Federal Reserve Bank of Minneapolis - Diamond and Dybvig's Classic Theory of Financial Intermediation: What's Missing? (p. 3)
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Federal Reserve Bank
of Minneapolis
Winter 2000
Diamond and Dybvig's
Classic Theory
of Financial Intermediation:
What's Missing? (p. 3)
Edward J. Green
Ping Lin
Bank Runs,
Deposit Insurance,
and Liquidity (p. 14)
Douglas W. Diamond
Philip H. Dybvig
1999 Contents (p. 24)
1999 Staff Reports (p. 25)
Federal Reserve Bank of Minneapolis
Quarterly ReviewVOL.24,NO.1
ISSN 0271-5287
This publication primarily presents economic research aimed
at improving policymaking by the Federal Reserve System and
other governmental authorities.
Any views expressed herein are those of the authors and
not necessarily those of the Federal Reserve Bank of Minneapolis
or the Federal Reserve System.
Editor: Arthur J. Rolnick
Associate Editors: Edward J. Green, Patrick J. Kehoe,
Warren E. Weber
Economic Advisory Board: V. V. Chari, Christopher Phelan
Managing Editor: Kathleen S. Rolfe
Article Editors: Kathleen S. Rolfe, Jenni C. Schoppers
Production Editor: Jenni C. Schoppers
Designer: Phil Swenson
Typesetter: Mary E. Anomalay
Circulation Assistant: Elaine R. Reed
The Quarterly Review is published by the Research Department Comments and questions about the Quarterly Review may be
of the Federal Reserve Bank of Minneapolis. Subscriptions are sent to
available free of charge. Quarterly Review
Quarterly Review articles that are reprints or revisions of papers Research Department
published elsewhere may not be reprinted without the written Federal Reserve Bank of Minneapolis
permission of the original publisher. All other Quarterly Review P. O. Box 291
articles may be reprinted without charge. If you reprint an article, Minneapolis, Minnesota 55480-0291
please fully credit the source—the Minneapolis Federal Reserve (Phone 612-204-6455 / Fax 612-204-5515).
Bank as well as the Quarterly Review—and include with the Subscription requests may also be sent to the circulation
reprint a version of the standard Federal Reserve disclaimer assistant at elaine.reed@mpls.frb.org; editorial comments and
(italicized above). Also, please send one copy of any publication questions, to the managing editor at ksr@res.mpls.frb.fed.us.
that includes a reprint to the Minneapolis Fed Research
Department.
Electronic files of Quarterly Review articles are available through
the Minneapolis Fed's home page on the World Wide Web:
http://www.minneapolisfed.org.
Federal Reserve Bank of Minneapolis
Quarterly Review Winter 2000
Bank Runs, Deposit Insurance, and Liquidity*
Douglas W. Diamondt Philip H. Dybvigt
Theodore 0. Yntema Professor of Finance Boatmen's Bancshares Professor of Banking and Finance
Graduate School of Business John M. Olin School of Business
University of Chicago Washington University in St. Louis
This article develops a model which shows that bank deposit for deposits above the insured amount.) It is good that de-
contracts can provide allocations superior to those of exchange regulation will leave banking more competitive, but policy-
markets, offering an explanation of how banks subject to runsmakers must ensure that banks will not be left vulnerable
can attract deposits. Investors face privately observed risks to runs.
which lead to a demand for liquidity. Traditional demand de- Through careful description and analysis, Friedman and
posit contracts which provide liquidity have multiple equilibria, Schwartz (1963) provide substantial insight into the prop-
one of which is a bank run. Bank runs in the model cause real erties of past bank runs in the United States. Existing the-
economic damage, rather than simply reflecting other problems. oretical analysis has neglected to explain why bank con-
Contracts which can prevent runs are studied, and the analysis tracts are less stable than other types of financial contracts
shows that there are circumstances when government provision or to investigate the strategic decisions that depositors
of deposit insurance can produce superior contracts. face. The model we present has an explicit economic role
for banks to perform: the transformation of illiquid claims
Bank runs are a common feature of the extreme crises that (bank assets) into liquid claims (demand deposits). The
have played a prominent role in monetary history. During analyses of Patinkin (1965, chap. 5), Tobin (1965), and
a bank run, depositors rush to withdraw their deposits be- Niehans (1978) provide insights into characterizing the
cause they expect the bank to fail. In fact, the sudden with-
drawals can force the bank to liquidate many of its assets
at a loss and to fail. During a panic with many bank fail- article is reprinted, with permission, from the Journal of Political Econom
ures, there is a disruption of the monetary system and a re- (1983,*This
vol. 91, no. 3, pp. 401-19). © 1983 by The University of Chicago. All rights
duction in production. reserved. The article was edited for publication in the Federal Reserve Bank of Min-
Institutions in place since the Great Depression have neapolis Quarterly Review. The authors are grateful for helpful comments from Milt
Harris, Burt Malkiel, Mike Mussa, Art Raviv, and seminar participants at Chicago,
successfully prevented bank runs in the United States since Northwestern, Stanford, and Yale.
the 1930s. Nonetheless, current deregulation and the dire Finance, fWhen this article was originally published, Diamond was Assistant Professor of
Graduate School of Business, University of Chicago and Dybvig was Assis-
financial condition of savings and loan associations make tant Professor of Finance, School of Organization and Management, Yale University.
bank runs and institutions to prevent them a current policy 'The aborted runs on Hartford Federal Savings and Loan (Hartford, Conn., Feb-
issue, as shown by recent aborted runs.1 (Internationally, examples.
ruary 1982) and on Abilene National Bank (Abilene, Texas, July 1982) are two recent
Eurodollar deposits tend to be uninsured and are therefore Bank (Oklahoma The large amounts of uninsured deposits in the recently failed Penn Square
subject to runs, and this is true in the United States as well of banks' current City, July 1982) and that failure's repercussions are another symptom
problems.
14
Douglas W. Diamond, Philip H. Dybvig
Bank Runs, Deposit Insurance, and Liquidity
liquidity of assets. This article gives the first explicit analy- time than the illiquid assets offer. These contracts have
sis of the demand for liquidity and the transformation multiple equilibria. If confidence is maintained, there can
service provided by banks. Uninsured demand deposit con- be efficient risk-sharing, because in that equilibrium a
tracts are able to provide liquidity, but leave banks vulner- withdrawal will indicate that a depositor should withdraw
able to runs. This vulnerability occurs because there are under optimal risk-sharing. If agents panic, there is a bank
multiple equilibria with differing levels of confidence. run and incentives are distorted. In that equilibrium, every-
Our model demonstrates three important points. First, one rushes in to withdraw their deposits before the bank
banks issuing demand deposits can improve on a com- gives out all of its assets. The bank must liquidate all its
petitive market by providing better risk-sharing among assets, even if not all depositors withdraw, because liqui-
people who need to consume at different random times. dated assets are sold at a loss.
Second, the demand deposit contract providing this im- Illiquidity of assets provides the rationale both for the
provement has an undesirable equilibrium (a bank run) in existence of banks and for their vulnerability to runs. An
which all depositors panic and withdraw immediately, in- important property of our model of banks and bank runs
cluding even those who would prefer to leave their depos- is that runs are costly and reduce social welfare by in-
its in if they were not concerned about the bank failing. terrupting production (when loans are called) and by de-
Third, bank runs cause real economic problems because stroying optimal risk-sharing among depositors. Runs in
even healthy banks can fail, causing the recall of loans many banks would cause economywide economic prob-
and the termination of productive investment. In addition, lems. This is consistent with the Friedman and Schwartz
our model provides a suitable framework for analysis of (1963) observation of large costs imposed on the U.S.
the devices traditionally used to stop or prevent bank runs, economy by the bank runs in the 1930s, although Fried-
namely, suspension of convertibility and demand deposit man and Schwartz assert that the real damage from bank
insurance (which works similarly to a central bank serving runs occurred through the money supply.
as lender of last resort). Another contrast with our view of how bank runs do
The illiquidity of assets enters our model through the economic damage is discussed by Fisher (1911, p. 64) and
economy's riskless production activity. The technology Bryant (1980). In this view, a run occurs because the
provides low levels of output per unit of input if operated bank's assets, which are liquid but risky, no longer cover
for a single period, but high levels of output if operated for the nominallyfixed liability (demand deposits), so deposi-
two periods. The analysis would be the same if the asset tors withdraw quickly to cut their losses. The real losses
were illiquid because of selling costs: one receives a low are indirect, through the loss of collateral caused by falling
return if unexpectedly forced to liquidate early. In fact, this prices. In contrast, a bank run in our model is caused by a
illiquidity is a property of thefinancial assets in the econ- shift in expectations, which could depend on almost any-
omy in our model, even though they are traded in competi- thing, consistent with the apparently irrational observed be-
tive markets with no transaction costs. Agents will be con- havior of people running on banks.
cerned about the cost of being forced into early liquidation We analyze bank contracts that can prevent runs and
of these assets and will write contracts which reflect this examine their optimality. We show that there is a feasible
cost. Investors face private risks which are not directly in- contract that allows banks both to prevent runs and to pro-
surable because they are not publicly verifiable. Under op- vide optimal risk-sharing by converting illiquid assets. The
timal risk-sharing, this private risk implies that agents have contract corresponds to suspension of convertibility of de-
different time patterns of return in different private infor- posits (to currency), a weapon banks have historically used
mation states and that agents want to allocate wealth un- against runs. Under other conditions, the best contract that
equally across private information states. Because only the banks can offer (roughly, the suspension-of-convertibility
agent ever observes the private information state, it is im- contract) does not achieve optimal risk-sharing. However,
possible to write insurance contracts in which the payoff in this more general case, there is a contract which achieves
depends directly on private information without an explicit the unconstrained optimum when government deposit in-
mechanism for information flow. Therefore, simple com- surance is available. Deposit insurance is shown to be able
petitive markets cannot provide this liquidity insurance. to rule out runs without reducing the ability of banks to
Banks are able to transform illiquid assets by offering transform assets. What is crucial is that deposit insurance
liabilities with a different, smoother pattern of returns over frees the asset liquidation policy from strict dependence on
15
the volume of withdrawals. Other institutions such as the One interpretation of the technology is that long-term
discount window (the government acting as lender of last capital investments are somewhat irreversible, which ap-
resort) can serve a similar function; however, we do not pears to be a reasonable characterization. The results would
model this here. The taxation authority of the government be reinforced (or can be alternatively motivated) by any
makes it a natural provider of the insurance, although there type of transaction cost associated with selling a bank's
may be a competitivefringe of private insurance. assets before maturity. See Diamond 1984 for a model of
Government deposit insurance can improve on the best the costly monitoring of loan contracts by banks, which
allocations that private markets provide. Most of the exist- implies such a cost.
ing literature on deposit insurance assumes away any real All consumers are identical as of period 0. Each faces
service from deposit insurance, concentrating instead on a privately observed, uninsurableriskof being of type 1 or
the question of pricing the insurance, taking as given the of type 2. In period 1, each agent's type is determined and
likelihood of failure. (See, for example, Merton 1977, revealed privately to the agent. Type 1 agents care only
1973; Kareken and Wallace 1978; Dothan and Williams about consumption in period 1, and type 2 agents care only
1980.) about consumption in period 2. In addition, all agents can
Our results have far-reaching policy implications, be- privately store (or hoard) consumption goods at no cost.
cause they imply that the real damage from bank runs is This storage is not publicly observable. No one would
primarilyfrom the direct damage occurring when produc- store between T = 0 and T = 1, because the productive
tion is interrupted by the recalling of loans. This implies technology does at least as well (and better if held until T =
that much of the economic damage in the Great Depres- 2). If an agent of type 2 obtains consumption goods at T=
sion was caused directly by bank runs. A study by Ber- 1, this agent will store them until T= 2 to consume them.
nanke (1983) supports our thesis; it shows that the number Let cT represent goods received (to store or consume) by
of bank runs is a better predictor of economic distress than an agent at period T. The privately observed consumption
the money supply. at T = 2 of a type 2 agent is then what the agent stores
The Bank's Role in Providing Liquidity
from T= 1 plus what the agent obtains at T = 2, or cx + c2.
Banks have issued demand deposits throughout their his- In terms of this publicly observed variable cT, the discus-
tory, and economists have long had the intuition that de- sion above implies that each agent j has a state-dependent
mand deposits are a vehicle through which banks fulfill utility function (with the state private information), which
their role of turning illiquid claims into liquid claims. In we assume has the form
this role, banks can be viewed as providing insurance that (2) U(cl9 c2; 0) =
allows agents to consume when they need to most. Our
simple model shows that asymmetric information lies at | w(Cj) if j is of type 1 in state 0
the root of liquidity demand, a point not explicitly noted
in the previous literature. | pw(cj+c2) if j is of type 2 in state 0
The model has three periods (T = 0, 1,2) and a single
homogeneous good. The productive technology yields R > where 1 > p > R~l and u : R++ —> is twice continuously
1 units of output in period 2 for each unit of input in period differentiate, increasing, and strictly concave and satisfies
0. If production is interrupted in period 1, the salvage value Inada conditions u(0) = °o and u (°°) - 0. Also, we as-
is just the initial investment. Therefore, the productive tech- sume that the relativerisk-aversioncoefficient -cu"(c) -r
nology is represented by u\c) > 1 everywhere. Agents maximize expected utility,
E[u(cx, c2; 0)1, conditional on their information (if any).
T= 0 T= 1 T= 2 Afraction t e (0, 1) of the continuum of agents are of
type 1, and conditional on t, each agent has an equal and
independent chance of being of type 1. Later sections will
allow t to be random (in which case, at period 1, consum-
ers know their own types but not t), but for now we take
where the choice between (0, R) and (1, 0) is made in pe- t to be constant.
riod 1. (Of course, constant returns to scale imply that a To complete the model, we give each consumer an en-
fraction can be done in each option.) dowment of one unit in period 0 (and none at other
16
Douglas W. Diamond, Philip H. Dybvig
Bank Runs, Deposit Insurance, and Liquidity
times). We considerfirst the competitive solution where The optimal insurance contract just described would al-
agents hold the assets directly, and in each period there is low agents to insure against the unlucky outcome of being
a competitive market in claims on future goods. Constant a type 1 agent. This contract is not available in the simple
returns to scale imply that prices are determined: the contingent-claims market. Also, the lack of observability
period 0 price of period 1 consumption is one, and the of agents' types rules out a complete market of Arrow-
period 0 and period 1 prices of period 2 consumption are Debreu state-contingent claims, because this market would
R~l. This is because agents can write only uncontingent require claims that depend on the nonverifiable private in-
contracts, since there is no public information on which to formation. Fortunately, it is potentially possible to achieve
condition. Contracting in period T= 0, all agents (who are the optimal insurance contract, since the optimal contract
then identical) will establish the same trades and will in- satisfies the self-selection constraints.3 We argue that banks
vest their endowments in the production technology. Giv- can provide this insurance: by providing liquidity, banks
en this identical position of each agent at T = 0, there will guarantee a reasonable return when the investor cashes in
be trade in claims on goods for consumption at T = 1 and before maturity, as is required for optimal risk-sharing. To
at T = 2. Each has access to the same technology, and illustrate how banks provide this insurance, wefirst exam-
each can choose any positive linear combination of cx = ine the traditional demand deposit contract, which is of
1 and c2 = R. Each agent's production set is proportional particular interest because of its ubiquitous use by banks.
to the aggregate set, and for there to be positive produc- Studying the demand deposit contract in our framework
tion of both cx and c2, the period T = 1 price of c2 must also indicates why banks are susceptible to runs.
be R~l. Given these prices, there is never any trade, and In our model, the demand deposit contract gives each
agents can do no better or worse than if they produced on- agent withdrawing in period 1 afixed claim of rx per unit
ly for their own consumption. Let clk be consumption in deposited in period 0. Withdrawal tenders are served se-
period k of an agent who is of type i. Then the agents quentially in random order until the bank runs out of as-
choose c\ = 1, c\ = c\ = 0, and c\ - R, since type Is sets. This approach allows us to capture the flavor of con-
always interrupt production but type 2s never do. tinuous time (in which depositors deposit and withdraw at
By comparison, if types were publicly observable as of different random times) in a discrete model. Note that the
period 1, it would be possible to write optimal insurance
contracts that give the ex ante (as of period 0) optimal
sharing of output between type 1 and type 2 agents. The
optimal consumption {clk*} satisfies 2The proof of this is as follows:
pRu\R) < Ru{R)
(3) cf = 4* = 0
= 1 • u\ 1) + (ti
(which says, those who can, delay consumption), = "'(!) + / * MY) + 7fc"(Y)]
< u'( 1)
(4) u\cf) = pRu\cf)
since u > 0 and (V y) -u"(y)y/u(y) > 1. Because «'(•) is decreasing and the resource
constraint (5) trades off c\* against cf, the solution to (3)-(5) must have c|* > 1 and
(which says, marginal utility is in line with marginal pro- c%
demand deposit contract satisfies a sequential service con- It is precisely the transformation of illiquid claims into straint, which specifies that a bank's payoff to any agent liquid claims that is responsible both for the liquidity can depend only on the agent's place in line and not on service provided by banks and for their susceptibility to future information about agents later in line. runs. For all rx > 1, runs are an equilibrium.6 If rj = 1, a We are assuming throughout this article that the bank bank would not be susceptible to runs because Vx(fj, 1) < is mutually owned (a mutual) and liquidated in period 2, V2(f, 1) for all values of 0
Douglas W. Diamond, Philip H. Dybvig
Bank Runs, Deposit Insurance, and Liquidity
plies that banks with pure demand deposit contracts will be Given this contract, no type 2 agent will withdraw at T =
very concerned about maintaining confidence because they 1 because no matter what the agent anticipates about oth-
realize that the good equilibrium is very fragile. ers' withdrawals, the agent receives higher proceeds by
The pure demand deposit contract is feasible, and we waiting until T = 2 to withdraw; that is, for all/and fj <
have seen that it can attract deposits even if the perceived f V2(.) > V^-). All of the type Is will withdraw every-
probability of a run is positive. This explains why the con- thing in period 1 because period 2 consumption is worth-
tract has actually been used by banks in spite of the dan- less to them. Therefore, there is a unique Nash equilib-
ger of runs. Next, we examine a closely related contract rium which has/= t. In fact, this is a dominant strategy
that can help to eliminate the problem of runs. equilibrium, because each agent will choose the equilibri-
um action even if it is anticipated that other agents will
Improving on Demand Deposits: choose nonequilibrium or even irrational actions. This
Suspension of Convertibility makes this contract very stable. This equilibrium is essen-
The pure demand deposit contract has a good equilibrium tially optimal
the good demand deposit equilibrium that achieves
risk-sharing.
that achieves the full-information optimum when t is not
stochastic. However, in its bank run equilibrium, the pure A policy of suspension of convertibility at /guarantees
demand deposit contract is worse than direct ownership of that it will never be profitable to participate in a bank run
assets. It is illuminating to begin the analysis of optimal because the liquidation of the bank's assets is terminated
bank contracts by demonstrating that there is a simple vari- while type 2s still have an incentive not to withdraw. This
ation on the demand deposit contract which gives banks a contract works perfectly only in the case where the normal
defense against runs: suspension of allowing withdrawal of volume of withdrawals, t, is known and not stochastic. The
deposits, referred to as suspension of convertibility (of de-more general case, where t can vary, is analyzed next.
posits to cash). Our results are consistent with the claim by
Friedman and Schwartz (1963) that in the 1930s, the newly Optimal Contracts With Stochastic Withdrawals
organized Federal Reserve Board may have made runs The suspension-of-convertibility contract achieves optimal
worse by preventing banks from suspending convertibility: risk-sharing when t is known ex ante because suspension
the total week-long banking "holiday" that followed was never occurs in equilibrium, and the bank can follow the
more severe than any of the previous suspensions. optimal asset liquidation policy. This is possible because
If banks can suspend convertibility when withdrawals the bank knows exactly how many withdrawals will occur
are too numerous at T = 1, anticipation of this policy pre- of typeconfidence
when is maintained. We now allow the fraction
vents runs by removing the incentive of type 2 agents to consider a general anclassunobserved
Is to be random variable, t. We
withdraw early. The following contract is identical to the to those who withdraw at T = 1 are any where
of bank contracts payments
function of fj and
pure demand deposit contract described in equations (6) payments to those who withdraw at 7 = 2 are any function
and (7), except that it states that agents will receive noth-
ing at T = 1 if they attempt to withdraw at T = 1 after a of/ ings
Analyzing this general class will show the shortcom-
of suspension of convertibility.
fraction / < rf of all deposits have already been with-
1
The full-information optimal risk-sharing is the same as
drawn. Note that we redefine V^-) and V2(-): before, except that in equations (3)-(5), the actual realiza-
tion of ~t = t is used in place of the fixed t. Since no single
r, if fj - f agent has information crucial to learning the value of t, the
(8) VU, r,) = arguments of footnote 2 still show that optimal risk-sharing
0 if fj>f is consistent with self-selection, so there must be some
(9) V2(f, r,) = max{(1 -frx )R/( 1 -/), (1 )RJ(l-f mechanism )} Wewhich
librium.
has optimalrisk-sharingas a Nash equi-
now explore whether banks (which are subject
where the expression for V2 assumes that l-frl> 0. to the constraint of sequential service) can do this too.
Convertibility is suspended when fj = / and then no From equations (3)-(5), we obtain full-information op-
one else in line is allowed to withdraw at T = 1. To dem- timal consumption levels, given the realization of ~t = t, of
onstrate that this contract can achieve the optimal alloca- c\\t) and cf(t). Recall that c\*(t) = c]*(t) = 0. At the op
tion, let rx = c}\ and choose any fe {t, [(7?-^ )/r, (/?-!)]}. timum, consumption is equal for all agents of a given type
19
and depends on the realization of t. This implies a unique result is consistent with contemporary views about sus-
optimal asset liquidation policy given t - t . This turns out pension in the United States in the period before deposit
to imply that uninsured bank deposit contracts cannot insurance. Although suspensions served to short-circuit
achieve optimal risk-sharing. runs, they were "regarded as anything but a satisfactory
PROPOSITION 1. Bank contracts {which must obey the they solution by those who experienced them, which is why
sequential service constraint) cannot achieve optimal risk-banking produced such strong pressure for monetary and
sharing when t is stochastic and has a nondegenerate dis-The mostreform" (Friedman and Schwartz 1963, p. 329).
important reform that followed was government
tribution. deposit insurance. Its impact is analyzed in the next
Proposition 1 holds for all equilibria of uninsured bank section.
contracts of the general form Vx(fj) and V2(f), where
these can be any functions. It obviously remains true that Government Deposit Insurance
uninsured pure demand deposit contracts are subject to Deposit insurance provided by the government allows
runs. Any run equilibrium does not achieve optimal risk- bank contracts that can dominate the best that can be of-
sharing, because both types of agents receive the same fered without insurance and never do worse. We need to
consumption. Consider the good equilibrium for any fea- introduce deposit insurance into the analysis in a way that
sible contract. We prove that no bank contract can attain keeps the model closed and assures that no aggregate re-
the full-information optimal risk-sharing. The proof is source constraints are violated. Deposit insurance guaran-
straightforward, a two-part proof by contradiction. Recall tees that the promised return will be paid to all who with-
that the place in line^ is uniformly distributed over [0, t] draw. If this is a guarantee of a real value, the amount that
if only type 1 agents withdraw at T = 1. First, suppose can be guaranteed is constrained: the government must
that the payments to those who withdraw at T = 1 is a impose real taxes to honor a deposit guarantee. If the de-
nonconstant function of fj over feasible values of t: for posit guarantee is nominal, the tax is the (inflation) tax on
two possible values of t, tx and t2, the value of a period 1 nominal assets caused by money creation. (Such taxation
withdrawal varies; that is, Vx(t^ Vx(t2). This immediate- occurs even if no inflation results; in any case, the price
ly implies that there is a positive probability of different level is higher than it would have been otherwise, so some
consumption levels by two type 1 agents who will with- nominally denominated wealth is appropriated.) Because
draw at T = 1, and this contradicts an unconstrained op- a private insurance company is constrained by its reserves
timum. Second, assume the contrary: that for all possible in the scale of unconditional guarantees which it can offer,
realizations of t = t, Vx(Jj) is constant for all fj e [0, t].we argue that deposit insurance probably ought to be gov-
This implies that c\(t) is a constant independent of the ernmental for this reason. Of course, the deposit guarantee
realization of 1, while the budget constraint, equation (5), could be made by a private organization with some author-
shows that c\(t) will vary with t (unless rx = 1, which is ity to tax or create money to pay deposit insurance claims,
itself inconsistent with optimalrisk-sharing).Constant c\(t) although we would usually think of such an organization
and varying c\(t) contradict optimalrisk-sharing,equation as being a branch of government. However, there can be
(4). Thus, optimalrisk-sharingis inconsistent with sequen- a small competitive fringe of commercially insured depos-
tial service. its, limited by the amount of private collateral.
Proposition 1 implies that no bank contract, including The government is assumed to be able to levy any tax
suspension of convertibility, can achieve the full-infor- that charges every agent in the economy the same amount.
mation optimum. Nonetheless, suspension can generally In particular, it can tax those agents who withdrew early in
improve on the uninsured demand deposit contract by pre- period T = 1, namely, those with low values of fj. How
venting runs. The main problem occurs when converti- much tax must be raised depends on how many deposits
bility is suspended in equilibrium, that is, when the point are withdrawn at T = 1 and what amount rx was promised
/ where suspension occurs is less than the largest possible to depositors. For example, if every deposit of one dollar
realization of t. In that case, some type 1 agents cannot were withdrawn at T= 1 (implying/= 1) and rx = 2 were
withdraw, which is inefficient ex post. This can be desir- promised, a tax of at least one per capita would need to be
able ex ante, however, because the threat of suspension raised because totally liquidating the bank's assets will
prevents runs and allows a relatively high value of rx. This raise at most one per capita at T= 1. As the government
20
Douglas W. Diamond, Philip H. Dybvig
Bank Runs, Deposit Insurance, and Liquidity
can impose a tax on an agent who has withdrawn, the gov-
ernment can base its tax on/ the realized total value of T = (11) W ) = c\(f) i f f < t
1 withdrawals. This is in marked contrast to a bank, which 1 i f / > t.
must provide sequential service and cannot reduce the The net payments to those who withdraw at T = 1
amount of a withdrawal after it has been made. This asym- determine the asset liquidation policy and the after-tax val-
metry allows a potential benefit from government interven- ue of a withdrawal at T = 2. Any tax collected in excess
tion. The realistic sequential service constraint represents of that needed to meet withdrawals at T = 1 is plowed
some services that a bank provides but which we do not back into the bank (to minimize thefraction of assets liq-
explicitly model. With deposit insurance, we will see that uidated). This implies that the after-tax proceeds, per dol-
imposing this constraint does not reduce social welfare. lar of initial deposit, of a withdrawal at T = 2, denoted by
Agents are concerned with the after-tax value of the V2(f\ are given by
proceedsfrom their withdrawals because that is the amount
that they can consume. A very strong result (which may be (12) V2(f) =
too strong) about the optimaiity of deposit insurance will
illuminate the more general reasons it is desirable. We
argue in the conclusion that deposit insurance and the Fed- f " [c\\f)f M l - / ) = c]\f) i f / < t
eral Reserve discount window provide nearly identical ser-
vices in the context of our model, but we confine discus- fl(l-/)/(l-/)
=R i f / > t.
sion here to deposit insurance. Notice that Vx(f) < V2(f) for all / e [0, 1], implying
PROPOSITION 2. Demand deposit contracts with govern- whatnotheytypeexpect
that 2 agents will withdraw at T = 1 no matter
ment deposit insurance achieve the unconstrained opti-0, implying that allothers type
to do. For all fe [0, 1], Vx(f) >
1 agents will withdraw at T= 1.
mum as a unique Nash equilibrium (in fact, a dominantTherefore, the unique dominant
strategies equilibrium) if the government imposes an op-t, the realization of t. Evaluatedstrategy at a
equilibrium is/ =
realization t,
timal tax tofinance the deposit insurance.
Proposition 2 follows from the ability of tax-financed (13) Vx(f= t) = c\\t)
deposit insurance to duplicate the optimal consumptions and
c\(t) = c\\t), c22(P = C22 (0, c\(t) = 0, c\(t) = 0from the
optimal risk-sharing characterized in equations (3)-(5). Let (14) V2(f= i) = [1 - tc\\t)]R/(l-t) = cf(t)
the government impose a tax on all wealth held at the be-
ginning of period T - 1, which is payable either in goods and the optimum is achieved.
or in deposits. Let deposits be accepted for taxes at the Proposition 2 highlights the key social benefit of gov-
pretax amount of goods which could be obtained if with- ernment deposit insurance. This insurance allows the bank
drawn at T - 1. The amount of tax that must be raised at to follow a desirable asset liquidation policy, which can be
T = 1 depends on the number of withdrawals then and the separated from the cash-flow constraint imposed directly
asset liquidation policy. Consider the proportionate tax as by withdrawals. Furthermore, deposit insurance prevents
a function of^ t: [0, 1] —> [0, 1] given by runs because, for all possible anticipated withdrawal poli-
cies of other agents, participating in a bank run never pays.
1 - [c\ (f)!r x ] i f / < t As a result, no strategic issues of confidence arise. This is
(10) T( / ) a general result of many deposit insurance schemes. The
1 - r -l
if/> t proposition may be too strong, since it allows the govern-
ment to follow an unconstrained tax policy. If a nonopti-
where t is the greatest possible realization of t. mal tax must be imposed, then when t is stochastic, there
The after-tax proceeds, per dollar of initial deposit, of a will be some tax distortions and resource costs associated
withdrawal at T= 1 depend on/through the tax payment with government deposit insurance. If a sufficiently per-
and are identical for all fj < f Denote these after-tax pro- verse tax provided the revenues for insurance, social wel-
ceeds by Vx(f), given by fare could be higher without the insurance.
21Deposit insurance can be provided costlessly in the sim- an institutional framework under which banks can operate
pler case where t is nonstochastic, for the same reason that smoothly, much as enforcement of contracts does more
there need not be a suspension of convertibility in equilib- generally.
rium. The deposit insurance guarantees that type 2 agents The riskless technology used in the model isolates the
will never participate in a run; without runs, withdrawals rationale for deposit insurance, but in addition it abstracts
are deterministic, and this feature is never used. In particu- from the choice of bank loan portfolio risk. If the risk of
lar, as long as the government can impose some tax to bank portfolios could be selected by a bank manager, un-
finance the insurance, no matter how distortionary, there observed by outsiders (to some extent), then a moral haz-
will be no runs and the distorting tax need never be im- ard problem would exist. In this case there is a trade-off
posed. This feature is shared by a model of adoption ex- between optimal risk-sharing and proper incentives for
ternalities in which a Pareto-inferior equilibrium can be portfolio choice, and introducing deposit insurance can in-
averted by an insurance policy which is costless in equi- fluence the portfolio choice. The moral hazard problem
librium. (See Dybvig and Spatt 1983.) In both models, the has been analyzed in complete market settings where de-
credible promise to provide the insurance means that the posit insurance is redundant and can provide no social im-
promise will not need to be fulfilled. This is in contrast to provement. (See Kareken and Wallace 1978 and Dothan
privately provided deposit insurance. Because insurance and Williams 1980.) But of course in this case there is no
companies do not have the power of taxation, they must trade-off. Introducing risky assets and moral hazard would
hold reserves to make their promises credible. This il- be an interesting extension of our model. It appears likely
lustrates a reason the government may have a natural that some form of government deposit insurance could
advantage in providing deposit insurance. The role of gov- again be desirable but that it would be accompanied by
ernment policy in our model focuses on providing an in- some sort of bank regulation. Such bank regulation would
stitution to prevent a bad equilibrium rather than a policy serve a function similar to restrictive covenants in bond
to move an existing equilibrium. Generally, such a policy indentures. Interesting but hard to model are questions of
need not cause distortion. regulator discretion which then arise.
Through its discount window, the Federal Reserve can,
Conclusions and Implications as a lender of last resort, provide a service similar to depos-
The model serves as a useful framework for analyzing the it insurance. The Fed would buy bank assets with (money
economics of banking and associated policy issues. It is creation) tax revenues at T = 1 for prices greater than the
interesting that the problems of runs and the differing ef- assets' liquidating value. If the taxes and transfers were set
fects of suspension of convertibility and deposit insurance to be identical to what is implicit in the optimal deposit
manifest themselves in a model which does not introduce insurance, the effect would be the same. The identity of de-
currency or risky technology. This demonstrates that many posit insurance and discount window services occurs be-
of the important problems in banking are not necessarily cause the technology is riskless.
related to those factors, although a general model will re- If the technology is risky, the lender of last resort can
quire their introduction. no longer be as credible as deposit insurance. If the lender
We analyze an economy with a single bank. The inter- of last resort were always required to bail out banks with
pretation is that it represents thefinancial intermediary in- liquidity problems, there would be perverse incentives for
dustry and that withdrawals represent net withdrawals from banks to take on risk, even if bailouts occurred only when
the system. If many banks were introduced into the model, many banks fail together. For instance, if a bailout is an-
then there would be a role for liquidity risk-sharing among ticipated, all banks have an incentive to take on interest
banks, and phenomena such as the federal funds market or rate risk by mismatching maturities of assets and liabilities,
the impact of bank-specific risk on deposit insurance could because banks will all be bailed out together.
be analyzed. If the lender of last resort is not required to bail out
The result that deposit insurance dominates contracts banks unconditionally, a bank run can occur in response to
which the bank alone can enforce shows that there is a changes in depositor expectations about the bank's cred-
potential benefit from government intervention into bank- itworthiness. A run can even occur in response to expecta-
ing markets. In contrast to common tax and subsidy tions about the general willingness of the lender of last
schemes, the intervention we are recommending provides resort to rescue failing banks, as illustrated by the unfor-
22Douglas W. Diamond, Philip H. Dybvig
Bank Runs, Deposit Insurance, and Liquidity
tunate experience of the 1930s when the Federal Reserve
misused its discretion and did not allow much discounting. References
In contrast, deposit insurance is a binding commitment
which can be structured to retain punishment of the bank's
owners, board of directors, and officers in the case of a
failure.
The potential for multiple equilibria when afirm's lia- Azariadis, Costas. 1981. Self-fulfilling prophecies. Journal of Economic Theory 25
(December): 380-96.
bilities are more liquid than its assets applies more gen- Bemanke, Ben S. 1983. Nonmonetary effects of thefinancial crisis in the propagation
erally, not simply to banks. Consider afirm with illiquid of the Great Depression. American Economic Review 73 (June): 257-76.
technology which issues very short-term bonds as a large Bryant, John. 1980. A model of reserves, bank runs, and deposit insurance. Journal of
part of its capital structure. Suppose one lender expects all Cass, David,
Banking and Finance 4 (December): 335-44.
and Shell, Karl. 1983. Do sunspots matter? Journal of Political Econom
other lenders to refuse to roll over their loans to the firm. 91 (April): 193-227.
Then it may be the lender's best response to refuse to roll Diamond, Douglas W. 1984. Financial intermediation and delegated monitoring. Re-
over its loans even if thefirm would be solvent if all loans view of Economic Studies 51 (July): 393-414.
were rolled over. Such liquidity crises are similar to bank Dothan,Journal
Uri, and Williams, Joseph. 1980. Banks, bankruptcy, and public regulation.
runs. The protection from creditors provided by the bank- Dybvig, Philip H.,ofandBanking and Finance 4 (March): 65-87.
Jaynes, Gerald D. 1980. Microfoundations of wage rigidity and
ruptcy laws serves a function similar to the suspension of unemployment. Manuscript. Yale University.
convertibility. Thefirm which is viable but illiquid is guar- Dybvig,Journal
Philip H„ and Spatt, Chester S. 1983. Adoption externalities as public goods.
of Public Economics 20 (March): 231-47.
anteed survival. This suggests that the transformation could Fisher, Irving. 1911. The purchasing power of money: Its determination and rela
be carried out directly by firms rather than byfinancial in- to credit, interest and crises. New York: Macmillan.
termediaries. Our focus on intermediaries is supported by Friedman, Milton, and Schwartz, Anna J. 1963. A monetary history of the United
the fact that banks directly hold a substantial fraction of the States, 1867-1960. Princeton, N.J.: Princeton University Press (for National
short-term debt of corporations. Also, there is frequently a Kareken,Bureau of Economic Research).
John H., and Wallace, Neil. 1978. Deposit insurance and bank regulation: A
requirement (or custom) that afirm issuing short-term com- partial-equilibrium exposition. Journal of Business 51 (July): 413-38.
mercial paper obtain a bank line of credit sufficient to pay Merton,loanRobert C. 1977. An analytic derivation of the cost of deposit insurance and
guarantees: An application of modem option pricing theory. Journal of
off the issue if it cannot be rolled over. A bank with de- Banking and Finance 1 (June): 3-11.
posit insurance can provide liquidity insurance to a firm, . 1978. On the cost of deposit insurance when there are surveillance costs.
which can prevent a liquidity crisis for afirm with short-
term debt and limit the firm's need to use bankruptcy to Journal of Business 51 (July): 439-52.
stop such crises. This suggests that most of the aggregate Niehans, Jiirg. 1978. The theory of money. Baltimore: Johns Hopkins University Press.
liquidityriskin the U.S. economy is channeled through its Patinkin, Don. 1965. Money, interest, and prices: An integration of monetary and v
insuredfinancial intermediaries, to the extent that lines of Tobin, theory. 2d ed. New York: Harper & Row.
James. 1965. The theory of portfolio selection. In The theory of interest rates,
credit represent binding commitments. ed. Frank H. Hahn and F. P. R. Brechling, pp. 3-51. London: Macmillan.
We hope that this model will prove to be useful in un-
derstanding issues in banking and corporate finance.
23You can also read
