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Bank runs and the suspension of deposit convertibility
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~
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.
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~
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.