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Federal Reserve Bank of Chicago
Preventing Bank Runs
David Andolfatto, Ed Nosal, and Bruno Sultanum
September 2014
WP 2014-19
WORKING PAPER
PREVENTING BANK RUNS
B Y D AVID A NDOLFATTO , E D N OSAL AND B RUNO S ULTANUM ∗
S EPTEMBER , 2014
Diamond and Dybvig (1983) is commonly understood as providing a formal rationale for the
existence of bank-run equilibria. It has never been clear, however, whether bank-run equilibria in
this framework are a natural byproduct of the economic environment or an artifact of suboptimal
contractual arrangements. In the class of direct mechanisms, Peck and Shell (2003) demonstrate
that bank-run equilibria can exist under an optimal contractual arrangement. The difficulty of
preventing runs within this class of mechanism is that banks cannot identify whether withdrawals
are being driven by psychology or by fundamentals. Our solution to this problem is an indirect
mechanism with the following two properties. First, it provides depositors an incentive to communicate whether they believe a run is on or not. Second, the mechanism threatens a suspension
of convertibility conditional on what is revealed in these communications. Together, these two
properties can eliminate the prospect of bank-run equilibria in the Diamond-Dybvig environment.
K EYWORDS: bank runs, optimal deposit contract, financial fragility.
JEL CLASSIFICATION: D82, E58, G21.
1.
INTRODUCTION
B ANKING is the business of transforming long-maturity illiquid assets into short-maturity
liquid liabilities. The demandable debt issued by commerical banks constitutes the quintessential example of this type of credit arrangement. The use of short-maturity debt to finance
long-maturity asset holdings is also prevalent in the shadow-banking sector.1 Demandable
debt or short-maturity debt in general has long been viewed by economists and regulators
as an inherently fragile financial structure—a credit arrangement that is susceptible to runs
or roll-over risk. The argument is a familiar one. Suppose that depositors expect a run—a
wave of early redemptions driven by fear, rather than by liquidity needs. By the definition of
illiquidity, the value of what can be recouped in a fire-sale of assets must fall short of existing
obligations.2 Because the bank cannot honor its promises in this event, it becomes insolvent.
In this manner, the fear of run can become a self-fulfilling prophecy.
If demandable debt is run prone, then why not tax it, or better yet, legislate it out of
existence?3 Bryant (1980) suggests that the American put option embedded in bank liabilities
is a way to insure against unobservable liquidity risk. In short, banking is an efficient
risk-sharing arrangement when assets are illiquid, depositors are risk averse, and liquidity
preference is private information. But if this is the case, then the solution to this one problem
seems to open the door to another. Indeed, the seminal paper by Diamond and Dybvig
(1983) on bank runs demonstrates precisely this possibility: Demandable debt as an efficient
risk-sharing arrangement is also a source of indeterminacy and financial instability.
∗ Federal
Reserve Bank of St. Louis, Federal Reserve Bank of Chicago and The Pennsylvania State University.
sector includes, but is not limited to, structured investment vehicles (SIVs), asset-backed commercial
paper (ABCP) conduits, money market funds (MMFs), and markets for repurchase agreements (repos).
2 If the underlying assets are not illiquid, the demand for maturity transformation would be absent.
3 This is essentially the recommendation recently put forth by Cochrane (2014).
1 This
D . ANDOLFATTO , E . NOSAL AND B . SULTANUM
2
Diamond and Dybvig (1983) is most often viewed as a theory of bank runs, but it also
offers a prescription for how to prevent bank runs for the case in which aggregate risk is absent.
The prescription entails embedding bank liabilities with a suspension clause that is triggered
when redemptions exceed a specified threshold. This simple fix prevents bank runs.
As Diamond and Dybvig (1983) point out, a full suspension of convertibility—conditional
on a threshold level of redemption activity being breached—is not likely to be optimal
in the presence of aggregate risk.4 In the absence of aggregate uncertainty, redemptions
exceeding the appropriate threshold constitutes a signal that a run is occurring. With aggregate
uncertainty, the optimal redemption schedule is state contingent.5 As a consequence, it is not
possible to confirm whether heavy redemptions are driven by fundamentals or by psychology.
Threatening full suspension is desirable in the latter case, but not the former.
Our proposed solution to the bank-run problem under aggregate uncertainty is to exploit
the idea that while the bank may not know whether a run is on, there are agents in the
economy that do. That is, in equilibrium, the beliefs of agents in the economy are consistent
with the reality unfolding around them. Can the bank somehow elicit this information in an
incentive-compatible manner? If it can, then might the threat of suspensions conditional on
such information—and not on withdrawals—serve to eliminate run equilibria?
We provide a positive answer for both these questions, and by so doing depart from the
direct mechanism approach usual in the literature. In a direct mechanism, a depositor in
the sequential service queue simply requests to withdraw or not. That is, the depositor
communicates only his type; impatient if he withdraws or patient if he does not. Our indirect
mechanism expands the message space to accommodate additional communications. In this
way, we permit a depositor to communicate his belief that a run is on. We can show that the
threat of suspension conditional on this communication eliminates the possibility of a run
equilibrium.
In practice, such information could be gleaned by introducing a separate financial instrument, the choice of which implicitly reveals what the depositor believes.6 Our mechanism
rewards the depositor for delivering such a message when a run is on. The reward is such
that his payoff is higher compared to the payoff associated with concealing his belief that a
run is on and making an early withdrawal—that is, misrepresenting his type and running
with the other agents. Upon receiving such a message, the mechanism fully suspends all
further redemptions. The design of our mechanism ensures that a patient agent never has an
incentive to either run when a run is on or announce that he believes a run is on when it is
not. At the end of the day, we are able to construct an indirect mechanism that implements
the constrained-efficient allocation in iterated elimination of strictly dominated strategies.
4 Diamond and Dybvig (1983) do not actually characterize the optimal contract for the case in which aggregate
risk is present.
5 This property was suggested by Wallace (1988) and later confirmed by Green and Lin (2003).
6 We elaborate on this in Section 8.
PREVENTING BANK RUNS
3
Literature Review
A number of papers have studied bank fragility under optimal arrangements in the Diamond and Dybvig (1983) setting. Green and Lin (2003) were the first to characterize an optimal
bank contract under private information, sequential service, and aggregate uncertainty. In
their version of the Diamond-Dybvig model, the first-best allocation is implementable as a
unique Bayes-Nash equilibrium of a direct revelation game.
The allocation rule in Green and Lin (2003) allows early withdrawal payments in the
sequential service queue to depend on the history of announcements—"I want to withdraw"
or "I do not want to withdraw"—and payments to that point. The maximum withdrawal
amount faced by an agent in the service queue is lower the larger is the number of preceding
withdrawals. This partial suspension scheme is in stark contrast to Diamond and Dybvig
(1983), who restrict the maximum withdrawal amount to be insensitive to realized withdrawal
demand, so that resources are necessarily exhausted in the event of a run.7
Peck and Shell (2003) modify the Green and Lin (2003) environment in at least two important ways. First, they alter the preferences so that incentive-compatibility constraints bind at
the optimum. This implies, among other things, that the first-best allocation cannot be implemented. Second, they assume that depositors do not know (or are not told) their position in
the service queue. If depositors do not know their queue position, then it is not possible to use
backward induction argument of Green and Lin (2003) to eliminate a bank-run equilibrium.
It also turns out—and this was not recognized at the time—not revealing queue positions to
depositors is part of an optimal mechanism when incentive-compatibility constraints bind.8
Peck and Shell (2003) use a direct revelation mechanism and demonstrate by example that the
optimal direct mechanism can have a bank-run equilibrium.
Ennis and Keister (2009b) modify the Green-Lin environment by assuming the distribution
of depositors types is correlated; Green and Lin (2003) assume independence. Using a direct
revelation mechanism, Ennis and Keister (2009b) demonstrate that a bank-run equilibrium can
exist. But, it is no longer obvious, that a direct revelation mechanism is the “best” mechanism
since it does not deliver a uniqueness result. Indeed, Cavalcanti and Monteiro (2011) examine
indirect mechanisms in the Ennis and Keister (2009b) environment and demonstrate that
the best allocation can be uniquely implemented in dominant strategies. Unfortunately, the
backward induction argument implicitly embedded in their mechanism—which is key to
their uniqueness proof—will not work in the more general Peck and Shell (2003) environment
since depositors do not know their positions in the queue.
There is a mechanism design literature that studies how indirect mechanisms can help to
implement optimal outcomes uniquely. Demski and Sappington (1984) examine a principal7 Wallace
(1990) reports that partial suspensions were prevalent in the banking panic of 1907, and that in one
form or another must have been a feature of other suspension episodes as well.
8 By not revealing or knowing queue positions, multiple incentive compatibility constraints can be replaced by
a single incentive compatibility constraint. As a result, the set of implementable incentive compatible allocations
expands.
D . ANDOLFATTO , E . NOSAL AND B . SULTANUM
4
two-agent setting where agents separately make production decisions and their costs are
private and correlated. The optimal direct mechanism has two equilibria: A truth-telling
equilibrium and a “cheating” equilibrium, where the cheating equilibrium leaves both agents
better off and the principal worse off compared to the truth-telling equilibrium. Ma et al. (1988)
shows how an indirect mechanism can prevent agents from misrepresenting their types—or
stop agents from cheating—in the Demski and Sappington (1984) model.9 Mookherjee and
Reichelstein (1990) generalizes this approach. Unfortunately, these results cannot be directly
applied to the banking problem because sequential service, which is absent in the mechanism
design models, complicates the analysis.
The paper is organized as follows. The next section describes the economic environment.
Section 3 characterizes the best weakly implementable allocation. In Section 4 we provide a
stripped down version of model to illustrative the key features of our mechanism. In Section
5 we construct an indirect mechanism and provide sufficient conditions for unique implementation of the best weakly implementable allocation. In Section 6, we examine examples for
which the sufficient conditions are not valid and Section 7 examines an alternative indirect
mechanism that addresses these examples. Some policy remarks are offered in the final
section.
2.
ENVIRONMENT
There are three dates: 0, 1 and 2. The economy is endowed with Y > 0 units of date-1
goods. A constant returns to scale investment technology transforms y units of date-1 goods
into yR > y units of date-2 goods. There are N ex-ante identical agents who turn out to be
one of two types: t ∈ T = {1, 2}. We label a type t = 1 agent “impatient” and a type t = 2
agent “patient”. The number of patient agents in the economy is drawn from the distribution
π = (π0 , . . . , π N ), where πn > 0, n ∈ N ≡ {0, 1, . . . , N }, is the probability that there are n
patient agents.10 A queue is a vector t N = (t1 , . . . , t N ) ∈ T N , where tk ∈ T is the type of the
agent that occupies the kth position or coordinate in the queue. Let Pn = t N ∈ T N | nt N = n
and Qn (t N ) = j | t j = 2 for t N ∈ Pn , where nt N denotes the number of patient agents in the
queue t N . Pn is the set of queues with n patient agents and Qn (t N ) is the set of queue positions
of the n patient agents in queue t N ∈ Pn .11 The probability of a queue t N ∈ Pn is πn /( N ),
n
where the binomial coefficient, ( N ), is the number of queues t N ∈ Pn . In other words, all
n
queues with n patient agents are equally likely. Agents are randomly assigned to a queue
position, where the unconditional probability that an agent is assigned to position k is 1/N.
Label an agent assigned to position k agent k. The queue realization, t N , is observed by no
one; not by any of the agents nor the planner. Agent k does not observe his queue position,
9 Postlewaite
and Schmeidler (1986) also produced an example where an indirect mechanism has a unique
equilibrium yielding the optimal outcome while the corresponding direct mechanisms possess multiple equilibria.
10 The full support assumption is not crucial to any result. It is imposed only for simplicity.
11 We omit the argument of Q ( t N ) throughout the paper to reduce notational burden.
n
5
PREVENTING BANK RUNS
k, but does privately observe his type t ∈ T. The utility function for an impatient agent is
U (c1 , c2 ; 1) = u c1 and the utility function of a patient agent is U (c1 , c2 ; 2) = ρu c1 + c2 ,
where c1 is date-1 consumption and c2 is date-2 consumption. The function u is increasing,
strictly concave and twice continuously differentiable, and ρ > 0 is a parameter.12 Agents
maximize expected utility.
The timing of events and actions are as follows. At date 0, the planner constructs a
mechanism that determines how date-1 and date-2 consumption are allocated among the N
agents. A mechanism consists of a set of announcements, M, and an allocation rule, c = (c1 , c2 ),
where c1 = (c1 , . . . c1 ) and c2 = (c2 , . . . c2 ). The planner can commit to the mechanism.13 The
N
N
1
1
queue t N is realized at the beginning of date 1. Then agents meet the planner sequentially,
starting with agent 1. Each agent k makes an announcement mk ∈ M.14 Only agent k and the
planner can directly observe mk . There is a sequential service constraint at date 1, which means
the planner allocates date-1 consumption to agent k based on the announcements of agents
j ≤ k, (mk−1 , mk ), where mk−1 = (m1 , . . . , mk−1 ), and each agent k consumes c1 (mk−1 , mk ) at
k
his date-1 meeting with the planner. Date 1 ends after all agents meet the planner. In between
dates 1 and 2 the planner’s resources are augmented by a factor of R. At date 2, the planner
allocates the date-2 consumption good to each agent based on the date-1 announcements, i.e.,
agent k receives c2 (m N ), where m N = (m1 , . . . , m N ) ∈ M N . Figure 1 depicts the sequence of
k
actions.
Realization
of t N ∈ T N
Bank announces
M, (c1 , c2 )
Period 0
announce mk and consume c1 (mk−1 , mk )
k
Sequential Service
Period 1
Return R realizes
and payment c2 (m N )
k
occurs
Period 2
Figure 1: Sequence of Actions.
3.
THE BEST WEAKLY IMPLEMENTABLE OUTCOME
An allocation is weakly implementable if it is an equilibrium outcome of a mechanism; it
is strongly or uniquely implementable if it is the unique equilibrium outcome of a mechanism. Among the set of weakly implementable allocations, the best weakly implementable
12 These
preferences are identical to the ones in Diamond and Dybvig (1983). In addition, they assume that
ρR > 1 and ρ ≤ 1.
13 For a discussion of bank fragility in a setting without commitment, see Ennis and Keister (2009a).
14 One could imagine that the planner makes announcement a to agent k before k makes his announcement.
k
For example, the planner could tell agent k his queue position, as in Green and Lin (2003), or the set of all
messages sent in the previous k − 1 planner-agent meetings, as in Andolfatto et al. (2007), or “nothing”, ak = ∅,
as in Peck and Shell (2003). The optimal mechanism, however, will have the planner announcing nothing. To
reduce notation, and without loss of generality, we assume that the planner cannot make announcements to
agents, unless otherwise specified. See footnote 16 for a discussion.
D . ANDOLFATTO , E . NOSAL AND B . SULTANUM
6
allocation provides agents with the highest expected utility. To characterize the best weakly
implementable allocation, it is without loss of generality to restrict the planner to use a direct
revelation mechanism, where agents only announce mk = tk ∈ T = {1, 2}. The welfare—which
we measure as the expected utility of an agent before he learns his type—associated with
allocation rule c when agents use strategies mk ∈ T is
N
(1)
N
πn
∑ (N) ∑ ∑ U
n =0
n
N
c 1 m k −1 , m k , c 2 m 1 ; t k .
k
k
t N ∈ Pn k=1
The allocation rule c = (c1 , c2 ) is feasible if for all m N ∈ T N
N
(2)
∑
Rc1 mk−1 , mk + c2 m M
k
k
≤ RY.
k =1
The best weakly implementable allocation has all agents k announcing truthfully, i.e.,
mk = tk . Allocation rule c must be incentive compatible in the sense that agent k has no
reason to announce mk = tk . Since impatient agents k only value date-1 consumption, they
always announce mk = 1.15 Patient agent k has no incentive to defect from the strategy mk = 2,
assuming that all other agents announce truthfully, if
N
(3)
ˆ
∑ πn ∑
n =1
t N ∈ Pn
1
∑ ρ u c 1 t k −1 , 2 + c 2 t N
k
k
n k∈Q
N
− u c1 tk−1 , 1 + c2 tk−1 , 1, tk+1
k
k
≥ δ,
n
j
where, for any vector x N = ( x1 , . . . , x N ), xi denotes xi , . . . , x j , δ > 0 is a parameter, and
ˆ
πn =
πn /( N )
n
N
N
∑ n =1 π n / ( n )
is the conditional probability that agent k is in a specific queue with n patient agents.16 The
1/n term that appears in (3) reflects that a patient agent has a 1/n chance of occupying each
of the patient queue positions in Qn .
The best weakly implementable allocation is given by the solution to
(4)
max (1) subject to (2) and (3),
15 This anticipates the result that the best weakly implementable allocation provides zero date-1 consumption
to agents who announce that they are patient, which implies that the incentive compatibility constraint for
impatient agents is always slack.
16 To characterize the best weakly implementable allocation, one wants to choose from the largest possible set
of incentive compatible allocations. This implies the planner should not make any announcements, as noted in
footnote 14. In particular, if the planner does not make any announcements, then there is only one incentive
compatibility constraint for all patient agents, (3). If, however, the planner did make an announcement ak
to agent k, there will be additional incentive constraints for the agent who received the announcement. For
example, suppose that ak = k for all k, i.e., the planner announces to each agent his place in the queue. Then
there would be N incentive compatibility constraints for patient agents, one for each queue position. Since an
appropriately weighted average of these distinct incentive constraints implies (3), the set of incentive compatible
allocations when the planner makes announcements is a subset of the set of incentive compatible allocations
when he does not. By not making any announcements, the planner is able to choose from a larger set of incentive
feasible allocations.
PREVENTING BANK RUNS
7
where mk = tk for all k ∈ N. We restrict δ > 0 to those values that admit a solution to
problem (4). Let c∗ (δ) = c1∗ (δ) , c2∗ (δ) be a solution to problem (4) and let W ∗ (δ) denotes
its maximum. We consider δ > 0 to guarantee that the incentive compatibility holds in
an open neighbourhood of c∗ . The existence of such neighbourhood is necessary for our
uniqueness result but δ > 0 can be made arbitrarily small. Therefore, we can apply Berge’s
maximum theorem, which says that W ∗ (0) is approximated by W ∗ (δ) when δ is close to zero.
The allocation rule c∗ (δ) has the following features: (i) an agent k who announces mk = 1
consumes only at date 1, that is, c2∗ (m1 , . . . , mk−1 , 1, mk+1 , . . . , m N ) = 0 for all k ∈ N; (ii) an
k
agent k who announces mk = 2 consumes only at date 2, that is, c1∗ (m1 , . . . , mk−1 , 2) = 0 for all
k
k ∈ N; and (iii) all agents j and k announcing m j = mk = 2 consume identical amounts at date
2, that is, c2∗ (m N ) = c2 (m N ) for all m j = mk = 2. The best-weakly implementable allocation is
j
k
c∗ (0), which corresponds to the allocation rule derived in Peck and Shell’s (2003) Appendix
B.
Define a bank run as a non-truthtelling equilibrium for the mechanism { M, c}, where some
and possibly all k ∈ Qn (t N ) announce mk = 1. Both Peck and Shell (2003) and Ennis and
Keister (2009b) demonstrate, by example, that the direct mechanism { T, c∗ (0)} can have two
equilibria: one where agents play truth-telling strategies, mk = tk for all k, and another where
all patient agents k play bank-run strategies, mk = 1.17 We claim that bank-run equilibria
arise in these examples because the direct revelation mechanism they use, { T, c∗ (0)}, is not
an optimal one; there exists an indirect mechanism that strongly implements the best weakly
implementable allocation. Before we demonstrate this result, we provide a simple example
that illustrates the basic intuition underling our optimal mechanism.
4.
A SIMPLE EXAMPLE
Consider a stripped-down version of a Diamond-Dybvig model where there are only 2
agents—column and row—and both agents are patient. Agents simultaneously announce
that they are either patient, m = 2, or impatient, m = 1. The payoffs to agents for this game
are given by
m=1
m=2
m=1
1, 1
2, 0
m=2
0, 2
3, 3
This simple normal form game captures two important insights of the Diamond-Dybvig
model. First, there are multiple equilibria: one where both agents announce the truth,
m = 2, one where both agents announce they are impatient, m = 1, and another where both
17
The Ennis and Keister (2009b) bank-run example is in section 4.2 of their paper. There, agents do not know
their position in the queue, as in Peck and Shell (2003), and the utility functions of patient and impatient agents
are the same, ρ = 1, as in Green and Lin (2003).
D . ANDOLFATTO , E . NOSAL AND B . SULTANUM
8
agents randomize between each strategy with probability half. And second, the truth-telling
equilibrium generates the higher payoffs for agents than a bank-run equilibrium.
Consider now a normal form game that simply augments the announcement space of the
original game from {1, 2} to {1, 2, g}, with associated payoffs
m=g
m=1
m=2
m=1
1, 1
2, 0
0, 1+
m=2
0, 2
3, 3
3, 2+
m=g 1+ , 0 2+ , 3
,
There are three features of the augmented game that we would like to highlight. First, when
agents restrict their announcements to {1, 2}, the payoffs they receive are identical to the
original game. Second, announcement m = g strictly dominates announcement m = 1. And
finally, the payoff to an agent who announces m = 2 is the same regardless if his opponent
announces m = 2 or m = g.
Since agents never play m = 1 in the augmented game—it is strictly dominated by playing
m = g—the relevant augmented game that agents play is
m=2
m=2
m=g
3, 3
2, 0
m=g 2+ , 3
,
But in this relevant augmented game, announcement m = g is strictly dominated by announcement m = 2. Therefore, the unique iterated strict dominant equilibrium to the augmented
game is one of truthtelling, m = 2. Hence, by modifying the game that agents play, we get rid
of the “bad” bank-run equilibria that existed in the original game.
The best weakly implementable allocation described in Section 3, c∗ (δ), is somewhat more
complicated than the payoff structure in the stripped-down example. Nevertheless, our
approach to eliminate the bad equilibria is the same: We construct an indirect mechanism
ˆ ˆ
ˆ
ˆ
{ M, c} with the properties: (i) M = {1, 2, g}; (ii) announcing mk = 1 is strictly dominated by
ˆ
ˆ
announcing mk = g for patient agents; and (iii) after announcement mk = 1 is eliminated for
ˆ
patient agents announcing mk = 2 strictly dominates announcing mk = g. The uniqueness
result is a bit more tricky to prove because we need enough resources to construct an allocation
ˆ
rule c that provides sufficiently high payoffs to patient agents so that announcing truthfully is
the unique rational strategy. In the subsequent section, we characterize an indirect mechanism
and provide sufficient conditions under which this mechanism uniquely implements the best
weakly implementable allocation using dominance arguments similar to the simple example.
9
PREVENTING BANK RUNS
5.
AN INDIRECT MECHANISM
ˆ ˆ
ˆ
ˆ
Consider an indirect mechanism { M, c}, where M = {1, 2, g} and c is described below. The
ˆ
ˆ
basic construction of the allocation rule c uses c∗ (δ). If agent j announces mk = 1, then
(5)
ˆk ˆ
c 1 m k −1 , 1 =
ˆ
c 1∗ ( δ ) m k −1 , 1
k
0
ˆ
if m j ∈ {1, 2} for all j < k
ˆ
if m j = g for some j < k
and
ˆk ˆ
ˆN
c2 mk−1 , 1, mk+1 = 0.
ˆ
An agent k announcing mk = 1 receives the date-1 consumption payoff under the direct
ˆ
ˆ
revelation mechanism { T, c∗ (δ)} only if all earlier agents j < k announce either m j = 1 or m j = 2;
otherwise he receives zero. That is, there is a suspension of first period payments after an agent
ˆ
j < k announces m j = g. The date-2 consumption payoff associated with the announcement
ˆ
ˆ
mk = 1 is zero, as in the direct revelation mechanism { T, c∗ (δ)}. If agent k announces mk = g,
then
(6)
ˆk ˆ
c 1 m k −1 , g = 0
and
ˆk ˆ
ˆN
ˆk ˆ
c2 mk−1 , g, mk+1 = c1 mk−1 , 1 + ,
where > 0 is an arbitrarily small number. To keep the presentation simple, we assume
throughout the paper that is taken small enough so all results hold. If agent k announces
ˆ
mk = g, then he receives a zero payoff at date 1. At date 2, he receives a payoff that is
ˆ
slightly bigger than the date-1 payoff he would receive by announcing mk = 1—see (5)—
ˆk ˆ
ˆN
ˆk ˆ
ˆ
which implies that c2 mk−1 , g, mk+1 = c1 mk−1 , 1 + . Hence, announcing mk = g strictly
ˆ
ˆ
dominates announcing mk = 1 for any patient agent k. Finally, if agent k announces mk = 2,
then
(7)
ˆk
c1
ˆ
m
k −1
,2 = 0
and
ˆk
c2
ˆ
m
k −1
ˆN
, 2, mk+1
=
R Y − ∑ N 1 c 1 ( m j ) − ∑ N 1 c 2 ( m N )1m j = g
ˆ
j = ˆj ˆ
j = ˆj ˆ
nm N
ˆ
ˆ
where nm N represents the number of agents who announced m = 2 in the announcement
ˆ
N and 1
ˆ
vector m
ˆ
ˆ
m j = g is an indicator function, where 1m j = g = 1 if m j = 1 and 0 otherwise. If
ˆ
agent k announces mk = 2, then he receives a 1/nm N share of the total date-2 output that
ˆ
remains after payments to agents j who announced either m j = 1 or m j = g are made. Since
ˆ
ˆ
the allocation rule c, given by (5)-(7), depends on δ and , we will denote it as c (δ, ).
Generally speaking, a patient agent j who announces m j = 1 adversely affects the payoffs
of truthfully announcing patient agents in two ways. First, the payments to an agent who
announces m j = 1 are made in period 1 which implies that these resources cannot benefit from
the investment opportunity, R, available between dates 1 and date 2. Second, if impatient
agents have a relatively high marginal utility of consumption compared to patient agents, i.e.,
ρ is small, then, due to risk-sharing considerations, payments to agents who announce m j = 1
can be quite high, leading to less resources available to the patient agents. Interestingly, the
ˆ
story is a bit different when patient agent j announces m j = g and impatient agents have
a relatively low marginal utility of consumption compared to patient agents. Following
a g announcement there is a suspension of date 1 payments and agents who announce g
D . ANDOLFATTO , E . NOSAL AND B . SULTANUM
10
receive their payments at date 2. Hence, all suspended payments benefit from the investment
opportunity that is available between dates 1 and 2, and patient agents who announced
truthfully will receive a fraction of the investment return, R. In addition, if ρ is relatively
large, then the date-2 payment to agent j will be relatively low, which benefits truth-telling
patient agents.
Patient agent k who announces truthfully will benefit from announcement m j = g if allocaˆ
tion rule c(δ, ) has the following property
ˆ
ˆN
ˆ
ˆN
ˆk
ˆ
ˆN
ˆk
(P1) c2 (δ, )(mk−1 , 2, mk+1 ) ≥ c2 (δ, )(tk−1 , 2, tk+1 ) = c2∗ (δ)(tk−1 , 2, tk+1 ),
k
ˆ
ˆ
ˆ
where ti ∈ T i (tiN ∈ TiN ) is a vector of length i (T − i) such that for each j ≤ i (i ≤ j ≤ N), t j = 1
ˆ
ˆ ˆ
ˆ
ˆ
ˆ
if m j = 1 and t j = 2 if either m j = 2 or m j = g. In words, vector ti (tiN ) is constructed from
ˆ ˆ
the message vector mi (miN ) by replacing all of the g’s with 2’s. The first term in (P1) is the
payoff to a truthfully announcing patient agent when some (patient) agents announce g. The
second term is the payoff to patient players when those g announcements are replaced by 2,
which, by construction, also equals the payment from the best implementable allocation. If
ˆ
the contract c(δ, ) is characterized by property (P1), then, clearly, a truthfully announcing
patient agent benefits if some other (patient) agent announces g. In fact, his payoff will exceed
that associated with the best weakly implementable allocation, c∗ (δ).
ˆ
Under what circumstances will the allocation rule c(δ, ) have property (P1)? The above
discussion suggests that truthfully announcing patient agents benefit from a m j = g announcement the larger is R and/or the larger is ρ. (Recall that the higher is ρ, the smaller will be the
payments to impatient agents.) Our first proposition verifies this intuition.
P ROPOSITION 1: If ρR > 1, then property (P1) holds.
P ROOF : See Appendix.
Property (P1) seems to imply that, since more resources are available to patient players
who announce truthfully and less to patient players who announce g, it is rational for patient
players to announce truthfully. Our main proposition demonstrates that this intuition is, in
fact, correct.
ˆ ˆ
P ROPOSITION 2: If property (P1) holds, then the indirect mechanism { M, c(δ, )} strongly implements allocation c∗ (δ) in rationalizable strategies.
ˆ ˆ
P ROOF : The mechanism { M, c(δ, )} induces a symmetric Bayesian game Γ = { T, S} where,
ˆ
T = {1, 2} is the set of types, st ∈ M is the player’s message contingent on his type t ∈ T and
ˆ
S = {(s1 , s2 ) ∈ M2 } is the set of pure strategies. We solve the game by iterated elimination of
strictly dominated strategies in two rounds.
Round 1 - Any strategy (s1 , s2 ) ∈ S, with s1 = 1, is strictly dominated by (1, s2 ) since,
contingent on being impatient, an agent only derives utility from period 1 consumption.
Additionally, any strategy (s1 , 1) is strictly dominated by (s1 , g) since, contingent on being
patient, agents are indifferent between period 1 or period 2 consumption and announcing
11
PREVENTING BANK RUNS
g always gives a total payment that is higher than announcing 1. Let S1 = {(1, 2), (1, g)}
denote the set of strategies that survive the first round of elimination of strictly dominated
strategies.
Round 2 - When strategies are restricted to S1 , impatient agents announce 1 and patient
agents announce either 2 or g. From property (P1), the lower bound on the expected payoff to
a patient player who announces 2 is
N
ˆ
∑ πn ∑
n =1
t N ∈ Pn
1
N
∑ ρu c2∗ (δ)(tk−1, 2, tk+1 ) .
k
n k∈Q
n
Since the payment to agent k who announces mk = g is either c1∗ tk−1 , 1 +
k
expected payoff to a patient player who announces g is bounded above by
N
ˆ
∑ πn ∑
n =1
t N ∈ Pn
1
∑ ρu c1∗ (δ) tk−1, 1 +
k
n k∈Q
or , the
.
n
Since u is continuous, there exists an > 0 sufficiently small so that
N
ˆ
∑ πn ∑
n =1
t N ∈ Pn
1
∑
n k∈Q
ρu c1∗ (δ) tk−1 , 1 +
k
− ρu c1∗ (δ) tk−1 , 1
k
< δ.
n
The incentive compatibility condition (3) can be rewritten as
N
ˆ
∑ πn ∑
n =1
t N ∈ Pn
1
N
∑ ρu c2∗ (δ) tk−1, 2, tk+1
k
n k∈Q
n
N
≥
ˆ
∑ πn ∑
n =1
t N ∈ Pn
1
∑ ρu c1∗ (δ) tk−1, 1
k
n k∈Q
+ δ.
n
Combining the above two inequalities, we get
N
(8)
ˆ
∑ πn ∑
n =1
t N ∈ Pn
N
1
1
N
ˆ
∑ ρu c2∗ (δ)(tk−1, 2, tk+1 ) > ∑ πn ∑ n ∑ ρu c1∗ (δ)(tk−1, 1) +
k
k
n k∈Q
n =1
k∈Q
tN ∈P
n
n
n
Therefore, the strategy (1, g) is strictly dominated by the strategy (1, 2) in S1 . Let S2 be the
set of strategies that survive the second round of elimination of strictly dominated strategies.
Since S2 = {(1, 2)} is a singleton, the game is iterated strict dominance solvable. The unique
equilibrium strategy is the truth-telling s = (1, 2), which implies the same outcome as the
truth-telling equilibrium of the direct mechanism { T, c∗ (δ)}.
ˆ
If allocation c(δ, ) has property (P1), then, just as in the stripped-down example from
ˆ ˆ
Section 4, mechanism { M, c(δ, )} admits only one equilibrium characterized by truthtelling
ˆ ˆ
for all agents. Hence, mechanism { M, c (δ, )} does not allow bank runs. In addition, the
ˆ
allocation delivered by the mechanism, c (δ, ), can be made arbitrarily close to the best weakly
∗ (0) by choosing δ arbitrarily close to zero.
implementable allocation c
Together, Propositions 1 and 2 imply that a sufficient condition for unique implementation
is ρR > 1. This is quite interesting and, perhaps, even remarkable. Diamond and Dybvig
(1983) construct a model where fractional reserve banks endogenously arise and use the
model to help us understand the notion that banks are inherently unstable. Their 1983 article
.
D . ANDOLFATTO , E . NOSAL AND B . SULTANUM
12
requires that ρR > 1. Propositions 1 and 2 in this article, however, indicates that for this
parametrization banks are always stable.
We want to emphasize that conditions stated in Propositions 1 and 2 are only sufficient
conditions. Regarding Proposition 1, one can see from the proof that if incentive compatibility
condition (3) does not bind, then condition the ρR > 1 is not necessary. This means that
ˆ
contract c(δ, ) can be consistent with property (P1) even if ρR < 1. In the subsequent section,
we provide an example of this (even when the incentive compatibility condition (3) binds).
Regarding Proposition 2, property (P1) allows us to derive a lower bound on the expected
payoff of a patient agent announcing m = 2 and, therefore, to use dominance arguments
to demonstrate uniqueness. But neither, such lower bound or dominance arguments, are
necessary for uniqueness. In the subsequent section we provide an example where contract
ˆ ˆ
ˆ
allocation c(δ, ) does not have property (P1) but the indirect mechanism { M, c(δ, )} uniquely
ˆ
implements c(δ, ).
6.
SOME EXAMPLES
In this section we provide some examples that show the sufficient conditions described in
Propositions 1 and 2 are not necessary for unique implementation of the allocation rule c∗ (δ).
The first example shows that property (P1) can hold when ρR < 1. A second example shows
that allocation rule c∗ (δ) can be uniquely implemented when property (P1) is violated.
Common to all examples are: (i) R = 1.05; (ii) Y = 6; (iii) ρR < 1; (iv) δ = 10−10 ; and (v) the
general structure of preferences is given by 18
(9)
u( x ) =
( x + 1 )1− γ − 1
, γ > 1.
1−γ
In the first example, N = 2, ρ = 0.9, γ = 1.01 and (π0 , π1 , π2 ) = (0.005, 0.4975, 0.4975).
Notice that ρR < 1. The best weakly implementable allocation, c∗ (0), which is obtained by
∗
∗
solving (4), has c1 1 (1) = 3.1487 and c2 1 (2, 1) = 3.1481. The other payments can be derived
from the resource constraint (2) holding at equality. It is straight forward to show that the
direct mechanism { T, c∗ (0)} admits a bank-run equilibrium for this example. For arbitrarily
small, property (P1) holds, even though ρR < 1. Therefore, although ρR > 1 is a sufficient
condition for property (P1), it is not a necessary one. Since property (P1) is satisfied in this
ˆ ˆ
example, Proposition 2 implies that { M, c(δ, )} uniquely implements allocation c∗ (δ) for δ
and small. In this example, constraint (3) binds. This implies that incentive constraints
in the Green and Lin (2003) environment—where agents know their queue positions—will
also bind and that the best implementable allocation from that environment is not equal to
c∗ (0).19 Hence, the Green and Lin (2003) mechanism is unable to even weakly implement the
allocation c∗ (δ), where δ is arbitrarily small.
18
Notice that u(0) = 0.
environment can be turned into the Green and Lin (2003) environment by allowing the planner to tell
agent k his queue position, k, before agent k makes sends his message.
19 Our
PREVENTING BANK RUNS
13
The second example replicates the Peck and Shell (2003) example in Appendix B. The only
difference between the examples is the specification of preferences. Peck and Shell (2003)
assume that u( x ) = c1−γ /(1 − γ), which implies that u(0) = −∞. For these preferences,
our mechanism trivially uniquely implements allocation c∗ (δ), since patient agent k will
never announce mk = g if there is a probability, however small, that some other agent j will
announce m j = g. The parameters for our second example are N = 2, ρ = 0.1, γ = 2 and
(π0 , π1 , π2 ) = (0.25, 0.5, 0.25). Notice that ρR < 1. The best weakly implementable allocation,
∗
∗
c∗ (0), is characterized by c1 1 (1) = 3.0951 and c2 1 (2, 1) = 3.1994. Allocation c∗ (0) features
bank runs and a binding incentive constraint (3). (This implies that a Green and Lin (2003)
mechanism cannot weakly implement c∗ (0).) It is straightforward to demonstrate that the
ˆ ˆ
mechanism { M, c(δ, )} uniquely implements allocation c∗ (δ), for δ and arbitrarly close
∗
to zero. For this example c1 1 (2, 1) + c∗2 (2, 2) > RY, which implies that property (P1) is not
ˆ
ˆ
satisfied for all m N ∈ M N . Hence, property (P1) is not necessary for unique implementation.
We are not aware of any mechanism in the literature that can implement the best weakly
implementable allocations from these two examples. We have experimented with many
combinations of model parameters. We are unable to find a set of parameters for which the
ˆ ˆ
indirect mechanism { M, c(δ, )} cannot uniquely implement an allocation that is arbitrarily
close to the best weakly implementable allocation. Our search, however, was restricted to
ˆ ˆ
N ∈ {2, 3}. It is, of course, possible that the indirect mechanism { M, c(δ, )} does not uniquely
implement the best weakly implementable allocation for some set of parameters—that we
were unable to recover—when ρR ≤ 1. In the next section, we propose an alternative indirect
mechanism to deal with this case.
7.
AN ALTERNATIVE MECHANISM
ˆ ˆ
The indirect mechanism { M, c(δ, )} uniquely implements allocation c∗ (δ) for the N ∈
{2, 3} examples we considered, but there may exist primitives for which it does not. To
address this issue, we construct an alternative mechanism that uniquely implements c∗ (δ)
in pure and symmetric strategies. The mechanism, however, does rule out the existence of
mixed or asymmetric Nash equilibria.20
˜
˜ ˜
˜
The alternative indirect mechanism is denoted by { M, c}, where M = {1, 2, g} and c is
k−1 ∈ M k−1 , define tk−1 ∈ T k−1 as a vector of length k − 1,
˜
˜
˜
described below. For a given m
˜
˜
˜
˜
˜
where for each j ≤ k − 1, t j = 1 if either m j = 1 or m = g; and t j = 2 if m j = 2. It is important
˜
ˆ
˜
ˆ
to emphasize that the relationship between m j and t j is different from that of m j and t j .
˜
˜
Specifically, the vector tk−1 is constructed from mk−1 by replacing any g’s with 1’s, while
ˆk−1 is constructed from mk−1 by replacing any g’s with 2’s.
ˆ
vector t
˜
The construction of the allocation rule c uses the best weakly implementable allocation
ˆ ˆ
Mechanism { M, c(δ)} does rule out these equilibria when ρR > 1. It is interesting to note, however, that in
the literature virtually all of the analyses of the Diamond-Dybvig model focus on pure and symmetric equilibria.
20
D . ANDOLFATTO , E . NOSAL AND B . SULTANUM
14
˜
rule, c∗ (0). If agent k announces mk = 1, then
˜
˜k ˜
(10) c1 (mk−1 , 1) = c1∗ (0) t j−1 , 1
k
and
˜j ˜
˜N
c2 (mk−1 , 1, mk+1 ) = 0.
˜
When agent k announces mk = 1 he receives the consumption associated with announcing
˜
mk = 1 in the direct revelation mechanism { T, c∗ (0)}, where announcement m j = g in the
˜
˜
indirect mechanism is treated as if it is m j = 1. If agent k announces mk = g, then
˜k ˜
(11) c1 (mk−1 , g) = 0
and
˜
c 1∗ (0 ) t k −1 , 1 +
k
0
˜k ˜
˜N
c2 (mk−1 , g, mk+1 ) =
˜
if m j = 1 for all j = k
,
otherwise
˜
where > 0 is arbitrarily small. If agent k announces mk = g, he receives a zero date 1 payoff.
˜
His date 2 payoff is slightly bigger than what he would receive by announcing mk = 1 but
only if all other agents j announce m j = 1; otherwise, he receives a payoff of zero. Finally, if
˜
agent k announces mk = 2, then
(12)
˜k
c1
˜
m
k −1
,2 = 0
and
˜k
c2
˜
m
k −1
˜N
, 2, mk+1
=
R Y − ∑ N 1 c 1 ( m k ) − ∑ N 1 c 2 ( m N )1m j = g
˜
j = ˜j ˜
j = ˜j ˜
nm N
˜
,
˜
where nm N represents the number of agents j who announced m j = 2. If agent k announces
˜
˜
mk = 2, then he receives an equal share of date-2 output net of any payments made to agent j
˜
˜
who announce m j = g. Since the allocation rule c given by (10)–(12) depends on δ, we will
˜
denote it as c(ε).
˜ ˜
When considering only pure and symmetric equilibria, the indirect mechanism { M, c(ε)}
is quite powerful. Specifically,
˜ ˜
P ROPOSITION 3: The indirect mechanism { M, c(ε)} uniquely implements the best weakly implementable allocation c∗ (0) in pure and symmetric Nash equilibrium.
˜
P ROOF : All impatient agents k announce truthfully since announcing mk = 1 results in a
˜
strictly positive date-1 payoff and announcing mk = 1 results in a date-1 payoff equal to zero.
˜
First, there cannot exist an equilibrium where all patient agents k announce mk = 1. Suppose
such an equilibrium exists. Then some patient agent j can defect from proposed equilibrium
˜
and announce m j = g. Agent j’s payoff is strictly greater than the payoff associated with
˜
announcing m j = 1 by the amount > 0; a contradiction.
˜
Second, there cannot be an equilibrium where all patient players k announce mk = g. To
˜
see this, note that if agent k announces mk = g, then his payoff will be zero if there are other
patient agents in the economy. The (proposed) equilibrium payoff, therefore, is
(13)
ˆ
π1 N
ˆ
∑ ρu c1∗ (0) 1k−1, 1 + ε + (1 − π1 ) ρu (0) .
k
N k =1
˜
If instead, agent k defects from proposed play announces mk = 1, his payoff will be
1 N
(14)
∑ ρu c1∗ (0) 1k−1, 1
k
N k =1
.
PREVENTING BANK RUNS
15
ˆ
Since π1 < 1, for > 0 sufficiently small (14) exceeds (13); a contradiction.
˜
Third, there is an equilibrium where all patient agents k announce mk = 2. By construction,
˜
patient agent j has no incentive to announce m j = 1 when all other agents announce truthfully,
∗ (0) is incentive compatible for patient agents when m is restricted to the
˜j
i.e., allocation rule c
set {1, 2}. Suppose, instead, that patient agent j defects from equilibrium play and announces
˜
m j = g. In this case, his payoff will be only slightly greater than the payoff associated with
˜
announcing m j = 1 if and only if he is the only patient agent in the economy—an event that occurs
ˆ
ˆ
with probability π1 . With probably 1 − π1 , there are other patient agents k who announce
ˆ
˜
mk = 2, which implies that patient agent j receives a zero payoff. For any π1 < 1, there exists
˜
an ε > 0 sufficiently small so that the expected payoff associated with announcing m j = g is
˜
strictly less than that associated with announcing m j = 1, when all other agents announce
truthfully.
˜ ˜
The unique symmetric and pure equilibrium strategy for mechanism { M, c} is character˜
ized by truth-telling, i.e., mk = tk for all k. By construction, these strategies implement the
best weakly implementable allocation in c∗ (0).
There is an interesting tradeoff between the two indirect mechanisms that have been
ˆ ˆ
studied. Mechanism { M, c} has a very weak equilibrium concept, rationalizability. However,
unique implementation is guaranteed only if the restriction ρR > 1 is satisfied. Unique
implementation is possible when ρR ≤ 1, as our examples demonstrate, but it has to be verified
˜ ˜
on a case-by-case basis. Mechanism { M, c} has a very strong equilibrium concept, pure and
symmetric Nash equilibria. However, no restrictions are required on model parameters to
guarantee unique implementation. Unique implementation is possible for when mix strategies
are allowed, but it has to be verified on a case-by-case basis.
ˆ ˆ
Finally, the indirect mechanism ( M, c) relies on both punishments and suspension for
unique implementation. Since strategies are restricted to be pure and symmetric, indirect
˜ ˜
mechanism ( M, c) only relies on punishments for unique implementation.
8. P OLICY DISCUSSION
The most common prescription for enhancing the stability of demandable debt is to
modify the contract to include a partial suspension clause. For example, Cochrane (2014),
suggests that if securities are designed so debtors have the right to delay payment, suspend
convertibility, or pay in part, then it is much harder for a run to develop. Santos and Neftci
(2003) recommend the use of extendable debt—which is a suspension in payments—in the
sovereign debt market to help mitigate the frequent debt crises that have afflicted emerging
economies and, recently, more advanced economies as well. In June 2013, the Securities and
Exchange Commission (SEC) announced a set of proposals to help stabilize money market
funds (MMFs). One of the key proposals recommends that the MMF board of directors have
the discretion to impose of penalty redemption fees and redemption gates—or suspsension of
payments—in times of heavy redemption activity.
D . ANDOLFATTO , E . NOSAL AND B . SULTANUM
16
The effect of such proposals is to render demandable debt more state-contingent. In this
sense, the proposals above are consistent with the properties of the optimal debt contracts
described in Diamond and Dybvig (1983), Green and Lin (2003), and Peck and Shell (2003).
But given that bank-run equilibria remain a possibility in the latter model, one is led to
question whether the the use of such measures constitute only necessary, and not sufficient
conditions, for stability.
The key question concerns the issue of precisely what information is be used to condition
the suspension/extension clause. In the Diamond and Dybvig (1983) model without aggregate
risk, suspension is triggered when “reserves” are reach a well-specified critical level. Evidently,
this conditioning factor is sufficient to prevent runs in that environment. Similarly, the partial
suspension schedules described in Green and Lin (2003) and Peck and Shell (2003) are
triggered by measures of reserve depletion (more precisely, the history of reported types). In
reality, the volatility of redemption rates varies across different classes of MMFs. Schmidt et al.
(2013), for example, report that MMFs with volatile flow rates prior to the financial crisis of
2008 were more likely to experience runs during the crisis. How are directors of these funds
to ascertain whether a spike in redemptions is attributable to fear rather than fundamentals?
Our indirect mechanism suggests that information beyond some measure of redemption
activity or resource availability is needed to prevent the possibility of a bank-run. We need
to know why depositors are exercising their redemption option. For better or worse, this
information is private and must therefore be elicited directly—as in our model—or inferred
indirectly—through some other means. Of course, information revelation must be incentive
compatible.
Just how realistic is this idea? There is, in fact, historical precedence for the practice of
soliciting additional information in periods of heavy redemption activity. For example, banks
would sometimes permit limited redemptions to occur for depositors that could demonstrate
evidence of impatience, e.g., a need to meet payroll. Gorton (1985, fn 7) reports that 19th
century clearinghouses would regularly investigate rumors pertaining to the financial health
of member banks.
As a practical matter, the spirit of our mechanism could be implemented in several different
ways. One way would be to permit depositors to pay a small fee for the right to have their
funds diverted to a segregated, priority account.21 Such an action could be interpreted as
a communication of an impending run. The priority debt differs from other debt only in
the event of failure and the ratio of priority to non-priority debt outstanding informs the
issuer on the degree to which depositors expect the bank to fail. In principle, the suspension
clause could be made conditional on this ratio hitting some specified threshold. It does not
need to be official as long as there is a mutual understanding that it will be used. And along
the lines suggested by our mechanism, if one knows that the bank will suspend before any
rumor-induced trouble affects their balance sheet, then depositors know that there will be no
21 This
is effectively what happens in our mechanism when a depositor reports m = g.
17
PREVENTING BANK RUNS
reason, in equilbrium, to actually exercise the option of converting their claims to priority
debt.
To summarize, current policy proposals designed to prevent, or at least mitigate, bank
runs in demandable debt structures focus on enhancing state-contingency, with contingencies
dictated by some measure of redemption activity or resource depletion. Our analysis suggests
that while state contingency is necessary, it may not be sufficient to prevent bank runs.
Suspension clauses should be conditioned on information relating to depositor beliefs about
what they perceive to be happening around them. The desired information could be elicited in
an incentive compatible manner through an appropriate modification of the deposit contract—
an example of which we described above. If we are wrong in our present assessment, the
inclusion of such a clause would be inconsequential. But if we are correct, then the inclusion
of such a clause may help to prevent bank runs in debt structures that are presently run prone.
APPENDIX : PROOF OF PROPOSITION
1
In order to prove proposition 1 we first establish the following result.
¯
¯
¯
L EMMA 1: If ρR > 1 then c1∗ (δ) tk−1 , 1 < c2∗ (δ) tk−1 , 2 N −k+1 for all k ∈ N and tk−1 ∈ T k−1 .
k
n denotes the n −dimensional vector of twos.
Where 2
P ROOF : Since c∗ (δ) solves problem (4), it satisfies the implied first-order conditions.22 Let λt N
denote the Lagrange multiplier of the feasibility constraint (2) for each t N ∈ T N and µ denotes
the Lagrange multiplier of the incentive compatibility (3). By simplicity, λt N is normalized by
πnt N /(nN ), where nt N denotes the number of type 2 players in queue t N . And µ is normalized
N
t
N
¯
by π = ∑n=1 πn /( N ). Since u (0) = ∞ the constraint c1 ≥ 0 and c2 ≥ 0 are not binding and the
n
respective Lagrange multipliers can be ignored. The first order conditions of the problem are
given below.
(15)
¯
c1 t k
k
N
πn
∑ (N) ∑
n =0 n t N ∈ P
:
¯
u c1 t k
k
N
− λt N R −
n =1
n
¯
tk =tk
πn
∑ (N) ∑
n
t N ∈ Pn
¯
tk =(tk −1 ,2)
µρ
¯
u c1 t k
k
nt N
=0
¯
¯
for all k ∈ N and tk−1 ∈ T k−1 such that tk = 1; and
(16)
c2 t N
:
πn
N
nt N
ρu c2 t N
− λt N +
µρ
u c2 t N
nt N
=0
for all t N ∈ T N such that nt N > 0. We can solve the above equations for λt N and obtain
λt N =
22
ρ 1+
µ
nt N
1
Ru
u c2 t N
c1 t N
N
if
nt N > 0
¯
if
nt N = 0
¯
.
From now on we will denote c∗ (δ) just by c in order to keep the notation short.
D . ANDOLFATTO , E . NOSAL AND B . SULTANUM
18
Note that c2 t N is not defined if t N = 1 N = (1, 1, . . . , 1)—there is no second period payments
when every depositor announces to be of type impatient in the first period. In order to keep
the notation short, let us define u c1 1 N = ρRu c2 1 N and 1/n1N = 0. Then, λt N is
N
given by
(17) λt N = ρ 1 +
µ
nt N
u c2 t N
.
¯
¯
After replace equation (17) in equation (15) we obtain that for all k ∈ N and tk = (tk−1 , 1) ∈
T k −1 :
N
πn
∑ (N) ∑
n =0
n
¯
u c1 t k
k
N
−
πn
∑ (N) ∑
n =1
t N ∈ Pn
¯
tk =tk
n
t N ∈ Pn
¯
tk =(tk −1 ,2)
N
µρ
¯
u c1 t k
k
nt N
=
πn
∑ (N) ∑
n
n =0
Rρ 1 +
t N ∈ Pn
¯
tk =(tk−1 ,1)
µ
nt N
u c2 t N
which is equivalent to
¯
P t k = ( t k −1 , 1 ) −
N
πn
∑ (N) ∑
n =1
n
t N ∈ Pn
¯
tk =(tk−1 ,2)
µρ
nt N
¯
u c1 t k
k
N
=
πn
∑ (N) ∑
n =0
n
Rρ 1 +
t N ∈ Pn
¯
tk =(tk −1 ,1)
µ
nt N
u c2 t N
.
We can also write the equation in expectations, which yields to the formula
(18)
¯
1 − γ t k −1
¯
u c1 t k
k
= Et N |tk =(tk−1 ,1) Rρ 1 +
¯
µ
nt N
u c2 t N
¯
¯
¯
where γ tk−1 = P tk = (tk−1 , 2) Et N |tk =(tk−1 ,2) [µρ/nt N ] P tk = (tk−1 , 1) .
¯
The result will be derived from equation (18). Let us use induction on k ∈ N starting from
k = N and going down until k = 1.
¯
¯
Proof for k = N: Fix any t N = (t N −1 , 1). From equation (18) we have that
¯
P t N = ( t N −1 , 2 )
µρ
¯
1−
×
u c 1 t N −1 , 1
N
N = ( t N −1 , 1 )
¯
n(t N −1 ,2)
P t
¯
= Rρ 1 +
µ
n(t N −1 ,1)
¯
¯
u c 2 t N −1 , 1
¯
¯
¯
¯
which implies that u (c1 t N −1 , 1 > u c2 t N −1 , 1 . Thus, c1 t N −1 , 1 < c2 t N −1 , 1 . We
N
N
know that the resources constraints holds at equality because u is strictly increasing. Therefore,
¯
¯
¯
¯
n(t N −1 ,2) c2 t N −1 , 2 = [nt N + 1] c2 t N −1 , 2 = nt N c2 t N −1 , 1 + Rc1 t N −1 , 1
¯
¯
¯
N
And after reorganize the equation above we have that
¯
c ( t N −1 , 2 ) =
2
¯
n ( t N −1 , 1 ) 2 N −1
1
¯
¯
¯
, 1) +
c (t
Rc1 (t N −1 , 1) > c1 (t N −1 , 1).
N
¯N −1 , 1) + 1
¯N −1 , 1) + 1 N
n(t
n(t
¯
¯
Hence, for the case k = N, we can conclude that c1 tk−1 , 1 < c2 tk−1 , 2 N −k+1 .
k
¯
¯
Proof for k < N: Assume the result holds for all j > k and t j = (t j−1 , 1) ∈ T j . That is, for
¯
¯
all j > k we have c1 t j−1 , 1 < c2 t j−1 , 2 N − j . Let us show it also holds for k. Fix some
j
.
19
PREVENTING BANK RUNS
¯
¯
tk = (tk−1 , 1) ∈ T k−1 , then equation (18) is given by
¯
u c1 t k
k
=
1
µ
Et N |tk =(tk−1 ,1) Rρ 1 +
¯
k −1
¯
nt N
1−γ t
u c2 t N
.
Note that, for any function X : T N → R, the conditional expectation can be decomposed as
EN
t
¯
tk =tk
N
X tN
∑
=
¯
¯
P t j = t k , 2 j − k −1 , 1 t k = t k E N
j = k +1
X tN
¯
t j = tk ,2 j−k−1 ,1
t
+
¯
¯
¯
P t N = tk , 2 N −k tk = tk X tk , 2 N −k .
Applying this decomposition to equation (18) we obtain
¯
u c1 t k
k
N
=
∑
¯
¯
P t j = t k , 2 j − k −1 , 1 t k = t k E N
t
j = k +1
Rρ 1 +
¯
t j = tk ,2 j−k−1 ,1
µ
nt N
1
.
¯
1 − γ t k −1
µ
¯
u c2 t k , 2 N − k
n(tk ,2N −k )
¯
¯
¯
P t N = tk , 2 N −k tk = tk Rρ 1 +
u c2 t N
By equation (18) we know that
¯
1 − γ t k , 2 j − k −1
¯
u c 1 t k , 2 j − k −1 , 1
j
=E N
t
¯
t j = tk ,2 j−k−1 ,1
Rρ 1 +
µ
nt N
u c2 t N
for j = k + 1, . . . , N. Hence,
¯
u c1 t k
k
N
=
∑
¯
¯
P t j = t k , 2 j − k −1 , 1 t k = t k
¯
1 − γ t k , 2 j − k −1
¯
u c 1 t k , 2 j − k −1 , 1
j
+
j = k +1
¯
¯
P t N = tk , 2 N −k tk = tk Rρ 1 +
µ
¯
u c2 t k , 2 N − k
n(tk ,2N −k )
¯
1
.
¯
1 − γ t k −1
¯
¯
By the inductive hypothesis we know that c1 tk , 2 j−k−1 , 1 < c2 tk , 2 N −k , which implies that
j
¯
u c1 t k
k
>
1
¯
1 − γ t k −1
N
∑
¯
¯
P t j = t k , 2 j − k −1 , 1 t k = t k
j = k +1
¯
¯
+ P t N = tk , 2 N −k tk = tk Rρ 1 +
=
1
¯
1 − γ t k −1
N
∑
1
¯
1 − γ t k −1
µ
n(tk ,2N −k )
¯
¯
¯
P t j = t k , 2 j − k −1 , 1 t k = t k
¯
u c2 t k , 2 N − k
¯
1 − γ t k , 2 j − k −1
j = k +1
¯
¯
+ P t N = tk , 2 N −k tk = tk Rρ 1 +
=
¯
1 − γ t k , 2 j − k −1
N
∑
j = k +1
µ
n(tk ,2N −k )
¯
¯
¯
P t j = t k , 2 j − k −1 , 1 t k = t k
¯
u c2 t k , 2 N − k
¯
u c2 t k , 2 N − k
+
D . ANDOLFATTO , E . NOSAL AND B . SULTANUM
N
−
∑
¯
¯
P t j = t k , 2 j − k −1 , 1 t k = t k
j = k +1
20
¯
P t j = ( tk , 2 j−k )
µρ
E N j ¯k j−k
¯
P t j = (tk , 2 j−k−1 , 1) t |t =(t ,2 ) nt N
¯
¯
+ P t N = tk , 2 N −k tk = tk Rρ 1 +
µ
¯
u c2 t k , 2 N − k
n(tk ,2N −k )
¯
After simplify the above equation we obtain
¯
(19) u c1 tk
k
>
1
¯
1 − γ t k −1
¯
¯
1 − P t N = tk , 2 N −k tk = tk
¯
P t j = (tk , 2 j−k )
µρ
− ∑
Et N |t j =(tk ,2j−k )
¯
¯
nt N
P tk = tk
j = k +1
N
¯
¯
+ P t N = tk , 2 N −k tk = tk Rρ 1 +
µ
n(tk ,2N −k )
¯
¯
u c2 t k , 2 N − k
.
The fact that the queue position is withdrawn uniformly implies that
¯
¯
¯
P t j = (tk , 2 j−k ) = P t j = (tk−1 , 1, 2 j−k ) = P t j = (tk−1 , 2 j−k , 1)
and
Et N |t j =(tk ,2j−k )
¯
µρ
µρ
µρ
= Et N |t j =(tk−1 ,1,2j−k )
= Et N |t j =(tk−1 ,2j−k ,1)
.
¯
¯
nt N
nt N
nt N
This implies that
¯
P t j = ( tk , 2 j−k )
µρ
Et N |t j =(tk ,2j−k )
¯
∑
k = tk
¯
nt N
P t
j = k +1
=
¯
P t j = (tk−1 , 1, 2 j−k )
µρ
∑ P tk = (tk−1 , 1) EtN |tj =(t¯k−1 ,1,2j−k ) n N
¯
t
j = k +1
=
¯
P t j = ( t k −1 , 2 j − k , 1 )
µρ
∑ P tk = (tk−1 , 1) EtN |tj =(t¯k−1 ,2j−k ,1) n N
¯
t
j = k +1
=
¯
P t j = ( t k −1 , 2 j − k , 1 )
µρ
Et N |t j =(tk−1 ,2j−k ,1)
¯
k = ( t k −1 , 1 )
¯
nt N
P t
j = k +1
N
(20)
N
N
N
∑
+
−
=
¯
P t k = ( t k −1 , 2 N − k )
µρ
k = ( t k −1 , 1 )
¯
n(tk−1 ,2N −k )
P t
¯
¯
P t k = ( t k −1 , 2 N − k )
µρ
k = ( t k −1 , 1 )
¯
n(tk−1 ,2N −k )
P t
¯
¯
P t k = ( t k −1 , 2 )
µρ
E N k ¯k−1
¯
P tk = (tk−1 , 1) t |t =(t ,2) nt N
−
¯
P t k = ( t k −1 , 2 N − k )
µρ
k = ( t k −1 , 1 )
¯
n(tk−1 ,2N −k )
P t
¯
¯
= γ t k −1 −
¯
P t k = ( t k −1 , 2 N − k )
µρ
.
k = ( t k −1 , 1 )
¯
n(tk−1 ,2N −k )
P t
¯
Replacing equation (20) in inequality (19) and reorganising the terms in the inequality, we
21
PREVENTING BANK RUNS
obtain
¯
(21) u c1 tk
k
>
1
¯
1 − γ t k −1
¯
1 − γ t k −1 +
¯
P t k = ( t k −1 , 2 N − k )
µρ
¯
n(tk−1 ,2N −k )
P t k = ( t k −1 , 1 )
¯
¯
¯
+ P t N = tk , 2 N −k tk = tk Rρ 1 +
µ
n(tk ,2N −k )
¯
−
1
Rρ
¯
u c2 t k , 2 N − k
.
Because Rρ > 1, the inequality (21) implies that
¯
u c 1 t k −1 , 1
k
¯
= u c1 t k
k
¯
> u c2 t k , 2 N − k
¯
= u c2 tk−1 , 1, 2 N −k
.
¯
¯
And since u is concave, it implies that c1 tk−1 , 1 < c2 tk−1 , 1, 2 N −k . The resources constraint
k
implies that
¯
¯
¯
n(tk−1 ,1,2N −k ) + 1 c2 tk−1 , 2 N −k+1 = n(tk−1 ,1,2N −k ) c2 tk−1 , 1, 2 N −k + Rc1 tk−1 , 1 .
¯
¯
k
And finally we can conclude that
¯
c 2 t k −1 , 2 N − k +1 =
n(tk−1 ,1,2N −k )
¯
n(tk−1 ,1,2N −k ) + 1
¯
¯
c2 tk−1 , 1, 2 N −k +
1
n(tk−1 ,1,2N −k ) + 1
¯
¯
¯
Rc1 tk−1 , 1 > c1 tk−1 , 1 .
k
k
We have shown that the result holds for k = N and that if it holds for all j ∈ {k + 1, . . . , N }
it holds for k. Therefore, by induction, we can conclude that the result holds for all k ∈ N.
Proposition 1
ˆ
ˆ
ˆ
P ROOF : We know that for any vector of announcements m N ∈ M N , if either m N ∈ T N or
ˆ
ˆ
ˆ
mk = 2 for all k, the result is trivial. Consider a realized vector of announcements m N ∈ M N ,
ˆ /
ˆ
with m N ∈ T N , mk = 2, and let j be the queue position of the first agent to announce g. As
ˆN ∈ T N denotes the vector m N we replace all g’s with 2’s. When agent j announced g
ˆ
before, t
the in the first period payments were suspended, hence, the total resources in the beginning
of period 2 is
N
j
i =1
i =1
ˆ
ˆi ˆ
ˆ∗ ˆ
R Y − ∑ c1 (mi ) = R Y − ∑ ci 1 (ti ) = n(t j−1 ,2N − j+1 ) c∗2 (t j−1 , 2 N − j+1 ).
ˆ
k
ˆ
Where n(t j−1 ,2N − j+1 ) is the number of 2’s in the vector (t j−1 , 2 N − j+1 ). Let dm N denote the number
ˆ
ˆ
of agents who have announced g and nm N the number of agents who announced 2. The total
ˆ
payments in the second period to agents who announced g is given by
N
ˆ
ˆk ˆ
ˆ
∑ c 2 ( m N )1m k = g = c ∗1 ( t j −1 , 1 ) + d m N
ˆ
j
.
k =1
Hence, payment to agent k is
ˆk
c2
ˆ
m
k −1
ˆN
, 2, mk+1
=
N
N
ˆk ˆ
ˆk ˆ
R Y − ∑ k =1 c 1 ( m k ) − ∑ k =1 c 2 ( m N )1m k = g
ˆ
nm N
ˆ
D . ANDOLFATTO , E . NOSAL AND B . SULTANUM
=
ˆ
ˆ
n(t j−1 ,2N − j+1 ) c∗2 (t j−1 , 2 N − j+1 ) − c∗1 (t j−1 , 1) − dm N
ˆ
ˆ
j
k
nm N
ˆ
22
.
¯
ˆ
By lemma (1) we know that c∗1 t j−1 , 1 < c∗2 (t j−1 , 2 N − j+1 ). Thus, by taking
j
j
enough, we have that,
ˆk
c2
ˆ
m
k −1
ˆN
, 2, mk+1
≥
ˆ
[n(t j−1 ,2N − j+1 ) − 1]c∗2 (t j−1 , 2 N − j+1 )
ˆ
k
nm N
ˆ
> 0 small
.
By construction we have that
ˆ
ˆN
ˆN
ˆk ˆ
c2 tk−1 , 2, tk+1 = c∗2 tk−1 , 2, tk+1
k
∗ ˆ
R Y − ∑iN 1 ci 1 (ti )
= ˆ
=
nm N + dm N
ˆ
ˆ
≤
j
ˆ∗ ˆ
R Y − ∑ i =1 c i 1 ( t i )
nm N + 1
ˆ
=
ˆ
n(t j−1 ,2N − j+1 ) c∗2 (t j−1 , 2 N − j+1 )
ˆ
k
nm N + 1
ˆ
Note that,
n(t j−1 ,2N − j+1 ) − 1
ˆ
nm N
ˆ
≥
n(t j−1 ,2N − j+1 )
ˆ
nm N + 1
ˆ
⇐⇒
n(t j−1 ,2N − j+1 ) nm N + n(t j−1 ,2N − j+1 ) − nm N − 1 ≥ n(t j−1 ,2N − j+1 ) nm N
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
⇐⇒
n(t j−1 ,2N − j+1 ) ≥ nm N + 1.
ˆ
ˆ
The last inequality holds because n(t j−1 ,2N − j+1 ) ≥ nt N = nm N + dm N . Hence,
ˆ
ˆ
ˆ
ˆ
ˆk
c2
ˆ
m
k −1
ˆN
, 2, mk+1
≥
≥
ˆ
[n(t j−1 ,2N − j+1 ) − 1]c∗2 (t j−1 , 2 N − j+1 )
ˆ
k
nm N
ˆ
∗2 ( t j −1 , 2 N − j +1 )
n(t j−1 ,2N − j+1 ) ck ˆ
ˆ
nm N + 1
ˆ
ˆN
ˆk ˆ
≥ c2 tk−1 , 2, tk+1 .
Which concludes the proof.
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William Roberds and François R. Velde
WP-14-03
Mandatory Disclosure and Financial Contagion
Fernando Alvarez and Gadi Barlevy
WP-14-04
The Stock of External Sovereign Debt: Can We Take the Data at ‘Face Value’?
Daniel A. Dias, Christine Richmond, and Mark L. J. Wright
WP-14-05
Interpreting the Pari Passu Clause in Sovereign Bond Contracts:
It’s All Hebrew (and Aramaic) to Me
Mark L. J. Wright
WP-14-06
AIG in Hindsight
Robert McDonald and Anna Paulson
WP-14-07
On the Structural Interpretation of the Smets-Wouters “Risk Premium” Shock
Jonas D.M. Fisher
WP-14-08
Human Capital Risk, Contract Enforcement, and the Macroeconomy
Tom Krebs, Moritz Kuhn, and Mark L. J. Wright
WP-14-09
Adverse Selection, Risk Sharing and Business Cycles
Marcelo Veracierto
WP-14-10
Core and ‘Crust’: Consumer Prices and the Term Structure of Interest Rates
Andrea Ajello, Luca Benzoni, and Olena Chyruk
WP-14-11
The Evolution of Comparative Advantage: Measurement and Implications
Andrei A. Levchenko and Jing Zhang
WP-14-12
5
Working Paper Series (continued)
Saving Europe?: The Unpleasant Arithmetic of Fiscal Austerity in Integrated Economies
Enrique G. Mendoza, Linda L. Tesar, and Jing Zhang
WP-14-13
Liquidity Traps and Monetary Policy: Managing a Credit Crunch
Francisco Buera and Juan Pablo Nicolini
WP-14-14
Quantitative Easing in Joseph’s Egypt with Keynesian Producers
Jeffrey R. Campbell
WP-14-15
Constrained Discretion and Central Bank Transparency
Francesco Bianchi and Leonardo Melosi
WP-14-16
Escaping the Great Recession
Francesco Bianchi and Leonardo Melosi
WP-14-17
More on Middlemen: Equilibrium Entry and Efficiency in Intermediated Markets
Ed Nosal, Yuet-Yee Wong, and Randall Wright
WP-14-18
Preventing Bank Runs
David Andolfatto, Ed Nosal, and Bruno Sultanum
WP-14-19
6
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