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ISSUES IN FINANCIAL REGULATION
Working Paper Series
Optimal Contingent Bank Liquidation
Under Moral Hazard
Charles W. Calomiris, Charles M. Kahn and Stefan Krasa
ECONOM ICS R E S ^R C H qLIBRARY
Df f i ™ S U Y T 0 F M IN N E S O T A
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FEDERAL RESERVE BANK
OF CHICAGO
W P - 1991/13
Optimal Contingent Bank Liquidation
Under Moral Hazard
Charles W. Calomiris
Charles M. Kahn
Stefan Krasa*
April, 1991
Abstract
This paper uses models of asymmetric information and incentive
compatibility to explain two distinctive features of the arrangements
used by banks to obtain financing—demandable debt and the sequen
tial service constraint. The option of early withdrawal and contin
gent liquidation of the bank serves as a disciplinary device to keep the
banker’s portfolio choice in line with depositors’ preferences. The pos
sibility of default and the first-come, first-served rule give depositors
the incentive to monitor the banker.
econpmicsresearchubwry
■Northwestern University, University of Illinois, Urbana-Champaign and University of
“
Illinois at Urbana-Champaign, respectively.
Support from the Gam Institute of Finance and the Herbert V. Prochnow Educational
Foundation of the Graduate School of Banking, Madison, Wisconsin is gratefully acknowl
edged. We thank David Bizer, V.V. Chari, George Pennacchi and Jay Ritter for useful
comments.
1
Introduction
A fundamental challenge for corporate finance is to understand the institu
tional structure of financial intermediaries. This paper uses models of asym
metric information and incentive compatibility to investigate two distinctive
features of the arrangements used by banks to obtain financing. These two
special features are demandable debt and the “sequential service constraint”
by which repayment is allocated to those first in line when demand for re
payment exceeds available reserves. For centuries the vast majority of exter
nally financed investments have been funded by banks, for which demandable
debt instruments (bank notes and demand deposits) have been the principal
source of funds. Demandable debt warrants explanation because it appears
more costly than available alternative contracting structures. When com
bined with a sequential service constraint, demandable debt carries the risk
of bank runs, as depositors rush to be first in line to withdraw funds. Since
bank runs are apparently costly, it becomes important to ask if there are
circumstances in which the bank’s institutional structure and the resultant
runs play a constructive role.
In our view demandable debt serves a useful role as a disciplinary device
to keep a bank manager’s interests in line with those of depositors. When
a depositor withdraws his deposits he is in effect voting for the replacement
of the bank’s management. If enough depositors take this action the bank is
liquidated. In general, monitoring the activities of the bank management is
costly. If there are many depositors, each would normally have an incentive
to free ride on the monitoring efforts of the other depositors. The sequential
service constraint helps eliminate free riding and makes it incentive compat
ible for depositors to monitor. This works as follows: Since non-monitors
only base withdrawal decisions on observations of runs already in progress,
monitors have a better chance than non-monitors of being at the front of the
line. By rewarding those who are first in line when a liquidation actually oc
2
curs, the structure gives depositors an incentive to be diligent in monitoring.
By providing an incentive compatible arrangement that effectively monitors
the management’s actions, the institution allows banks to acquire funds at
lower costs than they would face under alternate arrangements.
Although our work is the first to focus on a justification for these two
particular features,1 the general literature on the structure of banks is vast.
Some authors have investigated the relative merits of debt and equity fi
nancing (for example Diamond [1984] and Pennacchi [1985]). Others have
investigated the “maturity-mismatch” problem, considering the advantages
and disadvantages of financing long-term loans with short-term instruments.
Cone [1983] shows that mismatch can lead to excessive risk of liquidation in
a world of full information. McCulloch [1981] and Kareken [1985] argue that
maturity mismatch is an artifact of regulation, but such a view is hard to
reconcile with the historical importance of demandable debt in a variety of
regulatory regimes. On the other hand, Gorton [1985] and Diamond [1989]
describe incentive advantages from maturity mismatch.
Pennacchi [1983] and Flannery [1990] also develop models which demon
strate the incentive advantages that arise from demandability. These papers
are actually part of the maturity mismatch literature, since for these authors
short term debt and demandable debt are equivalent. This equivalence ul
timately is due to the assumption that it is costless for monitors to observe
the investments made by the intermediary; so that the market for the bank’s
short term debt is perfectly informed. By issuing short-term debt, the firm
commits itself to behave properly, since improper behavior would raise the
cost of debt service in the future. In our account, the costliness of monitoring
means that the bank’s short term debt would trade in an imperfect market.
In our account, demandable debt is not equivalent to relying on a secondary
market for short term (non-demandable) debt, since such a market might
lrThe implications of (but not the rationales for) the sequential service constraint are
investigated by Diamond and Dybvig [1983].
3
not allow monitors to reap the rewards of their monitoring.
Finally, the issue of the use of financial structure to ensure socially de
sirable liquidation is also a theme which has engaged the interest of both
finance and economics researchers in recent years. In papers in this liter
ature, liquidation, while costly, has advantages over allowing management
to remain in control when times become sufficiently bad. Harris and Raviv
[1990] use this insight to devise models which predict debt-equity mixes: the
inability to pay off debt holders becomes a signal that the firm should be
liquidated. Kahn and Huberman [1988] and Hart and Moore [1989] use the
insight to understand the role of collateral. In the banking context, Calomiris
and Kahn [1989] show the use of demandable debt to forestall ex post fraud
by bank management. When such fraud is likely, it becomes desirable to re
move the managers from control. While studies of bank failures historically
assign a prominent place to fraud in the list of causes, historical experience
also suggests that banks have much latitude in choosing the type of projects
they will finance, and that under some circumstances, banks may opt for “ex
cessively risky” loan portfolios. Banks may choose to do so because under
debt contracting with limited liability, they retain marginal upside gains and
share downside losses (cf., Black, Miller and Posner (1979)). In this paper,
unlike the others described above, the emphasis is on ex ante moral hazard
by bank management in choice of portfolio.
We show that this contracting structure induces some depositors to invest
in monitoring the bank’s portfolio, and to choose whether to force bank liqui
dation contingent on their observation of the signal. Monitoring is incentivecompatible because of potential gains from being first in line when excess risk
taking is detected. The potential for monitoring causes banks to reduce their
propensity to cheat. The social costs of implementing the equilibrium (costs
of monitoring) are justified by the benefit of reducing the costs associated
with default. In essence, demandable debt provides a beneficial tradeoff of
ex ante monitoring for the costs of asset transfer in default.
4
For the threat of liquidation to be an effective disciplinary device, it
must be credible that depositors desire reorganization of banks which have
managed their portfolios badly.
In our model—as in many other models
of intermediation—a bank derives its value from diversification.
A badly
diversified bank is less valuable to its depositors and they therefore have
less incentive to hold it together. This means that contingent liquidation of
badly diversified firms is in the interest of depositors, which in turn means
that contracts allowing contingent liquidation will dominate other contracts
in our setting.
The next section describes the physical assumptions of the model. Section
three examines as a special case the situation of a one-depositor bank and
derives circumstances under which contingent liquidation, demandable debt
contracts will be equilibria. Section four considers a multi-depositor case
and shows that the first-come, first-serve rule mitigates against free riding.
The final section shows that demandable debt as described dominates other
simple contracts and briefly considers extensions to the model and links to
historical reserve banking.
2
T he M odel
We begin with two types of agents: entrepreneurs and investors. Investors
each own one unit of capital; entrepreneurs each own a project requiring S
units of capital. All investors are risk neutral and derive utility only from
final consumption.2 If an investor does not lend his unit of capital to an
entrepreneur, he obtains an alternative return u.
The value of the matured project i is a random variable x,. The value is
observable by the entrepreneur, but not verifiable by the public. We assume
2This essentially means that there are no liquidity traders. In section 5.2 we consider
what happens if there is a certain percentage of liquidity traders, i.e. some consumers who
derive utility from early consumption.
5
that in such a world direct investment takes the following form: In return for
the necessary capital the entrepreneur promises to pay the investors a total
amount C .3 If the borrower defaults on that promise he can be forced to turn
over an equal fraction of the matured project to each investor. Collecting on
a defaulted investment is a costly procedure; we assume that a cost of v is
incurred by each investor when the matured project is handed over.4 Suppose
the distribution of payoffs from the project is denoted by the function D(.)
and let / be the probability that the payoff is at least C. If there are z direct
investors in project i and we represent the net expected payoff to a direct
investor by 'ip(z), then
0 (z )
~~J m in fan
C } dD(xi) -
(1 - f ) i
In this situation we would expect some agents to become intermediaries
so as to economize on the costs of asset transfers in default. Rather than
making this decision endogenous we simply assume that there is a third group
of agents who operate “banks,” which act as specialized intermediaries and
have lower costs (in fact zero costs) of collecting when a defaulted project
is handed over.5 Banks make direct investments in entrepreneurs’ projects.
Contracts between banks and investors take a similar form.
In a simple
debt contract each depositor is promised a constant payment R , which we
call the “face value” of the contract. If the bank reneges on its promise, each
depositor receives a share of the bank’s assets. (These assets are payments the
3C is determined by the supply of projects. We assume projects are in perfectly elastic
supply at a fixed required return for the entrepreneur. This has the effect of fixing C
exogenously. We can therefore treat C as a constant for the rest of this paper.
4These assumptions are slightly simpler than Townsend (1979) but have the same effect.
5In Diamond (1984) the lower costs of banks stem from eliminating duplicate liquidation
costs. For our analysis the source of the lower costs is immaterial. All that is important
is that banks be able to achieve low risk portfolios at less cost than individuals could on
their own. It will also be clear to the reader that we can easily make the choice to become
a bank endogenous. This extension is outside the main focus of the paper and is therefore
omitted.
6
bank has received from its debtors plus the matured projects of entrepreneurs
who have defaulted.) Again we assume that each depositor incurs a cost v if
the bank defaults. Thus the net-expected value to a depositor of the simple
debt contract with face value R is given by
[ y —v,
otherwise,
where y is the total value of the bank’s holdings divided by the number of
depositors.
More general forms of contracts are also possible and will be discussed
below. But throughout this paper, the terms of any contract will only depend
on observables. In particular monetary payments cannot depend on the value
of defaulted projects. Also, for simplicity, we will only deal with contracts
which are anonymous: depositors’ treatment in the contract depends on their
actions in the contract, but not on their names.6
Thus far, our framework captures the situation of Diamond (1984) while
avoiding many of its complications. In other words, intermediaries in the
model we have described thus far play a role identical to Diamond’s dele
gated monitors. By processing the defaulted investments themselves, banks
economize on the costs of default. In such a framework simple debt contracts
are optimal. The following two additional assumptions are what make demandable debt useful in this environment: First, depositors can, at a cost,
monitor the banker’s portfolio choice. Second, liability is limited; there is no
non-pecuniary cost depositors can impose on the banker. Limited liability
creates the need for penalizing bad portfolio choices by stripping the banker
of his assets. Monitoring makes that possible.
Specifically, we assume that the banker has the option of picking one of
two portfolios of projects. He can either pick a well-diversified portfolio X
or an undiversified portfolio Y . In a well-diversified portfolio, all projects
6This restriction simplifies the analysis without changing our main points.
7
are independent; in an undiversified portfolio, they are correlated. While the
two portfolios give the bank the same expected returns, they have different
variances.
In the main model in section 4, we will focus on banks which
invest in a large number of entrepreneurs; thus a well-diversified portfolio
will converge to a riskless portfolio. In section 3 we will examine a simplified
version of the story, and there we will treat the investment X as literally
riskless. Since the banker has limited liability, he will prefer the undiversified
portfolio. Unfortunately, because depositors bear the costs of default, this
portfolio is less valuable to them and socially inefficient.
Banks maximize their expected profits. A bank takes deposits. Some of
the deposits are placed in one of the two types of portfolios; the remainder
we will call reserves. The level of reserves is publicly observable. However,
depositors cannot costlessly observe which type of portfolio the banker has
chosen.
Depositors can learn a bank’s portfolio choice by paying a cost
m (which can be thought of as effort). Only the depositor himself knows
whether he has paid the costs.7
Finally it is possible to have a bank liquidated before its projects mature.
Liquidation is defined as the removal of the banker from control of the bank’s
assets, and the distribution of those assets among depositors. In other words
liquidation means reversion to a direct-investment equilibrium. We let l be
the transactions cost (per depositor) associated with a decision to liquidate
the bank.8
7The effect of this assumption is to rule out contracts in which compensation is explicitly
tied to monitoring. This assumption seems natural to us; even if the monitor is observed
to engage in monitoring, the quality of the monitoring and the information received in the
process will typically be private. If monitors are to be rewarded they must receive their
rewards in an incentive-compatible fashion. The sequential-service constraint will turn out
to be an incentive-compatible way to reward monitors.
8An explicit transaction cost is only necessary in the single depositor case examined in
Section 3. In the multiple depositor case, consumers can also have the option of directly
investing in projects initially. However, in equilibrium banks will drive out direct investors,
as in Diamond (1984), Williamson (1986) or Krasa-Villamil (1990).
8
The timing of these actions is as follows:
1st Period
2nd Period
3rd Period
i
------------------------- 1
------------------------- 1
Bank takes deposits;
Depositors can monitor;
Projects mature;
Portfolio decision.
bank can be liquidated
entrepreneurs pay up
or default.
A bank’s charter specifies the payments to be made to each depositor if
the bank is liquidated, the payments to be made to each depositor if the
bank defaults, and the payments to be made to each depositor if neither
liquidation nor default occurs. The charter also specifies conditions for early
liquidation. Contracts cannot be written on the portfolio structure of the
bank or on monitoring effort, neither of which is directly observable.9 The
decision to liquidate a bank must be a function of observable variables. The
only possible variable is an announcement by a depositor as to which signal
he has observed.
In our structure the “announcement” will come in the
form of an early withdrawal from the bank. Contracts will also specify the
payment to a depositor who chooses early withdrawal. Since reserves are
assumed observable, the contract can simply specify the liquidation decision
as a function of the level of reserves remaining after early withdrawals.
3
T he Case of a Single D ep ositor
In this section we examine the special case in which the banker deals with only
one depositor. Clearly, the economic interpretation of the single depositor
case is limited. However, it allows us to simplify the exposition by focusing
first on the incentive problem of the banker. We will show that demandable
9This assumption is natural since the portfolio is observed by all depositors individually,
and also if we generalize our model where their observations are subject to random errors.
9
debt solves this incentive problem. In the general model of Section 4, demandable debt works essentially the same way. In addition, however we have
to introduce the sequential service constraint in order to solve the incentive
problem for the depositors, i.e., to prevent depositors from “free riding” on
other depositors’ monitoring.
In this section, we specialize the analysis to deal with a situation in which
the risky portfolio has a two-point distribution. If the project is successful,
the bank receives C ; the probability of success is / . For further reduction
in notational requirements we assume that if the project is unsuccessful, its
value is exactly equal to v so that if the banker defaults when the project
is not successful, the residual to the investor is entirely eaten up by the de
fault proceedings. In the simplified model of this section, the well-diversified
portfolio is literally riskless: it gives a payoff of
x = /C + ( i - / ) »
with certainty. We assume that the required expected return u is greater
than v\ otherwise there is no incentive problem. We assume it is less than the
payoff from the riskless project, otherwise there is no profitable investment.10
When a simple debt contract has a face value lying between v and X , the
banker’s payoff is X — R if the banker chooses the riskless investment. His
expected payoff is (C —R)f if he chooses the risky investment. Thus, as long
as
X - u < { C - u)f,
the simple debt contract always has a default risk, since the banker will find
it tempting to invest in the risky asset. In such a circumstance, of course, the
10If the total payoff of the bank’s portfolio were observable then it would be possible
to distinguish the bank’s portfolio choice in this example since the two portfolios have
different supports. More general examples could be developed in which the supports
are identical so that this possibility does not arise. The reader may prefer to consider
this example as the limiting situation in which the well diversified portfolio converges in
distribution to the riskless portfolio.
10
payment to the depositor in the contract will have to incorporate a default
premium; the depositor will demand a face value of u / f .
Finally, suppose the cost of liquidation / is less than (1 —f ) v , the expected
cost of default if the risky investment is chosen. Then simple contracts are
dominated by a contract that always mandates liquidation, with the banker
receiving X — u — l. Since the asset passes to the depositor, the banker has
no incentive to pick the risky portfolio if the liquidation technology is used.11
Nonetheless the liquidation technology has a cost associated with it as
well, namely the amount l. Are there more complicated arrangements which
would reduce the transactions cost? After all, if the banker could be con
vinced to take the riskless choice, then the simple debt contract could be
used without suffering any default costs (as long as R < X ) .
In fact, the
simple liquidation contract is dominated by a stochastic liquidation contract
in which the liquidation occurs with some probability less than one. (In such
a contract, liquidation is determined by an exogenous random event - e.g., a
coin toss - not by either the depositor or the banker.) If liquidation does not
occur the depositor is paid the value u, the maximum promised payment the
banker can credibly make. If liquidation does occur, the depositor is given
the entire value of the liquidated project. By maximizing the payment to the
depositor in the default states, the banker is able to minimize the probability
of liquidation, and hence expected liquidation costs. This contract is optimal
among the stochastic liquidation schemes. The probability of liquidation A
is adjusted so that the weighted average of these two payments is equal to
the required return. Thus,
Since the banker never defaults under this scheme, the loss in profits stems en
11 Liquidation essentially means that the contracts are reversed, i.e., the bank now holds
the simple debt contract. The moral hazard problem disappears since the bank does not
have access to the upside gains of the distribution any more.
11
tirely from the costs associated with liquidation. Since liquidation no longer
occurs with certainty, the associated costs are reduced from / to IX.
While stochastic liquidation is the best that can be achieved without
monitoring, monitoring will in general improve the contract. If monitoring
were costless, the optimal arrangement would be the following: The depositor
always monitors. He is given the choice of calling for liquidation of the bank.
Whether or not the bank is liquidated, the depositor is promised a constant
amount. The banker receives the remnant.
The behavior of each party under such a contract is as follows: Since
monitoring is costless, the depositor always monitors. If he observes a risky
investment he calls for liquidation, and receives his payment. If he observes
the riskless investment he does not call for liquidation. With a riskless in
vestment, the banker will never choose to default, and the depositor will
receive his payment with certainty. Since the residual is greater when the
bank is not liquidated, the banker prefers to choose the riskless strategy.12
Note therefore that in the case where monitoring is costless, the outcome is
first best: The non risky investment is always chosen and all liquidation and
default costs are avoided.
When monitoring is costly, the situation is much more complicated. In
general, the first best will no longer be feasible. Nonetheless as long as mon
itoring is not too expensive, the optimal contract always involves a positive
probability of monitoring, as the following theorem shows. The theorem is
proved by finding a demandable debt contract which induces the depositor
to monitor and thereby induces the banker to choose the less risky invest
ment with positive probability. The contract we find gives the depositor his
required return and gives the banker a greater return than he can achieve in
any contract without monitoring.
T h e o r e m 1.
For positive but sufficiently sm all costs o f monitoring, the
12This is the essence of the models of Pennacchi or Flannery.
12
optim al contract involves a positive probability o f m onitoring. In the optim al
contract, the bank picks the risky outcome with positive probability.
P r o o f. We have already described the optimal contract with zero monitor
ing. It is therefore sufficient to demonstrate that the optimal zero-monitoring
contract is dominated by some contract with positive levels of monitoring.
The contract we will examine works as follows: The depositor is given
the option of demanding “early withdrawal.” If he does not call for early
withdrawal, then the contract becomes a simple debt contract with face value
R . If he does call for early withdrawal, then the bank is liquidated, the
depositor receives the payoff R — /, and the residual goes to the banker.13
In this contract the banker chooses the risky investment with some prob
ability p strictly between 0 and 1 and the depositor always monitors the
banker’s actions. The depositor calls for early withdrawal when and only
when the banker has chosen the risky investment. We begin by deriving the
necessary and sufficient conditions for this to be equilibrium behavior un
der the contract. First, note that conditional on the depositor’s behavior,
the banker’s payoff is X — R , regardless of his choice of investment. Thus
randomizing between them is indeed a best response.
The depositor’s behavior yields him a payoff of (1 —p )R + p (R — l) — m .
Suppose instead that the depositor always called for early withdrawal. Then
his payoff would be R — l. On the other hand, if he chose never to call for
early withdrawal, his expected payoff would be (1 —p )R + p f R .
Thus the prescribed behavior dominates the alternatives if the following
inequalities hold:
(1 - p ) l > m
(3.1)
13The banker’s residual under liquidation could be implemented by having him hold a
portfolio which includes bank debt. This residual is necessary to implement the contract
when there is a single depositor. As long as there are multiple depositors, demandable
debt can be implemented in an incentive compatible fashion with the banker retaining
none of the profits of the bank in the case of liquidation. See the next section for an
example.
13
and
p(R- l - fR )> m
(3.2)
It can be readily verified that if the prescribed behavior dominates these two
alternatives, it dominates all other possible behavior for the depositor (for
example, randomly monitoring or randomly calling for liquidation). Thus for
the contract to induce the prescribed behavior, conditions (3.1) and (3.2) are
necessary and sufficient. If in addition
R — pi — m = u.
(3.3)
then the contract yields the depositor the competitive payoff u.
In any contract the expected payoff to the banker is the expected return
on the investment less the expected payoff of the depositor less any costs of
liquidation or default. Thus the banker’s expected profits in the contract we
describe are
X —u — pi,
since the liquidation costs are borne whenever the banker chooses to engage
in the risky investment. The banker’s expected profits in the optimal contract
with no monitoring is
X -u-XI,
where A was defined above. Thus there exists a contract with monitoring
which dominates the optimal contract without monitoring as long as there
exists a pair (R , p ) satisfying equations (3.1)—
(3.3) with p less than A. It can
be verified that as long as m is sufficiently small, such a pair always exists.
Thus the contract with zero monitoring is not optimal.
Once we know that the optimal contract involves positive monitoring,
it is easy to see that the banker must randomize. For if in equilibrium he
chose one portfolio with certainty, it would not be in the depositor’s interest
actually to pay the cost of monitoring. This proves the theorem.
14
In the contract described above, monitoring occurs with certainty. It may
be that the globally optimal contract for this problem entails monitoring less
than one hundred per cent of the time. However, if this is the case, it can
be shown that the optimal contract specifies that when the monitoring does
not occur, liquidation will not be called for; in other words, the simple debt
contract will be the default option.
In summary, the optimal arrangement in the one-depositor case has the
following characteristics: The depositor monitors the portfolio choice of the
banker. If the monitor observes a risky choice, he calls for liquidation of the
bank. If he does not observe a risky choice, then he leaves the banker in
control of the bank’s assets. The monitor’s incentives to monitor are main
tained by the following considerations: If liquidation occurs, the monitor
receives less than he would had he allowed the banker to continue in control,
provided default does not occur. If the banker engages in a risky portfolio
choice, default becomes more likely, and the monitor prefers to force the bank
to liquidate rather than bear the losses default would entail. The banker him
self is indifferent between the two choices given the monitor’s response, but
he must choose the risky portfolio occasionally, just to give the depositor an
incentive to monitor. For if the banker knew the depositor was not moni
toring, the banker would actually prefer to choose the risky portfolio all the
time, wrecking the contract.
Thus we have shown that monitoring with contingent liquidation is op
timal given that investment is taking place through a bank.
As long as
the costs of monitoring are sufficiently low and the depositor cannot achieve
riskless portfolios on his own, the optimal bank contract dominates direct
investment.
15
4
T he Equilibrium W ith M ultiple D ep osi
tors
In this section we extend the analysis to a multi-depositor problem. In the
multi-depositor problem the sequential service constraint will prevent the de
positors form free-riding on other depositors’ monitoring. There are two key
differences between the model of this section and the model of the previous
section. First, in this section the well-diversified portfolio is only asymptot
ically riskless. Second, we no longer need to assume an exogenous cost to
liquidation. The breakup of the diversified portfolio into parcels for each de
positor in liquidation entails a cost which can take the place of the explicitly
described parameter /.
In the multi-depositor problem, the contract induces a game among the
depositors and the banker. An optimal contract and associated equilibrium
of the game must maximize the profit of the bank subject to the expected
utility of depositors being greater than the reservation level u. We propose a
demandable debt contract in which bank liquidation occurs when at least one
monitor observes that the bank has cheated. We will demonstrate that this
contract fulfills desirable properties in aligning the incentives of the banker
and all the depositors. In section 5 we show that this contract dominates
other simple contracts and dominates direct investment. It should be noted
that making the number of projects as large as possible is always desirable in
the contracts since it allows reduction in default costs through diversification.
Our strategy in this section is to treat the number of projects the banker can
manage as a technologically-determined constraint, and derive results as the
number of projects s becomes large.
In this section we proceed as follows: In 4.1 the game is formally speci
fied; 4.2 describes the demandable debt contract and the sequential service
constraint; in 4.3 we compute the payoffs to the bank and to the investors
for their choice of strategies; in 4.4 we finally show that demandable-debt is
16
incentive compatible (i.e, is a Nash equilibrium) for depositors and the bank.
All proofs for the sections 4 and 5 can be found in the appendix.
4.1
D escrip tion o f th e G am e
In the induced games we consider symmetric equilibria in mixed strategies.
The game works as follows. The bank offers deposit contracts to depositors
in period 1 and chooses a probability distribution over all feasible portfo
lio choices. Since the level of reserves is publicly observable, this probability
distribution can be simply described by the probability pq of choosing a port
folio of type q = X , Y . In what follows we will often denote the probability
of cheating (investing in the undiversified portfolio Y ) by p. In equilibrium
the bank will choose p to maximize expected profits.
Every depositor can choose a probability 7 of monitoring the bank. As
r
before, a monitoring depositor observes the banker’s portfolio choice with
out error. When the depositor monitors, his decision to withdraw will be a
function of which portfolio he observes. A depositor who does not monitor
can also randomize over withdrawing or not withdrawing. Every depositor
will choose withdrawing and monitoring strategies to maximize his expected
payoff, given the deposit contracts the bank offers, the bank’s portfolio choice
strategy p, and the monitoring-withdrawing strategy for the remaining de
positors. For a given monitoring-withdrawing strategy for all the depositors
other than depositor 0, let q{We\q) be the probability that We of the others
attempt to withdraw their deposits.14
4.2
D escrip tion o f th e D em andable D eb t C ontract
The demandable debt contract works as follows: depositors are given the op
tion of early withdrawal, with withdrawals backed by reserves. An agent who
14If every depositor monitors with the same probability and no depositor attempts to
withdraw without monitoring, then this is clearly a Binomial distribution (cf. Lemma 4).
17
withdraws early receives the fixed payment T. If there are more withdrawal
requests than reserve's, then only the first to request withdrawal receive early
payment. This is the “sequential service constraint,” or “first-come, firstserved rule.” If then' are more reserves than early withdrawal requests, then
other depositors, having observed early withdrawal can also be paid T as long
as there are reserves left. If reserves are depleted, then the bank is liquidated
and remaining depositors seize the remaining assets. Otherwise, the bank is
not liquidated, and remaining depositors have a simple debt contract with
payment R. Thus the terms of a demandable debt contract are described
by indicating the pair (71 R) and by specifying the number of initial depos
,
itors and the number of entrepreneurs the bank finances with its deposits.
(Reserves are the difference between deposits and loans.)
We describe an equilibrium in which monitors start a run on the bank
when they observe the banker has cheated. Once non-monitors observe re
serve depletion, they too will attempt to withdraw funds (we refer to late
queuing by non-monitors as “follow-up” withdrawals).15 Thus observing the
risky investment will always lead to full reserve depletion and bank liquida
tion, even if the number of monitors is small relative to the level of reserves.16
Suppose the bank receives deposits from zs depositors (each of whom
15These “follow-up” withdrawals are a non-essential feature of the model. One could,
equivalently, liquidate the bank immediately after the initial round of withdrawals. Under
risk neutrality, these two rules have the same expected payoffs to each depositor, and
therefore, under risk neutrality they are equally desirable. Under risk aversion, it would
be better to liquidate the bank before allowing follow-up withdrawals by non-monitors,
because follow-up withdrawals increase the riskiness of each depositors returns for the
same level of expected payoff. We allow follow-up withdrawals for algebraic simplicity,
and without loss of generality (that is, the desirability of demandable debt in no way
depends on this assumption).
The essential feature is that monitors have a better chance than non-monitors of being
first in line contingent on observation of Y . A non-monitor could of course choose always
choose to run—and thereby be first in line regardless of the banker’s portfolio choice—but
this strategy is dominated by other behavior.
16Again, we could equivalently and without loss of generality make bank liquidation
depend on the bank’s receiving one or more withdrawal requests.
18
provides one dollar of deposits) and invests
Ss
dollars in s projects; the
remnant of the bank’s deposits is held as reserves r. Thus,
Let
Fq
where
type
r
=
—S )s .
(z
denote the distribution of the value of the bank’s portfolio of projects
q = X ,Y
Y
determines whether the bank has invested in type
X
or in
projects.17
If the bank is liquidated, we revert to a direct investment equilibrium.
Each remaining depositor will receive a share of the liquidated portfolio and
will have to bear any transactions costs for defaults that result. Under the
circumstances transactions costs will be minimized if duplication is avoided—
that is, if liquidation is arranged so that each holder concentrates his holdings
in a single project rather than dispersing them. If
zs
depositors remain at
liquidation, then the net payoff to each is V>(z). Note therefore that the net
payoff in liquidation does not depend on which loan portfolio the banker
chose.18
If the bank is not liquidated the contract becomes a simple debt contract
with face value
x
R.
Since the per capita value of the bank is (a; + r ) / z s where
is the value of the loan portfolio, we conclude that if the banker chooses
portfolio
q,
a simple demandable debt contract with face value
R
gives a
depositor a net payoff of
f R ( ^ ^ ) d F q(x).
J
Let
£ (q )
ZS
denote the bank’s profit from a simple debt contract when the bank
has chosen portfolio
q.
Then
£(<?) = / max{x + r — z s R , 0}
d F q( x ) .
The first thing to note is that a under a simple debt contract the banker
always has an incentive to cheat.
17F x is the distribution of the random variable £V=1 min{C\;Ej})> and F Y is the dis
tribution of s minfC, £,}).
18Recall that individual projects all have the same distribution; once the bank is broken
up, benefits of diversification are lost.
19
Lemma 1. As s increases,
t w
f
t(Y)^
w here
0 < £
<
1
a s lo n g a s th e f a c e v a l u e R l i e s w i t h i n t h e s u p p o r t o f t h e
p e r - c a p i t a v a l u e o f t h e ban k.
Since a simple debt contract in which
R
does not lie between the maxi
mum and the minimum per-capita values of the bank is nor feasible (either it
would give negative profits to the bank, or would not give adequate returns
to the depositors, we get the following Corollary.
C orollary 1.
In a n y fe a s ib le s im p le d eb t c o n tr a c t th e b a n k e r p r e f e r s to in v e s t
in t h e u n d i v e r s i f i e d p o r t f o l i o .
4.3
C om p u tation o f th e Payoffs o f th e G am e
We now turn to computing the payoffs of the agents in the game given the
demandable debt contract we have described. First we derive the expected
payoff for a depositor
6.
We begin by calculating the payoff conditional on
(a) the depositor’s choice of attempting to withdraw or not; (b) the bank’s
portfolio; and (c) the other depositors’ strategies.
The depositor’s payoff
depends on other depositors’ strategies only through Wg, the number of other
depositors who attempt to withdraw in the first round. Call C
,w{q,Wg) the
payoff to a depositor who attempts to withdraw when Wg other depositors
are also attempting to withdraw. Call
£„(< Wg)
?,
the payoff to a depositor
who does not attempt to withdraw when Wg other depositors attempt to
withdraw.
First we calculate
£w .
If the depositor attempts to withdraw the bank is
liquidated. If the bank has r dollars in reserves, then it can only pay
r/T
of
its depositors with early withdrawals.19 All remaining depositors receive an
19We ignore integer problems. They can be accounted for without substantive change
but at the cost of a large increase in notational requirements.
20
equal share of the liquidated bank’s assets.20
Assume that the bank randomly pays the first r / T of the depositors who
request withdrawal, and that monitoring depositors who request withdrawal
in the first round are equally likely to be included in the first r / T requests.
If the depositor attempts to withdraw and W8 -f 1
<
r / T then he receives T
with probability one. Otherwise he receives T with probability
. After
the reserves are exhausted, z s — ( r / T ) depositors remain. Each receives an
equal share of the liquidated firm, for a net payoff of xp(z
— (r / T s
)). Thus
( w(q, Wj?) is a weighted average of T and xp.
Next we calculate £n, the payoff to a depositor who initially leaves his
deposit in the bank when W$ other depositors are attempting to withdraw.
Suppose W 8 is positive. Then the depositor attempts a “follow-up” with
drawal. His payment £n is xp with certainty if W8 > r / T . If W8
< r/T
then
his payment £n is T with probability ( r / T —W$)/(z —W 8) and xp otherwise.
On the other hand suppose W8 equals zero—that is, no one else attempts
to withdraw. Then the depositor receives the payoff of a simple debt contract
with face value
R
as calculated in the previous section.21
Now we use £n and
to analyze the expected returns to depositors, given
the bank’s portfolio choice and the equilibrium distribution of strategies of
all other depositors. The equilibrium strategies for all other depositors imply
that W 8 = 0 with probability one if q =
X;
it has a binomial distribution in
the symmetric equilibrium if q — Y .
There are four possible actions over which depositors randomize, i.e. any
combination of monitoring/not monitoring, and withdrawing/not withdraw 20We could generalize the set of contracts considered by allowing the possibility that
only a portion of the reserves are returned to the depositors in liquidation. But it should
be clear that it is always optimal to pay off some of the depositors with reserves if possible
since this cuts down the total costs spent for the transfer of assets. Aside from this there
is no loss of generality in returning to depositors the entirety of any residual reserves
remaining as a result of integer problems.
21 Note therefore that only in the case of W g = 0 does q affect
21
ing. We now describe the payoffs for each action. First, we consider the case
where depositor 6 monitors. Let (3wq be the expected utility of withdraw
ing given that the monitor observes the investment q = X , Y , and let /3nq
be the expected utility of not withdrawing given that the monitor observes
investment q = X , Y . These are given by the following formulas:
& , , = / U q ^ B ) d r , ( W 0]q).
(4.1)
Png = [ ( n(q, W e) d ^W e ^q ).
(4.2)
Finally, consider the cases in which the depositor does not monitor.
Let
a w be the expected utility of the depositor if he withdraws, and let a n the
expected utility if he does not withdraw. a w and a n are given by
^ v
^ ^
PqPwq
q —X , Y
PqPnq*
q —X y Y
Let we(q) be the function that describes the decision of depositor 0 given
that he observes q. Thus, let wo(q) = 1 if he withdraws, and wg(q) = 0,
otherwise.
Depositor 9 then chooses an optimal probability tt of monitoring, a func
tion wo(q), and a probability p of withdrawal without monitoring. Conse
quently
7r,/9, w$(q) G arg max(l - 7r){paw + (1 - p )a n} - n m
(4.3)
n , p ,w e ( q )
T
^ ^ ^Pq {^oi^Pwq T (1
^o{q)) 0nq} ‘
q —X , Y
The right side of (4.3) is the utility of a depositor for given demandable
debt contract (R , T) and for given strategies of the banker and the other
depositors.
Next, we derive the decision problem of the bank— that is the choice
of p. Let W be the number of consumers who choose early withdrawal. The
22
distribution of W depends on all depositors’ decisions and on q. If W > 0
then the bank is liquidated, and the banker’s payoff is zero. If W = 0 then
the banker’s payoff is i{q). Then for given R and n the bank solves
p € arg n r n x p £ { Y ) P{ ( W = 0)} + (1 - ? )£ ( * ).
(4.4)
p
The right side of (4.4) is the profit of the bank given a contract and strategies
for the depositors.
4.4
T h e Equilibrium o f th e G am e
Our next task is to show that in the class of demandable-debt contracts
described above, there are contracts and associated strategies which solve
the incentive constraints (4.3) and (4.4) and in which the depositor’s utility
(right side of (4.3)) is at least his reservation value u.
L em m a 2. I f the number of projects s is sufficiently large and the cost o f
m onitoring m is sufficiently low, then there exists a demandable debt contract
(R , T ) with zs depositors in which the following behavior occurs in equilib
rium:
(1) Each depositor monitors with probability n where 0 < 7 < 1.
r
(2) A depositor attempts early withdrawal if and only if he m onitors and ob
serves that the bank has chosen the risky portfolio. (All non-m onitoring
depositors attempt follow up withdrawals if they observe any early with
drawal.)
(3) The banker chooses the risky portfolio with a probability p where 0 < p <
1
.
Next we show that the demandable debt contract exists whenever direct
investment is feasible.
23
L em m a 3. Let m be sufficiently small and s be sufficiently large. Whenever
direct investm ent is feasible then the demandable debt contract is also feasible
- that is, the set o f incentive compatible demandable debt contracts yielding
depositors their reservation values and yielding a positive profit fo r the bank
is non-em pty. Further, there exists an optimal contract in this set.
5
O ptim ality of D em andable D eb t
Having shown that the demandable debt contract can be implemented, it
remains to show that it dominates other natural mechanisms for financial
intermediation. The set of contracts we will examine can be summarized as
follows: Contracts either utilize monitoring or they do not. Contracts with
out monitoring either always specify bank liquidation in the second period,
or they never do. Monitoring contracts are also of two types: (a) the de
mandable debt contract, in which liquidation occurs when the bad signal is
observed— where the incentive-compatibility of demandable debt is ensured
by the availability of reserves—or (b) non-liquidation contingent contracts
that allow varying payoffs conditional on the signal observed in period 2. We
will show that demandable debt is superior to the three alternative contracts.
The simple liquidation contract is equivalent to direct investment: The
assets are split up and returned to the depositors. Financial intermediation
dominates direct investment provided that the costs m of monitoring are not
too high and the number s of projects is sufficiently large that the benefits
of diversification apply.
As we have seen, in a simple debt contract bankers will choose project
Y . Since the bank is no better diversified than in the direct investment this
outcome is again dominated by schemes which induce diversification as long
as monitoring costs are not too high.
Finally, consider the non-liquidation contract with varying payoffs in pe
riod 2. This contract will never be optimal because liquidation will always
24
be preferred to risking default, given that depositors are aware that the bank
has chosen the Y projects. Again, once banks are known to have cheated,
direct investment will be superior because it avoids duplication of verification
costs.
The result of all this is the following Theorem.
T h e o r e m 2. There exists rh > 0 and an integer s such that fo r a bank with
at least s projects and a m onitoring cost m < rh there exists an incentive
compatible demandable debt contract with zs depositors and corresponding
probabilities o f monitoring it, and probability p o f moral hazard o f the bank
which gives higher profits than any other simple contract. A s s — oo, in the
►
demandable debt contract, the costs o f bank default per depositor converge to
p (l — £) > 0, and the total costs o f m onitoring the bank’ portfolio converge
s
to —m log£ > 0.
It is important to note that the ability to maintain demandable debt
contracting presumes an efficient legal system, which enforces some (socially
beneficial) contracts, and not others. In particular, our equilibrium requires
that banks and monitors not be able to collude privately to the detriment of
other depositors. To do so without depleting observable reserves (and thereby
initiating bank liquidation) bankers would have to offer special contracts to
monitors in period 2 that promised additional payments in period 3. If such
payments were enforced by the legal system, our equilibrium clearly would
be unsustainable. Thus it will be efficient (and necessary for our equilibrium)
for such side-payment contracts to be disallowed. Clearly, the legal system is
designed to prevent such abuses—for example, through enforcement of debt
covenants that preclude subordination or the accumulation of further debt.22
22For further examples of the effects of side trades in banking see Bizer and DeMarzo (1990) and Kahn and Mookherjee (1989).
25
5.1
T h e R ole o f R eserves in Im p lem entin g D em and able D eb t
In our model, reserves are useful because they provide a means for rewarding
monitors for their efforts through the first-come first-served rule. Given that
the bad signal has been received, monitors are better off accepting a lower
payment from reserves in period 2, than holding on to their original higher
promise of payment in period 3. The actions of monitors have positive ex
ternalities because of the observability of reserve withdrawals, which allows
the social decision of bank liquidation to be made easily.
Next, we show that the bank only has to hold a constant amount of
reserves independent of the number of depositors.
L em m a 4.
For every e > 0 there exists a C such that the probability that
more than C depositors m onitor in equilibrium is smaller than e independent
o f the size o f the bank.
Consequently, the probability that the demand of
m onitors fo r reserves can be covered by the bank can be made arbitrarily
sm all with a fixed amount of reserves.
Lemma 4 and (8) now imply that the bank will hold a strictly positive level
of reserves as s — oo, since the costs per depositor approaches zero, whereas
*
the gain per depositor due to a decrease of p remains strictly positive. This
economizing on reserves means that the demandable debt contract dominates
a similar contract in which depositors withdraw when they do not monitor.
This has two disadvantages.
It means that solvent entrepreneurs will be
liquidated unless ,large stocks of reserves are held. With sufficiently large
stocks of reserves this new contract could mimic a demandable debt, however,
holding such a large level of reserves is very costly.
One could imagine alternative means for implementing the mechanism
of contingent bank liquidation that would reward monitors for truthfully re
porting observed bad signals.
For example, suppose that depositors were
26
offered the opportunity to exchange their contracts at the beginning of pe
riod 2 for a new contracts (limited in number, as in the case of reserves) that
offered lower returns in period 3, but subordinated other depositors’ claims
on the bank to these new promised payments. The legal system could make
a special exception for such recontracting (to avoid encouraging collusion
between banks and monitors) by making the enforcement of such contracts
depend on their public observability, or equivalently, on the condition that
the bank is placed into liquidation in period 2. This arrangement, like reserve
banking, rewards monitors without encouraging non-monitors to announce
bad outcomes. As in the case of reserve pre-payment, the monitor will only
choose to convert to lower-paying “preferred” debt when he observes the bad
state. In this alternative arrangement, after liquidation, the depositors in
the new direct-investment equilibrium would each owe the monitors a fixed
share of the amount of the secured debt (new promised payment) received
by monitors in period 2.
Given the availability of this alternative, why would banks use reserves
to implement contingent liquidation, given that reserve holding entails op
portunity costs of foregone productive investments? One reason might be
that banks have other reasons for holding reserves (i.e., because of liquidity
traders, which we discuss below), and thus can use reserves to “kill two birds
with one stone.” Another reason might be the prohibitively high transaction
cost of writing and enforcing new contracts between the monitors and the
(now liquidated) bank.
5.2
T h e T ransactability o f Bank C laim s
An important property of demandable debt claims historically has been their
transactability. Bank deposits and notes often out-competed other financial
claims as media of exchange.
Calomiris and Kahn (1989) argue that the
transactability of these claims does not depend in a physical sense on the
27
withdrawal of reserves on demand. Rather, transactability depends on the
shared information of agents engaging in exchange.
While we do not model an exchange process among agents in this pa
per, our results do suggest reasons why demandable-debt bank claims might
be more transactable. In the contingent-liquidation equilibrium, the bank’s
portfolio decision is revealed to monitors, and their decision whether to run
the bank is observable and fully revealing to non-monitors. This implies that
information regarding the value of the bank claims is effectively symmet
ric among depositors, both before and after the critical moment in which
monitors receive and act on the signal.
Harris and Raviv [1990] make a similar argument for firms. They argue
that the fact that a firm is able to pay interest on its debt provides valuable
information to the market about the firm’s quality. In the case of a bank,
the criterion is more stringent and the resultant information is an assurance
of even higher quality. Since a bank would be closed if even a small per
centage of its depositors lacked confidence, information about low quality
banks becomes public much more easily than in the case of firms with a nondemandable financial structure. Hence, the fact that a bank’s doors remain
open provides the public with a strong indication of the bank’s strength.
We would argue that this strong indication itself becomes the basis of the
liquidity of demandable debt holdings.
The extreme result of full information sharing follows from the fact that
the bank can only choose two types of portfolios. Even in more complicated
contexts, however, one could argue that monitors are unable to make fine
distinctions regarding the state of the bank’s portfolio, and thus the simple
run/no-run rule may be still optimal, and the extent of shared information
between monitors and non-monitors may still be high (even though neither
is as informed as the banker).
28
5.3
Early C onsum ption D em and , L iquidity Traders
It is straightforward to add liquidity traders to our model. Assume that a
proportion q of the zs depositors will require reserves to finance consump
tion in period 2. The incidence of this demand for reserves is determined
randomly, and there is no aggregate uncertainty. Now the sequence of events
is as follows: The bank invests. After the signal is received, either monitors
demand reserves or not, and simultaneously liquidity traders demand early
withdrawal. Again the bank randomizes over which depositors are served.
Liquidity traders who are not served by the bank directly can trade their
claim on third period income for reserves with monitors who have withdrawn
from the bank.
In equilibrium the bank will have to hold enough reserves to cover the
demand of liquidity traders in period 2. Lemma 4 now immediately implies
that the probability that the demand of a monitor is not fulfilled approaches
zero as s — oo. As in the model without liquidity traders the change in the
►
level of reserves reveals the signal monitors have observed. Now, however,
the costs of holding reserves to implement demandable debt is zero for all
sufficiently large s, since this source of demand is not the binding constraint
on reserves. However, j3y will have to be lower than in the model without
liquidity traders, because now the second period payment T is not only a
“reward” for monitors.
The assumption of no aggregate uncertainty is of course unrealistic. But,
we would argue that historical experience shows that banks dealt with aggre
gate uncertainty through arrangement on the level of banking systems rather
than at the level of individual banks. As much recent work on banking history
has emphasized (for example, Gorton (1985), Calomiris and Gorton (1991),
and Calomiris and Schweikart (1989)) historically banks used suspension of
convertibility to forestall economy-wide liquidation.
Historical studies of
ten view panics as resulting from readily observable aggregate shocks to the
29
banking system, and suspension of convertibility or other mutual insurance
relations as the banking system’s response.
Extending the analysis to a multi-bank environment would allow investi
gation of many aspects of banking panics propounded in historical studies.
For example, prior to the Panic of 1857, the high-risk railroad bond market
collapsed, resulting in the insolvency of many bond brokers with large out
standing bank debt. In such circumstances, depositors may have incentives
to withdraw funds from banks that are actually unaffected by the distur
bance, and such disintermediation imposes costs on the banking system as a
whole.
To generate suspensions through this sort of signal-extraction problem in
our framework, one would have to begin with a multi-bank environment. Add
to our model a random disturbance that uninformed depositors can observe
in aggregate at no cost, but because they have not invested in information
about individual bank portfolios, they cannot tell which of their banks is
most at risk (cf. Gorton (1989)). This seems a fruitful means for integrating
our model’s emphasis on the disciplinary rule of contingent liquidation of
individual banks with the use of bank suspension in response to economy
wide shocks.
6
A p p en d ix
P r o o f o f L em m a 1. From the definition of £ we get
£(X )
f(V )
/ max{x + r — zsR , 0} d F x (x)
/ max{x + r — z sR , 0} d F Y (x)
f m ax{- + r — zR, 0} d F x (x)
= /m a x { f + r - z R , 0 } d F Y ( x ) '
^
Since the projects in a type Y portfolio are completely correlated it follows
that the denominator is equal to / max{r + r — z R, 0} d D( x ) , where D is
the distribution of one project.
Further, by the law of large numbers it
30
follows that the numerator converges to m a x {£ (x ) + r — 2# , 0}, where the
expected value is taken with respect to the distribution of D. Since x
►
m ax{x + r — z R } is a strictly convex function somewhere in the support of
D it follows that
max { E ( x + r — z R, 0} < E ( m a x {i + r — z R, 0}.
(6.2)
(6.1) and (6.2) imply Lemma 1.
P r o o f o f L em m a 2. If all zs depositors follow this strategy then the bank
solves
p E arg max (1 - p ) £ ( X ) + p( 1 - 7r)2S^(V').
(6.3)
pe[o,i]
As in the single-depositor case, the existence of the demandable-debt equilib
rium requires randomization by the banker in choosing the project. Certainty
of both monitoring and low-risk project choice cannot be an equilibrium. For
randomization and cheating to occur in equilibrium, the banker must be in
different over the choice of p. From expression (6.3) this implies that
(1~ ' r
=
fn '
(6'4)
For given z let 7 be the solution of (6.4). From the previous lemma recall
r
that the right side of this equation is less than one so the solution lies strictly
between zero and one.
The behavior specified for the depositors satisfies the depositors’ incentive
conditions if
<*n = (1 - P)PnX +
P0wY
~
m
>
Ctw .
By the definition of a n we therefore get
p{PwY - Pny ) = rn,
(6.5)
(1 ~ P)(PnX ~ PwX) >
(6.6)
and
31
where f3wy and 0ny depend on the distribution of W, which is binomial with
parameters n and zs. If m is small, it is possible to choose p so that (6.5)
and (6.6) hold provided
PwY
>
PnY
and
fin X
fiw X •
As s becomes large, fnx —(3wx — R —T . As s becomes large, the probability
i
►
of W = 0 approaches £, so fiwy > /?„y provided that T is greater than V
’-23 It
is always possible to choose R and T so that these constraints are satisfied.
This proves Lemma 2.
P r o o f o f L e m m a 3. By assumption, if liquidation does not occur, the con
tract becomes a simple debt contract. Lemma 1 of Krasa and Villamil (1990)
now immediately implies that ( n, ( w, and £ are continuous functions of R.
Therefore (3wq, (3nq, and a w, a n are all continuous functions of (p,ir, R , T ) ,
because they are linear combinations of £, ( n and £w.
Let C be the set of all combinations ( p , i r , R, T ) for which intermediation
is feasible—that means (6.4), (6.5), and (6.6) are fulfilled, and the right side
of (4.3), which is the utility of a depositor, is at least u. We have to show
that C is compact. Clearly C is bounded. Both sides of (6.4) and (6.5) are
continuous.
Furthermore (6.6) and the right side of (4.3) are continuous.
Therefore C is closed. It now remains to prove that C is non-empty for all
sufficiently large s and for all sufficiently small m: Clearly, it is possible to
find (p , tt, R , T ) that fulfill (6.3), (6.5) and (6.6). It remains to prove that
the utility of depositors is at least u for these contracts and strategies. If m
is small then the probability of cheating by the bank has to be small because
of (6.5). This implies that the delegated monitoring costs will be small as
s —>oo. Since the costs of lack of diversification remain large, intermediation
23T greater than i also assures that all non-monitoring depositors attempt follow-up
p
withdrawals.
32
will dominate direct investing (cf. Diamond (1984)). Since u is the utility of
a direct investor this proves that C is non-empty.
We now choose a contract from the set C which maximizes the profit of
the bank. This is possible since, as a consequence of the continuity of £, the
profit of the bank (right side of (4.4)) is also continuous in (p, 7r, R). We are
therefore maximizing a continuous function on a compact set. This proves
the Lemma.
P r o o f o f L em m a 4.
The probability that k of z depositors monitor is
(j^7rk(l — 7r)z~k. Hence,
pc = T ,
k—
C
(fcKf1 - * ) '
t-k
(6.7)
is the probability that at least C depositors monitor. For every e > 0 we get
\tt — 11 <
Since ( l —
for all sufficiently large z because of (6.4). Hence, (6.7) implies
< 1, and since (j^ <
(6.8) implies that
OO
lim Pc < lim
C -.0 0
— C —oo
Ac—
G
(a + e)k
k\
(6.9)
Since the series at the right-hand side of the inequality (6.9) converges, it
follows the lim c_ 00 Pc = 0. This proves the Lemma.
33
R eferences:
D. B izer and P . D eMarzo (1990), “Sequential Banking,” mimeo, Northwestern
University.
F. B lack, M. Miller and R.A. P osner (1978), “An Approach to the Regula
tion of Bank Holding Companies,”
J o u rn a l o f B u sin ess
51, 379-412.
C. C alomiris and G. G orton (1991), “The Theory of Banking Panics: Models,
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