MIT LIBRARIES
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3 9080 03317 5925
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Complexity and Financial Panics
Ricardo
J.
Caballero
Alp Simsek
Working Paper 09-17
May 15, 2009
RoomE52-251
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Cambridge,
02142
MA
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Complexity and Financial Panics
Ricardo
Caballero and Alp Simsek*
J.
May
15,
2009
Abstract
During extreme financial
rife
crises, all of
with profit opportunities
a sudden, the financial world that was once
for financial institutions (banks, for short)
becomes
exceedingly complex. Confusion and uncertainty follow, ravaging financial markets
and triggering massive
of this
phenomenon. In our model, banks normally
trading partners which assures
ever,
when acute
enough
In this paper
flight-to-quality episodes.
to
them
financial distress
collect
we propose a model
information about their
of the soundness of these relationstdps.
emerges
in parts of the financial
be informed about these partners, as
it
also
network,
becomes important
How-
it is
not
to learn
about the health of their trading partners. As conditions continue to deteriorate,
banks must learn about the health
of the trading partners of the trading partners
of the trading partners, and so on. At
becomes too unmanageable
for
some
point, the cost of information gathering
banks, uncertainty spikes, and they have no option
but to withdraw from loan commitments and
illiquid positions.
A
fiight-to-quality
ensues, and the financial crisis spreads.
JEL Codes:
Keywords:
EO, Gl, D8,
E5
Financial network, complexity, uncertainty, flight to quality, cas-
cades, crises, information cost, financial panic, credit crunch.
*MIT and NBER, and MIT,
respectively. Contact information: caball@mit.edu and alpstein@mit.edu.
thank Mathias Dewatripont, Pablo Kurlat, Guido Lorenzoni, Juan Ocampo and seminar participants
at Cornell and MIT for tlieir comments, and the NSF for financial support. First draft: .'^pril 30, 2009.
We
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium IVIember Libraries
http://www.archive.org/details/complexityfinancOOcaba
Introduction
1
The dramatic
and banks' perceived uncertainty'
rise in investors'
2009 U.S. financial
is
All of a sudden, a financial world that
crisis.
at the core of the 2007-
was once
rife
with profit
opportunities for financial institutions (banks, for short), was perceived to be exceedingly
Although the subprime shock was small
complex.
capital,
banks acted as
if
most of
relative to the financial institutions'
their counterparties
were severely exposed to the shock.
Confusion and uncertainty followed, triggering the worst case of flight-to-quality that we
have seen
in the U.S. since the
In this paper
spread panic
we
Great Depression.
present a model of the sudden rise in complexity, followed by wideIn the model,
in the financial sector.
banks normally
collect
information
about their direct trading partners which serves to assure them of the soundness of these
However, when acute financial distress emerges
relationships.
network,
it is
in parts of the financial
not enough to be informed about these partners, but
it
also
tant for the banks to learn about the health of their trading partners.
becomes impor-
And
as conditions
continue to deteriorate, banks must learn about the health of the trading partners of the
trading partners, of their trading partners, and so on. At some point, the cost of information gathering
becomes too
large
and banks, now facing enormous uncertainty, have no
option but to withdraw from loan commitments and illiquid positions.
A
flight-to-quality
ensues, and the financial crisis spreads.
The
starting point of our
framework
resenting financial institutions
against local liquidit}' shocks.
is
a standard liquidity model where banks (rep-
more broadly) have
The whole
financial
bilateral linkages in order to insure
system
ages which functions smoothly in the environments that
though no bank knows with certainty
work
(that
all
the
many
is
it
a complex network of linkis
designed to handle, even
possible connections within the net-
each bank knows the identities of the other banks but not their exposures).
is,
However, these linkages may also be the source of contagion when an unexpected event
of financial distress arises
literature
is
that
we use
somewhere
in the network.
this contagion
as the source of confusion
and financial
mechanism not
panic.
Our
point of departure with the
as the
main object
of study but
During normal times, banks only need to
understand the financial health of their neighbors, which they can learn at low
contrast,
when a
significant
problem
arises in parts of the
too complex
may
for
In
network and the possibility of
cascades arises, the number of nodes to be audited by each bank
that the shock
cost.
rises since it is possible
spread to the bank's counterparties. Eventually the problem becomes
them
to fully figure out,
uncertainty and they react to
it b}'
which means that banks now face significant
retrenching into liquidity-conservation mode.
This paper
related to several strands of literature. There
is
is
an extensive literature
that highlights the possibilitjf of network failures and contagion in financial markets.
An
incomplete
and Tirole
list
includes Allen and Gale (2000), Lagunoff and Schreft (2000), Rochet
and Rochet (2000), Leitner (2005), Eisenberg and Noe
(1996), Freixas, Parigi
(2001), Cifuentes, Ferucci
and Shin (2005)
survey). These papers focus mainly on the
shocks
may
and Babus (2008)
for
a recent
mechanisms by which solvency and
liquidity
(see Allen
cascade through the financial network. In contrast, we take these
phenomena
as the reason for the rise in the complexity of the environment in which banks
and focus on the
their decisions,
In this sense, our paper
is
effect of this
make
complexity on banks' prudential actions.
and Knightian
related to the literature on flight-to-quality
uncertainty in financial markets, as in Caballero and Krishnamurthy (2008), Routledge
and Zin (2004) and Easley and O'Hara (2005); and
investigates the effect of
and Wang
network
is
from the behavior of the financial
in endogenizing the rise in uncertainty
More
itself.
events and innovations in financial markets, e.g. Liu, Pan,
Brock and Manski (2008) and Simsek (2009). Our contribution relative
(2005),
to this literature
new
also to the related literature that
broadly, this paper belongs to an extensive literature on flight-to-
quality and financial crises that highlights the connection between panics and a decline in
the financial system's ability to channel resources to the real economy
and Kurlat
We
(see, e.g.,
Caballero
(2008), for a survey).
build our argument in several steps.
In Section 2
we
describe the normal envi-
ronment, characterize the financial network, and describe a rare event as a perturbation
to the structure of banks' shocks.
shock
for
which
it
Specifically,
was unprepared. In Section
one bank
3,
suffers
we show
that
an unfamiliar liquidity
if
banks can costlessly
gather information about the network structure, the spreading of this shock into precautionary responses by other banks
uncertainty
is
is
typically contained.
This scenario with no network
the benchmark for our main results.
In Section 4
we make information gathering
small, either because the liquidity shock
is
costlv-
In this context,
and we are back to the
is
limited or because banks' buffers are significant,
banks are able to gather the information they need about
liquidity shock
the cascade
if
full
their indirect exposure to the
information results of Section
3.
However,
once cascades are large enough, banks are unable to collect the information they need
to rule out a severe indirect hit.
their lending,
Their response to this uncertainty
which triggers a credit crunch. In Section
5
is
to retrench
we show that under
on
certain
conditions, the response in Section 4 can be so extreme, that the entire financial system
can collapse as a result of the
response
is
more
fiight to quality.
likely to take place in a
Somewhat
paradoxically, this extreme
developed financial market than
in
one with
limited precautionary options for banks.
The paper concludes with a
final
remarks section
and several appendices.
The Environment
2
we introduce
In this section
We
work.
the environment and the characteristics of the financial net-
describe the normal scenario in which the financial network facilitates
first
We
liquidity insurance.
which was unanticipated
then introduce a perturbation to this environment:
network formation stage
at the
A
shock
the financial network was
(i.e.
not designed to deal with this shock).
The Normal Environment
2.1
There are four dates
{
— 1,0,1,2}. There
is
a single good (one dollar) that serves as
numeraire, which can be kept in liquid reserves or
If
can be loaned to production firms.
it
kept in liquid reserves, a unit of the good yields one unit in the next date. Instead,
unit
is
loaned to firms at date -1,
The
or unloaded before this date.
and become
lose this option
At date
to the lender.
(e.g.
by settling
it
1,
it
bank
,
which
is
illiquid:
One
«
hit
/]
>
if
1
identical
of banks denoted by {b^}
banks and,
our unit of analysis.'
the depositor
is
by a liquidity shock. Let
bank
fr'
(i.e.
for simplicity,
we
but at date
yields
they
1
one unit
1
units.
To simplify
u)^
G
[0, 1]
'l-^-
Each of these continu-
refer to
each continuum
At the beginning of date —1, each bank
I
—
6^
b'
as
has
y units of loans, and liabilities
by a liquidity shock and
hit
/o
A demand
>
/i
if
deposit contract
the depositor
is
not
be the measure of liquidity-driven depositors of
the size of the liquidity shock experienced by the bank), which takes one of
the three values in {uj.lol^ujh} with loh
y
so that the
not recalled
throughout this paper.
which consist of a measure one of demand deposit contracts.
pays
if it is
unit of loan recalled at date
which consist of y units of liquid reserves and
assets
units at date 2
with the borrower at a discount) and receive r <
composed of
b^
1
loans have a recall option at date
The economy has 2n continuums
is
R>
a
the loan cannot be recalled, but the lender can unload the loan
the notation, we assume r
ums
then yields
if
bank has
'The only reason
=
for the
and
liuj
assets just
> ^l and
enough to pay
continuum
is
for
[1
/j
Gj
=
[ujh
+ tj^,)
—
y)
R=
I2CJ
/2,
and suppose
(resp. I2) to early (resp. late) depositors
banks to take other banks' decisions as given.
—
if
the size of the shock
is
u.
However, the hquidity needs
at date 1
may
There are three aggregate states of the world, denoted by
date
In state s (0)
0.
states s (r)
and
all
banks expect to receive at date
states are heterogeneous across banks.
More
and
s (0), s (r)
and the
(resp. green type)
specifically,
the banks in this
s (g) there is
enough aggregate
The
economy
(r) (resp.
s (g)),
are
the
expect to receive a high liquidity shock, Uf^, and
the other banks expect to receive a low liquidity shock, Ui. This
and
revealed at
liquidity shocks in these
divided half and half between two types: red and green. In state s
banks with red type
s {g),
the same liquidity shock O.
1
are realized with equal probability
s (g)
among banks.
not be evenly distributed
liquidity but there
means that
in states s (r)
a need to transfer liquidity across
is
banks, which highlights one of the (many) reasons for an interlinked financial network.
Given the financial network, banks make arrangements to transfer
s (r)
let
and
s {g)
G
{!,..,
i
liquidity in states
through bilateral demand deposit contracts signed at date —1. In particular,
2n} denote
a financial network and consider a permutation p
slots in
{1,..,277} -^ {l,...2n} that assigns
bank
6''^''
to slot
We
i.
:
consider a financial network
denoted by:
b
—
where the arc
>
(p)
=
^
(6^(^)
fe'^f^)
i
+
I
.
.
.
where we use modulo 2n arithmetic
forward neighbor of bank
bP^'+^^
We
6^^'^
i
bank
=
-
(w
^
_
(i.e.,
2
•
bank
^p(3)
denotes that the bank in slot
the bank in the subsequent slot
.
^
(i.e.,
bank
6''''''"^'
)
wl)
for the slot
f,p(2n)
_ ^.(D)
6''*'')
has a
.(J)
demand deposit
equal to
(2)
.
,
index
i.'
(and similarly, to bank
in
We
6''^*'
refer to
6''('+i^
bank
as the
as the backward neighbor of
).
say that the financial network
is
consistent
if all
odd
slots (resp.
all
even
slots)
contain banks of the same type, which means that red and green type banks alternate
around the financial
circle.
to consistent ones (as
For analytical simplicity, we restrict the set of feasible networks
opposed
to, for
be arbitrarily ordered around the
example, any circular network
circle), since
in
which banks may
these networks ensure that each
needs liquidity has deposits on a bank with excess
liquidity, facilitating bilateral liquidity
insurance (see below) with the minimally required level of cross-deposits
Next we introduce three features
^In particular,
integer
i.
i
—network uncertainty,
represents the slot with index
For example,
i
=
2n
-\-
1
i'
£ {l,..,2n} that
represents the slot with index
4
bank that
1.
~.
auditing, and loan recall
is
the
modulo
2?!
equivalent of
that play no role in the normal environment but that will prove central during a rare
event (defined as a perturbation to the normal environment).
Network uncertainty
Banks'
tj-pes
and the
network are realized at date —1 as follows: First the types
financial
of banks are realized at
random
banks become red type and the other half
(half of the
green type); then a particular consistent financial network h
types)
is
realized.
We
(with respect to these
[p]
define
B={h{p)
I
p
:
{1,
2n} -^
...
{1,
..,
2n}
is
a permutation}
,
as the set of consistent financial networks from an ex-ante point of view
types of the banks are realized) and
Assumption
its slot
and
2
=
?
+
and a consistent
are realized
p~^
(j)
and the
financial
over
B
with
support.
full
network forms, each bank
identities (and types) of the
banks
p{t-l)^p{t-l)
p{i) ^ p{i)
p{i + l) = p{i + l)
B^p) = {h{p)eB
Note
/-' (.)
in its
that
(^),rf{p(j_i)
the
p(i)
bank
p{j+i)}i
slots (see Figure 1).
i^or
The
does
b^
does
it
know
know how
the
all
types
t
=
the
of
observes
p
^
z
—
1
set:
{j)
remaining
banks
these banks are assigned to the remaining
means, and this
latter
these banks are connected to
not
where
IP
neighboring slots
This information narrows down the potential networks to the
1.
before the
we suppose:
(FS). Each bank has a prior belief
Once the types
(i.e.
is critical,
that banks do not
know how
other banks in the network.
Auditing Technology
Each bank
tP
can acquire more information about the financial network through an au-
diting technology.
state in {s [0)
audit
its
neighbor
,
At the beginning of date
s {r) ,s [g)},
forward neighbor
?/'('+2)
.
a bank
f^^'^^'^
LP
in slot
and
i
its
a-'
(i.e.
with
j
—
p
{i))
can exert
in order to learn the identity of this bank's
Continuing this way, a bank
sheets learns the identity of
after the realization of the aggregate
+
I
b^
that audits a number, a^
,
efi"ort
to
forward
of balance
forward neighbors and narrows the set of potential
Figure
1:
The
network and uncertainty. The
financial
actual financial network.
Each
circle
this realization of the network, each slot
contains bank
i
boxes show the other networks that bank
(i.e.
the set B^ (p)).
Bank
they are ordered in slots
i
b^
£
bottom-left box displays the
corresponds to a slot in the financial network, and in
cannot
{3, 4, 5}.
tell
b^ finds
b^
(i.e.
p
(?')
=
i).
The remaining
plausible after observing
its
the types of banks b^,b'^,b^, nor can
neighbors
it tell
how
financial networks to:
~p{i-l)=p{i-l)
BUp
a')
eB
= {h{p)
I
p{i
We
B^
+
a^
denote the posterior behefs of bank
(/9,
a^) gi\'en
narrow down the
in Figure
^fc.'-
+
tP
=
1)
p{i
with f^
+
+
a^
it
p
(j)
p,a^) which has support equal to
{.
\
would learn that bank
6^ is
networks to the two boxes at the
set of
=
i
1)
assumption (FS). In the example illustrated
one balance sheet, then
where
,
bank
b^
audits
assigned to slot 3 and
it
would
in Figure
hand
left
1, if
side of the
bottom row
1.
Loan Recalls
After learning about the aggregate state and
narrowng the
has updated information about
each bank
tP
rearrange
its
its liquidity
portfolio by recalling a portion of
its
financial
needs at date
loans yg €
units in liquid reserves. After this portfolio reallocation, each
invested in liquid reserves and
1
—
y
—
in loans.
yj^
any loans, while
if
=
yo
1
—
y,
[0, yo]
the banks can recall
all
If yo
=
0,
\
a^),
The bank can
1.
and by keeping these
bank has y + y^Q 6
The parameter
the flexibility of the banks in portfolio rearrangement.
network ioB^ [p
yo E
—
[0, 1
[y,
y
+
yo]
captures
y]
the banks cannot recall
made
of the loans they
at date
—1.
Bank Preferences
Consider a bank
by
q'-y
and
q^
and denote the bank's actual payments to early and
IP
(which
may
in principle
be
different
late depositors
than the contracted values
/i
and
/2).
Because banks are infinitesimal, they make decisions taking the payments of the other
banks as given. The bank makes the audit and loan
and
yo
€
[0, yo],
at date
0.
At date
1,
outstanding loans.
its liquidity
no benefit
The bank makes
bank, thus once
maximize the return to the
We
it
G
{0,
1.,
z^
€
[0,;:],
and
it
may
is,
until q\
—
li.
satisfies its liquidity
..,
its
also unload
these decisions to maximize g^ until
obligations to depositors, that
for the
a-'
the bank chooses to withdraw some of
on the neighbor bank, which we denote by
its
recall decisions,
Increasing
obhgations,
q[
it
it
2n — 3}
deposits
some
of
can meet
beyond
then
/i
has
tries to
late depositors g^.
capture this behavior with the followng objective function
V (1
{q>,
<h]q>, + l{cf,> h]ci) - d
[a^]
,
(3)
where v
R+ —
:
IR++
a strictly concave and strictly increasing function and d{.)
is
and convex function which captures the bank's non-monetary
increasing
auditing.
>
When
the bank
(which
for (g^, g^)
will
expression in (3) given
be the case
is
able to pay
posterior beliefs f^
(.
|
With
it
/i
at date 2
bank
date
1
takes one of two forms. Either there
which the bank
is
solvent
"^
insolvent, unloads
all
2;
all
late
a no-liquidation equilibrium in
(4)
or there
is
a liquidation equilibrium in which
outstanding loans, and pa3's
qi<luqi =
while
is
= h,qi>lu
while the late depositors withdraw at date
is
can refuse to pay the
and pays
gl
the bank
and a bank
this assumption, the continuation equilibrium for
if
early.
maximizes the expectation of the
p, a^).
depositors
IP at
from
disutility
liquidity shocks are observable,
depositors at least
its late
they arrive
then
in Section 4),
Suppose that the depositors' early/late
which
an
making a decision that would lead to an uncertain outcome
b^ is
its
is
,',/,
0,
(5)
,
depositors (including the late depositors) draw their deposits at date
1.
This completes the description of the normal environment with an uncertain financial
network. Figure 2 recaps the timeline of events in this economy.
do
in the
Appendix, that
the contracted values,
In states s (r)
and
qi
s [g],
in
=
li
and
q2
—
do not withdraw
in their
in
is
solvent
and pays
each state of the world
s (0)
,
its
s (r)
depositors
and
s (g).
demands by
forward neighbor banks, and the banks with excess
their deposits in order to receive higher returns at date 2.
Moreover, the banks do not audit or
facilitates liquidity
I2,
b^
can be checked, as we
the banks in need of liquidity meet their liquidity
withdrawing their deposits
liquidity
equilibrium each bank
It
recall
any loans at date
0.
The
network
financial
insurance and enables liquidity to flow across banks even though the
banks are uncertain about the network structure. In the next sections we show how things
change dramatically
in the
presence of a perturbation to this environment, especially
when
banks face uncertainty about the financial network.
•'Without this assumption, there could be multiple equilibria for late depositors' early/late withdrawal
decisions.
In cases with multiple equilibria, this assumption selects the equilibrium in which
depositor withdraws.
,
,
.,,
,
no
late
)
Date
Dnte-l
Liabilitiec;
Banlc:'
• y liquid
balance
1
—
y
loDJi:;.
tiio.t
paA'
i
Banlc malie the depodt
Measure one of
demand depodtc
receirec,
1
^'
tj'pec aje
(haJf of
bankc become red
realbed at
random
demand
the banls cannot promise
their depozitz.
condstent financial
net-u'orli
h{p]
unload
iz rea.li::ed.
realized
and only
if
aJl of
(
that pay q\
<
/j
their outstanding, loans.
Date 2
Do.te
NORMAL
if
qi i: /].
green)
t}-pe, half
Inooh-ent bankc
t.
decidon
Late depodtorc; demand their depodt:;
Eanf
A
withdravi-al
10, .],
Early depodtors
or ^,
/]
e
ENV:
and
State in {r{0),:{T),c{g)
obser-i-ed
PKKi'URBKD
by
EllY: State
\
banl;c.
i-'(O)
h
realized.
Banlcc pay ci to late depocitorc.
Banlc; malie the audit decidon a' 6 {0,1, ..,2n
Banlc maJie the loan
recall
decidon
j/^
6
[0,!/o]
Figure
2.2
A
Timeline of events.
2:
Rare Event
Henceforth we consider the equihbrium following an unanticipated change in the structure
of shocks:
At date
the banks learn that the actual state
except for the fact that one bank,
IP
is s^ (0),
becomes distressed and
,
which
loses
perturbed environment.
assets. Figure 2 describes the timeline in the
6'
is
<
We
just like s (0)
y of
liquid
its
formally define
the equilibrium in this econon\y as follows.
The
Definition
1.
drawal, and
payment decisions
equilibrium^ is a collection of bank auditing, loan recall, deposit with-
{a^ [p)
,
yl [p)
,
z^ [p)
each consistent realization of the financial network
b
[p) at date
the unanticipated aggregate state s^ (0) at date 0, each bank IP
in (3) given
its
prior belief f^
(.)
of their outstanding loans at date
only
if q2 (p)
As we
< h
(cf-
the liquidity
demand
and
the late depositors
for
that,
the realization of
maximizes expected
gj [p)
<
li)
utility
unload
withdraw deposits early
if
all
and
Eqs. (4) and (b)).
will see in the
useful in this scenario.
—1 and
over B, the insolvent banks (with
1
such
,q[[p), qi {p)\ \
subsequent sections, the loan
The
distressed
at date
1.
bank
tP
This bank
recall
and auditing options become
does not have enough
tries
licjuid
meet
but cannot obtain liquidity from cross-
deposits (since the other banks do not have excess liquidity either), thus
9
reserves to
it
benefits from
.
some
recalling
loans,
the
it will
of
pay
its
crisis will
less
loans at date
than
li
to
the distressed bank
If
0.
insolvent despite recalling
backward neighbor bank, and
depositors, including its
its
is
spread in this fashion to the other banks in the network. Anticipating
banks {V}j^j may also want to
some
recall
this,
These banks may acquire additional
loans.
information about the financial network to make more accurate loan recall decisions. Note
that each bank
know
IP
^
knows that the bank
b^
IP is
distressed, but
the slot
of the distressed bank. This
know how far removed
More
specifically,
exists a unique
6
/c
it is
key, since
is
from
which we define
it
means that a bank V ^ V does not
note that for each financial network b
{0,
..,
2n —
=
distressed
bank
The banks
b' will
bank
tP
from the distressed bank. As we
will
6-'
determine whether or not the
cascade to bank
5p('+i)^
fc''(*-i), 6^(0^
—
W
network and,
in particular,
crisis
We
3
We
/c
>
a-'
+
of
that originates at the
respectively with distances 1,0 and 2n
£
{2, ..,2n
—
^•
—
know
1,
their
2}) do not have this
G
{2,
..,
—
2n
2} (they
Note,
can use the auditing technology to learn about the financial
about
(wdth distance k) that audits
learns that
there
will see, the
by observing their forward and backwards neighbors).
1}
however, that the bank
,
how many
that decides
information a priori and they assign a positive probability to each
6 {l,2n
b^
V
distances, but the remaining banks (with distances k
rule out k
each bank
for
p(j,-k),
distance k will be the payoff relevant information for a bank
it
and
(p)
1} such that
as the distance of
the loans to recall since
necessarily
the distressed bank.
j
it
does not necessarily
it
a^
distance from the distressed bank.
its
>
1
banks either learns
its
distance k
[if
A
bank
k
<
a^
b'^^'"'^'^
+
1) or
2.
next turn to the characterization of equilibrium in the perturbed environment.
Free-Information Benchmark
first
auditing
study a benchmark case
a''^'"^^
= 2n —
3.
in
which auditing
is
free so
each bank
b''^^"^^
chooses
In this context banks learn the whole financial network
and, in particular, their distances
b
full
(p)
k.
All banks receive a liquidity shock,
cj,
and have
10.
liquid reserves equal to y
=
w/j, except
for
bank
=
IP
withdraws
its
which has hquid reserves y
b^^'''
—
At date
9.
1,
the distressed bank
As we show
deposits from the forward neighbor bank.
That
is
,
,
~j
bank
t'^''^
The bank
drawals.
_
Vie{l,.„2n}.
but cannot, obtain
tries,
also cannot obtain
obtain additional liquidity at date
some
of
loans at date
its
1,
w
0.
must have
1
an}' net liquidity
Anticipating that
the distressed bank
through cross with-
b''^'^
it
tries to
will
at
date
not be able to
obtain liquidity by
/],
a bank with no liquid reserves
The
deplete
all
of
level y^
is
its liquidity at
most
recall at
Combining the two
(T)
a natural limit on a bank's loan recalls (which plans to
date
any choice above
1) since
necessarily insolvent. If the actual limit on loan recalls
bank can
y^ loans while
yo'is
remaining solvent; or
cases, a bank's buffer
this
greater than
else
beyond
It
/3.
it
can
i/g,
recall
then the
j/o
loans.
•
A bank can accommodate losses in liquid reserves up to the buffer
losses are
would make the bank
given by
is
P = min{yo,yo}
when
left at
at least
l_,_5„"=ii^
units of loans.
1,
0.
In order to promise late depositors at least
the end of date
(6)
,
any liquidity by unloading the loans
since each unit of unloaded loan yields r
recalling
withdrawn.
;
,
In particular,
Appendix,
in the
this triggers further withdrawals until, in equilibrium, all cross deposits are
b^^'^
/?,
foUows that the distressed bank
but becomes insolvent
6^*''
will
be insolvent
whene\'er
e
that
is,
whenever
the case so bank
of
its
loans as
at date
early
1.
it
its
6'''''
can
>
(8)
/?,
losses in liquid reserves are greater
is
j/q
than
its buffer.
insolvent. Anticipating insolvenc}', this
=
Since the bank
yo (since
is
it
insolvent,
maximizes
all
q^
)
bank
and unloads
Suppose
will recall as
all
this
is
many
remaining loans
depositors (including late depositors) arrive
and the bank pays
11
.
where
to
h
recall that
bank
if
denotes bank
qf
bP^'+'^'^
Since bank
Partial Cascades.
5p(''-i)'s
to
We
K
6''^*~^\
bank
E
its
continuing
..,
crisis
banks have
Under
We
failed.
^p(0
g^j^i^
i^
5/'('-i)
^4th distance
pay
on
its
distressed
also go
—
cascades to bank
refer to
6''^'~-'^
severe enough, maj' cause
6^''~^', it
may
will
bank
then similarly cascade
)
>
if
is
1
gq^
=
.
will recall as
From
/c
many
1) is
fe''('+^\
^^^
j^g-j
—
which has a distance 2n
K
< 2n —
2
1, is
i)^('^~2)
but
—
if its
li
on
this
bank needs to
deposits from the
its
Hence, bank
in cross-deposits.
)
q^
losses
<
receives only q^
it
solvent. Therefore
Consider now the bank
calculated explicitl}^
]-)g
fo'^^'^i)
from cross-deposits are greater than
qt'^Kh-^.
1.
then the only insolvent bank
If this
loans as
it
:
is
its
will
buffer,
can,
i.e.
=
j/q
yo
(10)
:
the original distressed bank and the
condition holds, then bank
and
it
6^('~^)
anticipates insolvency,
pay
will
-
given by
it
depositors
all
.
(11)
,
onwards, a pattern emerges. The payment by an insolvent bank
b^'^^~'^^
'
'
'
'
p(r-fe)
this bank's
remain solvent. In other words, the
\,r'>=f(^f')- "^';^f"
this point
>
K
from the distressed bank. To remain solvent,
and only
fails,
K
>
are insolvent (there are
I
which can be rewTitten as
01
condition
K—
as the cascade size.
6^^'\ so it loses z (li
bankrupt
<f\
jj-^
K
bank
deposits to bank
bank
cascade size
and
if
banks with distance k <
all
..
(with
backward neighbor bank
cascade through the network but wiU be contained after
this conjecture,
_
^p('+i)
,
insolvent, its
while the banks with distance k
crisis will partially
If this
equal
cascade through the network in this fashion.
its
2n - 2} such that
K such banks)
z ill
is
conjecture that, under appropriate parametric conditions, there exists a threshold
{1,
^1
6^^'^ is
cross deposit holdings, which,
Once the
insolvency.
to early depositors (which
solvent).
is
experience losses in
payment
fe'^^'^-^^'s
backward neighbor
f (
C^'+i))
fe''*'
.
-
p(r-(fc-i))'
js
also insolvent
if
and only
if
7^
'
<
Zi
—
7
Hence, the payments of the insolvent banks converge to the fixed point of the function
12-
/
+
given by y
(.)
yo,
and
if^
+ yo>li--,
y
(12)
then (under Eq. (10)) there exists a unique A' > 2 such that
<h-'^
qf'""^
and gf-(^'-i))
If
2n —
2
K,
to this equation,
2n then, Eq.
l)P{'-k)
(13)
shows that
E
Y^ith distance k
{I,
from
K
—
deposits by recalling loans while
Since bank
/i).
6^('-^')
still
is
bank
In contrast,
solvent, since
it
banks
b''^'-''^
banks
(that receives
f^'"^^'
can meet
all
b''^^^)
from cross deposits
from cross
its losses
promising the late depositors at least
solvent, all
is
(14)
1} are insolvent since their losses
forward neighbor)
its
if
i.e.
> K,
2
are greater than their corresponding buffers.
Qi
(13)
addition to the trigger-distressed bank
(in
..,
-2}
{0,..J<
>l,-^.
greater than the solution,
is
ioreachke
{i.e.
li
with distance k 6 {A"
+
1,
>
q!^
-
..,2n
1}
are also solvent since they do not incur losses in cross-deposits. Hence these banks do not
recall
=
any loans y^
and pay
=
'
(qj
=
h.q^
'2), verifying
our conjecture
a partial cascade of size A' under conditions (12) and (14).
for
Since our goal
we take the
is
to study the role of network uncertainty in generating a credit crunch,
partial cascades as the
The next
benchmark.
above discussion and also characterizes the aggregate
as a
benchmark
Proposition
Suppose the financial network
conditions (8), (12) and (14) hold. Let
Then, the banks' equilibrium payments
from
spect to their distance k
K
< 2n —
2
where
K
the distressed bank k
(with distance k
and unload
all
>
K
and
it
is
<
)
i
—
condition (12)
which we use
—
fails,
Bank
(13).
fc''''"'^'
is
and
Banks
Banks
{6''*'"''"'},
a
1,
(weakly) increasing with re-
'^''^
)
there
is
a partial cascade of size
{t''*'~''''}^^_o
I) are insolvent while the
=
full cascade,
i.e.
13
all
'"
distance from.
(f'''^'"'''}?
~
do not recall
{^^''"''"'j^.^,.
=
)
)
;
f
/j
—
'~
~
g^
always remains below
banks are insolvent.
,,
can
recall all of the loans they
_„
while banks
f [Qi
(luith
remaining banks
recalls a level of loans y^
then the sequence iql
can be checked that there
,(?2
the distressed bank,
are solvent.
and
realized as h{p), auditing is free,
'"
"
((?i
defined by Eq.
K
is
p"^ (j) denote the slot of the distressed bank.
of the remaining loans at date
or unload any loans.
''if
level of recalled loans,
subsequent sections.
in
1.
proposition summarizes the
li
—
)
^,
'
5
4
So =0.123
3
J
2
.
-
1
1
"O
04
03
0.2
0.1
05
e
0.8
1*1=0.116
0.6
r
So=0.123
..—
fe,04
_
i
0.2
__....-.
02
01
)
05
0.4
0.3
e
Figure
The
3:
K
free-information benchmark. The top figure plots tlie cascade size
bank 6, for different levels of the flexibility
The bottom figure plots the aggregate level of loan recalls, J-, for the same
as a function of the losses in the originating
parameter
yo-
set of {yo}-
which
just
is
enough
to m^eet its losses
This bank pays
loans).
'
Q'l'
—
sequence
'^'
h,Q2^'
I
=
g^
=
(
h]-
=
gj
from
'^
li,q2
The payments of
p(l-(k-\))
/ Ui
cross deposits (and does not unload
while
h
all
any
other solvent banks pay
the insolvent banks are determined by the
A'-l
where the
Pil)
initial value q^
solved
is
from Eq
fc=0
p(r+i)
(9)
after substituting q^
The aggregate
h-
level of recalled loans is:
'
yo
Discussion.
bank
loans,
tP
=
T,
Proposition
6''(*~'^)
is
1
+
shows that the loan
only depends on
its
distance
k,
recall decisions
and the payments of a
and that the aggregate
roughly linear in the size of the cascade
for an)' given level of yo-
'15)
yo
K
(and
Figure 3 demonstrates this result
is
level of recalled
roughly continuous in 6)
for particular
parameterization
of the model.
The top panel
originating
bank
the cascade size
yo- Intuitivel)',
of the figure plots the cascade size
K
as a function of the losses in the
6 for different levels of the flexibility parameter yo- This plot
is
shows that
increasing in the level of losses 6 and decreasing in the level of flexibility
with a higher 9 and a lower
flexibility
14
parameter
yo,
there are
more
losses
.
.
to be contained
and the banks have
less
emergency reserves to counter these
losses, thus
increasing the spread of insolvency.
The bottom panel
plots the aggregate level of loan recalls J^, which
the severity of the credit crunch, as a function of
with 9 and
falls
with
yo-
This
an intuitive
is
9.
This plot shows that
result:
is
a measure of
T also increases
In the free- information
benchmark
only the insolvent banks (and one transition bank) recall loans, thus the more banks are
insolvent
(i.e.
T increases
These
the greater A') the
"smoothly" with
results offer a
Note also that
benchmark
K
and
There we show that once
for the next sections.
JT
may be non-monotonic
in yo
and can jump with
6.
Endogenous Complexity and the Credit Crunch
4
We
have now
realistic
laid out the
foundation for our main result.
assumption that auditing
can arise
is
are recalled in the aggregate.
9.
auditing becomes costly, both
small increases \n
more loans
in
is
costly
In this section
and demonstrate that a massive
we add
credit
the
crunch
response to an endogenous increase in complexity once a bank in the network
sufficiently distressed. In other words,
to figure out their indirect exposure.
and they eventually respond by
when
K
is
large,
it
becomes too costly
for
banks
This means that their perceived uncertainty
recalling their loans as a precautionary
measure
rises
(i.e.,
!F
spikes)
Note
rium
that, unlike in Section
for a particular financial
of the financial network
financial networks
b
(p)
is
3,
we cannot simphfy the
network b
analysis by solving the equilib-
[p) in isolation, since,
b(p), each bank also assigns a positive probability to other
G B.
As
such, for a consistent analysis
equilibrium for any realization of the financial network h{p)
Solving this problem in
even when the realization
full
generality
is
(E
B
we must
(cf.
describe the
Definition
1).
cumbersome but we make assumptions on the
form of the adjustment cost function, the banks' objective function, and on the
we
of the loan-recall limit, that help simplify the exposition. First,
flexibility
consider a convex and
increasing cost function d[.) that satisfies
(i(l)
=
and
d\2)
> v{h +
This means that banks can audit one balance sheet
12)
for free
-
v{{))
but
it is
(16)
very costly to audit
the second balance sheet. In particular, given the bank's preferences in
(3),
the bank will
never choose to audit the second balance sheet and thus each bank audits exactly one
15
balance sheet,
{a^ (p)
= !}
.
Given these audit decisions and the actual financial
•'ih(p)sB
L
network b(p), a bank
b^
has a posterior belief f^
with support B^
(-l/O;!)
(/5,1),
the set of financial networks in which the bank j knows the identities of
and
second forward neighbor. In particular, the bank
its
the distressed bank
6''^''
information from the outset of date
We
0).
distance from
learns
6'"^'"'"^)
which already have
its
is
neighbors
bP^'-~'^'>
6''^'~-'^ 6^''',
banks
(in addition to
its
which
this
denote the set of banks that know the slot of
the distressed bank (and thus their distance from this bank) by
Qknow
On
is
/
the other hand, each bank
at least 3
k G {3,
..,
2n —
>
k
(i.e.
We
2}.
3),
_
\
np{r-2) ^p(r-i)
^p{l)
denote the
/
\
set of
1) / (1
'
—
2} learns that
_
f ^p{T-3)
min
distance
to all distances
(0, 1)
j^,3(i-(2n-2)) \
f^/3(r-4)
in
{.)
with a —^ DO, so that the bank's objective
cr)
its
banks that are uncertain about their distance by
Second, we assume that the preference function v
—
—
with k € {3,..,2n
b^^'"'^^
but otherwise assigns a probability in
^unceTtain
(x^~°'
j^p(r+i)i
{l{qi{~p)<h}cf,{p)
(3)
is
Leontieff v (x)
=
is:
+ l{q\{p)>h}qi{p))-d{a^p)).
(17)
b(p)eej(p,i)
This means that banks evaluate their decisions according to the worst possible network
which they
realization, b{p),
The
third
and
last
find plausible.
assumption
is
that
'
yo<yoThat
is,
the actual limit on loan recalls
(which also implies that the buffer
is
in the continuation equilibrium at date
is
below the natural limit defined
given by
1,
(18)
V
/3
=
yo).
Eq.
(7)
This condition ensures that,
the banks that have enough liquidity are also
solvent (since they have enough loans to pay the late depositors at least
drop
in
/]
at date 2).
We
this condition in the next section.
We
next turn to the characterization of the equilibrium under these simplifying as-
sumptions. The banks make their loan recall decision at date
decision at date
realized).
1
At date
under uncertainty (before their date
1
the distressed bank
neighbor which leads to the withdrawal of
6^''^
all
16
1
withdraws
and deposit withdrawal
losses
its
from cross-deposits are
deposits from the forward
cross deposits (see Eq. (6)
and the Appen-
dLx) as in the free-information
benchmark. Thus,
to obtain additional liquidity at date
1 is
any distressed bank,
for
through ex-ante (date
only
tlie
way
which we
0) loan recalls,
characterize next.
A
bank
original distressed
make
from
its
if it
solvent), then
[p]
Qi
<
—
li
,3/z
also be insolvent), then
X
the bank
=
q[
=
0.
no loans
recalls
it
its
(i.e.
many
will
loans as
it
(p)
If it
can
0,
and
q2 [y'o.x] are
forward
—
its
bank
that this
More
Vo-
will
generally,
forward neighbor pays
(19)
characterized in Eqs. (25) and (26) in the
the bank does not necessarily
know x =
~
q1
[p)
and
it
has to
under uncertainty.
this observation along
can be simplified and
.t'"
its
That
y'^.
q-[\y'Q,x]
and
the bank's payment
is,
are
qily'^.x]
is
increasing
receives from its forward neighbor regardless of the ex-ante loan recall
it
max
(i.e.
qf^'\p)=q2[yix],
characterization in the Appendix also shows that
Using
li
qi[yi^]
qi [y'o^x]
loans
its
—
=
level of loan recalls
amount
receives in
it
will lose in cross-
qf~'\~P)
and
to
it
little
j/g
and
bank
knows with certainty that
pay so
at date
[0, yo]
(weakly) increasing in x for any given
s.t.
j/q
forward neighbor
recalls as
it
'"
then this bank's payment can be written as
Appendix. At date
decision.
amount
the
is
1,
where the functions
in the
which
/j,
date
Sit
[p)
The
sufficient statistic for this
knows with certainty that
chooses some y^ £
6''('~'^)
q^
choose the
<
(,5)
'
other than the
forward neighbor. In other words, to decide how many of
deposits. For example,
is
gj'
A
0).
'
bank only needs to know whether (and how much)
to recall, this
neighbor
is
6^^'"'^'
Consider a bank
Recalls.
suppose k >
(i.e.
the loan recall decision
equilibrium,
if
Loan
Sufficient Statistic for
its
optimization problem can be written as
(1 {q, [y^, x'"]
= min lx\x =
In words, a bank
6''''"'^'
neighbor the lowest
with Eq. (19), the bank's objective value in (17)
> h]
[y',,
x^']
qP'^''''~'>>
(with k
-possible
q,
(p)
>
,
+
1 {q,
[y^,
h(p) E B'
x^]
>
l,} q, [y'„
it
will receive
from
its
forward
payment x^.
Next we define two equilibrium
location notions that are useful for further characterization. First,
is
(20)
,
(/9,1)
0) recalls loans as if
Distance Based and Monotonic Equilibrium.
librium allocation
x^])
distance based
if
the bank's equilibrium
17
we say that the
al-
equi-
payment can be written only
as a function of its distance k from the distressed bank, that
functions Qi, (^2
{0,
:
..,
2n - 1}
R
-^-
is,
if
payment
there exists
such that
p{i-k)
ip),qf''Hp))-{Qi{k]
is
in
monotonic
—
2n
...
Second, we say that a distance based equilibrium
1}.
payment functions Qi
the
if
k e {0,
[k]
,
Q2
[k]
are (weakly) increasing in k. In words,
a distance based and monotonic equilibrium, the banks that are further away from the
bank
distressed
We
verif}'
X
B and
h{p) E
for all
=
yield (weakly) higher payments.
next conjecture that the equilibrium
below). Then, a bank
= Qi[k —
(p)
Qi
—
6''*'~''"''s
1]
distance based and monotonic (which
is
uncertainty about the forward neighbor's payment
reduces to
uncertainty about the forward neighbor's
its
its
own
can further be simplified by substituting gj
Qknow ^-^^ ^^^j, /^ >
since a bank b''^''~'^^ S
g) knows
^' ~
distance k
which
1,
equal to one less than
is
in (20)
with
= Qi
x"'
On
[/c-
distances k G {3, ...,2n
—
neighbor's payment Qi
We
^^) g^j^gg
are
now
distance
=
{p)
Qi
distance
its
Hence, the problem
k.
it
/c,
—
[k
In particular,
1].
solves
problem
(20)
1].
&''('"''=)
the other hand, a bank
^uncertain
we
/c
problem
^uncertain
assigns a positive probabihty to
f^p-^
Moreover, since the equilibrium
2}.
—
g
1
minimal
is
(20)
with
.t'"
in a position to state the
for the distance
=
Qi
main
is
=
/c
monotonic,
3,
its
all
forward
hence a bank
6^^''
G
[2].
result of this section,
which shows that
banks that are uncertain about their distances to the distressed bank recall loans as
if
all
they
are closer to the distressed bank than they actually are.
More
specifically, all
distance
A;
=
would
3
sufficiently large
(i.e.
benchmark would
k
>
K
banks
^""ce^*""'
(^) recall
recall in the free- information
K
recall
>
3) so that
many
many
also recall
in
the level of loans that the bank with
benchmark.
the bank with distance
of its loans,
banks
all
of their loans, even
in
/c
When
=
the cascade size
is
3 in the free-information
5""cer(am
^^^
with actual distances
though ex-post they end up not needing
liquidity.
To
state the result,
we
let
[y^jreeip)
decisions and
payments of banks
network h
<E
{p)
Proposition
B
2.
(8), (12), (14),
^^i, free (p) ^^i. free (p))
in the free-information
(characterized in Proposition
Suppose assumptions (FS),
and
(18) hold.
denote the loan recall
benchmark
for
each financial
1).
(16),
and
(17)
are satisfied
For a given financial network h{p),
let
i
=
and conditions
p~^
(j)
denote
the slot of the distressed bank.
(i)
For
the
continuation
equilibrium
(at
18
date
\):
The
equilibrium,
allocation
The cascade
size in the con-
h(p)eB
tinuation equilibrium
banks
and monotonic.
distance based
is
W(P)V2(P)},'
{^''^'~'^'}t_n
same
the
is
as in the free-information benchmark, that
insolvent while banks
^^^^
{&''*'"''"*
},_^ are solvent where
is,
at
date
K
is
defined
1,
in Eq. (13).
Each bank
(a) For the ex-ante equilibrium (at date 0):
level of loans y^ (p)
V
5""'='""""
6
=
.^^^
j/q
as
(p)
{p) recalls yl [p]
=
m
IP
the free-information
yJJ;/j {p) of
{p) recalls the
same
benchmark, while each bank
which
loans,
its
G B'^"°^
what bank
is
would
bP^''^^
choose in the free-information benchmark.
For
size
the aggregate level of recalled loans, there are three cases depending on the cascade
K:
If
K
<
then the crisis in the free-inform.ation benchmark would not cascade to bank
2,
which would
b'^''-^),
no loans
recall
no loans and the aggregate
recalls
y^[^~^^
=
(/s)
Thus, each bank
0.
e
b>
5""'^^^*'''"
(p)
benchmark Eq.
level of recalled loans is equal to the
(15).
If
at
K
bank
—
then the crisis in the free-information benchmark would cascade to and stop
2),
b^'^^'^^
5""'=^'"'°'"
each bankbP G
loans
which would
,
recall
an intermediate
level of loans yQ
and
(p) recalls y^r^f.^ {p) of its loans
l^J [p) €
[0, yo]
Thus,
•
the aggregate level of recalled
is:
-^-Eyo =
3yo
+ (2n-4)<(;-t^.
(21)
j
If
would
K
>
then in the free-information benchmark bank
4,
many
recall as
recalls as
many
of
its
loans as
can y^L^J
of their loans as they can
J'
The proof
it
of this result
is
and
—
fjo-
is
€
{1, 2, 271
payment Qi
[2]
1} (that a
tP
e
^^^
is:
(22)
1
is
Among
b^^'-^^
G
5'-'"°"'
[p)
distance based and monotonic, and
(ii)
since the
with ^ >
The date
payments Qi
[k
—
as their counterparts in the free-information benchmark.
1]
for
expects to receive) and the
(that the banks in ^""=6^''"" (p) effectively expect to receive) are the
19
is
other features, the proof
the same as in the free-information benchmark.
bank
^uncertain
relegated to the appendix since most of the intuition
loan recall decisions are characterized as in part
/c
Thus, each bank
and
be insolvent
= Y.yl = {2n-l)y,.
equilibrium allocation at date
that the cascade size
would
the aggregate level of recalled loans
provided in the discussion preceding the proposition.
verifies that the
6^''"'^'
same
3»=0.116
Jm=ai23_
0,1
Figure
The
4:
0.2
0.3
04
0.5
0,2
03
0.4
0.5
The top panel
costly-audit equilibrium.
a function of the losses
in the originating
bank
plots the cacade size
K
as
for different levels of the flexibility
yQ. The bottom panel plots aggregate level of loan recalls F for the same set
The dashed lines in the bottom panel reproduce the free-information benchmark
parameter
of {yo}-
in Figure 3 for
comparison.
Discussion.
The
plots in Figure 4 are the equivalent to those in the free-information
case portrayed in Figure
The top panel
3.
the losses in the originating bank
cascade
size in this case is the
characterized in Proposition
The key
same
1,
differences are in the
The
characterized in Proposition
2,
(i.e.
for
K
K switches from below 3
bottom
if
panel, which plots the aggregate level of loan
while the dashed lines reproduce the free-information
These plots demonstrate that,
for
to above
3,
it
is
low levels of
the same as the
increases roughly continuously with
the loan recahs in the costly audit equilibrium
is,
when the
losses
K
6.
As
make
(measuring the severity of
shock) are beyond a threshold, the cascade size becomes so large that banks
are unable to
act as
benchmark
solid lines correspond to the costly audit equilibrium
a very large and discontinuous jump. That
initial
satisfy condition (18) so that the
the aggregate level of loan recalls with costly-auditing
?)),
as a function of
figures coincide.
free-information benchmark, in particular,
the
K
as the cascade size in the free-information
also plotted in Figure 3.
<
The parameters
and both
recalls j^ as a function of 6.
benchmark
9.
plots the cascade size
tell
whether they are connected to the distressed bank. All uncertain banks
they are closer to the distressed bank than they actually are, recalling
more loans than
in the free-information
episode. This
our main result.
is
benchmark and leading
20.
many
to a severe credit crunch
Note
is
also that the aggregate level of loan recalls (and the severity of the credit crunch)
not necessarily monotonic in the level of flexibility in loan recalls
when
9
=
Figure 4 shows that providing more
0.5,
That
shock
sufficiently large. Intuitively,
is
the increase in flexibility
if
to contain the financial panic (by reducing the cascade size to
flexibilit}'
backfires since
at low levels of 6,
is,
may
increase in flexibility stabilizes the system but the opposite
For example,
the banks by increasing
flexibility to
actually increases the level of aggregate loan recalls.
i/o
yo-
take place
when
an
the
not sufficient enough
is
manageable
levels), niore
enables banks to recall more loans and therefore exacerbate
it
the credit crunch.
The Collapse
5
of the Financial System
Until now, the uncertainty that arises from endogenous complexity affects the extent of
number
the credit crunch but not the
show that
if
banks have "too much"
of banks that are insolvent. A'. In this section
flexibility, in
we
the sense that condition (18) no longer
holds and
e
yo
(which also implies
=
(3
tJq),
then the
(%", 1
-
(23)
y]
uncertainty
rise in
can increase the number of
itself
insolvent banks.
The reason
profitability
is
that a large precautionary loan-recall compromises banks' long run
by swapping high return
R
low return
for
In this context, even
1.
worst outcome anticipated by a bank does not materialize,
if
bank's large precautionary reaction improves
more benign
the additional
still
payment x
=
is
the
close to
vulnerability with respect to
of banks
number
may
find themselves in the
of insolvencies as a result of
very similar to that in the previous section.
depends on
q^
choice
its
{p) at
date
qf"''^ [p]
for
rise in
In other words, a
flexibility.
analysis
payment
can be a significant
number
its
the
become insolvent
outcome when very
liquidation
its
does so at the cost of raising
scenarios. Since ex-post a large
latter situation, there
The
it
still
than K) from the distressed bank.
sufficiently close (but farther
the distressed bank but
may
it
if
some functions
qi [y'o,x]
^
and
1.
j/q
€
[0, yo]
That
gj [y^, x]
92 [y'o,^]-
at date
and
In particular, a bank's
its
forward neighbor's
is:
and
g^''"'''
[p]
=
q2 [y'o,x]
However, the characterization of the piecewise
21
.
functions
and
[vq.x]
q-[
changes a
§2 [y'o^x]
when condition
little
and
particular, these functions are identical to those in (25)
now
Section 4) but
there
new element
critical
from cross-deposits and
(26) in the
>
li
.
is
We
That
refer to scenarios
(as in
bound
the
—
[(/j
j/q
x)
This
z].
a function of the losses
is
calculated as the level of loan recalls above which the bank's
is
Ril-y- yo"
Appendix
In
-x)z].
Vo [ih
remaining loans and liquid reserves (net of losses) would not
depositors at least
not satisfied.
is
an additional insolvency region:
is
y'o
The
(18)
—
x)
+
y^ [{h
-
—
x)
is, j/q [(/i
[(/:
-
where
x) z])
>
y^
pay the
late
the solution to
z] is
[(/j
j/q
be sufficient to
x)
-
z]
{h
-
x) z
for lack of
z] as,
=
h
{l
- O)
a better jargon, scenarios
of precautionary insolvency.
The
functions
gi [yo,x]
and
remain (weakly) increasing
^2 [y'o,^]
conjecture as before that the equilibrium
all
banks (except potentially bank
part (h) of Proposition
2.
monotonic and distance based (which we verify
is
below), so the banks' loan recall decisions
still
6''^'"'"-'^)
In particular,
Moreover, we
in x.
solve problem (20).
It
can be verified that
recall the level of loans as characterized in
banks
all
€
b^
5""<^erta2n
choose the level of
^-^^
insurance the bank with distance k
—
However, part
which characterizes the equilibrium at date
(i)
of the proposition,
once yo exceeds y^.
We
•
hmited.
rium
would choose
If
K
in this case
<
is
2,
is
no additional panic
each bank
V
e
K
as described in part
(i)
b^
€
transition
5""'=^^*'^'"
bank
be seen that
y^
(p) recalls y^
=
yg /7ee
<
2,
K
=
^^^ y:ecal\s y^
of Proposition
—
is
^o
>
equal to
=
2, in
A'.
K
>
4.
0.
z],
this, first
2,
so the banks in
with a cascade
note that y^
<
changes
j/q
;
size
first
flexibility
1
equilib-
particular, there are
Similarly,
if /("
=
3,
no
each
where the inequality follows since the
—
x)
1,
In the
The date
<
y^ [{h
benchmark.
where banks'
solvent in the free-information benchmark. Since yo
of Proposition
^To see
and
6^('~^) is
_g""certain
^^^
there are no precautionary insolvencies and the equilibrium
(i)
3,
relative to the case
^^"c'^rtain
precautionary insolvencies and the cascade size
bank
in the free-information
.
divide the cases by the cascade size:
two of these cases there
is
3
equal to A'
—
^^^.g
is
—
^O'
it
can
golvent.^ It follows that
again as described in part
3.
which impHes
{R-l)yi<Ry^-yi = {l-y)R-^{l-0)h-yi,
where the equality follows from Eq.
(7).
Combining
22
this inequality with the inequality
j/q
>
(Zj
—
x)
The new
—
i/q
>
yo
scenarios arise
Vo loans,
K
when
may
6''('+i)
bank
(which
recall a smaller
To analyze
is
Q^ncertam
6
IP
^^^ recalls
in this case, all
b^
e {b^^'K
...,
amount.
That
in x.
<
[0]
banks
losses
its
6''^'"^^""^^^} recall
the only informed bank far away from the distressed bank)
— y,
1
j/q
[(/i
—
x)
decreasing in
z] is
—
(/j
x)
z,
forward neighbor, the
its
experience a precautionary insolvency. Second, note
will
it
bound
the more a bank receives from
is,
bound above which
the inequality, y^
bank
In this case, each
this case, first note that the
and thus increasing
higher the
4.
and may experience a precautionary insolvency depending on
from cross-deposits. Note also that,
yo while
>
which follows from some algebra and using
<
I2/I1
R- Then,
there are two subcases to consider depending on whether or not the flexibility parameter
is
yo
greater than y^
Subcase
upper bound
the
less of
[f^'^
...,
If
1.
y^
i/o
[(/i
(which
[0]
in the interval
is
—
amount x
.t) z]
If yo
2.
€
from
insolvent.
some
insolvency by choosing
(j/q [0]
and a bank
receives
it
ii''^'-(2"-2)) j g^j.g
Subcase
the highest value of the
is
<
(yo-J/o [0]))
its
1
then yo
y],
is
[(^1
j/q
~
^) -])•
than the
alwaj^s greater
experiences a precautionary insolvency regard-
forward neighbor.
In particular,
banks
all
6''('+i)
can be verified that the informed bank
It
j/q
V
,
—
bound
in
averts
yj (see the Appendix).
then there exists a unique x
€ {h — y^/zJi) that
[yo]
solves
yo^[(/,
In this case, a
its
bank
V
that has recalled yo loans
< x
forward neighbor x
recalls yo).
By
(so that its
[yo]
is
=
(24)
yo.
insolvent
and only
if
upper bound y^
[{li
—
x)
z]
if it
is
receives
below
its
from
loan
a similar analysis to that in Section 3 for the partial cascades (which
carry out in the Appendix),
the banks
-,r[yo])c]
jfe^*'*,
it
can be checked that there exists
6''('-(^"^))
..,
K
[A', 2n —
€
| are insolvent while the banks jfoK'-'^).
are solvent. In other words, there
is
a partial cascade which
is
1]
we
such that
6''('-'2"-i))|
..,
at least as large as (and
potentially greater than) the partial cascade in the free-information benchmark.
We
summarize our findings
(since A'
<
3,
,
Note
2?""^'^'"*'''"
the banks in
vo
in the following proposition.*^
(p)
^
<
have sufficient hquid reserves at date
{l-y)R-{l-Q)h -(h-x):
^3^
also that condition (18) implies yg
—
^o
—
=
yo
[('1
1)
leads to
-x)z].
Vo ^"^^ thus rules out precautionary insolvencies
by
the above steps.
^Given the possibility of precautionary insolvencies, one maj' also wonder whether there could be
multiple equilibria due to banks' coordination failures. Suppose, for example,
contained after 3 banks
level of loans,
level of loans
fail.
Could there
also be a
bad equilibrium
in
which
all
K = 3, so that the crisis
banks
and their recall decisions are justified since their forward neighbors also
and experience a precautionary insolvency (thus paying a small ql)7
23
recall
the
recall the
is
maximum
maximum
Proposition
Suppose assumptions (FS), (16) and (17) are
3.
Suppose also that condition (23) (which
(8), (12), (14) hold.
For a given financial network h{p),
(18) J holds.
=
let i
and conditions
satisfied
the opposite of condition
is
p~^ (j) denote the slot of the
distressed bank.
For the ex-ante equilibrium
(i)
Qknow
information benchmark, while each bank
which
V
6
K
recalls y^
For
(p)
E
!
the
2n —
such that
1]
K
cascade size
K
<
3,
I
enough
•(f
to a
>
A'
recalls
^^-^
m
recall
=
y^ip)
the free-
Vofreeip)'
The bank
date
The
\):
and monotonic.
There
allocation
exists
a
unique
while
insolvent
are
>
K
size
equilibrium
banks
potentially larger than the
is
in the free-information benchmark. In particular, there are two cases:
V
then each bank
i.e.
4,
would
it
c
to avert insolvency.
The cascade
^^g solvent.
as
j^^^ {p)
<bP^^\ ..,bP^'~^^~^))
banks
E [bP^^ ,hP^-^\hP'''-'^'i]
free-information benchmark.
(at
based
QunceTtam
G
K=
some
chooses
^^^
precautionary insolvency. The cascade size
benchmark,
6
equilibrium
distance
^^
V
yfj
^uncertain
recall in the
y^ just
continuation
?2 (p)}
[A',
would
<
(p)
JljP[i-h)^^^^l^p{i-{2n-i))
If
6''('~'^)
what the bank
is
6''('+^)
(ii)
{'Ti
=
^p^ recalls the saine level of loans yj^{p)
V
Each bank
(at date 0):
m this
case
is
y^ {p)
<
y^,
and avoids a
identical to the free-information
K.
then each bank
G
b'
B^^^^^^"^^"^ (p)
chooses
j/q
=
(p)
y^
>
y^
.
which
may
lead
precautionary insolvency. There are two sub-cases:
U yo
£
size is given by
If yo
(yo
[0]
K=
1
,
2?!
€ (yo'^o
is
below x
Discussion.
a^^
A'
banks
>
V €
5""c«''*'^»"
G
(p)
5
""•=<='''<»*"•
yo.
[yo]
(p) is insolvent if
size is
£
(^i
K
<
yj,
maximum
ensure
if its
K
forward neighbor's
G
6, for
[A', 27T.
—
1],
different levels of
K
in
The top panel corresponds
to
For comparison, the dashed lines plot the cascade size
i.e.
This kind of coordination failure
(14). Tliese conditions
level
as a function of
the free-information benchmark for the same parameters.
the case in which yo
and the cascade
^TJo/PAi) characterized by
and only
an intermediate
Figure 5 plots the cascade size
the flexibility parameter
are insolvent
4.
The cascade
[yo]-
y
there exists a unique x
[0]]'
Eq. (24) such that bank
payment
— y],
— 1 >
tliat
when
is
condition (18) holds.
not possible
bank
6''*^'+'-^
in
is
By
Proposition
2, in this
case
our setup, precisely because of conditions (12) and
if all other banks choose the
always solvent, even
and experience precautionary insolvencies. To see this, note that the losses from
we move away from the distressed bank and eventually g''f'+2' > 'i — 13/ z.
Since bank fe''('+i) expects to receive at least /j — P/ z from its forward neighbor, it can avoid insolvency
by choosing an intermediate level of loan recalls. Hence, it is never optimal for bank 6''('+i) to undergo a
precautionary insolvency. But once we fix g^('+') = ci, the rest of the equilibrium is uniquely determined
level of loans
cross-deposits decrease as
as described above, that
is,
there
is
no coordination failure
24
among
banks.
03
02
01
04
05
0£
07
0.8
-
Figure
The
5:
K
of the four panels plots the cascade size
flexibility
parameter yo
plot the cascade size
K
size in the free-information
of yo that satisfies yo
sufficiently large 6
>
Vo-
Ii^
are insolvent in the costly audit
K
i.e.
may
> K. The
in
size of the
with an intermediate
fact, is stronger)
is
=
cascade
K
level of yo,
is
A'.
That
is,
rise,
(i.e.,
the effect of the flexibility parameter
non-monotonic: The whole financial system collapses
but the health of the financial system
with sufficiently high
levels of yo.
parameter yo reduces the cascade
this increase in flexibility
and the
we increase
then at some point the
The
is
As long
yo backfires and, in the current case,
precautionary insolvencies. However,
if
it
restored (and, in
J^.
size A' in the free-information
not sufficiently large. A" does not
financial panic remains.
is
intuition for this non-monotonicity
the same as the intuition for the non-monotonic effect of yo on
flexibility
in the
whole financial system
Figure 5 shows that as yo continues to
amplification disappears and again A'
on the
for
1).
The bottom panel
yo
level
and
benchmark than
third panel shows that, as
trigger a collapse of the
lines
the same as the cascade
is
this case, precautionary insolvencies are possible,
more banks
a sufficiently large shock 6
= 2n-
The dashed
benchmark. The second panel corresponds to a higher
free-information benchmark,
A'
runs. Each one
as a function of 9 for a different level of the
increasing order of yo from top to bottom).
in the free-information benchmark.
(i>^
there are no precautionary insolvencies and the cascade size
yo,
bank
costly-audit equilibrium with self-inflicted
as there
is
fall
benchmark.
manageable
If
levels
a financial panic, the increase in
also amplifies the cascade
the increase in yo
is
by generating more
sufficiently large,
the financial panic and restore the health of the financial system.
25
to
Increasing the
it
may end
Remarks
Final
6
Our model captures what appears
During
to be a central feature of financial panics:
severe financial crises the complexity of the environment rises dramatically, and this in
itself
The perception
causes confusion and financial retrenchment.
arises even in transactions
among apparently sound
of counterparty risk
engaged in long
financial institutions
term relationships. All of the sudden, economic agents are faced with massive uncertainty
The
as things are no longer business-as-usual.
current financial crisis
kets
is
In the
cascades.
Lehman Brothers during
one such instance, which froze essentially
and triggered massive run downs of
money market
collapse of
credit lines
all
the
private credit mar-
and withdrawals even from the
safest
funds.
model we capture the complexity of the environment with the
When
size of the partial
these cascades are small, banks only need to understand the financial
health of their immediate neighbors to
make
In contrast,
their decisions.
when
financial
conditions worsen and cascades grow, banks need to understand and be informed about a
larger share of the network.
At some point,
from intermediation rather than
risk
this
is
simply too costly and banks withdraw
exposure to enormous uncertainty, which triggers a
flight to quality.
We
also
loans or
but
if
showed that banks'
sell illiquid
flexibility,
defined as their ability to terminate long term
assets while in distress,
makes
it
harder for large cascades to develop,
they do develop they can trigger more severe credit crunches and even a collapse
in the financial system.
containing panic, but
it
Intuitively, a gain in flexibility
can be counterproductive
is
very useful
does not as
if it
it
if it
succeeds in
facilitates
banks'
withdrawal from intermediation.
An
aspect
in a related
we
did not explore in this paper but one which
work,
is
With
full
information, the distant banks
(i.e.,
0.
mechanism we
Our preliminary
highlight in this
the banks with k
natural buyers of the loans sold by the distressed banks.
face uncertainty
are currently pursuing
that of secondary markets for loans at date
findings point to yet another amplification aspect of the
paper:
we
and become worried that they may be too
> K)
are the
However, once distant banks
close to the distressed bank,
they cease to buy loans from these banks as they would rather hoard their liquidity, which
exacerbates the network's distress.
There are some obvious policy conclusions that emerge from our framework. For example, there
is
clearly scope for having
banks hold a larger
bufi'er
than they would be
privately inclined to do. Also, transparency measures, by reducing the cost of gathering
information, increase the resilience of the system to a lengthening in potential cascades.
26
There
even an argument to limit banks'
is
However, we are interested
in
flexibility to
undo
their financial positions.
going beyond these observations, and
in particular in ex-
ploring the impact of policies that modify the structure of the network.
there
OTC
an emerging consensus that the prevalence of bilateral
is
transactions
and that
compounded
it is
For example,
CDS
markets for
the confusion and complexity of the current financial
crisis,
imperative to organize these transactions in well capitalized exchanges to
Our frameM'ork can
prevent a recurrence.
help with the formal analysis of this type of
policy considerations. W'e leave this analysis for future research.
Appendix
7
We
Equilibrium in the Normal Environment.
ment described
in Section 2.1, the equilibrium
tates hquidity flow
in
and enables each bank
claim that, in the normal environ-
with the uncertain financial network
to pay the contracted values [ql
b^
each state of the world. Suppose that a consistent financial network, b
—1 and
at date
state s (r)
is
realized at date
0,
and suppose without
=
facili-
—
li,q^2
h)
realized
(p), is
loss of generality that
red type banks are assigned to odd slots (the case in which red type banks are assigned to
even
slots is
which
symmetric).
s {g) is realized is
to prove the statement for this case since the case in
It suffices
symmetric to the
s [r) case,
and the case
in
which
s (0) is realized
is tri^^al.
We
conjecture (and verify below) that each bank
positive audit costs d{.)
>
0)
and not to
Consider the equilibrium at date
odd
slot
bank,
A
chooses
in
B'''^-'^
^''(^i-i)
(p),
_
^_
drawing
p(20
92
p^j. q^q]^
;^*~'^
€
it
draws
(l-y)i? +
_
—
lo
> h and
+
c^C^"/^
'-.
(which
and
is
—
y^
deposits from the forward neighbor
6'''^'',
regardless of the financial
(j''^"''
=
/j
(c-z^(^-))/2
iOi
(.
p),
that liquidity flows through the network at date
equilibrium at date
1
(cf.
1
0.
It
even though each bank
is
uncertain
Eq. (4)) and the bank pays
27
and
draw
=
i.e.
In particular, for each
next consider the equilibrium at date
and
•
-
|
about the network structure.
0.
assigned to an
the preferences are given by (3), the green type banks do not
their deposits regardless of their beliefs /^(-')
We
=
a-'
deposits leads to the payments
1
Since
its
i.e.
6^'^'"-'\
green type bank,
[0, c]
chooses not to audit (for anj'
any loans,
recall
red type bank,
by assumption) needs liquidity so
i.e.
network
1.
bP
~''(2')
they choose
bank V, there
[ql
=
li.ql
—
is
follows
a no-liquidation
I2)
verify our conjecture that the
banks
choose not to audit and not to recall any loans. First note that a bank f^^^ in need of
liquidity at date
1 is
able to obtain
it
by withdrawing
deposits in the forward neighbor
its
at a cost of U/lj units at date 2 for
each unit of
liquidity bj' recalling loans at date
but this would cost
liquidity (since
any loans
at date
> h >
I2
in particular,
Therefore, each bank
I)-
Second note that a bank's,
0.
and deposit withdrawal
date
and
R>
b^^'\
R > h/h
6^^'^
units for each unit of
optimal actions
depend on
its slot
in
>
in the
normal environment, the financial network
and enables each bank
IP
parity),
its
Thus the bank
B''*'^ (p).
—
a''*''
(whenever
This completes the proof of our claim that,
thus verifying our conjecture.
0),
loan recall at
(for
(and only on
i
does not benefit from auditing and optimally chooses not to audit,
d{.)
also obtain
optimally chooses not to recall
independent of the financial network
is
it
at date 1) only
The bank could
liquidity.
banks
facilitates liquidity flow across
—
to pay the contracted values [ql
li,q2
=
h) hi each aggregate
state.
Proof of Eq.
(6)
withdrawn,
Eq.
tion 3
i.e.
and
Suppose
that, for
We
6'^('""'=).
is
insolvent thus
some k E
{0,
..,
claim that
cross-deposits are fully
all
both the free-information benchmark analyzed in Sec-
(6) holds, in
6''*'',
We
4.
and the costly audit model analyzed
distressed bank,
bank
for Sections 3
277
—
1},
6^('~''')
claim that bank
withdraws
it
By
in Section 4.
bank
also
all
6''''"''"""''^^^
condition
of its deposits,
withdraws
withdraws deposits,
all
the original
(8),
of
z.
deposits in
its
=
-''('"'"*)
i.e.
=
2^^'^
i.e.
which
~,
proves Eq. (6) by induction.
To prove the
we
As the
cases in turn.
insolvent
claim,
(i.e.
pays
it
first
first
it
cannot potentially
bail
further implies that the
^p{i-k)
_
solvent,
^_
i.e.
out
—
q'^
it
<
/^
and
/j.
g^^'""""^**
=
0).
Recall that bank
forward neighbor as given (see footnote
its
forward neighbor by withdrawing
bank withdraws
^g y^g ggcond
neighbor) and
/2//1
its
case,
all
of
its
deposits from
its
In this case,
bank
6^('~*-')
its
of its deposits
from
its
is
a cheaper
way
its
deposits,
to obtain liquidity,
4.
Recall that bank
backward
which costs
R>
forward neighbor, proving our claim that
Next consider the costly audit model of Section
i.e.
b^('-('=-i)), is
needs liquidity z (to pay
can obtain this liquidity either by withdrawing
This
z.
forward neighbor,
suppose that the forward neighbor bank,
per unit of liquidity. Since the former
all
in particular,
than
less
is
smaU
is
b''^''''^
1),
units at date 2 per unit of liquidity, or by recalling loans, which costs
withdraws
6^^'"''")
suppose that the forward neighbor of bank
case,
qf'^''-'^^'^
and takes the payment of
consider the free-information benchmark and analyze two
units
I2/I1
bank
6''('^'^)
;p('"'^~)
6'''^'"'''
=
-.
makes
the deposit \\'ithdrawal decision before the resolution of uncertainty for cross-deposits
(see
Figure
2).
As the
first case,
suppose that bank
28
6^('''^)
assigns a positive probability
b
to a network structure
such that q^
(/5)
assigns a positive probability tliat
takes the pa^mient of
its
^
forward neighbor as given and
(that
suppose bank
-^Qy.^
._
knows that
the bank
is,
bank
its
that
f^'-''^ believes
forward neighbor
its
(that
is,
suppose the bank
preferences are given
its
necessarily withdraws
information benchmark, the bank withdraws
to
< h
forward neighbor will be insolvent). Since the bank
its
LeontiefF form in (17), in this case the
r.p(i-k)
{p)
g^^'-C^-i')
all
=
(p)
[^
by the
of its deposits,
i.e.
with probability
1
solvent). In this case, as in the free-
is
=
z^^''^^
meet
z to
its
liquidity obligations
backward neighbor. This completes the proof of the claim and proves Eq.
by
(6)
induction.
Proof of Proposition
Contained
1.
in the discussion
Characterization of Banks' Payment Functions
4.
X
bank
If
=
~
~
by functions
Case
X £
If
1.
date
{p) at
(?i
giA^en
chooses some
b''<'~'^)
and
qj [yQ.x]
—
[/j
(and
1
,5/c,
= h; and
[Oi
The
first
not exceed
x
If
2.
<
li
- p/z
or
j/q
qi
=
this case, the
bank
the payment
when the bank's
losses
is
<
—
{li
-
{li
+
y
Section
payment
y'o
—
follows:
(25
UJ
x) z, then
+
zx
<
Ij
and
q-i
=
0.
(26)
losses
enough loans
recalled
solvent and pays according to (25).
losses
is
x) z, then
payment when the bank's
from cross-deposits do
to counter these losses.
The second
from cross-deposits exceed
In
case characterizes
its buffer,
or
when
the
do not exceed the buffer but the bank has not recalled enough loans to counter
these losses. In this case, the bank
Proof of Proposition
2.
depends only on
its
is
insolvent
and pays according
First consider part
of the loan recall decisions in part
banks
in
q2 [y'o-^]
forward neighbor pays
its
which are characterized as
>
y'^
and the bank has
buffer
and
= y'o-{h-x)z + 1il-y-y',)R ^^
> h-
92
case characterizes the
its
and
at date 0,
Vo]
1
Case
qi [i/o,x]
condition (18) holds), then this bank's
92 Wo-'^]
,
q-i
if
and
l\]
€
j/q
preceding the proposition.
(ii).
(i)
to (26).
taking as given the characterization
Note that the loan
recall decision of
each bank
distance from the distressed bank, which implies that the payments of
in the continuation equilibrium
that the equilibrium
is
distance based.
can be written as a function of their distances,
The
characterization in part
29
(ii)
i.e.
shows that each
bank
{bP^'\ bP^'-'^\
€
b"^'-'"''
benchmark chooses
=
y^
6''(^-*^-i»}
that would be insolvent in the free-information
..,
yo,
and thus
pays the same allocation
it
would pay in the
it
free-information benchmark:
Qi
Tlie
=
[fc]
bank
<
<liTree ip)
^1
^nd Q2
many
b''^^~^^ recalls at least as
benchmark, thus
it is
=
[k]
=
q',%'^ (p)
loans as
for
would
it
/c
6
{0,
..,
K-
1}
(27)
.
recall in the free-information
many
solvent given condition (18) (which ensures that recalling too
loans does not cause insolvency) and pays
(cf.
Eq. (25));
Qi{K] = h and Q2[K]>h.
The banks
|5p('-(/<+i)),6p(^-(a'+2))_
€
b^^'-''^
..^
bP(»-(2n-i))
j
(28)
and thus pay
^^^ solvent
(cf.
Eq. (25)):
Qi
[k]
=
h and Q2
[k]
^-
=
'—
z
—
1
>
[K +
h, for k e
1,
..,
2n - 1}
.
to
(29)
In particular, the size of the cascade
'ii
^^
free (p)
increasing in
A:
We
in part
it is
in the free-information
is
Qi
and Q2
[k]
[k],
monotonic.
Consider
are optimal.
through (29)
and proves that the
are increasing in k
and prove that the choices prescribed
next turn to loan recaU decisions at date
(ii)
benchmark. Since
(see Proposition 1), the characterization in (27)
also implies that the payments,
distance based equilibrium
K as
is
the banks in B^^°^
first
Comparing the charac-
(p).
terization of the continuation equilibrium in (27) through (29) to the characterization in
Proposition
1,
each bank
b'
E
5'="°"'
forward neighbor compared to what
(i.e.
each bank
recalls the
same
level of loans that
Next we consider a bank
We
claim that Qi
payment
b^
e
bank
recall in the free-information
either case
Qi
<
[2]
2.
—
would receive
problem
would
(20) with x""
=
(27)
through (29)
x""
= Qi
^i f^ee
[2].
(^^e
benchmark), which
in the free-information
same
also
it
benchmark.
equal to
is
its
benchmark
Thus
Qij'^ee'^''^)-
which solves problem (20) with
6''^*"'^^
recalls the
b^
level of loans y^ Z.^1 that
benchmark. To prove the claim that Qi
[2]
b^'^''"'^^
=
would
Qi flee^
^^^^
[2]
is
given by Eq. (28) or Eq. (29) and in
Note that by Proposition
1,
(?j
Note that
li.
bank
in the free-information
recall in the free- information
characterized in Eqs.
[2]
in turn proves that the
K
it
expects to receive the same payment from
S""^^'""'*" (p)
of the forward neighbor of
suppose that
it
6 Bknow{p) g^j^^g
b^
(^)
in this case
claim in this case. Next suppose
K
>
3
Qi
and note that
30
J^^g
=
^1
when
in this case
K
Qj
<
[2]
2,
is
proving the
given by Eq.
(27)
which shows Qi
The
and
K
=
[2]
g^'j^^e {p),
completing the proof of part
(ii).
'
characterization for the aggregate level of recalled loans for the cases
>
from part
4 then trivially follow
Proof of Proposition
the proposition. Here,
Most
3.
we
and Proposition
of the proof
text.
We
<
2,
K=3
thus completing the proof.
1,
contained in the discussion preceding
is
consider in turn the subcases
main
verify the claims in the
(ii)
K
and
1
2 (for case
K
>
A)
and we
also verify the conjecture that the equilibrium
is
distance based and monotonic.
Subcase
€
If yo
1.
(j/q [0]
1
,
'
-
.
any x E
for
-
y],
yo
>
then
yS
This implies that
[OJi].
all
>
[0]
yo
banks
-
[(^1
in
^)
~~]
{6''*'', ...,
they recall loans greater than their corresponding upper
6^''"^^""^^^} are insolvent since
These banks' payments
limits.
are characterized by the system of equations
qf-'^
where /
pluggmgm
By
(gf
/
-e--^^)
q['
=
'
each k €
~
=
''
(jj
+
yo
<
1
<
~
!-~gg
.
(?i
we have
(14),
Then, since bank
y^ j^^l
case can also avert insolvency by choosing
some
=
2n
—
l^
which
is
K
ized in Eq.
if
and only
2.
If yo
(24)
if
G iyo^Vo
is
<
the informed bank
J/q,
<
yg
It
j/o-
[0])i
there exists a unique x
forward neighbor x <
its
distance based and monotonic,
we
b^
of the banks in
^p,.-.,
^
J
<
f^'^
..,
b'^^''^
(gM-.«-.;;
)
fo^.
31
'~vj
^^^h
distance
q^
benchmark
6''('+i)
>
is
in this
follows that the cascade
K
(under
equilibrium in this case.
1
G
[yo]
such that a bank
in yo,
is
"~"''
< 2n — 2, which imphes
.^[yo].
(^i
— y^/zji),
j^uncertain
€
character-
^^^ jg insolvent
Using the conjecture that
further conjecture that the banks
hp^'^.^b^i'-i^-'))] are insolvent while the banks jft^^'-^),
The payments
given by Eq. (9) (after
greater than the free-information cascade size
and increasing
receives from
the equilibrium
is
increasing (and converges to
is
condition (14)), completing the characterization of the date
Subcase
'
(30)
,
in the free-information
b^'^''^''
some
K
2n - 2}
verifying our conjecture that the equilibrium
'i))
able to avert insolvency by choosing
is
..,
h).
based and monotonic. By condition
size
{l,
condition (12), the solution to the above system
the fixed point y
~
for
defined in Eq. (11) and the initial condition gf
is
(.)
=
6^('-(2"-i))|
..,
are solvent.
are characterized by
I
i^
e
^
1^
..^
A'
-
1
^
,
(31)
which
we
is
an increasing sequence (by condition
are back to subcase
1 (i.e.
K — 2n — 1),
(12)).
Then, either
"~
< x
(?f
or there exists a unique
K
6 [K, 2n —
[yo]
1]
and
such
that
5:<'-<''^^»<
In the latter case, the banks in
(since the}' receive less
is
J
solvent since
t)^l'~r^+^j)j
..,
it
than x
jfe''^'-^),
[yo]
I
g^j.g
g^^gQ
analysis implies that the equilibrium
1
(32)
..,
6K'-(^'-0)|
c
5""='="*°'"
[yo]
from
its
are insolvent
(/s)
forward neighbor.
solvent since the)' receive
ward neighbors. The informed bank
characterization of the date
<,:"-('--»
15.1
from their forward neighbor) but the bank
receives at least x
tp(i-(2n-2))
a-
6^^'"*"^^
is
is
li
> x
[yo]
also solvent as in subcase
6'''-'"
The banks
from their
1.
'
in
for-
Finally, this
distance based and monotonic, completing the
equilibrium in this case.
32
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