Fire Sales in a Model of Complexity

Ricardo J. Caballero and Alp Simsek

October 28, 2009

Abstract

Financial assets provide return and liquidity services to their holders. However,

during severe Önancial crises many asset prices plummet, destroying their liquidity

provision function at the worst possible time. In this paper we present a model of

Öre sales and market breakdowns, and of the Önancial ampliÖcation mechanism that

follows from them. The distinctive feature of our model is the central role played by

endogenous complexity: As asset prices implode, more ìbanksîwithin the Önancial

network become distressed, which increases each (non-distressed) bankís likelihood

of being hit by an indirect shock. As this happens, banks face an increasingly

complex environment since they need to understand more and more interlinkages

in making their Önancial decisions. This complexity brings about confusion and

uncertainty, which makes relatively healthy banks, and hence potential asset buyers,

reluctant to buy since they now fear becoming embroiled in a cascade they do not

control or understand. The liquidity of the market quickly vanishes and a Önancial

crisis ensues. The model exhibits a powerful ìcomplexity-externality.îAs a potential

asset buyer chooses to pull back, the size of the cascade grows, which increases the

degree of complexity of the environment. This rise in perceived complexity induces

other healthy banks to pull back, which exacerbates the Öre sale and the cascade.

JEL Codes: E0, G1, D8, E5

Keywords: Fire sales, complexity, Önancial network, cascades, markets freeze,

crises, information cost, Önancial panic, credit crunch, pecuniary externality.

MIT and NBER, and MIT, respectively. Contact information: caball@mit.edu and alpstein@mit.edu.

We thank David Laibson, Frederic Malherbe, Juan Ocampo and seminar participants at Harvard and

MIT for their comments, and the NSF for Önancial support. First draft: July 28, 2009.

1
1 Introduction

Financial assets provide return and liquidity services to their holders. However, during

severe Önancial crises many asset prices plummet, destroying their liquidity provision

function at the worst possible time. These Öre sales are at the core of the ampliÖcation

mechanism observed in severe Önancial crises: Large amounts of distressed asset sales

depress asset prices, which exacerbates Önancial distress, leading to further asset sales,

and the downward spiral goes on.

There are many instances of these dramatic Öre sales and the chaos they trigger in

recent Önancial history. As explained by Treasury Secretary Paulson and Fed Chairman

Bernanke to Congress in an emergency meeting soon after Lehmanís collapse, the main

goal of the TARP during the subprime crisis as initially proposed was, precisely, to put a

áoor on the price of the assets held by Önancial Örms in order to contain the sharp contractionary feedback loop triggered by the confusion and panic caused by Lehmanís demise.

And a few years back, after the LTCM intervention, then Fed Chairman Greenspan wrote

in his congressional testimony of October 1, 1998:

ìQuickly unwinding a complicated portfolio that contains exposure to all

manner of risks, such as that of LTCM, in such market conditions amounts

to conducting a Öre sale. The prices received in a time of stress do not reáect longer-run potential, adding to the losses incurred... ...a Öre sale may

be su¢ ciently intense and widespread that it seriously distorts markets and

elevates uncertainty enough to impair the overall functioning of the economy.

Sophisticated economic systems cannot thrive in such an atmosphere.î

In this paper we propose a model of Öre sales that builds on the idea that complexity,

a feature strongly disliked by investors during downturns for the uncertainty it generates,

rises endogenously during crises: As asset prices implode, more Önancial institutions

(banks, for short) within the Önancial network may go under, which increases each bankís

likelihood of being hit by an indirect shock from counterparty risk. This means that

banks face an increasingly complex environment since they need to understand more and

more interlinkages in making their Önancial decisions. Thus, perceived uncertainty rises

with complexity and makes relatively healthy banks reluctant to buy since they now fear

becoming embroiled in the cascade themselves, and no reasonable amount of research can

rule out this option in the time available. The liquidity of the market quickly vanishes

and a Önancial crisis ensues.

1
In Caballero and Simsek (2009) we show the basic interaction between the size of the

equilibrium cascade in a Önancial network and the degree of complexity of the portfolio

problem facing banks. In that model banks have bilateral linkages in order to insure each

other against local liquidity shocks. The whole Önancial system is a complex network

of linkages which functions smoothly in the environments for which it is designed, even

though no bank knows with certainty all the many possible connections within the network

(that is, each bank knows the identities of the other banks but not their exposures).

However, these linkages may also be the source of contagion when an unexpected event of

Önancial distress arises somewhere in the network. During normal times, banks only need

to understand the Önancial health of their neighbors, which they can learn at low cost. In

contrast, when a signiÖcant problem arises in parts of the network and the possibility of

cascades arises, the number of nodes to be audited by each bank rises since it is possible

that the shock may spread to the bankís counterparties. Eventually, the problem becomes

too complex for them to fully Ögure out, which means that banks now face signiÖcant

uncertainty and they react to it by retrenching into a liquidity-conservation mode.

This structure is the starting point of this paper. Our focus here is on secondary

markets and on the feedback between that environment and the Öre sales that arise in

these markets. In particular, this paper introduces a secondary market in which banks

in distress can sell their legacy assets to meet the surprise liquidity shock. The natural

buyers of the legacy assets are other banks in the Önancial network, which may also

receive an indirect hit. When the surprise shock is small, cascades are short and buyers

can inspect their neighbors to rule out an indirect hit. In this case, buyers purchase

the distressed banksílegacy assets at their ìfair" prices (which reáect the fundamental

value of the assets). In contrast, when the surprise shock is large, longer cascades become

possible which increases the complexity of the environment, and buyers cannot rule out

an indirect hit. As a precautionary measure, they hoard liquidity and disengage from

trades in (now illiquid) legacy assets and may even turn into sellers. The price of legacy

assets plummets to ìÖre-sale" levels.

A central aspect of our model is the dependence of the cascade size on the price

of legacy assets, which leads to multiple equilibria for intermediate levels of the surprise

shock. When legacy assets fetch a fair price in the secondary market, the banks in distress

have access to more liquidity and thus the surprise shock is contained after fewer banks are

bankrupt, leading to a relatively simple environment. When the environment is simple,

the natural buyers rule out an indirect hit and demand legacy assets, which ensures that

these assets trade at their fair prices. Set against this benign scenario is the possibility of

a Öre-sale equilibrium where the price of legacy assets collapses to Öre-sale levels, which

2
leads to a longer cascade and a greater level of complexity. As the level of complexity

increases, natural buyers become worried about an indirect hit and they hoard liquidity

and/or sell their own legacy assets, which reinforces the collapse of asset prices.

This ampliÖcation mechanism is exacerbated by a complexity-externality. As a potential asset buyer chooses to pull back, the size of the potential cascade grows and with it

the degree of complexity of the environment (each bank needs to explore larger segments

of the network to understand the risk it is exposed to). This rise in complexity induces

other healthy banks to pull back as a precautionary measure, which further exacerbates

the Öre sale and cascade.1

Our framework has two additional (and more standard) externalities: A networkliquidity externality that stems from the interlinkages of the Önancial system; when a

bank chooses to raise external liquidity rather than to generate it from its own resources,

it spreads the distress to other banks. And a Öre-sale externality that arises from the

negative e§ects that a bankís distressed asset sales have on other banksíbalance sheets.

These two externalities are present in many network and liquidity models. They interact

but are distinct from the complexity externality that we highlight, which stems from the

feature that any decision that lengthens the potential cascade, increases the complexity

of the environments that other banks need to consider.

This paper is related to several strands of literature. In the canonical model of Öre

sales, these happen because during Önancial crises the natural buyers of the assets (other

banks) also experience Önancial distress (cf. Shleifer and Vishny (1992,1997)). More

recently, Brunnermeier and Pedersen (2008) show that even if some potential buyers are

not distressed or constrained, these may choose not to arbitrage the Öre sale in the short

run because they anticipate a better deal in the future. Our model lies somewhere in

between these two views: Most potential buyers are unconstrained, as in Brunnermeier

and Pedersen (2008), but they are confused and hence fearful of going about their normal

arbitrage role (and in this sense they are distressed as in Shleifer and Vishny (1992)). It

is the complexity of the environment that sidelines potential buyers.

There is an extensive literature that highlights the possibility of network failures and

contagion in Önancial markets. An incomplete list includes Allen and Gale (2000), Laguno§ and Schreft (2000), Rochet and Tirole (1996), Freixas, Parigi and Rochet (2000),

Leitner (2005), Eisenberg and Noe (2001), and Cifuentes, Ferucci and Shin (2005) (see

1Although we do not pursue this avenue in the paper, complexity probably plays a role in creating

the incompleteness needed for a pecuniary externality to arise. If agents could sign contracts contingent

on the events that follow a cascade, then the externality would be greatly reduced. However, as potential

cascades lengthen, the number of contingencies that need to be written into contracts grow exponentially.

3
Allen and Babus (2008) for a recent survey). These papers focus mainly on the mechanisms by which solvency and liquidity shocks may cascade through the Önancial network.

In contrast, we take these phenomena as the reason for the rise in the complexity of the

environment in which banks make their decisions, and focus on the e§ect of this complexity on banksíprudential actions. It is also worth pointing out that the complexity

mechanism we emphasize in this paper is operational even for relatively small amounts

of contagion. The contagion literature is sometimes criticized because it is hard to believe that many Önancial institutions would be caught up in a cascade of bankruptcies.2

That is, even if there is a cascade, it is reasonable to expect that it would eventually be

contained (especially since banks take precautionary actions to Öght the cascade). But

as this paper illustrates, even partial cascades can have large aggregate e§ects, since they

greatly increase the complexity of the environment.3

Our paper is also related to the literature on áight-to-quality and Knightian uncertainty in Önancial markets, as in Caballero and Krishnamurthy (2008), Routledge and

Zin (2004), and Easley and OíHara (2005); and to the related literature that investigates

the e§ect of new events and innovations in Önancial markets, e.g. Liu, Pan, and Wang

(2005), Brock and Manski (2008), and Simsek (2009). Our contribution relative to these

literatures is in endogenizing the rise in uncertainty from the behavior of the Önancial

network itself. More broadly, this paper belongs to an extensive literature on áight-toquality and Önancial crises that highlights the connection between panics and a decline in

the Önancial systemís ability to channel resources to the real economy (see, e.g., Caballero

and Kurlat (2008), for a survey).

The organization of this paper is as follows. In Section 2 we describe the Önancial

network and the secondary market for assets, and we introduce a surprise shock (a rare

event) in the network. We also discuss a benchmark case without complexity e§ects

(because banks can understand the network at no cost). Section 3 contains our main

results. There, banks have only local knowledge about the Önancial network, and a

su¢ ciently large surprise shock increases the complexity of the environment and leads to

2See, e.g.,Brunnermeier, Crockett, Goodhart, Persaud, an Shin (2009) which argue that the domino

model of Önancial contagion is not useful for understanding Önancial contagion in a modern Önancial

system since ì.... It is only with implausibly large shocks that the simulations (of their model) generate

any meaningful contagion. The reason is that the domino model paints a picture of passive Önancial

institutions who stand by and do nothing as the sequence of defaults unfolds. In practice, however, they

will take actions in reaction to unfolding events, and in anticipation of impending defaults...î

3The role of cascades in elevating complexity was also highlighted in Haldaneís (2009) speech, who

nicely captures the mechanism when he wrote that at times of stress "knowing your ultimate counterpartyís risk becomes like solving a high-dimension Sudoku puzzle."

4
a breakdown in secondary markets. This section also highlights the dependence of the

level of complexity on asset prices and demonstrates the possibility of multiple equilibria.

In Section 4 we describe the three externalities in our setup and analyze their role in our

main results. The paper concludes with a Önal remarks section and several appendices.

2 Equilibrium without Complexity

In this section, we describe the basic environment and characterize the equilibrium for

a benchmark case in which Önancial institutions (banks, for short) have full knowledge

of the interlinkages between all the banks in the Önancial network. In this benchmark

we show that if the network is deep (i.e., there is a large number of banks) secondary

markets do not break down and the Önancial network is resilient to a perturbation. That

is, the size of Önancial cascades is contained and aggregate loan contraction is limited.

These relatively benign results contrast with those we obtain in the next section once we

introduce complexity.

2.1 The Environment

We consider an economy with three dates f0; 1; 2g in which a single good (one dollar)

can be kept in liquid reserves or it can be loaned to production Örms. If kept in liquid

reserves, a unit of the good yields one unit in the next date. Instead, if a unit is loaned

to Örms, it then yields R > 1 units at date 2. These new loans are completely illiquid.

The economy has n continuums of banks denoted by fb

jg

n

j=1. Each of these continuums

is composed of identical banks and, for simplicity, we refer to each continuum b

j as bank

b

j

, which is our unit of analysis.4 Each bank has initial assets which consist of 1  y

units of legacy loans, z units of demand deposits in one other bank, and y units of áexible

reserves set aside for making new loans at date 0. The bankís liabilities consist of a unit

measure of demand deposits held by retail depositors and z units of demand deposits held

by one other bank. The demand deposit contracts in this economy pay 1 unit at date 1

(resp. R units at date 2) if the depositor arrives early (resp. late). There are no liquidity

driven depositors. I.e. as long as the bank is solvent, all retail depositors would rather

arrive at date 2. Note that a bank that loans all of its áexible reserves has just enough

assets to pay its depositors at date 2. At date 0 (only) there is a secondary market for

legacy loans and the (equilibrium) price for these loans is r < 1 (see below). The central

trade-o§ in this economy will be whether the bank uses its áexible reserves to make new

4The only reason for the continuum is for banks to take other banksídecisions as given.

5
loans and to purchase legacy loans in the secondary market, or whether it hoards some

of this liquidity in response to a rare event (that we describe below).

The banksícross demand deposit holdings form a Önancial network denoted by:

b () = b

(1) ! b(2) ! b(3) ! :::: ! b(n) ! b(1)

(1)

where  : f1; ::; ng ! f1; ::; ng is a permutation that assigns bank b

(i)

to slot i in the

Önancial network. The arc ! denotes that the bank in slot i (i.e., bank b

(i)

) has z units

of demand deposits in the bank in the subsequent slot i + 1, and slot n has slot 1 as a

forward neighbor.5 A central feature of the model is the banksí uncertainty about the

Önancial network in (1), but we will shut down this ingredient until the next section. For

now, all banks know all the Önancial linkages (cross-deposits) in the network.

At date 0, the banks learn that a rare event has happened and one bank, b

j

, will

become distressed. Similar to Allen and Gale (2000), in order to remain solvent this bank

needs to make  dollars of payment (to an outsider) at date 1.

These losses might spill over to other banks via the Önancial network and may bring

them into Önancial distress at date 1. To prepare for date 1, at date 0 the banks take one

of the following actions A

j

0 = fHS; H; Bg, which are restricted to a discrete choice set for

simplicity (see Caballero and Simsek (2009) for a related model with unrestricted action

space). As the most extreme precautionary measure, the bank may choose A

j

0 = HS, to

hoard all of its áexible reserves y as liquidity and sell all of its legacy loans 1  y in the

secondary market, keeping a completely liquid balance sheet. As a less extreme measure,

the bank may choose A

j

0 = H, to hoard its áexible reserves as liquidity and to keep its

legacy loans on its balance sheet. Alternatively, the bank may choose A

j

0 = B; to keep

its own legacy loans on its balance sheet and to be a potential buyer of loans. That is,

this bank uses its áexible reserves either to make new loans or to buy legacy loans in the

secondary market (whichever is more proÖtable). At date 1, the bank chooses whether to

keep or withdraw its deposits on the forward neighbor bank, A

j

1 2 fK; Wg.

Given the rare event, a bank may not be able to pay its depositors the contracted

amount (1; R) (despite the precautionary measures it takes), but instead it ends up paying

some q

j

1

; q

j

2



to all depositors. Banks know that there are no liquidity driven retail

depositors, thus a bank which is able to pay its depositors at least q

j

1 at date 2 can refuse

to pay the retail depositors if they arrive early. With this assumption, the continuation

equilibrium for bank b

j at date 1 takes one of two forms. Either the bank is solvent, pays

5

In Caballero and Simsek (2009), we motivate the formation of the Önancial network for its role in

facilitating bilateral liquidity insurance, as in Allen and Gale (2000).

6
q

j

1 = 1; q

j

2  1, and the retail depositors withdraw at date 2; or the bank is insolvent, pays

q

j

1 < 1; q

j

2 = 0, and all depositors draw their deposits at date 1.

The bank chooses actions A

j

0

; Aj

1



to maximize q

j

1 until it can meet its liquidity obligations to depositors, that is, until q

j

1 = 1. Increasing q

j

1 beyond 1 has no beneÖt for the

bank, thus once it satisÖes its liquidity obligations, it then tries to maximize the return

to late depositors q

j

2

. Note that the banks make their decisions while facing uncertainty

about the network structure. For expositional simplicity we assume that banks are inÖnitely risk averse (rather than just risk averse) with respect to the Önancial network, i.e.

they evaluate their decisions according to the worst possible network realization which

they Önd plausible.

Secondary Market for Legacy Loans

Legacy loans are traded in a centralized exchange that opens at date 0. Given the loan

price r, the banks that choose Aj = HS sell all of their legacy loans (1  y units for each

bank) while the banks that choose Aj = B are potential buyers of legacy loans and may

spend up to y (their áexible reserves). There is no demand or supply for legacy loans

outside of the network.

A legacy loan traded in the secondary market is held by a bank di§erent than its

originator, and it yields returns R   < R at date 2. We assume  > 0 which simpliÖes

the subsequent analysis, but the economic results generalize also to the case  = 0.

6 Note

that there is an upper bound on endogenous loan prices. In particular, potential buyers

of legacy loans give up áexible reserves which they could also use to make new loans, thus

each loan has an opportunity cost of R units at date 2. Since secondary loans return R

units at date 2, no potential buyer would demand legacy loans at a price greater than

rfair  (R  ) =R:

Legacy and new loans can be freely disposed of at the scrap value rf ire 2 (0; rfair), so

rf ire is a lower bound on prices.

If r < rfair, potential buyers spend all of their áexible reserves y on legacy loans,

while if r = rfair, they are indi§erent between buying legacy loans and making new loans.

6

In other words, even though banks get returns slightly lower than R from legacy loans purchased

in the secondary market, they are the natural buyers of these loans in the sense of Shleifer and Vishny

(1992).

7
Figure 1: Timeline of events.

Thus, the market clearing condition for legacy loans can be written as

(1  y)

X

j

1



A

j = HS	



y

r

X

j

1



A

j = B

	

8

><

>:

 0 if r = rf ire

= 0 if r 2 (rf ire; rfair)

 0 if r = rfair

. (2)

The Örst term on the left hand side denotes the total supply of loans while the second

term denotes the maximum potential demand. If the left hand side of Eq. (2) is negative

for each r 2 [rf ire; rfair], then legacy loans trade at their fair value rfair, potential buyers

are indi§erent between buying legacy loans and selling new loans, and they buy just

enough legacy loans to clear the market. If the left hand side of Eq. (2)is 0 for some

r 2 [rf ire; rfair], then r is the equilibrium price. If the left hand side is positive for each

r 2 [rf ire; rfair], then there is excess supply of loans and the price is given by the scrap

value rf ire (note that this happens only if there are no potential buyers). We refer to this

situation as a breakdown in the legacy loan market.

Equilibrium

Figure 1 recaps the timeline of events in this economy. We can now deÖne equilibrium:

DeÖnition 1. An equilibrium is a collection of bank actions and payments

h

A

j

0

(); Aj

1

()

	

j

;



q

j

1

(); q

j

2

()

	

j

i

b()

and a price level r 2 [rf ire; rfair] for legacy loans

8
such that, given the realization of the Önancial network b () and the rare event, each bank

b

j

chooses its actions according to the worst case Önancial network that it Önds plausible,

the legacy loan market clears (cf. Eq. (2)), and the retail depositors withdraw deposits

early if and only if q

j

2

() < 1.

We next turn to the characterization of this equilibrium. Let { 2 f1; ::; ng denote the

slot of the distressed bank b

j=({)

. Note that, for each Önancial network b () and for each

bank b

j

, there exists a unique k 2 f0; ::; n  1g such that

j =  ({  k),

which we deÖne as the distance of bank b

j

from the distressed bank.7 The distance k is

the payo§ relevant information for a bank b

j and will play a key role in the analysis.8

2.2 The Benchmark without Complexity

We characterize the equilibrium in two steps: We start by describing the banksíactions

and payo§s for a given price level of legacy loans, r; and then solve for the equilibrium

price using the legacy loan market clearing condition (2).

Suppose the loan prices are Öxed at some r 2 [rf ire; rfair] and consider the banksí

optimal actions and payments in this setting. Consider a distressed bank that needs to

obtain liquidity at date 1 (e.g. the original distressed bank b

({)

). This bank could try to

obtain the required liquidity either by withdrawing its cross deposits at date 1 (i.e. by

choosing A

j

1 = W) and/or by taking a precautionary action at date 0 (i.e. by choosing

A

j

0 2 fH; HSg). It can be checked that the bank always weakly prefers ex-post liquidity

withdrawal to the ex-ante precautionary actions, thus we Öx banksíliquidity pecking order

as follows:

Assumption (LPO). The liquidity pecking order is such that a bank that will need

at least z units of liquidity at date 1 Örst chooses A

j

1 = W, and then (if there is need)

resorts to ex-ante precautionary measures.

Under assumption (LPO) and the parametric condition   z, the original distressed

bank b

({)

, which has to make a liquidity payment of  at date 1, withdraws its deposits

from the forward neighbor bank b

({+1). This puts bank b({+1) also in need of z units of

liquidity, which also withdraws its deposits on the forward neighbor bank. As in Allen

7We use modulo n arithmetic for the slot index i, e.g. {  k = 1 represents the slot n  1.

8Note that the deÖnition of equilibrium has built in a conservative feature which characterizes Önancial

markets in distress: Banks make their precautionary decisions based on the worst outcome they Önd

plausible. This feature plays no role in the benchmark since banks know the actual Önancial network.

9
and Gale (2000), this triggers further withdrawals until, in equilibrium, all cross deposits

are withdrawn, i.e. A

j

1 = W for all j. In particular, a bank in need of liquidity (including

the original distressed bank b

()

) tries, but cannot, obtain any net liquidity through cross

withdrawals. Anticipating that it will not be able to obtain additional liquidity at date 1,

the distressed banks try to obtain some liquidity by taking precautionary actions at date

0.

To simplify the characterization of the banksídate 0 actions, we assume

y + (1  y) rfair < 1 < (1  y) R. (3)

The left hand side of this condition is not new, as it follows from  > 0. It ensures that a

bank that sells its legacy loans (i.e. that chooses the most precautionary action A

j

0 = HS)

does not have enough resources to promise at least 1 unit to its depositors at date 2, and

thus it will be insolvent at date 1. The right hand side of this condition ensures that a

bank that keeps its legacy loans on its book (i.e. that chooses A

j

0 2 fH; Bg) can always

promise its depositors at least 1 unit at date 2. Under condition (3), a bank that expects

to make a net liquidity payment at date 1 Örst considers hoarding liquidity, i.e. the action

A

j

0 = H. If the bankís expected loss at date 1 is less than its áexible reserves y, then

it chooses A

j

0 = H and averts insolvency. If the bankís expected loss is greater than y,

then the bank expects to be insolvent at date 1 even if it chooses A

j

0 = H. Hence, this

bank believes that it cannot avoid insolvency and thus it chooses Aj = HS to improve its

liquidation outcome. We refer to y as the bankís bu§er, since this is the maximum level

of losses the bank can sustain while remaining solvent.

We next conjecture that, under appropriate parametric conditions (including condition

(3)), there exists a threshold K (r) 2 f1; ::; n  2g, which depends on loan prices, such

that all banks with distance k  K (r)  1 are insolvent (there are K (r) such banks)

while the banks with distance k  K (r) remain solvent. That is, the crisis will partially

cascade through the network but will be contained after K (r)  n  2 banks have failed.

We refer to K (r) as the cascade size.

The original distressed bank, b

({)

, is insolvent as long as its required payment is greater

than its bu§er y (and greater than z so that assumption (LPO) applies), i.e. if and only

if

 > max (y; z). (4)

Suppose this is the case so bank b

()

is insolvent. Anticipating insolvency, this bank

10
chooses A

({)

0 = HS, all retail depositors will arrive early at date 1, and the bank will pay

q

()

1 =

y + r (1  y)   + z

1 + z

< 1

to each depositor (where the inequality holds in view of condition (3)). Note that the

bank will have y + r (1  y) units of liquidity at date 1, it will make a payment of , and

it will receive z units in cross-deposits from its forward neighbor bank b

({+1)(which is

solvent in our conjectured equilibrium and thus will pay 1 unit for each unit of deposit).

Consider next the bank b

(1) with distance 1 from the distressed bank. To remain

solvent, this bank needs to pay 1 on its deposits to bank b

(2) but it receives only q

()

1 < 1

on its deposits from the distressed bank b

()

, so it loses z



1  q

()

1



in cross-deposits.

Hence, bank b

(1) will also become insolvent if (and only if) its losses from cross-deposits

are greater than its bu§er, z



1  q

()

1



> y, which can be rewritten as

q

()

1 < 1 

y

z

.

If this condition fails, the only insolvent bank is the original distressed bank and the

cascade size is K (r) = 1. If this condition holds, then bank b

(1) anticipates insolvency

and chooses A

({1)

0 = HS, and it will pay its depositors

q

(1)

1 = f



q

()

1





y + r (1  y) + zq()

1 + z

. (5)

From this point onwards, a pattern emerges. The payment by an insolvent bank b

({k)

(with k  1) is given by

q

({k)

1 = f



q

((k1))

1



and this bankís backward neighbor b

({(k+1)) is also insolvent if and only if q

({k)

1 < 1

y

z

.

Hence, the payments by insolvent banks converge to the Öxed point of the function f (:),

and if

y + r (1  y) > 1 

y

z

, (6)

then there exists a unique K (r)  2 such that

q

(k)

1 < 1 

y

z

for each k 2 f0; ::; K (r)  2g (7)

and q

((K(r)1))

1  1 

y

z

.

If n  2 is greater than the solution, K (r), to this equation, i.e. if

11
n  K (r) + 2; (8)

then, Eq. (7) shows that (in addition to the trigger-distressed bank b

({)

) all banks b

(k)

with distance k 2 f1; ::; K (r)  1g are insolvent since their losses from cross deposits are

greater than their corresponding bu§ers. These banks choose A

j

0 = HS to improve their

liquidation outcome. In contrast, bank b

({K(r)) is solvent, since it can meet its losses by

hoarding liquidity, i.e. this bank chooses A

({K(r))

0 = H. Since bank b

({K(r)) is solvent,

all banks b

(k) with distance k 2 fK (r) + 1; ::; n  1g are also solvent since they do not

incur losses in cross-deposits. These banks are potential buyers of legacy loans, i.e. they

choose A

j

0 = B. This veriÖes our conjecture for a partial cascade of size K (r) under

conditions (3), (4), (6); and (8).

Note also that Eqs. (5) and (7) imply that K (r) is decreasing in r: with a higher

loan price, the liquidation value of each bank is greater, thus the crisis is contained after

a smaller number of insolvencies. For future reference, we strengthen condition (6) to

y + rf ire (1  y) > 1 

y

z

, (9)

so that there exists a partial cascade for any r 2 [rf ire; rfair]. The next lemma summarizes

the above discussion.

Lemma 1. Suppose information is free so that banks know the Önancial network b (),

that the loan prices are exogenously Öxed at r 2 [rf ire; rfair], that assumption (LPO) holds,

and that conditions (3), (4), (8); and (9) are satisÖed. Then, there exists a partial cascade

of size K (r), where K (r) is deÖned by Eq. (7). Banks with distance from the distressed

bank k  K (r)  1 are insolvent and they choose A

j

0 = HS. The transition bank with

distance K (r) averts insolvency by choosing A

j

0 = H. The remaining banks with distance

k  K (r) + 1 are solvent and they choose A

j

0 = B.

Figure 2 displays the equilibrium cascade size K (r) when the loan prices are exogenously Öxed at some r 2 [rf ire; rfair] for a particular parameterization of the model,

illustrating that K (r) is decreasing in r. The negative dependence of the cascade size on

the price of loans will play an important role in the next section in which we consider the

equilibrium with endogenous complexity

We next consider the legacy loan market clearing condition and solve for the equilibrium level of prices in the free-information benchmark. We claim that if n is su¢ ciently

large, the endogenous loan price in the free-information benchmark is given by r = rfair.

To see this, note that the insolvent banks (there are K (r) of them) choose A

j

0 = HS

12
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0

1

2

3

4

5

6

Figure 2: Cascade size in the free-information benchmark. The plot displays the

cascade size K (r) when the loan prices are exogenously Öxed at r 2 [0; rfair] (for a given

level of ).

and sell all of their existing loans, hence the aggregate supply of loans is given by

K (r) (1  y). The transition bank b

({K(r)) chooses to hoard liquidity, while the remaining solvent banks 

b

({k)

	n1

k=K(r)+1 (there are n  K (r)  1 of them) choose A

j

0 = B, i.e.

they are potential buyers of loans. Suppose n is su¢ ciently large so that

(n  K (r)  1) y > r (1  y) K (r) for all r 2 [rf ire; rfair] . (10)

Under this condition, the demand from the potential buyers exceed the supply of loans

for any price level r < rfair, thus the loan market clearing condition (2) implies that

r = rfair.

Intuitively, if the cascade is only partial and banks know the Önancial network, then

there exist banks which will remain solvent and know that much. These banks do not

hoard liquidity and are ready to use their áexible reserves to purchase loans from distressed

banks. When there are su¢ ciently many banks, the demand from these solvent banks is

enough to absorb the supply from the distressed banks, ensuring that the secondary loans

are traded at their fair price rfair. We refer to condition (10) as the deep secondary market

assumption. The next proposition summarizes the above discussion and characterizes the

symmetric equilibrium for the free-information benchmark.

Proposition 1. Suppose information is free so that banks know the Önancial network

b (). Suppose assumption (LPO) holds, conditions (3), (4), (8); and (9) are satisÖed,

and the deep secondary market assumption (10) holds. Then:

13
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0

0.5

1

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0

2

4

6

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

10

10.5

11

Figure 3: Equilibrium in the free-information benchmark. The top, the middle,

and the bottom panels respectively plot the loan prices, the cascade size, and the aggregate

level of new loans as a function of the losses in the originating bank.

(i) The unique equilibrium price is given by r = rfair.

(ii) For the continuation equilibrium (at date 1): All banks choose A

j

1 = W. Banks

with distance k  K (rfair)  1 are insolvent while the banks with distance k  K (rfair)

are solvent.

(iii) For the ex-ante equilibrium (at date 0): The insolvent banks choose A

j

0 = HS,

the transition bank with distance K (rfair) chooses A

j

0 = H, and all other solvent banks

(i.e. banks with distance k  K (rfair) + 1) choose A

j

0 = B.

(iv) For the loan market: The insolvent banks sell all of their loans in the secondary

market, while the solvent banks with distance k  K (rfair) + 1 are indi§erent between

buying legacy loans and making new loans. These banks spend a portion of their áexible

reserves to buy the legacy loans sold by the insolvent banks and they make new loans with

the rest of their reserves. The aggregate level of new loans made in this economy is given

by:

Y = (n  K (rfair)  1) y  K (rfair) (1  y) rfair. (11)

Discussion. Figure 3 displays the equilibrium in the benchmark economy for particular

parameterization of the model. The Ögure demonstrates that the loan prices are Öxed at

rfair, the cascade size is increasing in , and the aggregate level of new loans is decreasing

14
in . Intuitively, as  increases, there are more losses to be contained, which increases

the spread of the insolvency. As the insolvency spreads, more banks hoard their áexible

reserves as liquidity instead of making new loans, which lowers Y. Note, however, that Y

decreases ìsmoothlyîwith . These results o§er a benchmark for the next section. There

we show that once auditing becomes costly, both K and Y may experience large changes

with small increases in .

3 Endogenous Complexity and Fire Sales

Let us repeat the steps of the previous section but now with our key assumption: Banks

only have local knowledge about the Önancial network b ().

Each bank observes its two forward neighbors but is otherwise uncertain about how the

remaining banks are allocated to the remaining Önancial slots (cf. Eq. (1)). In particular,

we assume this uncertainty takes a simple form: Banks with distance k  2 know their

distances from the distressed bank, and banks with distance k  3 are uncertain about

their distances and assign a positive probability to all distances ~k 2 f3; 4; ::; n  1g.

In this context, when the shock is small, the system behaves exactly as in the benchmark. But when the shock is large, banks need to understand distant (complex) linkages

in order to assess the amount of counterparty risk they are facing. Their inability to

Ögure out these complex linkages triggers a set of precautionary actions which overturns

the relatively benign implications of the benchmark environment.

As in the benchmark, the original distressed bank b

({) withdraws its deposits from

the forward neighbor bank, which triggers further withdrawals until, in equilibrium, all

cross deposits are withdrawn, i.e. A

j

1 = W for all j. Since the distressed bank knows that

it cannot obtain any net liquidity at date 1, it tries to obtain liquidity at date 0. Under

condition (4), it also knows that it cannot avoid insolvency, thus it chooses A

({)

0 = HS.

Next consider a bank b

j with distance k > 0, and note that a su¢ cient statistic for this

bank to choose action A

j

0 2 fHS; H; Bg is the amount it expects to receive in equilibrium

from its forward neighbor. In other words, to decide on the level of its precautionary

measure, this bank only needs to know whether (and how much) it will lose in crossdeposits. Formally, if the bank chooses A

j

0 at date 0 and its forward neighbor pays x

at date 1, then this bankís payment can be written as a function q1



A

j

0

; x; q2



A

j

0

; x.

However, the bank chooses A

j

0 while facing uncertainty about the Önancial network, and

consequently about x. Note also that q1



A

j

0

; xand q2



A

j

0

; xare increasing in x for any

choice of action. That is, the bankís payment is increasing in the amount it receives from

15
its forward neighbor regardless of the ex-ante precautionary measure it takes. Thus, since

the bank is inÖnitely risk averse, it will choose its precautionary action as if it will receive

from its forward neighbor the lowest possible payment x.

To characterize the bankís optimal action further, we deÖne a useful notion of equilibrium. We say that the equilibrium allocation is distance based and monotonic if the

banksí equilibrium payments can be written as an increasing function of their distance

from the distressed bank. That is, there exists weakly increasing payment functions

Q1; Q2 : f0; ::; n  1g ! R such that



q

(ik)

1

(); q

(ik)

2

()



= (Q1 [k] ; Q2 [k])

for all b () and k. We conjecture (and verify in Appendix A.1) that the equilibrium is

distance based and monotonic.

Under this conjecture, consider again a bank b

j with distance k > 0, and note that

the payment of the bankís forward neighbor can be written as x = Q1 [k  1]. Then,

the bankís uncertainty about the forward neighborís payment x = Q1 [k  1] reduces to

its uncertainty about the forward neighborís distance k  1, which is equal to one less

than its own distance k. If the bank knows its distance to the distressed bank, i.e. if

k 2 f1; 2g, then it chooses its optimal action A

j

0 2 fHS; H; Bg knowing that it will

receive x = Q1 [k  1] from its forward neighbor. On the other hand, if the bank is

uncertain about its distance, i.e. if k  3, then it assigns a positive probability to all

distances ~k 2 f3; :::; n  1g. Moreover, since Q1 [:] is an increasing function, the payment

Q1

h

~k  1

i

is minimal for the distance ~k = 3. Hence a bank b

j with distance k  3

chooses A

j

0 2 fHS; H; Bg as if it will receive x = Q1 [2] from its forward neighbor. In

words, the banks that are uncertain about their distances to the distressed bank choose

their precautionary action as if they are closer to the distressed bank than they actually

are.

Formally, our next lemma establishes that all banks with distance k  3 choose the

action that the bank with distance ~k = 3 would choose if the information was freely

available (characterized in Lemma 1). If the free-information cascade size is given by

K (r) = 3; so that the bank with distance ~k = 3 would take the precautionary action

A3

0 = H to avert insolvency, then all banks with k  3 also take the precautionary action

A

j

0 = H even though ex-post they end up not needing liquidity. If the free-information

cascade size is even greater (i.e. if K (r)  4) so that the bank with distance ~k = 3 would

be insolvent and choose the most precautionary action A3

0 = HS, then all banks with

k  3 fold by taking the most precautionary action A

j

0 = HS, which ensures that their

16
insolvency (even though some of these banks would be solvent in the free-information

economy). The proof of the following lemma is relegated to Appendix A.1 since most of

the intuition is provided by the above discussion.

Lemma 2. Consider the setup of Lemma 1, but now information is costly rather than

free, so that banks know only their two forward neighbors and they are otherwise uncertain

about the Önancial network b (). Let K (r) denote the cascade size that would obtain if

information was freely available (characterized in Lemma 1).

Then the equilibrium is distance based and monotonic. Each bank b

j with distance

k 2 f0; 1; 2g chooses the same action that it would choose in the free-information economy

(characterized in Lemma 1). Each bank b

j with distance k 2 f3; ::; n  1g chooses the same

action that the bank with distance 3 would choose in the free-information economy. In

particular, there are three cases to consider, depending on the cascade size:

If K (r)  2, then the crisis in the free-information economy would not cascade to

bank with distance 3, which would choose A

j

0 = B. Thus, each bank b

j with distance

k 2 f3; ::; n  1g chooses A

j

0 = B, and the equilibrium actions and payments are identical

to the free-information economy described in Lemma 1. In particular, the cascade size is

given by K^ (r) = K (r).

If K (r) = 3, then the crisis in the free-information economy would cascade to and

stop at bank with distance 3, which would choose A

j

0 = H to avert insolvency. Thus each

bank b

j with distance k 2 f3; ::; n  1g chooses A

j

0 = H. The equilibrium cascade size is

still the same as the free-information economy, i.e. K^ (r) = K (r), but the banksíactions

and payments are di§erent.

If K (r)  4, then bank with distance 3 would be insolvent in the free-information

economy and would choose A

j

0 = HS. Thus, all banks choose A

j

0 = HS and they are all

insolvent, i.e. the cascade size is given by K^ (r) = n.

Note that the loan trade decisions of the banks with distance k  3 depend on the

free-information cascade size K (r). In particular, if K (r)  2, these banks are potential

buyers in the loan market, while if K (r)  3, they are either neutral in the loan market

(if K (r) = 3) or they are potential sellers (if K (r)  4). This dependence of the banksí

loan trades on the free-information cascade size K (r) (which itself depends on the loan

price) plays a key role in subsequent analysis, where we endogenize the loan prices and

complete the characterization of the equilibrium.

We next solve the equilibrium level of loan prices in the costly information setting and

present our main results. There are three cases to consider depending on the cascade size

K (r) over the price range r 2 [rf ire; rfair].

17
Case (i). If K (rf ire)  2, then we conjecture that there is a unique symmetric equilibrium, that we refer to as the fair-price equilibrium, in which loan markets endogenously

clear and loans trade at their fair price r = rfair. To verify this conjecture, recall that

K (r) is a decreasing function so that K (r)  K (rf ire)  2 for all r 2 [rf ire; rfair]. Hence,

regardless of the endogenous price r, Lemma 2 implies that all banks with k  3 choose

A

j

0 = B; and as such, these banks are potential buyers of loans. Thus, under the deep secondary market assumption (10), the unique equilibrium price is given by r = rfair. Once

the equilibrium price is determined, the remaining equilibrium allocations are uniquely

determined as described in Lemma 2, proving our conjecture for this case.

Case (ii). If K (rfair)  3, we conjecture that there is a unique symmetric equilibrium,

which we refer to as the Öre-sale equilibrium, in which there is a breakdown in the loan

market and r = rf ire. To see this, note that 3  K (rfair)  K (r) for all r 2 [rf ire; rfair],

thus Lemma 2 implies that all banks with k  3 choose either A

j

0 = H (if K (r) = 3), or

A

j

0 = HS (if K (r)  4). In words, these banks are either neutral in the legacy loan market

or they are sellers, in particular, they are not potential buyers. Using Lemma 2 once more,

note that the banks with distance k  2 are necessarily insolvent, so they choose A

j

0 = HS

and become sellers in the legacy loan market. Thus, the legacy loan market features at

least 3 sellers but no buyers. The market clearing condition (2) implies that r = rf ire, i.e.

there is a breakdown in the secondary loan market.

Case (iii). If K (rfair)  2 < 3  K (rf ire), we conjecture that there are two stable

equilibria: one fair-price equilibrium and one Öre-sale equilibrium. To see this, Örst suppose that the price of loans is given by r = rfair so that the cascade size satisÖes K (r)  2.

Then, the analysis for case (i) applies unchanged, in particular, all banks with distance

k  3 are potential buyers and the price rfair clears the market, verifying that there is a

fair-price equilibrium. Next suppose that the price of loans is given by r = rf ire so that

the cascade size satisÖes K (rf ire)  3. Then, the analysis for case (ii) applies unchanged,

in particular, all banks with distance k  3 are either neutral in the loan market or they

sell their loans, and loan prices collapse to r = rf ire, verifying that there is a Öre-sale

equilibrium.

We summarize these results in the following, and main, proposition.

Proposition 2. Consider the setup of Proposition 1 with the di§erence here being that

banks only have local understanding of the network, so that banks know only their two

forward neighbors and they are otherwise uncertain about the Önancial network b ().

18
Let K (r) denote the cascade size in the free-information economy with price level r 2

[rf ire; rfair] (characterized in Lemma 1).

(i) Fair-price equilibrium: If K (rf ire)  2, there is a unique equilibrium in which the

loan market clearing condition (2) is satisÖed with equality and loans trade at their fair

price r = rfair.

The cascade size is the same as the free-information benchmark, i.e. K^ = K (rfair) 

2. The insolvent banks (with distance k  K^  1) choose A

j

0 = HS and sell their loans,

while the solvent banks with distance k  K^ +1 choose A

j

0 = B and are indi§erent between

using their reserves to make new loans or to buy legacy loans in the secondary market.

The aggregate level of new loans is equal to the benchmark Eq. (11).

(ii) Fire sale equilibrium: If K (rfair)  3, then there is a unique equilibrium in which

there is a breakdown in the secondary loan market, i.e. there is an excess supply of loans

and r = rf ire.

(ii.1) If K (rf ire) = 3, all banks with distance k  3 choose A

j

0 = H. These banks

remain solvent and the cascade size is K^ = K (rf ire).

(ii.2) If K (rf ire)  4, all banks with distance k  3 choose A

j

0 = HS. These

banks go under and the cascade size in this case is K^ = n.

In either sub-case, all áexible reserves are hoarded as liquidity and there are no new

loans, Y = 0.

(iii) Multiple equilibria: If K (rfair)  2 < 3  K (rf ire), there are two stable equilibria.

In the fair-price equilibrium, loans trade at their fair price r = rfair, the cascade

size is K^ = K (rfair)  2, the solvent banks are indi§erent between making new loans or

purchasing loans in the secondary market, and the aggregate level of new loans Y is given

by the benchmark Eq. (11).

In the Öre sale equilibrium, r = rf ire, the cascade size is given by either K^ = 3 (if

K (rf ire) = 3) or K^ = n (if K (rf ire)  4), all banks hoard liquidity and (if K (rf ire)  4)

dump their loans in the secondary market, and there are no new loans, Y =0.

Discussion. Figure 4 displays the equilibria with network uncertainty for a particular

parameterization of the model. The top panel reproduces the cascade size K (r)in the

free-information benchmark as a function of the losses in the originating bank , when

loan prices are Öxed at r = rf ire and r = rfair. The remaining three panels display

the equilibria with network uncertainty, illustrating the characterization in Proposition 2.

Note that there is a unique equilibrium for small and large levels of , however, there are

multiple equilibria for intermediate levels of .

19
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0

5

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

0

0.5

1

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

0

20

40

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

0

5

10

15

Figure 4: Equilibria with network uncertainty. The top panel plots the cascade size

K (r) in the free-information benchmark as a function of the losses in the originating bank

, when loan prices are Öxed at r 2 f0; rfairg. The remaining three panels display the

equilibria with network uncertainty: They respectively plot the loan prices, the cascade

size and the aggregate level of new loans as a function of , for both the fair-price and

the Öre-sale equilibria.

20
When  is su¢ ciently small so that K (rf ire)  2, i.e. when the free-information

benchmark features a short cascade even for the price level r = rf ire, there is a unique

fair price equilibrium. This equilibrium features a low level of complexity thus the banks

that are uncertain about their distance to the distressed bank can rule out an indirect hit.

Consequently, these banks use their reserves to make new loans and to demand assets.

The aggregate level of new loans is the same as in the free-information benchmark, and

assets trade at their fair prices.

In contrast, when  is su¢ ciently large so that K (rfair)  3, i.e. when the cascade

size in the free-information benchmark is su¢ ciently large, there is a unique Öre-sale

equilibrium. In this equilibrium, the aggregate level of new loans makes a very large

and discontinuous drop to zero. That is, when the losses (measuring the severity of the

initial shock) are beyond a threshold, the cascade size becomes so large that banks are

unable to tell whether they are connected to the distressed bank or not. All uncertain

banks act as if they are closer to the distressed bank than they actually are, hoarding

much more liquidity than in the free-information benchmark and leading to a severe

credit crunch episode. Moreover, these banks, some of which would be potential buyers

in the free-information benchmark, become sellers and this leads to a collapse in asset

prices. This result provides a rationale for the collapse of asset prices in an environment

in which complexity suddenly (and endogenously) rises. Furthermore, it is possible (if

K (rf ire)  4) that the uncertain banks panic so much that they take an extremely

precautionary action to increase their liquidation outcome. However, ex-post, this action

ensures that the uncertain banks become insolvent.

When  is in an intermediate range, the cascade size is manageable if price of loans

is high (i.e. K (rfair)  2), however it becomes unmanageable if loans trade at the Öresale price (i.e. K (rf ire)  3). In this case, the interaction between asset prices and the

endogenous level of complexity generates multiple equilibria.

In the fair-price equilibrium, loans trade at a higher price and the cascade size is

relatively small, which reduces the level of complexity. With the lower level of complexity,

the banks that are uncertain about their distance to the distressed bank become potential

buyers of loans, which ensures that loans trade at the higher price and that the cascade

is shorter.

Set against this benign scenario is the possibility of a Öre-sale equilibrium, in which

the price of loans collapses and there is a longer cascade, which increases the level of

complexity. As the level of complexity increases, banks that are uncertain about their

distances panic and sell their loans, which reinforces the collapse of loan prices.

Note also that, whenever there are multiple equilibria, the fair-price equilibrium Pareto

21
dominates the Öre-sale equilibrium for all banks and their depositors. Intuitively, the Öresale equilibrium entails two distinct social costs: fewer new loans and more banks become

insolvent. In the next section, we identify the negative externalities in our setup that

account for these social costs.

4 Three Externalities

In this section, we discuss the various externalities present in our setup and we highlight

the role they play in our main results. Our model features three distinct externalities,

which we call the network-liquidity externality, the Öre-sale externality and the complexity externality. The Örst of these emerges directly from the interlinkages between banks.

The second externality stems from the interaction of loan prices and the endogenous

determination of bank runs. These two externalities are relatively standard. Models of

interconnected banks typically feature externalities akin to our network-liquidity externality (see, for example, Rotemberg (2009)), while models based on the Diamond and Dybvig

(1983) setup typically lead to Öre-sale externalities whenever prices a§ect the likelihood of

bank runs (see, for example, Allen and Gale (2005)). The complexity externality is novel

to our analysis, and it emerges from the interaction of loan prices and the complexity of

the Önancial network.

4.1 Network-liquidity Externality

In our setup, a distressed bank can obtain liquidity through two distinct sources: It can

either withdraw its deposits on the forward neighbor bank, or it can hoard its liquid reserves (or it can sell its legacy loans, as a last measure). Since aggregate liquidity is scarce,

a bank that chooses the former option puts its forward neighbor bank also in distress,

imposing a liquidity externality on it. In particular, if the bank decides to withdraw its

cross deposits, the forward neighbor bank scrambles for liquidity and faces insolvency if it

is unable to meet this liquidity demand. A distressed bank in our setup always (weakly)

prefers to withdraw deposits (cf. assumption (LBO)).9In Appendix A.2, we study the

polar opposite case, enforced by the government, in which a bank that needs liquidity Örst

hoards its own liquid reserves, and thus avoids ináicting a liquidity externality whenever

it can. Our analysis shows that internalizing the liquidity externalities in this fashion

9The bankís preference for liquidity withdrawal is strict if the forward neighbor pays q

j

2 < R. In

the more general model analyzed in Caballero and Simsek (2009), the bank always strictly prefers to

withdraw deposits and thus always ináicts a liquidity externality.

22
leads to shorter cascades in aggregate (see Proposition 3 in the appendix).

Intuitively, the scramble for liquidity is like a hot potato which the banks can either

pass straight to their neighbors, or which they can cool down a bit using their resources

before passing it on. When all banks pass it without cooling it, the hot potato eventually

reaches a vulnerable bank, i.e. a bank which is su¢ ciently close to the original distressed

bank. This bank cannot pass the potato to its neighbor, which is already bankrupt.

Moreover, the resources of this bank alone are not su¢ cient to cool down the potato

hence the bank burns (i.e. it cannot meet the liquidity demand and becomes insolvent),

which lengthens the cascade. In contrast, when each bank cools down the potato before

passing it on, then the potato is cold before it reaches the vulnerable bank, leading to a

much shorter cascade.

4.2 Fire Sale Externality

Consider a bank that decides to sell some loans, leading to a small decline in loan prices.

This action has a small positive e§ect on the net budgets of the banks that buy legacy

loans, while it has a small negative e§ect on the net budgets of the banks that sell legacy

loans. Absent further e§ects, the welfare impacts of these budget changes would typically

ìnet out,îwhich is the content of the Örst welfare theorem. However, a small decline also

increases the likelihood of a bank run in our setup, leading to a further decline in the

welfare of these banks and their depositors. Consequently, the e§ects of price changes do

not necessarily ìnet out,îand loan prices feature a negative pecuniary externality.

To formalize this point, let us consider the benchmark economy analyzed in Section 2.2.

This setting shuts down the complexity externality (since the banks know the Önancial

network) which we will analyze in the next subsection.

Recall that the cascade size K (r) is decreasing in r: with a lower loan price, the

liquidation payment of each bank is greater (cf. Eq. (5)), thus the crisis is contained

after a smaller number of insolvencies (cf. Eq. (7)). In particular, there exists some

r 2 [rf ire; rfair] such that, for any " > 0 we have

K (r  "=2) = K (r + "=2) + 1:

That is, the cascade size increases by one in response to an arbitrarily small decrease

in loan prices. The small price drop from r + "=2 to r  "=2 leads to the insolvency of

one more bank, ináicting a discrete negative e§ect on the welfare of this bank and its

backward neighbor, while it has a continuous e§ect on the welfare of other banks. Hence,

23
for su¢ ciently small ", the net welfare e§ect of the price change from r + "=2 to r  "=2

is negative, demonstrating the Öre sale externality.10

The general intuition behind this externality is that a drop in asset prices lowers the

liquidation value of an insolvent bank, which increases the probability of a bank run in

the counterparties of this bank.

4.3 Complexity Externality

Consider next the model analyzed in Section 3 in which banks only have local information

about the Önancial network. Similar to the above analysis for the Öre sale externality, a

bank that sells a loan has a small negative impact on loan prices, which in turn increases

the length of the cascade. However, in this case, longer cascades also increase the complexity of the Önancial network. Since banks are averse to complexity (which we model

as inÖnite risk aversion with respect to the Önancial network), an increase in complexity

leads to a welfare reduction for banks that are uncertain about the Önancial network.

To formalize this point, consider the set of parameters that lead to multiple equilibria

in Proposition 2, K (rfair)  2 < 3  K (rf ire). Then, there exists r 2 [rf ire; rfair] such

that, for any " > 0 we have

K (r  "=2) = 3 > K (r + "=2) = 2.

That is, small decline in loan prices makes the cascade size in the free-information

benchmark exceed the critical threshold of 2. Recall that the banks with distance

k 2 f3; ::; n  1g mimic the bank with distance 3 in the free-information benchmark.

Then, when the loans trade at price r + "=2, these banks know that they will not lose

anything in cross deposits, thus they do not take any precaution and they pay out at least



q

j

1 = 1; q

j

2 = R



to their depositors (and perhaps more, if they can acquire legacy loans

at a discount).

In contrast, when the loans trade at price r  "=2, the banks with distance k 2

f3; ::; n  1g are worried that they might su§er an indirect hit from the cascade. Since

these banks are inÖnitely risk averse, their welfare is greatly reduced. Moreover, these

banks hoard liquidity to precaution against the worst case scenario, and they end up

paying q

j

1 = 1; q

j

2 < R

to their depositors. Hence, a small drop in prices ináicts a discrete

10The discrete nature of the Öre sale externality is due to our modeling assumptions, in particular, our

assumption that the good equilibrium is selected whenever there are multiple equilibria for bank runs.

However, the externality would also be present in other variants of the model, as long as the probability

of a bank run is increasing in the date 1 losses (i.e. the severity of the liquidity shock) of the bank.

24
negative e§ect on all of the banks with distance k 2 f3; ::; n  1g and their depositors,

demonstrating the complexity externality.

The complexity externality may also lead to multiple Pareto-ranked equilibria in our

setup, as we have already seen in Proposition 2. In particular, an increase in the level of

complexity due to a reduction in the price of assets not only lowers the welfare of many

banks and their depositors, but also induces these banks to take extreme precautionary

measures, which, in some instances, includes further asset sales. The sale of assets by

banks in panic mode reduces asset prices further, which leads to a vicious cycle culminating

in the Öre-sale equilibrium. In contrast, an increase in asset prices reduces the complexity

externality, which may mitigate the precautionary measures and turn more sellers into

buyers, leading to a virtuous spiral towards the fair price equilibrium.

Finally, note that the complexity externality is potentially much more potent than

the Öre sale externality analyzed in the previous subsection and highlighted in the literature. The reason is that the Öre-sale externality a§ects banks that are on the verge

of insolvency. In contrast, the complexity externality a§ects all banks that are uncertain

about the Önancial network, which, in practice, includes virtually all Önancial institutions.

The greater scope of the complexity externality also leads to widespread (precautionary)

actions, which has the potential to create aggregate e§ects, price changes, and multiple

equilibria.

5 Conclusion

In this paper we provide a model that illustrates how Öre sales and complexity can trigger a very powerful negative feedback loop within a Önancial network. More severe Öre

sales lengthen the potential cascades, and raise the complexity of the environment exponentially. This triggers confusion among potential asset buyers, which pull back and

exacerbate the Öre sale. In extreme scenarios these potential buyers can turn into sellers,

leading to a complete collapse in secondary markets.

In our model, the distressed institutionsí motive for Öre sales is to improve their

outcome in the worst case scenario of insolvency. This is an extreme assumption made

only to simplify the exposition. In reality, the distressed institutions may sell assets for a

multitude of reasons that do not imply anticipating insolvency. E.g. they may be forced

to do so by regulatory requirements, or they may hope to avoid a run by obtaining more

liquidity. These reasons can be incorporated into our framework while preserving our

main conclusions.

25
Having said this, the particular insolvency motive we consider also raises the question

of what would happen if the distressed institutions chose to gamble for resurrection by

not selling their assets, which would improve their outcome in good states at the cost of

a greater bankruptcy risk. Our model suggests that gambling for resurrection may be a

mixed blessing for the aggregate. Gambling by potential buyers, that is, institutions that

are far from the cascade but that do not know this, would limit the Öre sales and the

downward spiral of prices. On the other hand, gambling by institutions near the cascade

would increase the cascade size and trigger the complexity mechanism.

We did not explore policy questions, but it is apparent that our environment creates many policy opportunities during crises. Supporting secondary markets, insulating

Önancial institutions from counterparty risk, and stress testing (increasing transparency

and supporting if needed) systemically important Önancial institutions, are all policies

practiced during the current Önancial crisis and supported by our framework.

.

A Appendix

A.1 Proofs Omitted in the Main Text

Proof of Lemma 2

Case (i): K (r)  2. To prove that the conjectured actions and payments constitute

an equilibrium, Örst note that the original distressed bank optimally chooses A

({)

0 = HS

and pays out the same level Q1 [0] that it would pay in the free-information economy

(since it receives the full amount z from its forward neighbor, b

({+1), which is solvent

in the conjectured equilibrium). Next consider a bank with distance k 2 f1; 2g. Under

the conjectured equilibrium, this bank knows that it will receive x = Q1 [k  1] from its

forward neighbor, which is equal to what it would receive in the free-information economy.

Hence, it optimally chooses the same action it would choose when information is free.

Consider next a bank with distance k  3. This bank is uncertain about its distance,

thus it chooses its date 0 action as if it will receive x = Q1 [2] from its forward neighbor.

Under the conjectured equilibrium, the cascade size is K (r)  2, thus we have Q1 [2] = 1.

Since the bank makes no losses in cross deposits (even in the worst case scenario of ~k = 3),

it optimally chooses A

j

0 = B, verifying the optimality of the conjectured action.

Since the banksíactions are the same as the free-information economy, their payments

are also identical to that case, which veriÖes (by Lemma 1) that the equilibrium is distance

26
based and monotonic.

Case (ii): K (r) = 3. Similar to the previous case, the original distressed bank and

the banks with distance k 2 f1; 2g optimally choose the same action and make the same

payments as in the free-information economy.

Consider next a bank with distance k  3. This bank is uncertain about its distance,

thus it chooses its date 0 action as if it will receive x = Q1 [2] from its forward neighbor.

Since K (r) = 3, under the conjectured equilibrium, we have Q1 [2] (which is equal to its

level in the free-information benchmark) lies in the interval (1  y=z; 1). Hence, the bank

with distance k  3 expects (with some probability) to make losses less than its bu§er,

thus it chooses A

j

0 = H to counter these losses, verifying the optimality of its conjectured

action.

These banks with distance k  3 are solvent, but they pay potentially less than the

free-information economy. In particular, the bank with distance k = 3 pays the same

amount (Q1 [3] ; Q2 [3]) that it would pay in the free-information case, while the banks

with distance k  4 end up paying

Q1 [k] = 1 and Q2 [k] = y + (1  y) R 2 (1; R),

where Q2 [k]  1 follows from Eq. (3). It follows also that the equilibrium is distance

based and monotonic.

Case (iii): K (r)  4. In this case, note that the banksípayments, given the conjectured

actions, are the solutions to the following system of equations:

Q1 [0] = y + (1  y) r   + zQ1 [n  1]

1 + z

(12)

Q1 [k] = y + (1  y) r + zQ1 [k  1]

1 + z

, for k 2 f1; ::; n  1g .

Note that, by condition (3), the solution to this system satisÖes Q1 [k] < 1 for all k,

verifying that the banks are indeed insolvent given their actions. The system in Eq. (12)

also veriÖes that the equilibrium allocation is distance based and monotonic.

To verify the optimality of banksíactions, Örst note that Eq. (12) implies the payments

fQ1 [k]g

K(r)1

k=0 are lower than what they would be in the free-information economy, because

the forward neighbor of the original distressed bank pays Q1 [n  1] < 1 in this case (while

it would be solvent and would pay 1 unit per deposit in the free-information economy).

Consider a bank with distance k 2 f1; 2g, and recall that this bank chooses its date 0

27
action knowing that it will receive Q1 [k  1] from its forward neighbor. Since K (r) 

4, this bank would be insolvent in the free-information economy. In the present case,

Q1 [k  1] is even lower than what it would be in the free-information economy. Thus the

bank with distance k 2 f1; 2g is necessarily insolvent, and it optimally chooses A

j

0 = HS.

Consider next a bank with distance k  3. This bank is uncertain about its distance

and thus it chooses its date 0 action as if it will receive x = Q1 [2] from its forward

neighbor. Since K (r)  4, and since the payments fQ1 [k]g

K(r)1

k=0 are even lower than

the free-information economy, we necessarily have Q1 [2] < 1  y=z. In other words, the

bank with distance k  3 believes that it might experience losses beyond its bu§er and

go bankrupt. Consequently, it chooses the most extreme precautionary action A

j

0 = HS

to improve their liquidation outcome, verifying the optimality of the conjectured action.

A.2 Equilibrium without Liquidity Externalities

In the main text, the bank always considers deposit withdrawal as the Örst source of

liquidity (cf. assumption (LBO)), thus imposing a liquidity externality on its forward

neighbor bank. To clarify the role of liquidity externalities, in this appendix we analyze

the equilibrium under a government policy that prevents banks from withdrawing cross

deposits unless they have used all of their liquid reserves.

To facilitate the analysis, it helps to consider a slightly greater action space for banks:



A

j

0

; Aj

1



2 fH (~y); HS; Bg fW (~z); Kg , (13)

where A

j

0 = H (~y) denotes that the bank hoards y~ 2 (0; y] units of its áexible reserves,

and A

j

1 = W (~z) denotes that the bank withdraws z~ 2 (0; z] units of its deposits on

the forward neighbor bank. In other words, we allow the banks to partially hoard their

áexible reserves and to partially withdraw their cross deposits. Next, we introduce the

main assumption of this appendix: the government enforces the reverse of the liquidity

pecking order in assumption (LPO) so that banks do not impose a liquidity externality

on their forward neighbors, whenever they can avoid doing so.

Assumption (LPO-R). Consider a bank that needs liquidity at date 1 and whose

forward neighbor bank is solvent. The government imposed liquidity pecking order for this

bank is such that the bank Örst considers hoarding its áexible reserves, i.e. it considers the

action A

j

0 = H (~y). The bank withdraws some of its cross deposits (chooses A

j

1 = W (~z))

only if A

j

0 = H (y) is not su¢ cient to meet its liquidity demand.

28
Finally, we strengthen the right hand side of condition (3) to

1 + z < (1  y) R, (14)

so that a bankís own legacy loans are enough to promise all depositors (including the

backward neighbor bank) at least 1 unit at date 2.

Under these assumptions and when n is su¢ ciently large, we conjecture that there is

an equilibrium in which all banks with distance k  1 (i.e. all banks except potentially

the original distressed bank) are solvent. In other words, the cascade size is at most 1, in

stark contrast with the equilibria characterized in Propositions 1 and 2.

In the conjectured equilibrium, the original bank b

({) hoards all of its liquid reserves

and also withdraws its deposits on bank b

({+1), i.e. it chooses A

({)

0 = HS and A

({)

1 =

W (z), which puts bank b

({+1) in need of z units of liquidity. Under the new assumption

for the liquidity pecking order, bank b

({+1) (with distance n  1) Örst resorts to hoarding

its áexible reserves. If z  y, then this bank chooses A

({+1)

0 = H (z); A({+1)1 = K



,

that is, the bank meets its liquidity payments purely by hoarding its áexible reserves,

and it does not withdraw any cross deposits. Otherwise z > y and this bank chooses



A

({+1)

0 = H (z); A({+1)1 = W (z  y)



. In this case, consider bank b

({+2) with distance

n  2, which needs to Önd z  y units of liquidity. This bankís response is similar to bank

b

({)

: if zy  y, then the bank meets its liquidity payment purely by hoarding its áexible

reserves, and otherwise it withdraws an even smaller amount from its forward neighbor.

It follows that a pattern emerges for the banksícross withdrawal decisions. In particular, let n  1 denote the unique integer such that

yn  z > y (n  1)

and suppose n > n. Then, for each j 2 f1; ::; n  1g, the bank b

({+j) with distance n  j

chooses A

({+j)

0 = H (y); A({+j)1 = W (z  jy)



, while the bank b

({+n) with distance nn

chooses A

({+n)

0 = H (z  (n  1) y); A({+n)1 = K



. Since the bank with distance n  n

keeps its cross deposits, the remaining banks with distance k 2 f1; :::; n  n  1g do not

need any liquidity, and these banks withdraw their deposits if and only if their forward

neighbors are insolvent.

Going back to the original distressed bank b

({)

, note that, unlike in the previous

sections, this bank obtains a total of y + z units of liquidity: y units from its bu§er and z

units from cross deposits (if it remains solvent). Hence, this bank is insolvent if and only

29
if

 > y + z, (15)

which is a stronger condition than (4). Suppose this condition is satisÖed so that bank

b

({)

is insolvent and consider the backward neighbor bank b

({1). This bank chooses

A

({1)

1 = W (z) and receives some q

({)

1

z < z from the original distressed bank. Despite

incurring some losses, in view of condition (14), bank b

({1) is solvent and it can promise

its late depositors (including the backward neighbor bank) at least 1 unit at date 2, i.e.

it pays q

({1)

1 = 1; q

({1)

2 > 1



. Since the backward neighbor bank b

({2) does not need

liquidity, it chooses to keep its deposits in bank b

({1). Repeating this reasoning, all banks

with distances k 2 f1; ::; n  n  2g are solvent, and thus all banks with distances k 2

f2; ::; n  n  1g choose 

A

({k)

0 = B; A({k)1 = K



, in particular, keeping their deposits

in their forward neighbor banks.

Note also that there is at most one seller in the conjectured equilibrium (the original

distressed bank b

({)

) while the banks with distance k 2 f2; ::; n  n  1g are potential

buyers of loans. Hence, under the deep secondary market assumption (i.e. the analogue

of condition (10) for this setup),

(n  n  2) y > 1  y, (16)

the unique equilibrium price is given by r = rfair. Note also that this analysis applies

regardless of whether the banks know the Önancial network b () or whether they only

have local knowledge (i.e. they know only their two forward neighbors). We summarize

this result in the following proposition.

Proposition 3. Suppose the banksí action space is extended to (13), assumption (LPO)

for the liquidity pecking order is replaced by assumption (LPO-R) (which is imposed by

the government), conditions (15) and (16) hold. Then, regardless of whether the banks

know the Önancial network b () or just their two forward neighbors, there is a symmetric

equilibrium in which loan prices are given by r = rfair and the cascade size is equal to 1.

In particular, the original distressed bank is insolvent while all other banks are solvent.

At date 1, the banks with distance k 2 f1; 0; n  1; ::; n  n  1g withdraw all or some of

their cross deposits, while the banks with distance k 2 f2; 3; ::; n  ng keep their deposits

in the forward neighbor bank.

Proposition 3 establishes our main result in this appendix: If the banks avoid ináicting

a liquidity externality, then the cascade size is much shorter relative to the cases analyzed

in Propositions 1 and 2. To see the intuition, consider the equilibrium in Proposition 3

30
and consider what the banks would do if we removed the government imposed liquidity

pecking order in assumption (LPO-R). Consider bank b

({+n)

, which meets its liquidity

demand from its backward neighbor purely by hoarding its áexible reserves. For this

bank, hoarding reserves delivers 1 unit of liquidity at an opportunity cost of R (in period

2), while withdrawing cross deposits would deliver 1 unit of liquidity at an opportunity

cost of q

({+n+1)

2

. Given the characterization in 3, it can be checked that q

({+n+1)

2 < R

(as some of the losses in the original bank will spillover to bank b

({+n+1)). Hence, if

not restricted by government policy, bank b

({+n) would strictly prefer to withdraw its

cross deposits to hoarding liquidity, which would put bank b

({+n+1) in distress. Similarly,

absent government policy, bank b

({+n+1) would prefer to withdraw its cross deposits and

the scramble for liquidity would continue to cascade in similar fashion. Eventually, a

vulnerable bank, b

({1), which is su¢ ciently close to the original distressed bank (and

thus has incurred some losses) would become distressed. This bank might be unable to

Önd the required liquidity and might become insolvent. Hence, the liquidity externalities

have the potential to make vulnerable banks go insolvent. The government imposed

liquidity order in assumption (LPO-R) internalizes these liquidity externalities, which in

turn ensures that the cascade size is shorter.

31
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