Seediscussions,stats,andauthorprofilesforthispublicationat:
https://www.researchgate.net/publication/222378369
Bankrunsandthesuspensionof
depositconvertibility
ArticleinJournalofMonetaryEconomics·November1989
ImpactFactor:1.89·DOI:10.1016/0304-3932(89)90031-7
CITATIONS
READS
36
131
1author:
MerwanEngineer
UniversityofVictoria
34PUBLICATIONS292CITATIONS
SEEPROFILE
Availablefrom:MerwanEngineer
Retrievedon:14May2016
Journal
of Monetary
Economics
24 (1989) 443-454.
North-Holland
BANK RUNS AND THE SUSPENSION
DEPOSIT CONVERTIBILITY
Merwan
Unr~wrsr~v
Received
OF
ENGINEER*
of Guelph, Guelph, 0n1 Canadu NIG -7WI
September
1988, final version received May 1989
In a longer-horizon
version of Diamond and Dybvig’s (1983) model, suspending convertibrhty
of
bank deposits into cash does not always prevent a bank run. A bank run may occur even if the
bank can adJust new wrthdrawal payments after observing too many withdrawals
1. Introduction
Diamond
and Dybvig (1983) model a bank as a financial intermediary
which pools risk in an environment
where privately observed consumption
shocks are uncorrelated
across agents and longer-term
productive assets earn
greater fixed rates of return than shorter-term
assets. The pooling function
cannot
be performed
by insurance
markets because contracts
cannot
be
conditioned
on investors’ privately observed consumption
shocks. Diamond
and Dybvig show that an efficient equilibrium
exists if the bank employs a
standard
demand
deposit contract
and aggregate consumption
demand is
certain. However, they also show that there is a Pareto-inferior
equilibrium
that might be described as a bank run. The bank-run
equilibrium
can be
eliminated
by suspending convertibility
after observing too many withdrawals.
A bank run also can be averted by suspending
convertibility
if aggregate
consumption
demand is uncertain, but optimal risk sharing cannot be achieved
because some agents are prevented from withdrawing in the period they most
want to consume. This inadequacy
of the suspension
of convertibility
motivates government
deposit insurance which supports the efficient equilibrium.
This paper examines a longer-horizon
version of Diamond
and Dybvig’s
model. In the extended model, suspending
convertibility
is less effective: it
may not eliminate the bank-run equilibrium.
A run may occur even when the
bank can adjust new withdrawal
payments
after observing too many withdrawals in a bank run.
*I am grateful to Dave Backus, Dan Bemhardt, Michael
an anonymous
referee for their very helpful comments.
0304-3932/89/$3.500
1989, Elsevier Science Publishers
Hoy. Dave Nickerson.
B.V. (North-Holland)
Dan Peled, and
444
M Engrneer, Bank runs and the suspensron
ofdeposit converttbrlrty
In contrast,
in the shorter-horizon
(three-period)
model with aggregate
consumption
certainty, a bank run is averted simply by immediately
suspending convertibility.
The reasoning is straightforward.
After the initial investment period 0, all agents discover their type in period 1. A proportion
t’ of the
agents, called type 1 agents, experience a consumption
shock in period 1 and
want to withdraw all their deposit and consume in that period. The remaining
agents, type 2 agents, want to consume only in period 2. They would normally
prefer to withdraw in period 2 than in period 1 (and hoard their money to
period 2) since the payment is higher in period 2. A bank run occurs when all
type 2 agents panic and attempt to withdraw in period 1 forcing the bank into
insolvency.
Immediately
suspending convertibility
after 1’ withdrawals ensures
the solvency
of the bank since no extra assets have to be prematurely
liquidated.
More importantly,
it assures type 2 agents that they can withdraw
in period 2. Since type 1 agents always attempt to withdraw in period 1, there
is no excess demand for withdrawals in period 2. Hence, type 2 agents never
panic and bank runs are prevented.’
Under the more realistic assumption
that all agents do not discover their
type in period 1, bank runs may occur in a longer-horizon
model. In this
paper, a four-period
model is analyzed where type 2 and type 3 agents do not
discover their specific type until period 2.2 The immediate
suspension
of
convertibility
after the proportion
t’ withdrawals
in period 1 and t* withdrawals
in period 2 ensures the solvency of the bank (where t2 is the
proportion
of type 2 agents and t’ + t* < 1). Nevertheless,
bank-run
conjectures are self-fulfilling.
Nontype 1 agents run in period 1 out of fear that if
they turn out to be type 2 agents they will be in the bank queue when
convertibility
is suspended
in period 2. The excess demand for period 2
withdrawals
comes from type 1 agents displaced in a run. So many type 1
agents are displaced in a run because some nontype 1 agents who turn out to
be type 3 agents withdrew in period 1. If type was known in period 1, all type
3 agents would wait to withdraw in period 3, leaving no excess demand for
period 2 withdrawals.
The assumption
that agents discover their own prefer-
’ For drfferent perspecttves
Jagannathan
(1988).
on the suspensron
of convertibility
see Gorton
(1985) and Char-i and
“The reasons for a bank run in this model are quite different from those of Postlewaite and
Vives (1987) who also develop a four-period
banking model. They model the strategic game
between two depositors,
both of whom discover their type in period 1. If both agents turn out to
be type 2 agents a Prisoner’s Dilemma arises where each has a donunant
strategy to withdraw in
period 1. This bank run occurs not because agents condition
their behavior on an exogenous
sunspot but because aggregate preferences are uncertain.
In Bryant (1980) Chari and Jagannathan
(1988). and Jacklin and Bhattacharya
(1988) bank
runs occur because depositors
receive information
about the banks asset returns in the interim
penod.
In Diamond
and Dybvig and this paper bank runs occur in the absence of private
information
about asset returns.
M. Engmeer. Bunk runs and the suspensron of deposit convertrhilq
445
ences over time is in the spirit of the banking analysis based on agents having
unknown
liquidity demands.
A policy where the bank can freely liquidate assets and alter payments after
observing
too many withdrawals
in period 1 is also analyzed. The policy
involves offering a follow-up payment in period 1 that is attractive only to type
1 agents. If the remaining type 1 agents can be served cheaply in this way a
run is prevented.
However, if type 1 agents value period 2 consumption
sufficiently highly, the follow-up payment must be large and bank runs cannot
be prevented.
The paper proceeds as follows. The model is outlined in section 2 and the
optimal
risk sharing allocation
is described in section 3. Section 4 briefly
analyzes the equilibria under the standard demand deposit contract. Section 5
demonstrates
that the immediate suspension of convertibility
does not eliminate bank runs. A flexible payment policy designed to prevent bank runs is
developed
in section 6. Finally, other institutions
such as deposit insurance
and the exchange of dividend-paying
shares are briefly examined in section 7.
2. The model
Diamond
and Dybvig’s model is extended to four periods (T = 0,1,2,3) by
including
an asset which matures in period 3 and also a third type of investor
who most wants to consume in period 3. All agents are endowed with one unit
of a storable homogeneous
good in period T = 0. Any portion of the endowment can be either stored for subsequent
periods or invested at T = 0 in
productive
assets. One unit invested in the short-term productive asset earns a
certain gross rate of return of S, .I 1 if liquidated in period 1. or a return of
S, > 1 if held to maturity in period 2. One unit invested in the long-term
productive
asset yields L, < St, L, I S,. or L, 2 (S,)’ if liquidated in period
1, 2, or 3, respectively.
In Diamond and Dybvig assets are liquid: S, = L, = 1
and S, = L,. When S, = L, = L, = 0, the assets are completely
illiquid [for
example, Jacklin and Bhattacharya
(1988) and Peare (1988)J.
The infinite population
is divided into three preference
types (i = 1,2,3)
according to the period in which they receive their major consumption
shock:
Gpe
1
u(x:+e;x:+e:x:),
0<
e;< l/S,,
0 c
e: < e;s,p.,,
M Engrneer,
446
Bunk runs und the suspemon
of deposit convertrh1it.v
Type 3
where
x+- is the amount of goods consumed
by agent type i in period T,
and
u(x)
is
a
twice
differentiable,
increasing,
strictly concave funcP ’ s,-‘7
tion with relative risk aversion - xu”(x)/u’(x)
> 1 everywhere. The weights
ei, #, and 0-j are chosen so that type 1 agents prefer to consume in period 1
most and in period 3 least; type 2 agents prefer to consume in period 2, and
type 3 agents only receive utility from consuming in period 3.3
In period 0, agents perceive their future type to be chosen randomly
from
the infinite population,
and have ex ante utility:
tb( X; + e;X; + e;X;) + t*p~(X; + e;X;) + i3( p)*U(Xi),
Eu =
where t’, t’, and t3 = (1 - t’ - t2) are the known proportions
of type 1, 2, and
3 agents. Agent type is private information.
In period 1, type 1 agents discover
their type and types 2 and 3 learn they are nontype 1 agents; in period 2, type
2 and 3 agents discover their specific type. Unlike in the three-period
model,
all agents do not discover their type at the same time.
3. The optimal allocation
The ex ante
conditions:
l*=xl*=
X2
optimal
3
Xl
risk-sharing
2* =x2*
u’( xi*) = S*pu’(
tlxi*
+ t2(x:_*/S2)
3
allocation
= x3*
1
=x3*
2
= 0
is described
*
xi*)
by the following
04
(lb)
+ t3(X:*/L3)
= 1.
04
Type i agents only consume goods in period T = i. The restrictions
on asset
yields and preferences are sufficient to ensure that L, > x3’* > x,2* > xi* > 1.
To achieve the optimal allocation, a: = t’x:* of the deposits is invested in the
storage technology,
a2* = t*( x:*/S,)
is invested in short-term
asset and the
remainder
in the long-term asset. Alternatively,
if the long-term asset is liquid
all of the deposits may be invested in it.
3Dlamond
and Dybvig do not explicitly model type 1 agents valuing
Wallace (1988) agents have a constant
marginal rate of substitution
periods much hke above.
penod 2 consumption.
In
in consumption
between
M
Engtneer,
Bank runs and the suspensron of deposit concertdxlq
447
4. The bank equilibria under the demand deposit contract
The bank invests its deposits to achieve the optimal allocation.
Under the
standard demand deposit contract, the bank promises to pay out cT = xF* to
any agent withdrawing
his entire deposit in period T = 1,2. Agents that
attempt to withdraw in a particular period arrive in the bank line in random
order and are served sequentially.
If the bank faces a shortage of funds to
service withdrawals
in either period 1 or 2, it allocates on a first come, first
serve basis. In this case some demanders
are left with nothing in the period
that they most want to consume. In the last period the bank is liquidated and
the remaining
depositors receive their pro rata share of any remaining assets.4
The efficient bank equilibrium
emerges when all active deposit holders
believe that other agents intend to withdraw their deposits only in the period
that they most want to consume, ck = x;* for T = i and ck = 0 for T # i. With
these beliefs, the best response of any agent is to withdraw his deposit in the
period which he most wants to consume. At no stage does the bank have to
prematurely
liquidate productive assets to service withdrawal demands.
There also is a Pareto-inferior
bank-run equilibrium.
Suppose all agents in
period 1 believe that other investors are going to attempt to withdraw in
period 1. If an investor attempts to withdraw his deposit, he is successful with
< 1. On the other hand, if he does not
probability
( CX: + cy:S, + cu:L,)/x:*
attempt to withdraw his money, all of the bank’s assets are liquidated
and
distributed
in period 1 leaving him with nothing. For this reason all deposit
holders participate
in the bank run when they believe others also are going to
run.
5. A bank run with the immediate suspension
of convertibility
To prevent the premature
liquidation
of productive
assets, the bank can
suspend deposit convertibility
when faced with excess withdrawal
demands.
This is sufficient to avert bank runs in the three-period model.5 In this model,
the immediate
suspension
of convertibility
after t’ withdrawals
in period 1
and t’ withdrawals
in period 2 prevents the premature sale of assets, but as
the following proposition
shows it is not sufficient to prevent a bank run.
4Bank runs are harder to avert if the bank contract
deposit m period i- = 1.2.3 as long as funds last.
ts c-I = xF* to any agent who wtthdraws
hts
‘In the three-penod
model, declaring bankruptcy
to prevent the value destroymg sale of assets
in period 1, with the legal proceedings m period 2, has the same effect as tmmediately suspending
convertibility
and, therefore, also averts a run Note that wrth completely illiqmd assets. the bank
has no chorce but to declare bankruptcy
when faced with too many withdrawals m period 1 Thus,
the three-period
model suggests paradoxtcally
that banks, if they have the choice, should Invest m
completely
illiqmd rather than liquid assets to precommit and avoid the bad equilibnum.
448
M. Engineer, Bank runs and the swpens~on of deposrt conuertlbdq
Proposition.
With the immediate suspension of convertibility after t’ withdrawals in period 1 and t2 withdrawals in period 2, a bank-run equilibrium exists
if the expected utility of a nontype I agent withdrawing in period 1 is greater than
not withdrawing,
(t” + pt3)
1 - t’
PU( x:*> ’ 2,
where
u(x22*)
+
tlt3
+ mP+,zx:*)
&JPMX:.)I.
The equilibrium is characterized by all agents queuing for withdrawals in period 1
and the following rationed proportions of agents of each type withdrawing
c$ = XT’* in period T:
Period ( T)
Type (i>
1
2
3
(t’)Z
t’t2
1 - t3
t3( ty2
1
(t’)’
t’t2t3
l-t3
l-t3
2
3
et2
t’t3
0
l-t3
P(l
-t’)
Proof.
In period 7, type 1 agents have a dominant
strategy to attempt to
withdraw
their deposits. Suppose all nontype 1 agents believe that all other
agents are going to attempt to withdraw their deposits in period 1. If a
nontype 1 agent queues to withdraw his deposit, he receives xi* with probability t’. If he is successful in withdrawing
his funds, his expected utility in
period 1 is
t*
yqy+4*)
=“f’:f3’
p+:*>
+ -&p)*u(x;*)
If he fails, he is in the same position
as if he did not queue. The proportion
t1
M. Engineer, Bank runs and the suspensron of deposit convertrbd~ty
449
claims
of each type receive xi‘* , leaving 1 - t’ of each type with remaining
under the bank-run
scenario. In period 2, type 2 and 3 agents discover their
type. Now the remaining
type 1 and 2 agents have a dominant
strategy to
queue in period 2. Together they constitute the proportion
(t’ + t’)(l - rl) =
t2 + t’t3 of the population.
The bank distributes
xi* to t2 agents in period 2,
bumping
the remaining
t1t3 agents to period 3. Since there is no excess
demand
for deposits in period 3 [t’t3 + (1 - t’)t3 = t3], all the remaining
agents receive x33*. A nontype 1 agent p articipates in the bank run in period 1,
if his expected utility is greater than waiting to withdraw in later periods:
t’
(t’ + pt3)
( t2 + pf3)
1 - t’
pu(xi*)+(l-t’)Z>Z
or
1 - t’
pz+;*>
>z.
If this condition
is satisfied and all nontype 1 agents believe that all other
agents are going to run in period 1, they also run. The proportions
of agents of
each type that are able to withdraw is straightforwardly
derived from the
above sequence.
n
Bank-run
conjectures
are self-fulfilling
because each nontype 1 agent fears
that if he turns out to be a type 2 agent he may be one of the t1t3 unserved
agents in line when convertibility
is suspended in period 2. By joining in a run
in period 1, a nontype 1 agent can reduce the probability
of being a cashless
type 2 agent in period 2 by the factor t’. It is optimal for a nontype 1 agent to
run in period 1 (when all other agents run) if a type 2 agent’s utility of being
cashless in period 2 and consuming in period 3 is sufficiently low. Accordingly,
the condition
for a bank-run
equilibrium
is satisfied if t’, t2,t3 > 0 and
u( 0:x:* ) is small enough.
For example, let t’ = t 2 = t3 = i and U(X) = - l/x. A nontype 1 agent who
chooses not to run in period 1 has a + chance of being a type 2 agent who is
not served in period 2. In contrast, a nontype 1 agent who runs in period 1 has
a & chance of being a type 2 agent who is not served in period 2. The relative
importance
of consuming in period 2 over period 3 depends on 6):. As 13: + 0,
u( 0:x:*) -+ - cc, and the proposition
is satisfied.
A bank-run
equilibrium
is more likely to exist the smaller is the difference
between the optimal payments, because the relative gain to a nontype 1 agent
successfully
withdrawing
in the period of his consumption
shock becomes
smaller. In fact, as x22* and x:* approach xi*, the relative gain to successfully withdrawing
later goes to zero and the condition for a bank-run equilibrium is satisfied. Examples where close optimal payments lead to bank runs
are analyzed in the next section.
A bank run depends on type 2 and 3 agents not knowing their type in
period 1. The t’f3 excess demands for period 2 withdrawals originate from the
t’t3 type 3 agents who successfully withdrew in period 1 when they did not
450
M. Engmeer, Bunk runs and the suspension of deposit convertrbility
know their identity. If type 3 agents knew their type in period 1 they would
not run, because they are assured that they can withdraw more in period 3.
But then there is no excess demand for period 2 withdrawals and type 2 agents
are better off withdrawing in period 2 than running in period 1. It is because
all agents discover their type in period 1 in the three-period model that there is
no bank-run equilibrium with the suspension of convertibility. The assumption
that agents’ preferences are revealed over time is in the spirit of banking
analysis based on agents having unknown liquidity demands.
Also, a bank-run equilibrium does not exist if type 1 agents prefer to
consume in period 3 over period 2, 04 > O:S,/L,. With such preferences, there
is no excess demand for period 2 withdrawals, because if there were, there
would be an excess supply of period 3 payments (at rate x: * ) and type 1 and
3 agents could do better by withdrawing in period 3. Hence, nontype 1 agents
wait until period 2 to discover their specific type and no bank run occurs.
Nontype 1 agents face a greater temptation to participate in a bank run if
withdrawals can be reinvested in period 1 in a newly created bank. This bank
is assumed to be able to buy a two-period productive asset in period 1 that has
a gross rate of return S’ > 1 in period 3. The original bank is only viable if
nontype 1 agents are no worse off keeping their money in the old bank when
there is no run in period 1:
t2
t2+
tsP44*)+;r;-fit3 ( p)*u(x;*)
t2
’- ,,,,,PG)
+ gp~P~‘4~:),
where 2: and 22 are the solutions to the three-period optimal risk-sharing
problem starting in period 1 with deposits of x:* per capita. (Starting in
period 1, both banks have a three-period horizon so that the immediate
suspension of convertibility precludes runs.) Let S” be the rate of return the
new bank earns that makes nontype 1 agents indifferent between the above
alternatives.
A bank-run equilibrium exists if nontype 1 agents receive more utility
switching banks in the event of a run:
t2
t2+
GP43) + t2
*3 (
p )‘2.4($) >z.
This condition is always satisfied for S’ sufficiently close to S” (from below). If
S’ > S”, optimal payments must be abandoned by the original bank. If they
reduce the first-period payment just enough to satisfy the incentive compatibility condition, a bank-run equilibrium exists.
M. Engrneer, Bank runs and the suspension
of depositconvertiblht)
451
6. A flexible payment policy
This section considers a policy which allows the bank to liquidate assets and
alter payments
in a bank run after having observed too many withdrawals
(f > t’ withdrawals)
in period 1. To simplify the analysis, the long-run asset is
assumed to be perfectly liquid so that the bank invests all its deposits in the
long-run asset in period 0. Also, 63’ and 0: are assumed to be arbitrarily small
so that type 1 and 2 agents always prefer to consume in period 2 over period 3.
After t’ withdrawals
of xi* in period 1, the bank can either suspend
convertibility
or liquidate more assets to serve additional
agents in period 1.
Suppose the bank liquidates assets in a bank run to serve additional agents in
period 1. Since type is not observable,
it is optimal to pay the additional
withdrawers
in period 1 the same payments xi. The bank should offer a low
payment
xi to separate and cheaply serve the remaining
type 1 agents in
period 1. A necessary condition for type 1 agents to accept such a payment is
xi 2 8:x,. Therefore,
the bank sets xi = @x,. As 0: < l/S,,
the strategy of
separating
type 1 agents leaves more assets per capita for nontype 1 agents
withdrawing
in later periods. Therefore, this policy dominates
the policy of
suspending
convertibility
in period 1.
Under the separation policy the program to maximize the expected utility of
the remaining
nontype 1 agents in period 1 is
max
x2.x3
subject
L(r2~(~2)
1 - t’
+pt3u(x,)),
to
I -
tlx;* - (1 -
tl)tle:x2 =
(1 - tyx,
+ (1 -
s,
A nontype
t’)t’x,
L,
1 agent is made better off by running
’
when all others run if
(t2 + pt3)
1 - t1
pu(x:*)
’ $-+(Pu(P,)
+
Pf3d~3)),
where Zz and g2, solve (P).
An example that satisfies the condition
for a bank-run
equilibrium
is the
following:
L, = 1, L, = S, = 4, L, = 1.8, t’ = t2 = t3 = $, p = 0.76 u(x) =
-l/x,
and 6: = 0.735. In this example, the differences between the optimal
allocations
are small: xi* = 1.292, x2** = 1.301, and xi* = 1.318. In addition,
period 2 consumption
is a close substitute to period 1 consumption
for type 1
agents. The best the bank can do is set Zi = 0.821, P, = 1.125, and I, = 1.603.
The period 2 payment
is low in order to reduce the period 1 separation
payment. Nevertheless,
a bank-run equilibrium
exists in this case.
452
M. Engineer, Bank runs and the suspensmn of deposrr convertibd~<l:
With perfectly liquid assets and a flexible payment policy other examples
where bank runs cannot be prevented share similar features. The differences
between the optimal payments is small. Thus, xi* is relatively large and a
temptation
to period 1 runners. The large period 1 payments reduce the assets
available for later periods. The other feature is that 0: is large.6 This means
the bank cannot cheaply separate and serve the remaining
type 1 agents in
period 1. If restrictions
are put on policy or assets are illiquid, the condition
for the bank-run
equilibrium
becomes less restrictive. For example the condition in Proposition
1 is the best policy for preventing
a run when assets are
illiquid and payments are inflexible.
Finally, note that if the bank can anticipate a bank run and alter payments
to all agents including the first t’ agents, a run may be averted. For instance,
in Bental, Eckstein,
and Peled (1989) and Freeman
(1988) a bank run is
explicitly modelled as a ‘sunspot equilibrium’.
Runs in these papers can be
prevented
because the bank can observe and condition
payments
on the
sunspot.
However,
such a policy does not achieve optimal
risk sharing,
because type 1 agents receive too little in period 1.
7. Other institutions
In this model, government
deposit insurance,
similar to the type found in
Diamond
and Dybvig, can prevent all bank runs and achieve the optimal
allocation.
If f > t’ agents queue in period 1, the bank can serve all the queued
liquidating
assets by drawing ‘deposit insuragents xi* without prematurely
ance’. The deposit insurance
can be paid by levying a tax on period
1 withdrawers.
If f > t’, all agents withdrawing
in period 1 are taxed
(1 - t’/f )xi*, leaving each with (t’/f )xi* after tax. The capacity of deposit
insurance
to alter withdrawals
ex post effectively bypasses the sequential
service constraint.
A nontype 1 agent never runs because he knows that there
is no excess demand
for period 2 withdrawals.
Hence deposit insurance
achieves the optimal allocation
when there is uncertain
individual
liquidity
demands in period 1.
Diamond
and Dybvig show that even with uncertain aggregate consumption
demands deposit insurance can achieve the full-information,
optimal risk-shar6There is a tradeoff between 0: and x i* to satisfy the condition for a bank-run equilibrium.
This tradeoff can be reduced by increasing
the concavity
of U(X). For example, if u(x) =
-1/(2x2),
a smaller first-period
payment x1I* -- 1 290 is consistent with a smaller substitutton
coefficient 0: = 0.12.
‘Taxes can be pard wtth deposits. Diamond and Dybvig constder a scheme where the tax does
not always cover the entire amount of the excess withdrawals
and some assets are liquidated
prematurely.
The altemattve
mechanism
considered
here works even if assets are completely
illiquid.
M. Engrneer, Bank runs and the suspensron of deposit convertibility
453
ing allocation.
However, Wallace (1988) argues that government
deposit
insurance
cannot achieve optimal risk sharing if the implicit assumption
that
agents are isolated is added to the model.* In the model studied here two other
potential
problems
arise when there is aggregate uncertainty.
First, if the
long-term asset is illiquid, the optimal allocation requires that the bank invest
some of the deposits in the assets of shorter duration.
The exact amount
depends
on the realized aggregate demands.
But since this is unknown
in
period 0, the optimal allocation
cannot be achieved generally without an
infusion of external funds.
Second, if the aggregate uncertainty
is not resolved in period 1, there is not
enough information
in the economy to calculate the full-information
optimal
allocation.
For example, suppose u(x) = -l/x.
Then in the four-period
model the full-information
allocation xi* = l/[t’ + t2(~/S2)1/2 + t’p/L\“]
is
a function of the actual population
proportions
t’, t*, and t3. Thus, x:* can
take on different
values for a given 1’ depending
on t2 and t3, where
t2 + t3 = 1 - t’. Thus, even if the bank could infer t’ from the number of
withdrawals
in period 1, it cannot determine xi* with certainty when t 2 and
t3 are unknown
in period 1.9
Finally, it should be noted that bank deposits may not be the only way of
achieving
the optimal allocation.
Jacklin (1987) shows, using Diamond
and
Dybvig’s preferences,
that the optimal allocation
can be achieved by the
market exchange of dividend-paying
equity. This result generalizes
to this
model, even though all agents do not know their specific type in period 1. In
period 0 each agent buys one share with his endowment.
The firm invests the
endowment
and promises
to pay dividends
D’ = t’x:*,
D2 = t ‘x22*, and
D3 = t3X,3* in periods 1, 2, and 3, respectively.
Suppose that in period 1
nontype 1 agents exchange all their dividends for all the shares of the type 1
agents, and in period 2 type 3 agents exchange all their dividends for all the
shares of type 2 agents. Thus, each type 1 agent receives xi* and each nontype
1 agent has l/(1 - t’) shares entering period 2. In period 2, each type 2 agent
receives xi*
and each type 3 agent has l/(1 -t’) + t2/t3(1 - t’) = l/t3
shares entering period 3. The dividends to each type 3 therefore are xi*. With
these exchanges,
share prices in periods 1 and 2 are, respectively,
1 - t’xi*
and t3xz*. At these prices no agent has an incentive to deviate. Type 2 agents
prefer t3xf* per share in period 2 to t3x: * in period 3, and type 3 agents
prefer the opposite. Type 1 agents prefer (1 - t’)x:* per share in period 1 to
the most they can obtain in later periods: (1 - t’).xz* in period 2 or (1 - t’)x:*
“Without
the isolation of agents a credit market may exist. Jacklin
market is inconsistent
with bank deposits that provides liquidity.
(1987) shows that a credit
91n the three-period
model, r2 = 1 - ti. Thus, if t1 can be determined,
x:* can be calculated.
Diamond
and Dybvig use deposit insurance
to face agents with payoffs such that they have a
dominant
strategy to withdraw only in the period of their consumption
shock. Thus, t1 is inferred.
454
M. Engineer, Bank runs and the suspemon
of deposit convertibrht~
in period 3. Nontype 1 agents prefer the opposite in period 1. Thus, the
exchange of ex-dividend shares achieves the optimal allocation.”
References
Bental, B., Z. Eckstein, and D. Peled, 1989, Competitive
banking with confidence
crisis and
international
borrowing, Working paper (Technion-Israel,
Institute of Technology).
Bryant, John, 1980, A model of bank reserves, bank runs and deposit insurance,
Journal of
Banking and Finance 4,335-344.
Chari, V.V and Ravi Jagannathan,
1988, Banking panics, information,
and rational expectations
equilibrium,
Journal of Finance 63, 749-763.
Diamond,
D. and P. Dybvig. 1983, Bank runs, deposit insurance,
and liquidity,
Journal of
Political Economy 91, 401-419.
Freeman,
Scott, 1988. Bankmg as the provision of liquidity, Journal of Business 61, 45-64.
Gorton,
Gary, 1985, Bank suspension
of convertibility,
Journal of Monetary
Economics
15,
177-193.
Jacklin, Charles, 1987, Demand deposits, trading restrictions, and risk sharing, in: E. Prescott and
N. Wallace, eds., Contractual
arrangements
for intertemporal
trade (University of Minnesota
Press, Minneapolis,
MN) 26-47.
Jacklin, Charles and Sudipto Bhattacharya,
1988, Distinguishing
panics and information-based
bank runs: Welfare and policy implications,
Journal of Political Economy 96, 568-593.
Peare, Paula, 1988, The creation of liquidity by financial institutions:
A framework for welfare
and policy analysis, Working paper (Queen’s University, Kingston, Ont.).
Postlewaite,
Andrew and Xavier Vives, 1987, Bank runs as an equilibrium phenomena,
Journal of
Political Economy 95.485-491.
Wallace, Neil, 1988, Another attempt to explain an illiquid banking system: The Diamond and
Dybvig model with sequential service taken seriously, Federal Reserve Bank of Minneapolis
Quarterly
Review, Fall, 3-16.
“Jacklin
pomts out that with more general ‘smooth’ preferences (such that each agent receives
optimal payments
in more than one period) that the exchange of ex-dividend
shares generally
cannot achieve the optimal allocation whereas complex demand deposit contracts can.