Full text of DOI:10.1111/1468-0297.00491

The Economic Journal, 110 ( January), 69±91. # Royal Economic Society 2000. Published by Blackwell
Publishers, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA.
THE IMPACT OF LIQUIDITY CONSTRAINTS ON BANK
Ã
LENDING POLICY
David C. Webb
This paper examines banks' provision of liquidity to depositors and provision of loans. The
problem identi®ed is that banks may not be able to provide new funds for borrowers who are
short of cash, because either the return on investments is poor, or because depositors withdraw
more funds than expected. Banks subject to liquidity shortages may ration loans to good
borrowers. This problem is shown to depend upon the nature of the deposit contract and
banks' inability to issue subordinated deposits. State contingent renegotiation of loans and
matching of the duration of project returns and investment needs mitigates the problem.
This paper is concerned with banks' dual role in supplying loans to ®rms and
taking deposits from households. Lending arrangements are such that in some
circumstances loans need to be renegotiated. On the liability side of their
balance sheets, banks provide a way for consumers with liquidity needs to
smooth consumption. The paper integrates these two functions of banks and
in an environment with no deposit insurance or lender of last resort, illustrates
a problem for the banking system. The problem is that banks may not be able
to provide new funds for borrowers, particularly on renegotiated terms, if
withdrawals by depositors are too high. Supplying liquidity on the liability side
of the balance sheet may con¯ict with supplying liquidity on the asset side of
the balance sheet. Thus the paper shows how, in a ®nancial system where ®rms
are primarily ®nanced by bank debt and banks are primarily ®nanced by
demand deposits, there can be a knock-on effect whereby depositors' liquidity
demand means that a bank faced with a shortage of funds cannot renegotiate
loan terms for corporate borrowers, leading to under-investment relative to
the ®rst-best. The paper also shows how this affects entrepreneurs' project
choice, showing that it can cause a bias towards less risky short-term projects.
The focus of the paper is on the ®rst round effects of this problem so that
possible contagion effects are not examined.1
We assume that state contingent payments are not veri®able. Hence, ®nance
takes the form of debt, and because of the timing of project returns this is long-
term debt. However, at some intermediate date after the arrival of information,
a decision to continue or liquidate the ®rm must be made. Under some
circumstances the ®rm faces a classic debt overhang problem (see Myers
(1977)) so that the ef®cient investment decision requires that the debt contract
Ã
I thank Markus Brunnermeier, Vittoria Cerasi, Dennis Gromb and the three referees for very
helpful comments.
1
Rajan (1994) analyses a similar problem but from a different perspective. He examines the
interdependence of lending policies in an environment where banks have short horizons along the
lines examined by Stein (1989) and shows that if relative performance affects bank reputation and
hence stock price, banks may `jam' their earnings signal. Bank lending can be inef®ciently high or low.
The present paper in contrast examines the co-ordination problem of banks as suppliers of liquidity to
borrowers and depositors.
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be renegotiated. The nature of the information about the ®rm at this date and
the need to renegotiate the loan contract makes a bilateral relationship with a
single bank appropriate. Renegotiation is done on a bilateral basis and will not
just re¯ect the ®rm's prospects but also the strength of the bank's balance sheet.
Turning to the behaviour of banks in our model, Diamond and Dybvig
(1983) and Bryant (1980) provide the classic treatment of the deposit contract,
in which banks pool risks in an environment where privately observed con-
sumption shocks are uncorrelated across agents. Optimal insurance contracts
cannot be written on privately observed consumption shocks, but a bank can
provide insurance to depositors by arranging deposit contracts which support
a Pareto optimal allocation. However, their model has multiple Nash equili-
bria. One is the equilibrium which supports the ex-ante welfare maximising
allocation and, given the important institutional feature that the deposit
contract satis®es a sequential service constraint, the other is a bank run. The
run is triggered by events exogenous to the models fundamentals. The
investment technology which banks invest in is deterministic but illiquid, so
that a bank is unable to honour all of its liabilities at par if all depositors
withdraw and hence ultimately is liquidated. A second and perhaps more
satisfactory series of models concentrate on explaining the conditions under
which depositors might rationally change their beliefs about the riskiness of
banks. Gorton (1985), Chari and Jagganathan (1988) and Jacklin and Bhatta-
charya (1988) all offer information-based explanations of bank runs. In these
models depositors are incompletely informed about the true value of the
banks' portfolio and make inferences from available information about its
performance. Runs occur because a subset of depositors receive interim
information signalling a low future return on the bank's investments.
The present paper models the interaction of banks, ®rms and depositors
along similar lines to Jacklin and Bhattacharya (1988). The liability side of our
model is the same as in their paper, the banks' role being to provide liquidity
to depositors. In Jacklin and Bhattacharya banks can invest in a riskless short-
run technology or in a risky long-run technology that cannot be liquidated
before maturity. However, in the ®rst part of this paper we assume that bank's
income is only from ®nancing an overlapping sequence of risky long-run
investments, which require investment at two dates, but which can be liqui-
dated at the interim date when the quality of the investment is realised. Only
later do we examine the role of riskless investments. Firms' project selection
and ®nancing decisions play a crucial role in our model. In particular,
intermediate ®nancing decisions including liquidation and loan renegotiation
are of central importance.
Banks offer a menu of contracts to depositors and loans to ®rms which are
tailored to ®t with the expected liquidity needs of these agents. However, in
some circumstances, depositors who receive poor information about the per-
formance of the bank's loan book, choose to withdraw at a higher rate than
they would have in the absence of such information. If the withdrawal rate is
suf®ciently high, there is a run on the bank and the bank faces a liquidity
shortfall, in the sense that it cannot raise suf®cient ®nancing to take all positive
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net present value investments, while repaying depositors. The problem is the
deposit contract itself. This contract is not contingent upon the performance
of the bank's balance sheet or the state underlying it. Alternatively, given
depositors' withdrawals, the bank may suffer from low income relative to
withdrawals because of `bad luck', income is less than expected. The paper
explores the implications of these two types of liquidity shortfall for banks'
lending. The impact on lending policy is high because deposits are unsubordi-
nated and new deposits issued to ®nance loans may dilute existing depositors'
holdings, causing early withdrawal. This effect is more acute the larger the
liquidity shortfall. Moreover, the impact of a liquidity shortfall caused by a run
is always worse than one caused by bad luck.
An important feature of the model is that if the bank suffers a liquidity short-
fall of high magnitude, it rations even high-quality entrepreneurs and renego-
tiates loans with those that it does ®nance. In this case, the bank obtains the
full surplus from renegotiation. However, this is passed back to entrepren-
eurs through more favourable terms when they are high quality and are granted
®nance on unrationed terms. In other words, we have the following result: banks
suffer liquidity shortages which because of the way they are ®nanced (the deposit
contract) have real effects. The consequences of this are borne by bank's
borrowers who partially insure the bank through loan renegotation.
The paper goes on to consider the effects of the above problem on entrepre-
neurs' project choice. We show that in equilibrium, in order to offset the impact
of the liquidity shortfall, some entrepreneurs, but not all, will choose to under-
take safer projects of lower value which insure them against being affected by
the bank's potential liquidity problem. The results here complement those of
Dewatripont and Maskin (1995) and Von Thadden (1995) that illustrate short-
termism in project choice arising from inef®ciencies in ®nancial intermedia-
tion. Finally we draw conclusions and discuss the implications of the relaxation
of some assumptions, such as there being no interbank market.
The paper provides a theoretical model which is consistent with Bernanke
and Lown's (1991) ®nding, that limited bank capital, relative to loan demand,
did contribute to restrictive bank lending in the recession in the United States
in 1990. The model's predictions are also consistent with the ®nding of Berger
and Udell (1990), that, in slowdowns, banks deny credit to weaker borrowers
in favour of stronger borrowers. The model also makes predictions about
renegotiation of loans with banks that are short of capital. However, systematic
evidence linking the terms of renegotiated loans to the structure of bank
balance sheets is not yet available.
1. The Basic Model2
Firms with multiperiod investment projects obtain ®nance from banks. At each
date information is distributed symmetrically between borrowers and lenders.
2
Since the paper introduces quite a lot of notation, a glossary of terminology is given after the
Appendix.
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Banks themselves compete to supply loan contracts and for depositors' money
to fund them. In the basic model we think of banks as simple intermediaries
between lenders and borrowers. The key features we wish to capture are: a
continuation-liquidation decision for entrepreneurs and partial debt forgive-
ness. This latter feature takes the form of renegotiation of the terms of the
contract. Given the specialised nature of information and the problem of free-
riding in renegotiation, this type of ®nancial arrangement will most naturally
be implemented as a bilateral relationship with a given lender.
1.1. Entrepreneurs
Entrepreneurs are risk neutral wealth maximisers. They have a three date
horizon, t ˆ 0, 1, 2. They have no initial wealth, but each is endowed with a
single project which requires an investment of I0 at date t ˆ 0 and I1 at date
t ˆ 1. For simplicity, the project only yields a return at date t ˆ 2. If the
investment I1 is not made at date t ˆ 1 the project is liquidated for
0 < L1 , I0. Throughout the paper and without loss to the main points we
assume that L1 ˆ 0. At each date, if the entrepreneur is inactive he obtains an
alternative return on his human capital of u . 0.3
At date t ˆ 0, the entrepreneur's project type is uncertain but will be
publicly observable at date t ˆ 1 before the investment decision at that date.
However, project type is not veri®able. There are two project types: good and
bad (poorer); denoted respectively by g and b. These projects have the same
investment requirement but differ in their returns: a type g project returns Y 2g
at date t ˆ 2 with probability one, a type b project returns Y 2b with probability
0 , p2s , 1 and zero otherwise with p2s Y 2b , Y 2g . At date t ˆ 1, the returns on
the two different types of project satisfy the following inequalities:
Y 2g ꢀ I1 . u for typeg,
(1a)
(1b)
p2s Y 2b ꢀ I1 . u for typeb:
At date t ˆ 0, the entrepreneur assigns prior probabilities of each type of
project being realised, denoted, respectively, by p1g . 0 and p1b ˆ (1 ꢀ p1g ) . 0.
1.2. Unconstrained Banks
Our entrepreneurs must obtain ®nance from banks. Initially, banks are
assumed to have access to a perfectly elastic supply of deposits. We refer to
these banks as unconstrained. Throughout the paper our banks are risk
neutral value maximisers, who are engaged in Bertrand competition. Banks
receive applications for ®nance from identical entrepreneurs at date t ˆ 0. At
this time they, like entrepreneurs, do not know the project type but will
discover this at date t ˆ 1.
3
The utility from inactivity means that in some states unless the burden of debt is reduced an
entrepreneur with high debt will prefer inactivity to undertaking some positive value investments.
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1.3. The Contracting Problem with an Unconstrained Bank
We assume that ®nance takes the form of debt contracts. Because projects
generate no income at date t ˆ 1, if the entrepreneur is to be ®nanced at date
t ˆ 0 the bank must have a claim against date t ˆ 2 income. Let the contrac-
tual payment be denoted by D2. This ®xed payment can be enforced, if for
example in the event that it is not paid the borrower loses everything. An
important assumption we make is that participation in the borrower-lender
relationship at each date, t ˆ 0 and t ˆ 1, is voluntary. Hence participation at
each date requires that the parties have the incentive to do so.
There follows a summary of events:
Date t ˆ 0: Banks offer ®nance to entrepreneurs so as to undertake a project
whose type is unknown to both the bank and entrepreneur. Type contingent
contracts cannot be written. The borrower promises to repay D2 at date t ˆ 2.
The bank supplies I0 at date t ˆ 0.
Date t ˆ 1: Nonveri®able information about project type is observed by both
borrower and lender. Both parties can quit the relationship at this date, in the
event of which the bank recovers the liquidation value of the project but this is
assumed to be zero. The bank and entrepreneur can, however, renegotiate the
loan terms. In this renegotiation the entrepreneur receives a return at least
equal to his outside option u . 0. However, the entrepreneur cannot threaten
to switch to another bank because the costs of switching are by assumption too
high. We, assume the bank makes a take-it or leave-it offer, so the surplus from
renegotiation goes to the bank, and the entrepreneur receives not more than
u . 0. If the loan is not renegotiated, the original contract stands. If the
entrepreneur's project is not liquidated he is supplied with additional ®nance
of I1.
Date t ˆ 2: The returns from projects are realised and shared between the
bank and entrepreneurs.
1.4. The Contract
First we examine the entrepreneur's problem at date t ˆ 1. At this date he
discovers his type, then:
if he is type g, assuming Y 2g ꢀ D2 . u, the entrepreneur will want to
participate and make the investment I1;
if he is type b, p2s (Y 2b ꢀ D2) , u, but if ps Y 2b . u, there exists a reduced value
2
^
of D2 denoted by D2 for which the entrepreneur participates.
Now examine the bank's problem at date t ˆ 1. Having granted a contract
at date t ˆ 0 and having observed the entrepreneur's type, the bank chooses a
value maximising response to borrowers loan requests. If the entrepreneur is
type g it funds I1 and obtains a return of D2 . I1: The bank can demand no
more than D2, because it is not in the borrowers interests to pay more, and
renegotiation must be in both parties interests. If the entrepreneur is type b
and is to make the investment, D2 must be renegotiated. Since the bank gets
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^
the full surplus in renegotiation, D2 is cut to D2, which satis®es
b
s
s
^
^
p2(Y 2 ꢀ D2) ˆ u. Let p2 denote the actual proportion of type b entrepreneurs
re®nanced at date t ˆ 1 who turn out to be successful at date t ˆ 2. The
distribution of p2 has an unconditional mean of p2s and a positive, ®nite,
s
^
variance. The necessary and suf®cient condition for an unconstrained bank to
s
s
s
^
^
^
participate and supply I1 is that E1(p2)D2 ꢀ I1 . 0 where E1(p2) ˆ p2.
Given the above behaviour in the subgames at date t ˆ 1, we can now
examine the problem at date t ˆ 0: The net return to the entrepreneur at date
t ˆ 0 if he does the project is then given by
p1g (Y 2g ꢀ D2) ‡ p [p (Y ꢀ D2)] ꢀ 2u > 0,
(2)
b
1
s
b
^
2
2
with
Y 2g ꢀ D2 > u and p (Y ꢀ D2) ˆ u:
s
2
b
2
^
Let á1g and á1b ˆ (1 ꢀ á1g ) denote the respective proportions of type g and b
entrepreneurs realised at date t ˆ 1, who were ®nanced by the bank at date
^
t ˆ 0: The payments D2 and D2 satisfy the bank's pro®tability conditions:
g
b
s
^
E0[á1 (D2 ꢀ I1) ‡ á1(p2 D2 ꢀ I1) ꢀ I0] > 0,
(3a)
and
s
^
^
E1(p2)D2 ꢀ I1 . 0,
(3b)
we also assume that the bank's expectations of á1g and á1b satisfy the following
consistency conditions:
E0(ág ) ˆ p g ,
(3c)
1
1
E0(á1b) ˆ p1b,
where the distributions of á1g and á1b both have positive, ®nite variances.
Finally, Bertrand competition at date t ˆ 0 ensures that given D2, D2 satis®es
^
(3a) as an equality.
The solution to the contract problem illustrated above implements the ®rst
best. This follows because at each date all positive net present value invest-
ments are undertaken. Given the inequalities in (1), investment should take
place at date t ˆ 1 for ®rms of types g and b. Therefore investment should
take place at date t ˆ 0 if,
p1g (Y 2g ꢀ I1) ‡ pb(ps Y b ꢀ I1) ꢀ I0 ꢀ 2u > 0:
(4)
2
1
2
Implementation of the contract ensures that at date t ˆ 1 both type g and b
®rms apply for and receive ®nance. Then, substituting from condition (3) into
(2) yields (4), so we have the ®rst best.
2. The Contracting Problem with Constrained Banks
The unconstrained bank of the last section is an intermediary with access to a
perfect capital market. It has both the access to capital and incentives to
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implement the terms of the ®rst-best contract. We now change the model by
modifying bank behaviour. We assume that banks have the dual role of
®nancing entrepreneurs and providing liquidity services to depositors. Our
purpose then is a straightforward one of investigating the implications of this
dual function of banks for their lending policy and ®rms' investment beha-
viour. In particular, we wish to identify potential sources of inef®ciency in
lending policy.
Before embarking on this exercise we list a number of assumptions about
the market environment in which banks operate. There is no lender of last
resort, there is no deposit insurance, there is no secondary market for bank
loans and banks do not issue equity. Thus we restrict banks to be pure
intermediaries between the commercial loan market and depositors. Our
banks are owned by risk neutral wealth maximisers whose only initial asset is a
banking licence. These banks compete in attracting deposits and in making
loans to entrepreneurs. The only agents in our model who have any initial
wealth are depositors.
2.1. Depositor Behaviour
Banks attract funds from depositors by offering deposit contracts.4 The model
of depositor behaviour is in the spirit of Diamond and Dybvig (1983), although
the particular formulation used is based on Jacklin and Bhattacharya (1988).
The details of the depositor model are in the Appendix. The idea is that
depositors deposit at date t ˆ 0 and have consumption needs at dates t ˆ 1
and t ˆ 2. However, they are subject to shocks to their time preference
parameter at date t ˆ 1, so their liquidity needs are stochastic. Banks pool
different investments to generate income at different dates to supply deposi-
tors with higher returns than under autarky and provide insurance against
liquidity shocks. But depositors can withdraw when they want. The deposit
contract has to satisfy incentive constraints, which ensure that different types
of depositor on discovering their type (which is private information) make
withdrawals consistent with their type. This means that the deposit contract
can offer only partial liquidity shock insurance. In a non-run equilibrium, at
date t ˆ 0 depositors forecast that they will be high time preference, type 1,
with probability ð1 and low time preference, type 2, with probability ð2 ˆ
(1 ꢀ ð1); and the actual proportion of depositors depositing at date t ˆ 0, who
stick to the withdrawal pattern for type 1 at date t ˆ 1, denoted by f0, equals
the expected proportion, f0 ˆ ð1.
However, there is a problem with the above contract. The contract is
assumed not to be contingent on the realisation of f0 at date t ˆ 1 or the
variables that determine f0. Hence, as pointed out by Jacklin and Bhattachar-
ya, if some type 2 depositors receive interim information, ó, at date t ˆ 1, this
4
In this paper we restrict attention to deposit contracts. Jacklin and Bhattacharya show for particular
economies how for certain distributions of shocks, in terms of ex ante utility deposit contract allocations
dominate equity contracts, and for other distributions the reverse is true.
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may lead them to revise down the return on the bank's investments at date
t ˆ 2; this in turn causes a violation of the date t ˆ 1 incentive constraints
(given in the Appendix by (A4)). This leads to a higher rate of withdrawal than
expected so that f0 . ð1, which is called a run equilibrium.
There are four key points to note about deposit contracts. Firstly, deposits
are not contingent on the return from the bank's investments. They are claims
that are like debt, in the sense that they have a ®xed prior claim on the bank's
income. But this claim cannot be renegotiated. Secondly, deposit contracts
supply depositors with liquidity insurance; the depositors bear no risk when
the bank is solvent but share the bank's assets equally if it is insolvent. Hence,
unlike in the model in Hellwig (1994), date t ˆ 0 depositors cannot share the
risk of returns at date t ˆ 2 contingent upon date t ˆ 1 information. Thirdly,
depositors are free to withdraw early, on terms independent of their type,
thereby exposing the bank to further risk. Finally, we make the important
distinction between subordinated and non-subordinated deposits. If deposits
made at date t ˆ 1 are subordinated, they can only be paid at date t ˆ 2 after
date t ˆ 0 deposits have been paid. If deposits made at date t ˆ 1 are non-
subordinated, they have the same claim at date t ˆ 2 as those made at date
t ˆ 0, thereby diluting the income claim of the earlier deposits.
At each date banks attract two period depositors, with contracts of the type
described above. Our model has only four dates t ˆ ꢀ1, 0, 1, 2: At date t ˆ 0
new deposits per typical loan made at date t ˆ 0, are denoted by B0. Similarly,
at date t ˆ 1 new deposits per date t ˆ 0 loan, are given by B1. At date t ˆ 1, a
bank's deposits consist of two-period old deposits made at date t ˆ ꢀ1 and,
one-period old deposits made at date t ˆ 0: In the general case new deposits
made at date t ˆ 1 can be withdrawn at dates t ˆ 2 and t ˆ 3 (although our
example ends at date t ˆ 2). We de®ne aggregate deposit withdrawals at dates
t ˆ 1, 2 as the total value of depositor withdrawals per typical date t ˆ 0 loan.
At date t ˆ 2 the total withdrawal from the bank, denoted by C2, is
C2 ˆ f0C012 ‡ (1 ꢀ f0)C022 ‡ f1C111 ‡ (1 ꢀ f1)C211,
(5)
where f0C102 is the withdrawal by date t ˆ 0 type 1 depositors at date t ˆ 2, and
so on. Here the aggregate quantities C102 etc., are determined by the contract
described in the Appendix but the fraction f0 is determined by the actions of
date t ˆ 0 depositors at date t ˆ 1 based on their information, ó. However,
because our model ends at date t ˆ 2 deposits made at date t ˆ 1 must be
withdrawn at date t ˆ 2 so that for modelling reasons we set f1 ˆ 1.
The total withdrawal at date t ˆ 1 is given by
12
ꢀ1
C1 ˆ f
C
‡ (1 ꢀ fꢀ1)C2ꢀ21 ‡ f0C101 ‡ (1 ꢀ f0)C201:
(6)
ꢀ1
2.2. The Modi®ed Contract Problem
Banks supply capital to borrowers of the types examined earlier. To ®nance
their loans, banks supply two-period deposit contracts of the type outlined. At
each date information about entrepreneurs' projects and banks' balance
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sheets is symmetrically distributed between entrepreneurs and banks. New
depositors know the structure of the bank's balance sheet, but deposit
contracts cannot be contingent upon the realisation of the bank's investments.
However, at each date some one-period old type 2 depositors receive a signal
about the performance of the bank's portfolio and can withdraw early. The
bank can renegotiate with borrowers at date t ˆ 1 but deposit contracts are
not renegotiable.
The following is a summary of the sequence of events:
Date t ˆ 0: There are a suf®ciently large number of banks in the economy,
that no bank ®nds itself in a unique position at this date. Hence, any given
bank is engaged in Bertrand competition with at least one other bank. A bank
has an initial position resulting from making loans in the past and taking in
deposits. We assume it has serviced current withdrawals by depositors and has
re®nanced all existing borrowers. Clearly there could be rationing at date
t ˆ 0 of date t ˆ ꢀ1 borrowers but at no loss we assume that this is not the
case. Hence, at this date the bank's initial portfolio is taken as given and will
generate a random income of R1 at t ˆ 1. Banks offer ®nance to new
entrepreneurs. New entrepreneurs apply for ®nance. At this date neither the
entrepreneur nor the banks know the entrepreneur's project type which is
made known to both parties at date t ˆ 1. Banks do, however, forecast the
proportions of each type that they ®nance and entrepreneurs estimate their
probability of being each type. The bank supplies I0 to the entrepreneur at
this date and stands ready to supply I1 at date t ˆ 1: The entrepreneur
promises to repay D2 at date t ˆ 2 if he can, but this amount may be
renegotiated at date t ˆ 1: However, as will be shown, there is the possibility of
banks rationing ®nance at date t ˆ 1 according to a rationing scheme, so at
date t ˆ 0 both banks and entrepreneurs must estimate the probability of each
type of entrepreneur being rationed and, moreover, in the event of a type
being rationed the proportion who are denied ®nance. The parties also
anticipate that the terms of any renegotiated ®nance will depend upon
whether there is rationing or not. To ®nance these loans the bank competes to
attract new two-period deposits, B0, with claims (C101, C021) and (C102, C022).
g
m
s
^
^
Date t ˆ 1: The value of R1 ˆ á0 D1 ‡ á0 p1 D1 is realised; where D1 is the
^
payment made by date t ˆ ꢀ1 type g borrowers and D1 is the payment made
by date t ˆ ꢀ1 type b borrowers. R1 plays an important role in our model and
depends upon the realisation of two types of variable. It depends upon the
values of the proportions á0g and á0b determined at date t ˆ 0 and the
s
^
realisation of p1 at date t ˆ 1. Note that the proportion of type b entrepre-
neurs who are successful at date t ˆ 1 is the only uncertainty about R1 actually
resolved at this date. Type 2 depositors born at date t ˆ 0 receive a signal, ó.
This signal includes (precise) information on the bank's lending policy at date
t ˆ 1: If ó is high, the bank's return at date t ˆ 2 is forecast to be relatively
s
s
^
^
high. ó also includes information used to re-estimate p2 as E1(p2jó); for this
to have any effect on behaviour, ó must include information leading to a
s
^
revision of the forecast of p2 made at date t ˆ 0. Depositors then submit their
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withdrawal requests and the bank accommodates requests to withdraw C1 or is
liquidated. One period old borrowers, initially ®nanced at date t ˆ 1, who ®nd
out that they are either type g or b, apply for further ®nance.
The bank may have a liquidity shortfall. It decides how many new deposits to
take in, B1, with claim C111 and may ration ®nance to type g and b entre-
preneurs. Renegotiation of loans takes place, the form of which depends upon
whether there is rationing or not. By assumption borrowers cannot switch to
another bank.
At this date, a liquidity shortfall can occur for one of two reasons:
Bad Luck. The outcome of R1 given á0g and á0b, is assumed to be affected
s
^
by the actual proportion of type b date t ˆ ꢀ1 borrowers, p1, who are
successful at date t ˆ 1, which can differ from the expected value. If at
date t ˆ 1 the return from the bank's earlier investments, R1, is low
because of `bad luck'; then even if C1 is as expected with no type 2
depositors withdrawing money early, f0 ˆ ð1, the bank may have a liquidity
problem:
12
ꢀ1
R1 , f
C
‡ (1 ꢀ fꢀ1)C2ꢀ21
:
(7)
ꢀ1
In particular, income from past investments, R1, will not pay off date t ˆ ꢀ1
depositors. The probability of this occurring is higher, the higher the variance
of R1. The bank cannot make the depositors return at date t ˆ 1 contingent
upon R1 and cannot renegotiate with depositors. Hence, income from new
deposits, B1, will ®rst be used to pay this shortfall.
A Run. Now consider the alternative possibility that the bank has a liquidity
problem at date t ˆ 1 because there is a run, that is some date t ˆ 0, type
2 depositors, receive poor signals, ó. They use ó to estimate (precisely) a
low value of á1g or a low value of p given by E1(p jó). Thus they forecast
s
s
^
^
2
2
a poor return on the bank's portfolio at date t ˆ 2. Then if they withdraw
12
early f0(ó) . ð1. We could have R1 > f
(1 ꢀ f0)C201 is high.
C
‡ (1 ꢀ fꢀ1)C2ꢀ21 but f0C101
‡
ꢀ1
ꢀ1
Note that the ®rst type of liquidity problem depends critically upon a high
proportion of type b entrepreneurs turning out to have done poorly at date
t ˆ 1: The second type of problem can have two sources in our model
(although one will do). It can arise because at date t ˆ 1 the bank is found to
have a low proportion of good borrowers, or because the proportion of type b
loans that are expected to perform well at date t ˆ 2 is low.
It is important to emphasise here that the problems of liquidity shortfalls at
date t ˆ 1 arise, in both cases, from realisations of the bank's portfolio
differing from expectations.
Date t ˆ 2: Income from projects is realised and shared between depositors,
the bank's owners and entrepreneurs. At this date the only uncertainty to be
s
^
resolved is the proportion of type b entrepreneurs who are successful p2.
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2.3. Equilibrium Analysis
We develop the model by assuming that banks suffer from liquidity shortfalls
at date t ˆ 1 and that the banks ration loans at this date. We ®rst show the
properties of an equilibrium with rationing at date t ˆ 1 and then show that
rationing can exist in equilibrium. The contracting problem is examined
recursively. Thus we begin at date t ˆ 1 and examine the sub-game at this
stage. Entrepreneurs' types are realised and those of types g and b apply for
further ®nance. If the bank is not subject to a liquidity shortfall, it grants all
loan requests, with type g paying D2 at date t ˆ 2 and type b renegotiating
^
their loan and promising to pay D2 at date t ˆ 2. In the absence of rationing at
date t ˆ 1, the bank's return at date t ˆ 2 is
g
b
s
^
^
Z2 ˆ max(á1 D2 ‡ á1 p2 D2 ꢀ C2,0)
(8a)
Now suppose that the bank is subject to a liquidity shortfall and rations loans.
Clearly loans to type b are of lower value to the bank than loans to type g
entrepreneurs, so that this type will all be denied ®nance before any type g are
rationed.
Rationing Scheme
Let 0 < ã1 < 1 and 0 < ä1 < 1 be, respectively, the proportions of loan requests
granted to type b and g entrepreneurs. Then if rationing occurs, ®rst 0 < ã1 , 1, then if
ã1 ˆ 0 and further rationing is necessary set 0 < ä1 , 1.
Since all type b entrepreneurs are identical and at date t ˆ 1 need D2 to be
^
cut to D2 (at a maximum) if they are to continue, there is no scope for a type b
^
to manipulate the rationing scheme by offering to pay more than D2. However,
this is not so for a type g: If this group is rationed, a type g entrepreneur will
obtain ®nance with probability ä1. In the absence of rationing he would pay D2
at date t ˆ 2 and get a return of Y 2g ꢀ D2, but with rationing he expects
ä1(Y g ꢀ D2). But then he could offer to pay D2 ‡ å to the bank and be better-
2
off if ꢀä1å ‡ Ää1(Y g ꢀ D2) . 0, where Ää1 is the change in the rationing
2
parameter. Moreover, the bank will accept the offer. There will be competition
between type g entrepreneurs for the available funds. The bank will then
appropriate all the type g entrepreneurs' surplus, with D2 rising to D9 ˆ
2
Y 2g ꢀ u. Thus, if 0 < ã1 , 1, and ä1 ˆ 1 the bank's return at date t ˆ 2 is
g
b
s
^
^
Z2 ˆ max(á1 D2 ‡ á1ã1 p2 D2 ꢀ C2, 0)
(8b)
and if ã1 ˆ 0 but 0 < ä1 , 1,
Z2 ˆ max(ág ä1 D9 ꢀ C2, 0):
(8c)
2
1
The bank's owners net return at date t ˆ 1 depends upon whether the bank
continues at this date. If it does, then
Z1 ˆ R1 ꢀ á1g ä1 I1 ꢀ á1bã1 I1 ‡ B1 ꢀ C1 > 0,
(9)
where is R1 the bank's net income at date t ˆ 1 from earlier investments; the
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next two terms are the cost of ®nancing investments; B1 is the value of new
deposits at date t ˆ 1 and C1 is withdrawals by depositors.
Note that at date t ˆ 1 the bank may have positive net earnings. This arises
when, after having ®nanced all deposit withdrawals and loan requests out of
income, so no new deposits need to be attracted, B1 ˆ 0. Hence, we have
Z1 . 0. In this case it is assumed that Z1 is paid out as a dividend. Since this is
only done if the bank is solvent it does not affect the probability of a run.
Moreover, although not shown formally here, the assumption that surplus
funds are paid out is justi®ed in this model, since when there is no rationing at
date t ˆ 1 retained earnings can only raise depositor welfare at the expense of
the bank's owners. On the other hand, if the bank continues and new deposits
are issued to ®nance loans, B1 . 0, then Z1 ˆ 0. So the bank does not attract
deposits to increases retained earnings or to pay a dividend.
If on the other hand the bank is liquidated at date t ˆ 1, then B1 ˆ 0 and
R1 is distributed to depositors, with each depositor receiving R1=(C1 ‡ C2)
with C111 set equal to zero.
The bank's problem at date t ˆ 1 is to choose ã1, ä1 and to attract B1 to
maximise
Z1 ‡ E1 Z2,
(10)
where expectations are taken at date t ˆ 1 and Z2 as given by (8a, b or c),
depending upon the regime.
Given the above behaviour in the subgames at date t ˆ 1, we now examine
the problem at date t ˆ 0. At this date, the entrepreneur estimates the
probability that no type b projects are rationed at date t ˆ 1 as y1; and of no
type g being rationed as z1. The entrepreneur also estimates that if type b are
rationed, a proportion ã1 ˆ E0ã1 will receive ®nance, and if type g are
rationed, a proportion ä1 ˆ E0ä1 will receive ®nance. Therefore, the prob-
ability when his project is type b that he is funded is q1b ˆ y1 ‡ (1 ꢀ y1)ã1, and
if he is type g, q g ˆ z1 ‡ (1 ꢀ z1)ä1. Then the entrepreneur's expected net
1
return at date t ˆ 0 is
p g [z1(Y g ꢀ D2) ‡ (1 ꢀ z1)ä1(Y g ꢀ D9)] ‡ p q (p Y ꢀ D2)
b
1
b
1
s
2
b
2
^
2
1
2
2
‡ p1g (1 ꢀ q1g )u ‡ p1b(1 ꢀ q1b)u ꢀ 2u, (11)
^
where D2 and D9, respectively, satisfy
1
s
b
^
p2(Y 2 ꢀ D2) ˆ u,
(12a)
(12b)
(13)
and
Y 2g ꢀ D9 ˆ u:
2
The bank's expected return at date t ˆ 0 is
max[ꢀI0 ‡ B0 ‡ E0(Z1 ‡ Z2)],
where B0 are deposits attracted at date t ˆ 0 and Z1 and Z2 are given in (9)
and (8).
Using (8) and (9) and evaluating (13) over the different subgames, we obtain
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g
1
b
b
s
2
^
^
E0(maxfá [z1 D2 ‡ (1 ꢀ z1)ä1 D9] ‡ á q p D2 ꢀ C2, 0g
2
1
1
‡(R1 ꢀ á1g q1g I1 ꢀ á1b qb I1 ꢀ C1 ‡ B1)) ‡ B0 ꢀ I0 > 0, (14)
1
where á1g and á satisfy the consistency conditions in (3c) and E0(p ) ˆ p .
b
1
s
2
s
2
^
Finally, Bertrand competition at date t ˆ 0 ensures that (14) is set equal to
zero.
Lemma 1. Given that there is rationing at date t ˆ 1, entrepreneur participation
at date t ˆ 0 requires that there is a positive probability that type g projects are not
rationed at date t ˆ 1.
Proof. Suppose to the contrary, then type b are never given credit at date
t ˆ 1, y1 ˆ 0 and ã1 ˆ 0: Also for type g, z1 ˆ 0 and D9 ˆ Y g ꢀ u with the
2
2
bank gaining the full surplus from continuation. It follows by substitution into
(11), that the entrepreneur cannot be compensated for his human capital and
will not participate at date t ˆ 0: Hence, there must be a positive probability
that the entrepreneur is an unrationed type g.
h
The following proposition is immediate:
Proposition 2. If there is a positive probability that type g are not rationed at
date t ˆ 1, and there is a suf®ciently high return to type g entrepreneurs when this
type is not rationed, then participation at date t ˆ 0 occurs. Moreover, competition
between banks at date t ˆ 0 ensures that the surplus from renegotiation at date
t ˆ 1 is captured by borrowers at date t ˆ 0:
Proof. Consider the second part of the proposition ®rst. If the entrepreneur
is ®nanced at date t ˆ 0, competition between banks means that (14) is set
equal to zero. As we have seen, if the borrower is type b he receives a return at
date t ˆ 1 equal to the opportunity cost of his human capital. If he is type g
and there is rationing, renegotiation again gives the entrepreneur a return
equal to the opportunity cost of his human capital. Inspection of (14) shows
that zero pro®t at date t ˆ 0 requires that z1 D2 takes the appropriate value. It
is through the adjustment of this value in (11) that the full bene®t of
renegotiation is passed back to entrepreneurs. Inspection of (11) then shows
that if z2 is suf®ciently high and D2 suf®ciently low, participation occurs at date
t ˆ 0.
h
This proposition illustrates an important property of our model: Borrowers
bear some of the costs of liquidity shortfalls, by either being rationed or in
renegotiating loans. Entrepreneurs are compensated by obtaining better
returns at date t ˆ 1 when they are type g and there is no rationing. They
are also compensated by a reduced probability of being rationed if they
are type g.
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2.4. Existence of Rationing
We have examined the model with rationing at date t ˆ 1: Our task now is to
show that there will be rationing and to understand the properties of the
model at date t ˆ 1 when rationing occurs.
If the bank does not have a liquidity problem at date t ˆ 1, then depositors
are paid C1 and all type g and b entrepreneurs are granted further ®nance.
However, this is not so if there is a liquidity shortfall. We therefore investigate
what happens if the bank suffers bad luck or is subject to a run triggered by a
low signal, ó. In particular, we investigate the determination of ã1 and ä1. For
ã1 or ä1 to be set less than one is inef®cient. However, because withdrawal of
deposits used to ®nance earlier investments may not be paid for by the income
of these investments, our case of bad luck, the bank may face a type of debt
overhang problem (see Myers (1977)). Also the bank may be constrained
because of a run.
Bad Luck
First suppose the bank faces a liquidity shortage because of bad luck.
However, we assume that bad luck does not precipitate a bank run. Consider
the possibility that the bank issues either subordinated or non-subordinated
deposits to ®nance loans. Then we have the following two propositions.
Proposition 3. Suppose that at date t ˆ 1, the bank has a liquidity shortfall
because of bad luck. If the bank could issue subordinated deposits. Then either all
loan applications from entrepreneurs of types g and b will be granted, or none will
be granted.
Proof. Issuing subordinated deposits of value B1 at date t ˆ 1 with a claim of
C111 to ®nance entrepreneurs of types g and b, can only raise the value of type
2 deposits made at date t ˆ 0. Hence, f0(ó) ˆ ð1. Thus, if the bank issues
deposits, the optimal deposit contract for date t ˆ 0 depositors is implemen-
ted. The bank will only grant further loans if there exists a subset of type g and
b entrepreneurs who, if ®nanced, raise the value of the bank given in (10). It
then follows that the bank has overcome the liquidity shortfall and all other
type g and b entrepreneurs should and will be ®nanced. Hence ã1 and ä1 are
both set equal to one. On the other hand, if there does not exist a subset of
type g and b entrepreneurs who, if ®nanced at date t ˆ 1, will add to bank
owners' wealth, then all loan applications will be denied. If the bank makes no
further loans at date t ˆ 1 it is wound up.
h
In reality banks do not issue subordinated deposits, re¯ecting this we
assume that they cannot discriminate between depositors (new and old) at
the time of withdrawal. Issuing new, non-subordinated deposits, at date
t ˆ 1 with claim C111 dilutes existing deposits.
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Proposition 4. Consider a bank with the same liquidity shortfall at date t ˆ 1
as in Proposition 3. Assume that, type g and b entrepreneurs apply for loans. Then
a bank which issues non-subordinated deposits may supply more or fewer loans than
a bank that issues subordinated deposits.
Proof. (a) Suppose that under the conditions of Proposition 3, if the bank
issues subordinated deposits at date T ˆ 1, then ã1 and ä1 are set equal to zero.
Now, suppose the bank was instead able to issue non-subordinated deposits,
which dilute the claims of date t ˆ 0 depositors. Then, if this dilution effect is
suf®ciently large and the incentive constraint for type 2 depositors born at date
t ˆ 0, given by (A4), is not violated, the bank can increase its owners' wealth,
given in (10), by ®nancing some type g and b entrepreneurs. This will be done
according to the rationing scheme.
(b) Now suppose that if the bank were able to issue subordinated deposits,
all type g and b entrepreneurs are ®nanced at date t ˆ 1, ã1 ˆ 1 and ä1 ˆ 1: If
instead the bank needs to obtain a substantial number of new non-subordi-
nated deposits to ®nance these loans, there will be a signi®cant degree of
deposit dilution. But given the signal, ó, dilution of depositors may mean that
the incentive constraint (A4) is violated for some date t ˆ 0, type 2 depositors.
This means that f0(ó) . ð1, which means, using (5) and (6), that C1 rises
relative to C2. To mitigate this effect, the bank may restrict deposit issue and
thereby not ®nance some type g and b entrepreneurs. This will be done
according to the rationing scheme.
h
Bank Run
Now suppose the bank suffers a liquidity problems because of a potential
run. Then the bank suffers an actual run if nothing is done. Then we have two
propositions which parallel the above:
Proposition 5. Suppose that at date t ˆ 1 the bank has a liquidity shortfall
because of the behaviour of depositors. Then if the bank can issue subordinated
deposits as in Proposition 3, either the bank will ®nance all type g and b
borrowers, or none. In the latter case there will be a full run and the bank is
liquidated.
Proof. This proposition is analogous to Proposition 3. The key difference is
that drawing in new deposits to ®nance type g and b entrepreneurs with
subordinated deposits has to induce date t ˆ 0 type 2 depositors not to
withdraw early, so that f0(ó) ˆ ð1: If re®nancing a subset of type g and b
entrepreneurs has this effect, then all such requests will be granted. But
otherwise none will be ®nanced.
h
Now consider what happens if the bank can only issue non-subordinated
deposits. Then we have the following proposition which can be contrasted
with Proposition 4.
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Proposition 6. Suppose the bank is faced with the same liquidity problem as in
Proposition 5. Types g and b entrepreneurs apply for continuation loans. Then, if
the bank can only issue non-subordinated deposits, it will supply the same amount
or fewer loans than a bank that issues subordinated deposits.
Proof. (a) Suppose that if the bank can issue subordinated deposits at date
t ˆ 1, ã1 and ä1 and are set equal to zero. Hence with no dilution of
depositors, the value of the continuation loans is insuf®cient to overcome the
liquidity shortfall. Thus, if the bank is to ®nance any continuation loans it must
dilute existing depositors claims. Issuing non-subordinated deposits does this
but worsens the violation of the incentive constraint for date t ˆ 0, type 2
depositors, so that f0(ó) . ð1. Hence ã1 and ä1 are set equal to zero here also.
(b) On the other hand, if the bank could issue subordinated deposits it sets
ã1 and ä1 equal to one. But, if instead, it could issue non-subordinated
deposits, and at date t ˆ 1 sets ã1 and ä1 equal to one, the dilution of
depositors may prevent the date t ˆ 0, type 2 depositors incentive constraint
from being satis®ed, so that f0(ó) . ð1: If this happens the bank extends fewer
loans than if it could issue subordinated deposits, with rationing according to
the rationing scheme.
h
In comparing the two types of liquidity shortfall we note that at date t ˆ 1 a
potential run occurs because depositors receive a signal of poor returns at date
t ˆ 2: This information will raise the return required by new depositors there-
by increasing the impact on bank lending relative to a shortfall due to bad
luck.
3. Management of Liquidity and Project Choice
In the above we have seen how the liquidity of an incompletely diversi®ed bank
can affect its ability to implement the ef®cient lending policy. Diversi®cation
of the bank's loan portfolio reduces the risk depositors are exposed to along
the lines examined by Diamond (1984). In our model, diversi®cation will
reduce the variance of R1 and, other things equal, reduces the expected value
of the liquidity shortfall.
In developing the model, we did not allow banks to hold safe assets to insure
against liquidity shortfalls. Suppose that at date t ˆ 0, at the expense of
making productive investments, the bank can invest in one-period, safe govern-
ment bonds. The bene®t of holding safe bonds is that the expected value of
the liquidity shortfall at date t ˆ 1 is reduced. Hence, there is a trade-off for
the bank, with the bank limiting its holding of bonds and hence its insurance
against liquidity shortfalls. The question arises as to whether ®rms have an
incentive to insure against not being ®nanced by banks? To examine this
question we consider a simple example.
Assume that banks do not hold bonds. Suppose that at date t ˆ 0, entrepre-
neurs have a simple project choice. They can either choose a project of the
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type already examined, or a project which yields a return at date t ˆ 1 as well
as date t ˆ 2: This project, indexed by an asterisk, requires some initial
Ã
Ã
investment of I0 at date t ˆ 0 and a further investment of I1 at date t ˆ 1,
which are the same as for the alternative choice. However, for simplicity the
Ã
Ã
project yields a deterministic income stream of Y 1 at date t ˆ 1 and Y 2 at date
t ˆ 2. We assume that
Ã
Ã
Ã
Ã
Ã
Y 1 ‡ Y 2 > I0 ‡ I1 ‡ 2u,
(15a)
(15b)
(15c)
(15d)
Ã
Y 2 . I 1 ‡ u,
Ã
Ã
Y 1 ˆ I 1 ,
Ã
L1 ˆ 0:
This project is positive net present value at date t ˆ 0 and covers the
entrepreneur's opportunity cost (it is actually analogous to riskless productive
storage). The project generates enough income at date t ˆ 1 to ensure that
Ã
the entrepreneur can ®nance I1 at date t ˆ 1, without a further loan from the
bank. The crucial characteristic of this project for our analysis is not that it is
deterministic but that it is self ®nancing at date t ˆ 1: The bank will be
Ã
Ã
compensated with a payment of D2 at date t ˆ 2. Because L1 ˆ 0, it is in
neither party's interest to liquidate the project at date t ˆ 1: Since the
entrepreneur borrows I0 at date t ˆ 0, but will not borrow anymore at date
Ã
Ã
t ˆ 1, D2 ˆ I0 .
Assume that the uncertain project is better than the deterministic project in
the sense that the quantity in (4) exceeds the net return in (15a) given by
Ã
Ã
Ã
Ã
Y 1 ‡ Y 2 ꢀ I 0 ꢀ I1 ꢀ 2u > 0:
(16)
In the ®rst-best, all entrepreneurs are better-off doing the uncertain project.
Now consider what happens in our second-best environment to entrepreneurs
who undertake this project. In this environment, entrepreneurs of types g and
b at date t ˆ 1, receive ®nance for I1 with respective probabilities of q1g and qb.
1
Entrepreneurs' expected net return at date t ˆ 0 if they do the uncertain
project is given by (11). This can be compared to the expected return from the
deterministic project given by
Ã
Ã
Y 2 ꢀ D2 ꢀ 2u:
(17)
Now consider the deterministic project from the perspective of banks. At date
t ˆ 0, banks can ®nance entrepreneurs, who choose to undertake either the
uncertain or the deterministic project. In ®nancing some deterministic pro-
jects, there will be a reduction in both the expected return and risk of the
average project ®nanced. However, as a greater proportion of the projects that
the bank ®nances are deterministic, the variance of R1 declines and the
probability of liquidity shortfalls (due to differences between actual and
expected returns at date t ˆ 1) declines. By a similar argument, the greater
the proportion of deterministic projects ®nanced, the lower the impact at date
t ˆ 1 of differences between the actual and expected proportion of type b
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entrepreneurs on expected returns at date t ˆ 2 and hence the lower the
probability of a run. Then given a reduced proportion of entrepreneurs who
will demand continuation loans at date t ˆ 1, the values of q1g and q1b will be
higher.
We can now examine equilibrium when entrepreneurs have the above
project choice. Assume that banks do not hold bonds. Assume also that when
the bank ®nances no deterministic (or more generally, lower risk) projects,
there is a positive probability of a liquidity shortfall at date t ˆ 1. If the
expected cost of this liquidity shortfall is suf®ciently large, all entrepreneurs
would prefer to undertake the deterministic project if it were available. How-
ever, if on the other hand there was a zero probability of a liquidity short fall,
so there is no probability of rationing, the risky project dominates. Then by
assumption, when no deterministic projects are ®nanced, entrepreneurs' net
return from the uncertain project, as given in (11), will be less than would be
obtained from the deterministic project, as given in (17). Now if the determi-
nistic project is available, entrepreneurs will want to do it with the ®nancial
contract described. Banks will offer to supply this contract on competitive
terms. However, the greater the proportion of loans to ®nance deterministic
projects, the higher the values of q1g and q1b in (11). A bank will ®nance up to
the point that its portfolios of borrowers at date t ˆ 0 are indifferent between
the two types of project, so that (11) and (17) are equal. To summarise,
suppose that when no deterministic projects are ®nanced the deterministic
project is preferred to the risky project. However, at some given proportion of
deterministic projects ®nanced, the risky project dominates. Then we have the
following proposition:
Proposition 7. If the deterministic project is available, entrepreneurs will switch
to this project, thereby relaxing the rationing of risky project. In equilibrium the
probability of rationing with the risky project is determined such that entrepreneurs
are indifferent between the two projects.
It should be apparent that the higher the value of the deterministic project,
the greater the proportion of entrepreneurs who will undertake it and hence
the higher the values of q1g and q1b in (11), so there will be a lower probability
of rationing of entrepreneurs who undertake the risky project. Of course, if
even with no rationing the deterministic project dominates, no risky projects
would be ®nanced.
In establishing this proposition we assumed that the bank does not hold
bonds. But since our entrepreneurs have access to a riskless investment there is
no reason for the bank to hold the alternative riskless asset.
It should be apparent that the crucial feature of the deterministic project in
determining the above result is not that it is deterministic but that it is self
®nancing at date t ˆ 1. This insulates an entrepreneur who does this type of
project from the impact of rationing at the intermediate date. The determinis-
tic feature is not necessary for our result, what is needed is that this alternative
project has less risky intermediate cash ¯ows such that the probability of having
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to turn to the bank for additional ®nance at the intermediate date is reduced.
Hence, our result illustrates a trade-off in investment policy, that arises
naturally out of the liquidity shortfall, between the intrinsic value of the project
and its probability of obtaining intermediate ®nance.
The above result has some similarities to the ®ndings of Von Thadden
(1995). He argues that a commitment not to re®nance projects may be an
optimal screening device for creditors facing an adverse selection problem,
even though it can induce short-termism on the part of good entrepreneurs.
He shows that this problem is mitigated by ®nancial intermediation and in
particular by economies of scale in monitoring projects. Dewatripont and
Maskin (1995) on the other hand, argue in a similar set up to Von Thadden
that creditor illiquidity may be used as a commitment mechanism not to
re®nance poor projects but this may put too high a premium on short-term
returns. The present paper on the other hand sees the liquidity problem as
arising endogenously through the nature of ®nancial intermediation and in
particular the deposit contract. The bias towards short-term returns is then an
optimal response to this problem.
4. Conclusion
The paper has shown that banks ®nanced with deposit contracts will pass up
®nancing positive net present value investments, so there will be credit
rationing. Banks are ®nanced by an overlapping sequence of deposit contracts,
with, as is realistic, deposits returns not being made completely contingent on
the performance of the loans they are used to ®nance. The rationing arises for
two reasons: the bank can suffer low income, leading to a debt overhang
problem in ®nancing positive net present value continuation loans; a similar
problem arises if depositors demand for immediate liquidity is higher than
normal. The extent of credit rationing depends upon whether new deposits
can be issued to ®nance the liquidity shortfall and new loan requests. The
problem of rationing for a given liquidity shortfall is greater in the latter case
because new deposits are non-subordinated and therefore dilute existing
depositors, so their issue is restricted by the need to satisfy the incentive
constraint of these depositors.
It was shown that if the bank rations credit, it will ®rst cut ®nance to
borrowers who need to renegotiate loans in order to continue. Then if further
rationing is necessary, it will renegotiate with good borrowers who will `bribe'
the bank to ®nance them. We showed that in the event that there is no
rationing the bank improves loan terms to borrowers who do not need to
renegotiate. But when the bank has done poorly, although some good
borrowers will be denied credit, those given credit insure the bank through
higher loans repayments. Finally we showed that the existence of the above
type of rationing will cause some borrowers to switch to projects of lower net
present value because they have a lower (in our example, zero) probability of
being rationed at the intermediate date. This will take place up to the point
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that borrowers are indifferent between the different types of project, with the
probability of rationing serving as the equilibrating mechanism.
Our model involved competition between banks at date t ˆ 0 but has no
competition at date t ˆ 1. This meant that we did not need to specify whether
the shocks to the bank's returns at date t ˆ 1 were idiosyncratic or systemic.
However, if one bank is unlucky and has a poor draw of returns at date t ˆ 1,
others may be lucky. It follows that an interbank market (as in Bhattacharya
and Gale (1987)) will potentially improve matters. Then banks agree at date
t ˆ 0 that, in the light of asymmetric shocks at date t ˆ 1, there will be
liquidity transfers from lucky to unlucky banks. However, like any insurance
mechanism, its ef®ciency is limited if the bank's returns are unobserved by
other banks so the banks' can make false claims, and transfers are thereby
limited by the need to satisfy a further incentive compatibility constraint. Also,
it has the problem of creating a moral hazard, leading to greater risk taking by
banks.
Finally, in the paper we have not considered the roles of deposit insurance
and of a lender of last resort. These institutions will certainly have an effect on
the impact of liquidity shortages (see Rochet and Tirole (1996)). However,
their consideration shifts emphasis away from the central role of liquidity,
since the problem then becomes one of risk shifting between the banks and
the insurer or lender of last resort.
London School of Economics
Date of receipt of ®rst submission: February 1996
Date of receipt of ®nal typescript: May 1999
Appendix
Banks ®nance their loans by supplying deposit contracts. Our model of depositor
behaviour is based upon Jacklin and Bhattacharya (1988). First consider the behaviour
of a typical depositor, indexed by k. Each depositor is born at a given date, and then
has a three date horizon (0,1,2). He has initial wealth at time t ˆ 0 of unity. He desires
to consume at dates t ˆ 1 and t ˆ 2. Depositors are either type 1 (early consumers), or
type 2 (late consumers). At date t ˆ 1, with probability ðk(k ˆ 1, 2) their preferences
are:
U (ck1) ‡ rk U (ck2),
(A1)
where rk is the discount rate with r1 , r2 indicating that agents of type 1 discount
future consumption more heavily, and (ck1, ck2) is the consumption of agent k at dates
t ˆ 1 and t ˆ 2. We assume the utility function U is twice differentiable with
U 9(0) ˆ 1, U 9(c) . 0, U 0(c) , 0.
Individuals ®nd out their type at date t ˆ 1. However, this information is not publicly
observable, so that there remains asymmetric information. Hence, at date t ˆ 1 new
opportunities to trade may arise, since the information of the agents has increased.
Depositors could keep their wealth in a safe asset, however, higher returns are
available through ®nancing (indirectly) productive investment. The contract offered
by banks to individuals is a deposit contract. Agents who deposit one unit at date t ˆ 0,
are entitled to make a withdrawal at date t ˆ 1 and a second at date t ˆ 2: The amount
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that a depositor will be allowed to withdraw is contingent upon the number of
withdrawals that the bank has already serviced (sequential service constraint). This
constraint has been motivated by Wallace (1988) by the assumption that depositors are
suf®ciently isolated from each other at the intermediate date so that their early
withdrawal demands must be accommodated on a ®rst-come ®rst-served basis. This
means that banks cannot `cumulate' withdrawal requests and make payment contin-
gent on the total (Wallace 1988, p. 3). This has the following important implication: A
bank in facing a withdrawal demand from depositors at date t ˆ 1 is obliged to satisfy
this demand even if it has to liquidate investments inef®ciently (or forego making
ef®cient loans) and the bank may then be unable to ful®l its obligations to depositors
at date t ˆ 2.
Consider a bank which has a given portfolio of borrowers and which has supplied
deposit contracts to a large number of depositors. Let x1 and x2, respectively, be the
(uncertain) income of the bank at dates 1 and 2 per unit deposited. This income
depends upon the bank's investment policy and determines the maximum amount
payable on the average withdrawal at each date. In what follows the investment policy
of the bank is taken as given, though it has to be consistent with attracting deposits.
Then the optimal choice problem for a deposit contract, in the absence of interim
information, is solved by the vector of functions (c11, c12, c21, c22) and realisations x1
and x2. For an agent choosing the deposit scheme (c11, c12), the return will depend on
the fraction, f , of agents who choose the contract, the remaining fraction (1 ꢀ f )
choose (c21, c22). Importantly, the bank cannot offer contracts that are contingent on
f or the shocks that determine f . The optimal deposit contract is then the solution to
the following problem:
2
P
max E0
ðk[U (ck1) ‡ rk U (ck2)],
(A2a)
kˆ1
subject to:
x1 > ð1c11 ‡ ð2c21
,
,
(A2b)
(A2c)
x2 > ð1c12 ‡ ð2c22
and U (ck1) ‡ rk U (ck2) > utility obtained from misrepresenting true type for
k ˆ 1, 2:
(A2d)
The expectation in (A2a) is taken with respect to depositors' common prior informa-
tion on the distributions of x1 and x2. The inequalities in (A2b) and (A2c) are the
bank's budget constraints for dates t ˆ 1 and t ˆ 2. The condition in (A2d) is the
incentive compatibility constraint, which ensures that type k agents choose type k
deposits, provided the bank pays the promised amounts. The solution to this problem
has the important property of some preference shock insurance, however, the extent
of this insurance is limited if the incentive constraint is binding. It has the property
that c11 . c12 and c21 , c22 but involves less consumption smoothing than when the
incentive constraint is relaxed. Finally, the solution to the above optimisation problem
is an equilibrium, since the bank is solvent, and each agent is better-off choosing the
type of contract that has been designed for it, so that: f ˆ ð1.
Following Jacklin and Bhattacharya, assume that some (say a proportion â) of type 2
depositors receive information, ó, at date 1 on the performance of the bank's
portfolio, including knowledge of x1, which is used to recalculate the distribution of
x2. Type 2 depositors use the information ó to form a posterior belief about x2 denoted
by prob (x2jó). These beliefs are assumed to be consistent with the prior beliefs so
that:
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(A3)
P
prob(x2) ˆ
prob(ó)prob(x2jó) for all x2:
ó
We also assume that the conditional distribution of x2 for low realisations of ó is ®rst-
order stochastically dominated by distributions corresponding to high realisations of
ó. The contract solving (A2) is not contingent upon the interim information, ó.
However, this information causes a revision at the interim date of the `incentive
constraint' for type 2 depositors to:
E[U (c21) ‡ r2U (c22)jó] . E[U (c11) ‡ r2U (c12)jó]:
(A4)
For some low enough realisations of ó the revision of the distribution of x2 leads to a
violation of this constraint. If this happens, type 2 agents will prefer the type 1
allocation and the ex-ante ef®cient allocation will be upset, f . ð1.
Glossary of Notation
The initial risky project
I0, I1: investment in the risky project at dates t ˆ 0 and t ˆ 1;
L1: liquidation value of the risky project at date t ˆ 1;
u: alternative return on the entrepreneur's human capital per period.
g: good project;
b: bad project;
Y 2g : return on a type g project at date t ˆ 2;
b: return on a type b project if successful at date t ˆ 2;
Y 2
p2s : probability that a bad project is successful at date t ˆ 2;
D2: contractual payment on a debt contract agreed at date t ˆ 0 to be paid at date
t ˆ 2;
^
D2: payment on debt contract at date t ˆ 2 after renegotiation with an uncon-
strained bank at date t ˆ 1;
D9: payment on a debt contract at date t ˆ 2 after successful renegotiation with a
2
constrained bank at date t ˆ 1;
s
^
p2: the actual proportion of type b entrepreneurs re®nanced at date t ˆ 1 who turn
out to be successful at date t ˆ 2;
á1g and á1b ˆ (1 ꢀ á1g ) denote the respective proportions of type g and b entrepre-
neurs realised at date t ˆ 1, who were ®nanced by the bank at date t ˆ 0;
q1b, q1g : respectively, the probabilities that the entrepreneur is funded at date t ˆ 1,
when his project is type b or type g.
The deterministic project (in the ®nal part of the paper only)
Ã
Ã
I0 , I1 : investment at dates t ˆ 0 and t ˆ 1 in the alternative deterministic project
available at date t ˆ 1;
Ã
Ã
Y 1 , Y 2 , respectively the income from the deterministic project at dates t ˆ 1 and
t ˆ 2;
Ã
L1 : liquidation value of the deterministic project;
Ã
D2 , contractual payment at date t ˆ 2 on a debt contract agreed at date t ˆ 0 to
®nance the deterministic project.
Banks
ð1, ð2: are respectively depositors date t ˆ 0 forecasts that they will be high time
preference, type 1, and low time preference, type 2, at date t ˆ 1;
f0: the actual proportions of date t ˆ 0 depositors who withdraw at date t ˆ 1;
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B0, B1: new deposits attracted at dates t ˆ 0 and t ˆ 1 respectively, per typical loan
made at date t ˆ 0;
C1, C2: aggregate deposit withdrawals at dates t ˆ 1, 2 per typical date t ˆ 0 loan;
C102: the withdrawal by date t ˆ 0 type 1 depositors at date t ˆ 2;
ó: interim information received by type 2 depositors at date t ˆ 1;
Z1, Z2: the bank's owners net return at dates t ˆ 1 and t ˆ 2;
ã1, ä1: respectively, the proportions of loan requests granted to type b and g
entrepreneurs by a bank at date t ˆ 1;
y1, z1, respectively the bank's estimate of no type b and of no type g being rationed
at date t ˆ 1.
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