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Working Paper Series
Bank Runs and Investment Decisions
Revisited
WP 04-03
Huberto M. Ennis
Federal Reserve Bank of Richmond
Todd Keister
Centro de Investigación Económica
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Bank Runs and Investment Decisions Revisited *
Huberto M. Ennis
Research Department, Federal Reserve Bank of Richmond
Todd Keister
Centro de Investigación Económica, ITAM
Federal Reserve Bank of Richmond Working Paper No. 04-03
March 2004
Abstract
We examine how the possibility of a bank run affects the deposit contract offered and the
investment decisions made by a competitive bank. Cooper and Ross (1998) have shown
that when the probability of a run is small, the bank will offer a contract that admits a
bank-run equilibrium. We show that, in this case, the bank will chose to hold an amount of
liquid reserves exactly equal to what withdrawal demand will be if a run does not occur. In
other words, precautionary or “excess” liquidity will not be held. This result allows us to
determine how the possibility of a bank run affects the level of illiquid investment chosen
by a bank. We show that when the cost of liquidating investment early is high, the level of
investment is decreasing in the probability of a run. However, when liquidation costs are
moderate, the level of investment is actually increasing in the probability of a run.
JEL Classification Numbers: D84, E44, G21
Keywords: Bank runs; deposit contracts; bank reserves; sunspot equilibrium
________________________________________________
* We thank seminar participants at the Cornell-Penn State Macroeconomics Workshop, the Federal
Reserve Bank of Cleveland, ITAM, and the University of Guanajuato for useful comments. The
views expressed herein are those of the authors and do not necessarily reflect those of the Federal
Reserve Bank of Richmond or the Federal Reserve System.
E-mail addresses: huberto.ennis@rich.frb.org (H. M. Ennis), keister@itam.mx (T. Keister)
1 Introduction
The direct consequences of a crisis in the banking system are fairly well understood and have
been extensively documented.1 It seems likely, however, that the mere possibility of a banking
crisis will also have important macroeconomic consequences, even if a crisis does not occur. For
example, suppose it is believed that with some small but positive probability, all depositors will
suddenly rush to their banks and attempt to withdraw their money. This possibility will almost
certainly influence the level of cash reserves that a bank will choose to hold. One of the main roles
of the banking system is to perform maturity transformation, that is, to hold long-term assets while
issuing short-term liabilities. If the possibility of a crisis leads banks to hold a more liquid portfolio,
it will reduce the funds available for long-term investment and thereby have a substantial impact on
real economic activity. In addition, the possibility of a run on the banking system will likely change
the interest rate that a bank chooses to offer on deposits, thus changing the consumption profiles
available to depositors. Such indirect effects of banking crises are less well understood, and are
our focus in this paper. Specifically, we ask: How does the possibility of a bank run influence the
deposit contract offered and the investment decisions made by a competitive bank?
We build on the work of Cooper and Ross [3], who first analyzed this issue in an extension
of the classic Diamond and Dybvig [5] model. Diamond and Dybvig set up an environment with
idiosyncratic liquidity-preference shocks and private information. They showed that by offering
a contract that lets depositors choose when to withdraw their funds, a bank can implement the
first best allocation in this environment as a Nash equilibrium. However, they also showed that
there exists another equilibrium where all depositors attempt to withdraw their funds from the
contractual arrangement early, fearing that other agents will do the same and that as a result there
will be insufficient funds to cover all promised payments. In other words, the process of providing
liquidity insurance makes a bank illiquid, and thereby makes it susceptible to a bank run. The
Diamond-Dybvig analysis leaves open an important question: Why would an agent deposit in a
bank if she expected a run to occur? Cooper and Ross [3] address this issue by introducing a
sunspots-based equilibrium selection rule to the problem: If the banking contract is such that both
equilibria exist, then the run equilibrium will be selected with a fixed probability q. They restrict
the bank to offer demand deposit contracts, in which depositors are promised a fixed return and
1
See, for example, Caprio and Klingebiel [2] and Boyd et al.
1
[1].
are given this return as long as the bank has not run out of funds.2 Banks operate in a perfectly
competitive environment and hence choose the deposit contract that maximizes the expected utility
of depositors. Cooper and Ross show that if the probability of a run is above some cutoff level,
the bank will set the deposit contract such that waiting to withdraw is a dominant strategy for
depositors who do not need liquidity. Such a contract guarantees that a bank run will not occur,
and therefore is called run proof. However, they also show that if the probability of a bank run
is below the cutoff, the bank will choose a contract that is not run proof. In this case, bank runs
emerge as a truly equilibrium phenomenon; consumers are willing to deposit in the bank because
the improved consumption profile that the bank offers (if a run does not occur) outweighs the
possibility of losing one’s deposit if a run does occur.
Our interest is in characterizing the properties of the optimal deposit contract, including the
portfolio choice of the bank, and in examining how this contract changes when the probability of
a bank run changes. The Cooper-Ross model has some attractive features for this purpose. First,
the environment is well known and is a natural extension of that in Diamond and Dybvig [5].
We are adding nothing new in this dimension. In addition, the bank’s maximization problem is
conceptually easy to understand and allows one to gain a substantial amount of intuition about the
various ways in which the possibility of a run changes a bank’s incentives. Finally, the problem
has a small number of choice variables, which allows us to give a relatively sharp characterization
of the solution. In spite of this apparent simplicity, our analysis yields some surprising results. In
fact, we show that some of the conclusions drawn in Cooper and Ross [3] are misleading, and our
extension of their results helps clarify these issues. Cooper and Ross focus on the role of “excess
liquidity,” that is, liquid assets that a bank intends to hold over the long term, despite the fact that
these assets yield a lower rate of return than illiquid investment. There are in principle two reasons
why a bank might choose to hold excess liquidity. First, having a highly liquid portfolio would
minimize liquidation costs if a run were to occur. Second, if the bank decides to instead offer
a run-proof contract, holding a highly liquid portfolio might provide the best possible run-proof
consumption profile. Assuming preferences are of the constant-relative-risk-aversion variety, we
show that a bank will never hold excess liquidity for the first reason. That is, if a bank offers
This restriction is designed to mimic a sequential service constraint; suspension of convertibility is not allowed.
In the problem considered by Diamond and Dybvig [5], the restriction to deposit contracts was non-binding since the
first-best allocation was being implemented. Here, however, the constraint is binding. Nevertheless, Peck and Shell
[9] show that the same qualitative results can obtain when a broad class of possible contracts is considered.
2
2
a deposit contract that is not run proof, it will hold only enough liquid assets to exactly meet
withdrawal demand in the event that a run does not occur. The sole reason, then, that a bank would
hold excess liquidity is as a way of making the contract immune to runs. We provide an exact
characterization of the situations when the best run-proof contract does indeed involve holding
excess liquidity.
These results enable us to characterize the relationship between the probability of a run and
the level of illiquid investment undertaken by a bank. At first glance, it seems like the nature
of this relationship should be straightforward: the more likely is a run, the more likely it is that
the bank will have to liquidate all of its investment and therefore the less the bank should invest.
However, we show that there is another, more subtle, effect at work. Since we know that a bank
will use all liquid assets to meet short-term withdrawal demand, it follows that if the bank were
to choose a more liquid portfolio it would also offer a higher return to agents who withdraw their
deposits early. In other words, an increase in the liquidity of the bank’s portfolio will necessarily
be accompanied by an increase in short-term liabilities. In fact, we show that if the bank were to
choose a more liquid portfolio, it would be able to serve fewer depositors in the event of a run.
Decreasing the level of investment, then, leaves the bank with more resources if a run occurs, but
it also implies that the resources will be shared by fewer depositors (and more depositors will
receive nothing). We show that when liquidation costs are above some threshold, the first effect
dominates in expected utility terms and investment is decreasing in the probability of a bank run.
However, when liquidation costs are more moderate, the second effect dominates and an increase
in the probability of a run leads to an increase in investment. This latter result is fairly counterintuitive and has interesting implications. In this canonical model of bank runs, a more crisis-prone
economy could have a higher level of investment and hence could be expected to experience faster
growth, at least in periods where a run does not occur.
The remainder of the paper is organized as follows. In the next section, we briefly review the
environment of Cooper and Ross [3] and then set up the maximization problem of a competitive
bank. In Section 3 we describe the optimal deposit contract, examining the conditions under which
it is run proof and the conditions under which excess liquidity will be held. In Section 4 we study
how the level of investment responds to a change in the probability of a bank run, and in Section 5
we offer some concluding remarks.
3
2 The Bank’s Problem
The environment we study is that of Cooper and Ross [3]; we follow their notation as much
as possible. Consider an economy with a [0, 1] continuum of ex-ante identical agents, each of
whom lives for three periods. Each agent has an endowment (normalized to one) in period 0, and
no endowment in periods 1 and 2. In period 0, an agent decides whether or not to deposit her
endowment in the bank. With probability π ∈ (0, 1), at the beginning of period 1 she will discover
that she is impatient and only gets utility from consuming in period 1. With probability (1 − π)
she will discover that she is patient and derives utility only from consuming in period 2. Let u(c)
be the utility over consumption (in the appropriate period) and assume that u is strictly increasing,
strictly concave, and satisfies u(0) = 0. There are two technologies for saving, which we call
storage and investment. One unit of consumption placed in storage yields one unit of consumption
in the following period. One unit of consumption placed in investment in period 0 yields R units
of consumption in period 2, but only (1 − τ ) units if the investment is liquidated in period 1, where
τ ≥ 0 represents a liquidation cost.3
We study the problem of a bank that behaves competitively in the sense that it offers the contract
that maximizes the expected utility of depositors. Following Cooper and Ross [3], we restrict
the bank to offer deposit contracts of the following form. There is a fixed payment cE that is
promised to depositors who withdraw “early,” that is, in period 1. The bank must pay this amount
to each depositor arriving in period 1 unless it has completely run out of funds; no suspension of
convertibility is allowed.4 Whatever resources remain in the bank in period 2 are divided evenly
among the remaining depositors. Let cL denote the payment promised on these “late” withdrawals.
In addition to choosing these two payments, the bank must also decide how to divide its portfolio
between investment and storage. The bank will clearly place sufficient resources in storage to be
able to pay cE to all impatient depositors, since otherwise it would pay liquidation costs in period
Diamond and Dybvig [5] assumed τ = 0, in which case storage is a dominated technology and the bank’s portfolio
choice is trivial. Cooper and Ross [3] introduced the liquidation cost in order to address issues related to portfolio
choice.
4
As Diamond and Dybvig [5] show, when the fraction of impatient consumers is known with certainty a suspensionof-convertibility scheme can costlessly eliminate bank runs. However, issues of credibility and commitment might
make such a suspension difficult to implement. In addition, when the fraction of impatient consumers is random the
optimal contract from a broad class that includes total and partial suspensions of convertibility may permit runs to
occur (see Peck and Shell [9]). For the issues discussed in this paper, the important thing is that bank runs can occur
with positive probability in equilibrium, and the restrictions placed on the set of feasible contracts by Cooper and Ross
[3] are a useful way of generating this feature.
3
4
1 even if a run did not occur. Let i denote the fraction of the bank’s deposits placed into investment
and 1 − i the fraction placed into storage; then we know that cE ≤ 1 − i will hold. Define i2 to
be the difference between 1 − i and cE , so that if a run does not occur the fraction 1 − i − i2 of
deposits will be paid out in period 1. The variable i2 represents excess liquidity: resources that the
bank plans to keep in storage for two periods (if a run does not occur), even though investment
offers a higher two-period return.
At the beginning of period 1, each depositor learns whether she is patient or impatient. She then
decides whether to attempt to withdraw from the bank in period 1 or in period 2. If she is impatient,
she will clearly choose to withdraw in period 1. If she is patient, however, her optimal action may
depend on the choices of other patient depositors. We consider only symmetric outcomes. Suppose
that a patient depositor believes that all other patient depositors will try to withdraw in period 1.
The total amount of resources available to the bank in this case would be the funds held in storage
plus the amount of goods that can be obtained by liquidating all investment. If the deposit contract
is such that these resources are not enough to pay cE to all depositors, the bank will run out of
funds in period 1 (and have nothing left in period 2). The optimal strategy of a patient agent is
then to “run” and attempt to withdraw in period 1. In other words, if cE > 1 − iτ holds, then there
exists a bank-run equilibrium of the game played by patient depositors. If, on the other hand, the
bank has enough resources to meet all of its short-term obligations, a patient depositor knows that
the bank will have sufficient funds left in period 2 to pay her the promised amount no matter how
many early withdrawals take place. Waiting to withdraw is then a dominant strategy. That is, if
cE ≤ 1 − iτ holds the contract is run proof ; the only equilibrium of the game played by patient
depositors has all of them withdrawing in period 2.
If the deposit contract admits a bank run equilibrium, the bank and the depositors need to have
expectations about which equilibrium will be played. Cooper and Ross [3] introduce a sunspotsbased equilibrium selection rule: If the deposit contract is such that both equilibria exist, the run
equilibrium will occur with a fixed probability q. If the deposit contract is run-proof, of course, the
no-run equilibrium will obtain with certainty. The problem of the bank is therefore
5
max
{cE ,cL ,i,i2 }
s.t.
(1 − qIπb ) [πu (cE ) + (1 − π) u (cL )] + qIπb [πu
b (cE )]
(P )
πcE = 1 − i − i2
(1 − π) cL = iR + i2
i, i2 ≥ 0
1 − iτ
π
b =
c
½ E
1 if π
b<1
and Iπb =
0 if π
b≥1
.
The variable π
b gives the fraction of the bank’s depositors who would be served in the event of a
run; the run equilibrium exists if and only if this fraction is less than unity. The indicator function
Iπb reflects the equilibrium selection rule, whereby the run equilibrium occurs with probability q if
b is less than unity, but otherwise occurs with probability zero.5
π
We begin by looking at the solution to this problem when q is zero, that is, when a bank run is
ruled out by assumption. This solution generates the first-best allocation, which satisfies
u0 (cE ) = Ru0 (cL ) .
In order for the issue of bank runs to be relevant, we need for the deposit contract generating this
b < 1 holds and the bank-run equilibrium exists.
first-best allocation to satisfy cE > 1 − iτ , so that π
This condition, in turn, implies that the first-best allocation must satisfy
cE >
1−τ
1 − πτ
and
cL <
R
.
1 − πτ
In other words, the first-best allocation must be to the right of point A in Figure 1 (at point x).
Hence a run equilibrium will exist when the bank tries to implement the first-best allocation if and
only if
0
u
µ
1−τ
1 − πτ
¶
0
> Ru
µ
R
1 − πτ
¶
(1)
holds. We assume this condition throughout the analysis.
See Ennis and Keister [7] for a more detailed discussion of equilibrium selection in this type of model. Also note
that the presence of the indicator function implies that the problem (P ) is fundamentally different from problem (6)
in Cooper and Ross [3]. (P ) is the bank’s complete optimization problem, while their problem (6) is only intended
to be an auxiliary problem useful for characterizing the solution to the complete problem.
5
6
cL
R
(1−τπ)
A
45
o
y
x
1
(1−π)
B
1
i2 = 0
C
0
1
(1−τ)
(1−τπ)
i=0
1
π
cE
Figure 1: The set of feasible deposit contracts
Of course, the bank can always choose a contract such that cE ≤ 1 − iτ holds, in which case a
run will not occur. If the bank were to choose such a contract, it would obviously choose the best
contract in that category, which is called the best run-proof contract. In other words, if the solution
to (P ) is run proof, then it is also the solution to the following restricted problem
max
{cE ,cL ,i,i2 }
πu (cE ) + (1 − π) u (cL )
(RP )
s.t.
πcE = 1 − i − i2
(1 − π) cL = iR + i2
i, i2 ≥ 0
cE ≤ 1 − iτ .
The only difference between problems (P ) and (RP ) is the constraint set; (RP ) allows only
run-proof contracts. On this set, the objective functions in the two problems are identical. As
the statement of problem (RP ) makes clear, this objective function has standard properties: it
7
is continuous, strictly increasing, and strictly concave in the variables (cE , cL ). In other words,
the problem (RP ) is a fairly standard maximization problem whose constraint set consists of the
shaded region in Figure 1. Under condition (1), the slope of the indifference curve through point
A is steeper than the slope of segment AB, and hence the solution to (RP ) must lie on the line
segment AC. This solution may either be at the “kink” point A or at a point of tangency like y.
The objective function in the complete problem (P ), on the other hand, is not standard, both
b is a decreasing
because the indicator function leads to a discontinuity and because the variable π
function of cE . It is important to bear in mind that when q is positive, the indifference curves drawn
in Figure 1 do not apply in the non-run-proof region – the objective function may not be increasing
or even concave in this region. We show below how these non-standard properties generate some
surprising results. First, however, we address an issue that is ignored in the statement of the
problem given above: We need for the solution to satisfy the incentive-compatibility constraint
cE ≤ cL . If this inequality did not hold, withdrawing in period 1 would be a dominant strategy
for patient agents and hence a bank run would occur with certainty. Our first result shows that the
solution to the problem (P ) as stated above necessarily satisfies incentive compatibility, and hence
(P ) is indeed the correct problem for the bank to be solving.
Proposition 1 The solution to (P ) satisfies cE < cL .
The proof of this proposition is contained in the appendix. One way of interpreting the result is
as saying that depositors do not receive perfect insurance against the liquidity shock. Intuitively,
perfect insurance is not optimal because insurance is costly in this setting; payments to impatient
depositors must be made from relatively low-return assets. The proposition implies that the solution to (P ) must lie in the triangle ABC in Figure 1, not including the boundary segment BC. The
solution will lie on the segment AC if the optimal contract is run proof, and will lie either on the
segment AB or in the interior of the triangle if it is not.
Notice that the statement of the proposition is actually stronger than the requirement for incentive compatibility to hold, since it involves a strict inequality rather than a weak one. It is easy to
see from the figure that this stronger result directly implies that the solution to problem (P ) cannot
involve the bank holding only liquid assets.
Corollary 1 The solution to (P ) satisfies i > 0.
8
3 The Optimal Deposit Contract
We now turn our attention to characterizing the solution to (P ) , asking under what conditions
the optimal deposit contract is run proof and under what conditions it involves holding excess
liquidity.
3.1
When is the optimal contract run proof?
Suppose that the likelihood of a run, q, and the cost of liquidation, τ , are both large enough that
qτ > (1 − q) (R − 1)
(2)
holds. Our next result shows that this condition is sufficient to guarantee that the optimal deposit
contract is run proof.
Proposition 2 If (2) holds, the solution to (P ) is a run-proof contract.
The formal proof is contained in the appendix; the reasoning is as follows. Suppose the solution
to (P ) is not run proof, and consider the effect of decreasing i slightly while increasing i2 to keep
cE unchanged. That is, suppose we shift some of the resources set aside for period 2 withdrawals
from investment to storage. The benefit of such a change comes in the event of a run, since total
liquidation costs will then be τ units lower for each unit less of investment. For each additional
unit of resources the bank has during a run, it can give the fixed amount cE to (1/cE ) additional
depositors. Hence the value of having an additional unit of resources is measured by the average
utility of consumption u (cE ) /cE . Since a run occurs with probability q, the gain in expected utility
is therefore
qτ
u (cE )
.
cE
However, this proposed change causes a decrease in utility in the event that a run does not occur.
If there is no run, the bank will have earned a return of unity (instead of R) on the assets switched
from investment to storage, and therefore total resources in period 2 will be (R − 1) units lower
for each unit less of investment. The marginal value of a unit of resources in period 2 when a run
did not occur in period 1 is measured by u0 (cL ), the marginal utility of a depositor withdrawing in
period 2. Since the “no run” event occurs with probability (1 − q), the loss in expected utility is
9
equal to
(1 − q) (R − 1) u0 (cL ) .
From Proposition 1 we know that cE < cL holds at the solution point. Strict concavity of the
function u then implies that u (cE ) /cE is greater than u0 (cL ) . Therefore, when (2) holds the gain
in expected utility is necessarily greater than the loss, and hence the original contract under consideration cannot be optimal. Since this argument applies to any contract that is not run proof, the
solution to (P ) must be a run-proof contract.
Condition (2) was introduced by Cooper and Ross [3] for a different purpose, in an attempt to
give a sufficient condition for excess liquidity to be held. It is important to bear in mind that when
a run-proof contract is chosen, the objective function in (P ) reduces to that in (RP ) . In other
words, there is a discontinuity in the objective function along the line segment AC in Figure 1.
Hence the above argument does not imply that the solution to (P ) will have i2 > 0 when condition
(2) holds. It only shows that the solution to (P ) is run-proof; problem (RP ) can then be solved to
find the properties of this optimal contract. We show below that condition (2) is neither necessary
nor sufficient for i2 > 0 to hold.6
Proposition 5 in Cooper and Ross [3] establishes a cut-off probability q ∗ ∈ (0, 1) such that for
all q > q ∗ the solution to problem (P ) is run proof and for all q < q ∗ it is not. Our Proposition 2
gives an upper bound for q ∗ , which we state as a corollary.
Corollary 2 If q ∗ is the threshold value of q above which the solution to (P ) is a run-proof
contract, then
q∗ ≤
R−1
R − (1 − τ )
must hold.
3.2
When is excess liquidity held?
For the remainder of the paper, we assume that the utility function of depositors is of the constantIn other words, the statement of Proposition 3 in Cooper and Ross [3] is misleading. That proposition states that if
we consider the problem (P ) without the indicator function (their problem (6)), then under condition (2) the solution
to this problem has i2 > 0. Our Proposition 2 shows that under condition (2) the solution to their problem (6) is never
the solution to the bank’s complete optimization problem (P ) , and hence is irrelevant for characterizing the optimal
deposit contract.
6
10
relative-risk-aversion form
u (c) =
cα
α
with
0 < α < 1.
(3)
As discussed above, the benefit of an increase in the resources available to the bank in the event of a
run is measured by the average utility of consumption u (cE ) /cE , while this benefit is measured by
the marginal utility of consumption when a run does not occur. Finding the optimal deposit contract
when a run is possible, then, involves comparing average and marginal utilities of consumption.
With the utility function in (3), marginal utility is always a fraction α of average utility, and hence
sharp comparisons are possible. Most of our results below involve strict inequalities and hence are
robust to small perturbations of the function u.
We now investigate under what conditions the bank will actually choose to hold excess liquidity.
There are two distinct reasons why the bank might choose to do this. First, having more assets in
storage lowers liquidation costs and therefore makes depositors better off in the event of a run.
Second, holding a very liquid portfolio might be the best way to eliminate the possibility of a run
by satisfying the run-proof condition. Our next result shows that the first reason never applies. That
is, a bank will never hold excess liquidity for the purpose of mitigating the effects of a potential
run.
Proposition 3 If a solution to the bank’s problem is not run-proof, then the bank holds no excess
liquidity (i.e., i2 = 0).
The proof of this proposition consists of showing that the solution to (P ) does not lie in the interior
of the triangle ABC in Figure 1. The boundary segment AC can be ruled out by the hypothesis of
the proposition, while the boundary BC can be ruled out by Proposition 1. Hence the solution must
lie either in the interior of the triangle, or on the boundary AB. In the appendix we show that the
objective function cannot have an interior local maximum (although it may have a local minimum).
This establishes that if the solution is not run proof, it must lie on the boundary AB where i2 = 0
holds.
Proposition 3 shows that the bank will never hold excess liquidity for the purpose of providing
funds to depositors in the event of a run. The only reason it might set i2 positive, then, is as a way
of eliminating the run equilibrium. We now provide conditions under which the level of excess
liquidity in the best run-proof contract is positive. Under condition (1), the best run-proof contract
11
must lie on the segment AC in Figure 1. If the contract is at point A then the bank will not be
holding excess liquidity and the consumption allocation will be given by
cE =
1−τ
1 − πτ
and
cL =
R
.
1 − πτ
If the contract is to the right of point A, however, the bank will be holding excess liquidity. The
condition that determines whether or not the best run proof contract is at point A involves the slope
of the indifference curve of the objective function in problem (RP ) at that point. Specifically, the
best run proof contract is to the right of point A if
µ
¶ ·
¸ µ
¶
1−τ
R−1 0
R
0
u
> 1+
u
1 − τπ
πτ
1 − τπ
(4)
holds. Note that
1+
R−1
>R
πτ
holds, so that (4) is a stronger requirement than (1). When (1) holds but (4) does not, the best
run-proof deposit contract is at point A and has i2 equal to zero. We summarize this discussion in
the following proposition, whose formal proof is contained in the appendix.
Proposition 4 The best run-proof contract satisfies i2 > 0 if and only if (4) holds.
Proposition 2 provides a sufficient condition for the best deposit contract to be run proof. This
condition can be made to hold by choosing a sufficiently high value of q. At the same time,
Proposition 4 gives the condition under which the best run-proof contract involves holding excess
liquidity. Notice that this condition does not depend on q, so that it is clearly possible for both conditions to hold at the same time. Combining these two results therefore gives us a set of sufficient
conditions under which the solution to the bank’s problem (P ) involves holding excess liquidity.7
Corollary 3 If (4) and (2) hold, the solution to (P ) has i2 > 0.
It is important to bear in mind that the reason the bank holds excess liquidity is not to be able to
satisfy depositors’ demand during a bank run, but instead to avoid the possibility of a run altogether.
Under the assumption on preferences in (3), the Cooper-Ross model predicts that a bank will never
hold excess liquidity for the purpose of providing funds to depositors in the event of a run.
7
Note that Proposition 4 and Corollary 3 do not require that the utility function be of the form (3).
12
4 The Level of Investment
We now turn to the question of how the possibility of a run affects the amount of resources
that the bank places into investment. In the present model the level of investment affects the
consumption allocations of both types of depositors. In a dynamic setting where banks play an
important role in the capital formation process, it becomes even more important. For example,
Ennis and Keister [7] embed a similar banking environment into an endogenous growth model.8 In
such a setting, bank runs have two important effects on the growth rate of the economy. The first
is obvious: a bank run leads to the liquidation of investment and therefore a lower stock of capital
(and hence output) in the next period. The second effect is less direct but no less important: the
possibility of a run affects the level of investment chosen by banks, which in turn affects the growth
rate in periods when a run does not occur. Hence understanding how the investment decisions of
banks are affected by the variable q in the present model is an important part of understanding the
macroeconomic effects of bank runs.
At first glance, the answer to this question seems like it should be straightforward: when a run
is more likely, there is a higher probability that the bank will have to liquidate all of its investment
and therefore it seems optimal for the bank to invest less. In this section, we show that this intuition
is not necessarily correct. An increase in the probability of a run can either decrease or increase
the level of investment chosen by the bank, depending on parameter values. In particular, define
³ 1
´
π
(1 − α) R 1−α + 1−π
τ∗ ≡
.
1
π
(1 − α) R 1−α + 1−π
Note that 0 < τ ∗ < 1 holds. We then have the following result.9
Proposition 5 If τ > τ ∗ holds, i is strictly decreasing in q for q ∈ (0, q ∗ ) . If τ < τ ∗ holds, i is
strictly increasing in q for q ∈ (0, q∗ ) .
To see the intuition behind this result, suppose that q is equal to zero. In this case, the solution to
(P ) is to offer the first-best allocation, which is represented by point x in Figure 1. Now suppose
that q increases slightly. The objective function in a neighborhood of the first-best allocation is
See also Gaytan and Ranciere [8], which uses a different banking model to examine the relationship between bank
runs and economic development.
9
This proposition is similar in spirit to Proposition 4 in the working paper Cooper and Ross [4]. However, our result
characterizes the solution to the bank’s complete problem (P ) , whereas their result applied to an artificial problem
where i2 was required to be zero. In addition, our assumption on preferences (3) allows us to provide a more complete
characterization.
8
13
equivalent to
b (cE ) .
(1 − q) [πu (cE ) + (1 − π) u (cL )] + qπu
(5)
Increasing q generates a continuous change in this (local) objective and hence the solution to the
problem will move continuously, as long as q stays below q ∗ . Proposition 3 tells us that while
q < q ∗ holds, the solution has i2 = 0. In other words, as q increases the solution to the bank’s
problem must move along the line segment AB in Figure 1. The first term in (5) is maximized at
the point x, and hence there is no first-order loss in the value of this term from deviating in either
direction along the boundary. Therefore, whether the level of investment should be increased or
decreased when q is raised above zero is determined entirely by the last term in (5), which is the
expected utility of depositors in the event of a run.
If a run occurs, there is an obvious benefit to having less investment: total liquidation costs will
be lower and the bank will have more resources to give to depositors. However, there is a cost to
having less investment as well. Because we know i2 = 0 holds, the first constraint in problem (P )
sets πcE equal to 1 − i. Decreasing investment therefore implies raising the amount promised to
depositors who withdraw in the first period. In fact, it is straightforward to show that π
b is strictly
increasing in i, and hence decreasing i implies that fewer depositors will be served during a run. In
other words, lowering the level of investment implies that during a run a depositor will be given a
larger amount if she is served, but that she will be served with a lower probability. Which of these
two effects dominates in terms of expected utility depends on the size of the liquidation cost τ . If
τ is very large, then lowering investment leads to a large increase in the total resources available
to the bank and the first effect dominates. In this case, the answer is what one would (naively)
expect: The possibility of a run leads the bank to choose a more liquid portfolio. However, when τ
is relatively small, lowering investment leads to a modest increase in the total resources available
and the second effect dominates. In this case the answer is surprising: The possibility of a run
leads the bank to choose a less liquid portfolio. Applied to a growth setting, then, the Cooper-Ross
model predicts that when liquidation costs are small, a more crisis-prone economy would have
higher levels of investment and growth, at least in periods where a crisis does not occur.
It is important to bear in mind that the bank could decrease liquidation costs without the negative
side effect of decreasing the number of depositors served in the event of a run. By setting i2
positive, the bank could decrease i and leave cE unchanged, for example, guaranteeing that more
14
depositors would be served in the event of a run. However, Proposition 3 tells us that this response
would not be optimal for any q < q ∗ . The reason is that if a run does not occur, holding excess
liquidity is very costly. That is, while moving slightly away from the point x along the segment
AB in Figure 1 does not cause a first-order loss in expected utility in the no-run outcome, moving
strictly inside the constraint set does. Hence the optimal response of the bank to a small probability
of a run will always be to adjust only the level of investment, and the direction of this adjustment
will depend on the size of τ . The intuitive arguments we have given here are valid for q close to
zero, but the proof of the proposition in the appendix shows that they extend to all q less than q ∗ .
5 Concluding Remarks
We have studied what is in many ways a fairly simple model where a bank run is possible in
equilibrium. We have maintained the simplicity of the environment for several reasons. First, the
bank runs that we study here are exactly the kind that were first heuristically described by Diamond and Dybvig [5] in their seminal paper. The analysis of Cooper and Ross [3] can be seen as a
formalization of this argument and a fleshing out of its implications. We have been able to deepen
and clarify the results of Cooper and Ross [3] in important dimensions. Additionally, and perhaps
most importantly, the simplicity of the environment allows us to obtain a better understanding of
the various ways in which the possibility of a bank run affects the incentives of a bank offering
demand deposit contracts. In fact, some of our results are far from obvious, and hence this simple
environment seems to be a very useful step in understanding more general issues. There are many
ways in which extending the model presented here could give interesting new results. Just to mention one, modeling bank runs explicitly as an equilibrium phenomenon changes the willingness of
agents to participate in the banking system. In particular, when runs are possible, agents may not
want to deposit all of their resources in the bank (as is assumed in the Cooper-Ross framework).
Even though the main incentive effects that we study in this paper will not change much by permitting agents to keep some resources outside the banking sector, the macroeconomic implications
of such disintermediation could be significant (see Ennis-Keister [7]). In this sense, extending the
analysis presented here to richer environments seems to be a promising agenda. The analysis in
this paper should prove helpful in such a program, especially to the extent that it makes starkly
clear that the possibility of a run can change the incentives of a bank in relatively intricate ways.
15
Appendix A. Proofs
Proposition 1: The solution to (P ) satisfies cE < cL .
Proof: We break the proof into two parts: the case where the solution to (P ) is run proof and the
case where it is not. If the solution to (P ) has cE ≤ 1 − iτ (which implies cE ≤ 1) then this
solution must also solve (RP ). Since the objective function in (RP ) is strictly increasing in cE
and cL , we know that the solution must lie on the outer boundary of the constraint set, either at the
point A or along the segment AC in Figure 1. Hence cE ≤ cL necessarily holds. To show that this
inequality is strict, we only need to show that the point C cannot be the solution. The slope of the
indifference curve evaluated at any point on the 45-degree line is equal to −π/ (1 − π) . The slope
of the line segment AC is equal to
−
R − 1 + τπ
π
<−
.
τ (1 − π)
1−π
That is, the indifference curve through the point C is flatter than the boundary segment AC, and
hence C cannot be the solution.
More formally, let the multipliers on the first four constraints listed in problem (RP ) be given by
λE , λL , γ, and β, respectively, and let δ denote the multiplier on the run-proof condition cE ≤ 1 −
iτ . Then the necessary first-order conditions characterizing the best run-proof contract include:
cE : πu0 (cE ) − λE π − δ = 0
(6a)
cL : u0 (cL ) − λL = 0
(6b)
i : −λE + λL R − δτ + γ = 0
(6c)
(6d)
i2 : −λE + λL + β = 0.
where all of the multipliers must be non-negative. We also have the complementary slackness
conditions
γi = 0,
βi2 = 0,
and
δ(1 − iτ − cE ) = 0.
The first-order conditions can be combined to yield
u0 (cE ) − u0 (cL ) = β +
16
δ
.
π
(7)
Therefore, we need to show that any solution to problem (RP ) satisfies β + δ/π > 0, since the
strict concavity of u then implies that cE < cL then must hold. Assume to the contrary that both
β and δ are zero (note that by definition both must be non-negative). Then, from the first order
conditions we would have λE = λL = u0 (cL ). Using condition (6c) would then yield
γ = (1 − R)u0 (cL ) < 0
which is a contradiction because γ must be non-negative. Therefore it cannot be the case that both
β and δ are zero in the solution to problem (RP ), and hence we have β +δ/π > 0. This establishes
that the conclusion of the proposition must hold if the solution is a run-proof contract.
If the solution to problem (P ) is not run proof (i.e., has cE > 1 − iτ ), then in a neighborhood
of the solution point, the objective function is equivalent to
(1 − q) [πu (cE ) + (1 − π) u (cL )] + qπu
b (cE ) .
The first-order conditions characterizing a solution in this region are given by
·
¸
u(cE )
0
0
b
− u (cE ) − λE π = 0
cE : (1 − q)πu (cE ) − qπ
cE
cL : (1 − q)u0 (cL ) − λL = 0
u(cE )
i : −qτ
− λE + λL R + γ = 0
cE
i2 : −λE + λL + β = 0,
(8)
(9a)
(9b)
(9c)
(9d)
where all of the multipliers must be non-negative. Substituting (9a) and (9b) into (9d) yields
·
¸
π
b u(cE )
0
0
0
− u (cE ) .
β = (1 − q) [u (cE ) − u (cL )] − q
π
cE
Because β is non-negative, u(0) = 0, and u is strictly concave, this equation implies that u0 (cE ) −
u0 (cL ) > 0 holds and hence that we must have cE < cL . This establishes that the conclusion of the
proposition also holds if the solution to (P ) is not a run-proof contract, and therefore completes
¥
the proof.
Proposition 2: If (2) holds, the solution to (P ) is a run-proof contract.
Proof: Suppose the solution is not run proof. Then we have cE > 1 − iτ and π
b < 1, which implies
that there is an open ball around the solution point where the objective function is equivalent to
17
(8). As in the proof of Proposition 1, the first-order conditions characterizing a maximum of this
new objective subject to the feasibility constraints are given by (9a) - (9d). We also have the
complementary slackness conditions (7). By Corollary 1 we only need to look for a solution with
i > 0, so that γ = 0 holds. In this case, substituting (9b) and (9d) into (9c) yields
qτ
u (cE )
= (1 − q) (R − 1) u0 (cL ) − β.
cE
However, using the concavity of u and the result of Proposition 1 (that cE < cL holds), this
¥
condition contradicts (2).
Proposition 3: If a solution to the bank’s problem (P ) is not run-proof, then the bank holds no
excess liquidity (i.e., i2 = 0).
Proof: Suppose the solution to (P ) is not run proof. Then the first-order conditions for problem
(P ) are again (9a) - (9d). By corollary 1 we know that i > 0 and hence that γ = 0 holds. If i2 > 0
holds in the solution, then β = 0 must hold, which in turn implies that λE = λL = (1 − q)cα−1
L
holds. Combining this result with expression (9c) we obtain
(R − 1)(1 −
q)cLα−1
α−1
cE
= qτ
.
α
This expression tell us that when the solution to problem (P ) is not run-proof, if the bank is holding
excess liquidity (that is, if i2 > 0), then the optimal values of cE and cL lie on a ray from the origin
in the (cE , cL ) plane. In other words, we have cL = φcE where φ is a constant given by
φ≡
µ
α(1 − q)(R − 1)
qτ
1
¶ 1−α
.
We now show that these conditions cannot characterize a maximum of problem (P ). Using the
first two constraints in problem (P ) we can rewrite the level of investment as
i=
πcE + (1 − π)cL − 1
R−1
and hence we have
b=
π
1 − iτ
(R − 1) + τ [1 − πcE − (1 − π)cL ]
=
.
cE
(R − 1) cE
Using this expression for π
b and the relationship cL = φcE , we can rewrite the objective as a
18
function of the single variable cE . Calling this function f (cE ) we have
f (cE ) ≡ A
cα−1
cαE
+B E ,
α
α
where A and B are constants given by
A ≡ (1 − q) [π + (1 − π) φα ] −
qτ [π + (1 − π) φ]
R−1
and
B=q
R−1+τ
> 0.
R−1
Since the solution being considered is interior, the following two necessary conditions must hold
at the optimum: f 0 (cE ) = 0 and f 00 (cE ) ≤ 0. The first-order condition implies
cE = (1 − α)B/αA.
(10)
The second-order condition is given by
+
f 00 (cE ) = − (1 − α) Acα−2
E
(1 − α)(2 − α) α−3
BcE ≤ 0,
α
which can only hold if cE ≥ (2 − α)B/αA. This inequality contradicts (10) and therefore the
proposed solution cannot satisfy the necessary conditions for a maximum. Hence we must have
i2 = 0 at the solution. In other words, evaluated along the ray where cL = φcE , the objective
function cannot have a local maximum (although it may have a local minimum). If the solution
does not lie on the run-proof boundary, then it must lie on the other boundary, where i2 = 0 holds.
¥
Proposition 4: The best-run proof contract satisfies i2 > 0 if and only if (4) holds.
Proof: Condition (4) is stronger than condition (1). Hence cE = 1 − iτ must hold at the solution
to (RP ) and we can rewrite the constraints as
cE =
τ +R−1
(1 − π)τ
−
cL ,
πτ + R − 1 πτ + R − 1
19
and
1−τ
≤ cE ≤ 1.
1 − τπ
We have i2 > 0 whenever
1−τ
1 − τπ
cE >
(11)
holds (see Figure 1). Suppose we look at the first-order condition of the problem ignoring the
inequality constraints; this condition is given by
¸
·
R−1 0
u (cL ) = u0 (cE ).
1+
πτ
If (4) holds, the solution to the first-order condition satisfies (11). In this case, the solution to
the first-order condition is necessarily the solution to (RP ) , and hence that solution has i2 > 0.
Conversely, if (4) does not hold, then the solution to (RP ) is the boundary point
1−τ
1 − τπ
cE =
¥
where i2 = 0 holds.
Proposition 5: If τ > τ ∗ holds, i is strictly decreasing in q for q ∈ (0, q ∗ ) . If τ < τ ∗ holds, i is
strictly increasing in q for q ∈ (0, q ∗ ) .
Proof: Fix some qb < q ∗ . Then for values of q in a neighborhood of qb, we know that the solution to
(P ) is not run proof and, by Proposition 3, therefore has i∗2 = 0. In other words, the problem (P )
is locally equivalent to the problem of choosing the level of investment i to maximize (5), where
the consumption allocations are given by
πcE = 1 − i
and
(1 − π) cL = Ri.
These constraints can be substituted into the objective function to yield a maximization problem
in a single variable. Because the solution is not run proof, we know that it satisfies cE > (1 − iτ ),
which is equivalent to
i<
1−π
.
1 − πτ
20
¡ 1−π ¢
On the interval 0, 1−πτ
, the objective function is smooth. It may not be strictly concave on the
entire interval, but it can be shown to have a unique local maximum (which must be the global maximum because the solution is interior) and to be strictly concave at this maximum point. Therefore,
the first-order condition
" µ
¶α−1 µ
¶α−1 #
¶α−1
¸µ
·
Ri
q
1 − iτ
1−i
1−i
(1 − q) R
+
(1 − α)
−
−τ
=0
1−π
π
α
1−i
π
(12)
implicitly defines a differentiable function i (q) in a neighborhood of qb. Letting G (i, q) denote the
left-hand side of (12), the derivative of this function is given by
di
G2 (i, q)
=−
,
dq
G1 (i, q)
where subscripts on the function G represent partial derivatives. Strict concavity of the objective at
the solution point implies that the denominator of this expression is negative, and hence that di/dq
has the same sign as G2 .
The derivative G2 can be written as
¸
·
£
¤
1 − iτ
1
(1 − α)
− τ (cE )α−1 − R (cL )α−1 − (cE )α−1
G2 (i, q) ≡
α
1−i
(13)
Notice that this expression is independent of q. Suppose that qb is very close to zero. Then the
solution to (P ) will be very close to the first-best allocation, which implies that the second term in
(13) is very close to zero and the sign of G2 is equal to the sign of the first term. The value of i
will also be close to the first-best value
α
R 1−α
.
i =
α
π
R 1−α + 1−π
∗
If τ < τ ∗ holds, it is straightforward to show that the first term in (13) is negative. Therefore,
for values of qb close to zero we have that G2 is negative and i will be strictly decreasing in q at
qb. Furthermore, it is straightforward to show that the cross-partial derivative G21 is always strictly
positive. Thus for larger values of qb, the optimal value of i is lower than i∗ and hence G2 is again
negative, implying that i is still strictly decreasing in q. This analysis holds as long as the first-order
condition (12) characterizes the solution to (P ), which is for all q < q ∗ , and hence establishes the
first part of the proposition. The case of τ < τ ∗ is completely symmetric.
21
¥
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