# 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.

## 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:

Short-term and long-term investment opportunities are available for the consumption good. For a unit of input in *liquidation cost*.

Each consumer, *i*, is an element of *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, ^{[8]} Here,

An efficient allocation, say

where

### 2.2 Demand-deposit contracts and the bank-run problem

In this section, I consider the problem of uniquely implementing an efficient allocation *mechanism*. Let *i*, and *i*. For convenience, I denote a consumer’s message by *m*.

For example, the Cooper–Ross mechanism assumes that *sequential service constraint*: (1) ^{[9]} If *demand-deposit* if ^{[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

First, I consider the following problem. For an arbitrary

Let

**Lemma 1***For any*

**Proof**. See the Appendix.∎

Next, I define the following functions:

where

**Lemma 2***For any*

**Proof**. Lemma 1 and the definitions of

which shows the result.∎

I consider a mechanism that comprises two communication phases in *k*. The sequential service constraint is as follows: (1)

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 1***Suppose that**is public information. There exists a mechanism that uniquely implements the efficient allocation**in strictly dominant strategies*.

**Proof**. Let

**Day 1**. For all

**Day 2**. For all

**Day 3**. For all

Lemma 1 implies that

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

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

Following our three-day mechanism, the bank allows consumers to withdraw *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,

where

Obviously, inequality [4] is violated if

then any two-day mechanism must sacrifice efficient allocation

### 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 *shock*.

Consider the following functions:

Consumer *i* is served according to *i*. In contrast to most suspension mechanisms, *suspension of convertibility* independently of

Now, consider the case of a shock. Suppose that *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

for some

with complementary slackness conditions:

and

Eq. [6] implies that

Suppose that

# 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

©2015 by De Gruyter