DBJ Discussion Paper Series, No.0804

A Dynamic Equilibrium Model for
Relationship-Lending

Daisuke Miyakawa(Research Institute of Capital Formation,
Development Bank of Japan)

March

2009

Discussion Papers are a series of preliminary materials in their draft form. No quotations,reproductions or circulations should be made without the written consent of the authors inorder to protect the tentative characters of these papers. Any opinions, findings, conclusions or
recommendations expressed in these papers are those of the authors and do not reflect theviews of the Institute.

A Dynamic Equilibrium Model for
Relationship-Lending
Daisuke Miyakaway

Abstract
In this paper, we develop a dynamic equilibrium model of the banking industry
that takes into account for …rm and bank heterogeneity and the market for new
and existing loans with a full description for the demand and supply of bank loans.
The consequences of various shocks to the economy are considered through theequilibrium analysis. The model predicts that as the degree of interbank competition becomes higher, the number of loan relations for each …rm becomes smaller.
Our model is also consistent with several empirical regularities regarding the loan
structure.
Key words: Relationship-lending; Dynamic model; Stationary equilibrium
JEL Classi…cation: G21, G32, D53, C78

I thank Hugo Hopenhayn, Andy Atkeson, Lee Ohanian, Daniel Ackerberg, Isabelle Perrigne, Christian Hellwig, Pierre-Olivier Weill, Pierre-Yves Yanni, Shogo Hamasaki and seminar participants at
UCLA’s Monetary Economics Proseminar, Industrial Organization Proseminar, and DBJ Applied Economics Workshop for useful comments and suggestions.
yResearch Institute of Capital Formation, Development Bank of Japan, 1-9-3 Otemachi Chiyoda-ku,
Tokyo, 100-0004 JAPAN. E-mail: damiyak@dbj.jp.

11

Introduction

The dominant presence of bank …nancing in the capital and debt structures of …rms has
stimulated various theoretical and empirical studies about the role of banks as a …nancing
source. In the literature, one distinctive feature associated with bank loans is characterized as "relationship-lending"; that is, bank …nancing accompanied with some over-time
informational activity (Freixas and Rochet (2008)).1 Existing theoretical studies on
relationship-lending have provided interesting implications regarding, for example, thedynamics of loan prices (Rajan (1992)) and the implications of better credit information about borrowers (Padilla and Pagano (1997), Hauswald and Marquez (2003)). The
empirical literature has also examined the implications of relationship-lending (James(1987); Peterson and Rajan (1994); Berger and Udell (1995, 2006)).
The main mechanism and several implications originating from relationship-lending
are relatively well understood in the literature. Yet the question remains as to how di¤erent degrees of competitiveness in the banking industry a¤ect the …rms’loan structure in
the context of relationship-lending. Since traditional banking institutions have recently
become more competitive and subject to replacement by transaction-based institutions,
it is important to be able to answer this question. The motivation of this paper is todevelop a theoretical relationship-lending model to study this question.
To illustrate the main mechanism of our model, consider an economy where a …rm
has a set of young venture projects and established projects, both of which are …nanced
by separate banks. When the …rm faces an opportunity to upgrade one of its venture
projects to an established project (e.g., expanding their sales area or augmenting a new
manufacturing line for a speci…c product), it may need additional …nancing and expert
advice. Presumably, the bank holding an incumbent loan relation with this …rm (forthe original venture project) is in an advantageous position to provide such …nancing
and advice. We interpret this as relationship-lending. Rajan (1992) demonstrates thatbanks might even be willing to o¤er a discounted interest rate for the venture projectso that they can establish a loan relationship that eventually gives them some expertise.
After obtaining this expertise, banks can charge a higher interest rate. In contrast
to Rajan (1992), we develop a model in which the banks’ rent in the later stage ofthe relationship is determined as an equilibrium object. We do this by modeling thedemand and supply for the start-up loan as the solutions to the dynamic optimization
problems of …rms and banks. The loan spread between the start-up loan for a venture
project and the follow-up loan for an established project governs the demand and supply,and in equilibrium, the loan spread is endogenously determined to clear the bank loan
market. This is the basic mechanism incorporated in this paper. For this dynamic
model, we employ an analytically tractable dynamic programming problem with discrete
state variables developed in Klette and Kortum (2004). One important di¤erence is that
we have two sides corresponding to the demand and supply of the loan. In order to closethe model, we also employ a technique that is similar to Mortensen and Pissarides (1994)
1

Ex-ante screening and/or ex-post monitoring are the most typical informational activities and "overtime" implies that these activities are implemented at some point(s) over their relationships.

2

in order to characterize a stationary equilibrium in a dynamic search framework.
Compared to existing models, our model has two distinctive features. First, ourmodel contains the individual optimization problems for the demand and supply of bankloans. Most existing papers generally abstract from the demand side of bank loans byassuming that demand is inelastic (Boot and Thakor (2000)). Thus these models cannot
analyze demand-side repercussions caused by various economic shocks. As we will see
later, this channel generates interesting results. Second, our model employs a dynamic
equilibrium framework that complements the multi-stage game theoretic framework usedin existing studies. This approach allows us to endogenously determine the e¤ects ofseveral economic shocks to loan prices. Firms and banks choose their optimal project
and loan portfolios by considering the costs and bene…ts associated with their choices.
Our dynamic equilibrium framework explicitly captures this mechanism. Through theequilibrium analysis, we can study the individual behaviors of …rms and banks, as wellas analyze the implications of these behaviors at the industry-level.
In this paper, we also construct the model to be consistent with several empiricalregularities associated with the structure of bank loans. A number of existing empiricalstudies (Miyakawa (2008b)) have documented that …rms tend to diversify their loans
across banks and that loan relationships respond to various covariates (e.g., …rm size,…rm pro…tability, and bank size) in a systematic way. As far as we know, there is no
theoretical model that comprehensively considers these features. Our paper provides apotential underlying mechanism for such multiple loan structures.
Following the literature on relationship-lending, our model features an intangible
capital that accumulates between …rms and banks over time. This modeling strategy
re‡ects the traditional view that banks have the ability to search for better projects within
the …rms’pool of uncerti…ed projects (Greenwood and Jovanovic (1990))2 . It also re‡ects
a relatively new perspective that banks that have close and sustained relations with aparticular …rm has better visibility to screen the …rm’s projects (Hauswald and Marquez(2003)).3
This paper is structured as follows. Section 2 surveys the related literature. Sections
3 and 4 presents the model and describes the equilibrium. Section 5 implements comparative statics to study the implications of the model. Section 6 simulates the model anddemonstrates some quantitative comparative statics. Section 7 concludes and presents
future research questions.
2

In banking literature, this screening ability is recognized as an important feature of banks along witha monitoring ability. The former corresponds to an ex-ante activity and the latter is a ex-post activity.
3
There is an accumulation of empirical studies on this relationship-banking view. See Peterson andRajan (1994); Berger and Udell (1995); Degryse and Cayseele (2000).

3

2Related Literature

The role of banks has been discussed in two classic papers. First, Greenwood and Jovanovic (1990) takes the view that banks have an ex-ante screening ability. The mainidea is that since banks have a large pool of clients and investment opportunities, they
have an advantage over …rms in searching for pro…table projects. Second, Diamond(1984, 1996) points to the banks’ ex-post monitoring capabilities and argues that delegating this monitoring to the banks helps avoid the duplication of monitoring costs
among investors.4 Our model is based on the former view and abstracts from the one
associated with ex-post monitoring.
In the context of relationship-lending, there are two closely related studies.5 First,
Rajan (1992) models a bank’s dynamic loan provision. In his model, each bank needs
to pay out once-and-for-all sunk cost in monitoring for providing a loan to a …rm. The
existence of this sunk cost gives rise to a standard switching cost since the party incurs
a duplicated monitoring cost when they switch banks.6 If we associate lower monopoly
rents with a low degree of interbank competition, the model in Rajan (1992) has theimplication that higher inter-bank competition leads to smaller numbers of loan relations. Our model shares the persistent e¤ect related to switching costs and our main
result follows the perspective of his model. In our setup, the existing loan relation allows banks and …rms to implement a better project. This has a similar e¤ect to theonce-and-for-all sunk costs. Note that the main contribution of his paper is to model
the ex-post monopoly of information and the resulting dynamic price pattern, whichstarts at a discount and increases later in the relationship. This result is similar to conventional switching cost models. In contrast, we are mainly interested in the resulting
loan structure corresponding to several shocks. Moreover, unlike his two-period gametheoretic framework, our model employs a dynamic equilibrium setup which allows us toendogenously determine the loan price. Second, Hauswald and Marquez (2003) model
the emergence of relationship banking based on the perspective of incumbent lenders’accumulated ability to screen projects, which is also employed in our model. The main
contribution of their paper is to show that the e¤ects of better private information, obtained through relationship-lending, has the opposite e¤ect of better public information.
Intuitively, if the importance of private information is quite high, the ex-post monopoly
problem considered in Rajan (1992) becomes severe while better public information alleviates such a problem. Our model takes a similar view to their model about banks’
informational activity. The main di¤erence is that we allow for multiple loan relationships, which is one of the key issues that we want to consider.
4

Several papers (e.g., Boyd and Prescott (1990)) discuss the potential contracting mechanism allowing
this delegation to work. In this paper, we abstract from this contracting problem.
5
Also, there are many empirical papers that attempt to establish evidence for relationship lending(James (1987); Peterson and Rajan (1994); Berger and Udell (1995, 2006)). Considering the characteristics of this paper, we omit surveying these empirical papers. For a comprehensive survey, see Freixas
and Rochet (2008).
6
This informational activity modeled in Rajan (1992) is associated with the ex-ante cost. Although
this is called as a monitoring cost in his paper, we should categorize it as one variant of the ex-antescreening model.

4

3

Model

In this section, we construct a dynamic equilibrium model for relationship-lending. In
our model, we endogenously determine (i) the number of banks from which each …rm
borrows and (ii) the status of each bank loan relationship between …rms and banks.
Several important issues discussed in the existing corporate …nance and banking literature are omitted. First, we treat each loan contract as a given object and do notdiscuss a general optimal contract from which such loan contract emerges.7 Second, we
treat the bank loan as an exclusive …nancing channel and ignore other …nancing options(e.g., equity or any other hybrid securities).8 Third, there is no information asymmetry
between …rms and banks. The only friction we consider is that each …rm and bank needs
to incur search costs to initiate their loan relationships and can upgrade the project toa more pro…table one only if they continue the incumbent loan relation. This over-time
informational activity, originally introduced by Rajan (1992) and extended in Hauswald
and Marquez (2003), is the mechanism we employ.

3.1

Environment

Consider an economy consisting of a continuum of risk-neutral …rms and banks. Each…rm is represented by a set of projects. We assume that each …rm does not originallyhave any …nancial resources to implement a project and needs to use bank loans to …nanceeach project. For tractability, we also assume that each project needs to be …nanced bya di¤erent bank. This setup implies that a …rm’s project portfolio choice is equivalent
to its borrowing structure.
There are two types of projects: uncerti…ed venture-projects (U-P) and certi…ed
established-project (C-P), where (C-P) yields a higher revenue than (U-P). With somepositive probability, (U-P) can be upgraded to a (C-P) in the future. We assume that
this economy has a …xed unit mass of projects in total. The relative aggregate sharesof (C-P) and (U-P) in this economy is endogenously determined. One interpretation ofthis assumption is that …rms are competing for limited business opportunities. In orderto implement a project, each …rm needs to snatch another …rm’s current client.
We also assume that in order to implement one (C-P), each …rm needs to start fromone (U-P) and then use a loan from the bank that …nanced this particular (U-P). Thus,…rms need to take another loan from their incumbent bank to upgrade the project. This
corresponds to our notion of relationship-lending which requires time to exhibit a better
project screening ability.
7

Various frictions might be able to justify the debt contract as claimed in incomplete contract literature (Hart and Moore (1998)). Miyakawa (2008a) discusses this issue from a complete contract
context.
8
Coexistence of debt and equity in a …rm’s capital structure is one important issue as discussed in,for example, Holmstrom and Tirole (1997).

5

3.1.1

Technology

The initiation of a project requires inputs from both the …rm and bank as well as a loan
from the bank. This can be interpreted as the provisions of project idea from the …rmside and screening/monitoring capital from the bank side.9
Each (U-P) provides a deterministic outcome u > 0 and each (C-P) yields a strictly
larger deterministic outcome c > u over the next instantaneous moment. In this sense,
the di¤erence ( cu ) summarizes the value of the certi…cation.
We assume that each …rm incurs a cost to …nd a new (U-P). A new (U-P) arrives atPoisson rate (e; nc ). The variable e represents the …rm’s search cost, a choice variable,
and nc denotes the number of (C-P) held by the …rm. Intuitively, a …rm with a larger
number of (C-P) and/or exerts a larger search cost is more likely to …nd an additional(U-P) over the next instantaneous moment. We assume (e; nc ) is homogenous of degree
one in e and nc , and (0; 0) = 0.10
3.1.2

Bank Loan

As with …rms, each bank is represented by its loan portfolio. For tractability, we also
assume that each bank can o¤er at most one loan relation to each …rm. There are two
types of loans: Start-up loan for an uncerti…ed project (U-L) and follow-up loan for acerti…ed project (C-L). In this paper, we interpret (C-L) as relationship-lending.
To implement one (U-P), it su¢ ces for a …rm to use a start-up loan (U-L). For
simplicity, we assume that (U-L)’s face interest factor is an exogenous number R
1 + r < u . Since the project outcome is deterministic, R can be interpreted as arisk-free interest rate factor.11 To implement (C-P), however, each …rm needs to use
a follow-up loan (C-L) from its incumbent bank (i.e., the same bank provided (U-L) tothe (U-P)), which has a spread > 0 over the risk-free interest factor. This spread is
endogenously determined in equilibrium later. This setup also implies that each bankcannot directly provide (C-L) but needs to start from (U-L).
In order to …nd a new opportunity to provide (U-L), the bank needs to exert somesearch cost. A new (U-L) arrives at Poisson rate (d; mf ). The variable d denotes thebank’s search e¤ort, which is a bank’s choice variable, and mf denotes the number of(C-L) held by the bank. We assume (d; mf ) is homogenous degree one in d and mf ,and (0; 0) = 0.
9In this sense, we are assuming the Leontief production function which has the …rm’s idea and bank’s
screening/monitoring capital as its inputs.
10This speci…cation is widely used in the search literature. Klette and Kortum (2004) also employs
this assumption.
11
We abstract from the discount on the interest rate associated with (U-L), which is considered in
Rajan (1992), and focus on the spread between (U-L) and (C-L).

6

3.1.3Evolution of Match

Once a …rm and bank match, the project implemented by this match gives a stream ofpro…ts according to the following stochastic process: (i) One (U-P) match upgrades toone (C-P) match with an exogenous Poisson rate0, (ii) one (U-P) drops from the
project/loan portfolio with a Poisson rate
0 which is treated as given in an individual…rm/bank problem but is endogenously determined in equilibrium, and (iii) one (C-P)
match degrades into one (U-P) match with an exogenous Poisson rate
0. We caninterpret (iii) as a break-up of the long-term loan relationship.
Note that , , and are common to both …rms and banks. Thus, we can interpretthese three parameters as Poisson rates for each match. Once they are matched througha (U-P), these three parameters govern the dynamics of the match.
3.1.4

Firm’s Project Portfolio

Each …rm’s current project portfolio consists of nc (C-P) with project size 1 and nu (U-P)
with project size 1, both of which have already been held by the …rm at the beginningof a period. Since (nc ; nu ) are predetermined, they are treated as state variables in the…rm’s dynamic optimization problem. One interpretation of this timing assumption isthat the menu of projects has been established via past business activities and each …rmneeds to choose the search e¤ort for an additional uncerti…ed project for its future project
portfolio. We treat a …rm in state nc = 0 and nu = 0 as having permanently exited fromthe economy.
3.1.5

Bank’s Loan Portfolio

The structure of each bank’s loan portfolio takes a similar structure to the …rm’s project
portfolio. It consists of the …nancing of mc (C-P)’s with loan size 1 and mu (U-P)’s withloan size 1, both of which are state variables. We treat a bank in state mc = 0 and
mu = 0 as having permanently exited from the economy.
3.1.6

Market Structure

Without loss of generality, we assume that banks with lending capacities post o¤ers for
new contracts that involve the following rates of interest: (i) R when the project is
uncerti…ed, and (ii) R + when it becomes certi…ed. Perfect competition determines the
spread so as to equate the demand and supply of the loan for new uncerti…ed projects
as described later.1212Precisely speaking, the rate of interest R + determines how to split the surplus from (C-P) between
the …rm and bank. Since our model contains only one degree of freedom associated with the loan priceas we will see later, we …x R and endogenously determine .

7

3.2

Individual Optimization Problems

Each …rm maximizes its expected discounted life-time pro…t V (nc ; nu ) by choosing optimally the search e¤ort e. Here, from the assumption that is homogeneous of degree
one in (e; nc ), we can obtain the following convenient expression (1) in which we assumeeethat each …rm’s Poisson rate
; 1 is strictly increasing and concave in its
nc
nce
search intensity nc .
(e; nc ) = nc

e
nc

(1)

By using this expression, the dynamic optimization problem for each …rm can beconstructed. For given ( c ; u ; R; ; ; ; ), each …rm with (nc ; nu ) solves the following
dynamic programming problem.
8
9
[
(R
+
)]
n
+
[
R]
n
e
>
>
c
c
u
u
>
>
>
>
>
>
+
n
[V
(n
+
1;
n
1)
V
(n
;
n
)]
>
>
u
c
u
c
u
<
=
+
n
[V
(n
;
n
1)
V
(n
;
n
)]
u
c
u
c
u
rV (nc ; nu ) = max(F-P)
e > + n [V (n1; nu + 1) V (nc ; nu )] >
>
>
c
c
>
>
>
>
>
>
e
: +nc
;
[V
(n
;
n
+
1)
V
(n
;
n
)]
c
u
c
u
nc

Similarly, the dynamic optimization problem for each bank’s value W (mc ; mu ) can bedd; 1 where each bank’s Poisson rate is assumed to beconstructed. Letmcmc
strictly increasing and concave in its search intensity mdc . For given ( c ; u ; R; ; ; ; ),each bank with (mc ; mu ) solves the following dynamic programming problem.
9
8
m
d
>
>
c
>
>
>
>
>
>
+
m
[W
(m
+
1;
m
1)
W
(m
;
m
)]
>
>
u
c
u
c
u
=
<
+
m
[W
(m
;
m
1)
W
(m
;
m
)]
u
c
u
c
u
(B-P)
rW (mc ; mu ) = max
d >+ mc [W (mc 1; mu + 1) W (mc ; mu )] >
>
>
>
>
>
>
>
>
d
: +mc
;
[W
(m
;
m
+
1)
W
(m
;
m
)]
c
u
c
u
mc

Re‡ecting our consideration about exit, a …rm exits when V (0; 0) = 0 and a bankexits when W (0; 0) = 0. These two problems intend to model the choices of project
and loan portfolios, part of which require time to establish. The value of creating a new
project is determined by R, , and the rates of creation and destruction ( ; ; ). Banks
make pro…ts from establishing new relations under > 0.

3.3

Entry

Assume that there is an unbounded mass of potential entrant …rms. Each …rm must
invest at a rate of F f > 0 in return for a Poisson rate 1 of entering to this economy witha single (U-P). Also, we assume that there is an unbounded mass of potential entrant
8banks, which could also be interpreted as an alternative investment technology of existingbanks, and each entry bank must invest at the rate of F b > 0 in return for a Poisson rate
1 of entering to this economy with a single (U-L). As assumed above, each entry …rm
needs to displace an existing (U-P) in order to implement its own (U-P). This re‡ects
our assumption that the economy only has a limited number of clients for the products
made by the project.

3.4

Aggregate Flow

We are assuming that there is a unit mass of projects in total (i.e., the total numberof (C-P) and (U-P) is constant), and each project is taken care of by a single …rm and…nanced by a single bank.

Outside of the economy

λˆ

σU
δC
U

C

θU

λ ( E)C

Figure-1: In‡ow and Out‡ow
Let U and C denote the masses of (U-P) and (C-P). This assumption implies thatU + C = 1, which re‡ects our assumption about the limited number of clients/project
opportunities in this economy. Let b denote the ‡ow of new …rms. Figure-1 illustrates
the in‡ow to and out‡ow from the system. Note that the dashed arrow labeled (E)C
represents the in‡ow to U provided by the incumbent …rms. In the next section, wede…ne the equilibrium of this economy and characterize it.

4

Equilibrium Analysis

4.1

De…nition of Stationary Equilibrium

In this section, we establish the equilibrium of this economy and characterize it. First,
we de…ne the stationary equilibrium of this economy.

9

De…nition 1 A stationary equilibrium of this economy involves constant values for (1)the policy(E) ; b; (D) ; b , (2) the allocation (U; C), and (3) the price such that(i) potential entry …rms and banks break even in expectation, (ii) incumbent …rms andbanks maximize their values, and (iii) the loan market clears.
In order to characterize the equilibrium, we need to solve the individual optimization
problems, derive the break even conditions for entry banks and …rms, establish the steadystate condition of the economy and the market clearing condition for the loan market.

4.2

Equilibrium Conditions

4.2.1

Solutions for Individual Problems

The solutions for these two problems take convenient linear forms in terms of the statevariables. This makes it possible for us to analytically implement some comparative
statics. The following lemma summarizes the solutions for the individual problems foreach …rm and bank.
Lemma 2 (i) The solution for (F-P) is given by: (a) V = Anc + Bnu and (b) e = Enc
where (A; B; E) are determined by the following conditions81< B = 0 (E)(FOPT)(r + ) A
B(E) B f c (R + )g + E = 0:
A + (r + + ) B ( u R) = 0(ii) The solution for (B-P) is given by: (c) W = Gmc + Hmu and (d) d = Dmc where(G; H; D) are determined by the following conditions
81< H = 0 (D)(BOPT)(r + ) G
H
(D) H
+D =0
:
G + (r + + ) H = 0
Proof. See the appendix.

As important variables used for the later discussion, E nec and D mdc are de…ned.
Note that both of them do not depend on the state variables from the …rst conditions in(FOPT) and (BOPT).
4.2.2

Break-Even Condition for Potential Entrants

If there is positive entry into this economy in equilibrium, V (0; 1) = F f needs to hold so
as to make each entrant break-even in expectation, which implies B = F f . Also, if there
is positive entry into this economy in equilibrium, W (0; 1) = F b needs to hold, which
implies that H = F b .
10

4.2.3

Steady State and Market Clearing ConditionsFirst, the change in U can be expressed as follows.
U = min b + (E)C; b + (D)C + C

UUThe …rst term on the right-hand side is the aggregate new in‡ow to the pool of (U-P)(U-L) matches. Given the technological assumption that requires inputs from both the…rm and bank, it is similar to a Leontief production function. The second term accounts
for the in‡ow to the pool of uncerti…ed projects as a result of degraded certi…ed projects.
The third term corresponds to the out‡ow from the pool of uncerti…ed projects thatdrops out of the economy. The fourth term corresponds to the out‡ow from the pool of
uncerti…ed projects as a result of these projects being upgraded to certi…ed projects.
Also, the change in C can be summarized as follows. The …rst and second terms of
the right-hand side correspond to the in‡ow to C and out‡ow from C.
C= UC

We restrict our analysis to the steady state of this economy which is achieved whenU = 0 and C = 0. From these two steady state conditions along with the assumptionthat the total mass of projects is …xed, we can derive the steady state levels of the twomasses, Uss = = ( + ) and Css = = ( + ). Then, the following condition su¢ ces toclear the loan market.
b + (E)= b + (D) ++The condition guarantees that the size of the in‡ow to (U-P) in the …rm’s side andthe size of the in‡ow to (U-L) in the bank’s side are equal.13 This market clearing
condition can also be summarized by the following equation (MKT) which con…rms theequilibrium relationship between b and b. Since D and E are predetermined under the
break-even conditions for potential entrants, this market clearing condition works as one
equilibrium condition for our model in which ; b; b are endogenously determined.
b = b+ Z

where Z

+

f (D)

(E)g

(MKT)

Second, by using the steady state conditions and equation (MKT), we can obtain arelationship between the rate of destruction , which determines the total surplus made13

From the construction of the steady state, this also guarantees that the size of the out‡ow from
(U-P) and the size of the out‡ow from (U-L) are equal.

11

by each match, and b; b , where b denotes the ‡ow of new banks to U .
(= (1 + = ) b + (E) == (1 + = ) b + (D) =(STAT)

Then, the break-even condition for entry …rms with the conditions in (FOPT) and the…rst equation in (STAT) establishes a negative relationship between , which determines
the split of surplus from each project, and b.14 This is another equilibrium condition
for our model.
f(E)r = +(r+ + )r f
= (r+ )(1+ = )F b
F + ( u R) + ( cwhere E is determined by F f = 1= 0 (E)R)

E(FIRM)As the spread of (C-L) increases, each entry …rm decreases its Poisson rate b for (UP). This is because the relative pro…tability of searching becomes lower. The equation(FIRM) characterizes such a response to the change in loan spread .
Similarly, the break-even condition for entry banks with the conditions in (BOPT)
and the second equation in (STAT) establishes a positive relationship between and b
and this gives the last equilibrium condition. As the spread increases, each entry bankincreases its search intensity b for (U-L). The equation (BANK) characterizes such a
response of the bank side to the change in loan spread .
b= (r+ )(1+ = )F b + (D)r = +(r+ + )r F b + D
where D is determined by F b = 1= 0 (D)(BANK)

To summarize, the market clearing condition and the two break-even conditions forentry …rms and banks determine the stationary equilibrium of this economy which ischaracterized by ; b; b .

4.3

Existence and UniquenessWe now provide existence and uniqueness properties for this stationary equilibrium.
Proposition 3 If u + c > XF b + Y F f + (1 + ) R and Z 0 where Xr (D) +r (r + + ) and Yr (E)+Z + (r + Z) (r + + ), there exists a unique stationary
equilibrium with b > 0 and b > 0.
14

In order to derive this equation, we plug in the …rst equation in (STAT) into (FOPT) to eliminate, plug in F f = B into (FOPT) to eliminate B, and use the last two equations of (FOPT) to eliminate
A. The level of E is determined by the …rst equation of (FOPT) and F f = B.

12

Proof. See the appendix.
The equilibrium of this economy, summarized by ; b; b , can be determined as in
Figure-2. The proposition con…rms that if the pro…tability of the projects are highenough compared to the …xed entry costs and the risk-free interest factor, the economy
has positive in‡ows of …rms and banks as a unique equilibrium.15
γ(FIRM)(BANK)

γ(MKT) Z

λˆ φˆ

λˆ

φˆ

Figure-2: Equilibrium
We need to remark that the stationarity of the equilibrium hinges on the free entry(with cost) assumption. As with Klette and Kortum (2004), we cannot have a stationary
equilibrium if there is no entry to this economy. If the economy has no entry, thenumber of …rms and banks staying in the economy decreases while the number of projects
and loans held by …rms and banks increases on average. This is because a portion ofincumbent matches are broken with an exogenous Poisson rate and some …rms and banks
exit. In this paper, we limit our interest to the case where the stationary equilibrium isan equilibrium.

5Comparative Statics

In this section, we use the stationary equilibrium established in the previous section to
analyze the consequence of several shocks to this economy. In particular, how do di¤erent
degrees of interbank competition a¤ect the …rms’loan structure? Note that the …rm’s
15

In this paper, we assume Z > 0 and this does not a¤ect the qualitative implication of this model(unless noted otherwise). Intuitively, this implies that the incumbent banks’ Poisson rate for …nding
a new (U-L) is larger than the incumbent …rm’s Poisson rate for …nding a new (U-P). Consideringthat existing banks are likely to have a larger number of loan relations than …rms, this assumption isplausible.

13

loan structure can also be interpreted as …rm size since the loan structure represents the…rm’s project portfolio. This is a direct consequence of our assumption that each projectis …nanced by di¤erent banks. Thus, the primal interest is analyzing the e¤ect of higher
inter-bank competition on the loan structure or …rm size distribution. This distribution
can be summarized by (E); b .

To illustrate how these two parameters summarize the loan structure or …rm size
distribution, suppose (E) stays constant and b increases as a response to some speci…cshock. From the …rst equation in (STAT), this implies that increases. As a result,the measure of incumbent …rms with a large number of loan relations decreases since theincumbent …rms now face a larger Poisson rate of losing one of the (U-P)’s. This also
leads to a smaller number of (C-P)’s for the incumbent …rms. Hence, the …rms’ loan
structure, on average, now consists of a smaller number of loan relationships and is, thus,
more concentrated. This is a simple illustration of our comparative statics. In the next
subsections, we will change several parameters in the model and see what happens to(E); b and (D); b .

5.1

Interbank Competition

First, we study the impact of di¤erent degrees of interbank competition. Recall that theparameter F b denotes the …xed entry cost of each potential entry bank to the banking
industry. As this entry cost becomes lower, there will be more entry of banks, and thus,
higher competition. The next proposition summarizes the key result of this paper.
Proposition 4 As the cost of entry to the banking industry F b decreases (increases),(i) the spread decreases (increases), (ii) the incumbent …rms’search intensity (E) isconstant, (iii) the incumbent banks employ lower (higher) search intensities (D), (iv)the ‡ows of entering …rms and banks b; b increase (decrease), and (v) the overall e¤ect
on is positive (negative).
Proof. See the appendix.
We have already established that (E) is determined independently of F b . The
proposition con…rms that the equilibrium level of b increases as F b decreases. Thisimplies that the loan structure of each incumbent …rms become more concentrated andis associated with a smaller number of loan relations on average. To see this point moreclearly, suppose F b decreases. Each incumbent …rm keeps its search intensity constant
while the entry ‡ow increases. This decisively leads to higher .16 As a result of thishigher , each incumbent …rm’s (U-P) and (C-P) decrease.
From the bank’s side, as the banking industry becomes more competitive, the incumbent banks choose lower search intensity for additional (U-L) while the ‡ow of entry
16

This result also implies that the impact of a higher b dominates the impact of a lower (D).

14

increases. The large entry ‡ow decreases the value associated with each (U-P), which
induces the incumbent banks to reduce the search intensity. The loan spread becomes
lower (e.g., Japanese bank loan market in 1980s) as demonstrated in Figure-3.
To summarize, higher competition in the banking industry induces entry not only
into the banking industry but also into the …rm industry. The latter makes the …rms’
portfolio/borrowing structure more concentrated and reduces the number of loan relations
on average. One empirical implication of this model is that higher competition in thebanking market (via lower entry barriers) is accompanied by an increase in the numberof …rms and a reduction in the number of loan relations per …rm. This can partiallyexplain a recent phenomenon in the Japanese bank loan market. Deregulation over the
past two decades in the bank loan market leads to higher competition between banks,
which induces young and small …rms to enter the economy as predicted in our model.
This e¤ect would not arise without considering endogenous entry of …rms.
γ(FIRM)(BANK)(BANK)'

γ(MKT) Zγ'

(MKT) Z'λˆ φˆφˆφˆ' λˆ

λˆ'

Figure-3: Interbank Competition

5.2

Firm’s Pro…tability

In this section, we see how changes in …rm pro…tability a¤ect the loan structure. In
the model, ( c ; u ) and the search cost are the parameters of …rm pro…tability. For thesake of simplicity, we …x the costs and see how the equilibrium loan structure responds
to di¤erent levels of c and u . The following proposition summarizes the result of thiscomparative static.
Proposition 5 As c and/or u become larger (smaller), (i) the spreadincreases(decreases), (ii) the incumbent …rms and banks keep search intensities constant, (iii) the‡ows of entering …rms b and banks b increase (decrease), and (iv) the overall e¤ect on
15

is positive (negative).
Proof. (i), (ii), and (iii): A larger (smaller) c and/or u shift the equation (FIRM) to
the right (left). Considering that E and D are independent of ( c ; u ) and the equation(BANK) does not shift even if c and/or u change, such a shift in the equation (FIRM)
only generates a change in and b; b , as illustrated in Figure-4. (iv): Recall theequation (STAT). The results follow immediately.
γ(FIRM)'(FIRM)(BANK)

Z

γ
Z

λˆ φˆ
φˆ

φˆ'

λˆ

λˆ'

Figure-4: Firm’s Pro…tability
The proposition con…rms that the equilibrium levels of b and b are increasing in …rms
pro…tability. This implies that the loan structure of each incumbent …rm become moreconcentrated under a higher c and/or u through the same mechanism demonstrated in
the previous proposition.
From the …rm’s side, higher pro…tability induces a larger ‡ow of entrants and theseentering …rms take part of (U-P)’s originally held by incumbent …rms. As a consequence,
the loan structure of incumbent …rms becomes more concentrated and leads to a smaller
number of loan relations on average.
From the bank’s side, higher …rm pro…tability does not directly improve payo¤s sincebanks can only use straight debt. Nonetheless, each bank’s value improves due to ahigher equilibrium level of . This induces an increase in the ‡ow of new entry banks,
thus accommodating the increased demand for (U-L)s from …rms.
To summarize, an increase in …rms’pro…tability induces entry in both the …rm andbank industries. The additional entry of …rms makes the …rms’ portfolio/borrowing
structure more concentrated and leads to a smaller number of loan relations. A positive
correlation between …rms’ pro…tability and the concentration of loan structure are es16

tablished in several empirical studies (Degryse and Ongena (2001); Harho¤ and Korting(1998); Foglia et al. (1998); Gordon and Schmid (2000); Machauer and Weber (1999)).17
Our model successfully replicates these empirical …ndings.

5.3

Search Technology for Incumbent Banks

Consider a technological improvement that makes it easier for incumbent banks to …nd
a new client. In the setup of the base-line model, this technological progress can besummarized by a change in the shape of the function ( ).
Suppose (D) shifts up while 0 (D) is held constant. Note that so long as 0 (D)for each level of D is unchanged, the equilibrium level of D does not change. The higher(D) with a constant D results in the shift of the equation (BANK) to the left and Z
increases.18 We can con…rm that improvements in bank’s search technology leads to asmaller b. As a result, the incumbent banks hold a larger number of loans and this leads
to a larger number of loan relations on average.
This feature can be interpreted as an illustration of a recent trend in which …nancial
institutions have become larger as a result of progress in information technology. As theincumbent banks obtain more e¢ cient search technologies relative to potential entrants,
the size of incumbent banks becomes larger.
γ(BANK)'(BANK)(FIRM)

γ'
Z'

γ

Z

λˆ φˆ
φˆ'

φˆ

λˆ'

λˆ

Figure-5: Technological Progress in Bank’s Searching Technology
17

Weinstein and Yafeh (1998) claims the opposite result.
This can be veri…ed by (BANK). On the right-hand side of (BANK), the terms associated with(D) can be summarized as (D) r= , which implies (BANK) shifts to the left as (D) shifts up.
18

17

5.4

Empirical Regularities

Is the model consistent with empirical regularities? First, our model is designed to beconsistent with the empirical fact that large …rm size (asset and/or debt) leads to a larger
number of loan relations (Ogawa et al. (2007), Miyakawa (2008b)). Second, we havecon…rmed that, in equilibrium, higher …rm pro…tability leads to more concentrated loan
relations (Degryse and Ongena (2001)). Third, loan relations are modeled to exhibit
persistence (Miyakawa (2008b), Tachibanaki and Taki (1991)) in this paper.

6

Quantitative Discussion

6.1

Simulated Loan Structure/Firm Size Distribution

In the previous section, we have established the existence of a stationary equilibrium.
Unfortunately, it is di¢ cult to analytically characterize the loan structure or …rm size
distribution over the two state variables under the current setup. In this section, we
simulate this …rm size distribution and consider quantitative comparative statics.1920
F b = 0:6
F b = 0:3
(C-P)
(U-P)
(C-P)
(U-P)
(i)
Number of Firms
12998
15374
(ii)
Mean
1:0488
1:4807
1:0268
1:2662(iii)
Standard Deviation
0:2237
0:63950:1692
0:5056(iii)’ (Coe¢ cient of Variation) (0:2133) (0:4319) (0:1648) (0:3993)(iv)
Skewness
4:6626
1:2295
6:63401:8037(v)
Kurtosis
22:2274
1:6174 47:1659
2:8216
Table-1: Quantitative Comparative Statics
Table-1 summarizes the summary statistics of the simulated distribution under twolevels of entry cost to the banking industry (F b = 0:6 and 0:3).21 As our analyticalresult suggests, when the entry cost to the banking industry decreases from 0:6 to 0:3(i.e., higher inter-bank competition), (i) there are more …rms in this economy, (ii) the
average number of (C-P)’s and (U-P)’s held by each …rm decrease. Moreover, we can seethat (iii) the dispersion of the distribution becomes smaller, (iv) the distribution becomes
more skewed, and (v) the distribution exhibits higher kurtosis.
19

The simulation algorithm is summarized in the appendix.
Bank distribution over the two state variables can be obtained through a similar simulation algorithm. We omit this object and focus on the …rm distribution in this paper.
21
The summary statistics are conditional on positive project numbers. For example, the mean of(C-P) is the average of the (C-P)s number among the …rms with at least one (C-P).
20

18

6.2

How to Bring the Model to Data?

As detailed in the previous section, our model describes the initiation of loan relations
and the evolution of each match in a stationary equilibrium. The model further implies
a distribution of the duration of loan relations between …rms and banks. For example, if
the bene…ts of relationship-lending considered in our model are signi…cant, matches withlonger relations are expected to break up less often. Thus, the parameters governing theevolution of each match lead to a speci…c distribution for the duration of loan relations.
One possibility towards sophisticated quantitative analyses is to match the distribution of the duration of loan relations generated by our model with the data. For example,
Miyakawa (2008b) constructs a unique data set that contains the duration of long-termloan (i.e., original maturities are greater than 1 year) relations between all Japanese listed…rms and all Japanese banks. By using such data sets, we can structurally estimate themodel’s parameters, for example through the simulated method of moments, and use theestimated model to quantitatively evaluate various shocks to the economy. We leave thisto our future research.

7

Concluding Remarks

In this paper, we develop a dynamic equilibrium model for relationship-lending thatis consistent with empirical regularities established in the literature. In our model,
the sustained loan relation makes it possible to implement relatively pro…table projects
through the provision of relationship-lending. The loan spread associated with such
relationship-lending, which determines how to allocate the surplus between …rms andbanks, is endogenously determined so as to clear the bank loan market. The novel featureof our paper is that it employs a dynamic equilibrium model to study the implications ofdi¤erent degrees of competition for bank loan structures or …rm size distribution. This
is in contrast to the existing literature which uses multi-period game theoretic models toanalyze this issue.
The model predicts that higher interbank competition leads to a more concentrated
loan structure and smaller number of loan relations for each …rm. It also implies that(i) an increase in …rm pro…ts leads to a more concentrated loan structure with a smaller
number of loan relations for each …rm, and (ii) improvements in the bank’s client searchtechnology (e.g. progress in informational technology) leads to a larger number of loan
relations for each incumbent bank.
This paper motivates several questions for future research. First, as mentioned in thelast section, we plan to structurally estimate the model. This allows us to examine theimpact of …nancial shocks (e.g., sudden break-down of incumbent banks) to the economy
quantitatively. When a bank becomes insolvent or bankrupt, governments often takegreat interest in ensuring the survival of …rms that previously had loan relations withthese banks. This policy sometimes takes the form of providing liquidity to these …rms
19

and/or …nancial markets. However, aside from the usual systemic risk considerations forthe economy, it is not clear how we can justify such a policy intervention. Our model
can potentially provide one plausible framework to answer this question. We conjecturethat the termination of an incumbent relation leads to some loss of economic value if
there exists a relationship-capital, as modeled in this paper. The quanti…cation of sucha loss could be informative for policy discussions. Second, a potential extension of ourmodel is to incorporate some matching frictions. Our current model essentially assumes
that …rms and banks can match in the market instantaneously. Considering that thebank loan market has been considered not to necessarily …t the Walrasian market, it
is fruitful to consider this friction and study its implications. Third, the assumptionthat each project for a …rm needs to be …nanced by di¤erent banks should be relaxed.
The empirical literature shows that the actual loan relations are highly asymmetric.
This asymmetry has not been taken into account in our current model. By relaxing
these assumptions, we can analyze the implications of various economic shocks (e.g.,competitiveness in bank loan market) more precisely. Fourth, one drawback of ourmodel is that for given parameters ( ; ), we have no variation of the aggregate share of(C-P) and (U-P) in the stationary equilibrium. Endogenizing such share can help us
to understand the impact of several economic shocks on the share of (C-P) and (U-P).
One possibility to accomplish this extension is to introduce an e¤ort choice for upgrading(U-P)’s to (C-P)’s.

20

8

Appendix:

8.1

A. Proof

Proof. (Lemma-2): (i) Guess the functional form for the value function and policy
function for the …rm’s problem as (a) V = Anc + Bnu and (b) e = Enc . Then, theBellman equation (F-P) takes the following form:89[ c (R + )] nc + [ u R] nu Enc
>
>
>
>
>
>
>
>
+
n
[A
(n
+
1)
+
B
(n
1)
An
Bn
]
>
>
u
c
u
c
u
<
=
+ nu [Anc + B (nu 1) Anc Bnu ]rAnc + rBnu =>+ nc [A (nc 1) + B (nu + 1) Anc Bnu ] >
>
>
>
>
>
>
>
>
e
: +nc
;
[An
+
B
(n
+
1)
An
Bn
]
c
u
c
u
nc

First, this Bellman equation can be rewritten as[(r + ) A
B
(E) B f c
+ [ A + (r + + ) B(R + )g + E] nc( c R)] nu = 0

In order to hold this equation for all the possible cases of (nc ; nu ), we need(r + ) A
B(E) B f c (R + )g + E = 0
A + (r + + ) B ( c R) = 0

(1
(1

1)
2)

Furthermore, from the …rst-order-condition with respect to e, we have0=(11) to (1

1+

0

(E) B

(1

3)

3) are the three conditions characterizing (A; B; E).

(ii) Guess the functional form for the value function and policy function for the bank’s
problem as (a) W = Gmc + Hmu and (b)d = Dmc . Then, the Bellman equation (B-P)
takes the following form:
8
9
m
Dm
>
>
c
c
>
>
>
>
>
>
+
m
[G
(m
+
1)
+
H
(m
1)
Gm
Hm
]
>
>
u
c
u
c
u
<
=
+
m
[Gm
+
H
(m
1)
Gm
Hm
]
u
c
u
c
u
rGmc + rHmu =>+ mc [G (mc 1) + H (mu + 1) Gmc Hmu ] >
>
>
>
>
>
>
>
>
d
: +mc
;
[Gm
+
H
(m
+
1)
Gm
Hm
]
c
u
c
u
mc

First, this Bellman equation can be rewritten as[(r + ) G
H(D) H
+ D] mc+ [ G + (r + + ) H] mu = 0

21In order to hold this equation for all the possible cases of (nc ; nu ), we need(r + ) GH(D) H
G + (r + + ) H = 0

+D =0

(1
(1

1)0
2)0

Furthermore, from the …rst-order-condition with respect to d, we have0=(11)0 to (1

0

1+

(D) H

(1

3)0

3)0 are the three conditions characterizing (G; H; D).

Proof. (Proposition-3): The equations (FIRM), (BANK), and (MKT) imply themonotonic relations between (i) , b, (ii) , b, and (iii) b , b, which guarantee
the uniqueness of the equilibrium. Suppose the equations (FIRM) and (BANK) are
expressed as = M1 b + M2 and = N1 b + N2 . Recall that the equilibrium b can
be determined from the three equations (FIRM), (BANK), and (MKT) b = b + Z asfollows.
b = M 2 M 1 Z N2
M1 +N1Sine the denominator is always positive, M2 + N1 Z N2 0, which can be summarized
as the condition in the proposition, is su¢ cient to obtain b 0. The equation (BANK)
implies that0 as far as b 0. Moreover, if Z is positive, b 0 as far as b 0.

Proof. (Proposition-4): (i) and (iv) First, the equation (BANK) shifts to right andbecomes ‡atter as F b decreases. This can be veri…ed from the positive correlation
between D and F b and the equation (BANK).
b= (r+ )(1+ = )F b +(D)r = +(r+ + )r

Fb + D

Second, Zf (D)(E)g becomes smaller as F b decreases. Then, we can verify+
that the size of horizontal shift of the equation (BANK) is always greater than the sizeof decline in Z. From the equation (BANK), we can obtain the following relationshipbetween b and F b for a …xed .
n bo00 (D)
r 0 (D) 00 (D)(D)r = +(r+ + )r
db =
+
+
dF b
(BHOR)(r+ )(1+ = )Fb(r+ )(1+ = )F b
(r+ )(1+ = )F b

00
We used the result that dD =(D)dF b , which is derived from the …rst condition in(BOPT) and the break-even condition for the potential entry banks. We can also obtainthe following relationship between Z and F b from the de…nition of Z.

dZ =

+

0

(D) 00 (D)dF b(ZDEC)

Note that the second and the fourth terms on the right-hand side of the equation (BHOR)
can be summarized as follows.
rF b 0 (D)+(r+ )F b 0 (D) +0

22(D) 00 (D)dF b

Here, F b 0 (D) = 1, which follows from the …rst condition in (BOPT) and the break-evencondition for the potential entry banks, gives the following result.
rF b 0 (D)+=1(r+ )F b 0 (D)This implies that the second and the fourth terms in the right-hand side of the equation(BHOR) has the same magnitude as the right-hand side of the equation (ZDEC), which
con…rms that the rightward horizontal shift of the equation (BANK) corresponding to
the decline in F b is always greater than the decline in Z corresponding to the samedecline in F b in terms of the absolute values (signs are di¤erent). Further considering
the equation (FIRM) does not shift even if F b changes, the new equilibrium level of b
increase as demonstrated in Figure-3. This is accompanied with a lower level of new. (ii) and (iii) We have already established that D decreases (increases) but E stays
constant when F b decreases (increases). (v) Recall the equation (STAT). The result
immediately follows.

8.2

B. Simulation Algorithm

We use the following algorithm to obtain the stationary …rm distribution over its state
space.
Step-1: Set R, c , u , , , F f , F b , (e; nc ), and (d; mc ).2223 Step-2: Solve forthe optimal level of E and D. Step-3: Compute (E) and (D). Step-4: Computethe equilibrium levels of b, b, and from the equations (FIRM), (BANK), and (MKT).
Compute from the equation (STAT). Step-5: Set I …rms indexed by i = 1;
;I
j 4
24
with one (C-P) and one (U-P) for each.
Step-6: Draw zi j=1 for each …rm independently from a unit uniform distribution. Step-7: Assign the individual dynamics by thefollowing rule:(1) Consider the segments in Figure-6 and follow the rule described below.
θnu

σnu

δnc

nc λ (E )

(a)

(b)

(c)

(d)

(e)
1

0

Figure-6: Segmentation for Individual Dynamics
For this exercise, we use (e; nc ) = Jea n1c a and (d; mf ) = Kdb m1f b where (a; b) 2 (0; 1) (0; 1)
and J; K > 0.
23
As a baseline case, we use R = 1 + r = 1:04, c = 1:5, u = 1:1, = 0:05, = 0:45, F f = 0:3,b
F = 0:3 or 0:6, a = 0:6, b = 0:5, J = 1, and K = 1.
24
Let (nc (i); nu (i)) denote the state variables for …rm-i. We use I = 10000 in this exercise.
2223

(2) If zi1 falls (a), revise the state to (nc (i) + 1; nu (i) 1). (3) If zi1 falls (b), (i) re2
vise
the
h
n state to (nc (i); nou (i) 1), and (ii) add one entry …rm with one (U-P) if zi 20; b= b + (E) C + C
or (ii)’ choose one incumbent …rm to replace (U-P) basedon fnc (i)g and zi3 otherwise. (4) If zi1 falls (c), revise the state to (nc (i) 1; nu (i) + 1).
(5) If zi1 falls (d), (i) revise the state to (nc (i); nu (i) + 1), and (ii) choose one incumbent…rm losing (U-P) based on fnu (i)g and zi4 . (6) If zi1 falls (e), stay at the same state(nc (i); nu (i)).

Step-8: Drop …rm-i if nc (i) = 0 and nu (i) = 0. Step-9: Repeat Step-7 and -8 until
the distribution converges. In order to con…rm the convergence, check if the di¤erence
between the revised and pre-revised means, variances, and skewness of the (C-P)’s and(U-P)’s distributions becomes smaller than threshold values.

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