory in which the bank’s incentive to keep its commitment is derived as a

function of (1) its reputation, (2) the number of competing banks and their

reputation, and (3) the competition from bond markets. The type of borrower

that is offered bank commitments is also determined.

A bank can provide arm’s length lending in which the bank makes no

commitment to future reﬁnancing if the borrower experiences ﬁnancial dis-

tress. A bank with a good reputation can also provide relationship lending

in which the bank promises reﬁnancing, which may be costly to the bank

in the short term. The difference between the future (discounted) expected

return from arm’s length lending and that of relationship lending determines

the bank’s incentive to incur any short-term cost to keep its commitment and

maintain a good reputation.

An increase in credit market competition may decrease the bank’s return

from relationship lending. Whether this decrease weakens the bank’s incen-

tive to maintain a good reputation depends, however, on how the same

increase in competition affects the bank’s return from arm’s length lend-

ing, which does not require a good reputation. In particular, if the increased

competition decreases the bank’s return from arm’s length lending more than

it decreases its return from relationship lending, the additional competition

strengthens the bank’s incentive to maintain a good reputation. Thus it can

be misleading to conclude that any increase in credit market competition that

decreases the bank’s return from relationship lending is necessarily harm-

ful for the effectiveness of a reputation mechanism. This article shows that

whether the additional competition is beneﬁcial or harmful depends both on

the source of competition and the level of competition.

To understand why the source of an increase in credit market competition

can be important, consider the case in which borrowers that could previ-

ously borrow only from banks gain access to bond markets. Bonds are a

closer substitute for arm’s length lending than relationship lending. Hence

when a bank faces competition from bond markets, its proﬁt from arm’s

length loans decreases more than its proﬁt from relationship lending. This

asymmetric effect increases the bank’s incentive to keep its commitment in

782

Bank Reputation, Bank Commitment, and the Effects of Competition

relationship lending (i.e., maintaining a good reputation), although its proﬁts

from it decrease.

The level of competition is also important because an increase in the num-

ber of banks may have a nonmonotonic effect on the feasibility of a reputa-

tion mechanism. If the banks have large market power, they already earn so

much from arm’s length loans that any additional return they can capture does

not justify the cost of a commitment. On the other hand, reputational rents

ultimately decrease with the number of banks that have a good reputation,

which makes the reputation mechanism most effective with an intermediate

number of banks.

This article also makes predictions about the type of borrower to which

a bank offers relationship lending. The bank offers relationship lending to

borrowers with the highest credit quality because only these borrowers have a

sufﬁciently high net return from their projects to cover the commitment costs.

The bank offers to give only arm’s length loans to medium-quality borrowers,

and it refuses to lend to the lowest level altogether. Whether a borrower is

offered relationship lending or not also depends on the competition the bank

faces. In particular, since the funds raised in the bond market are a closer

substitute for arm’s length loans, the competition from bond markets forces

the bank to lower the threshold above which it offers relationship lending.

This article sheds light on several trends in banking. One is the deregu-

lation of capital markets that is taking place in many countries, including

Japan and the European Union. The impact of deregulation is reinforced

by another trend, the closer integration of ﬁnancial markets. An important

concern is how the resulting increase in competition will affect relation-

ship lending. Indeed, as argued by Allen and Gale (1998), the importance

of relationship lending—or lack thereof—is an important difference among

ﬁnancial systems. This article shows that not only do the bank’s incentives

to invest in these lending relationships survive the increased competition, but

they may even be strengthened, provided that borrowers continue to value

these relationships.

3

Disintermediation is a trend in which borrowers increasingly use security

markets to raise funds instead of borrowing directly from banks. An interest-

ing aspect of this trend is that banks continue to play a role even if they do

not provide the funds, because the securities the borrowers issue are often

“backed up” by loan commitments from banks. How the capital market com-

petition affects the bank’s incentive to monitor the borrower, and ultimately

honor its commitment, is an open question. This article shows that capital

market competition may actually enhance a bank’s incentive to honor its

commitment, although the competition decreases its proﬁt from doing so.

3

See Aoki and Din¸c (2000) for a discussion of the Japanese case.

783

The Review of Financial Studies/v13n32000

Finally, bank mergers, some of them of unprecedented magnitude, are an

important trend worldwide. An important issue is whether a bank will con-

tinue to honor its commitments if the increase in its market power following

a merger makes its borrowers reluctant to “punish” that bank by deserting

it if it does not keep its promises. This article demonstrates that the bor-

rowers’ easier access to capital markets decreases the minimum number of

banks necessary to sustain a reputation mechanism and hence mitigates the

potential negative effects of mergers.

Petersen and Rajan (1995) study the borrower’s commitment problem to

share future surplus with the bank; inability to commit may prevent the bor-

rower from obtaining funds for a project. The market power of a bank in

the credit market mitigates this commitment problem by allowing the bank

to capture in relationship lending some of the borrower’s future surplus.

Petersen and Rajan show that an increase in the credit market competition,

which decreases the bank’s market power and weakens the bank’s incentive

to offer funds at the beginning, has a negative effect on relationship lending.

Alternatively, this article studies the bank’s commitment problem to lend to a

borrower in temporary distress after ﬁnancing that borrower in “good” times.

In the model, the bank can offer relationship lending or arm’s length lending

to a borrower based on the borrower’s credit quality. Unlike the borrower’s

commitment problem in Petersen and Rajan, an increase in the credit market

competition may help mitigate the bank’s commitment problem in relation-

ship lending. This article demonstrates that the effect of any increase in credit

market competition is not uniform, but depends both on the source and the

level of competition. It also provides empirically testable predictions about

the type of borrowers that are offered relationship lending and how their

characteristics change with an increase in credit market competition.

Boot and Thakor (1999) also study the viability of relationship lending

under increasing competition. They show that initially the expected amount

of relationship lending increases, but it decreases as competition escalates. In

addition, they ﬁnd that capital market competition causes the bank to increase

its relationship lending relative to its arm’s length lending. Although these

results are similar, different empirical implications about the type of borrower

that is offered relationship lending are obtained in this article.

4

This article

predicts that (1) higher-quality borrowers are offered loans with commitment,

but the threshold of creditworthiness above which a loan with commitment is

offered decreases with competition, and (2) bond markets decrease the mini-

mum number of banks necessary to sustain a reputation mechanism. Boot and

4

Among the major differences that lead to different predictions are (1) entry restrictions in banking and the

bank’s credit screening ability are the sources of bank rents in this article (in addition to any reputational

rents) while it is the access to cheaper core deposits in Boot and Thakor (1999); (2) the bank’s return from

relationship lending depends only on the number of other banks in Boot and Thakor, while it also depends

on their reputation and the borrower’s access to bond markets in this article. The second difference leads to

more precise predictions about the effects of increasing competition in this article.

784

Bank Reputation, Bank Commitment, and the Effects of Competition

Thakor offer opposite predictions for the ﬁrst and have no counterpart for the

second, as the existence of relationship lending (but not its importance rela-

tive to arm’s length lending) is independent of the borrowers’ access to bond

markets in their article. The predictions in this article appear to be consistent

with the existing empirical literature, as discussed in the conclusion.

The rest of the article is organized as follows. Section 1 presents the model.

Section 2 provides a preliminary analysis, which shows that the assumption

of decreasing market power with increasing competition has, by itself, only

ambiguous implications. Accordingly, the model is extended in Section 3

to derive a bank’s market power as a function of the number of competing

banks. The effects of an increase in the credit market competition are exam-

ined in Section 4, where both an increase in the number of banks as well

as the borrowers’ access to bond markets are considered. Section 5 discusses

the robustness of the ﬁndings. The ﬁnal section concludes.

1. The Model

Entrepreneur: Consider a risk-neutral entrepreneur who has a two-subperiod

project.

5

The project requires one unit of capital at date 0 and returns are

obtained at date 2. At date 1, the project can be at one of three states: suc-

cess, distress, or failure with probability p

S

, p

D

, and p

F

, respectively. In suc-

cess, the project returns a cash ﬂow R

S

and nontransferable control beneﬁts

C

S

to the entrepreneur. These beneﬁts can be prestige, perks, or rents to the

entrepreneur from an accumulated knowledge about the project as well as the

expected return from being able to undertake future projects upon the success-

ful completion of the current one. In failure, both cash and nontransferable

returns are zero. In distress, the returns are the same as in failure, unless

the entrepreneur further invests one unit of capital at date 1. In that case,

cash and nontransferable returns are R

D

and C

D

. This additional investment

will be referred to as rescue investment. Any such investment in other states

is wasted. The project has no liquidation value and there is no discounting

between subperiods. The following parametric assumptions are made.

Assumption 1.

(i) R

D

< 1

(ii) R

D

+ C

D

> 1

(iii) p

S

R

S

− p

D

(1 − R

D

)>1

(iv) p

S

R

S

+ p

D

R

D

< 2

Assumption 1(i) implies that investing an additional unit in a distressed

project has a negative net present value (NPV) if only cash returns are con-

sidered. Assumption 1(ii) states that a distressed project has positive NPV

5

This game will be the stage game when its inﬁnitely repeated version is studied. The term subperiod is used

now since the term period in repeated games is traditionally reserved for the time during which the stage

game takes place.

785

The Review of Financial Studies/v13n32000

if the nontransferable returns to the entrepreneur are also included. Assump-

tion 1(iii) implies that the project has a positive expected NPV in cash returns

at date 0, including the (cash) losses from further investing at the distressed

state. Assumption 1(iv) states that the borrower cannot borrow rescue funds

at date 0 to use them in case of distress at date 1.

Lenders: The entrepreneur has no funds, but can borrow from a type of

institutional lender, or, simply, a “bank.” To model the market power of

banks in the credit market, it is assumed that the banks have a riskless lending

opportunity at date 0 that gives them a net return M>0 at date 2. The model

will be extended in Section 3 to determine endogenously a bank’s market

power and its lending strategy as a function of the number of competing

banks. The bondholders will be introduced in Section 4.

Information and contracts: The uncertainty about the project state is

revealed at date 1 and, while it can be publicly observed, it cannot be ver-

iﬁed in courts. The cash returns are veriﬁable at date 2 without cost. This

article focuses on debt contracts. One way to justify this focus is that equity

contracts may be too risky for the lenders.

The time line of this two-subperiod game is summarized as follows:

Date 0: The bank determines the interest factor (one plus the interest

rate). The entrepreneur borrows funds and undertakes the project.

Date 1: The project state is revealed and publicly observed. In the dis-

tressed state, the bank decides whether to offer a rescue credit.

If provided with funds, the entrepreneur undertakes the rescue

investment.

Date 2: The returns are obtained; payments are made.

2. Preliminary Analysis

2.1 One-shot lending

In a subgame perfect equilibrium, the bank must be provided with incentives

to offer rescue ﬁnancing in the distressed state. Since the project state is not

veriﬁable, these incentives cannot be provided by court-enforced contractual

clauses that oblige the bank to give a rescue loan in a distressed state but

not in others. The bank therefore provides rescue ﬁnancing only if the cash

returns in a distressed state are sufﬁcient to pay back the loan. By Assump-

tion 1(i), they are not; hence the bank does not provide rescue ﬁnancing in

a distressed state in a subgame perfect equilibrium.

Proposition 1. The following strategy proﬁle is the subgame perfect equi-

librium: At t = 0, the bank lends one unit with the interest factor (one plus

the interest rate) b, where:

b =

1 + M

p

S

(1)

At t = 1, if the project is in distress, the bank does not rescue the entrepreneur.

786

Bank Reputation, Bank Commitment, and the Effects of Competition

It is important to consider whether an equity contract could induce the

bank to provide rescue funds. If the equity contract allows the bank to cap-

ture some of the entrepreneur’s control beneﬁts, the bank might indeed have

an incentive to rescue the entrepreneur. The control beneﬁts may, however,

include future returns to the entrepreneur, who may not be forced to dis-

tribute dividends [Hart and Moore (1989, 1994)].

6

Furthermore, the bank is

unlikely to capture any managerial perks.

More complicated contracts, even if they are feasible,

7

are not likely to

improve efﬁciency either, as long as the project state at date 1 and the

entrepreneur’s control beneﬁts remain unveriﬁable in court. However, it is

useful to examine why the contracts that give the entrepreneur the option to

borrow one unit at date 1 are not feasible. The entrepreneur would use that

option not only in distress, but also in failure. Thus any such option would

also have to give the bank the discretion to refuse the loan in the failure state

but to impose costs to the bank if it also refused to lend in the distressed

state. Since the project state is not veriﬁable, such discretion and costs cannot

be contracted upon; hence that discretion must be provided within a different

institutional arrangement. The following subsection analyzes one of the most

common of such institutions, namely, bank reputation.

2.2 Bank commitment in repeated lending

Banks are not one-time, anonymous participants in the credit market; they

engage in repeated lending with many borrowers. This repeated, nonanony-

mous nature of bank lending might make a reputation mechanism feasi-

ble to enforce bank commitments.

8

Accordingly, an inﬁnitely repeated game

framework is used to analyze whether a bank’s commitment to rescue an

entrepreneur in distress can be enforced by bank reputation. In the repeated

games terminology, the investment game of the previous section becomes the

stage game of the repeated game. The length of the stage game is referred to

as a period. A common discount factor δ<1 is assumed between the peri-

ods, with no discounting within a period. The entrepreneurs exit the economy

6

The entrepreneurs can provide themselves with large salaries instead of distributing dividends or can do

business with companies that are self-owned; these possibilities also rule out preferred shares as a way to

mitigate the bank commitment problem.

7

See Hart (1995) for why such contracts may not be feasible.

8

Reﬂections of a bank’s concern for its reputation are often seen in the popular press. Yoh Kurosawa, the

deputy president of Industrial Bank of Japan, explains their incentive as the main bank to rescue a distressed

borrower by stating that “[o]ur reputation is that we never let a client go bust,” [Economist (October 17,

1987)]. Indeed, banks may go to great lengths to keep a good reputation. After real estate prices in Japan

suddenly and drastically declined in 1991, the banks continued to support even the companies that had clearly

gambled. The banks’ motives were interpreted by one analyst with Nikko Research Center Ltd. as “[b]anks

have been supporting deadlocked debtors to save face as main banks, but current public opinion is that

banks are better off disposing of non-working assets for their health,” [Nikkei Weekly (August 2, 1993)]. The

difference between the main bank loans that include a rescue commitment and the bank loans without such

commitment also seems to be clearly recognized. When Azabu Tatenomo Co., a real estate company that

faced large debts after investing heavily during the 1980s, rejected the Mitsui Bank’s conditions for a rescue,

Mitsui Bank declared that “[they] will have to change from being the main bank to a legalistic relationship

of creditor and debtor,” [Japan Economic Newswire (March 11, 1993)].

787

The Review of Financial Studies/v13n32000

after one period, while the banks live inﬁnitely long. This assumption allows

for a focus on the bank’s commitment problem by abstracting from the pos-

sibility of the entrepreneur’s commitment to borrow from the same bank in

the future. It is also assumed that the history of the economy is common

knowledge and that a bank, for simplicity, only lends to one entrepreneur in

a given period.

Consider now a reputation mechanism with the following features: At

t = 0 in a given period the bank commits to a rescue if the project is in

distress at t = 1 (in the same period). If it does not rescue the entrepreneur

in the distressed state, it loses its good reputation and no other entrepreneur

will ever take a loan with a rescue commitment from that bank again. Since

an entrepreneur can borrow from another bank, this threat is credible. By not

rescuing, the bank saves the rescue costs but loses future return from lending

with a rescue commitment. If the present value of such losses is greater than

the rescue cost, this reputation mechanism can enforce a bank’s commitment.

Some additional terminology and notation will facilitate the formal state-

ment of a repeated game equilibrium. The term relationship lending is used

interchangeably with loans with commitment, and arm’s length lending with

loans without commitment. Let M

G

, with M ≤ M

G

, denote the market power

of a bank with a good reputation when the bank lends with the promise of

rescue. Although M

G

depends on the number of competing banks with the

same reputation, as shown in the next section, it is taken as given in this

section in order to focus on the equilibrium features. To avoid triviality, it

is assumed that the entrepreneur prefers to borrow with a bank commitment,

that is,

Assumption 2. p

D

C

D

≥ M

G

− M.

Proposition 2. Consider the following strategy proﬁle of the repeated game:

At t = 0 in a given period the entrepreneur borrows from a bank that has

never shirked from rescuing. At t = 1, if the project is in distress, the bank

rescues the entrepreneur. If it fails to do so, future entrepreneurs do not

borrow from that bank; if an entrepreneur borrows, the bank does not rescue

the entrepreneur. If

M

G

− M>p

D

(1 − R

D

) (2)

then there exists a

¯

δ, where

¯

δ =

1 − R

D

M

G

− M + (1 − p

D

)(1 − R

D

)

(3)

such that for δ ≥

¯

δ this strategy proﬁle is a subgame perfect equilibrium of

the repeated game.

788

Bank Reputation, Bank Commitment, and the Effects of Competition

Proof. Let W

G

(δ) and W(δ) be the present value of a bank’s net return

on the equilibrium path and on the punishment path while being punished,

respectively. Then

W

G

(δ) =

M

G

− p

D

(1 − R

D

)

1 − δ

and W(δ) =

M

1 − δ

. (4)

It is only necessary to verify that a one-shot deviation on the equilibrium path

is not beneﬁcial. The bank’s incentive constraint to rescue an entrepreneur in

distress is given by

R

D

− 1 + δW

G

(δ) ≥ δW (δ). (5)

Equation (2) is a necessary and sufﬁcient condition for Equation (5) to be

satisﬁed for some δ. Equation (3) then follows. The interest factor b

G

is then

given by

b

G

=

1 + M

G

p

S

. (6)

On the punishment path, the entrepreneur has no incentive to borrow from

the deviant bank because a loan can be obtained with a commitment from

another bank. Finally, the deviant bank has no incentive to rescue a distressed

borrower, as no future entrepreneur will believe its commitment once it has

shirked. Q.E.D.

The bank does not rescue every entrepreneur who is unable to meet debt

obligations; the genuine failures are not provided with credit. This feature

of the equilibrium is similar to the material adverse change clause that is

observed in virtually all loan commitment contracts [Shockley and Thakor

(1997)]. This clause gives the bank the right not to honor its commitment.

If the public state is not perfectly and publicly observed, the use of this

clause may have reputational costs to the bank. However, the banks and the

entrepreneurs may still prefer loan commitment contracts to other feasible

contracts [Boot, Greenbaum, and Thakor (1993)].

The assumption about the perfect public observability of the distressed

state facilitates the analysis, but the equilibrium is robust to that assumption.

Even if the distressed state cannot be distinguished from the failure state by

outsiders, a bank rescue can still be observed because it is likely to include a

signiﬁcant restructuring in the ﬁrm with possible asset sales. If a bank denies

the occurrence of the distressed state to avoid rescue costs, its rescues—

and all the observable activities associated with it—will have a different

frequency from the rescues by a bank that keeps its commitment. A ﬁnding

789

The Review of Financial Studies/v13n32000

by Fudenberg, Levine, and Maskin (1994) implies that this difference is

sufﬁcient for a similar repeated game equilibrium to exist.

9

Proposition 2 gives the conditions for a reputation equilibrium. As is

typical in repeated games, this is not the only equilibrium. The stage game

equilibrium given in Proposition 1 is also an equilibrium when it is repeated

every period. In this article I abstract away from how a reputational equilib-

rium is selected or how the banks build their reputation. The focus here is

on the feasibility of building a reputation and on the effects of competition

once banks have built their reputation.

Equations (2) and (3) hint at the nontrivial impact of bank competition on

the reputation mechanism. Equation (3) indicates that whether a reputation

mechanism is feasible or not does not depend on the bank’s market power

per se, but on the difference between the bank’s market power when it has a

good reputation and its power without such a reputation. In particular, even

if an increase in competition decreases the bank’s market power, it may be

easier to sustain a reputation mechanism if this difference increases. This

gives the following corollary.

Corollary 1.

¯

δ decreases—and the reputation mechanism becomes easier to

sustain—as M

G

− M increases. In particular,

¯

δ decreases if an increase in

competition decreases both and M

G

and M but increases M

G

− M.

It may seem that Equation (2) can be easily satisﬁed if the entrepreneurs

voluntarily leave more of the project return to the bank whenever the bank’s

market power is not enough to induce it to keep its commitment. The

entrepreneur can, after all, increase the bank’s return by purchasing addi-

tional services from the bank and/or by concentrating all their borrowing

needs on one bank to give information rents. Although these practices are

indeed observed in reality, they will have only a limited effect for at least

two reasons. First, the asymmetric information problems between a borrower

and a lender limit the effectiveness of such practices; borrowers with the

highest probability of requiring a rescue will be the most willing to offer

such concessions to a bank. Second, what the entrepreneur can voluntarily

leave to the bank is limited by the total cash return from the project. This

is likely to be a problem when a bank’s market power in lending without a

commitment is already large.

Thus the mere assumption that a bank’s market power decreases with an

increase in competition has ambiguous implications. For a theory with more

9

Technically speaking, suppose the project state is observable only by the bank but not by the public. There

are two pure actions the bank can take: rescue and not rescue. The probability distributions induced by these

actions on the publicly observable outcome (date 2 returns) are linearly independent of each other. Hence the

action proﬁle that prescribes rescue when the entrepreneur is in distress but not otherwise has individual full

rank [Deﬁnition 5.1 in Fudenberg, Levine, and Maskin (1994, p. 1014)]. Consequently, Condition 2 (p. 1021)

is trivially satisﬁed and the Nash-threat folk theorem 6.1 (p. 1022) follows. Unfortunately this theorem only

gives the existence of an equilibrium; it does not give the equilibrium strategy proﬁle.

790

Bank Reputation, Bank Commitment, and the Effects of Competition

powerful implications, a bank’s market power and its lending strategy must

instead be determined endogenously as a function of the competition the

bank faces.

3. Equilibrium with Bank Competition

To determine a bank’s market power and its lending strategy endogenously

as a function of the number of its competitors, the model is extended to

incorporate one of the main characteristics of banks as institutional lenders:

information processing capabilities that mitigate the asymmetric information

problem in creditor-debtor transactions.

10

This article focuses on the credit

screening activities of banks before they offer a loan. The analysis below

follows Din¸c (1997), showing that the credit screening abilities of banks are

enough to give them market power in the credit market when there are entry

restrictions in banking.

Consider a second type of entrepreneur who always fails. These

entrepreneurs are referred to as “bad” and those described earlier as “good.”

An entrepreneur is of the good type with probability λ. The entrepreneur’s

type is her private information at date 0. Each bank screens the entrepreneur

by obtaining a costless signal at date 0. These signals are correlated with

the entrepreneur’s type but are subject to errors that are independent across

banks. In particular, it is assumed that bank i obtains the signal x

i

that is

real valued and has full support over [x

, ¯x] for any given type θ of the

entrepreneur. Given the nontransferable nature of the information obtained in

the credit screening, it is assumed that each bank’s signal is its private infor-

mation. The signals are identically and, conditional on the entrepreneur’s type

θ, independently distributed across banks with conditional density function

f(x|θ). The standard assumption that f(x|θ) satisﬁes the monotone likeli-

hood ratio property (MLRP) is adopted, that is,

Assumption 3 (MLRP). f(x|θ = G)/f (x|θ = B) increases in x.

Therefore if a bank obtains the signal x, its probability estimate µ(x) that

the entrepreneur is of the good type is given by

Pr(θ = G|x) =

λf (x |θ = G)

λf (x |θ = G) + (1 − λ)f (x|θ = B)

≡ µ(x). (7)

Notice that µ(x) is increasing in x by Assumption 3. It is assumed that

lending and rescuing an entrepreneur is a positive-NPV project for the bank

10

For theoretical arguments see, for example, Leland and Pyle (1977), Campbell and Kracaw (1980), Diamond

(1984, 1991), and Fama (1985). For empirical evidence, see, for example, James (1987), Lummer and

McConnell (1989), Hoshi, Kashyap, and Scharfstein (1990), James and Wier (1990), Petersen and Rajan

(1994), Berger and Udell (1995).

791

The Review of Financial Studies/v13n32000

if its signal indicates sufﬁciently good credit quality, that is,

Assumption 4. µ( ¯x)(p

S

R

S

− p

D

(1 − R

D

)) − 1 > 0.

At date 0, the entrepreneur asks each bank the interest rate it demands for a

unit loan. Each bank quotes its interest rate without observing the quotations

of other banks; the entrepreneur selects the bank that offers the lowest rate.

The bank has the option to refuse to lend. The analysis that follows focuses

on the symmetric equilibrium that banks with the same reputation have the

same strategies.

11

3.1 One-shot lending

The equilibrium in the one-shot lending game has the same features as stated

earlier, except the determination of the interest rate. A bank with a low esti-

mate µ(x)—or with a low signal x—refuses to lend; hence there is a thresh-

old of x below which the bank does not lend. However, the calculation of

this threshold and of the interest rate the bank quotes when it lends is not

immediate due to the “winner’s curse.” Technically this bank competition

model is a sealed-bid, ﬁrst-price, common-value auction; thus the insights

developed in other contexts are valid [see Milgrom and Weber (1982)].

To gain intuition about the winner’s curse and a bank’s strategy in deter-

mining the interest rate quoted at equilibrium, suppose that the rate the bank

quotes decreases with µ(x)—hence with its signal x (this is indeed the case

in equilibrium). Consequently a bank lends only if it quotes the lowest inter-

est rate. In a symmetric equilibrium, this implies that the bank’s loan offer is

taken if it has the highest estimate µ(x), or equivalently, the highest signal

x, among all the banks. Therefore the bank that gets to lend is the most

optimistic bank about the entrepreneur’s prospects as its signal provides an

upper bound on the signals of all other banks. Consequently, a bank chooses

its quote to maximize its expected proﬁt based on not only its own signal,

but also the fact that winning the competition gives an upper bound on the

other banks’ signals.

Proposition 3 (Arms’s length lending). The following strategy proﬁle is the

(symmetric) subgame perfect equilibrium of the (stage) game: At t = 0, the

interest factor demanded by a bank is given by the decreasing function b

N

(x)

for x ≥ x

N

—derived in the appendix—with no loan offered for x<x

N

,

where x

N

decreases with λ.

12

The entrepreneur borrows from the bank that

11

This article shares the same bank competition model with Rajan (1992) with the important exception of

the information structure. In Rajan, one bank (the insider) knows everything—and more—the other banks

(outsiders) know about the borrower. In this model, all the banks are symmetric at the time of competition

in the sense that (i) none of them has access to what the other banks know about the borrower; and (ii) all

the banks have the same credit screening technology. The information structure of this model is closer to

Broecker (1990) and Thakor (1996).

12

“N” mnemonic for no commitment.

792

Bank Reputation, Bank Commitment, and the Effects of Competition

demands the smallest interest factor. If the loan is not obtained, the project

is not undertaken. At t = 1, if the project is in distress, no bank rescues

the entrepreneur. The expected proﬁt of each bank at date 0 is positive; it

decreases with the number of banks and converges to zero as the number of

banks goes to inﬁnity. There is no subgame perfect equilibrium in which the

bank rescues the entrepreneur in distress.

Proof. See the appendix.

An important feature of this model is that it derives both a bank’s market

power and its lending strategy with respect to the number of banks. Although

the economic intuition behind the positive proﬁts of the banks is the same as

those of the bidders in mineral rights auctions [Milgrom and Weber (1982)],

it is worth discussing how the banks with the same screening technology and

no inside information make positive proﬁts. Each bank bases its strategy on

its signal about the entrepreneur; consequently the probability of winning the

competition for a given bank, and thus the bank’s return, depends on other

banks’ signals as well. However, each bank’s signal is its private information.

The private nature of these signals gives the bank a rent to private informa-

tion. As the bank always has the option of not offering a loan when it expects

losses, this information rent leads to positive expected proﬁts when the entry

into banking is restricted.

13,14

3.2 Bank commitment in repeated lending

The equilibrium strategy proﬁle presented in Proposition 2 is maintained

when the bank commits to rescuing in the distressed state, but the lending

strategy of the banks is modiﬁed. To allow a bank to offer loans both with a

rescue commitment and without, it is assumed that whether a loan carries a

rescue commitment or not is publicly observed.

15

The lending strategy of banks that commit to rescue is similar to their strat-

egy when they do not commit except for one important difference. A bank’s

13

For a more technical intuition about why banks earn positive proﬁts, suppose that they do not. If the bidding

function b

N

leaves the bank with zero expected proﬁts before obtaining signal x, then the bank must also

make zero proﬁts for any signal x (otherwise, for the range of x where the bank makes a negative proﬁt, the

bank could do better by not bidding). Suppose for some x>x

N

the bank bids slightly higher than b

N

(x).

The probability that it wins the bidding then decreases, but it would make a positive proﬁt when it wins.

Therefore the bank could do better by deviating from b

N

, which contradicts the fact that b

N

is an equilibrium

bidding function.

14

The different information structure in this article gives a very different result from the one in Rajan (1992),

who adopts the same competition model. In Rajan, the insider bank not only has better information about

the entrepreneur than other banks, but it also knows what the others know about the entrepreneur. Since the

information of each outside bank is not its private information, each makes zero proﬁt and their number has

no effect for the inside bank’s rent.

15

An important lending practice in which a bank’s reputation enforces its rescue commitment is the main bank

lending in Japan. Although this commitment is implicit, it is publicly known which bank acts as a main bank

for a given company [see Aoki, Patrick, and Sheard (1994)]. The observability of the bank commitment is, of

course, not an issue for loan commitment contracts in the United States.

793

The Review of Financial Studies/v13n32000

lending strategy now has a three-tiered structure with respect to its signal

instead of two. At the top range, it offers a loan with commitment, while

its loan offer does not carry a commitment at the middle range. The bank

refuses to lend at the lower range. For intuition, suppose that the entrepreneur

always prefers to be rescued even if all the cash return has to be left to the

bank in the success state. If the bank lends to a good type with a rescue

commitment for a given interest factor b, its expected net return at date 0 is

p

S

b − p

D

(1 − R

D

) − 1, (8)

which is less than what it would be without a rescue commitment. The inter-

est factor the bank can demand is naturally bounded from above by the

available cash return, so the threshold below which the bank does not lend

with a commitment is higher than the threshold for a loan without a com-

mitment. This three-tiered lending strategy is consistent with the empirical

evidence on loan commitment contracts in the United States [see Avery and

Berger (1991), Qi and Shockley (1995)].

However, the lending strategies of banks on the punishment path (i.e.,

the deviant bank cannot lend with a commitment while all the others can)

are very complicated to derive explicitly for generic parameters because the

symmetry among the banks is lost.

16

To simplify the derivation, the private

beneﬁts an entrepreneur obtains from reﬁnancing in the distressed state are

assumed to be sufﬁciently high. Thus even if a bank offers relationship lend-

ing in exchange for all the cash return in the success state, its offer cannot

be undercut by a bank that offers only an arm’s length loan.

Assumption 5. p

D

C

D

≥ p

S

R

S

− 1.

Proposition 4 (Relationship lending). Consider the following modiﬁcations

in the strategy proﬁle given in Proposition 2 (b

C

and x

C

are derived in the

appendix).

17

Equilibrium path: At t = 0 in a given period, the bank lending strategies

are given by

b

C

(x) and the bank commits to rescue, for x ≥ x

C

b

N

(x) and the bank does not commit to rescue, for x

N

≤ x<x

C

no loan, for x<x

N

where b

N

and x

N

are as given in Proposition 3.

The entrepreneur considers an offer with a rescue commitment only if the

bank has never shirked from rescuing. At t = 1, if the project is in distress

16

The derivation of equilibrium strategies in a sealed-bid, ﬁrst-price, common-value auction with n bidders who

have asymmetric payoff functions is an open question in auction theory. Thus reputation building by banks is

not modeled explicitly here, as reputation building inherently involves asymmetries among banks.

17

“C” mnemonic for commitment.

794

Bank Reputation, Bank Commitment, and the Effects of Competition

and the loan carries a rescue commitment, the bank that lends at t = 0

rescues the entrepreneur.

Punishment path: The deviant bank offers only loans without a commitment

while all the other banks adopt a similar three-tiered lending strategy as on

the equilibrium path. (The exact lending strategies are given in the appendix.)

There exists ¯n ≥ 2 such that for n ≥¯n a subgame perfect (repeated game)

equilibrium with features given earlier exists for δ ≥

¯

δ(n). For all n< ¯n no

subgame perfect equilibrium with these features exists for any δ<1.

Proof. See the appendix.

In three-tiered lending strategies, banks offer a rescue commitment—and

incur rescue costs in distress—only if a borrower is a good credit risk.

Accordingly the bank’s proﬁt shows a qualitative difference between x

C

and

x

N

. Although the reason for the existence of a middle range [x

N

,x

C

) in bank

strategies is the rescue cost, x

C

is not the threshold below which the bank

makes expected loses if it lends with a commitment. Instead, x

C

is the level

at which the bank is indifferent between lending with a commitment for all of

the cash return in the success state and lending without a commitment at the

(lower) equilibrium bid. Since the bank makes positive expected proﬁts from

lending without a commitment when x>x

N

, it also makes positive proﬁts

if it lends with a commitment at x = x

C

. However, the bank is indifferent at

x = x

N

to lending at all, for the bank’s proﬁt is zero at x = x

N

.

A deviant bank can still make positive proﬁts on the punishment path due

to information rents discussed earlier. However, its expected proﬁt is less

than the proﬁt of a bank with a good reputation on the equilibrium path.

This difference, which induces the bank to keep a good reputation for high

δ, can be interpreted as rents to reputation.

An important aspect of the equilibrium is that it exists only if there is a suf-

ﬁcient number of competing banks. In fact, the reputation mechanism cannot

be sustained with a monopolist bank and it may take more than a duopoly

to sustain it. For example, if the monopolist bank judges an entrepreneur

creditworthy, it sets the interest rate such that it receives all the cash return

in the success state. Since it already receives all it can, it has no incentive to

commit to a rescue and incur costs; therefore lending without a commitment

is more proﬁtable.

Corollary 2. The monopolist bank has no incentive to commit to a rescue.

By the same reasoning, the condition ¯n ≥ 2 may hold as a strict inequal-

ity.

18

The proﬁt from arm’s length lending decreases to zero as the number of

banks increases; after a threshold, relationship lending becomes more prof-

itable if the entrepreneur is sufﬁciently creditworthy.

18

By making the left-hand side of Assumption 4 arbitrarily small, ¯n can be made arbitrarily large.

795

The Review of Financial Studies/v13n32000

4. The Effects of Competition

4.1 Entry of new banks

An important issue is whether new banks can provide relationship lending.

Depending on δ, there exist multiple equilibria that differ as to whether new

banks can provide relationship lending; the repeated game theory is silent

in equilibrium selection. Since reputation building by banks is not modeled

explicitly,

19

the entry of banks is studied under two different assumptions.

4.1.1 Entry of new banks that can commit to rescue. It is helpful to start

with a study of the limit behavior when the number of banks that can provide

relationship lending goes to inﬁnity. At the limit, the minimum discount

factor that sustains a reputation mechanism converges to 1. Intuitively, as

the number of banks with a good reputation increases, the rent to a good

reputation converges to zero; hence it becomes more difﬁcult to sustain the

reputation mechanism.

Proposition 5. lim

n→∞

¯

δ(n) = 1.

Proof. See the appendix.

This ﬁnding is consistent with the view that commitment is difﬁcult to

sustain by a reputation mechanism with ﬁerce competition. Notice, however,

that this is a limit result. It does not imply that the reputation mechanism

becomes monotonically more difﬁcult as the number of banks increases. In

fact, Propositions 4 and 5 together establish the lack of any such monotonic-

ity. They imply that additional competition may be beneﬁcial to sustain a

reputation mechanism when the number of banks is small; hence an inter-

mediate number of banks is optimal.

Corollary 3. It is easiest to sustain a reputation mechanism with an inter-

mediate number of banks, that is,

¯

δ(n) is minimized at n

∗

where 1 <n

∗

< ∞.

4.1.2 Entry of new banks that cannot commit to rescue. At the other

extreme, the entry of banks that cannot commit affects directly the compe-

tition for arm’s length loans. This is sufﬁcient, however, for the banks that

can provide relationship lending to change their equilibrium behavior. As the

market for arm’s length loans becomes more competitive with new banks, the

return from such loans decreases. Thus relationship lending becomes more

proﬁtable for the banks that give these loans relative to arm’s length lending.

19

The model can be modiﬁed where rescues require certain banking skills, for example, intense monitoring, but

the banks have heterogeneous skills and their skills are their private information. The capable banks would

build a reputation by gradually revealing their type. The analysis of bank competition in that context is very

complicated, however, because the symmetry among banks is lost (see note 16). A natural conjecture would

be that whenever the competition enhances the reputation mechanism in the model, it would also facilitate

reputation building of capable banks.

796

Bank Reputation, Bank Commitment, and the Effects of Competition

These banks decrease the minimum credit quality they demand to extend

relationship lending and the share of relationship lending increases in their

loan portfolios. Even though new entrants only provide arm’s length lend-

ing, relationship lending becomes easier to obtain from banks with a good

reputation.

The entry of banks that cannot commit also extends the range of param-

eters where the reputation mechanism can be sustainable. Recall that if a

bank with a good reputation does not keep its commitment, it tarnishes its

reputation and it can then provide only arm’s length loans. The entry of new

banks decreases the return from such loans, and, hence, it decreases the return

from not keeping a commitment. This enhances a bank’s incentive to keep its

commitment and the reputation mechanism becomes easier to sustain with

the entry of banks that cannot provide relationship lending. Thus an increase

in competition increases the effectiveness of a reputation mechanism.

Proposition 6. The entry of banks that cannot commit makes the reputa-

tion mechanism easier to sustain and lowers the minimum credit quality

demanded for relationship lending, that is,

¯

δ(n), ¯n, and x

C

decrease with the

number of banks that cannot commit.

Proof. See the appendix.

4.2 Competition from security markets

Banks face increasing competition from bond and commercial paper markets,

since the funds raised in these markets are close—but not perfect—substitutes

for bank loans. The effect of this competition is likely to be different on

different types of loans. In fact, one of the conditions for a reputation mech-

anism to be effective is the identiﬁability of the lender that makes a com-

mitment. The anonymous and diffuse nature of security markets does not

allow the reputation mechanism to be an enforcement mechanism for a secu-

rity holder’s commitment [Chemmanur and Fulghieri (1994)]. Funds raised

through these markets are a closer substitute for arm’s length loans than for

relationship loans. This difference makes the competition from security mar-

kets similar to the competition from the entry of banks that cannot commit;

it decreases the return to a deviant bank more than it does to a bank that

keeps its commitment.

The model is modiﬁed to incorporate bond issues by entrepreneurs. At

date 0, the entrepreneur decides on the fraction to be raised by bond issues.

This may be different for different types of loans. Banks make offers for

loans. To minimize the signaling issues and focus on bank reputation, it is

assumed that bondholders observe only whether the loan is obtained and

the type of loan—arm’s length or relationship—but not the default premium

on loans. Bondholders do not have any credit screening capabilities, unlike

797

The Review of Financial Studies/v13n32000

banks, so they make zero expected proﬁts. Although the debt priority struc-

ture is not essential for the results, it is assumed for concreteness that the

bonds have seniority over bank loans.

20

Proposition 7 (Relationship lending with bond markets). There is a rela-

tionship lending equilibrium at which the entrepreneurs raise a fraction of

their needs by issuing bonds; the bank provides the rest through a loan with

a commitment to rescue. The borrower’s access to bond markets makes the

relationship lending easier to sustain and lowers the minimum credit quality

demanded for loans with commitment, that is,

¯

δ(n), ¯n, and x

C

decrease with

the borrower’s access to bond markets.

Proof. See the appendix.

This ﬁnding indicates that the idea that bond markets are not compati-

ble with banking relationships is not necessarily correct as far as the bank’s

incentives to commit to a rescue are concerned. First, it shows that raising

funds through the bond market and maintaining a lending relationship are not

mutually exclusive for a borrower. The borrower raises some funds by issu-

ing bonds while also obtaining relationship lending. Second, the borrower’s

access to a bond market does not limit the bank’s ability to commit to a

costly rescue, but, in fact, it enhances its incentives to do so. This is true,

although the bank’s return from relationship lending decreases with compe-

tition from the bond market. Relationship lending becomes easier to sustain

and the necessary number of banks for an effective reputation mechanism

declines. Finally, competition from bond markets forces banks to offer rela-

tionship lending to borrowers with lower credit quality.

5. Robustness

5.1 Bank competition

While this article adopts a speciﬁc bank competition model to analyze bank

strategies and returns, the ﬁnding that banks make positive proﬁts under

restricted entry depends not on the speciﬁc game form, but on the private

information a bank obtains by credit screening. This suggests that the ﬁnd-

ing about the effects of competition on bank reputation is valid for differ-

ent bank competition models, including when the entrepreneur approaches

banks sequentially, as long as a bank’s screening results are not observed—

or inferred—by other banks before they make their own loan offers [see Din¸c

(1997) for a more detailed discussion].

20

The model can also be modiﬁed in such a way that the entrepreneur uses the bank commitment to back up

its obligations to bondholders.

798

Bank Reputation, Bank Commitment, and the Effects of Competition

5.2 Costly screening

Although this article assumes zero screening costs for banks, the results

are robust as long as screening costs are not too large. If a bank incurs a

(ﬁxed) screening cost, the size of the cost determines whether the bank makes

ex ante positive proﬁts. If the screening cost is less than the expected proﬁt,

every bank screens and participates in the bidding; the qualitative results

remain unchanged. If the cost is high with respect to the loan size and the

number of competing banks, each bank plays a mixed strategy in screening

and participating in the bidding. The mixing probability is determined such

that each bank makes zero expected proﬁts. Which case is relevant is, of

course, an empirical question. However, one immediate implication of large

screening costs is that an increase in the number of banks does not change the

(average) number of banks that bid for the loan. It only decreases the prob-

ability with which a bank screens and participates in the bidding process;

the equilibrium (average) interest rate remains constant. Thus the empirical

evidence about the effects of an increase in the number of banks on bank

lending is not consistent with screening costs that are high enough to erase

bank proﬁt.

21

5.3 Bank risk taking

Agency problems between bank managers and shareholders are not a part of

the model. Although this omission is necessary to keep the model tractable,

it has to be kept in mind in determining the overall effects of the changes

in credit market competition. For example, while an increase in competition

may enhance the bank’s incentive to keep its reputation, it may also induce

bank managers to take risks to keep their perks [Gorton and Rosen (1995)].

Indeed, Din¸c (1999) ﬁnds that Japanese banks increased their real estate lend-

ing in the 1980s after the capital market deregulation, and that the keiretsu

ties between banks and some of their shareholders provided protection to

bank managers from the discipline of other shareholders. Such risk taking

may affect relationship banking in Japan more substantially than any change

in the bank’s incentive to keep its commitment due to increased competition.

5.4 Long-lived entrepreneurs

In order to focus on the problem of bank commitment, it was assumed that

each entrepreneur exits the economy after one period. If there are long-lived

entrepreneurs, then this affects R

D

, the bank’s return at the end of a rescue.

Such an extension to the model does not change the qualitative results, but

it does raise further issues. One issue is the inside information the bank

would have in the following rounds of lending. In that case, R

D

can include

the information rent to the bank after a rescue. Rajan (1992) shows that

the bank’s rent from this inside information is independent of the number of

21

See, for example, Hannan (1991), Petersen and Rajan (1995), as well as the references cited in Din¸c (1997).

799

The Review of Financial Studies/v13n32000

competing banks. Thus the results reported here on the effects of competition

on bank reputation would be the same. Another issue is that repeated lending

allows the entrepreneur to commit to borrowing from the same bank and/or

not to raise more than a certain fraction by issuing bonds. This has the same

effect as increasing R

D

; it facilitates the bank’s commitment. It does not

change the ﬁnding about the effects of competition.

With long-lived entrepreneurs, the positive effects of competition may be

offset by the negative effects identiﬁed by Petersen and Rajan (1995). Banks

are likely to face this tension as credit markets are deregulated because, while

their ability to capture rents from a borrower in future lending will decrease,

they will have to focus on the type of lending that distinguishes them from

the security markets.

22

6. Conclusion

This article studies the bank’s commitment problem to lend to a borrower

in distress after ﬁnancing the borrower in good times. It shows that a rep-

utation mechanism may mitigate this problem in a repeated bank lending

setting. The effectiveness of this reputation mechanism depends on the credit

market competition. Unlike the borrower’s commitment problem, an increase

in credit market competition may enhance the bank’s incentive to lend to a

borrower in distress and maintain a good reputation. Whether an increase in

competition is beneﬁcial or not depends on both the source and the level of

competition. For example, bonds are a closer substitute for arm’s length bank

lending (loans without commitment) than for relationship lending (loans with

commitment). Hence the borrower’s access to bond markets decreases the

bank’s return from arm’s length lending more than that of relationship lend-

ing. This enhances the bank’s incentive to keep its commitment and maintain

a good reputation.

The effect of an increase in the number of banks depends on the number of

banks already in the market. An increase in banks may decrease reputational

rents too much to sustain a reputation mechanism if there is already a large

number of banks. On the other hand, if banks have large market power, they

can capture so much of the borrower’s surplus that they do not have incentive

to offer costly commitments. In that case, an increase in the number of banks

is beneﬁcial for the reputation mechanism.

The theory presented has further empirical implications. These include

Entrepreneurs with higher credit quality are more likely to be offered

bank commitments.

22

The comments of Reese Harasawa, a corporate planner at Mitsubishi Bank, made at a time when Japanese

companies drastically increased the amount they raised from security markets after deregulation, seem to

reﬂect these opposing effects in main bank lending: “Banks are still lenders of last resorts. [However,] banks

used to endure bad times in the hope of better deals later. That idea is changing now.” [Financial Times

(September 23, 1988)].

800

Bank Reputation, Bank Commitment, and the Effects of Competition

The interest rate charged decreases with credit quality.

The average credit quality of the entrepreneurs who are offered relation-

ship lending decreases if the banks face competition from bond markets.

Faced with competition from bond markets, the share of relationship lend-

ing in a bank’s loan portfolio increases.

The minimum number of banks necessary to sustain a reputation mech-

anism decreases with the borrowers’ access to bond markets.

The entry into a banking market by banks that cannot offer relationship

lending has implications similar to those of the competition from bond

markets.

Some of these empirical implications already have support in the empirical

literature. Avery and Berger (1991) observe that the performance of the loans

extended through loan commitments is better than that of noncommitment

loans, implying that borrowers with higher credit quality are offered loan

commitments. Qi and Shockley (1995) show that better quality ﬁrms tend

to ﬁnance with loan commitments. Shockley and Thakor (1997) ﬁnd that

interest rates and fees paid on loan commitment contracts decrease with the

borrower’s credit quality.

The available empirical evidence about the effect of bond markets on

bank lending incentives is also consistent with the theoretical predictions

provided in this article. Horiuchi (1994) and Din¸c (1999) observe that large

Japanese banks substantially increased the share of loans made to small and

medium-size companies after the capital market deregulation in the early

1980s. Anderson and Makhija (1999) ﬁnd that the proportion of bond debt

of Japanese companies in the late 1980s was inversely related to their growth

opportunities, which is consistent with the prediction that bond markets

strengthen a bank’s incentive to keep its commitment not to hold up its

borrowers in relationship banking. Gande, Puri, and Saunders (1999) ﬁnd

that smaller companies beneﬁted more from the increasing competition in

bond underwriting, which is similar to this article’s prediction that increased

competition induces banks to offer services to lower-rated borrowers than

they previously did.

The theory presented has additional implications for empirical work on the

effects of credit market competition on bank lending. One of the main points

is that the effects of credit market competition are different for different types

of bank lending. Consequently any empirical study on credit market compe-

tition has to be precise about the type of bank lending studied. Such studies

also have to be explicit about the source of the increase in competition, espe-

cially whether this increase is due to better access to security markets or to

the entry of new banks. Finally, the impact of a change in competition may

not be monotonic but may show qualitative differences at different levels.

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Appendix

Proof of Proposition 3. The symmetric equilibrium of the competition game is analyzed with

a more general payoff function for banks than implied in Proposition 3, for this competition

game will be the main building block for the analysis to come. Let X = (X

1

,...,X

n

) denote a

vector, the components of which are real-valued signals of the banks, and Y

1

be the maximum

signal among X

−1

= (X

2

,...,X

n

). Let f

Y

1

and F

Y

1

denote the density and the distribution

functions of Y

1

and

v(x,y) = E[V(θ)|X

1

= x, Y

1

= y], (A1)

respectively, where V(θ) is the identity function that takes the value 1 if θ = G, and 0 oth-

erwise. Notice that v(x, y) depends on the number of banks n through y. Milgrom and Weber

(1982) show that, if the density f(x

i

|θ) has MLRP for all i, then variables θ,X

1

,...,X

n

are

afﬁliated,

23

and that v(x, y) increases in both x and y.

Lemma A1. Let the bank’s payoff function π be given by

π(b) = V (θ)(p

S

bL − κ) − L, (A2)

where L is the loan size, κ ≥ 0 is any possible cost incurred in lending to a good type, and b is

the interest factor (interest rate plus one). Let T denote the maximum cash return the bank can

obtain. The equilibrium bidding strategies in bank competition are then given by

b

∗

(x) =

x

x

L

v(α, α)κ + L

v(α, α)p

S

L

dM(α|x) for x ≥ x

0

(A3)

where M(α|x) = exp

−

x

α

v(s,s)f

Y

1

(s|s)

H(s,s)

ds

with H(x, s) =

s

x

v(x, α)f

Y

1

(α|x) dα,

x

L

= sup

x

x

0

x

v(α, α)κ + L

v(α, α)p

S

L

dM(α|x) ≥ T

and

x

0

= inf

x

{+(T ; x) ≥ 0}.

The bank does not offer a loan for x<x

0

. The bank’s expected proﬁt is positive and decreasing

in n.

Proof. The symmetric equilibrium (b

∗

,...,b

∗

) is studied, and, without loss of generality, the

bidding strategy of bidder 1 is examined. The analysis follows Milgrom and Weber (1982).

The necessary conditions are derived for a symmetric equilibrium by assuming that b

∗

is a

decreasing function of x. A bidding function that satisﬁes these necessary conditions is then

found and veriﬁed that it is indeed an equilibrium strategy if all the other banks bid according

to that strategy.

23

Recall that two random variables X and Y are afﬁliated if f (x, y)f (x

,y

) ≥ f(x

, y)f (x, y

) for any x

<x

and y

<y.

802

Bank Reputation, Bank Commitment, and the Effects of Competition

If all other banks bid according to b

∗

, bank 1’s return at Equation (A.2) given its signal

becomes

+(b; b

∗

−1

,x) = E

(V (θ)(p

S

bL − κ) − L)1

{b

∗

(Y

1

)>b}

|X

1

= x

= E

E

(V (θ)(p

S

bL − κ) − L)1

{b

∗

(Y

1

)>b}

|X

1

,Y

1

|X

1

= x

= E

(v(X

1

,Y

1

)(p

S

bL − κ) − L)1

{b

∗

(Y

1

)>b}

|X

1

= x

=

b

∗−1

(b)

x

(v(x, α)(p

S

bL − κ) − L)f

Y

1

(α|x)dα. (A4)

Twoofthenecessary conditions for the equilibrium are

b

∗

(x) ≤ T, for all x (A5)

+(b

∗

; b

∗

−1

,x) ≥ 0, for all x. (A6)

Equations (5) and (6) imply that the bank does not offer a loan for x<x

0

, where

x

0

= inf

x

{+(T ; b

∗

−1

,x) ≥ 0}. (A7)

Differentiating Equation (A4) with respect to b and using the inverse function theorem,

+

b

(b;b

∗

−1

,x) =

1

b

∗

(b

∗−1

(b))

v(x,b

∗−1

(b))(p

S

bL− κ)−L

f

Y

1

(b

∗−1

(b)|x)

+

b

∗−1

(b)

x

v(x,α)p

S

Lf

Y

1

(α|x)dα. (A8)

Setting +

b

(b; b

∗

−1

,x) = 0 and arranging terms gives the linear differential equation:

b

∗

(x) =

−

v(x,x)(p

S

b

∗

(x)L − κ) − L

f

Y

1

(x|x)

x

x

v(x, α)p

S

Lf

Y

1

(α|x)dα

, for x ≥ x

0

. (A9)

This ﬁrst-order linear differential equation is a necessary condition. Equations (A5) and (A6)

give the boundary condition for Equation (A9):

b

∗

(x

0

) =

T

L

. (A10)

The solution [Equation (A3)] to the differential equation [Equation (A9)] with the boundary

condition [Equation (A10)] gives the bidding strategies in the symmetric equilibrium. Finally,

notice that M(α|x), regarded as a probability distribution on (x

,x) is stochastically increasing

in x, that is, M(α|x) decreases in x. Hence b

∗

(x) is (strictly) decreasing in x.

Veriﬁcation of b

∗

(x): The change of variable

dM(α|x) = M(α|x)

v(α, α)f

Y

1

(α|α)

H (α, α)

dα

gives

b

∗

(x) =

(v(x, x)κ + L)

p

S

L

f

Y

1

(x|x)

H(x, x)

+

x

x

L

v(α, α)κ + L

p

S

L

M(α|x)

−

v(x,x)f

Y

1

(x|x)

H(x, x)

f

Y

1

(α|α)

H (α, α)

dα.

Equation (A9) then follows from Equation (A3).

803

The Review of Financial Studies/v13n32000

Sufﬁciency: To show that b

∗

is a best response when all the other players play b

∗

,itis

sufﬁcient to consider only the bids in the range of b

∗

.Forb

∗

(z) to be an optimal bid when

X

1

= z, it is sufﬁcient for +

b

(b

∗

(x); z) to be nonnegative for x>zand nonpositive for x<z.

From Equation (A8),

+

b

(b

∗

(x); z) =

1

b

∗

(x)

(v(z, x)(p

S

bL − κ) − L)f

Y

1

(x|z)

+

x

x

v(z, α)p

S

Lf

Y

1

(α|z) dα

=

1

b

∗

(x)

(v(z, x)(p

S

bL − κ) − L)

v(z, x)p

S

L

+

x

x

v(z, α)f

Y

1

(α|z) dα

v(z, x)f

Y

1

(x|z)

. (A11)

The right-hand side of Equation (A11) is, of course, zero for z = x. The following working

lemma will be used to show {+

b

(b

∗

(x); z) − +

b

(b

∗

(x); x)}=sgn{x − z}.

Lemma A2. (

x

x

v(z, α)f

Y

1

(α|z)dα)/v(z, x)f

Y

1

(x|z) is decreasing in z.

Proof. The afﬁliation property implies that

v(z, α)f

Y

1

(α|z)v(z

,x)f

Y

1

(x|z

) ≤ v(z

,α)f

Y

1

(α|z

)v(z, x)f

Y

1

(x|z)

or

v(z, α)f

Y

1

(α|z)

v(z, x)f

Y

1

(x|z)

≤

v(z

,α)f

Y

1

(α|z

)

v(z

,x)f

Y

1

(x|z

)

. (A12)

Integrating both sides of Equation (A12) with respect to α gives the desired result. Q.E.D.

Note that

sgn

v(z, x)(p

S

b

∗

(x)L − κ) − L

v(z, x)p

S

L

−

v(x,x)(p

S

b

∗

(x)L − κ) − L

v(x,x)p

S

L

= sgn{z − x}. (A13)

As b

∗

(x) < 0, it follows from Equation (A13) and Lemma A2 that

sgn

+

b

(b

∗

(x); z) − +

b

(b

∗

(x); x)

= sgn{x − z}. (A14)

Positive proﬁts: For x>x

0

,

+(b

∗

(x); x) > +(b

∗

(x

0

); x)

>+(b

∗

(x

0

); x

0

)

= 0,

where the ﬁrst inequality follows from the equilibrium property while the second follows from

Equation (A4) and the fact that v(x, y) increases with x.

24

Hence the bank’s expected proﬁt

before it obtains its signal is given by E

θ

[E

X

[+(b

∗

(x); x)|θ ]] > 0.

24

The strict inequality for all x>x

0

follows from the full support assumption for bank signals. Without

that assumptions, the inequalities hold strictly only for some x>x

0

. However, the qualitative results are

unaffected.

804

Bank Reputation, Bank Commitment, and the Effects of Competition

Proﬁts decreasing with n: Note that

v(x, α)f

Y

1

(α|x) =

f

X

1

Y

1

(x, α|θ = G) Pr(θ = G)

f

X

1

Y

1

(x, α)

f

X

1

Y

1

(x, α)

f(x)

=

f

X

1

Y

1

(x, α|θ = G) Pr(θ = G)

f(x)

.

Therefore Equation (A4) becomes

+(b

∗

(x); x) =

F

X

1

Y

1

(x, x|θ = G) Pr(θ = G)

f(x)

p

S

b

F

− F

Y

1

(x|x). (A15)

Note that

F

X

1

Y

1

(x, x|θ = G) = [F(x|θ = G)]

n

,

F

Y

1

(x|x) =

[F(x|θ = G)]

n−1

Pr(θ = G) + [F(x|θ = B)]

n−1

Pr(θ = B)

f(x)

.

By MLRP F(x|θ = B)>F(x|θ = G). Thus

∂

∂n

+(b

∗

(x); x) < +(b

∗

(x); x) ln[F(x|θ = G)]

< 0. (A16)

This concludes the proof of Lemma A1. Q.E.D.

With L = 1, κ = 0, and T = R

S

in Lemma A1, the bidding function b

N

and the threshold

level x

N

in one-shot lending follow. The rest of the equilibrium is as given in Proposition 1.

Q.E.D.

Proof of Proposition 4. As in Proposition 3, only the bidding strategies will be stated and

veriﬁed; the rest follows from Proposition 2. Let +

N

(·) and +

C

(·) be the bank’s proﬁt function,

as given in Equation (A4) with κ = 0 and κ = p

D

(1 − R

D

), respectively. Suppose

+

C

(R; x) ≥ +

N

(b

N

(x); x) for some x< ¯x. (A17)

Let

x

C

≡ inf

x

{+

C

(R; x) ≥ +

N

(b

N

(x); x)}. (A18)

Let b

N

(x) be as in Proposition 3 and b

C

(x) be the bidding function given in Lemma A1 with

κ = p

D

(1 − R

D

), T = R

S

, and x

0

= x

C

.

Punishment path: If a bank does not keep its rescue commitment, future entrepreneurs do

not borrow a loan with a rescue commitment from that bank. Lending strategies of the deviant

bank are

b

N

(x

C

) and the bank does not commit to rescue, for x ≥ x

C

b

N

(x) and the bank does not commit to rescue, for x

N

≤ x<x

C

no loan, for x<x

N

.

Lending strategies of other banks are

b

CP

(x) and the bank commits to rescue, for x ≥ x

CP

b

N

(x) and the bank does not commit to rescue, for x

N

≤ x<x

CP

no loan, for x<x

N

.

805

The Review of Financial Studies/v13n32000

Let +

(n)

(·) and b

(n)

(·) be the bank’s proﬁt function and the equilibrium bidding function when

the number of banks is n. Let

x

CP

≡ inf

x

+

(n−1)

C

(R; x) ≥ +

(n)

N

b

(n)

N

(x); x

. (A19)

b

CP

(x) is then the bidding strategy b

(n−1)

(x) with κ = p

D

(1 − R

D

) and x

0

= x

CP

in Lemma A1.

¯

δ(n): Let w

C

and w

P

be the expected proﬁt per period of a bank with a good reputation on

the equilibrium path and of a deviant bank on the punishment path, respectively that is,

w

C

= E

θ

x

C

x

N

+

N

(b

N

(x); x)f (x|θ)dx

+

¯x

x

C

+

C

(b

C

(x); x)f (x|θ)dx

(A20)

and

w

P

= E

θ

x

CP

x

N

+

N

(b

N

(x); x)f (x|θ)dx

+

¯x

x

CP

+

N

(b

N

(x

CP

); x)f (x|θ)dx

. (A21)

Notice that w

C

and w

P

are a function of n. The bank’s incentive constraint to rescue is

R

D

− 1 +

δ

1 − δ

w

C

≥

δ

1 − δ

w

P

(A22)

or

δ ≥

1 − R

D

w

C

− w

P

+ 1 − R

D

≡

¯

δ(n) (A23)

¯n: Note that

lim

n→∞

ν(x, α) = 0 for all x<¯x, α ≤ x. (A24)

Hence, for sufﬁciently high n, it follows from Assumptions 4 and 5 that a bank has an incen-

tive to offer a loan with a commitment—and keep its commitment—if its signal about the

entrepreneur is good enough, that is, there exists ¯n, such that for all n ≥¯n,

µ(x)

p

S

R

S

− p

D

(1 − R

D

)

− 1 ≥ +

N

(b

N

(x); x) for some x<¯x. (A25)

To see that Equation (A25) is sufﬁcient for the existence of the equilibrium described, suppose

all the banks except one are “forced” to lend without a commitment. Then an equilibrium exists

with one bank offering both types of loans—depending on its signal—while n−1 banks offering

only loans without commitment (Proposition 6 gives an equilibrium in which n banks offer both

types of loans and m banks can offer only loans without commitment). The expected return to

the bank that can lend with a commitment is higher than the return of the other banks and those

banks are better off if they are also “allowed” to lend with a commitment. Hence an equilibrium

can be constructed with two banks offering commitment loans, while n − 2 banks are still forced

to lend without a commitment, and the rest follows by induction.

Note that a monopolist bank always demands all the cash return from the project; hence

rescuing a distressed borrower only adds a rescue cost without increasing its return. Therefore

¯n ≥ 2. Finally, by making the right-hand side of Assumption 4 sufﬁciently low, ¯n can be made

arbitrarily large. Q.E.D.

806

Bank Reputation, Bank Commitment, and the Effects of Competition

Proof of Proposition 5. Equation (A24) implies that

lim

n→∞

+

C

(b

C

(x); x) = lim

n→∞

+

N

(b

N

(x); x) = 0 for x<¯x.

Hence, from Equations (A20), (A21), and (A23),

lim

n→∞

¯

δ(n) = 1.

Proof of Proposition 6. As before, the symmetric equilibrium of the bank competition where

the same type of banks using the same strategies is considered. The bidding strategies in this

equilibrium are similar to those on the punishment path of bank lending with a rescue commit-

ment, as described in Proposition 4. The banks that can lend with a commitment will be referred

to as established and the other banks as new. Let the number of established banks be n and that

of the new banks be m. The bidding strategies of established banks are

b

(n)

C

(x) and the bank commits to rescue, for x ≥ x

CF

b

(n+m)

N

(x) and the bank does not commit to rescue, for x

NF

≤ x<x

CF

no loan, for x<x

NF

where

x

CF

≡ inf

x

+

(n)

C

(R; x) ≥ +

(n+m)

N

b

(n+m)

N

(x); x

(A26)

and x

NF

is same as what x

N

would be with n + m banks.

25

The lending strategy of a new bank with an estimate x ≥ x

CF

is different from that of the

deviant bank in Proposition 4 because it has to compete with other new banks. Let X, Y, Z be

a (new) bank’s own estimate, the highest estimate among all other new banks and the highest

estimate among all established banks, respectively. Similar to Equation (A1), let

ν

F

(x,y,z)= E

V(θ)|X = x, Y = y,Z = z

. (A27)

Thus the expected proﬁt of a new bank playing the equilibrium bidding strategies b

F

when

x ≥ x

CF

is given by, from Equation (A4),

+

F

(b

F

; x) =

x

x

(ν

F

(x,α,x

CF

)p

S

b

F

− 1)g

Y

(α|x, x

CF

)dα, (A28)

where

g

Y

(α|x, x

CF

) = f

Y

(α|X = x, Z ≤ x

CF

). (A29)

b

F

can be obtained from Lemma A1 by substituting g

Y

(α|x, x

CF

) for f

Y

1

(α|x) and determining

x

L

such that b

F

(x

CF

) = b

(n+m)

N

(x

CF

). Therefore a new bank’s bidding strategies on the equilib-

rium path are

b

F

(x) and bank does not commit to rescue, for x ≥ x

CF

b

(n+m)

N

(x) and bank does not commit to rescue, for x

NF

≤ x<x

CF

no loan, for x<x

NF

.

25

Notice that x

CF

<x

NF

for large m, in which case the established banks compete by offering only loans with

a rescue commitment.

807

The Review of Financial Studies/v13n32000

The bidding strategies on the punishment path are the same as those with n − 1 established

banks and m + 1 new banks, with the deviant established bank playing the same strategies with

the new banks. Let x

CFP

be the threshold on the punishment path, that is,

x

CFP

≡ inf

x

+

(n−1)

C

(R; x) ≥ +

(n+m)

N

b

(n+m)

N

(x); x

. (A30)

¯

δ decreases with m: Let w

E

(n, m) and w

N

(n, m) be the expected proﬁt per period of an

established bank and a new bank, respectively, when n established banks and m new banks

compete. The bank’s incentive constraint is analogous to Equation (A23), so

¯

δ decreases with

m,if

w

E

(n, m) − w

E

(n, m + 1)<w

N

(n − 1,m+ 1) (A31)

−w

N

(n − 1,m+ 2).

Note that

w

E

(n, m) − w

E

(n, m + 1)<w

N

(n, m + 1) − w

N

(n, m + 2). (A32)

The proof is then complete if it can be shown that the right-hand side of Equation (A31) is

greater than the right-hand side of Equation (A32) or (∂

2

+

F

(b

F

; x))/∂m∂n > 0 for all x.

From Equation (A15),

+

F

(b

F

; x) =

F

Y

(x,x,x

CF

|θ = G) Pr(θ = G)

f(x,x

CF

)

p

S

b

F

(A33)

−G

Y

(x|x, x

CF

).

Lemma A3.

∂

2

∂m∂n

F

Y

(x,x,x

CF

|θ = G) Pr(θ = G)

f(x,x

CF

)

=

ACD

(A + C)

2

ln F(x

CF

|θ = G) − ln F(x

CF

|θ = B)

(A34)

and

∂

2

G

Y

(x|x, x

CF

)

∂m∂n

=

AC(D − E)

(A + C)

2

ln F(x

CF

|θ = G)

− ln F(x

CF

|θ = B)

, (A35)

where

A ≡ f(x|θ = G)[F(x

CF

|θ = G)]

n−1

Pr(θ = G) (A36)

C ≡ f(x|θ = B)[F(x

CF

|θ = B)]

n−1

Pr(θ = B) (A37)

D ≡ [F(x|θ = G)]

m−1

ln F(x|θ = G) (A38)

E ≡ [F(x|θ = B)]

m−1

ln F(x|θ = B) (A39)

808

Bank Reputation, Bank Commitment, and the Effects of Competition

Proof. Note that

F

Y

(x,x,x

CF

|θ = G) = f(x|θ = G)[F(x|θ = G)]

m−1

× [F(x

CF

|θ = G)]

n−1

(A40)

f(x,x

CF

) = f(x|θ = G)[F(x

CF

|θ = G)]

n−1

Pr(θ = G)

+ f(x|θ = B)[F(x

CF

|θ = B)]

n−1

Pr(θ = B) (A41)

G

Y

(x|x, x

CF

) =

F

Y

(x,x,x

CF

|θ = G) Pr(θ = G)

f(x,x

CF

)

+

F

Y

(x,x,x

CF

|θ = B)Pr(θ = B)

f(x,x

CF

)

. (A42)

Hence

∂

∂m

F

Y

(x,x,x

CF

|θ = G) Pr(θ = G)

f(x,x

CF

)

=

AD

A + C

;

∂

2

∂m∂n

F

Y

(x,x,x

CF

|θ = G) Pr(θ = G)

f(x,x

CF

)

=

D

(A + C)

2

A(A + C) ln F(x

CF

|θ = G)

−A

A ln F(x

CF

|θ = G) − C ln F(x

CF

|θ = B)

, (A43)

and Equation (A34) follows. Similarly,

∂

2

G

Y

(x|x, x

CF

)

∂m∂n

=

ACD

(A + C)

2

ln F(x

CF

|θ = G) − ln F(x

CF

|θ = B)

+

ACE

(A + C)

2

ln F(x

CF

|θ = B) − ln F(x

CF

|θ = G)

and Equation (A35) follows. Q.E.D.

By Lemma A3,

∂

2

+

F

(b

F

; x)

∂m∂n

=

AC

(A + C)

2

D(p

S

b

F

(x) − 1) + E

×

ln F(x

CF

|θ = G) − ln F(x

CF

|θ = B)

. (A44)

Notice that D<0 and E<0. From Equation (A6), p

S

b

F

(x) − 1 > 0. Finally, MLRP implies

that F(x

CF

|θ = G) < ln F(x

CF

|θ = B); hence ln F(x

CF

|θ = G) − ln F(x

CF

|θ = B) < 0. This

establishes that

¯

δ decreases with m.

Finally, +

(n+m)

N

(b

(n+m)

N

(x); x) decreases with m; therefore ¯n and x

CF

decreases with m.

Q.E.D.

Proof of Proposition 7. Although it is not essential for the results, the game has a signaling

component at date 0, when the entrepreneur sets the amount to be raised from the bond market.

Notice, however, that a bad entrepreneur always mimics the good one. Let β denote the amount

the entrepreneur raises by issuing bonds. The loan size L will be explicit in the notation in this

section, for example, b(x, L) and +(b; x, L), respectively. Let Z

1

denote the highest estimate

among banks.

809

The Review of Financial Studies/v13n32000

Entrepreneur’s strategy: Raise β = β

N

if a loan without commitment is offered and β = β

C

for a loan with commitment. Collect offers from banks for a loan of size 1 − β

N

without

commitment and 1 − β

C

with commitment.

Beliefs of banks (before credit screening) and bondholders: If β = β

N

for a loan without

commitment and β = β

C

for a loan with commitment, set Pr(θ = G) = λ. Otherwise Pr(θ =

G) = 0.

Bondholders update their beliefs after observing the loan type as follows: The probability

that the entrepreneur is good is

Pr(θ = G|x

NB

≤ Z

1

<x

CB

) if the loan comes without a

rescue commitment,

Pr(θ = G|x

CB

≤ Z

1

≤¯x) if it comes with a rescue commitment.

Bondholders’ strategy: They do not lend if no bank offers to lend. When they lend, the

repayment D they demand depends on both the type and the amount of the bank loan the

entrepreneur obtains. If the lending bank does not commit to rescue, the repayment D

N

satisﬁes

Pr(θ = G|x

NB

≤ Z

1

<x

CB

)p

S

D

N

(β

N

) − β

N

= 0. (A45)

If the lending bank commits to rescue, the repayment D

C

satisﬁes

Pr(θ = G|x

CB

≤ Z

1

≤¯x)

p

S

D

C

(β

C

)

+p

D

min{R

D

,D

C

(β

C

)}

− β

C

= 0, (A46)

with

x

NB

= inf

x

+

N

(R − D

N

(β

N

); x,1 − β

N

) ≥ 0

and (A47)

x

CB

= inf

x

+

CB

(R − D

C

(β

C

); x,1 − β

C

)

>+

N

(b

N

(x; 1 − β

N

); x,1 − β

N

)

, (A48)

where +

CB

is the proﬁt function + in Lemma A1 with κ = p

D

(1 − max {0,R

D

− D

C

(β

C

)}).In

equilibrium, Equations (A45)–(A48) are satisﬁed simultaneously.

Access to bond markets decreases

¯

δ

: Suppose β

C

= 0 and β

N

= 1. This case is equivalent to

the entry of inﬁnitely many new banks (m =∞) in Proposition 6. Hence, by continuity, there

exists an equilibrium with 0 <β

C

,β

N

< 1, which decreases

¯

δ. Q.E.D.

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