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Accessible Published by De Gruyter July 17, 2014

On Run-preventing Contract Design

Yoshihiro Ohashi

Abstract

This study considers how to implement an efficient allocation of a financial intermediation model, including liquidation costs. The main result shows that there is a mechanism such that, for any liquidation cost, an efficient allocation is implementable in strictly dominant strategies. There is no need for third-party assistance, such as deposit insurance. In addition, the mechanism is tolerant of a small, unexpected shock caused by premature withdrawals.

JEL Codes: D82; G21

1 Introduction

Diamond and Dybvig (1983) point out that a demand-deposit contract can achieve a socially efficient allocation, but can also bring about an inefficient bank run outcome in equilibrium. Following this study, a number of other studies have addressed bank runs.[1] From a theoretical perspective, it is interesting to investigate the unique implementability of efficient allocations.[2]

Cooper and Ross (1998) extend the model of Diamond and Dybvig (1983) by introducing a liquidation cost. Their extended model has enriched the analyses of banking systems, making them more realistic.[3]Cooper and Ross (1998) provide a necessary condition for which a direct mechanism relative to an efficient allocation prevents bank runs in all Nash equilibria.

However, their mechanism only offers a one-time opportunity to provide a good to consumers in each period. This restriction generally makes it difficult to achieve the optimum outcome. In fact, their run-preventing mechanism fails to implement an efficient allocation if the liquidation cost is sufficiently high.

In this study, I consider a more general mechanism and revisit the problem of implementing an efficient allocation. The mechanism I consider allows, at most, two opportunities to provide a good in one period.[4] The proposed mechanism can be defined regardless of liquidation costs. The main result shows that the mechanism uniquely implements an efficient allocation in strictly dominant strategies (see Section 3).

My mechanism enforces a degree of illiquidity on consumers in that there is a delay before they can complete premature withdrawals. Consumers can withdraw their deposits up to a limit. However, the remainder of their deposits and interest remains frozen for a time.[5] Without this illiquidity, some efficient allocations are not implementable (see Sections 2.2 and 3.1).

My result requires no deposit insurance. Although deposit insurance is useful to prevent bank runs, some researchers consider it to be controversial because it can cause other problems, for example, a moral hazard.[6]

Without aggregate risk or estimation error for withdrawals, the obtained outcome from my mechanism coincides with that of the mechanism of suspension of convertibility (Diamond and Dybvig 1983). Although the suspension mechanism appears attractive, some studies have shown its limits.[7] My mechanism is superior to the suspension mechanism in terms of the continuity of outcomes. That is, my mechanism makes moderate changes to a provision level, depending on the number of unexpected premature withdrawals. In contrast, the suspension mechanism makes drastic changes to a provision level if the number of unexpected premature withdrawals is sufficiently large (see Section 3.2).

2 Banking model

2.1 Preliminaries

This section describes the setting for Cooper and Ross’s (1998) extended model. Consider an economy with a single consumption good, homogeneous competitive banks, and a continuum of consumers. The economy has three periods: t0,t1, and t2.

Short-term and long-term investment opportunities are available for the consumption good. For a unit of input in t0, a short-term investment yields one unit of the consumption good in t1. For the same input, a long-term investment yields R>1 units of the good in t2. The long-term investment can be liquidated, but this yields 1κ in t1 per unit of input, where κ[0,1] denotes a liquidation cost.

Each consumer, i, is an element of I=[0,1] and has one unit of the consumption good as an endowment. At the commencement of t1, a fraction of the consumers, θ(0,1), face a liquidity shock and can only obtain utility from consumption in t1. The remaining consumers can obtain utility in both t1 and t2. I refer to the former type as early consumers and the latter as late consumers. Each consumer’s type is known only to that consumer (i.e. it is private information).

Let u denote the consumers’ utility function over consumption, where u is a von Neumann–Morgenstern utility function, u:R+R. This function is strictly increasing, strictly concave, twice differentiable, and satisfies limc0u(c)=.[8] Here, u(c1+c2) applies to late consumers, where ck is consumption in tk, and u(c1) applies to early consumers.

An efficient allocation, say (ce,cl), is defined as the solution to the following optimization problem:

[1]maxce,cl,ψθu(ce)+(1θ)u(cl)s.t.θce1ψ,(1θ)clRψ,

where ψ[0,1] is the ratio of long-term investment to bank deposits. In the optimum solution, ψ=1θce and cl=R(1θce)(1θ)1. In addition, the first-order condition implies that ce<cl.

2.2 Demand-deposit contracts and the bank-run problem

In this section, I consider the problem of uniquely implementing an efficient allocation (ce,cl) by constructing a mechanism. Let M=((Mi),(gi)) denote a mechanism, where Mi is the message space of consumer i, and gi denotes a provision for consumer i. For convenience, I denote a consumer’s message by miMi and a profile of all consumers’ messages by m.

For example, the Cooper–Ross mechanism assumes that Mi={0,1}, for all iI, where mi=0 shows the willingness to withdraw. It further assumes that gi is described as gi=(gi1,gi2), where gik represents a provision in tk. The mechanism is subject to the following sequential service constraint: (1) gik=gjk, for all k{1,2} and i,jI; (2) gi1=gi1(mi) and gi2=gi2(m), for all iI.[9] If Mi={0,1}, a contract is said to be a demand-deposit if gi1(0)1, for all iI. Under demand-deposit contracts, if mi=0 for all iI, the bank has no asset in t2, so it is plausible to assume that gi2(0)=0.[10]

Then, as Diamond and Dybvig (1983) show, any demand-deposit contract may cause a bank run under the sequential service constraint. Cooper and Ross (1998) consider a run-preventing contract, which prevents bank runs in all Nash equilibria, although their mechanism no longer embodies a demand-deposit contract if consumers are less risk-averse.

3 Results

This section describes how the efficient allocation (ce(θ),cl(θ))=(ce,cl) is uniquely implementable in strictly dominant strategies.

First, I consider the following problem. For an arbitrary θˆ(0,1):

[2]maxce,clθˆu(ce)+(1θˆ)u(cl)s.t.θˆce1ψ(1θˆ)clRψ+1ψθˆce.

Let (ae(θˆ),al(θˆ)) denote the solution to problem [2].

Lemma 1For anyθˆ(θ,1), ae(θˆ)<ce<cl<al(θˆ).

Proof. See the Appendix.∎

Next, I define the following functions:

[3]f1(θ^)=(ce*ifθ^θae(θ^)ifθ^>θ,f2(θ^)=(dl(θ^)ifθ^θal(θ^)ifθ^>θ,

where dl(θˆ)(1θˆ)1[(1θ)cl+(θθˆ)ce], ae(1)limθˆ1ae(θˆ), and al(1)M1[R(1θce)]>ae(1), for some M(0,1).

Lemma 2For anyθˆ[0,1], f1(θˆ)<f2(θˆ).

Proof. Lemma 1 and the definitions of f1(1) and f2(1) imply that f1(θˆ)<f2(θˆ), for all θˆ(θ,1]. Thus, we only have to prove the case for each θˆ[0,θ]. The fact that ce<cl implies that:

(1θˆ)dl(θˆ)=(1θ)cl+(θθˆ)ce>(1θ)ce+(θθˆ)ce=(1θˆ)ce,

which shows the result.∎

I consider a mechanism that comprises two communication phases in t1 on, say, Day 1 and Day 2, and one phase in t2 on, say, Day 3. Let M=((Mi),(gi)) denote the mechanism, where Mi={0,1} and gi=(gi1,gi2,gi3), where gik represents a provision on Day k. The sequential service constraint is as follows: (1) gik=gjk, for all k{1,2,3} and i,jI; (2) gi1=gi1(mi), gi2=gi2(m), and gi3=gi3(m), for all iI.

Here, I consider a mechanism with the following characteristics: (1) the bank does not allow early consumers to withdraw their deposits all at once and (2) consumers can be served all their deposits with a delay.

Theorem 1Suppose thatθ(0,1)is public information. There exists a mechanism that uniquely implements the efficient allocation(ce(θ),cl(θ))=(ce,cl)in strictly dominant strategies.

Proof. Let c_einfθˆ(θ,1]ae(θˆ). Then, c_e>0 because limc0u(c)=. Consider the following provision functions:

Day 1. For all iI, gi1(0)=c_e and gi1(1)=0.

Day 2. For all iI, if mi=0, then gi2(m)=f1(θ1)c_e, where θ1 is the number of consumers whose message mi is zero; otherwise, gi2=0.

Day 3. For all iI, if mi=1, then gi3=f2(θ1); otherwise, gi3=0.

Lemma 1 implies that gik0 for all k{1,2,3} and iI. Note that, for all θ1[0,1], the bank has Rψ on Day 3. Hence, providing f2(θ1) to 1θ1 consumers is feasible. Lemma 2 implies that all late consumers prefer to be served in t2, regardless of θ1. In contrast, all early consumers prefer to be served in t1, because f(θ1)>0 for all θ1[0,1]. Hence, we obtain θ1=θ and the mechanism implements (ce,cl) in strictly dominant strategies.∎

This three-day mechanism has three important characteristics. First, it prevents bank runs. Second, the unique equilibrium outcome is efficient. Third, the mechanism is defined independently of the liquidation cost. Note that this mechanism sacrifices some liquidity for the benefit of stability. In particular, it does not embody a demand-deposit contract, as the Cooper–Ross’ mechanism does.

3.1 Example

Suppose that consumers have the utility function u(c)=(1γ)1c1γ, with a parameter γ>0. Then, we can easily derive that:

ce=1θ+(1θ)R1γγ,cl=R1γce.

In the optimum solution to problem [2], we obtain

ae(θˆ)=θceθˆ,al(θˆ)=(1θ)R1γce1θˆ.

Following our three-day mechanism, the bank allows consumers to withdraw θce at once. In general, θce<1 holds, hence our three-day mechanism does not represent a demand-deposit contract. However, this type of illiquidity is inevitable because a single bank has to avoid bank runs while achieving an ex ante Pareto-efficient outcome without any assistance, such as deposit insurance.

Here, we confirm that some efficient allocations fail to be implementable with any “two-day” mechanism, such as that of Cooper and Ross (1998). To achieve both the run-preventing property and efficiency, any efficient consumption, ce, must satisfy

ce*1ψ*+ψ*(1κ)=1κψ*,

where ψ=1θce, which is equivalent to

[4]ce1κ1κθ.

Obviously, inequality [4] is violated if κ is sufficiently large. In our example, for any θ(0,1) and γ>0, if κ satisfies

θ+(1θ)R1γγ1θ+(1θ)R1γγθ<κ1,

then any two-day mechanism must sacrifice efficient allocation ce=ce(θ) to prevent bank runs.

3.2 Mechanisms under an unexpected shock

If there is an estimation error for the number of early consumers, our three-day mechanism has better properties than the classic two-day mechanism of suspension of convertibility.[11]

In this section, I consider the following situation. The bank and consumers estimate the probability of a consumer being early as θ in t0. However, the true value is θ=θ+ε, which remains unknown to the bank and consumers until the end of t1. Here, ε[0,1] is an unexpected term, referred to as a shock.

Consider the following functions:

[5]h1(θ^)=(ce*ifθ^θ0ifθ^>θ,h2(θ^)=(dl(θ^)ifθ^θal(θ^)ifθ^>θ.

Consumer i is served according to hk(θˆ) in period tk, where θˆ is the number of consumers who have been served before i. In contrast to most suspension mechanisms, (h1,h2) describes the suspension of convertibility independently of κ[0,1]. Without a shock, that is, θ=θ, (h1,h2) implements the efficient allocation (ce,cl) in strictly dominant strategies.

Now, consider the case of a shock. Suppose that θ=θ+ε is realized, where ε>0 is sufficiently small. Then, h1(θ)=0, while f1(θ)ce and f2(θ)=h2(θ)cl, because (f1,f2) are continuous. While our three-day mechanism provides goods that are sufficiently close to being efficient, the suspension mechanism (h1,h2) ignores the consumption by ε early consumers. Hence, our three-day mechanism is more prudent than the suspension mechanism in the case of unexpected shocks caused by premature withdrawals.

4 Conclusions

This paper proposes a provision mechanism in a deposit contract. The mechanism has the following strengths: (1) it is bank run-proof; (2) it uniquely implements an efficient allocation; (3) it is defined independently of liquidation costs; (4) it is defined without any third-party assistance, such as deposit insurance; and (5) it is tolerant of a small unexpected shock, as such a shock does not cause drastic changes in the outcome.

Appendix

Appendix: Proof of Lemma 1

For notational convenience, let aj=aj(θˆ), for j=e,l. The Lagrangian of the problem described in eq. [2] is

L=θˆu(ce)+(1θˆ)u(cl)+λ(θceθˆce)+μR(1θce)+θceθˆce(1θˆ)cl,

for some λ0 and μ0. Using the Kuhn–Tucker theorem, we obtain

[6]u(ae)=u(al)+λ,u(al)=μ,

with complementary slackness conditions:

[7]λ(θceθˆae)=0

and

[8]μR(1θce)+θceθˆae(1θˆ)al=0.

Eq. [6] implies that aeal and μ>0.

Suppose that ae=al. In this case, eq. [6] implies that λ=0. Then, in eq. [8], we have ae=R(1θce)+θce. As R(1θce)=(1θ)cl and (1θ)cl+θce>ce, we must have ae>ce. Then, the first inequality of eq. [2] implies that θˆ<θ, which contradicts the assumption that θ<θˆ. Hence, we obtain ae<al and λ>0 whenever θ<θˆ. Then, in eq. [7], θce=θˆae holds and, in eq. [8], as μ>0, al=(1θˆ)1(R(1θce))=(1θˆ)1(1θ)cl>cl holds. Thus, we obtain ae<ce<cl<al. This result is valid for any θˆ(θ,1).

Acknowledgments

This work was supported by MEXT KAKENHI Grant Number 24730278. I gratefully acknowledge the comments of anonymous referees. I take full responsibility for all possible errors.

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Published Online: 2014-7-17
Published in Print: 2015-1-1

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