Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium Member Libraries
http://www.archive.org/details/collectiveriskmaOOcaba
1B31
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Collective Risk
Management
in a
Flight to Quality Episode
Ricardo
J.
Caballero
Arvind Krishnamurthy
Working Paper 07-02
January 29, 2007
RoomE52-251
50 Memorial Drive
Cambridge,
02142
MA
This
paper can be downloaded without charge from the
Network Paper Collection at
Social Science Research
httsp://ssrn.com/abstract=960520
I
MASSACHUSETTS
TITUTE
Kit-
Management
Collective Risk
in
a Flight to Quality
Episode
Ricardo
J.
Arvind Krishnamurthy*
Caballero
January
2007
29,
Abstract
We
son
present a model of optimal intervention in a flight to quality episode.
for intervention
model make
their
own
risk
stems from a
management
shocks, but are uncertain of
how
is
They understand
correlated their shocks are with systemwide
shocks, treating the latter uncertainty as Knightian.
uidity
We
show that when aggregate
liq-
low, an increase in uncertainty leads agents to a series of protective actions -
decreasing risk exposures, hoarding liquidity, locking-up capital - that reflect a
to quality.
flight
However, the conservative actions of agents leave the aggregate economy
over-exposed to negative shocks. Each agent covers himself against his
own
scenario, but the scenario that the collective of agents are guarding against
A
rea-
Agents in the
collective bias in agents' expectations.
decisions with incomplete knowledge.
The
lender of last resort, even
if
less
is
worst-case
impossible.
knowledgeable than private agents about individual
shocks, does not suffer from this collective bias and finds that pledging intervention in
extreme events
JEL Codes:
is
valuable.
The
intervention unlocks private capital markets.
E30, E44, E5, F34, Gl, G21, G22, G28.
Keywords: Locked
collateral, liquidity, flight to quality,
Knightian uncertainty, collateral shocks,
collective bias, lender of last resort, private sector multiplier.
'Respectively:
MIT and NBER;
Northwestern University. E-mails: caball@mit.edu, a-krishnamurthy@northwestern.edu.
We
are grateful to Marios Angeletos, Olivier Blanchard, Phil Bond, Jon Faust, Xavier Gabaix, Jordi Gali, Michael Golosov, William
Hawkins, Burton
Hollifield,
Bengt Holmstrom, Dimitri Vayanos, Ivan Werning and seminar participants
Imperial College, LBS, Northwestern, MIT, Wharton,
Conference,
Bank
of England,
Fed Liquidity Conference,
and the Central Bank of Chile
research assistance. Caballero thanks the
replaces) "Flight to Quality
NY
NSF
for their
for financial support.
and Collective Risk Management,"
NBER MIDM
at
Columbia, DePaul,
Conference,
UBC Summer
comments. David Lucca and Alp Simsek provided excellent
This paper covers the same substantive issues as (and hence
NBER WP #
12136.
Introduction
1
Policy practitioners operating under a risk-management paradigm may, at
"...
times, be led to undertake actions intended to provide insurance against especially
When
adverse outcomes
uncertainty,
human
term commitments
confronted with uncertainty, especially Knightian
beings invariably attempt to disengage from
in favor of safety
and
medium
to long-
The immediate response on
liquidity...
the part of the central bank to such financial implosions must be to inject large
quantities of liquidity..." Alan Greenspan (2004).
Flight to quality episodes are an important source of financial
bility.
Modern examples
of these episodes in the
US
include the
and macroeconomic
Penn Central
insta-
default of 1970;
the stock market crash of 1987; the events of the Fall of 1998 beginning with the Russian
default and ending with the bailout of
LTCM;
Behind each of these episodes
of 9/11.
lies
as well as the events that followed the attacks
the specter of a meltdown that
prolonged slowdown as in Japan during the 1990s, or even a catastrophe
pression.
1
In this paper
we
much
Great De-
as
needed to prevent a meltdown.
present a model to study the benefits of central bank actions during
flight to quality episodes.
Our model has two key
ingredients: capital/liquidity shortages
Knightian uncertainty. The capital shortage ingredient
and theoretical
is
like the
lead to a
In each of them, as hinted at by Greenspan, the Fed intervened early and stood
ready to intervene as
tainty
may
literature
commonly
less
on financial
crises
a recurring theme in the empirical
is
and requires
and
little
studied, but practitioners perceive
it
motivation. Knightian uncer-
as a central ingredient to flight
to quality episodes (see Greenspan's quote).
Most
flight to
quality episodes are triggered by unanticipated or unexpected events.
In
1970, the Penn-Central Railroad's default on prime rated commercial paper caught the market
by surprise and forced investors
to re-evaluate their
models
of credit risk.
The ensuing dynam-
temporarily shut out a large segment of commercial paper borrowers from a vital source
ics
of funds.
In
October 1987, the speed of the stock market decline took investors and market
markers by surprise, causing them to question their models. Investors pulled back from the
market while key market-makers widened bid-ask spreads. In the
:
See Table
1
(part
A)
during the 20th century.
in
Barro (2005)
for
a comprehensive
list
fall
of 1998, the
comovement
of extreme events in developed economies
of Russian
government bond spreads, Brazilian spreads, and U.S. Treasury bond spreads was
a surprise to even sophisticated market participants. These high correlations rendered stan-
dard risk management models obsolete, leaving financial market participants searching
new models. Agents responded by making
decisions using "worst-case" scenarios
testing" models. Finally, after 9/11, regulators were concerned that commercial
and
for
"stress-
banks would
respond to the increased uncertainty over the status of other commercial banks by individually
2
hoarding liquidity and that such actions would lead to gridlock in the payments system.
aspects of investor behavior across these episodes - re-evaluation of models,
The common
conservatism, and disengagement from risky activities - indicate that these episodes involved
Knightian uncertainty and not merely an increase in
risk.
The emphasis on
tail
outcomes and
worst-case scenarios in agents' decision rules suggests uncertainty aversion rather than simply
risk aversion.
It is also
seldom repeats
itself.
noteworthy that when
it
comes to
flight to quality episodes, history
Similar magnitudes of commercial paper default (Mercury Finance in
1997) or stock market pullbacks (mini-crash of 1989) did not lead to similar investor responses.
Today, as opposed to in 1998, market participants understand that correlations should be
expected to
rise
in lending to
during periods of reduced liquidity. Creditors understand the risk involved
hedge funds.
the recent default of the
The
While
in
1998 hedge funds were
still
Amaranth hedge fund barely caused a
a novel financial vehicle,
ripple in financial markets.
one-of-a-kind aspect of flight to quality episodes supports the view that uncertainty
important ingredient
is
an
in these episodes.
Section 2 of the paper lays out a model of financial crises based on liquidity shortages
and Knightian uncertainty.
in
We
analyze the model's equilibrium and show that an increase
Knightian uncertainty or decrease in aggregate liquidity can reproduce
effects.
When
Knightian uncertainty leads agents to make decisions based on worst-case scenarios.
the aggregate quantity of liquidity
is
limited, the Knightian agent grows concerned that
he will be caught
in a situation
available to him.
In this context, agents react
of safe
liquidity,
but there
by shedding
is
not enough liquidity
risky financial claims in favor
Investment banks and trading desks turn conservative
They
The main
2
where he needs
and uncontingent claims. Financial intermediaries become self-protective and hoard
liquidity.
capital.
flight to quality
lock
up
capital
and become unwilling
to flexibly
results of our paper are in Sections 3
and
4.
in their allocation of risk
move
it
across markets.
As indicated by Greenspan's
See Calomiris (1994) on the Penn-Central default, Melamed (1988) on the 1987 market crash, Scholes
(2000) on the events of 1998, and Stewart (2002) or Ashcraft and Duffie (2006) on 9/11.
comments, the Fed has
We
historically intervened during flight to quality episodes.
analyze
the macroeconomic properties of the equilibrium and study the effects of central bank actions
environment.
in our
First,
we show
that Knightian uncertainty leads to a collective bias
Each agent covers himself against
in agents' actions:
own
his
scenario that the collective of agents are guarding against
so despite agents' uncertainty about the environment.
is
We
worst-case scenario, but the
known
impossible, and
to be
show that agents' conservative
actions such as liquidity hoarding and locking-up of capital are macroeconomically costly as
they exacerbate the aggregate shortage of
We
to alleviate the over-conservatism.
and improve outcomes.
capital
when
show that a lender
same incomplete knowledge that
facing the
agents' liquidity
is
It
Central bank policy can be designed
liquidity.
of last resort
(LLR), even one
triggers agents' Knightian responses, can unlock
does so by committing to intervene during extreme events
depleted.
Importantly, because these extreme events are unlikely,
the expected cost of this intervention
is
minimal.
If credible,
the policy derives
its
power from
a private sector multiplier: each pledged dollar of public intervention in the extreme event
matched by a comparable private
bailout
was important not
sector reaction to free
for its direct effect,
but because
readiness to intervene should conditions worsen.
The
model
value of the
of
bank
environment
is
LLR
facility is usually
The LLR
runs.
up
The
capital. In this sense, the
LTCM
served as a signal of the Fed's
it
signal unlocked private capital markets.
analyzed in terms of
Diamond and Dybvig's
rules out the "bad" run equilibrium in their model.
a variant of Diamond and Dybvig's,
it
(1983)
While our
does not include the sequential service
constraint that leads to a coordination failure and informs their discussion of policy.
benefit of the
LLR in
in agent decisions
is
our model derives from a different mechanism, which
is
The
the collective bias
caused by an increase in Knightian uncertainty. Our model also provides a
clear presciption to central
banks on when to intervene
in financial markets:
The
benefit of the
LLR is highest when there is both insufficient aggregate liquidity and Knightian uncertainty.
We also show that the LLR must be a last-resort policy: If liquidity injections take place too
often, the policy exacerbates the private sector's mistakes
and reduces the value of intervention.
This occurs for reasons akin to the moral hazard problem identified with the LLR.
Holmstrom and
Tirole (1998) study
liquidity provision (see also
ment can
issue
how
a shortage of aggregate collateral limits private
Woodford, 1990). Their analysis suggests that a credible govern-
government bonds which can then be used by the private sector
provision.
The key
Woodford,
is
that
difference
between our paper and those
we show how even aggregate
collateral
of
Holmstrom and
may be
for liquidity
Tirole,
and
inefficiently used, so that
private sector liquidity provision
is
In our model, the
limited.
only adds to the private sector's collateral but
also,
and more
government intervention not
centrally,
it
improves the use of
private collateral.
Routledge and Zin (2004) and Easley and O'Hara (2005) are two related analyses of Knightian uncertainty in financial markets.
3
Routledge and Zin begin from the observation that
financial institutions follow decision rules to protect against a worst case scenario.
They
de-
velop a model of market liquidity in which an uncertainty averse market maker sets bids and
asks to facilitate trade of an asset.
quality:
Their model captures an important aspect of
flight to
uncertainty aversion can lead to a sudden widening of the bid-ask spread, causing
agents to halt trading and reducing market liquidity. Both our paper and Routledge and Zin
share the emphasis on financial intermediation and uncertainty aversion as central ingredients
But each paper captures
in flight to quality episodes.
different aspects of flight to quality.
Easley and O'Hara (2005) study a model where ambiguity (uncertainty) averse traders focus
on a worst case scenario when making an investment decision. Like
us,
Easley and O'Hara
point out that government intervention in a worst-case scenario can have large effects. Easley
and O'Hara study how uncertainty aversion
while the focus of our study
Finally, in our
is
affects investor participation in stock markets,
on uncertainty aversion and financial
crises.
model agents are only Knightian with respect to systemic
(2001) explores the
home
more uncertain about
events. Epstein
bias in international portfolios in a related setup, where agents are
foreign than local markets. 4
As Epstein points
highlights the difference between high risk aversion
out, this modelling also
and aversion to Knightian uncertainty.
Moreover, our modelling shows that max-min preferences interact with macroeconomic conditions in
We
ways that are not present
show that when aggregate
differ,
3
liquidity
However when there
identically.
in
is
models with an invariant amount of
is
plentiful,
risk aversion.
Knightian and standard agents behave
an aggregate liquidity shortage, the actions of these agents
leading to flight to quality in the Knightian model.
There
is
a growing economics literature that aims to formalize Knightian uncertainty (a partial
contributions includes, Gilboa and Schmeidler (1989),
Dow and Werlang
(1992), Epstein
and
Wang
list
of
(1994),
Epstein (2001), Hansen and Sargent (1995, 2003), Skiadas (2003), Epstein and Schneider (2004), and Hansen,
et al. (2004)).
As
in
much
of this literature,
treatment of Knightian uncertainty
case
4
among a
is
we use a max-min device
to describe agents expected utility.
most similar to Gilboa and Schmeidler,
in that agents
class of priors.
Epstein's model
is
closely related to our
model
in
Caballero and Krishnamurthy (2005).
Our
choose a worst
The Model
2
We
study a model conditional on entering a turmoil period where liquidity risk and Knightian
Our model
uncertainty coexist.
is
silent
on what triggers the episode. In practice, we think
Penn Central
that the occurrence of an unexpected event, such as the
agents to re-evaluate their models and triggers robustness concerns.
model to study the
default or 9/11, causes
Our
goal
to present a
is
role of a centralized liquidity provider such as the central bank.
The environment
2.1
PREFERENCES AND SHOCKS
The model has a continuum
An
may
agent
of competitive agents,
specialist (e.g. a trading desk,
first- wave
00,(1),
and note
With
With
wave
The
first
randomly chosen group
denote the probability of agent
u>
wave
on average, across
is
probability 0(2|
1),
We
uj
by a
of one-half of the agents
^.
all
first
wave
(1)
agents, the probability of an agent receiving a
of being in this second
—
0(1)
>
wave
is
Let 0(2)
W (2), which
=
0(1)0(2|1).
satisfies,
<w = *®.
(2)
l
the economy experiences no liquidity shocks.
note that the sequential shock structure means that,
0(1)
>
0(2)
>
0.
This condition states that, in aggregate, a single-wave event
event.
hit
a second wave of liquidity shocks hits the economy. In the second
n
1
is
^p.
probability for agent
probability
economy
receiving a shock in the
of liquidity shocks, the other half of the agents need liquidity.
With
We
that,
(1) states that
shock in the
[0, 1].
liquidity immediately.
probability 0(1), the
Ul)dw =
Equation
Q =
€
hedge fund, market maker).
of liquidity shocks. In this wave, a
We
u>
a sudden need for capital by a financial market
for
are correlated across agents.
have liquidity needs.
by
some
receive a liquidity shock in which he needs
view these liquidity shocks as a parable
The shocks
which are indexed by
We
will refer to the
(3)
is
more
likely
than the two-wave
two-wave event as an extreme event, capturing an unlikely but
6
many
severe liquidity crisis in which
Relation
agents are affected.
(3),
deriving from the
sequential shock structure, plays an important role in our analysis.
We
Agent
1983).
ui
receives utility:
U w (c
With
Q'i
=
1,
Q2
=
second wave; and,
a.\
shock date as "date
u
:
1Z+
—
s-
x
,
c 2 ,c T )
if
the agent
=
0,
1,"
if
+ a 2 u(c 2 +
ot\u{c\)
)
in the early wave; a\
is
a2 =
=
the agent
is
=
and the trader
c 2 reflect the potentially
ENDOWMENT AND
of one,
is
= 1 if the agent is in the
shock. We will refer to the first
and the
date as "date TV"
final
c\
and
in the future,
effectively risk neutral.
c2
and
linear over Cf.
when market conditions
The concave
preferences over
c\
and
higher marginal value of liquidity during a time of market distress.
SECURITIES
endowed with
is
(4)
.
a2
0,
Preferences are concave over
oo.
view the preference over ct as capturing a time,
Each agent
2,"
Bc T
twice continuously differentiable, increasing, strictly concave and satisfies
72. is
are normalized
=
not hit by a
the second shock date as "date
the condition lim c _>o u'(c)
We
Diamond and Dybvig,
liquidity shock as a shock to preferences (e.g., as in
model the
and then liquidated
if
Z
units of goods. These goods can be stored at gross return
an agent receives a liquidity shock.
the model as financial traders,
we may think
of
Z
If
we
interpret the agents of
as the capital or liquidity of a trader.
Agents can also trade financial claims that are contingent on shock realizations. As we
show, these claims allow agents
who do not
receive a shock to insure agents
who do
will
receive a
shock.
We
will
assume
all
shocks are observable and contractible. There
is
no concern that an agent
pretend to have a shock and collect on an insurance claim. Markets are complete. There
are claims on
all
contingent claims
histories of shock realizations.
when we analyze the
We
will
be more precise in specifying these
equilibrium.
PROBABILITIES AND UNCERTAINTY
Agents trade contingent claims to insure against their liquidity shocks.
In
making the
insurance decisions, agents have a probability model of the liquidity shocks in mind.
We
assume that agents know the aggregate shock
probabilities,
(f)(1)
and
(f>(2).
We may
think that agents observe the past behavior of the economy and form precise estimates of
these aggregate probabilities. However, and centrally to our model, the
not reveal whether a given
u> is
more
likely to
treat the latter uncertainty as Knightian.
be in the
first
wave
same past data does
or the second wave.
Agents
Formally,
we use
<fi
u (l) to denote the true probability of agent
and 0^(1) to denote agent-u/s perception
and
the
</£j(2)).
first
We
assume that each agent
or second wave,
</>
+
w (l)
receiving the
first
shock,
of the relevant true probability (similarly for
to
knows
his probability of receiving a
W (2)
shock either in
5
W (2), and thus the perceived probabilities satisfy:
C(l) + C(2) =
We
uj
Ml) + MV =
0(1)
(2)
t^
(5)
-
define,
% = C(2) - ^).
That
is,
9^ reflects
how much agent
(6)
a/s probability assessment of being second
is
higher than
the average agent in the economy's true probability of being second. This relation also implies
that,
-%
= C(i) - ^y~-
Agents consider a range of probability-models 9"
(K <
4>{2)/2)),
and design insurance
in the set
0, with support
portfolios that are robust to their
follow Gilboa and Schmeidler's (1989)
Maximin Expected
[—K,+K]
model uncertainty.
We
Utility representation of Knightian
uncertainty aversion and write:
max
(ci,c 2> c T
where
mm£
)0£ee
K captures the extent of agents'
[[Hci,c2 ,cT )|^]
(7)
uncertainty
In a flight to quality event, such as the Fall of 1998 or 9/11, agents are concerned about
systemic risk and unsure of
how
this risk will
impinge on their
good understanding of their own markets, but are unsure
other markets
in the
may
affect
of
activities.
how
They may have a
the behavior of agents in
them. For example, during 9/11 market participants feared gridlock
payments system. Each participant knew how much he owed to others but was unsure
whether resources owed to him would arrive
(see, e.g.,
Stewart, 2002, or Ashcraft and Duffle,
2006). In our modeling, agents are certain about the probability of receiving a shock, but are
uncertain about the probability that their shocks will occur early relative to others, or late
relative to others.
5
For further clarification of the structure of shocks and agents' uncertainty, see the event tree that
detailed in the Appendix.
is
We
view agents' max-min preferences
The
in (7) as descriptive of their decision rules.
widespread use of worst-case scenario analysis in decision making by financial firms
example
an
is
of the robustness preferences of such agents.
SYMMETRY
To
simplify our analysis
agent's true Q w
may be
The symmetry
we assume that the agents
common
applies in other dimensions as well:
We
common
is
drawn from the same 9.
is
<p
u
While each
0.
,
K, Z, and u(c) are the same
knowledge. As noted above,
cf>(l)
for all
and 0(2) are
also
knowledge.
A
2.2
every agent
different, the 9^ for
u. Moreover, this information
are symmetric at date
benchmark
begin by analyzing the problem
insurance that
is
when
economy
valuable in our
K=
0.
as well.
This case
We
a planning problem, where the planner allocates the
will clarify the
nature of cross-
derive the equilibrium as a solution to
Z
across agents as a function of shock
realizations.
Figure
1
below describes the event tree of the economy. The economy
two waves
one, or
two wave
case, or
of shocks.
may
An
agent
u>
may be
is
denoted by
C
s
affected in the first or second
be affected or not affected in the one wave event.
Agent w's allocation
of waves, u/s shock) as the state for agent u.
where
in the event of agent
may
We
receive zero,
wave
denote s
in the
=
(
#
as a function of the state
being affected by a shock, the agent receives
u>
a consumption allocation upon incidence of the shock, as well as a consumption allocation
For example,
at date T.
affected
C
by the
= {C } is
We note
s
if
the
economy
is
wave, we denote the state as
first
the consumption plan for agent
that
c\
is
the
same
in
ui
An
by two waves
s
=
(2, 1)
of shocks in
which agent
and agent w's allocation
u>
is
as (c\, cf ).
(equal to that for every agent, by symmetry).
both state
sequential shock structure in the economy.
at that time,
hit
(2, 1)
agent
and state
who
(1,1).
This
is
because of the
receives a shock first needs resources
and the amount of resources delivered cannot be made contingent on whether
the one or two wave event transpires.
Figure
and
K=
0w(2)
wave
=
is
1
0,
also gives the probabilities of each state
s.
Since agents are ex-ante identical
each agent has the same probability of arriving at state
0(2)/2, which implies that the probability of
one-half.
Likewise, the probability of
u
u)
s.
Thus we know
that
being hit by a shock in the second
being hit by a shock in the
first
wave
is
one-half.
These computations lead to the probabilities given
= (#
s
2
waves, w's shock)
1.
C
s
p
s
1
(2.1)
#2)/2
(d.4'
(2.2)
0(2)/2
(c 2
Waves
Prob
)
4>{2)
Prob
1 -
No Shocks
0(1)\
Figure
The
in Figure
planner's problem
is
1:
(1,1)
fo(l)-#2))/2
(l,no)
(0(1)
(0jno)
x
-
_
0(2))/2
,4
)
1
(d.4-
(c^"
o,n
(c
0(1)
2
)
)
0)
Benchmark Case
to solve,
max^p
s
UJ
t/
(C s )
subject to resource constraints that for every shock realization, the promised consumption
amounts are not more than the
total
1
-
/
\C\
endowment
I
(Ci
+
1>1
CT
"
+
I
of Z:
„0,no
<
z
l,nO\
<
*
Z
Cr
'
J
2,2\
+ C 2.1
T + C 2 + Cy
.
,
'
J
-.
<
rj
ry
Z,
Zj
as well as non- negativity constraints that for each
5,
every consumption amount in
C
s
is
non-negative.
It is
for
We
obvious that
consumption
if
shocks do not occur, then the planner will give
at date T.
rewrite the problem
max
0(1)
—
0(2)
Thus.c T'"°
=Z
and we can drop
Z
to each of the agents
this constant
from the objective.
as:
,
(u(ci)
no
+ pcyn + /Jc^ J
!
H
subject to resource and non-negativity constraints.
10
—
0(2)
v
,
(u(ci)
+
u(c 2 )
o 2,2\
+ /3c2,1
r + /?c^ J
,
'
Observe that
we
of generality
constraints.
1
c^'
set c^
Without
{Cl,C2,Cj
enter as a
c^'"
=
1
sum
+
4>{l)u{ Cl )
we
both objective and constraints. Without
in
and
Likewise, cj
0.
loss of generality
max
\,no
I
subject
and
set
Cj.
T
'
—
0.
sum
in
both objective and
The reduced problem
2
<K2)(u(c2)
2,2\
,C
enter as a
cT
+ /3c /) +
(0(1)
-
loss
is:
0(2)) f3cY°
)
to:
c\
= oIZr/
1,710
+
,
C\
Cj.
+ c2 +
l,no
'
Cl,C 2 ,C T
= 2Z
Cji
2,2
n
v^
>
,C T
(J.
The
Note that the resource constraints must bind.
solution hinges on whether the non-
negativity constraints on consumption bind or not.
If
the non-negativity constraints do not bind, then the
first
order condition for
c\
and
c2
yield:
=
c2
=
2(Z
—
c*),
c1
The
u'-\P)
=
c*.
solution implies that,
=
Cy
Thus the non-negativity
lo
Cr/
constraints do not bind
sufficient aggregate liquidity.
When Z
plan in which marginal utility
is
is
if
= 2Z —
Z >
c*.
We
c*.
refer to this case as
large enough, agents are able to finance a
equalized across
all states.
the marginal utility of early consumption with that of date
one of
consumption
At the optimum, agents equate
T
consumption, which
is (5
given
the linear utility over ct-
Now
consider the case in which there
to achieve full insurance.
case
(i.e.
that c 2
=
use
cj
no
all
first
is
— 2Z —
Ci
Z <
towards shock
and the problem
0(l)n(c 1 )
insufficient liquidity so that agents are not able
the case where
of the limited liquidity
max
with
This
is
c*
.
It is
states).
obvious that c^
Thus,
for
2
a given
=
c\
in this
we have
is,
+ 0(2)u(2Z-c
order condition,
11
1 )
+ (0(l)-0(2))/3(2Z-c
1 )
(8)
Since u'(2Z
—
c\)
>
(3 (i.e.
c2
f3
The
who
last inequality
are affected by the
<
<
c*)
we can
<
u'(ci)
u'(2Z
on the right of (10)
first- wave
by the second-wave. There
is
is
-
the important result from the analysis. Agents
more
first
of the limited liquidity to the
more
shocks. This
because there
Proposition
•
1
is
is
scarce (small Z)
On
of each agent
is
are affected
because the
(or,
a chance the economy
it is
optimal to allocate
the other hand,
when
liquidity
is
not contingent on the order of the
is
enough liquidity to cover
all
shocks.
results as follows:
The equilibrium in
The economy has
is
likely shock.
Z) the liquidity allocation
summarize these
who
than that of the first-wave
wave having taken place there
plentiful (large
We
liquidity than agents
strictly smaller
is
spared of a second wave). Thus, when liquidity
is
(10)
cross-insurance between agents. Intuitively, this
equivalently, conditional on the
more
ci>Z.
=»
ci)
of shocks receive
probability of the second-wave occurring
is
order:
the
benchmark economy with
insufficient aggregate liquidity if
c*
Agents are partially insured against
>
c\
> Z >
Z<
c2
c*
.
K=
has two cases:
In this case,
.
liquidity shocks. First
wave
liquidity shocks are
more
insured than second wave liquidity shocks.
•
The economy has
sufficient aggregate liquidity if
Cj
and agents are
=
c2
c*
.
In this case,
c*
fully insured against liquidity shocks.
Flight to quality effects, and a role for central
(insufficient aggregate liquidity).
2.3
=
Z>
This
is
the case
bank intervention,
we analyze
arise only in the first case
in detail in the next sections.
Implementation
There are two natural implementations of the equilibrium: financial intermediation, and trading in shock-contingent claims.
12
In the intermediation implementation, each agent deposits
and receives the
right to
withdraw
>
c\
Z,
if
Z
in
an intermediary
initially
he receives a shock in the first-wave.
Since
shocks are fully observable, the withdrawal can be conditioned on the agents' shocks. Agents
who do
—
[Z
not receive a shock in the
<
C\
The second group
Ci).
wave own claims
first
T and
at date
all
redeem
of agents either
second wave of shocks, or at date T. Finally,
if
to the rest of the intermediary's assets
their claims
upon incidence
no shocks occur, the intermediary
is
of the
liquidated
agents receive Z.
In the contingent claims implementation, each agent purchases a claim that pays 2(ci
Z) >
the event that the agent receives a shock in the
in
— Z)
identical claim to every other agent, paying 2(ci
that this
If
is
no shocks occur, agents consume their own Z.
wave), to net
first
later agent has
T
as date
wave.
in case of the first
Z—
If
— Z) = 2Z —
(cj
if
there
also
is
c\
units of liquidity to either finance a second shock, or
is
sufficient aggregate liquidity either the intermediation or
Moreover,
implementable by self-insurance. Each agent keeps his
to finance a shock.
The
self-insurance implementation
is
Z
in this case,
and liquidates
not possible when
because the allocation requires each agent to receive more than his endowment of
is
2.4
robustness case
turn to the general problem
when
K
>
0.
Once
again,
by solving a planning problem where the planner allocates the
shocks.
Z < c*,
Z if the
hit first.
K>
We now
first
consumption.
the allocation
agent
an
wave shock. Note
an agent receives a shock in the
contingent claims implementation achieves the optimal allocation.
< Z
sells
— Z) and pays out c\ — Z (since one-half of the agents are affected in
c\ — Z. Added to his liquidity of Z, this gives total liquidity of c\. Any
Finally, note that
c*
The agent
a zero-cost strategy since both claims must have the same price.
wave, he receives 2(c\
the
first
—
When
K
>
0,
agents
This decision making process
is
make
Z
we
derive the equilibrium
to agents as a function of
decisions based on a "worst-case" for the probabilities.
encompassed
planning problem by altering the planners
in the
objective to,
raaxmmyp
The only change
the worst-case
in the
min
problem relative to the
rule.
13
s ' a,
K
[/(C
=
s
(11)
)
case
is
that probabilities are based on
Figure 2 redraws the event tree
the notation that 0^(2)
is
is
unsure
indicating agent's worst-case probabilities.
when the economy
is
use
going through a two-wave event in which
other agents' shocks are going to occur before or after agent w's.
if
s
2
We
agent u/s worst-case probability of being hit second. In our setup,
this assessment only matters
the agent
now
= (#
waves, o/s shock)
6
b
V
(2^)
Waves
Prob 0(2)
(2,2)
Prob 1-^(1)
simplify the problem following
be equal to Z. Since the problem
observe that c^
1
(0(l)-0(2))/2
(l,no)
(0(l)-0(2))/2
1-0(1)
(0,no)
Figure
We
(1,1)
and c^"° enter
some
in the
as
Robustness Case
2:
sum
The reduced problem
is
V(CM = maxmin
0-(l)u( Cl )+(0(2)
O.no
of the steps of the previous derivation. c^"°
one-wave node
is
must
=
both objective and constraint and choose c^
in
we
the same as in the previous case,
0.
then,
-
0^(2))
/?# +0^(2)
(n(c 2 )
+
^
2
)
^ (1)
+
0(2)
l.no
/3
(12)
The
is
first
two terms
in this objective are the utility
hit first (either in the
consumption bundle
when
6
the agent
We
is
if
one wave or two wave event). The third term
the agent
is
derive the probabilities as follows, p
-0£ (2) = #S(1)-
Substituting forp
2
'
1
The
hit second.
last
term
is
is
the agent
if
the utility from the
the utility from the bundle
not hit in a one- wave event.
since the probabilities have to
0(2)
from the consumption bundle
-",
sum up
* (1) ~ 0(2)
we
which we can use to solve
2" 2-"
=
2
<^(2) by definition. This implies that p
to the probability of a
using relation
(5).
The
can rewrite this to find that?
for
p
1
'
710
two wave event
probability of w being hit
1 ' 1 '"
^
14
=
&ii±&2l. Finally,
(0(2)).
'
1
'"
We
=
4>{2)
=
p
=
0£(1)
'
- <j%{2)
rewrite p
'"+p 1 no w
first is
1 1
-
'
2
'
1 <"J
2
<
l
+p
<
1
w
'
1
=
'
w
.
0(l)-^(2),
The
resource constraints for this problem are,
Cl
c\
The optimization
Proposition 2
+
+
2,1
,
c2
+4' no < 2Z
Cni
'
+
.
^
<
2,2
c^
r,n>
2z.
also subject to non-negativity constraints.
is
Let:
R^
f(l)
-
0(2)
4
/V(Z) V
/?
u 'i z )
K
Then, the equilibrium in the robust economy depends on both
•
When
there
— For
is
Z
as follows:
insufficient aggregate liquidity, there are two cases:
K
<
and
< K,
agents' decisions satisfy:
C(lK(ci) = ^(2)u'(q
>
)-+/?
* (1) ~
where, the worst-case probabilities are based on 6^
m
'
.
(13)
= +K:
In the solution,
c2
< Z <
c\
<
c*
We
with C\{K) decreasing and C2(K) increasing.
refer to this as the
robust" case.
— For
K
>
K
,
agents' decisions are as if
Ci
We
•
When
K=K
= Z=
C2
<
,
and
C*
refer to this as the "fully robust" case.
there
is
sufficient aggregate liquidity (Z), agents' decisions satisfy,
C\
=
C2
15
=
C*
<
Z.
"partially
The formal
proof of the proposition
to account for
show
controls, the
Appendix, and
complicated by the need
is
when
possible consumption plans for every given &£ scenario
all
max-min problem. But,
We
in the
is
in the
there
is
a simple intuition that explains the
Appendix that c^
problem simplifies
max
_
min
and c^
,
solving the
results.
Dropping these
are always equal to zero.
to:
0£(l)u(ci)
+ ^(2)u(c 2 +
)
m
~
^~
4>(1)
-
3
-(3c
,
l,no
C\,C2,Crp
For the case of insufficient aggregate
c2
Then the
first
liquidity, the resource constraints give:
= 2Z-c 1}
order condition for the
4'
no
max problem
= 2Z-c
a
.
for a given value of &£
is,
>(1)"0(2)
^(1K( Ci = ^(1K(c2 )+/32
)
benchmark
In the
case, the uncertain probabilities are </>"(l)
yields the solution calling for
K>
When
0,
liquidity to
decisions. Thus, they bias
K
is
whoever
is
affected
by the
</£J(2)
first
=
^jp,
shock
[c\
which
>
C2).
agents axe uncertain over whether their shocks are early or late relative to other
Under the max-min decision
agents.
When
more
= ^- and
agents use the worst case probability in making
rule,
up the probability
small, agents' first order condition
of being second relative to that of being
first.
is,
(m- K M =(m +Ky, M+p t>m - «2)
y
As
K
becomes
larger, c 2 increases
toward
This defines the threshold of K. In this
c\.
For
K
is
set equal to c\.
"fully robust" case, agents are insulated against their
uncertainty over whether their shocks are likely to be
2.5
sufficiently large, c 2
first
or second.
Flight to quality
A
flight to quality
K
or Z.
Let us
fix
episode can be understood in our model as a comparative static across
a value of
K
and
Z
and suppose that
contractual arrangement as dictated by Proposition
(non-contracted) event that increases
We may
K
(or
2.
at a date
At date
0,
— 1,
there
agents enter into a
is
an unanticipated
reduces Z) and leads agents to rewrite contracts.
think of a flight to quality in terms of this rewriting of contracts.
16
In this subsection
we
discuss three concrete examples of flight to quality events in the
Our
context of our model.
first
two examples identify the model
in
terms of the financial
intermediation implementation discussed earlier, while the last example identifies the model
in
terms of the contingent claims implementation.
The
the
example
first
of 1998.
fall
We
one of uncertainty-driven contagion and
is
interpret the agents of our
Each trading desk concentrates
bank.
model
drawn from the events
is
as the trading desks of
in a different asset market.
of
an investment
At date —1 the trading
desks pool their capital with a top-level risk manager of the investment bank, retaining C2 of
capital to cover
They
any needs that
may
arise in their particular
also agree that the top-level risk
manager
shocks that hit whichever market needs capital
agent in an unrelated market -
An
defaults.
then before, so that
likely
</>£(l)
+
0£(2)
other trading desks will suffer shocks
i.e.
first
an extra
c\
("trading capital").
—
capital").
>
C2
At date
a market in which shocks are
to cover
0,
Russia
now no more
unchanged - suddenly becomes concerned that
is
first
will provide
market ("committed
and therefore the agent's trading desk
will
not
have as much capital available in the event of a shock. The agent responds by lobbying the
top-level risk
manager to increase
every trading desk
now has
became
up more
committed capital up to a
less capital in
argues that during the 1998
desks)
his
crisis,
the
(likelier)
level of ci
=
c\.
As a
result,
event of a single shock. Scholes (2000)
the natural liquidity suppliers (hedge funds and trading
liquidity demanders. In our model, uncertainty causes the trading desks to tie
of the capital of the investment bank.
shocks, suggesting reduced liquidity in
all
The average market has
less capital to
absorb
markets.
In this example, the Russian default leads to less liquidity in other unrelated asset markets.
Gabaix, Krishnamurthy, and Vigneron (2006) present evidence that the mortgage-backed
securities market, a
and wider spreads
effect
if
Z
is
market unrelated to the sovereign bond market, suffered lower
in the 1998 crisis.
Note
also that in this
example there
large as the agents' trading desk will not be concerned
capital to cover shocks
when
Z >
c*.
is
no contagion
about having the necessary
Thus, any realized losses by investment banks during
the Russian default strengthen the mechanism
Our second example
is
liquidity
we
highlight.
a variant of the classical bank-run, but on the credit side of a
commercial bank. The agents of the model are corporates. The corporates deposit
commercial bank at date
receive a shock.
wave
— 1 and
The corporates
of shocks) the
bank
sign revolving credit lines that give
are also aware that
will raise lending
if
them the
right to c\
Z
in a
if
they
banking conditions deteriorate (a second
standards/loan rates so that the corporates will
17
effectively receive only c 2
suffering losses
<
The
C\.
flight to quality
than
when
periods
by the commercial bank
will transpire.
credit lines, effectively leading all firms to receive
Gatev and Strahan (2006) present evidence
C\.
triggered
is
and corporates becoming concerned that the two-wave event
They respond by preemptively drawing down
less
event
of this sort of credit-line run during
the spread between commercial paper and Treasury
bills
widens
(as in the fall
of 1998).
The
last
example
is
agents of the model are
one of the interbank market
all
commercial banks who have
Each commercial bank knows that there
from
reserve account, which
its
that bank
A
is
for liquidity
can
it
is
some
by
offset
and the payment system. The
Z Treasury bills at
possibility that
selling
Treasury
it
will suffer
To
bills.
advantageous terms (worth
at
worse terms of c 2
A
is
a bank outside
(say,
C\ of T-bills).
If
a second set of shocks
the discount window). At date
New York
City which
is
0,
first,
hits,
its
which
and
possible shock at
4pm. In
first.
this
banks receive credit
A
is
becomes concerned
may
arise before
4pm.
Rather,
it
will
hoard
its
Treasury
Bills of
Z
to cover
example, uncertainty causes banks to hoard resources,
often the systemic concern in a
is
receives credit on
interbank lending arrangements and be unwilling to provide
will renegotiate its
to any banks that receive shocks
own
suppose
not directly affected by the events, but which
that other commercial banks will need liquidity and that these needs
c\
fix ideas,
9/11 occurs. Suppose that bank
concerned about a possible reserve outflow at 4pm. However, now bank
A
a large outflow
worried about this happening at 4pm. At date —1, the banks enter into an
interbank lending arrangement so that a bank that suffers such a shock
Then, bank
the start of the day.
payments gridlock
(e.g.,
Stewart, 2002, and Ashcraft
Duffie, 2006).
The
can be
different interpretations
mapped
into the actors
financial system.
As
is
we have
offered
show that the model's agents and
and actions during a
apparent, our environment
is
flight to quality
their actions
episode in a
a variant of the one that
modern
Diamond and
Dybvig (1983) study. In that model, the sequential service constraint creates a coordination
failure
and the
possibility of a
In our model, the crisis
realization of the
is
bad
crisis
equilibrium in which depositors run on the bank.
an unanticipated
rise in
bad equilibrium. Our model
of the rewriting of financial contracts triggered
behavior of a bank's creditors.
7
There
is
Knightian uncertainty rather than the
also offers interpretations of a crisis in
terms
by the uncertainty increase, rather than the
7
an underlying connection between the
crisis in
our model and that of
Diamond and Dybvig
(1983).
As Diamond and Dybvig emphasize, a deposit contract implements an optimal shock-contingent allocation
18
of
Collective Bias
3
In this section,
we study the
and the Value of Intervention
benefits of central
bank actions
in the flight to quality episode of
our model. Since the agents of the model are not standard expected utility maximizers
(i.e.
adhering to the Savage axioms), there are delicate issues that arise when defining the central
bank's objective.
On
the one hand,
the planner in (11), then since
the central bank's objective
if
is
the
min-max
we have previously derived the equilibrium
a planner's problem, the central bank cannot improve on the allocation.
throughout this section, we argue
for
objective of
as the solution to
On
the other hand,
a paternalistic central bank objective which does not
We
incorporate agents' worst-case probability assessments.
paternalistic policies that arise in our setting.
argue that there are "natural"
8
Collective bias
3.1
In the fully robust equilibrium of Proposition 2 agents insure equally against
shocks.
To
conditions
first
and second
arrive at the equal insurance solution, robust agents evaluate their first order
(equation 13) at conservative probabilities:
« (1) . ffi
Suppose we compute the probability
.iM(S]
of
(14)
one and two aggregate shocks using agents' con-
servative probabilities:
0(1)
The two
by each
=2 f
0(2)
<f>Z(l)dw,
=
2
in front of these expressions reflects the fact that
of the shocks. Integrating equation (14)
liquidity. In the fully robust equilibrium of
/
<%(2)dw.
only one-half of agents are affected
and using the definitions above, we
our model, agents choose an allocation that
is
find that
non-contingent on
each other's shocks instead of the contingent liquidity allocation of the benchmark economy. In this sense,
robust agents' preference for shock-independent liquidity allocations
is
related to the behavior of panicked
depositors in a bank run.
8
The appropriate notion
the literature.
choices
The debate
and welfare
of welfare in
models where agents are not rational
is
subject to
centers on whether or not the planner should use the
(see, e.g.,
19
in
same model to describe
Gul and Pesendorfer (2005) and Bernheim and Rangel (2005)
the argument). See also Sims (2001) in the context of a central bank's objective.
some debate
for
two
sides of
agents' conservative probabilities are such that,
m
The
- #2) =
(0(1)
-
0(2))
(£[§j)
last inequality follows in the case of insufficient
he receives a shock
last
Since each agent
and there
is little
-
0(1)
0(2).
aggregate liquidity (Z
<
c*).
an agent's chances of being affected
Implicitly, these conservative probabilities overweight
second in the two-wave event.
<
concerned about the scenario in which
is
liquidity left, robustness considerations lead each
agent to bias upwards the probability of receiving a shock later than the average agent.
However, every agent cannot be
probabilities violate the
known
later
than the "average." Across
all
agents, the conservative
and second wave events, implying that
probabilities of the first
agents' conservative probabilities are collectively biased.
Note that each agent's conservative probabilities are individually
range of uncertainty over 6^,
of being second.
it is
u
possible that agent
plausible.
Given the
has a higher than average probability
Only when viewed from the aggregate does
it
that the collective of conservative agents are guarding against
become apparent the
is
scenario
impossible.
These observations motivate us to study how a central bank, which
is
interested in maxi-
mizing the collective, can improve on outcomes.
Central bank information and objective
3.2
The
central
bank knows the aggregate
drawn from a common
is
common
distribution for
probabilities
all
u.
We
(f)(1),
0(2),
and knows that the
w 's are
have previously noted that this information
knowledge, so we are not endowing the central bank with any more information
than agents. The central bank also understands that because of agents' ex-ante symmetry,
agents choose the same contingent consumption plan
C We denote pfj CB
s
.
that the central bank assigns to the different events that
central
The
may
affect agent
as the probabilities
u>.
bank does not know the true probabilities pw Additionally, pj CB may
s
s
.
central
bank
is
concerned with the equally weighted ex-post
all
Like agents, the
differ
s
from pJ^
utility that agents derive
from their consumption plans:
V CB =
=
[
J^pf
s
5> c/(c*)
20
B
U(C s
)
(15)
where the second
line follows
from exchanging the integral and summation, and using the fact
that the aggregate probabilities are
One can view
common
knowledge.
this objective as descriptive of
how
As noted above,
central banks behave:
central banks are interested in the collective outcome, and thus
natural that the objective
it is
adopts the average consumption utility of agents in the economy.
The
arise
objective can also be seen as corresponding to agents' welfare.
when making the
consumption
ignores any utility costs that agents
"anxieties" about ex-post outcomes.
economy, they
may
suffer
Second,
if
we
in this
ignore the
first
because of date
suffer
we introduce some
from the type of policies we discuss
address the second concern below. In contrast,
may
potential issues
based on ex-post
First, since the objective is
latter identification.
utility, it
Two
rational agents into the
and the next
one, which
is
section.
We
what makes
ours a paternalistic criterion.
3.3
management and wasted
Collective risk
Starting from the robust equilibrium of Proposition
agents' decisions
by increasing
the same amount.
The
First, note that
if
by an
infinitesimal
there
is
m
amount, and decreasing
sufficient aggregate liquidity, c\
Aci)
c-i
and c^
Turning next to the
=
C2
=
c*
=
u'~
l
((3).
no
by
is:
For this
.m AC2) .msm,^
(16) implies that there there
robustness equilibrium
consider a central bank that alters
value of the reallocation based on the central bank objective
case,
and equation
c\
2,
liquidity
is
no gain to the central bank from a
reallocation.
insufficient liquidity case, the first order condition for agents in the
satisfies,
C(iK(ci) - C(2K(c2 ) -
^ffl-M = o
or
(m- K
y
Xci)
-(m +K y,
Rearranging this equation we have that,
f»'W - ^<*> -
{C2)
J«
21
_
4>(l)-0(2)
- AV(c) +
.
Ac,)).
Substituting this relation into (16),
bank
+ u'(c 2 ))
K{u'(c\)
is
Summarizing these
which
is
it
follows that the value of the reallocation to the central
positive for
all
K
>
0.
results:
K>
Proposition 3 For any
if the
0,
economy has
insufficient aggregate liquidity
[Z <
c*),
agent decisions are collectively biased.
Agents choose too much insurance against receiving
shocks second relative to receiving shocks
first.
The
central bank that
economy can improve outcomes by
post) utility of agents in the
toward the
A
maximizes the expected
reallocating agents
ities.
reallocation
central
valuable to the central bank because from
is
bank reaches
this conclusion requiring only
As we have remarked, the
about individual
O^s.
central
its
perspective agents are
In fact, going
beyond
of the central
bank objective
is
2,
knowledge of aggregate probabil-
bank may be the
least
informed agent of the
that the central bank does not suffer from collective bias. 9
local perturbations, the linearity
of the planner in Section
risks.
bank may be more confused than individual agents
In this sense, the central
economy. The important point
3.4
insurance
first shock.
wasting aggregate liquidity by self-insuring excessively rather than cross-insuring
The
'
with respect to individual probabilities
in (15) implies that the central
regardless of
how uninformed
it
bank
may
solves the
K=
problem
be about individual 9w s.
Welfare and paternalism
To what extent does the
arguments
central bank's objective correspond to welfare?
We
offer
in favor of our paternalistic welfare criterion in this subsection. First,
two further
suppose that
a fraction of the agents in the economy are rational and had probabilities such that 6U
these agents
i.e.
Suppose the
agents
all
is
know
that the worst-case probabilities are truly their
rest of the agents are
zero.
it
If
the central bank
is
utility cost to the rational agents is
uncertain about the values of
we may imagine a
<p(l)
situation in which the central
objective function overweights liquidity crises
first
shocks hurts the
(i.e.
and
bank
is
second order, while, since the
0(2), then
we could overturn the
result.
In
uncertain about these probabilities, and
its
the incidence of both shocks occurring). In this case, the
central bank will also be subject to the "overinsurance" bias of agents. However, this "bias"
nature than the one
probabilities.
helps the average Knightian agent. However, note that the envelope
theorem implies that the
particular,
own
= +K;
Knightian agents with 6Js such that the average #w across
In this case, reallocating insurance from second to
rational agents while
9
(ex-
we emphasize
as
it
would not be
is
of a different
collectively inconsistent with conditional probabilities.
22
envelope theorem does not apply to the average Knightian agent, the policy results in a
Thus, although the central bank's policy
order utility gain to the Knightian agents.
Pareto improving,
involves asymmetric gains to the Knightian agents.
it
the asymmetric paternalism criterion that Camerer,
policy
when some agents
Another
justification
the liquidity episode
agent
u>
al.,
is
policy satisfies
(2003) propose for evaluating
through the following thought experiment: Suppose that we repeat
we have described
infinitely
many times. At the beginning
i.i.d.
of each episode,
across episodes, and the agent
be zero. In each episode, since agent
his 8^ will
not
are behavioral.
draws a 8^ G 0. These draws are
on average
et.
Our
is
first
episode, the agent's worst-case decision rule will have
u
does not
know
knows that
the 8U for that
him using #w = +K. The
central bank's
ex-post utility criterion corresponds to the expected consumption utility of an agent across
where the expectation
of these episodes,
all
taken using agents' known probabilities. 10
is
Risk aversion versus uncertainty aversion
3.5
Given the centrality
of collective bias in Proposition 3,
increased risk aversion
may
should be apparent that while
it
generate positive implications
(flight to quality)
to those of Knightian uncertainty, its normative implications are not.
and regardless
of the agent's degree or
to reallocate liquidity toward the first
change
wave
in risk aversion,
of shocks
ti'(ci)
=
= j—. Then
the
first
Cl
y
7
we think
C\ falls
and
of a flight to quality event in
c 2 rises as
7
rises.
sector's choices.
order condition in the
is,
<P{2)
is,
W
0(2)
si)
If
beyond the private
order condition
'
collective bias,
0(1)'
u'(ca)
Suppose that u(c)
Without
our central bank sees no reason
We can make this point precise by returning to the agents' first
= case of Proposition 1. Equation (9) simplified by setting (3 =
K
that are similar
terms of increased risk aversion then
However, since
K=
0,
there
is
no
it is
clear that
role for the central
bank
in
this case.
10
9U
.
In living through repeated liquidity events, an agent will learn over time about the true distribution of
However,
it
is still
the case that along this learning path,
and hence the qualitative features of our argument
will
23
K
go through
remains
for
strictly positive (while shrinking)
a small enough agent discount
rate.
We
conclude that there
a role for the central bank only in situations of Knightian
is
Of course not
uncertainty and insufficient aggregate liquidity.
these ingredients.
and the
The
are
many
recessionary episodes exhibit
scenarios where they are present, such as October 1987
Fall of 1998.
An
4
But there
all
Application: Lender of Last Resort
abstract reallocation experiment considered in Proposition 3
flight to quality
episodes the central bank will find
against second shocks and
of this result
more against
and consider a lender
first
it
clear that during
desirable to induce agents to insure less
shocks. In this section
of last resort
makes
(LLR) policy
we
discuss an application
in light of the gain identified
in Proposition 3.
As
in
Woodford (1990) and Holmstrom and Tirole
to collateral that private agents
do not
(or at least,
(1998),
we assume the LLR has
has access at a lower cost). Woodford
it
this collateral.
Our
that, because of the reallocation benefit of Proposition 3, the value of the
LLR
and Holmstrom and Tirole focus on the direct value of intervening using
novel result
is
access
exceeds the direct value of the intervention. Thus our model sheds light on a new benefit of
the LLR.
The model
also stipulates
when the benefit
will
be highest. As we have remarked previously,
the reallocation benefit only arises in situations where
directly to our analysis of the
show that the
LLR must
LLR
>
benefits are highest
be a last-resort policy. In
often, the reallocation effect
4.1
LLR: the
K
and
when
Z <
K>
c*.
and
This carries over
Z<
fact, if liquidity injections
works against the policy and reduces
c*.
We
also
take place too
its value.
policy
Formally, the central bank credibly expands the resources of agents in the two-shock event by
an amount
Z G That
.
is,
agents
who
are affected second in the two- wave event (s
have their consumption increased from
c 2 to c 2
agents are affected by the second shock)
problem)
.
The
+ 2Z°
(twice
ZG
=
(2, 2)), will
because one-half measure of
resource constraints for agents (for the reduced
are:
Cl
+
ci
4'"°
+
c2
<
2Z
< 2Z +
24
(17)
2Z G
.
(18)
may
In practice, the central bank's promise
be supported by a credible commitment to costly
ex-post inflation or taxation and carried out by guaranteeing, against default, the liabilities
who have
of financial intermediaries
sold financial claims against extreme events.
we study an
are interested in computing the marginal benefit of intervention,
intervention of
ZG
Since
we
infinitesimal
.
the central bank offers more insurance against the two-shock event, this insurance has
If
a direct benefit in terms of reducing the disutility of an adverse outcome. The direct benefit
of the
LLR
is,
yCB,direct
=
f
2
^u'
=
(c 2 ^) dlO
0(2) W '(c 2 ).
Jn
The
anticipation of the central bank's second-shock insurance leads agents to reoptimize
their insurance decisions.
Agents reduce their private insurance against the publicly insured
The
second-shock and increase their first-shock insurance.
total benefit of the intervention
includes both the direct benefit as well as any benefit from portfolio reoptimization:
trCB, total
v g
n
,
s
,/
,dci tU
first
M - ^w +
simplify the expression for
substituting for u'{c\) from the
we
Last,
v
0(1)
^' dZ G
^M
-
no
I,
4>{2)
dcj
dZ G
2
du.
order condition for agent decisions, from (13), in the robust equilibrium gives,
*®*
We
dc 2 ,u
,
rww
dZ G
z
The
,
y G ®>
first
total
+ KMci) + Aci))
.
by integrating through 0^(1) and ^(2) and then
order condition. These operations yield,
differentiate the resource constraints, (17)
_
dci
dc 2
dZ G
dZ G
and
dci
dZ G
'
(18) with respect to
dc^
no
ZG
to yield,
_
dZ G
Then,
CB,
V^
r
total
tatM
=
i/ n \ //
\
0(2)u'(c 2 )
rst ti
+ ^(u'(
Cl +
= V^GB direct +
'
The
additional benefit
we
identify
is
\
.
.
)
K(u'( Cl )
+
ii
dc\
w
u'( C2 ))-
dZ G
dc\
u'(c 2 ))-
dZ G
'
due to portfolio reoptimization: Agents cut back on
the publicly insured second shock and increase
25
first
shock insurance, thereby moving their
what the central bank would choose
decisions closer to
for
them.
policy can help to implement the policy suggested in Proposition
We
also note that without Knightian uncertainty
direct benefit)
from the
Moreover,
policy.
it
(K =
0),
In this sense, the
3.
there
is
no gain (beyond the
straightforward to see that
is
LLR
if
Z >
c*
then
agents will not use the additional insurance to cover their liquidity shocks, but will reoptimize
in
a way as to use the insurance at date T. In this case also there
public insurance (since ^fc-
Proposition 4 For
exceeds
its direct
K
>
=
).
and
We
summarize these
Z <
,
total
important to note that under the
As we
only rarely.
failure in our
t
rCB
Z°
V
LLR
,
direct
policy the central
bank does not promise many
Diamond and Dybvig's
differences in the
>
bank
associate the second-shock event with an extreme
in expectation the central
similar to
results as follows:
the total value of the lender of last resort policy
c* ,
rC B
V Z°
is
no gain to offering the
value:
<
It
is
(1983) analysis of a
mechanism through which the
resources.
injects resources
and unlikely event,
This aspect of policy
LLR. However there are a few important
policies work.
As
there
is
no coordination
model, the policy does not work by ruling out a "bad" equilibrium. Rather,
the policy works by reducing the agents' "anxiety" that they will receive a shock last
the
is
economy has depleted
its liquidity
a high 0^(2) in their decision rules.
ingredient in the policy
is
resources.
From
anxiety that leads agents to use
It is this
this standpoint,
when
it is
also clear that
an important
that agents have to believe that the central bank will have the
necessary resources in the two-event shock in order to reduce their anxiety. Credibility and
commitment
LLR
11
policy.
Moral hazard and early interventions
4.2
The
are central to the working of our
policy
we have suggested
interventions.
cuts against the usual moral hazard critique of central bank
The moral hazard
of public insurance
critique
by cutting back on
is
their
predicated on agents responding to the provision
own insurance
activities. In
our model, in keeping
with the moral hazard critique, agents reallocate insurance away from the publicly insured
n In
this sense, the policy relates to the
government bond policy of Woodford (1990) and Holmstrom and
Tirole (1998) who argue that government promises are unique because they have greater
than private sector promises.
26
collateral backing
shock.
However, when
outcomes on average.
the concern, the reallocation improves (ex-post)
flight to quality is
12
Public and private provision of insurance are complements in our
model.
This logic suggests that interventions against
shocks
first
may
be subject to the moral
hazard critique as agents' portfolio reoptimization would lead them toward more insurance
against the second shock.
bank credibly
from
c\
to c\
To consider the
consumption of agents who are affected
offers to increase the
+ 2Z° The
'
.
in the first
shock
resource constraints for agents (for the reduced problem) are:
Ci
+ 4'"° < 2Z + 2Z G
ci
The
"early intervention" case, suppose that the central
+
< 2Z + 2Z G
c2
direct benefit of intervention in the first
yCBJirectJirst
=
is:
j^^fa = 0(1)^).
f
g
shock
.
JQ.
We
compute the
total benefit as previously except that
we
substitute agents'
first
order con-
dition using,
Also, using the fact that,
we
dc\
dc2
dZ G
dZ G
dZ G
'
no
dZ G
find that,
trCB, total
VZG
The expected
= -trCB,
Vz a direct, first - A
t,*/
bank
to the two-shock event,
which
We
is
resort policy
is
if
.
"'-l
1/
is
much
.
t
larger than the second shock
the direct effect of intervention
time inconsistent. This result
is
the private sector reaction, not from the policy
cost of insurance
exactly the opposite of what the central bank
in intermediate events are subject to the
conclude that the lender of
private financial markets, has to be a last
Note that
\\
.direct, first
+ u (C2))^c < V-CB
za
Agents reallocate the expected resources from the
wants to achieve. In this sense, interventions
hazard critique.
\
bank rather than the private sector bears the
against the (likely) single-shock event.
central
1/
(u (ci)
cost of the early intervention policy
intervention, since the central
12
dch
dc\
last resort facility, to
and not an intermediate
is
moral
be effective and improve
resort.
insufficient to justify intervention,
then the lender of last
not surprising as the benefit of the policy comes precisely from
itself.
27
Multiple shocks
4.3
It is
LLR
clear that the
should not intervene during early shocks and instead should only
pledge resources for late shocks; but
realistic
we move away from our two-shock model
context with multiple potential waves of aggregate shocks,
To answer
the
if
we extend
this question
economy may experience n
probability of the
=
for all
1...N waves of shocks, each affecting
economy experiencing n waves
is
policy takes the following form:
The
denoted
focusing on the fully robust case where c„
Z <
the intervention
The
c*
(i.e.
is
N).
injected to a measure
effects
>
M
n
=
is
r
with
late?
We
all
<fi(n)
N _^- +1
<
—
</>(n
1).
The
Also,
= ^~.
units of liquidity
also simplify our analysis
n and by
on c^
assume
of the agents.
^{njduj
injects
We
-^
more
no
.
setting
(3
=
c n rises to c n
0,
by
thereby
+ N 3 +i
in
of agents).
jj
direct value of the intervention as a function of j
vzcGB
bank
the same for
and allowing us to disregard
N 2-+\
<
(j
4>(n),
satisfies J^
en
central
shocks after (and including) the jth wave
assuring that
late
the model to consider multiple shocks.
each w's probability of being affected in the nth wave
The LLR
how
to a
is,
N
j^j^JJZUny^)^
J
n=j
Agents reduce insurance against the publicly insured shocks and increase their private
insurance for the rest of the shocks.
direct benefit as well as
The
total benefit of the intervention includes
both the
any benefit from portfolio reoptimization:
N
-,/CB, total
VZ°
From the
resource constraint
=
f
V^
^^\ n U
J Q, „
we have
/
i
\
1/
)
\.
dCn,LO
\
j
C n^)-T^Ga.UJ.
1
that
N
dcni
dZ G
In the fully robust case, c n ^ and
dc n
^§-
are the
same
^
V™*** = «'(ci)^£>(n)
=
rv
dZ G N
n=l
28
Then,
for all n.
'
i)4E^)
«'( c l
v
'
n=l
(
19 )
Note that this expression
yCB> direct
that
decreasing with respect to j since the
is
Thus, the
independent of the intervention rule
is
V-CB, direct
it is
apparent
are monotonically decreasing.
</>(n)'s
strictly greater
than one
Of course, the above
shock. Instead,
it
for all j
>
pledge resources to the
it
1
\—
i/„\
^ nn
±i
is
\
)
increasing with respect to
j.
of resources available for intervention,
Nth shock and
where the shadow cost of the
continue to do so until
N—
1st shock,
it
and so
LLR
LLR become
increasingly less likely.
resources for the central bank
is
If
completely
on.
also clarifies another benefit of late intervention.
events that are being insured by the
falls
amount
should then move on to the
The multiple shock model
policy
Y^ w
result does not suggest that intervention should occur only in the iVth
replaces private insurance,
LLR
and
suggests that for any given
LLR should first
cost of the
1
1
J^j+1 En=j
ZO
the
In contrast,
ratio:
trCB, total
is
j.
As
j rises,
we take the
case
constant, the expected
as j rises, while the expected benefit remains constant.
In other words, as j rises,
it is
the private sector that increasingly improves the allocation
of scarce private resources to early
and more
thereby reducing the
likely aggregate shocks,
extent of the flight-to-quality phenomenon. In contrast, the central bank plays a decreasingly
small role in terms of the expected value of resources actually disbursed, as j increases.
Thus, while a well designed
unlikely events, the policy
is
LLR
policy
not irrelevant
may
indeed have a direct effect only in highly
for likely
unlocking private markets to insure more likely and
less
Its
main
benefits
come from
extreme events.
Final remarks
5
Flight to quality
and
outcomes.
is
a pervasive
financial accelerators.
phenomenon that exacerbates the impact
The
starting point of this paper
is
a model of this phenomenon
We
based on Knightian uncertainty aversion among financial specialists.
aggregate liquidity
when
there
is
is
plentiful, agents' uncertainty
of recessionary shocks
show that when
does not affect the equilibrium. However,
both Knightian uncertainty and agents think that aggregate liquidity
they take a set of protective actions to guarantee themselves
safety,
is
scarce,
but which leave the
aggregate economy overexposed to negative shocks.
In this context, a
Our model
Lender
of Last Resort policy is valuable
prescribes that the benefit of the
LLR
29
is
when used
highest
when
to support rare events.
there
is
both insufficient
Many
aggregate liquidity and Knightian uncertainty.
this
paper (1987 market crash,
are times
when
examples we have discussed
of the
Fall of 1998, 9/11) satisfy these criteria.
call for central
by a hedge fund - even one that
is
bank
action. Likewise,
we
large - should not elicit central
prescribe that a default
bank reaction unless the
default triggers considerable uncertainty in other market participants
financially weak.
The model
also suggests that
it is
understanding of whether a given market dynamic
is
just as important
the "dog did not bark." For example, a recession in which the ingredients are
not both central, does not
There
But
in
important
is
currently considerable uncertainty over
affect the credit derivatives
for
and hedge funds are
the central bank to have an
driven by risk or uncertainty.
how
the downgrade of a top
name
will
market. Our model suggests that ex-ante actions to reduce the
extent of uncertainty during a flight to quality episode are valuable. For example, recent moves
to increase transparency
and
risk assessment in this
market
as well as streamline back-office
settlement procedures can be viewed in this light (Geithner, 2006).
The
implications of the framework extend beyond the particular interpretation
given to agents and policymakers. For example, in the international context one
of our agents as countries
prescribes that the
IMF
and the policymaker as the
not support the
to intervene once contagion takes place.
first
The
IMF
we have
may
or other IFI's, Then, our
think
model
country hit by a sudden stop, but to commit
benefit of this policy
comes primarily from the
additional availability of private resources to limit the impact of the initial pullback of capital
flows.
30
References
[1]
Ashcraft,
Adam
and Darrell
Duffie, 2006,
"Systemic Dynamics in the Federal Funds
Market," Mimeo, Stanford University.
[2]
Barro, R.J., 2005, "Rare Events and the Equity Premium," Mimeo, Harvard University.
[3]
Bernheim, B.D. and A. Rangel, 2005, "Behavioral Public Economics: Welfare and Policy
Analysis with Non-Standard Decision-Makers," Stanford Mimeo, June.
[4]
Caballero, R.J. and A. Krishnamurthy, 2005, "Financial System Risk and Flight to Quality,"
[5]
NBERWP #
Calomiris, Charles
11996.
W.
1994, "Is the Discount
spective," Review, Federal Reserve
[6]
Camerer, C,
S. Issacharoff,
for Conservatives:
Dow,
J.
and
S.
Law Review,
Economy
1983,
Penn Central Per-
pp. 31-55
vl51, 1211-1254, 2003.
"Bank Runs, Deposit Insurance, and
Liquidity,"
91, pp. 401-419.
Werlang, 1992, "Uncertainty Aversion, Risk Aversion, and the Optimal
Choice of Portfolio," Econometrica
[9]
(May/ June),
A
G. Loewestein, T. O'Donaghue, and M. Rabin, "Regulation
Diamond, D.W. and P.H. Dybvig,
Journal of Political
[8]
of St. Louis
Necessary?
Behavioral Economics and the Case for 'Asymmetric Paternalism',"
University ofPennsylvania
[7]
Bank
Window
60, p. 197-204.
Easley,
David and Maureen O'Hara, 2005, "Regulation and Return: The Role of Ambi-
guity,"
working paper, Cornell University.
[10]
Epstein, L., 2001, "Sharing Ambiguity,"
[11]
Epstein, L. and
American Economic Review,
M. Schneider, 2004, "Recursive Multiple
91(2), 45-50.
Priors," Journal of
Economic
Theory, 113(1), 1-31.
[12]
Epstein, L. and T.
tainty,"
[13]
Wang,
Econometrica
1994, "Intertemporal Asset Pricing
Under Knightian Uncer-
62, 283-322.
Gabaix, Xavier, Arvind Krishnamurthy, and Olivier Vigneron, 2006, "Limits of Arbitrage:
Theory and Evidence from the Mortgage-Backed Securities Market," forthcoming,
Journal of Finance.
31
[14]
Gatev, Evan, and Phil Strahan, 2006, "Banks' Advantage in Hedging Liquidity Risk:
Theory and Evidence from the Commercial Paper Market," Journal of Finance,
61(2),
867-92.
[15]
Growth
Geithner, Timothy, 2006, "Implications of
Remarks
Stability,"
at the
New York
in Credit Derivatives for Financial
University Stern School of Business Third Credit
Risk Conference.
[16]
Gilboa,
I.
and D. Schmeidler, 1989, "Maxmin Expected Utility with Non-unique Priors,"
Journal of Mathematical Economics,
[17]
18, 141-153.
Greenspan, A, 2004, "Risk and Uncertainty in Monetary Policy," Remarks
AEA
for the
Meetings, January.
[18]
Gul, F. and
W.
Pesendorfer, 2005, "The Case for Mindless Economics," Princeton Mimeo,
November.
[19]
Hansen,
"L.
IEEE
Control,'
[20]
Transactions on Automatic Control 40, 968-971.
Hansen, L.P. and T.J. Sargent, 2003, "Robust Control of Forward-looking Models," Journal of
[21]
and T. Sargent, 1995, "Discounted Linear Exponential Quadratic Gaussian
Monetary Economics
50(3), 581-604.
Hansen, L.P., T.J. Sargent, G.A. Turmuhambetova, N. Williams, 2004, "Robust Control
and Model Misspecification," April, Mimeo.
[22]
Holmstrom, Bengt and Jean
Political
[23]
Economy,
Melamed,
L., 1998,
and the Remedies,"
[24]
Tirole, "Private
and Public Supply of Liquidity," Journal of
106(1), 2/98, pp. 1-40.
"Comments
in
to Nicholas Brady's:
The Crash, the Problems Exposed,
Brookings-Warton Papers on Financial Services, Brookings
,
D.C.
Routledge, Bryan and Stanley Zin, 2004, "Model Uncertainty and Liquidity," working
paper, Carnegie Mellon University.
[25]
Scholes,
Myron
Vol. 90, No.
2,
S.
2000, "Crisis and Risk
Management,"
American Economic Review,
Papers and Proceedings of the One Hundred Twelfth Annual Meeting of
the American Economic Association (May), pp. 17-21
32
[26]
Sims, Christopher A., 2001,
"Pitfalls of
a Minimax Approach to Model Uncertainty,"
working paper, Princeton University.
[27]
Skiadas, Costis, 2003, "Robust Control and Recursive Utility," Finance and Stochastics
7,
[28]
475-489.
Stewart
11."
Jr.,
Jamie
Remarks by
B., 2002, "Challenges to the
First Vice President
Payments System Following September
Jamie B. Stewart,
Jr.
before the Bankers' Associ-
ation for Finance and Trade.
[29]
Woodford, M., "Public Debt as Private Liquidity," American Economic Review, Papers
and Proceedings
80,
May
1990, pp. 382-88.
33
A
Event Tree and Probabilities
Prob
2
Waves
1
Wave
2nd Wave
(f)(2)
1st
Wfcve
No More Shocks
Prob
The event
(f)(2);
of
tree
is
1
pictured above.
-
No Shocks
0(1
The
probability of two waves affecting the
the probability of one wave affecting the economy
no waves
economy
affecting the
is 1
—
(f)(1).
We
is (f)(1)
—
economy
is
0(2); and, the probability
used the dashed box around "1st wave"
to signify that agents ate unsure whether they are in the upper branch (one
more wave
will
occur) or the middle branch (no further shocks).
Consider an agent
u who may be
when
being affected by a shock
affected in these waves.
the event
is
the middle branch ("1 wave")
that his probability of being of being affected by a
branch ("2 waves")
is
if)
is
shock when the event
first
uncertain about
first
t/>
is
agent's probability of being affected by a
first
=
shock
is,
1
-(f)(2))
-.
is,
0(2)(l-&,).
that,
1
&,(!)
+ <M2) =0(2) +
(^(1)
34
the upper
is 1
,
agent's probability of being affected by a second shock
&,(2)
Suppose
—
ip^.
w which we interpret as the agent
or second, in the case of a two
^(l) = ^(2)^ + ^(1)
Note
one-half.
,
uncertain about his likelihood of being
The
is
his probability of
u while his probability of being affected by a second shock
Moreover, suppose that the agent
The
Suppose that
-0(2))
_
(f)(1)
+
(f)(2)
wave
event.
is
and,
<Mi)
—y
=
—
=
&/(2)
These expressions show that the event tree
^ 2 W"
—
+
-<f>{2)il> u
2~
—
consistent with agents being certain about their
is
probability of receiving a shock, but being uncertain about their relative probabilities of being
or second. In the text,
first
terms of
in
B
il)
u
we describe the uncertainty
terms of
(f) UJ
{2)
— ^~
rather than
.
Proof or Proposition
We
in
2
focus on the case of insufficient aggregate liquidity (Z
same
logic as the
(12).
We
K
—
We
case.
<
The other
c*).
case follows the
are looking for a solution to the problem in equation
can describe this problem in the game-theoretic language often used in max-min
problems. The agent chooses C to maximize V(C; &£) anticipating that "nature" will choose
9% to minimize V(C; &£) given the agent's choice of C.
The
solution (&£,C) has to satisfy a pair of optimization problems. First, 6%
That
is,
That
is,
We
Let us
cT
'
5
=
the agent chooses
C optimally given nature's choice
compute:
dV
—
=
\
-
I
\
m(cx)
+
,
this value of 9% let us consider the agent's
0.
With
To
2,2
-
problem
utility gain of
we
'
<
)
0.
If so,
first
> 0. Then we can reduce c^' by
when 9% >
5(<^(2) - 0(2) + </£(2)) >
+ PC?2,2 <
u{c\)
=>
J^-
C\
<
>
C before nature chooses 9" does not
only gives the agent an advantage
if
set
5
= +K.
First note that
and increase c^ by
0.
as,
c2
affect
our problem. To see
this,
note
the agent can induce nature to choose a 6% different
/ C to increase V('). Clearly this choice reduces V below V(0£,C).
6% = 9" and make the agent strictly worse off than at the choice C = C.
than 6%. Suppose the agent chooses C
Thus, nature can always choose to
then clearly 0"
in equation (12).
1
rewrite the condition that
fact that the agent chooses
13
2,1\
cT
'
this knowledge,
that choosing
at
p{cf
of #£.
see this, suppose that c r
u{c 2 )
The
I
u{c2 )
ask whether there exists a solution in which J^r
and produce a
13
9^
nature chooses &£ optimally given the agent's choice of C. Second, C € argmaxcV(C; 0£).
first
Taking
£ argmineu.V(C;
35
If C\
>
and
c2
Z<
c* it follows
from the resource constraint that
2
from the agent's problem, we must have that
T
if
=
c^'
(i.e.
c*.
But
if
c2
<
do not save any resources
c*
then
for
date
first
order condition for the agent at &£
+A
=
in (12) for values of c-i,C2,c^
We
note that for
K=
0,
As
K
This
>
0.
<
the unique solution to the agent's problem
small values of A", a solution exists in which the agent chooses
= +A.
n0
is,
(m_ Ky. iCl)= (m +Ky, {C2)+p
&£
<
these resources could be used earlier).
Thus, we only need to consider the agent's problem
The
c2
>
c\
>
is c 2
c2
c2
.
Thus,
for
and nature chooses
the partially robust solution given in the Proposition.
is
becomes
larger, Ci/c 2 falls
and
at
some point
C\
=
c2
=
Z.
This occurs when
K
solves,
(m_ Ry, {z)= (m +Ry, {z)+0 m^m.
which gives the expression
Note that
if
K
> K,
the solution 6%
The
nature's optimization problems.
nature
is
K defined in the Proposition.
= +K and = c
solves both
for the value of
c\
agent's choice
is
uniquely optimal at 6^
G [—A, +A]. This
indifferent over values of &£
the agent's and
2 still
is
= +K,
while
the fully robust solution given in
the Proposition.
We
have thus
Proposition
is
far
shown that considering the case where -^ <
the only solution to the problem in (12).
are no other solutions to the problem.
which
exists a solution in
Suppose there does
^>
To do
this,
We
0,
the solution given in the
conclude by showing that there
we only need
to consider whether there
0.
exist such a solution.
If
J^ >
0,
then 6%
= -A. We
09
through arguments similar to those previously offered to show that c^ and
zero in this case.
Then the condition
that
Jjj£
<
c\
The
first
order condition for the agent
>
is
solution to this problem
exist a solution in
which
is
-^ >
that
c\
can go back
i
cT
'
must both be
to,
c2
is,
(m +K M ±(m- K
y
The
equivalent
9
>
c2
,
Ae )+0
which
0.
36
i
.
)
is
a contradiction. Thus there does not
521