This is a three-date (dates 0, 1, and 2) model. There are two banks, banks A and B, located in different geographical areas; each bank is owned and controlled by its manager. For each bank, there are numerous potential depositors living the area where the bank is located. At date 0, each potential depositor receives an endowment of $1.
A potential depositor can either deposit her endowment at the bank in her
neighborhood, or stores the endowment herself. There are no storage costs if potential depositors store their endowments. Depositors face liquidity shocks. Some of them die at date 1, so have to consume before they die. The others die at date 2, and can
consume at either date 1 or date 2. We will call those who die at date i type-i depositors, i = 1, 2. The fraction of type-1 depositors is denoted by t. The liquidity shocks are realized at date 1. At date 0, depositors do not know whether they will die early, and they have an equal chance of becoming type-1 depositors.
Depositors are risk-neutral. However, if a type-1 depositor consumes less than y at date 1, she will suffer a liquidity loss X, where y and X are constants with y > 1 and
X > 0.
1 Let Ui denote the utility function of a type-i depositor, and cj denote adepositor’s consumption at date j. The depositors’ utility functions can be written as
≥
<
= −
, ) ,
, (
1 1
1 1
2 1
1 c if c y
y c if X c c
c U
and U2(c1,c2)=c1+c2.
At date 0, each bank offers a deposit contract (d1
, d
2) to depositors in its
1 The purpose of making this assumption is to simplify the discussions on the optimal deposit contract.
It allows us to concentrate on the depositors’ response to public information about their banks.
neighborhood.2 For each dollar deposited at date 0, the bank promises to pay d1 if the depositor withdraws at date 1, and pay d2 if the depositor withdraws at date 2. When serving depositors, banks cannot distinguish between type-1 and type-2 depositors.
The sequential service constraint is imposed, which means depositors are served according to the time they arrive at the bank. For now, assume that banks do not have any capital, and there is no deposit insurance. We will relax these two assumptions in Sections 4 and 5, respectively. Convertibility suspension is not allowed. A bank has to keep open at date 1 unless it runs out of money. When determining the deposit
contract, a bank’s manager maximizes depositor welfare subject to the bank’s zero-profit constraint.
For each bank, if potential depositors deposit their endowments at the bank at date 0, then the bank invests these endowments in a long-term paper that matures at date 2. Both banks’ papers have the following features. The paper either succeeds or fails. For each dollar invested, a paper yields R if it succeeds and yields nothing if it fails. The probability that a bank’s paper will succeed depends on the prospects of the banking industry. If the prospects are favorable, then the paper will succeed with probability pg and will fail with probability 1 – pg. If the prospects are unfavorable, then the paper will succeed with probability pb and will fail with probability 1 – pb. Both pg and pb are constants with 1 ≥ pg
> p
b≥ 0.5.
3 At date 0, the prior probability that the banking industry’s prospects are favorable is θ. Let p0 denote the date 0 probability that a bank’s investment will succeed. We havep
0≡ [θ p
g+ (1 – θ) p
b]. (1)
The paper can be liquidated at date 1; for each dollar invested at date 0, earlyliquidation yields one dollar. Assume that p0
R + (1 – p
0) > 1, so the net present value
of the paper is positive if it is continued to date 2 when it will succeed and isliquidated at date 1 when it will fail.
The two banks invest in different papers. Assume that, given the prospects of the banking industry, the returns of the two banks’ papers are independent. Since both papers’ probabilities of success are affected by the prospects of the banking industry, at date 0 the returns of the banks’ papers are positively correlated. It is easy to show
2 Different banks may offer different deposit contracts. However, as mentioned below, we focus on the behavior of bank B’s depositors. Therefore, we only study the deposit contract offered by bank B.
3 The justification for assuming pb ≥ 0.5 is that, even during the Great Depression, only about one fifth of the banks in the United States failed. Therefore, it is not likely that the chance of a bank failure will exceed 0.5.
that the correlation coefficient between them is4
) 1 (
) (
) 1 (
0 0
2
p p
p p
g b−
−
≡
θ
−θ
ρ
> 0. (2)If a bank invests at date 0, then a public signal about its paper will be revealed at date 1. Let sA and sB denote the public signals about bank A’s and B’s papers,
respectively. For bank i, i = A, B, if bank i’s paper will succeed, then si
= H with
probability qi and si= L with probability 1 – q
i, where qi is a constant and qi≥ 0.5. On
the other hand, if Bank i’s paper will fail, then si= H with probability 1 – q
i and si= L
with probability qi. The qi can be explained as the precision of si; the larger the qi, the more precise si is. All the depositors of the two banks can observe both signals when they are revealed. The public signals sA and sB are the only information depositors receive. That is, they cannot observe whether their banks’ investment will fail, neither can they observe the prospects of the banking industry.As mentioned above, both public signals and the depositors’ liquidity shocks are revealed at date 1. For each bank, the signal about its paper and the liquidity shocks of its depositors are revealed simultaneously. To explore issues on contagious bank runs, we assume that sA and the liquidity shocks of bank A’s depositors are revealed first, and sB and the liquidity shocks of bank B’s depositors are revealed later. We will say that a contagious run occurs to bank B if the revelation of sA triggers a bank run on bank B. In our model, a contagious run is inefficient because depositors of bank B forego the information about their own bank when they make the withdrawal decisions. We will study whether and when a contagious run will occur, and how it can be eliminated.
Given our focus on contagious runs, we will analyze only the behavior of bank B’s depositors. The sequence of events about bank B is summarized as follows.
Date 0. Bank B’s manager offers a deposit contract (d1
, d
2) to potential depositors
in its neighborhood. Depositors decide whether to deposit theirendowments.
Date 1. (i) Signal sA is revealed. If depositors deposit their endowments at bank B
4 The variance of each bank’s return is p0(1 – p0)R2, and the covariance of the returns of the two banks’
papers is θ(1 – θ)(pg – pb)2 R2. From these results, we can get the expression for ρ.
at date 0, then decide whether to withdraw.
(ii) Signal sB and the liquidity shocks of depositors who live near to bank B are revealed. Depositors decide whether to withdraw if depositors deposit their endowments at bank B at date 0 and a contagious run does not occur when sA is revealed.
Date 2. Bank B’s paper matures if the investment is made at date 0 and the bank is not closed at date 1. Depositors who have not withdrawn at date 1
withdraw.