ELSEVIER Journal of Monetary Economics 41 (1998) 27- 38
JOURNALOF Monetary ECONOMICS
Bank runs: Liquidity costs and investment distortions
R u s s e l l C o o p e r a'*, T h o m a s W . R o s s b
"Department of Economics, Boston University, Boston, MA 02215, USA b Facul~ of Commerce and Business Administration, Universi~ of British Columbia, Vancouver, B.C.
Recent events have cal led into quest ion the heal th of cer ta in sectors of the Amer ican f inancial industry . In par t icu lar , the Savings & Loan crisis led to the t ransfer of a huge a m o u n t of bad debt , es t imated recently at a b o u t $200 bill ion,
* Correspondence address: Department of Economics; Boston University, 270 Bay State Road, Boston, MA 02215, USA. Tel.: 617 353 7082; fax: 617 353 4449; e-mail: [email protected].
28 R. Cooper, 72 IF. Ross~Journal of Monetary Economics 41 (1998) 2 ~ 38
onto taxpayers shoulders. 1 Commercial banks too seemed vulnerable. Central to many discussions of these problems is the role of deposit insurance, created in the United States during the Great Depression (1934) to restore depositor confidence in financial institutions.
Diamond and Dybvig (1983) present a model which highlights the insurance gains. Banks provide liquidity to depositors who are, ex ante, uncertain about their preferences over consumption sequences. The deposit contract provides insurance to depositors and supports a Pareto-optimal allocation of the risk. However, a second, inefficient equilibrium exists in which bank runs are driven by pessimistic depositor expectations. Diamond-Dybvig find that deposit insur- ance rules out these Pareto-inferior Nash equilibria.
Despite the clear insurance benefits, federal deposit insurance may encourage excessive risk taking by financial intermediaries. 2 These concerns have led to calls for reform of deposit insurance and even demands that it be abolished. Before proceeding with deposit insurance reform, it is critical to understand how parties will respond to the strategic uncertainty that may reappear. Thus, our goal in this paper is to analyse the response of the private banking sector to the prospect of bank runs.
To study this, we modify and extend the Diamond-Dybvig model in two important ways. First, we introduce a non-trivial investment choice into the bank's optimization problem. While Diamond-Dybvig talk about the role of liquid and illiquid investments, in their actual model, the liquid investment technology is completely dominated by the illiquid technique. In fact, their model demonstrates the insurance aspect of intermediaries and not the role of intermediation in providing liquidity: the investment portfolio of the intermedi- ary is identical to that which private agents would select in autarky. Second, we characterize optimal contracts when runs are possible. Diamond-Dybvig do not analyse the impact of runs on the behaviour of banks, either in terms of the optimal deposit contract or their investment portfolio. 3
These modifications permit an evaluation of the optimal contract given that runs are possible in the absence of deposit insurance or other intervention. 4 If the likelihood of a run is sufficiently high, the optimal contract will avoid runs
1 Volumes have been written about the S&L crisis. Interesting discussion of the sources of the problem and proposed resolutions can be found in Feldstein (1991) and Kormendi et al. (1989).
2 Americans are not alone in this regard. Canada established a similar system of desposit insurance in 1967 and there is concern that excessive risk-taking and insufficient monitoring plague the Canadian financial industry as well. See, e.g., Smith and White (1988).
3 Diamond-Dybvig do note that in the ex ante contracting stage, the chance of runs could be modeled as a sunspot but never pursue the implications of this for the design of the contract.
4 Of course, the possibility of runs depends on the contract offered, as we make clear below.
R. Cooper, T.W. Ross~Journal of Monetary Economics 41 (1998) 27-38 29
a l together . Otherwise , the op t ima l con t r ac t will a l low the poss ib i l i ty of equi l ib- r ium bank runs. Banks r e spond to the poss ib i l i ty of runs by ad jus t ing their inves tment por t fol ios . If l iqu ida t ion costs are sufficiently high, in te rmediar ies will desire more l iquidi ty ex ante and will thus hold excessive a m o u n t s of the l iquid investment . This is consis tent with the obse rva t ion in F r i e d m a n and Schwar tz (1963), (pp. 176 and 333) tha t dur ing per iods of ins tabi l i ty in the bank ing system, the depos i t reserve ra t io tended to fall.
2. Model
The mode l is a modi f ica t ion of the s t ructure explored in D i a m o n d and Dybv ig (1983). 5 Cons ide r an e c o n o m y in which N agents live for, at most , three per iods. In pe r iod 0, all agents decide whether to depos i t funds in an in te rmedi - ary or to invest their unit e n d o w m e n t themselves. At the s tar t of pe r iod 1, a p r o p o r t i o n 7~ of the agents learn tha t they ob ta in uti l i ty from per iod 1 con- sumpt ion only while the o ther agents ob ta in uti l i ty from per iod 2 consumpt ion . These agents are referred to as ear ly and late consumers , respectively. Assume that ~ is non-s tochas t i c and k n o w n to all agents so that there is ind iv idua l uncer ta in ty over tastes but no aggrega te uncer ta inty . 6 Let CE and Ce be the c o n s u m p t i o n levels for ear ly and late consumers , respect ively and U(c) is their uti l i ty funct ion over consumpt ion . Assume tha t U( . ) is s tr ict ly increas ing and str ict ly concave, U'(0) = o~ and set U(0) = 0.
There are two technologies avai lab le to agents. The i l l iquid inves tment provides a p roduc t ive means of mov ing resources from per iod 0 to 2, with a re turn of R > 1 over the two periods. However , l iqu ida t ion of projects using this technique yields 1 - r in per iod 1 per unit of pe r iod 0 investment , where r ~ [0, 1]. D i a m o n d - D y b v i g assume tha t r = 0, thus ignor ing these l iqu ida t ion costs. 7 Whi le l iqu ida ted projects are not necessar i ly worthless , there are cer ta in- ly some costs associa ted with convers ion to their or ig inal state. 8 The l iquid technology yields one unit in pe r iod 1 per unit of per iod 0 investment . Whi le not
s Some of these modifications also appear in the related paper by Freeman (1988).
6 This follows the first part of Diamond-Dybvig. In the last part of their paper they consider the importance of aggregate uncertainty to argue further in favor of deposit insurance.
v At the other extreme, Jacklin and Bhattacharya (19881 and Haubrich and King (1990) consider the case of r = 1. More recent versions of this model, such as Wallace (1990), dispense with two technique specfication altogether and just assume that the return on storage between periods 0 and 1 differs from that between periods 1 and 2.
s In fact, one could imagine that the magnitude of r would be market determined in a more general economic model. This would provide an interesting link between the state of the aggregate economy and the optimal contract.
30 R. Cooper, T. W. Ross~Journal of Monetary Economics 41 (1998) 27-38
as productive as the illiquid technology over two periods, the liquid technology provides for a higher one-period return.
If there are no intermediaries in the economy, individual agents will allocate their endowment across the two types of investment before knowing their preferences. Agents choose the level of illiquid investment (i) to maximize
Let V A be the expected utility from autarky. If R is large enough relative to the costs of liquidation, then the agent will invest in the illiquid technology knowing that costly liquidation might be necessary ex post. Specifically, if rg < (1 - ~z)(R - 1), then i > 0 and costly liquidation will occur with prob- ability ~.
One means of avoiding costly liquidation is through an ex ante insurance arrangement. We represent this as a planning problem and then discuss the decentralization of the resulting allocation. To begin, assume that agents' types are ex post observable so that consumption can be made contingent upon tastes. The planner's problem is simply to determine consumption levels for each type, an allocation we denote as 6 = (CE, C0, subject to a resource constraint in each period. The planner chooses the per capita level of investment in the illiquid investment to solve
max rtU(cr) + (1 -- ~z)U(cL) i
s.t. ~ C E = ( 1 - - i ) and ( 1 - - 7 0 c L = i R . (2)
The optimal allocation 6* satisfies
U'(CE) = RU'(CL) (3)
Note that this allocation is independent of t . In the first best equilibrium, there is no aggregate uncertainty and hence no liquidations. Since R > 1, the strict concavity of U( ') implies that CE < CL for Eq. (3) to hold.
In general, the utility obtained from 6", denoted by V*, will exceed the expected utility under autarky, V A. The autarkic allocation is certainly feasible for the planner but is clearly never chosen as the feasible (CE, Cr) frontier under autarky is everywhere inside the frontier facing the planner. The planner can achieve higher utility by avoiding costly liquidations and pooling risks. Thus for r > 0 , V * > VAil
This allocation can be decentralized through an intermediary which takes deposits in period 0 and offers a contract stipulating a type-specific return per
9 If preferences exhibit constant relative risk aversion (parameterized by a) and ~ = 0, the autarkic consumpt ion profile of (1,R) will solve the planner 's problem iff a = 1.
R. Cooper, T.W. Ross~Journal of Monetary Economics 41 (1998) 27-38 31
unit of period 0 investment of 6* * * = (cE, co) to depositors. In this contract, (c*, c*) solves Eq. (2). This intermediary breaks even in equilibrium and no other intermediary could offer a contract which is preferred by the agents in period 0 and yields non-negative expected profits. It is also possible to think of 3" as being the contract offered by a large number of intermediaries: due to constant returns to scale the number of active intermediaries is not determined.
The interesting part of the analysis occurs when tastes are private informa- tion. That is, suppose that at the start of period 1, when agents learn their preferences, tastes are not observable to the planner. The problem now has three stages. First, the planner sets a contract that specifies a consumption level for each type of consumer independent of the number of consumers claiming to be each type. Second, agents learn their preferences and these are announced to the planner. Without loss of generality, we employ a direct revelation mechanism in which the agents report their taste types to the central authority. Finally, the allocation of goods to agents is determined by the contract. 1°
Consider first the implementation of 3". In the game, truthtelling is a domi- nant strategy for early consumers since they have no value for goods delivered in period 2. From the viewpoint of the late consumers, truthtelling is a best response to truthtelling by all other late consumers. If a single late consumer misrepresents preferences, then this agent will receive, at most, cE in period 1 which can be privately stored until period 2. However, under 6", cc > cE so that this misrepresentation by a single agent is not optimal. Thus, the optimal allocation can be achieved under 3" as a Nash equilibrium in which all agents honestly report their true tastes.
Under 6", there may also exist equilibria in which all late consumers misrepre- sent their tastes and announce that they are early consumers. This can be an equilibrium if the planner does not have sufficient resources (including liquidated illiquid investments) to provide cE to all agents.
In the decentralized model, agents arrive sequentially at an intermediary to obtain funds and no direct revelation arises. 1~ Agents claiming to be early consumers are able to withdraw c~: until the intermediary has no further resources. Finally, late consumers obtain the minimum of cc and the capital of the intermediary (the value of all remaining investment). ~2
~o In the discussion that follows, we call the first stage the ""contract" and the second stage the "'game".
~ For the planner's problem, the restriction that the consumption level for each type is indepen- dent of the number announcing that type is a means of mimicking sequential service.
2 Note that we do not consider the suspension of convertibility under which withdrawals in the decentralized environment could depend on the number of agents who had already withdrawn funds. Diamond and Dybvig recognize that suspension of convertibility can avoid runs and modify their model so that the fraction of early consumers is random.
32 R. Cooper, T.W. Ross~Journal of Monetary Economics 41 (1998) 27-38
The N a s h equ i l ib r ium with mis represen ta t ion is t e rmed a ' b a n k run' . Let N r be the n u m b e r of depos i to r s receiving p a y m e n t under a run and p(z) be the ra t io o f N r to N. W h e n N r ~< N, p('c) is the p robab i l i t y of being served if all agents run as a funct ion of the l iquidi ty cost. N r and p(z) are defined relat ive to an exist ing con t rac t (which is suppressed in the nota t ion) . The resource cons t ra in t for the i n t e rmed ia ry implies NrCE = (1 -- i)N + Ni(1 - z) = N(1 -- it) so tha t p(z) = (1 - iZ)/CE. U n d e r the first-best cont rac t , ~*, p(z) = (1 - iz)rc/(1 - i) where i is de t e rmined by Eq. (3). By defini t ion, the first-best a l loca t ion has a runs
equ i l ib r ium iff p(z) < 1. N o t e tha t the cond i t ion for runs depends on two i m p o r t a n t variables: the size
of the l iqu ida t ion cost, r, and the level of inves tment in the i l l iquid inves tment (or, equivalent ly , CE). Clear ly p(z) is a decreas ing funct ion of • since, from Eq. (3), i* is i ndependen t of z. At z = 1, there will be a runs equ i l ib r ium for all concave U ( ' ) since p(1) = rt < 1.
In con t ras t to the results r epo r t ed by D i a m o n d - D y b v i g , runs do not require very r isk-averse agents. To see this point , suppose tha t U(c) = c t ~/1 - a, so tha t the degree of relat ive risk avers ion for this agent is a. F o r these preferences, the op t ima l a l loca t ion satisfies:
( C L / C E ) a = R. (4)
At a = 1, the so lu t ion to Eq. (4) entai ls CE = 1, CL = R and i = 1 - n. Hence, p(r) = (1 - (1 - r0z). W h e n z = 0, as in the D i a m o n d - D y b v i g model , the econ- o m y will no t have a bank runs equ i l ib r ium since p(0) = 1. At r = 0, increas ing
above 1 implies tha t CE mus t increase relat ive to CL SO tha t p(0) is less than 1 and a runs equ i l ib r ium will exist, as a rgued by D i a m o n d - D y b v i g . 13 At a = 1, z > 0 implies p(r) < 1 so tha t runs m a y occur. Fur the r , for a < 1, p(z) will be less than 1 for large enough z so that , in con t ras t to D i a m o n d - D y b v i g , a runs equ i l ib r ium exists even if consumers are no t too risk averse. ~4
So, i n t roduc ing the n o n - d o m i n a t e d l iquid asset has an i m p o r t a n t effect on the runs condi t ion . The requi rements on the degree of r isk avers ion can be weakened , relat ive to those repor ted by D i a m o n d - D y b v i g , if there is a cost of l iqu ida t ing the h igh- re turn project . In general , runs will occur for sufficiently large l iqu ida t ion costs or if consumers are sufficiently risk
averse.
13 From the resource constraint, c E is a decreasing function of i and CL increases with i. Hence, in order for Eq. (3) to hold, an increase in ~r must be offset by a reduction in i so the CE increases and CL decreases.
14 Cooper and Ross (1991) includes a proposition which shows that p(z) is a decreasing function of consumer risk aversion.
R. Cooper, 1~ W. Ross~Journal of Monetary Economics 41 (1998) 2~38 33
3. The implications of runs
Given that runs may occur, it is na tura l to return to the first stage of the opt imizat ion problem and consider the implicat ions of this possibility for the design of the contract . Here, as in m a n y abstract mechanism design problems, an impor t an t issue arises: how is an opt imal contract designed given that there may be mult iple equil ibria in the stage game that ensues once the terms of the contract have been set? t5
At one extreme, one might choose an al locat ion for which there is a un ique Nash equi l ibr ium in the game. These contracts are feasible but, as we shall see, may create costly distortions. Alternatively, one might construct a model of the equi l ibr ium selection process and solve for the opt imal contract. One simple
model relies on the existence of publicly observable, but not contractible, variables (sunspots) that correlate agents behaviour at a part icular equi l ibr ium of the game. 16 Instead of prevent ing runs, the in termediary adjusts the contrac t and its portfolio to reduce the impact of runs in the event they arise.
Th roughou t this analysis we focus solely on deposit contracts. As suggested by Jacklin (1988), opt imal risk sharing in the D i a m o n d - D y b v i g model may be obta ined using equity shares rather than deposit contracts. Doing so, avoids the
possibility of runs. However, Jacklin does note that in other versions of the D i a m o n d Dybvig model d e m a n d deposit contracts may be desirable. Fur ther , as argued by Wallace (1990), other a r rangements may be inconsis tent with sequential service. Our results indicate the opt imal use of demand deposit contracts in the face of strategic uncer ta in ty and thus lay the basis for further compar isons across al ternative insti tutions, t v
s See the discussions of this class of problems in Palfrey and Srivastava (1987), Ma (1988) and Ma et al. (1988).
16 Bental et al. (1990) and Freeman (1988) also adopt a sunspots approach. In contrast to our work, those papers allow for sunspot contingent contracts. While it is convenient ot think of sunspots as determining which equilibrium of the subgame will be observed, contracts contingent on these events are assumed to be infeasible. In recent independent work, Alonzo (1996) also studies the adaptation of banks to the possibility of runs. In contrast to the expectations-driven runs of our model, bank runs in Alonzo's model are due to the poor performance of banks.
17 Jacklin (1987), Jacklin and Bhattacharya (1988) and Postlewaite and Vives (1987) discuss other, more general contracts, that may not have some of the problems associated with deposits contracts. For example, Postlewaite and Vives mention contingent contracts in which the payment could depend on the amount withdrawn by others in that period. This is, in fact, a variant of the suspension of convertibility. Jacklin (1987) and Jacklin and Bhattacharya (1988) argue that equity contracts may also support the optimal allocation. Calomiris and Kahn (1991) argue that demand deposits are a desirable form of bank liability as they, along with sequential service, provide an incentive for monitoring by the depositors. Haubrich (1988) and Haubrich and King (19901 discuss optimal financial contracts in a setting with random endowments.
34 R. Cooper, T.W. Ross~Journal of Monetary Economics 41 (1998)27-38
In analysing these contracts two issues are important. First, what is the implication of the possibility of a liquidation on the portfolio of the intermedi- ary? In particular, is there more or less investment in the illiquid technique relative to the first-best allocation? Second, when is it optimal to choose a contract that prevents runs with probability one?
3.1. R u n s p r e v e n t i n g con t rac t s
Consider first a runs p r e v e n t i n 9 c o n t r a c t , hereafter RPC, for which there is no equilibrium with runs. To make the analysis interesting, assume that a runs equilibrium exists under 6*. The set of RPC contracts is obtained by making two modifications to the constraints in Eq. (2). First, we add the condition that p(z)/> 1 to ensure that truthtelling is the only Nash equilibrium for the portfolio and the consumption allocation chosen in this problem. Second, we allow the planner or intermediary to hold liquid investment, denoted by i2, over two periods: this was feasible but clearly not desirable when types were observable.
Formally, the set of RPC is characterized by three conditions:
~C E = 1 -- i - i 2 , (1 - - ~ ) C L = i R + i2, C E <~ 1 -- iz. (5)
The first two are resource constraints and the third is the no-runs condition. Fig. 1 illustrates feasible consumption opportunities and depicts the RPC set
as the shaded region. Since i ~> 0 and z >~ 0, CE cannot exceed 1 so there is a vertical segment at CE = 1. Then there is a segment from (1,1) to ((1 - z)/(1 - ~zz), R / ( 1 - ltr)), denoted by point A in Fig. 1. This segment of the boundary of the set lies below the resource constraints from Eq. (2). I.e., there
CL
R
1-7 l "
R
1-7"
I - ~ T 7
°, '. ..
F ig . 1.
-°.
1 )
%
R. Cooper, T.W. Ross~Journal of Moneta~ Economics 41 (1998)27-38 35
are al locat ions which satisfy the resource constraints , but are not runs proof. In this segment, i2 > 0; al locat ions here require that excess liquidity be held to suppor t a relatively high level of CE while avoiding runs. A third segment coincides with the resource constra int for CE < (1 -- Z)/(1 -- nz). In this region, the runs prevent ion const ra in t is not binding as cE is sufficiently low. It is s t ra ightforward to check that the al locat ions that were feasible under au ta rky are also in the set of runs prevent ing contracts .
The degree to which excess liquidity is employed in the R P C will depend upon the l iquidation costs as demons t ra t ed in the following pair of proposi t ions .
Proposition 1. l f r = 0, the optimal R P C entails cE = 1 and CL = R and there will be no excess liquidity (i.e. i2 = 0). 18
Proposition 2 . / f z = 1, the optimal R P C involves excess liquidity (i.e. i 2 > 0).
The best runs prevent ing cont rac t will always weakly domina te autarky. When r is sufficiently large, the a m o u n t of liquid inves tment will exceed CE.; that is, there will be excess liquidity. This allows the in termediary to provide CE without violating the condi t ions for no runs. As suggested by Fig. 1, excess liquidity will be held when the consumer ' s margina l rate of subst i tut ion at point A exceeds the slope of the middle segment of the feasibility constraint .
3.2. Contracts with runs
In contras t to the best RPC, now consider the opt imal contrac t with runs. Suppose that with probabi l i ty q, there is a wave of economy-wide pessimism that determines the beliefs of depositors. If the outs tanding contrac t has a runs equil ibrium, then the pessimism leads to a bank run. With probabi l i ty 1 - q, there is op t imism and no runs. In this way, the beliefs of deposi tors are tied to a move of nature which determines their actions. The in termediary recognizes this dependence in designing the opt imal contract .
Tak ing the probabi l i ty of l iquidation, q, as given, the contrac t solves
max (1 - - q)[TzU(CE) + (1 -- 7~)U(CL) ] ~- qU(CE)((1 - - iz)/cE) cl, e l , i , i2
s.t. 7rCE= l - - i - - i 2 , ( l - - ~ ) c c = i R + i 2 and i2 ~>0. (6)
Let ~J(q) be the contract solving this problem and V(q) denote the resulting expected utility.
One interesting aspect of Eq. (6) is whether the in te rmediary will set i 2 > 0 to avoid l iquidation costs. This opt ion is par t icular ly valuable for large values of
~8 Proofs of all propositions are in an appendix which is avilable from the authors on request.
36 R. Cooper, T.W. Ross~Journal of Monetary Economics 41 (1998)27-38
z and q since the gain to i 2 > 0 arises when runs are likely and liquidation costs are large. We find that:
Proposition 3. I f qz > (1 - q)(R - 1), then i 2 > 0 in the solution to Eq. (6).
This proposit ion captures the gains to excess liquidity: it allows the banks to separate the level of early consumption from the proport ion of agents that can be served in the event of a run. If we set i 2 = 0 by assumption, then the intermediary would avoid high liquidation costs by raising CE. 19 This is costly in two ways. First, the number of agents who can be served in the event of a run is lower. Second, if runs do not occur, the fact that U'(CE) < RU'(CL) implies that there is a distortion in consumption. When the bank is allowed to hold excess reserves, it does so if the expected liquidation costs (qz) are large relative to the difference in returns between the two types of investments (R - 1). The holdings of excess reserves are likely when the runs are more probable and liquidation costs are high.
At the other extreme, when T is near 0, there are no liquidity gains so that i2 = 0. Formally, we find
Proposition 4. I f z is near O, then i 2 = O.
In this case, the effect of the possibility of runs is to reduce CE to serve more agents in the event of a run.
From an empirical perspective, as discussed by Friedman and Schwartz (1963), accounts of panics indicate a drive toward liquidity for banks (through the holding of excess reserves) and by depositors (through increases in the currency/deposit ratio). These types of effects arise in this model because of the dominance of the liquid technology for short-term investments. Hence, for values of r and q near 1, we find that banks allocate more of their funds to the liquid investment and may hold excess liquidity to provide funds to depositors in the event of a run. While our model is static in that runs can occur in only one period, thinking about the implications of increasing q in our framework is useful for understanding the dynamic implications of runs if the chance of a run in period t + 1 increases in the event of a run in period t.
Given this contract, should the bank adopt a runs preventing contract? The relationship between the expected utility from the contract that prevents runs (V "r) and the optimal contract when liquidations occur with probability q, V(q), is given by
Proposition 5. There exists q*e(O, 1) such that V"r> V(q) if q > q* and V nr < V(q) i f q < q*.
19 This is shown formally as Proposition 4 in Cooper and Ross (1991).
R. Cooper, T.W. Ross~Journal of Moneta~ Economics 41 (1998) 27-38 37
This proposition provides a characterization of the maximum level of utility that depositors can receive from intermediaries if runs occur with probability q and there is no deposit insurance. If the probability of a run is low, then 6(q) will be the optimal contract. Since V nr is the best runs preventing contract, V(q) > V "r implies that 6(q) cannot be runs proof so that runs will be observed under the optimal contract. When the probability of runs is sufficiently high, the solution will be to adopt a runs preventing contract. In either case, the process of intermediation will not stop due to the possibility of bank runs.
An important effect of runs under 6* will be to alter the consumption profile and the investment strategy of the bank. If the probability of a run is sufficiently high, then a runs preventing contract will be adopted which will lower early consumption and, perhaps, require the holding of excess liquidity. If a contract allowing runs is chosen, then for z large enough, excess liquidity will be held.
4. Conclusions
The goal of this paper has been to extend the Diamond-Dybvig framework to understand the implications of runs and to predict how banks would respond to the probability of bank runs in an economy without deposit insurance or other such regulation. In our model, there is a true liquidity role for intermediaries which provides new conditions for runs and some implications for the response of intermediaries to the possibility of runs. The expected utility from the optimal contract provides a benchmark of the best the private sector can provide if runs are possible and there is no deposit insurance.
This modification of the basic Diamond-Dybvig model might be useful in the ongoing debate over the design of deposit insurance. In particular, knowing the response of banks to the possibility of runs is needed to fully understand the consequences of reduced deposit insurance to the extent that this policy change may again leave banks vulnerable to runs. Of course, other aspects of deposit insurance, such as moral hazard considerations, can interact with the effects highlighted in this paper. Some of these points are explored in Cooper and Ross (1991).
Acknowledgements
We have benefited from discussion on this topic with Paul Beaudry, Fanny Demers, Jon Eaton, Alok Johri, Arthur Rolnick, Thomas Rymes, Fabio Schian- tarelli, David Weil and Steven Williamson. Suggestions from Robert King and an anonymous referee were much appreciated as were the comments of seminar participants at Boston University, Brown University, Carleton University, the Federal Reserve Bank of Minneapolis, the University of British Columbia and the University of Maryland. Financial support from the National Science
38 R. Cooper, T.W. Ross~Journal of Monetary Economics 41 (1998)27-38
Foundation, the Social Sciences and Humanities Research Council of Canada and the Carleton Industrial Organization Research Unit is gratefully acknow- ledged. The first author is grateful to the Institute for Empirical Macroeco- nomics at the Federal Reserve Bank of Minneapolis for providing a productive working environment during preparation of this manuscript. Some of this work was done while the second author was with the Canadian Competition Bureau and he is grateful for the Bureaus hospitality. The views expressed here are not necessarily those of the Federal Reserve Bank of Minneapolis nor of the Competition Bureau.
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