Bank Loan Search

Murray Z. Frank

ABSTRACT

This paper provides a search model of bank loans in which it is costly for firms and

banks to find each other. The negotiated loan interest rate must compensate both

parties for their search costs. In the model, if the original interest rate is low enough,

then an increase in the risk-free rate leads to an increase in the loan rate. But if the

original interest rate is high enough, then an increase in the risk-free rate leads to a

drop in the loan rate due to the impact on the present value of future profits. The

model provides a rationale for the sluggish adjustment of the aggregate volume of

bank loans. The main empirical puzzle relative to the model is why loan defaults

increase after the end of a recession, even as demand for new loans is increasing.

JEL classification: G21

Keywords: bank loan, search theory, financing constraint.

Helpful comments by Raj Singh and Pedram Nezafat are appreciate. I am responsible for any errors.

c©2011 Murray Z. Frank. All rights reserved. Pipper Jaffray Professor of Finance, Carlson School of

Management, University of Minnesota, Minneapolis, MN, USA 55455.

“Big banks in recent months eased standards on small-business lending for the first

time since late 2006, a Federal Reserve survey found, but customers of all sizes showed

little appetite for loans with the economy slowing. ... Indeed, bankers insist that they are

booking all the good loans they can find.” Wall Street Journal, August 17, 2010.

I. Introduction

When a bank extends a loan to a borrower, they sign a contract. A rich literature has

studied many aspects of bank lending in an effort to understand how the contractual

terms are determined. Theories have been developed based on risk-sharing, costly state

verification, screening, monitoring, managerial moral hazard, and credit rationing. Ob-

served contracts are often much simpler than predicted by such theories, which helped

stimulate interest in incomplete contracting models. There is also significant work (eg.

Bolton and Scharfstein (1990)) on the firm’s dynamic incentive to repay a bank loan that

has already been extended. The surveys by Gorton and Winton (2003) and Freixas and

Rochet (2008) provide valuable overviews of the extensive literature.

In this paper, the focus is on the impact of the logically prior problem: the borrower

and the lender must first find each other. Because the bank loan market is decentralized,

finding a new trading partner is neither free nor immediate. These up front costs must

be reflected in the market equilibrium loan terms. But this creates complexity because

it implies quasi-rents. Accordingly each loan that is actually observed must have been

expected to provide at least enough positive profits to offset the upfront search costs.

Exactly how the quasi-rents are expected to be split will in turn affect the still earlier

decisions of banks and firms to search.

The main contribution of this paper is to examine the effect of search on the bank loan

market equilibrium. Most previous studies implicitly assume that the bank loan market

is centralized and so there is no need to search. An important exception is Bizer and

DeMarzo (1992). They study an externality that arises when there is moral hazard and

borrowers borrow from more than one bank. Fresh lending reduces the chances that a

1

previous loan will be repaid. They focus on the impact of such further borrowing. Duffie

and Manso (2007) and Duffie et al. (2009) analyze the effect of dynamic information

revelation with search – ‘percolation’. They find that this can also create an interesting

externality. In contrast the current paper assumes away such dynamic externality effects,

instead the analysis highlights the role of loan search costs on loan terms. In particular, the

current paper characterizes the steady state equilibrium interest rate, and the aggregate

quantity of loans.

A basic bank loan search/negotiate model is presented. This model is in the spirit of

Duffie et al. (2005) and Duffie et al. (2007) which in turn draws on the economics search

literature as in Diamond (1982), Mortensen and Pissarides (1999) and Rogerson et al.

(2005). In the model banks and firms engage in costly search for each other. Once a

partner has been found, loan terms must be negotiated. The lending relationship will

endure unless the borrower is hit by a very bad shock. A very bad shock causes the firm

to default on the bank loan and enter bankruptcy.

In a steady state equilibrium the flow of new matches must balance the flow of matches

that are ending. This implies that the firm failure rate is a crucial determinant of the

rate at which banks enter and the rate at which inactive firms start looking for a bank.

The loan terms are negotiated in a forward-looking manner. But the volume of loans

can only adjust to a demand shock more slowly because it may take time to find a

good match. Default shocks have an immediate impact on the volume of loans.1 The

equilibrium interest rate on loans reflect several forces. As usual loan default risk and the

bank’s opportunity cost of funds are important. Beyond these familiar effects, the loan

rate also serves to split the present value of the expected surplus created by the loan.

This third effect means that loan interest rates can differ from traditional models.

In the model the impact of an increase in the risk-free rate depends on the level of

current interest rates. If the original risk free rate is low enough then an increase, results

in an increase in the bank loan rate. But if the risk free rate is high enough, then an

1In reality default on a bank loan is not always so abrupt. Missing a payment can lead to renegotiationsof the terms of the loan. Empirically non-performing loans, loan loss provisions, and charge-offs are highlycorrelated in the aggregate quarterly data.

2

increase reduces the bank loan rate. This may help explain why empirically the spread

on a bank loan is not a simple monotonic function of the prime lending rate.

Suppose that more businesses start looking for a loan. The bank’s bargaining position

improves because it is easier for a bank to find an alternative borrower. Both in the model

and empirically, the negotiated loan spreads are higher when demand is higher. However,

looking does not mean finding immediately. Finding can take time. Thus there is no tight

connection between the demand for loans at a moment in time, and the volume of loans

at that same moment. Instead there is a lag.

When the loan failure rate is higher the interest rate on loans increase to reflect the

extra risk of loss. The bank faces an extra expected cost of looking for new borrowers.

Empirically when charge-offs are high, banks tend to reduce their lending standards but

charge a higher spread as befits the riskier environment.

As observed by Santos and Winton (2008) bank lending standards do change signifi-

cantly over the business cycle. The standards tend to peak during recessions and fall back

during recoveries. The lending standards can be viewed as a proxy for the bank’s bar-

gaining power.2 A bank with a great deal of bargaining power can charge a high spread.

Consistent with this perspective, there is a strong positive correlation between the loan

spreads and lending standards.

Much of the prior literature focuses on cross sectional differences across banks. The

current paper focuses on aggregate fluctuations in loan volume and loan terms across time.

In the model the quantity of bank loans is adjusted both by extending new credit as well

as by canceling existing credit. This distinctions is, of course well known. Dell’Ariccia and

Garibaldi (2005) observed that there is a great deal of reshuffling of credit between banks.

As a result there is more volatility at the individual bank level and more persistence in the

volume of bank loans at the aggregate level. Craig and Haubrich (2006) make interesting

comparisons between the bank loan market and the job market. Loans exhibit much less

seasonality than do jobs.

2An alternative interpretation is provided by Dell’Ariccia and Marquez (2006). They suppose thatbanks have private information about the creditworthiness of the borrowers. The current paper assumessymmetric information about creditworthiness.

3

Variation in bank lending over the business cycle has been studied by Santos and

Winton (2008) and Santos and Winton (2010). In contrast to the current paper’s focus

on aggregate fluctuations, they are more interested in cross sectional differences. In Santos

and Winton (2008) the focus is on the rates charged to bank dependent borrowers relative

to less dependent borrowers over the business cycle. Firms with access to public bond

markets are less susceptible to exploitation by their bankers when the credit markets

are soft. In Santos and Winton (2010) the focus is on the capital position of the banks

extending the loans.

II. The Model

In the model there are banks and firms. The firms have real investment opportunities,

but no money to make use of them. The banks have money but no direct access to real

investment opportunities.3 If the bank does not find a firm, then the money can be left

on deposit with the Central Bank earning a risk-free rate of return, ρ. The number of

banks is determined by free-entry, while the number of firms is fixed.

Time is continuous and denoted by t. The time horizon is infinite. The order of events

is that searching for a business partner must come before the negotiations. So first the

bank looks for a borrower, and at the same time the borrowers look for banks. When a

borrower is matched to a bank, they bargain over the terms of the loan. If they agree,

then a loan is extended and the money is put into production. Production produces

revenue continuously from which the agreed interest is paid to the bank continuously.

This continues until something bad happens to the firm.

When something bad happens to the firm it is bankrupt and loan is defaulted. When

a bankruptcy takes place, the bank replenishes capital from the owner. If it is profitable

to do so, the bank looks for a new borrower.

Two bankruptcy codes are considered. If bankruptcy is ‘terminal’, a new firm joins

the pool of unmatched firms when an existing firm is bankrupt. If bankruptcy is ‘fresh

3For a study of the impact of bank capital heterogeneity see Santos and Winton (2010).

4

start’ then the bankrupt firm joins the pool of unmatched firms. The main results are

the same under either bankruptcy code.

A. Matching Flows

The requirements for a steady state flow equilibrium are essentially the same as in any

other search/matching model. There is a matching function that takes the current flows

of firms and banks and creates matches. Since matches are mutually beneficial, matches

result in deals. So the basic structure can be described before spelling out the details of

the firm and the bank’s Bellman equations.

Suppose that over a short period of time δt there are fδt firms that are looking for

a loan, and bδt banks looking for borrowers. Since the transition rate for an unmatched

firm is a, the total flow of firms getting a loan is afδt.

The matching function between firms and banks is µ = µ(f, b). Here µ is the number of

matches. Accordingly µ = af . This function is assumed to be continuous, differentiable,

with positive first and negative second derivatives, and have constant returns to scale.

It is conventional to define the ‘tightness’ of the bank loan market as θ = bf. When

θ gets large (a ‘loose credit market’) there are many banks and few firms looking. Firms

easily find banks, but banks have trouble finding firms. When θ gets small (a ‘tight

credit market’) there are few banks and many firms looking. Banks easily find firms, but

firms have trouble finding banks. Because banks have free entry, in an equilibrium θ is a

constant.

Let m(θ) denote the rate at which a bank that is looking for a borrower finds one.

The total loan flow is bm(θ). A loan requires both a lender and a borrower, so

af = bm(θ). (1)

As a result m = µ/b. The rate at which a bank finds a firm is,

m = m(θ) = µ(f

b, 1) = µ(θ−1, 1). (2)

5

The rate at which a firm finds a bank is

a = µ(1,b

f) = θm(θ). (3)

Assume that µ takes the Cobb-Douglas form, µ(bt, ft) = m0b1−αfα, where, m0 > 0,

and 0 < α < 1. Since m = µ/b,

m(θ) = m0θ−α. (4)

At times it is more convenient to think about the mass of firms without a loan (f),

while at other times it is more convenient to think about the mass of firms that have a

loan (l). Due to loan size being fixed, this is also the total mass of bank loans. Because

the number of firms is fixed, it can be normalized to 1, and then write 1 = f + l.

The mass of loans evolves according to

dltdt

= θm(θ)(1− l)− sl. (5)

The first term on the right hand side is the increase in loans that is due to previously

unmatched firms finding a bank. The second term is the loss of loans due to bankruptcies

at existing firms that had a loan.

Much of the analysis focuses on the steady state. In a steady state dltdt

= 0. Thus the

steady state volume of loans must be

l =θm(θ)

s+ θm(θ)=

m0θ1−α

s+m0θ1−α. (6)

Condition 6 is the first key condition that is required for the analysis of a steady state

equilibrium.

6

In the short run there may be many reasons not to be in the steady state. In that

case 5 needs to be solved using a given initial condition, denoted by l(0) = l0. Using the

initial condition, the solution of 5 is

l(t) =θm(θ)

s+ θm(θ)+ (l0 −

θm(θ)

s+ θm(θ))e−(s+θm(θ))t. (7)

Notice that if the initial condition is at the steady state, then the system stays there. The

rate at which departures from the steady state vanish depends in a simple manner on the

matching rate and the shock rate.

This subsection provides the steady state mass of loans. It also shows that if for some

reason the mass of loans is not at the steady state value, then it gradually returns to

the steady state. This depends on the fact that θ is kept constant by bank free entry.

In a more general setting, in which θ itself fluctuated, richer loan dynamics might be

observed.4

B. Firm Problem

A firm either has a lender or else it does not. If the firm does not have a lender, it can

look for one, or it can decide not to bother looking. Looking is costly. Getting a suitable

loan is beneficial since it permits production. Let Vfu be the expected present value of a

firm that is unmatched to a lender. Let Vfm,t be the expected present value of a firm that

is matched to a lender as of t.

Consider a short time interval δt. Over the short time interval δt the firm incurs a

search cost (‘hunt’) of −hδt and gets a loan at rate aδt. If the firm does not search then

4A potentially interesting extra feature to consider is the fact that the loan money may not be usedfully at the moment that the agreement is reached. Thus many firms have unused loan commitments atbanks. Using a loan commitment is crucially different from using a new loans. A loan commitment canbe drawn down immediately. A new loan which requires finding a lender which takes time. Interestinganalysis of lines of credit is provided by Sufi (2009) and by Acharya et al. (2010). Incorporating linesof credit into a loan search model is potentially quite interesting, but goes well beyond the scope of thecurrent paper. Figure 1 shows that loans continue to fall even after the end of a recession. To what extentis this do to finally recognizing poor performance? TO what extent is this due to ‘excessive’ drawingdown of lines of credit during the recession?

7

a = 0. If the firm does search then a > 0. Because search is costly −h < 0. If a bank

is found during δt then the firm can negotiate a loan and get Vfm,t+δt. Or, the firm can

keep looking and get Vfu,t+δt. If no bank is found then the firm gets Vfu,t+δt.

First consider an unmatched firm. The Bellman equation is,

Vfu,t =−hδt

1 + ρδt+ aδt

max{Vfm,t+δt, Vfu,t+δt}1 + ρδt

+ (1− aδt)Vfu,t+δt1 + ρδt

(8)

This can be rewritten as

ρVfu,t = −h+ a(max{Vfm,t+δt, Vfu,t+δt} − Vfu,t+δt) +Vfu,t+δt − Vfu,t

δt. (9)

Take the limit as δt→ 0, and drop the time subscripts,

ρVfu = −h+ a(max{Vfm, Vfu} − Vfu) +dVfudt

. (10)

If the firm is in a steady state,dVfudt

= 0.

There will be surplus to be split between a firm and a bank that are matched. The

negotiation over loan terms are solved using generalized Nash bargaining. This means

that the surplus will be captured. Accordingly Vfm > Vfu, and so

ρVfu = −h+ a(Vfm − Vfu). (11)

Next consider a matched firm. A matched firm borrows M from the bank which is

immediately put into production, and agrees to pay ongoing interest rate r. As long as

nothing bad happens to the firm, over a short time period δt the production generates a

revenue flow of AMδt , and the firm uses the revenue to honor its loan agreement with

the bank. With probability s (‘shock’) something very bad happens, and so the firm is

bankrupt. Since bankruptcy is forever the firm gets zero from then on.

The derivation of the Bellman equation for a matched firm follows the same steps as

for an unmatched firm. Thus,

8

ρVfm = (1− s)M(A− 1− r). (12)

At times it is useful to rewrite the flow conditions in present value form as,

Vfu =−h+m0θ

1−αVfmρ+m0θ1−α

(13)

Vfm =(1− s)M(A− 1− r)

ρ. (14)

C. Bank Problem

A bank is in one of three conditions: inactive, unmatched and looking for a borrower,

matched with a borrower. The present value of being an unmatched bank is denoted Vbu.

The present value of being a bank that is matched with a borrower is Vbm. Each bank

has M dollars and will make either no loans or one loan to a firm. Any profits or losses

are absorbed by the owner on an ongoing basis to maintain M . This gets rid of the need

to keep track of the bank’s capital position.5

Following the same procedure as for the firm, the bank’s payoffs can be written in flow

form. For an unmatched bank that is looking for a borrower, the Bellman equation is,

ρVbu = −k +Mρ+m(θ)(Vbm − Vbu). (15a)

For a matched bank, the Bellman equation is,

ρVbm = −M + (1− s)M(1 + r) + s(Vbu − Vbm). (16)

The left hand side give the flow of gains to the bank according to the state. An

unmatched bank with no borrower (state Vbu) is spending money to search, and leaving

money on deposit earning the risk-free rate. With probability m this bank will transition

from the current state to state Vbm which gives a gain of Vbm − Vbu.5Since everyone is risk-neutral, keeping track of the gains or losses would not affect the inferences. All

banks make the same decisions. It would simply add extra notation.

9

An inactive bank invests M at the risk-free rate. Thus the inactive bank has a present

value of M .

A bank with a borrower (state Vbm) has the money loaned out to a firm. As long as

nothing bad happens to the firm, the bank receives a flow of interest payments. However,

if something bad happens to the firm (probability s), then both the principal and the

interest are lost. The bank transitions back to state Vbu.

The bank conditions can also be expressed in present value form

Vbm =−M + (1− s)M(1 + r) + sVbu

ρ+ s(17)

Vbu =−k +Mρ+m(θ)Vbm

ρ+m(θ). (18)

D. Free Entry of Banks

The next step is examine the aggregate willingness of banks to supply loans. This is

determined by free entry. There are an infinite number of inactive banks that all deposit

their money with the Federal Reserve earning the risk-free rate. An inactive bank may

choose to become active by looking for a borrower. Since there are more potential banks

than firms, free entry limits the number of banks. By free entry, any entry date must give

the same expected payoff, and this value must be M , and so

Vbu = M. (19)

Free entry together with the first flow conditions gives,

Vbm =k

m(θ)+M (20)

This says that the value of being a matched bank is given by the value of the capital

plus the cost of search divided by the matching probability. Notice that the matching

probability will adjust in order to maintain this condition.

10

From free entry and the second flow condition,

Vbm =Mr(1− s)ρ+ s

. (21)

The using this condition together with (Vbu = M) and the first flow condition gives

r(1− s)− (ρ+ s)− k(ρ+ s)

m(θ)M= 0. (22)

This says that the probability of a match is in effect pinned down by the bank’s entry

decision.

Condition 22 is the second key condition that is required to solve for the steady state

equilibrium. The number of banks, or equivalently the volume of bank loans is controlled

by the entry decisions. Since m is a function of θ, 22 can also be viewed as a loan supply

relationship between θ and r, r = s+ρ1−s + k(s+ρ)θα

(1−s)Mm0. This version of the loan supply curve

in θ− r space has an intercept of s+ρ1−s . The slope is positive and the curve is concave, with

the degree of concavity controlled by α. As α increases towards 1, the curve gets flatter.

Bank willingness to lend depends on the interest it expects to be able to earn. If r is

greater, then more banks enter the loan market causing θ to increase. At times it may

also be convenient to reexpress 22 as θ = ( (r(1−s)−(ρ+s))Mk(ρ+s)

)1/α. For this to have an interior

solution requires r(1− s)− (ρ+ s) > 0. The following proposition follows from taking the

appropriate partial derivatives.

Proposition 1 Assume that r(1− s)− (ρ+ s) > 0, so that there at least some banks that

are trying to find borrowers. Then credit market tightness, θ is increasing in r, M , and

decreasing in s, ρ, k. If r is low enough (r(1 − s) − s − ρ − 1 < 0), then θ is increasing

in α, but if r is high enough (r(1− s)− s− ρ− 1 > 0), then θ is decreasing in α.

In the bank problem it is assumed that there are a suitable measure of infinitessimal

banks, each of which are making a single loan. As long as nothing else is changed,

the model could have banks with multiple loans. In that case it would still need to

be assumed that the banks are infinitessimal, that there are no economies in search,

11

bargaining, operations, etc. As long as such conditions are satisfied, nothing important

changes in the equilibrium if banks have more capital and make more loans.

E. Bargaining

When there is a match, the two parties bargain over the terms of the loan. For

simplicity the size of the loan is fixed at M . The only thing left to negotiate is the loan

interest rate. The outcome of the bargaining depends on the relative bargaining strengths

of the bank and the firm. A generalized Nash bargaining solution is assumed to hold.

The bank’s bargaining power is denoted β, and the firm’s bargaining power is (1− β).

When a bank and a firm are matched they must agree on the terms of the loan, i.e.

the interest rate. In order for there to be anything to negotiate, both parties must agree

to participate. The bank’s participation constraint is

Vbm − Vbu ≥ 0. (23)

If this were not true, the bank could simply refuse to make the loan. Using 21, and the fact

that Vbu = M , this provides a lower bound on the interest rate, r ≥ s+ρ1−s . This constraint

shows that, quite naturally, the lower bound increases when the risk-free interest rate (ρ)

increases, and also when the risk of bankruptcy (s) increases.

The firm’s participation constraint is

Vfm − Vfu ≥ 0. (24)

Using 14 and 13, this provides an upper bound on the interest rate, r ≤ A− 1 + h(1−s)M .

Accordingly any negotiated interest rate must satisfy

s+ ρ

1− s≤ r ≤ A− 1 +

h

(1− s)M. (25)

12

If the parameters are such that these cannot hold simultaneously then there is no mutually

satisfactory negotiated loan deal that is feasible. In what follows it is assumed that the

parameters satisfy this restriction. High productivity (A) and low risk-free rate (ρ), and

low risk of bankruptcy (s) are helpful to making a mutually beneficial deal feasible.

Assume that bargaining satisfies the generalized Nash bargaining solution. The bank’s

bargaining power is denoted β. The bargaining problem is

rN = arg max(Vbm − Vbu)β(Vfm − Vfu)1−β. (26)

In the model the size of the loan is taken to be exogenous. There are several plausible

ways to endogenize the loan size. The easiest is to let it also be chosen by the Nash

bargaining. In that case instead of one first order condition from the negotiation, there

are two. This complicates the algebra without enough extra insight to be worth it.6

To solve this problem note that this problem assumes that the bargaining is a con-

tinuous process. In this case Vbu and Vfu reflect the payoff that would be obtainable by

leaving the match. Clearly these do not depend on the value of r. Accordingly neither

Vbu nor Vfu are functions of the current interest rate.7

The first order condition can be written as

Vfm − Vfu =(1− β)(ρ+ s)

ρβ(Vbm −M) (27)

Then using 20

Vfm − Vfu =k(1− β)(s+ ρ)

ρm(θ)β. (28)

6An alternative approach is to allow the loan size to be pinned down by liquidity constrains in themanner of Holmstrom and Tirole (2011). This is being examined in a separate paper. A key implicationis that because search is costly, the search process can interact with the firm’s liquidity. This can resultin firms searching for a bank at first, but giving up after a while. They might no longer have enoughliquidity even if they find a bank – a discouraged entrepreneur effect.

7The first order condition is

β(Vbm − Vbu)β−1 ∂Vbm∂r

(Vfm − Vfu)1−β + (Vbm − Vbu)β(1− β)(Vfm − Vfu)−β∂Vfm∂r

= 0.

Note that∂Vfm∂r = −M(1−s)

ρ and, ∂Vbm∂r = M(1−s)

s+ρ .

13

Using the firm’s participation constraint, the negotiated interest rate on a loan is

rN = A− 1 +h

M(1− s)− k(1− β)(s+ ρ)(ρ+ θm(θ))

ρm(θ)βM(1− s), (29)

This is the third key component need to compute the steady state equilibrium. The

properties of this condition are collected as follows.

Proposition 2 The negotiated loan interest rate, rN , is increasing in A, h, m, β and

decreasing in k. The effect of ρ is given by ∂rN

∂ρ=

(1−β)(smF−ρ2)k(1−s)Mβρ2m

and so it has the

same sign as smF − ρ2. The effect of an increase in the separation rate is given by

∂rN

∂s= hmβρ−k(1+ρ)(ρ+mθ)(1−β)

(s−1)2Mmβρ, which will be negative if k (1 + ρ) (ρ+mθ) (1− β) > hmβρ,

and positive in the reverse case. The impact of credit market tightness is ∂rN

∂θ=

(β−1)(θm0+αρθα)(s+ρ)k(1−s)Mθβρm0

< 0, and ∂2rN

∂θ2= (β−1)(α−1)(s+ρ)(θα)kα

(1−s)Mθ2βm0> 0

The more costly it is for the firm to search, the higher the interest rate that the bank

can charge. The more costly it is for the bank to search, the lower the interest rate that

the bank will be able to charge since everyone knows that the bank’s implicit threat to

walk away is less attractive.

It is interesting that the matching probabilities, the bargaining power, and the risk free

rate all operate through the impact on the compensation for the bank’s cost of search.

If it does not cost the bank anything to search, then the bank is able to claim all of

A− 1 + hM(1−s) .

The impact of an increase in the separation rate depends on, among other things, the

relative search costs of the two parties. If the bank’s search cost (k) is high, then an

increase in the separation rate tends to have a negative impact on the loan interest rate.

If the firm’s search cost is high (h) then an increase in the separation rate tends to have

a positive impact on the loan interest rate.

14

F. Steady State Equilibrium

The steady state equilibrium is a solution for θ, r, l. Collecting the crucial conditions

l =m0θ

1−α

s+m0θ1−α

r =s+ ρ

1− s+

k (s+ ρ) θα

(1− s)Mm0

r = A− 1 +h

M(1− s)− k(1− β)(s+ ρ)(ρ+m0θ

1−α)

ρm0θ−αβM(1− s)

The equivalent to setting loan supply equal to loan demand is to simultaneously solve 22

and 29 to find θ∗. Hence,

s+ ρ

1− s+

k (s+ ρ) θα

(1− s)Mm0

= A− 1 +h

M(1− s)− k(1− β)(s+ ρ)(ρ+ θm0θ

−α))

ρm0θ−αβM(1− s)

This simplifies to s + ρ + k(s+ρ)θα

Mm0= (1 − s)(A − 1) + h

M− k(1−β)(s+ρ)(ρ+θm0θ−α))

ρm0θ−αβM. This

equation is easily solved numerically when there are specific parameter values. If θ = 1/2

it also has a closed form solution.8

Consider plotting with θ on the x-axis, and r on the y-axis. Then the free entry (loan

creation) condition is an increasing function that is concave down. The loan interest rate

condition is a decreasing function that is convex. The intersection of these two curves

solves for θ and r. Then the value of θ is substituted into the loan market steady state

condition to get the steady state volume of loans.

With the notable exception of firm productivity (A) many of the parameters affect

both the free entry condition and the loan interest rate condition. Thus the impact of

8(s+ ρ)M + k(s+ρ)θα

m0= M(1− s)(A− 1) + h− k(1−β)(s+ρ)(ρ+θm0θ

−α))ρm0θ−αβ

((s+ ρ)M −M(1− s)(A− 1)− h)ρm0θ−1/2β = ρβk (s+ ρ)− k(1− β)(s+ ρ)(ρ+m0θ

1−1/2)Define, a1 = ((s + ρ)M − M(1 − s)(A − 1) − h)ρm0β, a2 = ρβk (s+ ρ), a3 = k(1 − β)(s +

ρ). Then write, a1θ(−1/2) = a2 − a3(ρ + m0θ

1/2). The solution is given by, θ = ± a1a3m0

−1

a23m20

(−a2 + ρa3)(

12a2 −

12ρa3 + 1

2

√−2ρa2a3 − 4a1a3m0 + a22 + ρ2a23

)Clearly only the positive solution for θ makes economic sense.

15

most shocks will depend on which of the curves is affected more strongly, and in which

direction they move.

The traditional real business cycle models interpret cycles as shocks to productivity.

A recession is then a drop in A. That will result in a drop in θ, and a drop in r. The

drop in θ will also translate into a drop in l since ∂l∂θ> 0. A recovery is an increase in A.

This will have the reverse effects.

III. Numerical Example

Consider the following set of parameter values: s = 0.05, ρ = 0.04, k = 0.25, m0 =

0.50, M = 1, α = 0.50, A = 5, h = 0.25, β = 0.50.

Then the bank loan creation/free entry curve is an increasing function with a small

amount of curvature close to the origin. However it is actually fairly flat at about 0.13.

The loan interest rate is essentially a straight line with a negative slope. If A shifts

around, there will be a great deal of movement in the tightness of the credit market, but

almost no change in the rates charged on loans.

IV. Fresh Start Bankruptcy

Bankruptcy is sometimes said to provide an insolvent borrower an escape from the

debt burden in order to have a ‘fresh start.’ The borrower loses the assets, but is able to

start again without any debt overhang. In contrast to terminal bankruptcy, is not stuck

with a payoff of zero for ever after. The purpose of this subsection is to trace out the

equilibrium impact of this extra benefit to a borrower in the case of a bad event.

The bank’s problem is not changed. The bank still gets paid if and only if the borrower

is solvent. The bank is only affected in equilibrium if the agreed upon interest rate is

affected.

16

The firm’s problem does change. The firm’s flow payoff conditions are now

ρVfu = −h+m(θ)(Vfm − Vfu) (30)

ρVfm = (1− s)M(A− 1− r) + s(Vfu − Vfm) (31)

In the event of bankruptcy the firm gets to start again in state Vfu. This shows up in the

flow payoff when a loan is obtained. There is an extra term +s(Vfu − Vfm).

The present value conditions are

Vfu =−h+ θm(θ)Vfmθm(θ) + ρ

Vfm =(1− s)M(A− 1− r) + sVfu

ρ+ s

The only change is to the expression for Vfm.

Solving these simultaneously gives

Vfm =(1− s)M(A− 1− r)(ρ+ θm(θ))− hs

ρ(ρ+ s+ θm(θ))

Vfu =(1− s)M(A− 1− r)θm(θ)− h(ρ+ s)

ρ(ρ+ s+ θm(θ))

The negotiation problem has the same structure as in the terminal bankruptcy problem.

The bargaining problem is

rN = arg maxr

(Vbm − Vbu)β(Vfm − Vfu)1−β. (32)

The bank’s problem is unchanged, and so Vbm − Vbu is the same as before. The firm’s

participation constraint simplifies to

Vfm − Vfu =(1− s)M(A− 1− r) + h

(ρ+ s+ θm(θ))≥ 0.

The denominator does differ from the terminal bankruptcy expression. However, in both

cases the denominator is positive. Thus the critical issue is the numerator. The numerator

17

is the same as in the terminal bankruptcy case. Since this is key for participation, it turns

out that the upper bound on the interest rate is

r = A− 1 +h

(1− s)M.

This is exactly the same as the upper bound on the interest rate as in the terminal

bankruptcy case. Using the same steps as for terminal bankruptcy, the next proposition

is readily derived.

Proposition 3 The steady state equilibrium interest rate with fresh start bankruptcy

(rN = A − 1 + hM(1−s) −

k(1−β)(s+ρ)(ρ+θm(θ))ρm(θ)βM(1−s) ) is exactly the same as under terminal

bankruptcy.

Since the bank’s problem is unchanged, and the equilibrium interest rate is unchanged,

so too is the overall bank loan market equilibrium.

V. Bank Loan Market Facts

The data is all aggregate quarterly US data. As described in the Appendix, the main

source of the data is from the Federal Reserve. Lending standards and loan spreads

data are from the Survey of Senior Loan Officers that is carried out by the Federal Re-

serve Board (http://www.federalreserve.gov/boarddocs/snloansurvey/). This is a survey

of about 60 large domestic banks and 24 branches of foreign banks. Accordingly it is

weighted towards the larger banks.

Some data series such as the total volume of business loans, have been collected from

the start of 1947 (255 quarters). Other data series such as the number of new businesses

formed are only available since the 1990s (67 quarters). All the data ends as of the 3rd

quarter of 2010. For consistency with the macroeconomic literature the data is filtered

using the usual Hodrick-Prescott filter. The use of the filter seems to reduce the volatility,

but otherwise does not cause major changes to the inferences.

18

As illustrated in Figure 1. the volume of business loans is strongly nonstationary. So

it is first differenced before the usual Hodrick-Prescott filter is applied.9 In the raw data

two fact jump out. First, as expected business loans peak during recessions. Second, the

decline persists even after the economy comes out of the recession. In other words, the

volume of business loans lags the cycle. The average quarterly growth in business loans

from April 1947 to July 2010 was 1.88% a bit below the 2.2% for all loans and leases.

Figure 2 shows the typical loan rate spread above the bank’s cost of funds. Like the

total volume of loans, the spread peak during recessions. Unlike the volume of business

loans, the spreads seem to start dropping earlier, perhaps even before the recession has

fully ended. The lending standards behave similarly to the spreads as depicted in Figure

3.

There are several closely related measures available for troubled loans. A commercial

loan that worries the bank induce an addition to the loan loss reserve, or it can already be

nonperforming, or finally the bank can take a charge-off. These stages of trouble are high

correlated empirically, and it makes little difference for current purposes, which is used.

The charge-offs seem to be the most closely related to the search model, and so that is the

focus here. However, in a more refined analysis, these three stages of loan trouble could

be distinguished. The charge-offs are depicted in Figure 4. The charge-offs help explain

the total loan volume. In particular charge-offs seem to peak after a recession is already

over. Similar patterns are found in the other measures of problem loans. Apparently this

pattern does not depend too much on exactly how problem loans are measured. There

are lingering effects of a recession on existing loans.

To examine whether there are significant dynamics, Table 1 reports the results from

running an AR-1 model separately on each series. All of the series are highly auto-

correlated even after being filtered. This differs from the individual bank level results

reported by Craig and Haubrich (2006). At the individual bank level they report much

less autocorrelation.

9The use of first differencing makes only small differences to the inference to be drawn from the data.

19

To see whether there are cross lagged effects a VAR with one lag in each series was

run. Many of the cross effects are reasonably small. Mostly they seem fairly easy to

understand. Apart form the lagged own effects, the following cross effects were observed:

• DLoan: negative impact from lagged standards.

• Standards: positive effect from lagged spreads, negative effect from lagged DS&P

500.

• Spreads: nothing significant.

• Prime Rate: negative effect from lagged DLoans, positive effect from lagged demand.

• Demand: positive effect from lagged DLoans, negative effect from lagged prime rate,

positive effect from lagged DS&P 500.

• DS&P 500: nothing significant.

• Charge-offs: positive effect from lagged spreads, fairly strong negative effect from

lagged demand, positive effect from lagged non-performing loans.

• Non-performing: negative effect from lagged DLoans, positive effect from lagged

spreads.

Individually each of the cross effects can be interpreted in terms of the search model.

But effect by effect verbal interpretations does not easily capture the consistency of ex-

planations. Hence, these observations are recorded here, but not interpreted for now.

Table 2. reports the pairwise correlations. Several of the correlations are quite strong.

Lending standards and loan spreads are highly positively correlated as illustrated in Figure

5. Both are involved in clearing the loan market. They play a complementary role. If

standards can be correctly interpreted as a measure of bank bargaining power, this is

saying, quite reasonably, that when the bank has more bargaining power, it charges a

higher mark-up. When there is an increase in the stock market (DS&P500) there is a

reduction in the bank’s bargaining power. When bank standards are high, loan demand

is low.

20

Troubled loans do have rather strong correlations with other aspects of the market.

Many of these are independent of which measure of troubled loans is used. When there

are more charge-offs, the volume of loans falls. This is almost mechanical. For this not

to be true would require extra effort on the part of banks to very rapidly replace the

failed loans. When charge-offs are high, the prime rate tends to be low. Presumably this

reflects a policy response function by the Federal Reserve. When the economy is weak,

charge-offs will tend to be high, and loan demand will tend to be low. When the economy

is weak, the Federal Reserve will commonly try to reduce interest rates in an effort to

stimulate the economy. This will show-up in the data as a low prime lending rate.

In the survey data the loan demand variable requires a bit of care. In a search model

there is a distinction between many searchers and many matches. Either could be what the

Loan officers have in mind. These are likely to be correlated. But they do have somewhat

different interpretations for the connection between the model and the empirical data.

The connection between demand and both spreads and standards will be discussed at

greater length in relation to the numerical results.Numerical Results

To calibrate the model requires taking a stand on parameter values. For some of these

parameters appropriate values are fairly clear. For other parameters it is less clear.

The total number of firms in the USA is about 6 million. The total number

of banks is about 8000 currently, and falling. So somehow I guess that I need to

standardize. Similarly I suppose we standardize at loan size of M = 1. For a

quarterly risk-free interest rate about 0.013. For the bargaining power start with

β=0.5 One source for firm survival there is no accepted number. Some data is here:

http://www.bls.gov/bdm/us age naics 00 table7.txt. So it seems that maybe 6% of firms

fail each year. So maybe 1.5% per quarter fail. Other data on firm birth and death is

here: http://www.sba.gov/advo/research/dyn us tot.pdf. I do not have any strong idea

about the values of h and k. Maybe 10% of the size of the loan I suppose. (This is a very

wild guess. Presumably it is higher for the firm than it is for the bank – at least for small

firms.) The value of A must be bigger than the costs or else there will not be an interior

loan solution. I guess 25% might be a number to start with. Most likely that will prove

to be too high eventually.

21

VI. Implications for the Bank Loan Market

In the model each bank has the same amount of capital and is looking to make (or

has made) a single bank loan. As long as there are constant returns to scale, no search

economies, and each bank remains small relative to the market this is innocuous. Suppose

that the government suddenly gave each operating bank an extra M dollars. Each bank

would pass the extra cash back to the owner leaving the original loan market equilibrium

unchanged.

Suppose that the government recognized this, and so passed a law forbidding an op-

erating bank from passing back the extra cash. Entry of new banks would cease. If it

caused the credit market tightness to become excessively loose, then that would enhance

the bargaining power of the existing borrowers, and so the interest rates on existing loans

would drop. But that requires that the banks spend extra cash looking for the (suddenly

unprofitable) borrowers. A sensible bank would not spend the money hunting for such

a borrower. Instead they would deposit the extra cash back with the Federal Reserve,

leaving θ unchanged.

The implication is that within this kind of search model context, giving each bank

extra cash would benefit the bank shareholders, lead to extra cash on deposit at the

Federal Reserve, and leave the loan market equilibrium unchanged.

Suppose that the government wants to encourage banks to make more good loans.

Within the context of the model such a policy needs to focus on equation 22 and proposi-

tion 1. The natural place to focus is on k the cost of searching for borrowers. For example

The government could in effect provide a tax subsidy to cover those costs. That would

increase the number of banks looking to place loans. Of course, proposition 2 warns that

there are further equilibrium effects to consider. Decreasing the bank’s search cost will

also translate into a windfall for existing borrowers as the interest rates on existing loans

will also tend to drop.

This highlights a general policy issue that goes well beyond the specifics of the current

model. Attempts to change the quantities of loans are likely to have important implica-

tions for the terms of other existing loans. Exactly how this works out will depend on the

22

specifics of a particular model. This creates a concern that the secondary effects can be

quantitatively more significant, and much harder to predict, than the intended primary

effect of a policy.

The bank loan literature (eg. Gorton and Winton (2003), Freixas and Rochet (2008))

has paid a great deal of attention to both adverse selection and moral hazard in the loan

making process, see Petersen and Rajan (1995), and, Diamond (1991). Acharya et al.

(2010) consider the impact of aggregate risk. The search friction considered in this paper

is complementary to the usual incentive effects as in Diamond (1984), Rajan (1992).

The potential importance of search is already suggested by the importance of distance

in lending that was documented by Petersen and Rajan (2002).

Weil and Wasmer (2004) and Petrosky-Nadeau and Wasmer (2010) study Diamond-

Mortensen-Pissarides style models models with two frictions, one in the financing, and

the other in the labor market. In contrast to the current paper, their main interest is in

the impact of a financial friction on the labor market. A stylized financing search friction

is used to address puzzles about the job market.

The history of the Senior Loan Officer survey is described by Schreft and Owens (1991).

The survey goes back to late 1964, and the questions asked have changed somewhat

over the years. They observe that: credit standards and willingness to lend are very

closely connected, banks are less willing to lend during recessions, the Loan Officers report

increasing standards, but rarely report a lowering of standards.

Lown and Morgan (2006) use a VAR analysis to show that credit standards as reported

in the Senior Loan Officer survey dominates loan interest rates in terms of the ability

to explain the variation in business loans outstanding, and overall output. It is well

understood that loan contracts are multidimensional, and so in general ‘tightening’ could

refer to adjustments on a variety of alternative loan contractual terms. Lown and Morgan

(2006) suggest that ‘tightening’ and ‘unwillingness to lend’ be interpreted in terms of

informational frictions.

23

VII. Conclusion

The analysis in this paper may provide a start on consideration of the corporate finance

implications of search. But the lack of centralized loan markets is a broad topic, with

many aspects that go far beyond the analysis in this paper. For example recently there

has been some surprise in policy circles at the relative non-response of bank loans to the

increases in bank liquidity engineered by the Federal Reserve. From a search perspective

this is not surprising. Part of the impact would be on the entry and exit decisions of

banks, and empirically we are observing some banks shutting down. Part of the impact

would be on the bank’s own savings behavior which also seem to have taken place. The

marginal impact on actual bank lending could easily be zero or close to zero in a natural

equilibrium search model.

There are potentially extremely interesting potential interaction effects between or-

dinary bank loans and other forms of corporate financing. Presumably search implies a

greater need for a firm to hold cash reserves. But what is the mix between actual cash

holdings and lines of credit? How do each of these respond to aggregate shocks? How

does the use of equity affect the theory? If some equity is publicly traded how does

this cross over to the terms in the debt market. Presumably effects akin to those in

Duffie and Manso (2007) will arise. But how do they interact with more familiar tax

and bankruptcy cost considerations? Clearly there is a great range of potential corporate

finance implications that stem from the need to find willing investors. There is much to

do.

24

VIII. Appendix: Data Sources

The source of data is FRED: http://research.stlouisfed.org/fred2/, except where noted

below. All data from that source was extracted at quarterly frequency. The following

series were extracted:

• Loans. BUSLOANS (Commercial and Industrial Loans at All Commercial Banks).

The data was extracted at quarterly frequency with units ‘billions of dollars’.

LOANS (Total Loans and Leases at Commercial Banks).

• Standards. DRTSCILM (Net Percentage of Domestic Respondents Tightening Stan-

dards for Commercial and Industrial Loans Large and Medium Firms). Quarterly,

Percentage.

• Spread. DRISCFLM (Net Percentage of Domestic Respondents Increasing Spreads

of Loans Rates over Banks’ Cost of Funds Large and Medium Firms). Quarterly,

Percentage.

• Prime. MPRIME (Bank Prime Loan Rate), Quareterly, Percentage.

• Demand. DRSDCILM (Net Percentage of Domestic Respondents Reporting

Stronger Demand for Commercial and Industrial Loans Large and Medium Firms).

• New Firms. Quarterly data on the births and deaths of establishment is taken from

http://www.bls.gov/web/cewbd/table9 1.txt. It starts in Septmber 1992 and ends

in December 2009.Related data is available at annual frequency up to 2005 here:

http://www.ces.census.gov/index.php/bds/bds database list. Further links can be

found at: http://www.sba.gov/advo/research/data.html.

• Market. SP500 (S&P 500 Index), Quarterly, Index Level.NFCPATAX (Nonfinancial

Corporate Business: Profits After Tax), MVEONWMVBSNNCB (Market Value of

Equities Outstanding - Net Worth (Market Value) - Balance Sheet of Nonfarm

Nonfinancial Corporate Business).

25

• Charge-offs. NCOCMC (Commercial Net Loan Charge-offs), Quarterly, Ratio.

USLSTL (Net Loan Losses / Average Total Loans for all U.S. Banks), NPCMCM

(Nonperforming Commercial Loans), NPTLTL (Nonperforming Total Loans), NCO-

TOT (Total Net Loan Charge-offs).

In each case the first listed data series is taken to be the main definition. The remaining

series were extracted for use as robustness checks.

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27

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