Market Liquidity and Creditor Runs: Feedback, Ampli cation, and Multiplicity - FBE HKU
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Market Liquidity and Creditor Runs:
Feedback, Ampli…cation, and Multiplicity
Xuewen Liuy
Hong Kong University of Science and Technology
This draft: August 2018
Abstract
In the global games framework, this paper studies bank runs in a …nancial system, in which
there are many banks and they share a common asset market. Our model 1) endogenizes market
liquidity, 2) demonstrates two-way feedback between market liquidity and creditor runs, and 3)
shows the possibility of multiple equilibria in the system: even when the precision of creditors’
private signals approaches in…nity, multiple equilibria (i.e., a self-ful…lling systemic crisis) can
still emerge. Our model helps explain ampli…cation and multiplicity in …nancial crises.
JEL classi…cation: G01; G21; D82; D53
Keywords: Systemic crises, bank runs, strategic complementarities, equilibrium multiplicity
I am grateful to Deniz Okat for valuable discussion and suggestions. I thank Liang Dai, John Kuong, John Nash,
Ming Yang, and seminar participants at HKU, HKUST, Society for Economic Dynamics (SED) Annual Meeting 2018,
and International Conference on Economic Theory and Applications 2018 for helpful comments.
y
Associate Professor of Finance, Department of Finance, Hong Kong University of Science and Technology
(HKUST), Clear Water Bay, Hong Kong. Tel: (+852) 2358 7679. Email: xuewenliu@ust.hk.1 Introduction
At the heart of the recent …nancial crisis of 2007-2009 was a series of bank runs that caused imme-
diate failures of a number of major …nancial institutions (Gertler et al. (2017)). Remarkably, the
systemic bank runs coincided with the “sudden dry-up” of asset market liquidity.1 Fire-sale dis-
counts of asset-backed securities and repo rates for securitized bonds exhibited jump discontinuities
at the climax of the runs.2 Commentators have attributed the sudden discrete drops in market
liquidity in the absence of any apparent large exogenous disturbance to the economy to multiplicity
jumps from one equilibrium to another under self-ful…lling beliefs (see, e.g., Brunnermeier (2009),
Brunnermeier and Pedersen (2009)).
Why do systemic runs occur simultaneously with market liquidity dry-ups? What gives rise
to jump discontinuities (multiple equilibria)? These phenomena seem di¢ cult to explain with
current banking theory which was built on the global games framework. In fact, the seminal works
of Rochet and Vives (2004), Goldstein and Pauzner (2005), and Morris and Shin (2009) have
shown the existence of a unique equilibrium for a Diamond-Dybvig (1983) bank run model.3 They
demonstrate that as long as the precision of private signals of depositors (creditors of a bank) is
su¢ ciently high, the equilibrium is guaranteed to be unique. However, their models include only
one bank and so do not consider the interaction between di¤erent banks in the system. In this
paper, we further advance these works to study a …nancial system, in which there are many banks
and they share a common asset market. We show that when an asset market is present with market
liquidity being endogenous, equilibrium multiplicity can emerge: even if the precision of creditors’
private signals approaches in…nity, multiple equilibria (i.e., a self-ful…lling crisis) in the system can
exist.
The driving source of multiplicity in our model is that the presence of an asset market gives
rise to strategic complementarities between creditors of di¤erent banks, besides complementarities
between creditors of the same bank. Speci…cally, when the creditors of one bank run, that bank
is forced to conduct …re sales in the asset market; this decreases market liquidity for all other
banks’ assets because of the limited risk-absorbing capacity of investors in the asset market. As
lower market liquidity increases coordination risk, the creditors of the other banks thus also have
incentives to run. The increased degree of strategic complementarity results in a higher likelihood
of equilibrium multiplicity. To put it di¤erently, equilibrium multiplicity under self-ful…lling beliefs
can arise in our model because of the two-way feedback between market liquidity and creditor runs
(coordination risk). If creditors think market liquidity to be low, they will optimally choose to run
1
See evidence in Gorton and Metrick (2011), Covitz et al. (2013), and Krishnamurthy et al. (2014) among others.
2
The e¤ect of securitized borrowing (against assets as collateral) is similar to the e¤ect of asset sales/liquidation:
both concern an exchange of cash ‡ows across time. In this sense, a repo rate is essentially a …re-sale discount.
3
Carlsson and van Damme (1993) and Morris and Shin (1998) are the pioneering works on global games.
1more often because lower market liquidity increases coordination risk for them; then more banks
will fail which reduces market liquidity, con…rming the creditors’initial beliefs.
Our model thus has implications for …nancial fragility at the system level. When the funda-
mentals of the aggregate state worsen continuously, the equilibrium outcome may change discon-
tinuously. That is, when the fundamentals of the aggregate state worsen to a certain point, there
is a regime change from uniqueness to equilibrium multiplicity. Around that critical point, a small
decline in the fundamentals can lead to a big jump (to the “bad” equilibrium), which corresponds
to a discontinuous drop in market liquidity simultaneously with systemic creditor runs on …nancial
institutions.
Our paper contributes to understanding ampli…cation and multiplicity in …nancial crises by 1)
endogenizing market liquidity, 2) demonstrating two-way feedback between market liquidity and
creditor runs, and 3) showing the possibility of multiple equilibria in the system.4 One distinct
feature of our model is that we study bank runs in a …nancial system, in which the interim liquidation
price of a bank is endogenous and depends on its fundamentals as well as on the market liquidity
in the system which in turn endogenously depends on the status of other banks.5 Goldstein and
Pauzner (2005) and Rochet and Vives (2004) treat the liquidation price or market liquidity as
exogenous and Eisenbach (2017) models the liquidation price as fundamentals–independent. Our
model demonstrates the feedback loop between market liquidity and coordination risk, which is
related to, but di¤erent from, the feedback loop between market liquidity and margin requirements
in Brunnermeier and Pedersen (2009). The recent papers of Liu (2016) and Eisenbach (2017) study
the interplay between asset prices and creditor runs. However, Eisenbach’s (2017) paper implicitly
assumes the existence of a unique equilibrium and does not mention equilibrium multiplicity at all.
Liu’s (2016) model, on the other hand, shows equilibrium multiplicity only under the condition that
creditors’private information is su¢ ciently noisy, as in the standard global games literature. Our
current paper develops a simple and tractable model framework showing the existence of multiple
equilibria even when the precision of private signals approaches in…nity.
The paper is organized as follows. Section 2 describes the model setting. Section 3 presents the
model equilibria. Section 4 includes robustness analyses. Section 5 concludes.
4
See the recent survey by Brunnermeier and Oehmke (2013) on …nancial crises and systemic risk.
5
Diamond and Rajan (2005) study banking crises and aggregate liquidity shortages, and there are no coordination
issues (panic) among creditors in their model. Allen and Gale (2000) study …nancial contagion through interbank
claims and assume that banks face exogenous idiosyncratic liquidity shocks (see also Bhattacharya and Gale (1987)).
22 Model Setting
There are three dates in the model: t = 0, 1 and 2. We discuss banks, the asset market, and
creditor runs, in order.
2.1 Banks
There is a continuum of banks with unit mass, indexed by i 2 [0; 1]. At t = 0, each bank invests in
one unit of its own assets with the cost being 1. The cost is …nanced from two sources: an amount
F comes from a continuum of its creditors with F mass, each one contributing 1, and an amount
1 F comes from its equityholder (bankowner).6 At t = 1, a creditor of a bank has the right to
decide whether or not to roll over his lending to the bank. If he decides not to roll over, his claim
is the par value 1 at t = 1; if, instead, he decides to roll over, the (promised) notional claim to him
is R at t = 2, where R > 1 is the gross interest rate.7 Creditors and bankowners are risk-neutral.
The payo¤ of bank i’s assets at t = 2 is
vi = i + ei ,
which follows a normal distribution as in Grossman and Stiglitz (1980). Speci…cally, ei is a random
variable with distribution ei N (0; 2) and resolves its uncertainty at t = 2. For simplicity and
e
without loss of generality, we assume that ei e is perfectly correlated across banks.8 The term
i, interpreted as asset quality, has its realization at t = 1, which is independently drawn from an
identical distribution N( ; 2) across banks. Denote 1= 2 and 1= 2.
i e e
Although the asset quality of a bank is realized at t = 1, its creditors are not informed of it.
Nevertheless, a creditor of a bank receives imperfect information (a signal) at t = 1 about the asset
quality of the bank. Speci…cally, the signal for creditor h of bank i (about asset quality i) at t = 1
is shi = i + s
h, where s > 0 is constant, and the individual-speci…c noise h N (0; 1). h is
independent across creditors of a bank, and each is independent of 2.
i. Denote s 1= s
2.2 Asset Market
If a bank su¤ers a creditor run (to be elaborated), its assets must be liquidated or under …re sales
at t = 1 in a competitive asset market, which consists of a continuum of competitive investors with
6
We assume that each bank has its own creditor base (for example, these banks are regional banks).
7
Without loss of generality, we normalize the interim notional claim to 1. What matters to the model is the
interest rate between t = 1 and t = 2, i.e., the R.
8
As long as ei is correlated across banks to some degree (i.e., not perfectly diversi…ed away), our model results
will hold true. In reality, the number of banks is not in…nite; therefore, ei cannot possibly be diversi…ed away even
when ei is independent across banks.
3unit mass. Investor j has utility function
U (W j ) = exp Wj ;
where W j is the end-of-period wealth at t = 2, and is the risk-aversion (CARA) coe¢ cient. The
risk-free (gross) interest rate between t = 1 and t = 2 is normalized as 1.
Investors have private information (signals) about banks’asset qualities. Speci…cally, the signal
j
for investor j about asset quality i at t = 1 is i = i + "ji , where 0 is constant, and the
individual-speci…c noise "ji N (0; 1). "ji is independent across assets for a given j and independent
across investors for a given i, and each "ji for a given i is independent of i.
Suppose that the system has, in total, a mass of ' (2 [0; 1]) of banks su¤ering creditor runs.
Then, there are ' units of assets in the system under …re sales.9 Denote by li the liquidation
(…re-sale) price of bank i’s assets.
2.3 Creditor Runs
li
Let us consider a typical bank i. If it has greater than F proportion of its creditors declining to
roll over their lending at t = 1, its liquidation value will not be su¢ cient to cover these creditors’
claims, leading to its failure (we call this scenario a “creditor run”). Alternatively, one may think
of li as the collateral value of the bank’s assets. This means that the bank can raise at most li
amount of cash at t = 1 by using its assets as collateral. If the demand for cash exceeds li at t = 1,
the bank will fail.
Following the work of Rochet and Vives (2004), we use a simpli…ed payo¤ structure of the
creditor-run game. Speci…cally, as in Rochet and Vives (2004), we assume that each creditor of
a bank is an institutional investor (a fund), run by its fund manager. A fund manager has the
following compensation scheme. If the fund manager calls his fund’s investment at t = 1, his payo¤
is a constant w0 , or the face value 1 multiplied by proportion w0 . This could be because a creditor
who calls loans at t = 1 will either fully recover the face value of investment 1 (in the case of
bank survival) or su¤er a small loss (in the case of bank failure). If, instead, the fund manager
holds the investment at t = 1, he obtains compensation w conditional on his fund’s investment not
defaulting (i.e., the investment return is no less than R), where w > w0 . Compensation contingent
on non-default captures the reality that “breaking the buck” has severe consequences.
A creditor’s payo¤ depends crucially on the actions of other creditors of the same bank. Let
denote the proportion of creditors of a bank who choose not to roll over (i.e., choose to call). Then,
the payo¤ for a particular creditor is given in Table 1.
9
We will focus on the case of s ! 0. In equilibrium, then, if a bank su¤ers a creditor run, it will liquidate its
assets entirely (i.e., no partial liquidation).
4Total calling proportion 2 [0; Fli ) Total calling proportion 2 [ Fli ; 1]
(bank survives) (bank fails )
!
F
1 li i (1 )F R
Hold w F
0
1 li e
Call w0 w0
Table 1: Creditor-run payo¤ structure
If 2 [ Fli ; 1], a creditor run occurs and the bank fails at t = 1; its staying creditors get
nothing and thus their fund manager’s compensation is 0 because of the default. If 2 [0; Fli ), the
F
bank must liquidate li units of its assets to raise cash to pay its F calling creditors. Thus, at
F
t = 2, 1 li units of assets will remain, the payo¤ distribution of which, conditional on i, is
2
F F F 2
1 li vi N 1 li i; 1 li e . Since the number of staying creditors at t = 2 is
(1 ) F , these creditors’total notional claim is (1 ) F R. Hence, the probability !
that the bank
F
1 li i (1 )F R
will not default to these creditors at t = 2 conditional on i is F
, where ()
1 li e
stands for the c.d.f. of the standard normal and ( ) denotes its p.d.f.
By introducing a third party, fund managers, we have a discrete-state payo¤ structure of the
creditor-run game. This simpli…ed structure captures the key feature of the creditor-run game: if
li
the proportion of creditors of a bank calling is higher than F, the optimal strategy for an individual
li
creditor is to also “call;”if, however, the proportion of creditors calling is less than F, the optimal
strategy for an individual creditor is likely to also “hold.”The simpli…cation in the payo¤ structure
is a convenient way to deal with the fact that the property of global strategic complementarities
fails to exist in a creditor-run game (see Goldstein and Pauzner (2005)).10
For a cleaner and simpler analysis, we follow Morris and Shin (2009) to simplify the payo¤
structure of the creditor-run game in Rochet and Vives (2004). Morris and Shin (2009) assume
that “if there is not a run, new creditors will eventually be found and the balance sheet reverts to
its initial state after the failed run.” Basically, they are assuming that “the partial liquidation of
assets has no long-run e¤ect”(in the language of Vives (2014a)). Concretely, if a bank has less than
li
F proportion of its creditors calling, partial liquidation will occur but the bank can still survive to
t = 2, in which case Morris and Shin (2009) assume that the bank’s balance sheet reverts to its
initial state. Essentially, after an unsuccessful run, the asset side of the balance sheet of the bank
is restored to vi and the liability side reverts to the total notional debt value F R claimed by F
creditors.11 In short, the assumption of Morris and Shin (2009) gives the simpli…ed payo¤ structure
10
See also Dasgupta (2004). Relatedly, He and Xiong (2012) use a staggered debt structure.
11
For example, as long as the bank is still alive (after an unsuccessful run), it can buy back its (partially) liquidated
5in Table 2.
Total calling proportion 2 [0; Fli ) Total calling proportion 2 [ Fli ; 1]
(bank survives) (bank fails )
FR
Hold w i
e
0
Call w0 w0
Table 2: Simpli…ed creditor-run payo¤ structure
Notice that Table 2 is identical to Table 1 if we set = 0 for the payo¤ of holding in the case
li li
of 2 [0; F ). That is, as long as 2 [0; F ), the bank continues as if it had not experienced any
withdrawals in Table 2.
In the main model, we use the simpli…ed payo¤ function in Table 2. In section 4, we will show
that our model results are robust under the alternative payo¤ structure in Table 1.
2.4 Timeline
At t = 0, the liability side of the balance sheet of a bank is given by (F , 1 F ) and the contract
term (1; R) with the creditors is …xed. At t = 1, creditors move …rst by making their rollover
decisions based on their private information shi , and banks move later by conducting asset sales in
the asset market based on the total withdrawals requested by their creditors.
3 Equilibrium
We focus on the equilibrium at t = 1. At t = 1, creditors need to make their rollover decisions. We
are interested in the equilibrium where every creditor uses a threshold (monotone) strategy. The
strategy is given by (
Call shi < s
shi 7 ! ,
Hold shi s
where shi is the signal received by creditor h of bank i and s is the rollover threshold. Because
banks are identical ex ante, we naturally consider the symmetric equilibrium in which creditors of
all banks use a common strategy, i.e., the threshold s is not bank-speci…c.
We will show that an upper dominance region exists for the bank-run game in our model. The
assets with re…nancing from its original withdrawing creditors or new creditors. Alternatively, as long as the total
amount of early withdrawals at t = 1 is less than li , the bank is able to raise this amount of cash temporarily (from,
for instance, some outside deep-pocketed investors) by using its assets as collateral; new creditors will then eventually
be found to replace the withdrawing creditors and the re…nancing amount from the new creditors is used to repay
the temporary borrowing.
6existence of an upper dominance region is because in our model the interim liquidation value of a
bank is fundamentals-dependent as in Rochet and Vives (2004).12
3.1 Solving the Equilibrium
Formally, we de…ne the equilibrium at t = 1 as follows.
De…nition 1 The equilibrium at t = 1 is characterized by triplet (s ; fli g ; '), where s is the
rollover threshold for creditors, li is the liquidation price of bank i’s assets, and ' is the total mass
of banks under …re sales, such that 1) given creditors’ rational expectations of ', they set their
rollover threshold as s ; 2) given the rollover threshold s , the mass of failed banks in the system is
'; and 3) given the total …re sales ', the equilibrium price of bank i’s assets in the asset market is
li .
We derive the equilibrium in three steps.
Asset market equilibrium The portfolio choice of an investor in the asset market at t = 1
is given by
h n o i
j
max E exp Wj j i ; fli g (1)
fxji g
Z
s.t. W = j
xji (vi li )di,
where xji is the quantity of demand for asset i for investor j. As long as the risk of vi (i.e., the
term ei ) is correlated across i to some degree or the number of banks is …nite, the risk of payo¤ of
the portfolio cannot be completely diversi…ed away. For simplicity and without loss of generality,
we have assumed that ei e is perfectly correlated across i.
For simplicity, we follow the trading game (mechanism) in Vives (2014b) and Benhabib, Liu and
Wang (2016) to focus on the fully-revealing equilibrium of the asset market, i.e., the equilibrium
in which …nancial prices fully reveal the fundamentals of the trading assets in the spirit of Hayek
(1945). In our context, it is the equilibrium in which f i g is fully revealed to the investors through
the …nancial prices.13
12
Goldstein and Pauzner (2005), following the original model of Diamond and Dybvig (1983), assume that the
interim liquidation value of a bank is fundamentals-independent, so an upper dominance region does not exist in
their model.
13
Alternatively, instead of f i g being fully-revealed through the …nancial prices, we can assume that the precision
of investors’ private signals approaches the limit ! 0 as in Morris and Shin (2004b), just as s ! 0 for the
precision of creditors’private signals; this would also lead to f i g being perfect information for investors. Note that
this part of the model is not a focus of our paper, and we can adopt either alternative.
7The …rst-order condition of (1) implies that xji = xi for any j (i.e., a representative investor
exists and demands xi ), and that Z
i li
xi di = 2
.
e
Given that the total mass of banks su¤ering creditor runs is ', the market clearing condition
dictates Z
xi di = ' .
Lemma 1 follows.
Lemma 1 The liquidation price of bank i’s assets is given by li = 'k, where k 2 and 'k
i e
measures market liquidity.
Proof. See Appendix.
The result of Lemma 1 is in the spirit of Grossman and Miller (1988). When the risk-averse
market maker sector is forced to absorb more risky assets, the price of every risky asset is a¤ected
and reduced because of the limited risk-absorbing capacity of the market maker sector. It is worth
noting that every bank is a price-taker in the liquidation market. In fact, there is a large number of
investors in the asset market and each individual investor only holds a tiny amount of a particular
bank’s assets in his portfolio, as shown in (1).
As in Brunnermeier and Pedersen (2009), market liquidity is measured as the degree to which
the market price of an asset is depressed away from its fundamental value. Market liquidity in our
model is thus measured by the term 'k. That is, …re sales conducted by other banks reduce market
liquidity for a particular bank.
Creditor-run equilibrium for an individual bank Considering that li is fundamentals
( i )-dependent by Lemma 1, when i is su¢ ciently high, bank i will survive even if every creditor
of it withdraws. That is, an upper dominance region exists. Therefore, we only need to focus on
threshold equilibria (see Morris and Shin (2003) and Vives (2014a)).
Given that all other creditors of a bank use threshold s , the bank when realizing asset quality
h s
as i has a ( i ; s ) = Pr( i + s < s !) = s
i
proportion of its creditors withdrawing.
Moreover, the bank with realized asset quality i will have its asset liquidation value as li = i 'k.
Hence, by the nature of credit runs, the threshold of the bank’s failure, denoted by , is given by
'k s
= : (2)
F s
That is, the bank fails if and only if i < and individual creditors rationally anticipate this.
Given the bank’s failure threshold , what is the optimal strategy of an individual creditor? He
rolls over if and only if his signal is above threshold s , where s is given by the following indi¤erence
8condition:
0 1
Z +1 i + s
s
i FR 1 @ + s + s
Ad
w q q i = w0 : (3)
e 1 1
+ s + s
When receiving the signal exactly equal to s , the fund manager of the investor has the expected
payo¤ on the right-hand side (RHS) of (3) if he chooses to withdraw, and has the payo¤ on the
left-hand side (LHS) of (3) if he chooses to roll over: when i < , the bank will fail and thus
the fund manager will get nothing; conditional on i , his expected compensation at t = 2 is
FR
w i
e
for a given i.
We focus on the limiting case of signal precision: s ! 0. Under the limiting case, we prove
that the system of equations (2) and (3) is transformed into = s and
s 'k s FR w0
= : (4)
F }
| {z e w
| {z }
Illiquidity (coordination) risk Insolvency (fundamental) risk
The …rst term on the LHS of (4) characterizes the interim illiquidity risk and the second term gives
the insolvency risk at the …nal date. The limit s ! 0 also implies that in equilibrium all creditors
of a bank are in the same position ex post, i.e., either all of them decide to roll over or none of them
does so. This in turn implies that in equilibrium a bank either completely liquidates its assets or
does not liquidate any fraction, i.e., there is no partial liquidation.
Bank failures in the system Recall that the asset quality distribution across banks in the
system at t = 1 is N( ; 2 ). Banks with realized asset quality s survive at t = 1 while
i i
all others fail. Hence, the total measure of failing banks in the system is given by
s
'= . (5)
Lemma 2 The creditor-run equilibrium in the system at t = 1, given by (s ; '), solves the system
of equations (4)-(5) under the limiting case of s ! 0. Two-way feedback exists between market
@s @'
liquidity ('k) and the creditor-run threshold (s ): @' > 0 in (4) and @s > 0 in (5).
Proof. See Appendix.
The two-way feedback highlighted in Lemma 2 is intuitive. When creditors run on banks with
a higher threshold, more banks in the system will fail, resulting in market liquidity drying up for
every bank. Creditors of a bank have rational expectations on this and thus have higher incentives
to run in the …rst place.
93.2 Characterization of the Equilibrium
It is helpful to start by examining two benchmark cases. First, consider the case where the liqui-
dation value li is exogenous. In this case, it is easy to show that the creditor-run equilibrium is
given by
li s FR w0
= ; (6)
F
|{z} e w
| {z }
Illiquidity (coordination) risk Insolvency (fundamental) risk
which clearly has a unique equilibrium. Proposition 1 follows.
Proposition 1 (A single bank with exogenous liquidation value) Under the limiting case
of s ! 0, when li is exogenous, the creditor-run equilibrium at t = 1, given by (6), has a unique
(threshold) equilibrium.
Proof. See Appendix.
The uniqueness of equilibrium for runs on a single bank under global games has been well
established in the literature. The result in Proposition 1 is essentially the result in Goldstein and
Pauzner (2005).
Second, consider the case where a bank’s liquidation value li depends on its own fundamentals i,
but not on other banks’situation; that is, ' is exogenous. In this case, the creditor-run equilibrium
is given by (4) but with an exogenous '.
Proposition 2 (A single bank with fundamentals-dependent liquidation value) Under the
limiting case of s ! 0, when ' is exogenous, the creditor-run equilibrium at t = 1, given by (4),
has a unique equilibrium.
In Proposition 2, our paper studies bank runs with a fundamentals-dependent liquidation value
and a fundamentals-dependent payo¤ structure (i.e., the system of equations (2) and (3)) and shows
the uniqueness of equilibrium under s ! 0. Solving the equilibrium is nontrivial (see the proof of
Lemma 2 in Appendix).14
Now we move to study the equilibrium in Lemma 2, that is, there are many banks with a linkage
through the asset market. Combining (4) and (5) yields one equation:
1 s s FR w0
s k = : (7)
F e w
14
Similar to Morris and Shin (1998), Rochet and Vives (2004) use a fundamentals-dependent liquidation value
(with an exogenous market liquidity or “…re-sales penalty”) but a fundamentals-independent payo¤ structure (i.e.,
the payo¤ for a fund manager conditional on the bank’s survival or failure does not depend on the bank’s asset
fundamental value). The model of Eisenbach (2017) assumes physical liquidation, where an asset’s liquidation value
does not depend on its fundamentals.
10The equilibrium at t = 1 is fully characterized by equation (7). Mathematically, the …rst term on
the LHS of (7) can be decreasing in s while the second term is increasing in s , so the function
on the LHS with respect to s may be non-monotonic and thus multiple solutions to equation (7)
are possible. Write the LHS of (7) as function V (s ; k). Figure 1 plots the function under a set of
parameter values, where = 1:4, = 0:4, F = 0:8, R = 1:1, and e = 1.
Figure 1: Function V (s ; k)
Proposition 3 (Banking system with an asset market) Under the limiting case of s ! 0,
when k is low enough, the creditor-run equilibrium in the system at t = 1, given by (4)-(5), is
unique. When k is su¢ ciently high, there may exist multiple (typically three) equilibria.
Proof. See Appendix.
Even under the limit s ! 0, multiple equilibria (a self-ful…lling crisis) can still exist, which
is in contrast to the classical result of the bank-run game for a single bank. The intuition is
the following. The presence of a common asset market gives rise to strategic complementarities
between creditors of di¤erent banks, besides the complementarities between creditors of the same
bank. That is, there is an increased degree of strategic complementarity between the creditors in
the system, which makes equilibrium multiplicity more likely. To put it di¤erently, equilibrium
multiplicity under self-ful…lling beliefs can arise because of the two-way feedback between market
liquidity ('k) and the rollover threshold (s ) discussed in Lemma 2.
The recent papers of Liu (2016) and Eisenbach (2017) study the interplay between asset prices
and creditor runs. However, Eisenbach (2017) implicitly assumes the existence of a unique equilib-
rium and does not mention equilibrium multiplicity at all. Liu’s (2017) model, on the other hand,
having other focuses and purposes, shows equilibrium multiplicity under the condition that private
11information is su¢ ciently noisy, as in the standard global games literature, and cannot show the
existence of multiple equilibria when the precision of private signals approaches in…nity.
The mechanism of equilibrium multiplicity in our model is di¤erent from that in Angeletos and
Werning (2006) and Hellwig et al. (2006). Importantly, in our model, the timeline is that creditors
move …rst and banks move later. At the time when creditors make rollover decisions, they do not
observe the liquidation prices. In Angeletos and Werning’s (2006) model, there is a …nancial market
in the …rst stage which aggregates the dispersed private information of agents, and the endogenous
…nancial price is observable by and serves as a public signal for agents who play the currency-crises
game in the second stage. When the precision of the private signals of agents increases, the precision
of the endogenous public signal can increase even faster, which results in equilibrium multiplicity.
Similarly, in Hellwig et al. (2006), the …nancial price — the interest rate — aggregates private
information, as well as having direct e¤ects on the traders’payo¤s. The mechanism in our model is
instead that the presence of an asset market with endogenous market liquidity increases strategic
complementarity between creditors.
We have comparative statics with respect to .15 Corollary 1 follows.
Corollary 1 The lower the , the more likely the equilibrium multiplicity.
Proof. See Appendix.
3.3 Implications for Ampli…cation Multiplicity
For simplicity and to illustrate the point, here we can borrow the setup of Liu (2016). Basically,
there are three aggregate states at t = 1 with a positive probability for each: “Normal” state,
“Slightly Bad” state (small-sized shock), and “Bad” state (medium-sized shock). The better the
aggregate state, the higher the realization of (i.e., …rst-order stochastic dominance). Speci…cally,
N > S > M, where N, S and M denote the realization of for “Normal” state, “Slightly
Bad” state and “Bad” state, respectively. Ex ante, at t = 0, the contract terms with creditors as
well as other arrangements are set based on the preparation for the occurrence of the “average” of
the three states at t = 1. Now we ask what happens if the “Slightly Bad”state or the “Bad”state
actually occurs at t = 1.
Proposition 4 (Ampli…cation by Multiplicity) Under the limiting case of s ! 0, it is possible
that for the “Slightly Bad” state ( = S) there exists a unique equilibrium while for the “Bad”
state ( = M) there are multiple equilibria.
Proof. See Appendix.
15
may measure the heterogeneity of assets across banks. Liu (2017), studying bank diversi…cation, endogenizes
the heterogeneity across banks.
12Proposition 4 gives the implication for ampli…cation multiplicity. When a negative shock to
is small enough ( = S ), ampli…cation (with the feedback loop) is on a given equilibrium.
When the negative shock is big enough ( = M ), there is also a regime change from uniqueness
to equilibrium multiplicity, so there may be a jump from one equilibrium to another. Note that
function V is increasing in , so a decrease in shifts the solid curve in Figure 1 downward.
Figure 2 illustrates ampli…cation multiplicity. Rewrite the LHS of (7) as V (s i ; s ; ) =
n h io
1 i s s i FR
F s k e
. Hence, the solution with respect to s i to equation V (s i ; s ; ) =
w0
w gives the best response function s i = r (s ; ). Figure 2 plots the best response function, where
di¤erent curves correspond to di¤erent . Suppose that the equilibrium is initially at Point A.
When the “Slightly Bad” state ( = S) is realized, the equilibrium is still unique. In contrast,
when the “Bad”state ( = M) is realized, the equilibrium goes from A to either B (through the
0
feedback loop of ampli…cation) or B (through a multicity jump).16 That is, when the fundamentals
of the aggregate state worsen continuously, the equilibrium outcome may change discontinuously.
Figure 2: Ampli…cation Multiplicity
When the “Bad” state ( = M) is realized, multiple self-ful…lling equilibria can exist, which
gives rise to the role of an ex-post regulatory intervention. For example, to eliminate the ine¢ cient
equilibrium, the government can announce and commit to supporting asset prices in the asset
market as long as the proportion of failing banks in the system exceeds ' (or equivalently the risk
premium is higher than 'k).17 This would su¢ ciently change creditors’ expectation and thereby
16
The new curve after the shock has three intersections with the 450 line. The middle intersection corresponds to
an unstable equilibrium. The other two correspond to stable equilibria.
17
In the recent crisis, the Federal Reserve adopted unconventional policies through lending to …nancial institutions,
providing liquidity directly to key credit markets, and purchasing long-term securities (Bernanke (2009)).
13coordinate the agents to the e¢ cient equilibrium.
4 Robustness
In this section, we conduct two robustness analyses.
4.1 A Single Bank with a Downward-sloping Liquidation Price
Alternatively suppose there is only one bank, and when it liquidates its assets, it faces a downward-
sloping demand curve. We show that in this case the creditor-run equilibrium is unique.
Speci…cally, assume that the demand function of the bank’s assets (totally one unit) is p = f (q),
where q is the quantity of selling and p is the price. Let the revenue function of selling be (q) =
pq = f (q)q. Without loss of generality, it is reasonable to assume that 0( ) > 0 for q 2 [0; 1].
Then, (2) is replaced by
(1) s
= : (8)
F s
(8)-(3) together gives the creditor-run equilibrium. Hence, like the case of the liquidation value li
being exogenous in Proposition 1, the equilibrium is unique.
4.2 Alternative Creditor-run Payo¤ Structure
Suppose we use the payo¤ structure in Table 1 instead of the one in Table 2. Under this alternative
payo¤ structure, we prove in Appendix that (4) is replaced by
0 1
Z s 'k
1 F
s (1 )FR
s 'k
A d = w0 .
F
@ (9)
0 1 F w
s 'k e
Notice that if we set = 0 inside the integral, (9) becomes identical to (4). We prove in Appendix
that the following key property of (4) in Lemma 2 remains for (9):
@s
> 0:
@'
Therefore, under the alternative payo¤ structure, the results of the model change only quantita-
tively, not qualitatively.
145 Conclusion
This paper presents a simple global-games model that features a banking system with endogenous
market liquidity. We show that multiple equilibria can exist even when the precision of creditors’
private signals approaches in…nity. Our paper aids in explaining ampli…cation and multiplicity in
…nancial crises and also contributes to the theoretical literature on bank runs with global games.
Our model framework may also be useful for future work studying bank runs in a …nancial system.
15Appendix
A Proofs
Proof of Lemma 1: Considering that the price li fully reveals the fundamentals i (see the
trading mechanism in Vives (2014) and Benhabib, Liu and Wang (2016)), an investor does not rely
on his private information in trading. Hence, all investors are basically the same (as a representative
investor). Thus, the objective function of (1) can be transformed into maximizing
Z Z
1
xi ( i li )di V ar e xi di
2
Z Z 2
1 2
= xi ( i li )di xi di :
2 e
The …rst-order condition with respect to any xi implies
Z
2
( i li )di e xi di di = 0;
R li
R
that is, xi di = i
2 . Because xi di = ' by the market clearing condition, we have li = i 'k,
e
where k 2.
e
Proof of Lemma 2: Combining (2) and (3), we have
0 1
Z +1 i + s
s
i FR 1 @ + s + s
Ad w0
q q i = , (A.1)
i= e 1 1 w
+ s + s
'k s 1 'k
where is given by the implicit function F = s
or s = + s F .
Write the LHS of (A.1) as Y (s ; s ). We transform Y (s ; s) by changing variables to z =
+ s s
i + s + s
q
1
and obtain
+ s
0q 1
Z 1
1
z+ + s
s FR
+ s + s + s
Y (s ; s) = @ A (z) dz,
z=z0 e
16where z0 satis…es the joint equations
1 'k
s = + s
F = i
i + s
+ +
s
s
s
z= q :
1
+ s z=z0
That is, z0 satis…es
1 i 'k
s = i + s ; (A.2)
F
q
1
with i = + s
+ s
+ s
s + z0 + s
. Plugging the above expression of i into (A.2), we
can express z0 by an implicit function
0 q 1
1
+ s
s + z0 'k
+ s (s ) s 1@ + s + s + s
A.
z0 = q q
1 1 F
+ s + s
So it follows that
1 s 'k
lim z0 = :
s !0 F
Thus, under the limit s ! 0 for a given , we have = s and
Z 1
s FR
lim Y (s ; s) = (z) dz
s !0 e 1 s 'k
F
s FR s 'k
= :
e F
Hence, (4) is proved.
We also need to prove that a creditor rolls over when his signal is higher than s and otherwise
withdraws. An individual creditor takes as given. Writing the LHS of (A.1) as Y~ (s ) and
+ s s
i + s + s
changing variables to z = q
1
, we obtain
+ s
0q 1
Z +1
1
z+ + s
s FR
+ s + s + s
Y~ (s ) = + s @ A (z) dz.
+ s + ss
z= r e
1
+ s
We have @ Y~ (s ) =@s > 0.
@' @s
It is straightforward to show that @s > 0 in (5). As for @' > 0 in (4), write the LHS of
@ Y^ s 'k @ Y^
(4) as function Y^ (s ; ') and we have @s = F
1 s FR
e
+ F
s FR
e
1
e
> 0 and @' =
k s FR @s @ Y^ @ Y^
F e
< 0, and thus @' = @' = @s > 0.
17Proof of Proposition 1: If the liquidation value li is exogenous, (2) is replaced by
li s
= : (A.3)
F s
Similar to the prove of Lemma 1, under the limit s ! 0 for a given , (2)-(3) together will give
= s and (6).
Proofs of Proposition 3 and Corollary 1: Write the LHS of (7) as function V (s ), where
1 s s FR
V (s ) = s k .
F e
It is easy to show that
@V (s )
@s
1 s k s FR 1 s s FR 1
= 1 + s k .
F e F e e
(A.4)
@V (s ) @V (s )
When k ! 0, @s > 0 holds for any s ; when k is small enough, @s > 0 also holds for any
w0
s , so V (s ) = w has a unique solution with respect to s .
@V (s ) w0
When k is high enough, @s < 0 at some s , so V (s ) is non-monotonic and thus V (s ) = w
can admit multiple (typically three) solutions with respect to s . Speci…cally, consider the case
where k is high enough, is low enough, and is slightly higher than F R, ceteris paribus. So
@V (s ) @V (s )
@s js = < 0, and @s > 0 when s is su¢ ciently far away from (i.e., lower or higher than)
. That is, V (s ) is increasing in s initially, and then decreasing in s around s = , and
w0
then increasing in s again, as the solid curve in Figure 1 depicts. Thus, for some w , equation
w0
V (s ) = w admits three solutions.
1 @V (s )
Clearly, from (A.4), has a similar e¤ect to k. That is, when is big enough, @s >0
w0
also holds for any s , so V (s ) = w has a unique solution with respect to s . When is small
@V (s ) w0
enough, @s < 0 at some s , so V (s ) is non-monotonic and thus V (s ) = w can admit multiple
(typically three) solutions with respect to s .
Proof of Proposition 4: Note that V (s ; ) is increasing in . Hence, when is high enough,
w0 w0
it is possible that V (s ) = w has a unique solution; when is low enough, V (s ) = w can have
multiple solutions.
18Proof in Section 4.2: Under the alternative payo¤ structure in Table 1, equation (3) is replaced
by the following equation:
!
+ s s
0h i h i 1 i + s
q
+ s
Z +1 1 F s i
i 1 s i
FR 1
li s s + s
w @ h i A q d i = w0 ;
F s 1
1 li s
i
e + s
(A.5)
h s
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