Financial Regulation Lecture 9Based on notes by Marlena El.docx

Financial Regulation Lecture 9
Based on notes by Marlena Eley
March 22, 2019
1
1 Concept Review
We have been studying Cooper and Ross’s paper, which
demonstrates how
deposit insurance causes moral hazard. Their work uses the
Diamond and
Dybvig model and adds:
1. Risky Technology
2. Monitoring by Households (HH)
The introduction of deposit insurance reduces HH’s incentives
to monitor.
HHs would have to pay a fixed cost Γ, which is measured in
utils (an effort
cost rather than a resource cost), to monitor the bank. When
HHs do monitor,
they are able to FORCE the bank to invest in the safe
technology. We know
that HHs always prefer the safe long term technology
investment to the risky

technology investment because they are risk averse (u′′ < 0).
1.1 LEMMA
Let q denote the probability of a run - this q is exogenously
given. A Bank
that maximizes HH’s utility solves the following problem:
if q ≤ q∗ the deposit contract offered allows runs
if q > q∗ the deposit contract offered is a run preventing
contract
When q ≤ q∗ , banks are providing liquidity insurance, cE > 1,
which means
that a run is a potential equilibria. When q > q∗ , banks provide
contracts
similar to autarky allocations, cE = 1,cL = R.
Intuitively, if runs are not very likely, then the bank prefers to
offer a contract
that allows for runs but offers some liquidity insurance, which
is valuable to
households.
We’re studying when q ≤ q∗ because we want deposit insurance
to be offered
so that moral hazard is an issue.
1.2 Risky Technology
Risky technology is defined as:
(−1, 0,
{
λR with probability ν

0 with probability 1 −ν
Where:
1. λ > 1
2. νλ ≤ 1
2
Sidenote: when νλ = 1, we say this technology has a mean
preserving spread.
Moral hazard is introduced here and this implies that rather than
strictly
maximizing HH’s utility, Banks are now maximizing profit. The
“managers” of
the Banks get whatever is leftover after feeding the consumers.
Banks then maximize profit as follows:
max
i∈ [0,1]
ν[iλR + (1 − i−πcE)R− (1 −π)cL]
+ (1 −ν)max(i∗ 0 + (1 − i−πcE)R− (1 −π)cL, 0)
We see that the objective function is linear in i which tells us
that there may
be corner solutions where i∗ = 0 (investment is only made in
the safe
technology) or i∗ = 1 (investment is only made in the risky
technology).

1.3 The Threshold ī
We study this decision by establishing a threshold of
indifference, ī - when
they are indifferent between investing everything in the risky
technology and
investing in the safe technology.
We can establish ī by looking at the investment decision in the
low state,
(1 −ν) when the risky technology fails.
Conditional on being in the low state (1 −ν):
ī = {i ∈ [0, 1] : (1 − i−πcE)R− (1 −π)cL = 0}
∀i > ī, (1 − i−πcE)R− (1 −π)cL < 0, max = 0 therefore invests in
risky.
∀ i < ī, (1 − i−πcE)R− (1 −π)cL > 0, max = (1 − i−πcE)R− (1
−π)cL therefore invests in safe.
Our next step is to plug these values back into the objective
function for each
case.
If i ≥ ī, hence the max = 0, then the objective function becomes:
ν[iλR + (1 − i−πcE)R− (1 −π)cL] which is increasing in i
⇒ i∗ = 1
If i < ī, max = (1− i−πcE)R−(1−π)cL and the objective function
becomes:
ν[iλR + (1 − i−πcE)R− (1 −π)cL] + (1 −ν)[(1 − i−πcE)R− (1
−π)cL]
νiλR + (1 − i−πcE)R− (1 −π)cL which is decreasing in i

⇒ i∗ = 0
Now, we need to know which investment strategy results in the
higher profit
and therefore which investment strategy will be picked?
3
1.3.1 i∗ = 0
Well when i∗ = 0, the profit becomes: (1 −πcE) − (1 −π)cL.
We have been solving the Bank’s problem under the assumption
that they
would be offering the First Best contracts of c
sp
E and c
sp
L . We know at the First
Best the resource constraint faced by the Social Planner
resembles very closely
(exactly) the profit of the bank. The resource constraint being:
(1 −π)cL = (1 −πcE)R. This then tells us the the Bank’s profit
will be 0
when i∗ = 0!
1.3.2 i∗ = 1
Now we have to evaluate the profit when i∗ = 1, which is
νλR−πcER− (1 −π)cL. Again, part of this profit, πcER− (1
−π)cL
resembles very closely, actually exactly, the Resource

Constraint faced by the
Social Planner at the First Best solution, where (1 −π)cL = (1
−πcE)R. This
then tells us the profit when i∗ = 1 is νλR, which we know is
greater than 0!
Comparing these two profits, we know that the Bank will invest
in the risky
technology for the larger profit! This is the risky technology, so
i∗ = 1.
2 Monitoring
Given what we know:
1. δ = (cE,cL) is the first best allocation and
2. I(cE) and I(cL) are the deposit insurances offered if the bank
fails in t=1
and t=2, respectively
If the bank is investing in the safe technology, HH won’t
monitor. Why would
they pay the fixed cost Γ if they know they will get the safe
payout?
Given we know that the Bank will invest in the risky asset
(which we proved
in part 1), we want to find the Γ for which a HH will choose to
monitor.
Remember, by monitoring, HH FORCE the Bank to invest in the
safe
technology. We can find the Γ for which HH will monitor by
finding the
following:

EUtility without monitoring < E Utility with monitoring
2.1 E (Utility without monitoring)
The Expected utility without monitoring is as below:
π{(1 −q)[U(cL) + qU(I(cE))}+
(1 −π){[νU(cL) + (1 −ν)U(I(cL))] + qU(I(cE))
Broken down we have:
4
1. π{(1 −q)U(cE) + qU(I(cE))}, which is the expected utility of
impatient
agents. This multiplies π, the fraction of impatient agents by (1
−q), the
probability of no run, times the utility they would get in this
equilibria,
U(cE), plus q, the probability of a run times the utility they
would get in a
run U(I(cE)).
2.(1 −π){1 −q)[νU(cL) + (1 −ν)U(I(cL))] + qU(I(cE)), the
expected utility of
patient agents. This multiplies (1 −π), the fraction of patient
agents, by
(1 −q), the probability of no run, by the payoff received in no
run which is
νU(cL) + (1 −ν)U(I(cL)), which itself is an expectation of the
payoff based on
the success or failure of the risky technology, and q, the
probability of a run in
t = 1, multiplied by U(I(cE)).

2.2 E(Utility with monitoring)
The expected utility with monitoring is as below:
−Γ + [π((1 −q)U(cE) + q(I(cE)))] + (1 −π)[(1 −q)U(cL) +
qU(I(cE))]
Remember, by monitoring and incurring the effort cost Γ, HH
force the Bank
to invest in safe technology so we don’t even consider payoffs
of risky
technologies!
Broken down we have:
1. π((1 −q)U(cE) + q(I(cE))), which is the utility of impatient
agents and
2. (1 −π)[(1 −q)U(cL) + qU(I(cE)), the utility of patient agents.
2.3 E(Utility without monitoring) < E (Utility with
monitoring)
This inequality is as below:
π{(1 −q)[U(cE) + qU(I(cE))} + (1 −π){(1 −q)[νU(cL) + (1
−ν)U(I(cL))] +
qU(I(cE))}≤−Γ+[π((1−q)U(cE)+q(I(cE)))]+(1−π)[(1−q)U(cL)+q
U(I(cE))]
This equation greatly simplifies, namely terms for early
consumers cancel out,
which makes sense as they withdraw at t = 1 anyway so they
don’t care about
whether the bank invests in the risk technology (they are not
around at t = 2
anyway as they consume before and their preferences are not
defined over

consumption at t = 2). 1 In other words, if the bank invests in
risky
technologies, they are not the ones who face an insolvent bank.
This then simplifies to:
(1 −π){(1 −q)[νU(cL) + (1 −ν)U(I(cL))]}≤−Γ{(1 −π)[(1
−q)U(cL)]}
⇒ Γ ≤ (1 −π)(1 −q)(1 −ν)[U(cL) −U(I(cL))]
Here the tradeoff between deposit insurance and monitoring is
greatly
apparent, look at the term U(cL) −U(I(cL))!
1This is equivalent to saying they exit the economy, for our
purposes.
5
So far, we have been considering deposit insurance to be
complete, that is that
I(cL) = cL. If this is the case then,
Γ < 0 for HH to monitor the banks!
Therefore monitor will never take place under complete deposit
insurance for
any value of Γ > 0.
2.4 Finding Γ
Well, lets try to figure out if there is a way to have HH monitor
AND still
implement the First Best allocation.

Let’s consider the assumptions that we’re making:
1. cL = c
sp
L
2. If there was no insurance at all monitoring would take place
if we make
the following assumption:
Γ ≤ (1 −π)(1 −q)(1 −ν)U(cspL )
Now we have to consider whether we can implement the first
best allocation.
Well, if we do, there is a potential for a run equilibrium and HH
will monitor.
But by monitoring, HH incur the cost Γ and there is a potential
for a run.
Well, what happens with complete deposit insurance? HH don’t
monitor and
Banks end up investing in risky technology. HH are paid even in
the event of a
run or a Bank failure. Can the First Best be implemented in this
situation?
The solution offered by Cooper and Ross is Capital
Requirements. They
suggest that if Banks are solely gambling with depositors money
what would
happen if Banks were forced to invest some of their own
resources?
3 Capital Requirements
In the case of Capital Requirements we have to make an
additional

assumption.
1. Banks’ shareholders have ”deep pockets”
This just says that Banks get some money from outside the
model. These
shareholders have a large endowment of the good.
Cooper and Ross then introduce Capital Requirements which
require banks
to invest at least K units of their own endowment per unit of
deposit. With
the introduction of Capital Requirements the Banks problem
becomes:
max
i∈ [0,1+K]
{ν[iλR + (1 + K − i−πcE)R− (1 −π)cL]
+ (1 −ν)max((1 + K − i−πcE)R− (1 −π)cL, 0)}
6
We solve this the exact same way that we solved the previous
problem!
∀ī ∈ [0, 1 + K] such that the payoff in the low state is the same
regardless
whether the bank is solvent or not.
(1 + K − ī−πcE)R− (1 −π)cL = 0
∀ (1 + K − ī−πcE)R = (1 −πcL)

For i ≥ ī: max (1 + K − ī−πcE)R− (1 −π)cL, 0) = 0
⇒ i∗ = 1 + K (invest in the risky technology)
For i ≤ ī ⇒ i∗ = 0 (invest only in the safe technology)
The question then becomes, is there a value of K such that the
First Best
allocation can be implemented as an equilibrium outcome?
3.1 Finding K
There are three steps to finding this K:
1. Use the First Best allocation
2. Evaluate the Bank’s profit from investing all in safe or all in
risky
technology
3. Solve for the smallest K such that i = 0 (investment in all
safe
technology) yields the highest profit.
Solving for K in this way is solving for the minimum capital
requirement
necessary for the bank to be indifferent between investing in
risky technologies
and safe technologies. We know how to solve this: we evaluate
the investment
strategies at the two corners!
Step 1 - First Best Allocation
c
sp
E > 1 ⇒ c

sp
L < R
From the resource constraint at the First Best, (1 −π)cspL = R(1
−πc
sp
E )
Step 2 - The Bank’s Payoffs
max
i∈ [0,1+K]
ν[iλR + (1 + K − i−πcE)R− (1 −π)cL]
+ (1 −ν)max((1 + K − i−πcE)R− (1 −π)cL, 0)
Plugging in the Resource Constraint for (1 −π)cL = R(1 −πcE),
this
simplifies to:
max
i∈ [0,1+K]
{ν[iλR + (K − i)R] + (1 −ν)max((K − i)R, 0)}
The payoff at i = 0 is then KR.
The payoff at i = 1 + K is then ν[(1 + K)λR−R].
7
Step 3 - Finding the Smallest K
This then sets us up with the inequality KR ≥ ν[(1 + K)λR−R],
from which

we can solve for a K.
KR = ν[(1 + K)λR−R]
KR = ν(1 + K)λR−νR
K = νλ + νKλ−ν
K −νλK = νλ−ν
K(1 −νλ) = ν(λ− 1)
∴ K =
ν(λ− 1)
(1 −νλ)
For K =
ν(λ−1)
(1−νλ) , there is no monitoring AND we are implementing the
First
Best allocation! We are disciplining the moral hazard by
implementing these
Capital Requirements.
Because we know what λ and ν are, we can talk about how K
changes when
the technology becomes riskier.
3.2 How K Changes with λ and ν
If we assume that the risky technology has a mean preserving
spread, i.e.
λ̂ν̂ = λν and λ̂ > λ (the technology has higher returns) and ν̂ < ν
(the
technology has a lower probability of success), then:
K
̂ = ν̂λ̂−ν̂
1−ν̂λ̂

= λν−ν̂
1−λν ∀ K
̂ > K
This tells us that the riskier the technology, the higher the
capital requirement
needed to discipline the moral hazard and implement the First
Best allocation.
8
Institute of Social and Economic Research, Osaka University
Bank Runs: Deposit Insurance and Capital Requirements
Author(s): Russell Cooper and Thomas W. Ross
Source: International Economic Review, Vol. 43, No. 1 (Feb.,
2002), pp. 55-72
Published by: Wiley for the Economics Department of the
University of Pennsylvania and
Institute of Social and Economic Research, Osaka University
Stable URL: https://www.jstor.org/stable/827056
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INTERNATIONAL ECONOMIC REVIEW
Vol. 43, No. 1, February 2002
BANK RUNS: DEPOSIT INSURANCE AND CAPITAL
REQUIREMENTS*
BY RUSSELL COOPER AND THOMAS W. ROSS1
Boston University, U.S.A. and University of British Columbia,
Canada

Diamond and Dybvig provide a model of intermediation in
which deposit
insurance can avoid socially undesirable bank runs. We extend
the Diamond-
Dybvig model to evaluate the costs and benefits of deposit
insurance in the
presence of moral hazard by banks and monitoring by
depositors. We find that
complete deposit insurance alone will not support the first-best
outcome: de-
positors will not have adequate incentives for monitoring and
banks will invest
in excessively risky projects. However, an additional capital
requirement for
banks can restore the first-best allocation.
1. INTRODUCTION
The publicly supported deposit insurance plans of a number of
countries, most
notably the United States and Canada, have recently come
under intense public
scrutiny as concerns have mounted about the substantial
contingent liabilities they
have created for taxpayers. In the United States the savings and
loan (S&L) crisis led
to the transfer of a huge amount of bad debt, estimated recently
at about $130
billion, onto taxpayers' shoulders.2 Created originally to
support the banking sector
by building depositor confidence, there is recognition that the
insurance provided by
* Manuscript received November 1998; revised October 1999.
1This is a considerably expanded version of Section IV of our
NBER Working Paper, #3921,

November 1991. We have benefited from discussions on this
topic with Paul Beaudry, Fanny Demers,
Jon Eaton, Alok Johri, Arthur Rolnick, Thomas Rymes, Fabio
Schiantarelli, David Weil, and Steven
Williamson, and from helpful comments received from 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. The
extensive comments provided by three
referees and the editor of this journal are gratefully
acknowledged. Financial support for this work
came from the National Science Foundation, the SFU-UBC
Centre for the Study of Government and
Business, and the Social Sciences and Humanities Research
Council of Canada. The first author is
grateful to the Institute for Empirical Macroeconomics at the
Federal Reserve Bank of Minneapolis
for providing a productive working environment during
preparation of parts of this manuscript. Some
of this work was done while the second author was visiting the
Canadian Competition Bureau and he
is grateful for the Bureau's assistance. The views expressed
here are not necessarily those of the
Federal Reserve Bank of Minneapolis or of the Canadian
Competition Bureau. Please address
correspondence to: Russell Cooper, Department of Economics,
Boston University, 270 Bay State
Road, Boston, MA 02215. Fax: 617-353-4449. E-mail:
[email protected]
2 There is a considerable literature on the S&L crisis; see, for
example, Feldstein (1991),
Kormendi et al. (1989), and White (1991).

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COOPER AND ROSS
these plans has encouraged excessive risk taking by financial
intermediaries.3 These
concerns have led to calls for the reform of deposit insurance
and even suggestions
that it be abolished.
This paper attempts to evaluate the trade-offs between risk
sharing and moral
hazard associated with the design of banking regulations. In
particular, we focus on two
policy instruments: deposit insurance and bank capital
requirements. We are interested
in how these instruments can be used (and misused) to control
bank runs in an envi-
ronment in which banks can make imprudent investments and
depositors can monitor
bank behavior.
Reflecting ongoing problems in the financial services sector,
there has been a great
deal of research recently on lending behavior, bank stability,
and optimal banking
regulation. While a number of publications have considered
parts of the problem
addressed here, no individual contribution tackles the joint
determination of optimal

deposit insurance and capital requirements within a bank runs
model with risk-
averse depositors, depositor monitoring, and moral hazard.4
Given the ongoing
public debate over deposit insurance and capital requirements
and the attention paid
to the supposed trade-off between bank runs and moral hazard,
a structure is needed
that contains these elements.
With its emphasis on bank runs, the model of Diamond and
Dybvig (1983) provides a
convenient starting point for studying these issues. In the
absence of any moral hazard
considerations, Diamond and Dybvig argue that publicly
provided deposit insurance
can be effective as protection against expectations-driven bank
runs.5 However, their
3 Deposit insurance was created in the United States during the
Great Depression (1934) to
restore depositor confidence. It came to Canada in 1967.
Concerns about the Canadian system are
expressed in Smith and White (1988).
4 Some of this literature is reviewed in the recent books by
Dewatripont and Tirole (1994) and
Freixas and Rochet (1997). The articles closest in purpose to
this one include Giammarino et al.
(1993), Matutes and Vives (1996), Besanko and Kanatas
(1993), Holmstrom and Tirole (1993),
Kupiec and O'Brien (1997), and Peck and Shell (1999). Each
considers some aspect of our problem,
but none combines the elements we view as important here. For
example, Giammarino et al. (1993)
consider optimal deposit insurance premia in markets with bank

moral hazard but no bank runs.
Matutes and Vives (1996) study the effect of competition on
bank fragility with deposit insurance.
Besanko and Kanatas (1993) consider the provision of funds to
firms from both banks (through
loans) and capital markets in a model with bank moral hazard
but no bank runs. Studying bank
lending behavior (without deposits or bank runs), Holmstrom
and Tirole (1993) find that borrower
moral hazard can be controlled by requiring that borrowers
contribute some of their own funds-a
requirement not unlike the capital requirements that banks face.
Finally, Peck and Shell (1999) also
examine policies that might influence the probability of bank
runs, but focus on deposit contracts that
permit the suspension of convertibility and on government
restrictions on banks' portfolios of loans.
5 Further, Wallace (1988) has argued that there is an
inconsistency in the Diamond-Dybvig
model's treatment of deposit insurance. The spatial separation
that motivates banking appears
inconsistent with the ability of governments to provide deposit
insurance. However, Wallace goes on
to point out that "...this argument does not say that any kind of
deposit insurance is infeasible. It only
says that the policy that Diamond and Dybvig identify with
deposit insurance is infeasible..." (p. 13).
We are in complete agreement; clearly the financing of deposit
insurance must be credible to
eliminate certain equilibria. Therefore, in contrast to Diamond
and Dybvig, we rely on the presence
of an outside group of agents ("taxpayers") as a tax base.
Essentially, the government has enough
information to tax labor income without needing to overcome
any spatial separation constraints.

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DEPOSIT INSURANCE AND INCENTIVES
model does not incorporate the moral hazard considerations
seen to be central to
recent policy debates. Deposit insurance avoids bank runs but
has adverse incentive
effects: it implies less monitoring by depositors, which allows
banks to hold riskier
portfolios. In fact, if deposit insurance is complete enough,
depositors' and banks'
interests are aligned: both types of agents are eager to hold
high-risk portfolios,
effectively gambling with taxpayers' money. Thus a trade-off
emerges between
providing insurance against bank runs and monitoring
incentives.
By characterizing this trade-off, our model permits a derivation
of the optimal
degree of deposit insurance. In general, deposit insurance with
depositor monitoring
is not sufficient to support the first-best outcome. However,
appropriately designed
capital requirements can eliminate the incentive problem
caused by deposit insur-
ance and support the first-best allocation.

From the perspective of our model, the experience in the U.S.
during the 1980s
suggests two forms of regulatory failure. First, capital
requirements were inadequate.
Second, the relaxation of Regulation Q allowed banks to more
aggressively compete
for deposits, which, along with deposit insurance, led to
excessively risky investment.
This is certainly not a novel story but one that appears here in a
consistent, formal
framework.
2. MODEL
The model is a modified version of Diamond-Dybvig (1983).
There are N,
ex ante identical, agents in the economy who are each born
with a unit endow-
ment, which they deposit with an intermediary in period 0.6 At
the start of period
1, agents are informed about their taste types. A fraction r learn
that they obtain
utility from period 1 consumption only (early consumers),
while the others obtain
utility exclusively from period 2 consumption (late consumers).
As in the first part
of Diamond and Dybvig (1983), assume that n is nonstochastic
and known to all
agents.7 Denote by CE and CL the consumption levels for early
and late con-
sumers, respectively, and let U(c) represent their utility
function over consump-
tion. Assume that U(') is strictly increasing and strictly
concave, U'(0)= oo, and
U(0) =0.
There are two technologies available for transferring resources

over time. First,
there is a productive technology that is not completely liquid.
This technology
provides a means of shifting resources from period 0 to 2, with
a return of R > 1 over
the two periods. However, liquidation of projects using this
technique yields only one
unit in period 1 per unit of period 0 investment. Second, there
is a storage tech-
nology, available to both intermediaries and consumers, that
yields one unit in
6 In Cooper and Ross (1998) we allow consumers to make their
own investments rather than using
an intermediary and prove that using an intermediary in this
structure always weakly dominates
autarky.
7 In the last part of their article they consider the importance
of aggregate uncertainty to argue
further in favor of deposit insurance instead of policies that
suspend convertibility.
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COOPER AND ROSS
period t + 1 per unit of period t investment, t = 0, 1. While not
as productive as the

illiquid technology over two periods, storage provides the same
one-period return.8
The intermediary operates in a competitive environment, which
compels it to offer
contracts that maximize consumers' ex ante expected utility
subject to a break-even
constraint. If the ex post consumer taste types were costlessly
verifiable it would
therefore offer a contract 6* = (c, cL) solving
(1) max 7rU(cE) + (1 - 7)U(cL)
CE,CL
(1 - 7)cL s.t. 1 = CE +1
From the first-order conditions, the optimal contract satisfies
(2) U'(c) = RU'(cL)
Since R > 1, the strict concavity of U(.) implies that cE < c4
for (2) to hold.
Diamond and Dybvig establish that when consumer tastes are
private information,
multiple equilibria may exist. The contracting problem can be
formulated with three
stages. First, the contract is set by the intermediary, which
specifies a consumption
level for each type of consumer independent of the number of
consumers claiming to
be each type.9 Second, agents learn their preferences and these
are announced to the
intermediary. Finally, the allocation of goods to agents is
determined by the contract.
The first-best outcome with the contract b* will be one

equilibrium of this game.
Truth telling is a dominant strategy for early consumers while
truth telling by late
consumers is a best response to truth telling by all other late
consumers.
Under b* there may also exist an equilibrium in which all late
consumers mis-
represent their tastes and announce that they are early
consumers. This can be an
equilibrium if the intermediary does not have sufficient
resources (including liqui-
dated illiquid investments) to provide cE to all agents. As in
the Diamond-Dybvig
model, the late consumers who do not withdraw in period 1
obtain a pro rata share of
the bank's period 2 assets. This equilibrium with
misrepresentation is termed a
"bank run."
The first-best allocation is vulnerable to runs iff cE > 1:
otherwise, the interme-
diary would have sufficient resources to meet the demand of cE
by all agents in
period 1. Diamond and Dybvig (1983) show that if agents are
sufficiently risk averse,
then cE will exceed 1.
8 In this setup, which comes from Diamond and Dybvig,
returns on investments made in this
productive technology are always (weakly) greater than those
in the alternative (storage). In Cooper
and Ross (1998) we extend the model by adding a liquidation
cost to these illiquid projects that

renders the one-period return to liquidated investments less
than the alternative. This expands the
set of conditions under which bank runs can occur and
influences agents' investment and contract
choices. It does not, however, have implications for the results
described below so we have chosen to
work with the simpler model here.
9 Thus, in particular, it is not feasible for the bank to
accumulate information about withdrawals
and make payments to depositors contingent on this
information. Further, agents are unable to meet
at a common location after period 0, thus eliminating the types
of ex post markets considered in, for
example, Jacklin (1987).
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As described in Alonso (1996) and Cooper and Ross (1991,
1998), there are
essentially two ways the intermediary can optimally respond to
the possibility of
multiple equilibria. One is to find the best contract available
that is not vulnerable to
runs. This best runs-preventing contract comes from solving (1)
with the added
constraint CE < 1 so that there are always sufficient resources
available in period 1 to

pay all consumers. Concavity arguments demonstrate that if the
first-best contract is
vulnerable to runs (i.e., c3 > 1), the best runs-preventing
contract will involve CE= 1
and CL = R.
As an alternative, one might construct a model of the
equilibrium selection pro-
cess and solve for the optimal contract. One simple model
relies on the existence of
publicly observable, but not contractible, variables (sunspots)
that correlate agents'
behavior at a particular equilibrium of the game.10 Instead of
preventing runs,
the intermediary adjusts the contract to reduce the impact of
runs in the event they
arise.
Suppose that with probability q there is a wave of economy-
wide pessimism that
determines the beliefs of depositors. If the outstanding contract
has a runs equilib-
rium the pessimism leads to a bank run. With probability (1 -
q), there is optimism
and no run occurs. In this way, the beliefs of depositors are
tied to a move of nature
that determines their actions. The intermediary recognizes this
dependence in de-
signing the optimal contract.
Taking the probability of liquidation, q, as given, the contract
solves (assuming
CE> 1)
(3) max(1 - q)[nU(cE) + (1 - m)U(CL)] + qU(cE)(1/cE)
CE,CL

(1 - t)cL
s.t. 1 = 7cE + -
Let 6(q) be the contract solving this problem.1 Cooper and
Ross (1998) show the
existence of a critical q* C (0, 1) such that the best runs-
preventing contract domi-
nates the best contract with runs if q > q* and the reverse holds
if q < q*.
3. SUPPORTING THE FIRST-BEST: DEPOSIT INSURANCE
AND CAPITAL REQUIREMENTS
The previous section characterizes the optimal response of a
private bank facing
the prospect of a run. Regardless of whether the intermediary
optimally adopts a
runs-preventing contract or allows runs, the possibility of bank
runs clearly lowers
expected utility below that attainable in the first-best solution.
This naturally raises
10 Bental et al. (1990) and Freeman (1988) also adopt a
sunspots approach. In contrast to our
work, those articles allow for sunspot-contingent contracts.
While it is convenient to think of sunspots
as determining which equilibrium of the subgame will be
observed, contracts contingent on these
events are assumed to be infeasible.
1 Here an agent receives CE with probability liCE in the event
of a run, which occurs with
probability q. Note that if the solution to (3) involved CE < 1 it
would in fact be runs-preventing and
therefore be dominated by the best runs-preventing contract

(CE = 1 and CL = R).
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COOPER AND ROSS
the question of whether some government intervention in the
form of deposit in-
surance or other instrument could prevent runs and thus
improve welfare.12
Deposit insurance is a contract set by the government that
provides a payment to
depositors in the event that the bank is unable to meet its
obligations.13 Diamond
and Dybvig argue that a simple deposit insurance scheme will
eliminate bank runs in
their model. However, their argument leaves aside the adverse
incentive effects of
deposit insurance on both the investment strategy of the
intermediary and the
monitoring decisions of depositors. We study this by adding
both moral hazard and
monitoring by depositors to our model. Our main result in this
section is that an
appropriately designed capital requirement coupled with
deposit insurance can avoid
bank runs without creating severe moral hazard problems.
3.1. Extended Model. We modify the basic model in a number

of ways, detailed
in the subsections that follow. First, we introduce a richer
investment choice for the
banks. Second, we allow for a monitoring decision by
depositors. Third, we introduce
both deposit insurance and capital requirements as policy
instruments for the gov-
ernment.
The sequence of events in period 0 is as follows: First, the
government sets a
deposit insurance policy. In general, the government contract
stipulates payments to
early and late consumers as a function of the deposit contract
in the event the
intermediary is unable to make its promised payments. We
denote the payments to
early and late consumers as I(CE) and I(CL) respectively. Since
the government is
unable to observe the types of private agents, it too must rely
on the agents' an-
nouncements. Put differently, those agents who appear at the
intermediary in period
1 are termed early consumers and are eligible for the
government insurance over CE
in the event the intermediary is unable to meet its obligations.
Likewise, an agent
who makes the announcement of being a late consumer is
eligible for government
insurance over CL if the bank fails in period 2. Importantly, if
a bank fails in period 1,
then late consumers will not receive insurance over CL.
Instead, they will receive the
same payment as early consumers if a bank fails in period 1.
Note that we assume
that government insurance policy depends on the deposit

contract offered by the
intermediary.
Second, the competitive banks offer a contract, 6. Depositors
then decide on
the allocation of their endowment and whether to monitor the
bank. If the bank
12 For the purposes of this exercise, we do not consider private
deposit insurance schemes.
13 For simplicity, assume that the tax obligations to finance
deposit insurance fall upon agents who
are not depositors. Hence we do not consider the possibility
that intermediaries make payments into
a deposit insurance pool but rather focus on the obligations of
taxpayers to the system. Here we
imagine a government policy that provides deposit insurance to
agents who arrive at the bank after
the bank has exhausted resources and then taxes, say, the
endowment of a group of agents in the
economy not involved with the intermediary or even the
endowment of the next generation of
depositors, as in Freeman (1988), to finance these transfers. We
assume that the social welfare
function is such that providing this insurance is desirable. The
key point is that there must be a
government taxation scheme that is not inconsistent with
isolation that is capable of generating the
needed revenues.
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is monitored, investment decisions are observable to all agents.
The depositors
then learn their taste types. Finally, the bank manager allocates
the funds to the
two investments. Our choice of timing here is not very
restrictive: the outcome of
this model and that with simultaneous moves by the monitor(s)
and the banker
are the same though it is important that the monitoring occurs
before the types
are realized.
3.1.1. Richer technology. To allow the bank an avenue for
moral hazard, as-
sume that there exists a second, multiperiod technology that
yields a second period
return of 2R with probability v and 0 otherwise. Further,
assume that 2 > 1 and
v2 < 1 so that this risky technique has a higher return if it is
successful but a lower
expected return than the riskless illiquid investment. Thus, the
riskless two-period
investment is preferred to the risky illiquid investment by all
risk averters.14 As with
the riskless illiquid technology, this alternative technology also
yields one unit in
period 1 per unit invested in period 0.
The bank's investment policy is chosen by a risk-neutral
manager who represents
the bank's owners (shareholders). We assume that any funds

remaining after the
payment of cL to the late consumers are retained by the
shareholders of the bank. As
before, if the intermediary does not have sufficient funds for
the late consumers,
then these agents (and not the shareholders) have rights to a
pro rata share of the
bank's resources.
As we shall see, under some contracts, the manager may have
an incentive to
invest using the risky technology. In particular, in the absence
of a minimum capital
requirement, the risky investment is preferred by the manager
since 2 > 1 gives him
(i.e., the shareholders) a chance at a high return.15 When
deposit insurance is suf-
ficiently generous, depositors will not care that the bank
undertakes risky invest-
ments.
More formally, suppose that the bank offered depositors the
first-best contract 6*
and that the government provides depositors with complete
deposit insurance; i.e.,
I(cL)= CL. Let i denote the amount of resources (per unit of
deposit) that the in-
termediary places in the risky illiquid investment. Then i is
chosen to
max[v(i2R + (1 - i - nrc)R - (1 - n)c) + (1 - v)max((1 - i- rc* )R
- (1 - n)cj, 0)]
The max operator appears here since the bank may not have
enough resources to
meet the needs of depositors when the risky investment fails.

Since the inter-
mediary earns zero profits in the first-best contract when it
invests all of its funds
in the riskless illiquid technology, for any i > 0, the
intermediary has zero return
in the state in which the risky investment fails. Further, with
vA < 1, the inter-
mediary's expected return is positive and increasing in i. Thus
the solution is
for the intermediary to place all funds in the risky illiquid
investment. Since
14 That is, vU(RX) + (1 - v)U(O) < U(vRA) < U(R) for any
concave U(.).
15 "For example, if a bank's liabilities are deposits insured
with fixed-rate Federal Deposit
Insurance Corporation (FDIC) insurance, it is well known that
the bank may have an incentive to
select very risky assets since the deposit insurers bear the brunt
of downside risk but the bank owners
get the benefit of the upside risk" (Diamond and Dybvig, 1986,
p. 59).
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COOPER AND ROSS
depositors receive full deposit insurance, they have no

incentive to oppose this
investment strategy.
3.1.2. Depositor monitoring. The second change to our model is
the inclusion of
a monitoring decision on the part of depositors. Any depositor
who monitors incurs a
cost r (modeled as a utility loss) and can force the bank to
adopt the depositor's
desired portfolio.16 Given the moral hazard problem outlined
above, depositor
monitoring is a potentially important element in overcoming
the incentive of banks
to invest in risky ventures.
We begin the analysis by studying the monitoring decisions by
the depositors given
the investment choices by the manager, the level of deposit
insurance provided by
the government, and a deposit contract, (CE, CL). We consider
here the case of a
single depositor, but the qualitative results can be extended to
the multidepositor
case.17 If the bank has an incentive to invest in the risky
technique, then monitoring
will occur iff
(4) (1 - 7r)(1 - v)(1 - q)[U(cL)- U(I(cL))] > r
The left-hand side is the expected gain to the depositor from
turning the problem
into one of full information for a given value of CL and the
right-hand side is the
monitoring cost. Note that this condition incorporates the
assertion that if moni-
toring did not occur, the bank would invest in the risky

technology that would yield
the depositor CL with probability v. Further, as the monitoring
decision is made in
period 0, the individual values the information only if he is a
late consumer, which
happens with probability (1 - n). Finally, the gains to
monitoring are lost if there is a
bank run since both the risky and riskless illiquid techniques
generate equal returns
over the first period. So, the left-hand side of (4) includes (1 -
q), the probability of
optimism.
The influence of deposit insurance on monitoring is apparent
from this condition.
If I(CL) is close to CL for all levels of late consumption, then
the single agent has no
incentive to monitor. However, for small levels of insurance,
monitoring will take
place. For this analysis, we assume that when there is no
deposit insurance, a single
depositor will monitor if CL = CL.
16 Calomiris and Kahn (1991) model monitoring as a private
activity though the outcome of
monitoring is made public. The incentives to monitor are
created by sequential service in which the
agents who monitor are "first in line." Our results are robust to
assuming that the information
generated by monitoring is private.
17 The existence of multiple depositors creates a number of
interesting complications due to
free riding on the monitoring of others. One possibility of
resolving this is via a cooperative

agreement on monitoring: an accounting firm is retained as part
of the deposit arrangement.
Alternatively, in the noncooperative game between depositors
to determine the level of
monitoring by each, there will be asymmetric equilibria in
which one depositor monitors and the
others free ride. There may also be equilibria in which
monitoring costs are shared by a subset of
the depositors. Finally, there may also be a mixed-strategy
equilibrium that each agent monitors
with some probability. Such a model is considered in an
expanded version of this article available
from the authors.
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3.2. Capital Requirements. Consider a second instrument of
government policy:
a requirement on the ratio of debt to equity financing for an
intermediary. To be
precise, suppose that the shareholders of the intermediary are
required by the
government to contribute K units of the numeraire good per
unit of deposit to the
intermediary's capital account.
Let i again denote the funds (per unit of deposit) that the
intermediary places in

the risky investment. Then the portfolio choice of the
intermediary is determined
from
(5) max[v(i2R + ((K + 1) - i - 7cE)R - (1 - 7)CL)
+ (1 - v) max((1 + K - i - 7CE)R - (1 - 7t)CL, 0)]
The first part of this expression applies to the case of a
successful risky invest-
ment outcome, in which case the shareholders of the bank earn
a high return of
AR on the i units placed into the risky illiquid investment.
With probability
(1 - v), however, the risky investment fails and the bank's
resources are limited to
(1 + K -i- - i CE), which earns a return of R. These funds are
then used to
meet the demands of late consumers, given by (1 - 7t)CL. It is
possible that the
intermediary does not have sufficient resources to meet these
demands by late
consumers so that the bank's shareholders obtain 0. Hence the
max operator
in (5).
In fact, the nonlinearity created by the possibility of
bankruptcy is central to the
moral hazard problem faced by a bank. In particular, suppose
that the terms of the
contract offered depositors are such that there exists a level of
risky illiquid in-
vestment (i') satisfying
(1 + K - i' - 7CE)R = (1 - 7)CL

At this critical level of risky investment, the firm has zero
profits in the second
period when the risky project fails. It is easy to see that the
expected payoff of
the intermediary is higher at i = 0 than for any i c (0, i') since
shareholders bear
all of the downside risk from investing more resources in the
risky illiquid project
for i in this interval. For any i > i', shareholders do not bear the
risk of this
investment, so it is profitable to put more funds in the risky
investment. Thus,
from this optimization problem, the choice of the intermediary
is reduced to
placing either all of the funds in the risky investment or all of
the funds in the
riskless investment.
3.3. Supporting the First-Best Allocation. The point of the
following proposi-
tion is that if the capital requirement is sufficiently large,
shareholders will no
longer prefer to gamble with depositors' funds and thus the
moral hazard problem
is solved. Further, with complete deposit insurance, bank runs
are eliminated.
Finally, depositors will have no need to monitor the bank since
they are completely
insured.
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COOPER AND ROSS
Formally:
PROPOSITION 1. If I(CL) = CL for CL < C, I(CL) = CL for
CL > C, I(CE) = CE for
CE < CE, I(CE) = CE for CE > CE, and K > K* - [v(2 - 1)]/[1 -
2v], then the first-best
allocation of (c , cL) is achievable without bank runs and
without monitoring.
PROOF. Since deposit insurance is complete up to (c4, cL), if
the first-best con-
tract is offered, bank runs will be eliminated.
Using the first-best contract, (5) becomes
(6) max[v(i)R + ((K + 1) - i - rcc)R - (1 - n)c*)
+ (1 - v) max((1 + K - i - rnc)R - (1 - n)c*, 0)]
Using the resource constraint of R = (1 - n)c4 + R7rc, this
reduces to
(7) max[v(i2R + (K - i)R) + (1 - v) max((K - i)R, 0)]
Clearly, i will be set to 0 or to its maximal value of (1 + K)
since any interior choice of
i is dominated by one of these extremes. The profits of the
intermediary are higher at
i=O than at i = 1 + K iff
RK > v(l + K)XR - Rv

which reduces to the condition given in the proposition.
Finally, from the definition of the first-best, there is no other
contract that can
increase the expected utility of the consumer. Thus, if capital
requirements meet the
bound given in the proposition, banks will offer the first-best
contract to depositors
and will not have any incentive to invest in the risky
technology. Depositors will
therefore have no incentive to monitor and, given the presence
of complete deposit
insurance, there will be no bank runs. i
The point of this proposition is that an adequate equity capital
base can provide
sufficient incentive to owners managers to overcome the moral
hazard problems
without the need for monitoring by depositors. In this case,
deposit insurance can
prevent bank runs without creating incentive problems and the
first-best allocation,
given as the solution to (2), can be supported.18
Note that the capital requirement does not specify how the
intermediary must
invest the funds that shareholders provide. In the proof of
Proposition 1, we find that
if the intermediary has an incentive to invest depositors' funds
in the risky illiquid
technology (which occurs iff K < K*), then the intermediary
will invest shareholders'
funds in the risky venture as well. If it did not do so, the
intermediary would be
forced to pay depositors all of the shareholders' funds in the
event that the risky

venture failed. Hence the incentive to gamble with depositors'
funds will spill over to
the allocation of shareholders' funds as well.
18 As a referee has correctly pointed out, the assumption of
risk neutrality on the part of the bank
is important to this result. If the bank manager and
shareholders were risk averse, there would be
additional costs associated with investing own-capital in a bank
with uncertain returns.
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The effects of parameter changes on the critical level of capital
K* are of interest.
For example, a mean-preserving spread on the returns from the
risky asset, as rep-
resented by a combination of increasing i and decreasing v that
leaves 2v constant,
will increase K*. That is, as the probability of the risky asset
succeeding falls, holding
the expected return constant, more capital will be needed to
deter morally hazardous
investment behavior. This is a fairly intuitive result. However,
if we increase either i
or v while holding the other fixed-in either case increasing the
efficiency of the risky
investment-the minimum capital requirement actually rises. As

the risky asset is
more attractive, we need to impose tighter minimum capital
requirements.
4. THE SAVINGS AND LOAN CRISIS
The model developed here is also useful in understanding the
role that sub-
optimal regulatory policies played in the S&L crisis in the
United States in the
1980s.19 This crisis, almost certainly one of the most
important events in
American banking history, has imposed costs on taxpayers that
continue to
mount.
In the late 1970s and early 1980s interest rates climbed
substantially, and S&Ls
and some banks were squeezed as depositors withdrew funds to
put them into
higher-yielding Treasury Bills and money market funds while
the long-term
mortgages that provided much of the S&L income were fixed at
interest rates far
below market rates. Regulatory reforms introduced to help
S&Ls compete (e.g.,
flexible rate mortgages), the relaxation of controls on interest
rates paid (Regu-
lation Q), and the expansion of deposit insurance protection
combined with a
lack of regulatory oversight to introduce severe problems of
moral hazard. Thrifts
with low levels of net worth now had the opportunity to gamble
with other
people's (i.e., taxpayers') money and insured depositors had
little incentive to

monitor their thrifts. Indeed, if taxpayers were going to cover
the downside,
depositors shared the thrift owners' interest in risky
investments with high upside
potential, even if the expected yield was low. For a time, this
strategy led to rapid
growth of S&Ls, but eventually the poor quality of their
investments brought
many down.
To see how our model can explain important aspects of the
S&L crisis, we focus on
two key aspects of White's (1991) description of the S&L
crisis: (i) the removal of
Regulation Q and (ii) the inadequacy of capital requirements.
Removing Regulation
Q allowed banks more flexibility in competing for depositors,
that is, greater latitude
in setting CE and CL. One view of Regulation Q was that it
essentially mandated runs-
preventing contracts, and its repeal allowed banks to offer
contracts that were vul-
nerable to runs. When squeezed by the new pressure to offer
higher interest rates to
attract deposits even while many of their loans (often
mortgages) were set at very low
rates, many smaller institutions became seriously
undercapitalized-a deficiency not
19 For background on the crisis and its causes, see, for
example, White (1991), Grossman (1992),
and Dewatripont and Tirole (1994, Chapter 4).
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COOPER AND ROSS
always noticed by regulators failing to measure the values of
assets at current market
prices.20
While the inadequate capitalization may have changed the
incentives of banks
to avoid risky projects, the existence of deposit insurance
implied that depositors
were still willing to place funds in these institutions. It is
important to recognize
that, in our model, deposit insurance does not create the moral
hazard problem:
the manager's interest in the risky asset would exist in the
absence of insurance.
What the deposit insurance does is reduce the incentive of
depositors to monitor
banks. In the case of many of the failed S&Ls, the interests of
these agents
became aligned with those of the banks and jointly they
gambled with taxpayers'
money.21
To formalize this point, we consider the implications of
suboptimal deposit in-
surance and inadequate capital requirements. In particular, we
assume that no
capital requirements are in place. This assumption simplifies
the analysis and
captures the theme that a key aspect of this experience was

inadequate capital
requirements. While outside our model, one could imagine that
a period of de-
flation led to a reduction in the value of capital and thus the
inadequacy of existing
capital requirements.22 Further, we consider a relatively
simple deposit insurance
scheme, in which the government provides a fraction c of the
resources owed to
depositors (both early and/or late types) when a bank fails.23
In particular, recall
that we assume that if a bank fails in period 1, both early and
late consumers
receive a fraction of CE. Essentially, the government insures
current deposits rather
than promised payments.
While admittedly quite crude, this configuration of policy
choices and market
conditions matches the description of the savings and loan
industry in the 1980s
provided by White (1991). Consider first the extent of deposit
insurance coverage
in the United States. Note that partial insurance is ostensibly a
component of
U.S. policy through limits on coverage. However, it is well
understood that in a
large number of cases, such as Continental Illinois in 1984, the
U.S. government
did provide deposit insurance to individuals with accounts in
excess of the
20 White (1991) admits that the regulators had a very difficult
job in this new environment and that
they even suffered from some very bad luck. For example, a
key Texas office was moved at just the

wrong time-disrupting the work of regulators just when their
oversight was needed the most.
21 In related research, Grossman (1992) studies the risk-taking
behavior of insured and uninsured
thrift institutions in Milwaukee and Chicago during the 1930s.
He finds evidence of moral hazard in
that after a few years of deposit insurance coverage, thrifts
would move toward holding riskier
portfolios. He also finds that the level of regulatory oversight
influenced the degree of risk taking.
Federally insured thrifts were the most heavily regulated and
took on less risk than their state-
chartered counterparts. And the stricter regulation in Wisconsin
led thrifts in that state to build less
risky portfolios than those of state-chartered thrifts in Illinois,
where the regulations were less
stringent.
22 Mechanisms such as this, where deflation leads to incentive
problems, are often discussed in the
literature on financial frictions.
23 By assumption, the deposit insurance covers the same
fraction of early and late consumption.
Hence, more sophisticated policies that might prevent runs
without creating a moral hazard problem
by insuring early consumption only are not considered.
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$100,000 cap.24 Diamond and Dybvig (1986) suggest that
since the government
did not credibly commit ex ante to pay off all depositors
(which might have
protected the bank from the capital flight it experienced) but
then covered those
losses ex post, "they incurred the expense of deposit insurance
without the
benefits" (p. 64).
With regard to capital requirements, the losses suffered by
many S&Ls had ef-
fectively reduced their capital to levels so low that
shareholders had relatively little
to lose from making high-risk investments. With these
investments, they were es-
sentially gambling with taxpayers' money. Hence it is of
interest to determine the
model's predictions under this scenario, to see if we have a
structure that can explain
what actually happened.
There is an obvious concern associated with this
characterization of deposit in-
surance: taking a as given, an intermediary has the incentive to
make outrageous
promises to depositors, given that the government is insuring
these offers. While the
removal of Regulation Q certainly gave the intermediaries more
latitude, some
constraint on the choice of 6 = (CE, CL) must be imposed. In
our analysis, we assume
that the government will provide insurance iff the terms of 3

solve the contracting
problem given the level of deposit insurance and under the
presumption that the
bank will not invest in the risky illiquid technology. Given that
the environment is
public information, there is no reason for the government to
insure contracts that are
only reasonable if the bank commits moral hazard and invests
in risky projects. As a
consequence, the bank is unable to pass along gains from
excessive risk taking to
depositors.25
To make the role of monitoring clear, we make use of (4) and
assume that there is
effectively only a single agent who can either monitor the bank
or not. Since the cost
of monitoring has been assumed to take the form of a utility
loss, the contracting
problems specified above do not change as we vary the cost of
monitoring.
Further, following Proposition 1, the bank chooses to invest
funds in either the
risky illiquid investment project or the riskless illiquid
investment project.
24 The Federal Deposit Insurance Company employs two
strategies to deal with failed institutions:
deposit payoff and deposit assumption. In the former case,
depositors simply receive their funds and
the bank is closed. In the latter case, the bank is taken over by
another institution and FDIC funds
are used to compensate the acquiring bank. In this case, large
and small depositors are protected.
Since a large fraction of the resolution of bank failures has

been through deposit assumption, large
depositors have, in effect, received insurance. The FDIC
Annual Report provides a more complete
explanation and data on the frequency of use of these policies.
We are grateful to Warren Weber and
Art Rolnick for discussions of this point.
25 One could add an element of unobservable side payments
from the bank to depositors into the
model to allow the sharing of these gains. As discussed below,
this would certainly influence the
characterization of the critical value of deposit insurance in
Proposition 2 but not change the results
qualitatively. An alternative approach that would permit some
of the gains from this risk taking to be
passed on to depositors would allow the bank to offer
depositors contracts that it would have the
resources to fully honor only if the risky investment was
successful. That is, suppose regulators could
not observe v and believed the bank's claim that v = 1. In this
case with full deposit insurance the
bank can-indeed competition will force it to-provide more
generous deposit contracts knowing
that the deposit insurer will certainly be needed if the risky
project fails. So again we have the
depositors and shareholders both wanting to invest in the risky,
inefficient project.
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COOPER AND ROSS
This discrete choice highlights the moral hazard problem for
the bank and its
depositors.
Finally, we assume that q is sufficiently small so that the
contract with runs
dominates the runs-preventing contract in the absence of
deposit insurance. Hence,
when we characterize the optimal contract in the presence of
deposit insurance, the
assumption that q is small implies that the private sector will
not adopt runs-
preventing contracts. We comment below on the robustness of
our results to the
alternative assumption that q is large enough to warrant the
adoption of runs-
preventing contracts, at least for some levels of deposit
insurance.
With complete deposit insurance (c = 1), the intermediary will
prefer to invest
in the risky illiquid technology and depositors will not care
since, in effect, they
are gambling with other agents' money. At the other extreme of
no deposit
insurance (a = 0), there is no moral hazard problem if
monitoring costs are low
enough so that depositors monitor the intermediary and thus
force the inter-
mediary to invest in the riskless illiquid technique. From this,
it is not surprising
that there exists a critical level of deposit insurance, denoted
a', at which
depositors are indifferent between investment in the risky and

riskless ventures.
This leads to the following characterization of the optimal level
of deposit
insurance, c*.
PROPOSITION 2. The optimal level of deposit insurance will
be at one of two
levels, Ca* E {a', 1}.
PROOF. To understand the possibility of a* = o', consider first
the design of the
best contract allowing for runs in the presence of deposit
insurance assuming that the
intermediary uses the riskless illiquid technique. This is (3)
modified to include
deposit insurance, i.e.,
(8) max(1 - q)[7U(cE) + (1 - T)U(CL)] + q U(CE) ( + U(OCCE)
( 1 CE,CL CE CE
s.t. (1 - n)cL = R(1 - 71CE)
Let b(c) = (cE(x), cL(a)) denote the solution to this contracting
problem.
Now consider the optimal contract allowing for runs in the
presence of deposit
insurance assuming that the intermediary uses the risky illiquid
technique. This is
clearly preferred by the intermediary, given that its
shareholders benefit when
the risky project succeeds and bear no risk if it fails. Put
differently, with no
capital requirement, the bank will have an incentive to invest in
the risky
technique.

Will the depositors monitor? Given 65(a) and the assumption of
a single monitor,
(4) becomes
(9) (1 - 7)(1 - v)(1 - q)[U(cL(x)) - U(0c(cL(a)))] > r
Let a' be the level of deposit insurance such that (9) holds as an
equality. We
assume that r is small enough so that monitoring will occur at a
= 0 but not for a
near 1. So by continuity of U(.) and hence continuity of the
solution to (8), a' e (0, 1).
In sum, if the government sets the level of deposit insurance at
a', it would anticipate
68
2019 02:57:00 UTC
that the optimal contract would solve (8) and the depositor
would be indifferent
between monitoring and not monitoring.
At a = 1, there will be no bank runs and no monitoring. Hence,
investment will be
in the risky illiquid investment.
It is straightforward to see that only two values of deposit
insurance are relevant.

For values of a c [0, c'], there is no moral hazard as the bank is
always monitored.
Starting at a 0, increases in a would just change the level of
insurance given to
depositors. As the best contract does not eliminate runs, this
insurance may have
social value. For values of a c (c', 1), there is a moral hazard
problem but there are
no additional incentive problems created by increasing the
level of deposit insur-
ance from o' toward 1. Hence, it is sufficient to compare social
welfare at c' with
that at 1. U
Intuitively, the reduction of the optimal deposit insurance rate
to the two possible
outcomes reflects the trade-off between insurance and moral
hazard. For c < a',
consumers monitor and prevent the risky venture. At a = a', the
incentives change and
for a > c', depositors are unwilling to monitor and thus
intermediaries choose risky
illiquid investments. Thus, a key aspect of the proof concerns
the existence of a'.
Thus, with inadequate capital requirements, the government is
forced to choose
between the insurance gains from deposit insurance and its
adverse incentive effects.
In our model, this trade-off is reflected in the choice between
a* = 1 and a* = c'. By
continuity, if q is sufficiently close to zero so that the prospect
of runs is infinitesimal,
then the best policy is to adopt partial deposit insurance and
thus avoid bank moral
hazard problems. Alternatively, if the moral hazard problem is
itself small, say

because vi is near 1, then it is best to offer full deposit
insurance.
Regardless of whether deposit insurance is full or partial, it is
important to note
that the first-best outcome is not achieved. In the case of
partial deposit insurance,
depositors will monitor the bank but they face either strategic
uncertainty or the
inefficiencies created by a runs-preventing contract. Full
deposit insurance clearly
creates an incentive problem since the interests of the bank and
its depositors are
aligned. Thus, even if monitoring costs are 0, the first-best is
not obtained.
We consider the robustness of these results with respect to two
important assump-
tions. First, if monitoring was the outcome of the interaction of
multiple agents rather
than just one, then the conditions for monitoring would not be
given by (4) and the
optimal action by the government would change. We show
elsewhere that the critical
value of a characterized in Proposition 2 is relevant for the
case of multiple depositors.
In particular, if a > ', then the Nash equilibrium is for no
depositor to monitor. That is,
if no other depositor monitors, then the remaining agent uses
(4) to determine whether
or not monitoring is desirable so that c' is again the critical
level of insurance. For
a < a', the symmetric Nash equilibrium will entail monitoring
and the probability that
any individual agent monitors will increase as a falls.26
Second, suppose that q was large enough so that, in the absence

of deposit
insurance, banks would have elected to offer runs-preventing
contracts. In such a
26 These results are contained in an earlier version of this
article, available from the authors upon
request.
69
2019 02:57:00 UTC
COOPER AND ROSS
case, the provision of partial deposit insurance can have the
perverse effect of
increasing the probability of runs. The insurance can make it
optimal to abandon
the runs-preventing contract and if the insurance is not
complete runs can still
happen. Thus, while with zero monitoring costs the best runs-
preventing contract
will involve neither runs nor moral hazard, adding partial
deposit insurance can
lead to both.
5. CONCLUSIONS
The goal of this article has been to extend the Diamond-Dybvig
framework to
understand the implications of runs and moral hazard for the
evaluation of the costs

and benefits of deposit insurance. In our analysis, as in that of
Diamond-Dybvig,
there is a clear benefit to the provision of deposit insurance as
it prevents runs. The
costs modeled here are associated with a reduction in the
incentives for depositors to
monitor, giving rise to riskier investments by intermediaries.
From the perspective of our model, the first-best allocation is
achievable with a
combination of policies. Deposit insurance is needed to avoid
bank runs. Capital
requirements are needed to overcome the adverse incentive
problems associated
with the provision of deposit insurance.
The article has demonstrated that one potential consequence of
the combination
of an inadequate capital requirement, say due to regulatory
failure, with a generous
deposit insurance fund is the type of banking instability
observed in the U.S. during
the 1980s. We therefore believe that our article contributes to
an understanding of
what happened to many of the failed S&Ls.
This work leaves a number of interesting avenues for future
research. For ex-
ample, we do not explicitly consider here the implications of
risk-based deposit
insurance plans. The 1991 FDIC Improvement Act mandated a
move toward risk-
based premia in the United States and a similar program
appears to be coming to

deposit insurance in Canada. While it might appear that such
policies would solve
the runs problem without introducing moral hazard, much
depends on the timing of
moves. If bank owners can adjust their portfolios after premia
have been paid, then
the problems we analyze here may remain. The premia, once
paid, become a sunk
cost that will not influence future investment behavior. Of
course, in a multiperiod
environment, "punishments" can be administered in the future
in the form of higher
premia, but in the case of banks with depleted capital bases,
and therefore nothing
much to lose, the punishment might come too late.27
While the model developed here does present simple conditions
to achieve
the first-best outcome, potential limitations of this solution
arise from the presence
of moral hazard between bank owners and managers and
difficulties in raising
27 This point is also made by Freixas and Rochet (1997, p.
270). Chan et al. (1992) demonstrate
the impossibility of fairly priced deposit insurance in a model
with asymmetric information and
incentive compatibility constraints. Dewatripont and Tirole
(1994) argue that "it is extremely
difficult to devise proper risk-based premiums, especially if
those are to be determined in a
transparent, nondiscretionary manner" (pp. 60-61). They
characterize the American approach with
premiums ranging from 23 to 31 cents per $100 of deposits as
"very timid."

70
2019 02:57:00 UTC
sufficient equity capital. Further, the equity capital requirement
must be adjusted
in response to changes in the economic environment. These
adjustments and the
continued monitoring of compliance with this requirement
might be costly. We
leave the question of the second-best policies in this
environment for future
research.
REFERENCES
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BESANKO, D., AND G. KANATAS, "Credit Market
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CALOMIRIS, C., AND C. KAHN, "The Role of Demandable
Debt in Structuring Optimal Banking
Arrangements," American Economic Review 81 (1991), 497-
513.
CHAN, Y., S. GREENBAUM, AND A. THAKOR, "Is Fairly
Priced Deposit Insurance Possible?,"
Journal of Finance 47 (1992), 227-45.
COOPER, R., AND T. W. Ross, "Bank Runs: Liquidity and
Incentives," NBER Working Paper
#3921, November 1991.
AND , "Bank Runs: Liquidity Costs and Investment
Distortions," Journal of Monetary
Economics 41 (1998), 27-38.
DEWATRIPONT, M., AND J. TIROLE, The Prudential
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AND , "Banking Theory, Deposit Insurance and Bank
Regulation," Journal of
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FELDSTEIN, M., "The Risks of Economic Crisis:
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Economic Crisis (Chicago: University of Chicago Press, 1991).
FREEMAN, S., "Banking as the Provision of Liquidity,"
Journal of Business 61 (1988), 45-64.
FREIXAS, X., AND J.-C. ROCHET, Microeconomics of

Banking (Cambridge, MA: MIT Press, 1997).
GIAMMARINO, R., T. LEWIS, AND D. SAPPINGTON, "An
Incentive Approach to Banking Regu-
lation," Journal of Finance 48 (1993), 1523-42.
GROSSMAN, R. S., "Deposit Insurance, Regulation, and Moral
Hazard in the Thrift Industry: Evi-
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800-21.
HOLMSTROM, B., AND J. TIROLE, "Financial
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JACKLIN, C., "Demand Deposits, Trading Restrictions and
Risk Sharing," in E. Prescott and
N. Wallace, eds., Contractual Arrangements for Intertemporal
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Minnesota Press, 1987).
KORMENDI, R., V. BERNARD, S. C. PIRRONG, AND E.
SNYDER, Crisis Resolution in the Thrift
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KUPIEC, P., AND J. O'BRIEN, "Deposit Insurance, Bank
Incentives, and the Design of
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1997.
MATUTES, C., AND X. VIVES, "Competition for Deposits,
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PECK, J., AND K. SHELL, "Bank Portfolio Restrictions and
Equilibrium Bank Runs," mimeo, Ohio
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SMITH, B., AND R. WHITE, "The Deposit Insurance System
in Canada: Problems and Proposals for
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72 COOPER AND ROSS
WALLACE, N., "Another Attempt to Explain an Illiquid
Banking System: The Diamond-Dybvig
Model with Sequential Service Taken Seriously," Federal
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2019 02:57:00 UTC
Project can be of any length you like, from a few pages to 20
(although I doubt any of you will have 20 pages to write). I do
NOT want you to write random stuff just to fill the pages, I

want to read essential information.
The logic to follow is similar to what you see when you read the
papers we have studied: abstract (very short), introduction
(where you say what your question is and why it is interesting,
and how you will study it) and then the body of the paper.
Finally a conclusion that can be again very short and usually it
is just saying the same things as the abstract.
If you do not have a mathematical model that’s also ok,
although I would encourage you to think through the
frameworks we have seen carefully and you might find useful
material.

Financial Regulation Lecture 9Based on notes by Marlena El.docx

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