Under our assumption that depositors cannot invest themselves, bank B can easily offer a deposit contract that gives depositors a strictly higher payoff than what they can get from storing the endowments themselves.5 This means depositors always deposit their endowments at the bank in equilibrium. To simplify the exposition, we consider only the cases where the optimal deposit contract satisfies the depositors’
liquidity needs, that is, the cases where d1
≥ y. This condition implies that the amount
a type-1 depositor gets is larger than the liquidation value of her deposits. As will be seen, this condition may induce depositors to have too much incentive to withdraw, thus leading to a contagious run.Also, for simplicity, we study only symmetric pure-strategy subgame-perfect Nash equilibria.6 Given this criterion, there are two equilibrium candidates in each date 1 subgame. For bank B, when a public signal (sA or sB) is revealed, either all depositors withdraw or no depositor withdraws. Also, we assume that depositors choose the Pareto dominant equilibrium when there are multiple equilibria.7 As in Diamond and Dybvig (1983) and Chen (1999), it can be shown that (i) a bank run is always an equilibrium phenomenon in all the date 1 subgames, and (ii) a bank run is the Pareto dominated equilibrium when there are multiple equilibria.8 Therefore, in our model a bank run will occur if and only if it is the only subgame-perfect Nash
5 For example, he can set d1 = 1 and d2 = R. In this case, bank B serves as an agent who invests for depositors. Obviously, given (d1, d2) = (1, R), depositors strictly prefer depositing to storing the endowments themselves.
6 That is, depositors of the same type will adopt the same pure strategy.
7 The purpose of making this assumption is to illustrate the point that information-based bank runs are still inefficient even if depositors choose the Pareto dominant equilibrium.
8 If ‘no depositor withdraws’ can be sustained as a Nash equilibrium in a date 1 subgame, the depositors’ equilibrium payoff must be no lower than y. It can be shown that the depositors’ expected payoff in the bank run equilibrium is strictly less than 1.
equilibrium. The equilibrium selection criterion we impose is not critical. All the main results hold if we assume there is a sunspot random variable that determines which equilibrium is realized in case of multiple equilibria.
The model is solved backwards. Sections 3.1 and 3.2 study the subgames when
s
B and sA are revealed, respectively. Section 3.3 determines the optimal deposit contract.3.1 When s
Bis revealed (at date 1)
First consider the subgame when sB is revealed. Let p2
(s
A, s
B) denote the
probability that bank B’s investment will succeed given sAand s
B. Given sB, the probability that bank B’s paper will succeed is higher when sA= H than when s
A= L
because the two banks’ returns are positively correlated. Therefore, we know that p2(L, L) < p
2(H, L) and p
2(L, H) < p
2(H, H). In addition, we assume
P
2(H,L) <
R
1 <R ty
y t
) 1 (
) 1 (
−
−
< p
2(L,H). (3)
The assumption p2
(H, L) < p
2(L, H) means that s
B contains more information about bank B’s investment than sA.9 Assuming p2(H, L) < 1/R < p
2(L, H) implies that bank
B’s investment should be liquidated at date 1 if and only if sB= L. To see this, note
that given sA and sB, the continuation and liquidation values of bank B’s per dollar investment are p2(s
A, s
B) R and 1, respectively. As a result, the paper should be
liquidated if and only if p2(s
A, s
B) < 1/R. Given (3), we have
p
2(L, L) < p
2(H, L) <
R
1 < p2(L, H) < p
2(H, H),
which implies bank B’s investment should be liquidated at date 1 if and only if sB
= L.
The reason for assuming P2
(H,L) <
R ty
y t
) 1 (
) 1 (
−
−
< p
2(L,H) will be explained in Section
3.3.
Suppose that no run has occurred to bank B yet before sB and the liquidity shocks of bank B’s depositors are revealed. When information about bank B arrives, type-1 depositors will withdraw. Whether type-2 depositors will withdraw depends on the updated probability that their bank’s paper will succeed. For a type-2 depositor who believes that no other type-2 depositors will withdraw at date 1, her payoff for waiting
9 Note that if qA is high and qB is low, then sA may contain more information about bank B’s investment than sB. Condition (3) excludes this possibility.
until date 2 is p2
(s
A, s
B) d
2 and her payoff for withdrawing now is d1. So, she will not withdraw if and only if p2(s
A,s
B) d
2≥ d
1, or2 1
2( , )
d s d s
p
A B ≥ . (4) The no-run equilibrium can be sustained if and only if (4) holds. Therefore, if no bank run has occurred before sB is revealed, then sB will trigger a run on bank B if and only if (4) is violated.3.2 When s
Ais revealed (at date 1)
Now back to the time when sA is revealed. Suppose that no depositor at bank B has withdrawn before information about bank A is revealed. Obviously no depositor will respond to sA and withdraw if sA
= H. Now consider the case of s
A= L. If d
1/d
2≤ p
2(L, L), then by (4) a bank run will never occur when either s
A or sB arrives. On the other hand, if d1/d
2> p
2(L, H), then a contagious run will always occur when depositors of
bank B learn that sA= L.
10 Finally, consider the case where p2(L,L) < d
1/d
2≤ p
2(L,H).
Define
X d t d d
V
BR1 1 1
1 1 )
( ≡ − − , (5) which is the depositors’ expected payoff when a bank run occurs.11 If sA
= L and a
depositor of bank B believes that all the other depositors will not withdraw before sBis revealed, then her payoff for not responding to sA is12
V
Wait(d
1,d
2)≡π
[t d
1+(1−t
)p
2(L
,H
)d
2]+(1−π
)V
BR(d
1), (6) where π is the conditional probability that sB= H given s
A= L. The depositor will
respond to sA and withdraw if VWait< d
1. Given our equilibrium selection criterion and (6), a low value of sA will trigger a run on bank B if and only if VWait< d
1.By equations (5) and (6), VWait is increasing in d2 and is decreasing in d1. There results are intuitive. When d2 increases, a late withdrawer gets more if a bank run does
10 When d1/d2 > p2(L,H) > p2(L,L) and sA = L, the ‘no depositor withdraws’ equilibrium cannot be sustained whatever the value of sB is. Therefore, all the depositors of bank B will withdraw once they learn sA = L.
11 Remind that we discuss only the cases where d1 ≥ y > 1. When a bank run occurs, some type-1 depositors cannot get their money from the bank, and have to suffer the liquidity loss X. The fraction of depositors who will suffer this loss is t (1 – 1/d1). Therefore, the depositors’ expected payoff when a bank run occurs is 1 – t X (d1 – 1)/d1.
12 In this case, given sA = L, a bank run will occur if and only if sB = L. If sB = H, only early diers withdraw at date 1, so the depositors’ expected payoff is t y + (1 – t) p2(L, H) d2. If sB = L, a bank run occurs, so the depositors’ expected payoff is VBR.
not occur at date 1, so a depositor’s payoff for waiting until sB is revealed increases.
On the other hand, when d1 increases, more type-1 depositors have to suffer the liquidity loss X in case a bank run occurs, so a depositor’s payoff for waiting until sB
is revealed decreases.
3.3 When the deposit contract and bank capital are chosen (at date 0)
At date 0, the manager of bank B determines (d1
, d
2) to maximize depositor welfare
subject to the bank’s zero profit constraint. The following proposition states the optimal deposit contract and the equilibrium. The proofs of all the propositions and the lemma are available from the author upon request.Proposition 1. Suppose that banks do not have capital, and that there is no deposit
insurance.Part (a) of Proposition 1 says that bank B will set d1 just enough to cover type-1 depositors’ liquidity needs. Increasing d1 over y is never optimal because doing so not only increases the amount of early liquidation (which is not efficient since p0
R > 1),
but also induces depositors to have more incentive to withdraw early. The optimal d2in equation (7) follows from the bank’s zero profit constraint. By (3), )
This means that given the deposit contract (y, d20), if a contagious run does not occur, then a bank run will occur to bank B if and only if sB
= L. By the assumption that P
2(H,L) <1/R < p
2(L,H), such a bank run is efficient.
Part (b) of Proposition 1 states the conditions under which a contagious run will occur. It suggests that an improvement in information transparency may either raise or reduce the chance of a contagious bank run. To see this, note that both qA and qB
increase as banks disclose more precise information. When qA is higher, sA contains more information about the return of bank B’s investment, so depositors of bank B have a stronger incentive to respond to sA and withdraw. On the other hand, when qB
increases, sB contains more information about bank B’s investment, so depositors are more willing to wait until sB arrives. Whether depositors of bank B will be more or less eager to respond to sA depends on the relative sizes of these two effects. Part (b) of Proposition 1 predicts that banks with the least transparent information are likely to suffer a contagious run problem. It also implies that contagious runs are more likely to occur when the banking industry is weak (that is, when pg and pb are low).