MIT LIBRARIES
3 9080 02617 9298
.M415
tiMimumuiUi'ifiiiitu^itiiitniiiti''
no.06-07
2006
i
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium IVIember Libraries
http://www.archive.org/details/flighttoqualitycOOcaba
OEWEV
HB31
.M415
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Flight to Quality
and Collective Risk Management
Ricardo
J.
Caballero
Arvind Krishnamurthy
Working Paper 06-07
March 15,2006
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:/7ssrn.com/abstract=891856
I
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
LIBRARIES
Flight to Quality
Ricardo
and
Collective Risk
Arvind Krishnamurthy*
Caballero
J.
Management
This draft: March 15, 2006
Abstract
We
present a model of flight to quality episodes that emphasizes systemic risk and the Knightian
uncertainty surrounding these episodes. Agents
edge.
They understand
their
own
systemwide shocks. Aversion to
make
risk
management
decisions with incomplete knowl-
how
correlated their shocks are with
shocks, but are uncertain of
them
this uncertainty leads
to question whether their private risk
man-
agement decisions are robust to aggregate events, generating conservatism and excessive demand
safety.
We
wasteful in
show that agents' actions lock-up the
capital of the financial system in a
the aggregate and can trigger and amplify a financial accelerator.
lective of conservative agents are
incomplete knowledge.
A
guarding against
impossible, and
is
lender of last resort, even
less
if
known
The
for
manner that
is
scenario that the col-
to be so even given agents'
knowledgeable than private agents about
in-
dividual shocks, does not suffer from this collective bias and finds that pledging intervention in extreme
events
is
The
valuable.
benefit of such intervention exceeds
its
direct value because
it
unlocks private
capital markets.
JEL Codes:
E30, E44, E5, F34, Gl, G21, 022, 028.
Keywords: Locked
collateral, flight to quality, insurance, safe
and risky claims,
financial intermediaries,
collective bias, lender of last resort, private sector multiplier, collateral shocks, robust control.
'Respectively:
We
MIT
and
NBER;
Northwestern
Universitj'.
are grateful to Marios Angeletos, Olivier Blanchard, Phil
William Hawkins, Burton
Hollifield,
Conference,
Bank
of England,
issues as (and hence replaces) "Financial
NY
Fed Liquidity Conference,
and the Central Bank of Chile
provided excellent research assistance. Caballero thanks the
more abstract model.
caball@mit.edu, a-krishnamurthy@northwestern.edu.
Gali,
Michael Golosov,
Bengt Holmstrom, Dimitri Vayanos, Ivan Werning and seminar participants
DePaul, Imperial College, LBS, Northwestern, MIT, Wharton,
Summer
E-mails:
Bond, Jon Faust, Xavier Gabaix, Jordi
NSF
System Risk and Flight
for their
for financial
to Quality,"
comments.
NBER MIDM
at
Columbia,
Conference,
UBC
David Lucca and Alp Simsek
support. This paper covers the same substantive
NBER WP #
11996 (2005) with a simpler but
1
Introduction
Polic}' practitioners operating
"...
under a risk-management paradigm may, at times, be led to
When
undertake actions intended to provide insurance against especially adverse outcomes
confronted with uncertainty, especially Knightian uncertainty,
medium
human
beings invariably attempt
liquidity....
The
immediate response on the part of the central bank to such financial implosions must be to
inject
to disengage from
to long-term
commitments
in favor of safety'
and
Alan Greenspan (2004).
large quantities of liquidity..."
Modern
Flight to quality episodes are an important source of financial and macroeconomic instability.
examples of these episodes
US
in the
include the
Penn Central
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;
as well as the events that followed the attacks of 9/11.
Behind each of these episodes
meltdown that may lead
Japan during the 1990s, or even a catastrophe
slowdown as
to a prolonged
in
lies
the specter of a
like
the Great Depression.^ In each of them, as hinted by Greenspan, the Fed intervened early and stood read}'
to intervene "as
Two
much
as needed" to prevent a
questions immediately arise
to quality episodes.
First,
how can
when
meltdown.
comments on
flight
private wealth create so
much
considering these examples and Greenspan's
events that appear small relative to
US
havoc? Second, how can interventions by the Fed, that also appear small relative to
in preventing a
meltdown?
answers are that during
its ability
we develop a model
wealth, be effective
to address these questions.
episodes private capital becomes locked up, and
flight to quality
of intervention derives from
Our model
In this paper
US
much
In short, our
of the
power
to unlock this capital.
builds on two observations about flight to quality events. First, at a basic level these episodes
are about an increase in perceived "risk."
However, the
risk does not circumscribe a purely
fundamental
shock, and instead centers around the nerve center of the economy, the financial system. For example, the
Russian default in the summer of 1998 eliminated a small fraction of the
trillions of dollars of
Although small, the default created circumstances that severely strained the
illiquid assets fell, losses
grew
in
and moved
into short-term
and
'See Table
century.
1
(part
A)
The primary
in
Barro (2005)
risk
withdrew
liquid assets.
as sophisticated parts of the financial system were
relatively unaffected.
As
wealth.
prices of
commercial banks, investment banks, and hedge funds, leading investors
to question the safety of the financial sector. Investors
institutions
financial sector.
US
risk-capital
Bottlenecks in the
from the affected markets and
movement
emerged
compromised while other sectors of the economy were
during this episode was financial system
for a
of capital
comprehensive
list
of
extreme events
in
risk.
developed economies during the 20th
Second, the "risk" in these episodes
of an extreme nature.
is
event that causes agents to re-evaluate
tlreir
Most episodes
models of the world. In the
are triggered
by an unexpected
Fall of 1998, the
comovement
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 standard
market participants searching
obsolete, leaving financial
decisions using "worst-case" scenarios
uncertainty.
Our model
more than
in their markets.
and
theirs
just risk.
Agents contract with financial intermediaries to cover liquidity shocks that may
These
management
risk
decisions are
treat this uncertainty as Knightian.
grow concerned that shocks may
in
models, in a manner suggestive of Knightian
"stress-testing"
arise in other
their ability to deliver
Knightian uncertainty or a
risk as well as the
on
fall in
intermediaries' assets.
own
else receives shocks.
will
deplete the resources of intermediaries and
Such
riskiness rises either
through an increase
Thus our model combines the
financial S3'stem
flight to quality episodes.
for safety.
Second, we show that Knightian agents respond to
markets by requiring financial intermediaries to lock-up some capital to
markets' shocks, regardless of what happens in other markets.
to dedicate capital to cover their
markets different from
aspects of flight to quality episodes. First, the increase in perceived
and demand
their uncertainty regarding other
devote to their
markets that
in
intermediaries have limited resources, agents
If financial
Knightian uncertainty that underlies
riskiness generates conservatism
model describing shocks
their financial contracts.
The model captures two important
arise
made with incomplete knowledge. Agents understand
shocks, but are uncertain about the probability
compromise
Agents responded by making
focuses on agents' perceptions regarding the ability of financial intermediaries to deliver on
their financial contracts.
own
new models.
management models
Indeed, the extreme disengagement from risky activities that Greenspan highlights suggests
that agents respond to
their
and
for
risk
own
By
forcing intermediaries
shocks, agents ensure that their shocks are covered regardless of
However, once locked-up, intermediaries' capital
is
not free to
move
who
across markets in
response to shocks, resulting in bottlenecks and market segmentation.
The
central
capital bottlenecks have
bank action during
macroeconomic consequences, and lend support to Greenspan's
flight to quality.
call for
While each Knightian agent covers himself against an extreme
shock, collectively these actions prevent intermediaries from moving capital across markets to expediently
offset
shocks as they arrive.
We
show that
this inflexibility leaves the
economy overexposed
aggregate shocks that are manageable by the private sector in the absence of
the scenario that the collective of conservative agents are guarding against
so even given agents' incomplete knowledge. Collectively, agents
make poor
is
to (moderate)
flight to quality.
impossible, and
risk
management
Moreover,
known
to
be
decisions that
lead to avoidable losses.
In this context, a lender of last resort, even
own market, can unlock
if less
informed than private agents about each agent's
private capital and stabilize the
economy during moderate
shocks.
It
does so
by committing to intervene during extreme events wliere the financial intermediaries' capital
is
depleted.
Importanth', because these extreme events are highly unlikely, the expected cost of this intervention
minimal.
If credible,
the pohcy derives
its
public intervention in the extreme event
is
power from a private sector
matched
bj'
but because
it
each pledged dollar of
a comparable private sector reaction to free flexible
LTCM
resources to deal with moderate shocks. In this sense, the
effect,
multiplier;
is
bailout
was important not
for its direct
served as a signal of the Fed's readiness to intervene should conditions worsen, which
unlocked private collateral.
The model we develop and our
Literature review.
related to the
Diamond and Dybvig
Diamond and Dybvig model
relation to
is
Our paper
bank
(1983) model of
runs.
An
in greater
depth
after
We
will discuss the
we introduce the model.
Papers such as
a gap in the literature on financial frictions in business cycle models.
fills
closely
important difference relative to the
that ours does not not involve a coordination failure.
Diamond and Dybvig
most
prescriptions over the lender of last resort are
Bernanke, Gertler, and Gilchrist (1998) and Kiyotaki and Moore (1997) highlight how financial frictions
firms amplify aggregate shocks.
Instead,
uncertainty in response to shocks, and
Our paper
also studies the
sense, our paper
is
closer to
we emphasize how
how
in
financial frictions lead to greater (Knightian)
back into the financial accelerator.
this rise in uncertainty feeds
macroeconomic consequences
of frictions in financial intermediaries.
Holmstrom and
and Krishnamurthy (2003) that emphasize that
Tirole (1998)
with complete financial contracts, the aggregate collateral of intermediaries
is
In this
ultimately behind financial
amplification mechanisms.
In terms of the policy implications,
Holmstrom and
collateral limits private liquidity provision (see also
government can
The key
issue
difference
limited. In our model, the
is
but
and more
There
is
in the aggregate
centrally,
it
may be
As
in
much
for liquidity pro\'ision.
and Woodford,
Tirole,
is
that
we show how
inefficiently used, so that private sector liquidity
government intervention not only adds to the private
improves the use of private
sector's collateral
collateral.
a growing economics literature that aims to formalize Knightian uncertainty (a partial
contributions includes, Gilboa and Schmeidler (1989),
Hansen and
of aggregate
Woodford, 1990). Their analysis suggests that a credible
between our paper and those of Holmstrom and
provision
how a shortage
government bonds which can then be used by the private sector
even a large amount of collateral
also,
Tirole (1998) study
Dow and Werlang
Sai-gent (1995, 2003), Skiadas (2003), Epstein
of this literature,
Knightian uncertainty
is
we use a max-min device
(1992), Epstein and
and Schneider (2004), and Hansen,
to describe agents expected utilit}^
most similar to Gilboa and Schmeidler,
in that agents
Wang
et
al.
list
of
(1994),
(2004)).
Om- treatment
choose a worst case
of
among
a class of priors.
Our paper
applies
These firms typically
max-min expected
stress-test their
utility
theory to agents
who we
interpret as running financial firms.
models to various extreme scenarios before formulating investment
The widespread
policies.
making
Corporate liquidity management
in practice.
We
mind.
in
use of Value-at-Risk as a decision
is
making
also
criterion
is
an example of robust decision
done with a worst case scenario
for cash-flows
view the max-min preferences of the agents we study as descriptive of decision rules that
overweight a worst-case rather than as stemming from a deeper psychological foundation. In
paper we
refer to agents as robust decision makers.
making process of the
much
of the
This terminology most closely corresponds to the decision
financial institutions that concern us in this paper.
Routledge and Zin (2004) also begin from the observation that financial institutions follow decision rules
to protect against a worst case scenario.
maker
averse market
sets bids
and asks
They develop a model
of
market liquidity
to facilitate trade of an asset. Their
in
which an uncertainty
model captures an important
aspect of flight to quahty: 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
episodes.
But each paper captures
in flight to quality
different aspects of flight to quality.
Easley and O'Hara (2005) study a model where ambiguity averse traders focus on a worst case scenario
when making an investment
in a worst-case scenario
Finally, in
the
home
than
Like us, Easley and O'Hara point out that government intervention
decision.
can have large
effects.
our model agents are only Knightian with respect to systemic events. Epstein (2001) explores
bias in international portfolios in a related setup, where agents are
local markets.^
As Epstein points
out, this modelling also highhghts the difference
aversion and aversion to Knightian uncertainty.
interact with
macroeconomic conditions
We
of risk aversion.
to flight to
quahty
In Section 2
to quality.
we
in
in the
foreign
between high
risk
Moreover, our modelling shows that max-min preferences
ways that are not present
show that when aggregate intermediary
agents behave identically. However
more uncertain about
when aggregate
collateral
in
is
models with an invariant amount
plentiful,
Knightian and standard
collateral falls, the actions of these agents differ, leading
Knightian model.
describe the environment, while in Section 3
we describe
decisions, equilibrium
Section 4 derives the value of a lender of last resort in our economy.
interaction between aggregate collateral and robustness considerations.
problems and other critiques associated with the lender of
last resort.
and
flight
Section 5 illustrates the
Section 6 discusses moral hazard
Section 7 concludes.
The Model
2
We
study a model conditional on entering a period of turmoil where Knightian uncertainty
model
is
silent
on what
^Epstein's model
is
triggers the episode.
closely related to our
model
in
In practice,
we think that the occurrence
Caballero and Krishnamurthy (2005).
is
high.
Our
of an unexpected
LTCM
event, such as the
We focus
concerns.
or
Enron
crises,
on the mechanisms that play out during the
concerns affect prices and quantities?
What
is
causes agents to re-evaluate their models and triggers robustness
How
do these robustness concerns interact with aggregate constraints?
a variant of a bank-run model
is
The agents
agents and intermediaries.
(e.g.
"eeurly" in
Diamond and Dybvig,
which case he needs some
liquidity immediately.
by a financial market
We
sudden need
The
agents sign financial contracts with the intermediaries to provide
are "early."
An
Diamond and
view the liquidity shocks as a parable
specialist (e.g. a trading desk,
hedge fund, market maker).
them with
liquidity insurance
they
if
intermediary can be thought of as a bank which extends a credit line to the specialists, or
as the top-level capital allocator in
may
in the canonical
which case he has no special liquidity needs, or he
"late" in
for a
for capital
1983). It has a set of competitive
As
are subject to liquidity shocks.
Dybvig (1983) banking model, an agent may be
We
do agents' robustness
the role of an outside liquidity provider such as the central bank?
The model
may be
How
liquidity episode:
an investment bank.
where the agents perceive that there
are interested in situations
not deliver the contracted liquidity insurance.
is
some chance that intermediaries
bank run model, a central concern
In a
of agents
is
that they will arrive too late to claim liquidity from an intermediary, and possibly not receive the liquidity.
We
introduce this type of concern by assuming that early agents are further divided into those
shocks
first
and those who receive shocks second. Each intermediary
of liquidity/collateral, Z, which
(or collateral)
also Z,
is
it
delivers to the agents.
We
is
who
receive
assumed to have a limited amount
normalize things so that
in
aggregate liquidity
and no longer distinguish between one intermediary and the aggregate.
SHOCKS AND PROBABILITIES
Formally, there
is
U'^
u
TZ+
:
u'(0)
=
—
*
7^
=
{
b)' tj
u
if
late
u{ci)
if
early
and
u(c2)
if
early
and second
£
fi
=
[0, 1].
Agent w receives
utility:
Probabilitj^
Probability: 0.(1)
first
Probability:
twice continuously differentiable, increasing, strictly concave and satisfies the condition
cx).
Unlike the
occur, there
()!)(1),
is
a continuum of agents indexed
is
Diamond and Dybvig model, we assume
a
first
that the shocks are correlated across agents.
wave, and then possibly a second wave.
The
first
and the second wave takes place with conditional probability
wave
0(2|1).
If
shocks
of shocks occurs with probability
We
assume that 0(2|1) <
1,
so
that
1
with 0(2)
=
>
0(1)
>
0(2)
>
(1)
0(2|1)(/)(1).
This condition states that, in aggregate, a single-wave event
is
more
likely
than the two-wave event.
We
will refer to the two- wave event as
many
an extreme event, capturing an unlikely but severe liquidity
crisis in
which
agents are affected.
The shocks
are aggregate in that they simultaneously affect a
wave occurs, then one-half
shock
of the agents
conditional on the
first,
first
is
hit
mass
by the shock. For agent
wave of shocks occurring,
u>,
equal to
is
We
of agents.
assume that
if
the
first
the probability that he receives a
While
Xn)
for
conditional probability need not be one-half, the average conditional probability across
any given w, the
agents must be
all
equal to one-half:
^
n
If
the second wave occurs, the remaining one-half of the agents
hit
is
by the shock:
^
n
SECURITIES
There
a contracting time, date
is
We
insurance to the agents.
an agent
s{uj) is
shock
• r(w)
not
We
if
offer financial contracts to provide liquidity
shocks are observable and contractible. There
during the
first
w
or second wave.
The claim
a risky claim. This claim pays one to agent
he receives
its
We
define
is
no concern that
two types of
financial
offer:
a safe claim. This claim pays one to agent
is
is
all
where intermediaries
pretend to have a shock and collect on an insurance claim.
will
contracts that intermediaries
•
assume
0,
if
he receives a shock regardless of whether the
costs p{s,{jj).
u) if
he receives a shock during the
first
wave, but
shock during the second wave. The claim costs p(r, w).
refer to the r claim as risky
In the example of a bank-run,
it
because
it
depends on the order of the agent
reflects the feature
where an agent
may need
in the sequence of shocks.
liquidity but
be second
in line
to claim the liquidity.
These two
securities that
financial securities
span the uncertainty
depend on the shock
in
our economy.
realizations over every subset of
fi.
In principle one could introduce
But given the
securities
we have
defined, such securities will not be traded.'^
INFORMATION
Agents purchase financial claims from the intermediaries to insure against their liquidity shocks.
making the insurance
decisions, agents have a probability
assume that agent-w knows
for the
indexation.
We
=<^J1)+^J2).
example, we could imagine a security that pays one
bj'
of their liquidity shocks in mind.
his probability of receiving a shock:
0,
•'For
model
In
if
both
ui
and
uj'
receive a
first
shock. However, agent
w
has no use
However, and centrally to our model, agent
uncertain about the correlation between his
ui is
know
aggregate conditions. In particular he does not
own shock and
among
the relative likelihood of being
the
first
wave
or second wave. Agents treat the latter uncertainty as Knightian.
We
define,
= -JW'
^->
Agent
UI
does not
know
the true probabilities 0tj(l) and d>^{2).
unknown
We
use the notation 0J^(1),
know
Since agents
agent-w's perception of the relevant true probability.
agent-w's
^
etc.,
to denote
a sufficient statistic for
4>^, 6'^ is
Normalizing these probabilities by the aggregate shock probabilities
probabilities.
'
is
convenient in the analysis.
Agents consider a range of probability-models
insurance portfolios that are robust to their model uncertainty.
Maxmin Expected
Utility representation of the
max min
K captures the extent of agents'
J5o[t/'^|^"]
problem and
We
wq
is
[1
impinge on their
how
activities. The}'
may have
the behavior of agents in other markets
feared gridlock in the
unsure whether
payments
money owed
to
sj'stem.
,
1
(3)
the initial
endowment
of agents.
how
aggregate conditions
own markets, but
are unsure of
them. For example during 9/11, market participants
affect
arrive. In
K], and design
< Wq
Each participant knew how much money he owed
him would
+
write:
a good understanding of their
may
/\
and Schmeidler's (1989)
In a flight to quality event, such as the Fall of 1998 or 9/11, agents are unsure of
will
—
follow Gilboa
p(s,a;)s(w) -|-p(r, w)r-(w)
s.t.
uncertainty, while
with support
in the set 0,
9'^
to others but
was
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.
We
view agents' max-min preferences in
(3) as descriptive of their decision rules.
worst-case scenario analysis in decision making by financial firms
of such agents.
is
The widespread
use of
an example of the robustness preferences
"*
SYMMETRY
To
may
simplify our analysis
be different, the
The symmetry
this information
We
is
9ui
wc assume that the agents
for every agent
is
0.
While each agent's true
6,^
drawn from the same 9.
applies in other dimensions as well:
common
are sj'mmetric at date
4>^,
K, wq, and u{c) are the same
for all
ui.
Moreover,
knowledge.
Eissume that the aggregate shock probabilities,
cf){l)
and
(p{2),
are
known by
all
agents.
We may
think that agents observe the past behavior of the economy and form precise estimates of these aggregate
"See also Routledge and Zin (2004)
for
a discussion of max-min behavior
among
financial specialists.
probabilities.
However the same past data does not reveal whether a given
more
ui is
likely to
be
in the first
wave or the second wave.
INTERMEDIARIES
Each intermediary
Financial intermediaries have a limited capacity to deliver liquidity insurance.
endowed with some goods
to deliver to
its
liquid funds or as the collateral/capital of
may have more goods and can
much
of these goods. For
varying aggregate collateral
Z
claimholders. These goods can be thought of as the intermediarj^'s
A
an intermediary.
thereby credibly
of our analysis,
sell
more
better capitalized/coUateralized intermediary
liquidity insurance. In aggregate, there are
we assume that
Z
is
and
safe claims sold
The
the intermediaries pay out
^^.^
units
effect of
R
and S the aggregate amount
Agents can observe total
respectively.
liabilities
and
collateral
agents require that each intermediary has sufficient assets to cover the
the world.
liabilities in all states of
Z
and Moore, 1997).
(as in Kiyotaki
by intermediaries,
assets for each intermediary.
we study the
constant. In Section 5
Agents recognize that intermediaries have limited resources. Denote by
of risky
If
If
the
wave occurs, one-half of the agents
first
will
wave occurs, the remaining one-half
the second
intermediaries pay out ^. Then, the collateral constraint for the intermediaries
is
be affected and
affected
and the
is:
§ + S<Z
A representative
less
intermediary's objective
is
U' =
of <p{l), 0(2),
ci
It
maximize the date
first
and ex-ante symmetry of agents are
at the rate
P>0
(5)
a standard expected utihty maximizer,
is
sufficient to
in selling insurance.
compute the expectation
Knowledge
in equation (5).
(1983)
important distinction between our model and the Diamond and Dybvig (1983) model of bank runs
constraint creates a possible coordination failure.
runs and policy intervention.
to write
^
and second shock agents, discounted
The intermediary
our assumption of complete state-contingent markets.
agents
revenue from the sale of the insurance
does not need to know the true 9^ of each agent
RELATION TO DIAMOND AND DYBVIG
An
(4)
+ Eo[P{ci+4)],
collateral constraint in (4).
with linear preferences.
is
to
a cost of the insurance resources disbursed to both
and subject to the
is
is
Since
In
Diamond and Dybvig
The coordination
Our model does not impose a
the sequential service
failure informs the discussion of
bank
sequential service constraint. Agents are able
complete state-contingent contracts on the order of the shocks. The amount of early liquidation by
predetermined by date
it
is
common knowledge
the shock in the
continuum of
first
clients,
wave,
it
R/2
faces
contracts rather than by coordination failure
that agents are symmetric, agents
know
tiiat r(a))
=
among
agents.
R. Since one-half of the agents receive
of risky claims are settled by the intermediaries. Also note that since each intermediary has a
no uncertainty on the quantities to be delivered
in
each aggregate state.
Diamond and Dybvig,
In
the bank run triggers unanticipated withdrawals and default by the bank.
may
In our model, agents recognize that intermediaries
default because they have limited
the collateral constraint of equation (4). However, in our complete markets
This
equilibrium.
intermediary's
is
Z
and impose
model default never occurs
in
because, with complete markets, agents anticipate the state of the world where the
liabilities
lead
it
and rewrite contracts so as to reduce the
to default,
point where no default occurs. Thus default
is
liabilities
central in the calculations of agents but,
and
exactly to the
as a result, does
not occur in equilibrium.^
Knightian Uncertainty and Flight to Quality
3
In this section
we
describe agents' decisions and equilibrium, and connect robustness concerns with flight-
to-quality episodes.
The
3.1
cost of locking collateral
The problem
and
of a financial intermediary
q{r,Lj), in order to
is
maximize revenue
to sell r(aj) risky claims
less
/
[{p{r,uj)
safe claims, at prices p(r, w)
the actuarial cost of providing liquidity insurance, subject to
the collateral constraint. Following the objective in
max
and ^(w)
(5),
- Pcfjl))^^) +
intermediaries solve:
{pis,io)
-
Picf^Jl)
+
^J2)))s{uj)]d^.
Because agents are ex-ante symmetric, their decisions are identical and intermediaries
same
R
prices to
and
S
all
it
affected. Likewise,
intermediary
objective
is
is
if
common knowledge
that
if
in the decisions
and price functions and
the
let
aggregate shock
hits,
one-half of the agents are
it
is
affected. Since the
has sold, we can rewrite
as:
(
practice, liquidity shocks trigger
p(r)
-
/?
p(.) - ^^^i^-M
/3^
'^)-('
dynamic problem
objective
is
-f
)
dynamic trading
features from our framework through a complete
(
in asset
and not
)
5.
martcets - for example, by shedding risky assets and
all
ex-ante.
While
interesting,
Arrow-Debreu markets assumption which,
we
moving
deliberately removed these
as usual, allows us to state a
as a static one.
to provide an
endogenous explanation
for
why
an exogenous segmentation/incomplete markets assumption.
isolates the
first
only concerned about the aggregate paj'out on the financial claims
to Treasury Bills - so that agents' responses are ex-post
Our
u>
the second aggregate shock hits, the other one-half of the agents
rnax
''In
we omit the argument
insurance at the
denote the aggregated claims sold to agents.
Recall that
its
agents. For brevity,
offer
mechanism we
highlight.
As
capital does not
The assumption
move
across markets, as opposed to making
of complete liquidity-shock-contingent claims
a matter of interpretation, the ex-ante insurance contracts can be thought of as
collateralized contingent credit-lines.
10
The intermediary must
satisfy the collateral constraint in (4):
Because of the linearity of the objective function,
to the intermediary's
problem yields that
R
and S are
P{'r)=P^For example, when
If
/?
=
if
the collateral constraint does not bind the solution
an
at
and p(s)
=
interior only
/3
if,
.
prices are actuarially fair at an interior solution.
1,
the collateral constraint
is
binding, then:
.M^/sf^ + f a„d.W.^*l+ie)+,.
where
/i
>
is
the Lagrange multiplier on the collateral constraint.
A
,6)
positive multiplier in the cost of
insurance reflects that the intermediary has limited resources and hence the opportunity cost of selling one
type of insurance claims
is
selling less of the other.
Safe claims use up twice the
risky claim, given our assumption that one-half of the agents are affected in a
sells
one
less safe claim,
then
it
can credibly
sell
factor twice the cost of the collateral constraint
We
are
most interested
writing liquidity insurance.
drop the
in the
when
To simphfy some
is
The
logic leads the
intermediary to
we assume
for
now
that
/?
=
and thereby
(6) to yield:
=
^.
(7)
the main result from modelhng the supply side of the economy.
equilibrium cost of "locking" collateral. In Section 5
3.2
shock. If the intermediary
selling safe claims over risky claims.^
of our expressions,
pM
This pricing expression
of collateral as a
opportunity cost of using collateral, as opposed to the actuarial cost of
term on the right-hand-side of
first
two more risky claims. This
first
amount
we
revisit the case of
/3
>
It
captures the
0.
agent's decision problem
Facing prices p{r) and p(s) each agent solves the decision problem in
max min cPZC^Mci) + Ci^Mco) +
r,s
(l
(3):
- 4,Ju
(8)
fl";ee
^Note that
to
if
the
first
and second waves were mutually exclusive events, then the intermediary could use the same
back liquidity insurance
in
each event
(i.e.,
it
would separate the
first
and second wave
collateral constraints).
collateral
In this case
the tradeoff of collateral would only be one for one between risky and safe claims, and the only binding constraint would be
that for claims sold for agents hit on the
first
wave. In contrast,
mutually exclusive.
11
in
our models waves take place sequentially and hence are not
where,
The
off
=
r
j.'-n^
"^-(^^
=
0(1) 0(1) + 0(2)
"^0(1)+O(2)'
expressions for cj and C2
only in the
first
first
+
C2-S,
s,
come from our
let
= 0„-0Jl).
'^w(2)
earlier definitions of risky
first
and
safe claims: the risky claim pays
and second shocks.
us consider the problem in (8) for a given value of
6*^ (i.e
we drop the min
operator).
order conditions are,
cf>^{l)u'{c^)
where
+p(r)r <wo,
p{s)s
shock, while the safe claim paj's off in both
For the moment,
The
ci
tp is
+
(PZilWici)
= Mr)
cj>^{2)u'{c2)
=
Ms)
the Lagrange multiplier on the agent's budget constraint.
Dividing the second equation by the
first
equation yields:
C(2)u^^p(£2
+ 0-(l)V(ci)
From
the definition of
9'^
in (2),
we
rewrite this expression as,
0(2)
_
^.'(C2)
Xl)u'(ci)
The
right
hand
right
hand
side
is
An
affects the agent's decisions
agent
who
considers a higher value of
fully collateralized
to set aside
more
by the intermediary,
We now
when he does
is
alternative values of
6'^.
first
analysis of the intermediary's problem,
C
^
in
9'^
chooses higher cn and lower
safe claims
and
demanding more
likely to
first
Ci.
less risky claims.
'
or second.
He can
Since
all
achieve this
claims have to
safe claims, the agent also forces the
bank
happen second. In terms of the bank-run example we
be
last in line at the
bank, requires the bank to
set aside
arrive at the bank.
turn to the robustness step.
theory, the agent
0(2)
collateral to cover shocks that will
have used, an agent who thinks that he
resources for
From our
by altering the perceived odds of being
consumption pattern by holding relatively more
be
p(r)
equal to one. Thus
u'(ci)
is, 6"^
p(.)
side of this expression are equilibrium prices.
we know that the
That
^''>
p(r)
The agent makes
his insurance choices while being robust to
In the game-theoretic language often used to describe
chooses r and
s,
then "nature" chooses
given his choices.
12
6*^
€
max-min expected
utility
to minimize the utility of agent A,
If
is,
>
u{ci)
when the agent
his true 9'^
9'^
u{c2),
=
is
u{ci)
The agent makes
that ^"
is
(8),
we
see that nature will set
more insured against being
equal to
is
— K. At
1
from
=
1
+
On
K.
u(c2), the expected utihty
Ci
>
C2 so that
u{c2), nature will
for the agent
behavior by nature. First, suppose that
nature will choose
9'^
=
I
is
that
do the opposite and
from the agent's decisions are independent of
his decisions anticipating this
near one. In this case
<
u(ci)
if
That
as high as possible.
than second, the worst-case
hit first
the other hand,
and thus 6",
(l)'^{2),
+ K. The
K
set
9'^.
small so
is
agent chooses
cj
and
Co to satisfy:
U'(C2)
^
0(1)
li'(ci)
As
K rises,
1
the right-hand side of this expression
+
1
0(2)
A'
and ci/co
falls
0(2)
the right-hand side
is
equal to one, and
=
ci
C2.
We
ci
<
C2,
if
refer to this situation as the fully robust case since
=
then nature will set 0^
2p(r)
=
2p,
=
1
— K.
economy
rises.
<
K<K=
^Lt —
1,
The
,^
-
if
the agent chooses
refer to this as the "partially robust" case.
For
K > K,
if
_
w'(ci)
We
agents' decisions are as
ci
=
=
C2.
price,
and decrease
an equilibrium in the robust economy:
constitute
U'{C2)
if
=
9"
0(1)
0(2)
1
+ K:
1
1
+
^
K
^
K = K:
U\C2)
^
u'ici)
3.3
because
Agents' demands are infinite at p
agents' decisions are as
0(2)
We
is
following proposition summarizes these results:
0(1)
•
This
Given the supply determined relative
unique.
is
effects.
Proposition 1 The following insurance decisions
For
0(^(2).
Anticipating this action, the agent will prefer to choose
a change in p only has income
monotonically toward zero as p
•
and
K rises past K, the agent will not change his decision.
Finally, note that equilibrium in this
p{s)
toward one. When,
'
agents' decisions are robust to their uncertainty over ^^^(l)
Notice that
falls
0(1)
1
<t>{2)l
+
K
=
1
refer to this as the "fully robust" case.
Flight to quality
In the partially robust solution, every agent chooses
choose r
=
ci
—
C2
and
s
=
C2.
As
K rises,
ci
>
C2.
To achieve
this
consumption pattern, agents
so that the uncertainty for agents rises, ci
13
and
co
converge to
each other. Agents lower r and increase
The pattern
At the
s.
fully robust solution, agents set ci
of shedding risky claims in favor of safe claims
is
The
facilities.^
The concern
of each agent
bank's limited credit
from
when
that his shock
is
facility will
^^'ill
=
0.
K
generates a "run" on
lines.
occur after the average agents' shocks, at which point the
By
have been depleted. °
prearranged at date
is
demands towards
can also interpret the
By
decision.
own
agents'
choosing r
"committed capital"
As
)
=
effect of
>
0, s
,
in the
a
rise in
setting r
=
0,s
>
0,
agents insulate themselves
by an agent preemptively drawing down a
when
r falls
of capital
where
think of these actions as agents shifting
unlikely that the intermediary's resources
it is
in flight to quality episodes.
K on equilibrium in terms of intermediaries' capital allocation
and there
is
is
free
is less
(
"trading capital" ) to allocate to the
first
is
locked-up
agents
who
trading capital to allocate flexibly. In the fully robust
locked-up and there
latter interpretation captures a feature shared
movement
we can
Alternatively,
are complete in our model, the future
these shocks occur, s of the capital of an intermediary
and r of the capital
6'^ rises,
As markets
credit line
agents force intermediaries to reserve some resources to cover the
0,
case, all of the intermediary's capital
The
0.
common
shift is
shocks, regardless of
receive shocks.
reflected
well capitalized intermediaries,
be depicted. This portfolio
We
may be
the fear of other agents' shocks arises.
action by each agent
(
and r
s
this concern.
at a date
will
=
agents recognize that banks have limited resources to back up their credit
In practice, the robustness action
their
C2
central to flight to quality episodes.
Interpreting the financial claims of agents as credit lines from banks, the rise in
banks' credit
=
is
no capital mobility.
by most
Bottlenecks arise
flight to quality episodes.
and markets appear segmented. For example,
markets where hedge funds specialized were particularly affected by the
in the Fall of
crisis,
1998 episode the
as (abundant) capital did not
flow into these markets.
For a fixed
case in
more
K
,
an alternative way to generate
flight to quality in
detail in Section 5, but the intuition
risk as agents
is
clear enough.
our model
A
fall in
Z
terms of financial intermediary
'Although the actions of agents
level, agents'
to reduce Z.
in
We
explain this
increases flnancial intermediary
grow concerned that intermediaries maj' not have enough resources
Agents worry about being second and raise 6^. Some episodes, such as the
in
is
to cover their shocks.
Fall of 1998,
can be thought of
risk.
our model are most naturally interpreted as a run on banks' credit
actions are also related to the
more commonly analyzed run on banks'
deposits.
facility, at
a deeper
As Diamond and Dybvig (1983)
emphasize, a deposit contract implements an optimal shock-contingent allocation of liquidity. In the fully robust equilibrium
of our model, agents choose an allocation that
their
own
allocations
'We
is
non-contingent on each other's shocks. Agents each hoard liquidity to cover
shocks, independent of other markets' shocks. In this sense, robust agents' preference for shock-independent liquidity
is
related to the behavior of panicked depositors in a
can also interpret robustness
of an investment bank.
Each
u>
in
terms of the internal
lobbies for
more
risk capital,
risk
bank run.
management
of an investment bank.
because of fear that other agents
the bank's total risk capital.
14
Each
tj is
a trading desk
will suffer losses soon,
reducing
Collective Bias
4
and the Lender
In the fully robust equilibrium of Proposition
1
of Last Resort
agents insure equally against
and second shocks. To
first
arrive at the equal insurance solution, robust agents evaluate their first order conditions
(equation 9) at
conservative probabihties:
C(l) = C(2).
Suppose we compute the probabihty of one and two aggregate shocks using agents' conservative
probabilities.
Then,
^(1)
=
2
/C(l)'i^in
where the two
in this expression reflects the fact that
only half of agents are affected by the
first
aggregate
shock. Likewise,
0(2)
=
2
/ C(2)f^.
Together, these computations imply that agents' conservative probabilities are such that,
0(1)
Implicit in the conservative probabilities, the
aggregate shocks. But we know (and
all
=
0(2).
economy
is
perceived as equally likely to receive one or two
agents know) that actually 0(1)
>
0(2),
which implies that agents'
conservative probabilities are collectively biased.
Since each agent
is
concerned about the scenario
in
which he receives a shock
last
and the intermediary's
resources have been depleted, robustness considerations lead each agent to bias upwards the probability
of receiving a shock later than the average agent.
But
collectively,
every agent cannot be later than the
"average."
These points carry over to the partially robust equilibrium.
1
+
min[A',
A'],
we
K
>
0,
robust agents set
B'^
=
implying that,
0-(2)
If
For any
=
(l
+ min[A^A'])^0::;(l).
take the average of the conservative probabilities on the
left
and
(11)
right
hand
side of equation (11),
we
find.
When
A"
>
the last inequality
is
strict,
showing that agents conservative probabilities are collectively
biased.
Note that each agent's conservative probabilities are individually
over
6uj, it is
possible that agent
from the aggregate does
it
ui
plausible.
Given the range of uncertainty
has a higher than average probability of being second. Only
becomes apparent that agents'
15
when viewed
collective probabilities are implausible.
These
observations motivate us to study
how a
central bank, which
is
interested in maximizing the collective, can
improve on outcomes.
Central bank objective and information
4.1
The
central
average,
also
bank maximizes the average
we assume
that the central
utility that
bank uses
its
own
assume that the central bank equally weights the
because agents are observationally identical at date
When
computing
this
probability assessments rather than the agents'.
We
agents derive from their choices.
utility of
every agent.
The
The former assumption
0.
is
latter
more
In
assumption follows
delicate.
agents have non-Savage preferences, at least two approaches seem defensible:
The
central
bank
objective uses both the agents' preferences and the agents' conservative probability assessments; Or, the
central
bank takes a more
paternalistic approach in
from consumption, but not evaluated at the agents'
We
which
it is
concerned with agents' ex-post average
utility
collectively biased conservative probability assessments.
follow the latter path but conclude the section by showing that both approaches have similar implications
for the welfare gains associated to the presence of a lender of last resort.
As noted
earlier in the paper,
decision rules of financial specialists
we think
(e.g.,
they reproduce the behavior of agents in
uncertainty.
Prom
function that
Thus,
let
this perspective,
may
it is
of the robustness preferences as a realistic depiction of the
worst-case scenario analysis). These preferences are useful because
flight to quality situations
not obvious that a central bank should build biases into
lead to obvious average losses, just because agents exhibit these biases.
us consider the foUowing central
V=
ci,wiC2,w are the
where they are faced with Knightian
J
bank objective
objective
^°
function:^'
+ 0^^(2)«(C2,„)]
[t'^'^ilMc:,^)
its
doj.
consumption resulting from agents' insurance decisions, while
bank's assessments of the probabilities of the relevant events.
We
4>^
(1), etc., are
the central
note again that the central bank uses
agents' utility function of u{c) to evaluate their consumption decisions, but discounts u{c) using
its
own
probability assessments, rather than the agents' conservative probability assessments.
Importantly, we do not assume that the central bank knows the true
biases stem from a lack of knowledge of
central
bank
how
O^^s of agents.
aggregates impinge on their activities,
lacks knowledge over 0^. In fact, for
all
of our results in this section
we
Since the agents'
also
assume that the
we can assume that the
'"Sims (2001) has made a related point in questioning the application of robust control to central bank policymaking. He
argues that max-min preferences are simply shortcuts to generate observed behavior of economic agents, but should not be seen
as deeper preferences.
robust control for
its
''We have omitted
Note, however, that the collective bias we identify would be solved by a central bank, even
decisions.
the constant term involving u in this objective.
16
if it
uses
bank knows
central
strictly less
than the agents;
that the
i.e.
K of the central
bank
is
larger than the
K of
agents.
The
central bank, like the agents,
affected
knows both aggregate
and that one-half of the agents
is
by each aggregate shock:
U^^\l)±o =
t^
U^J{2)d^
and,
^
Jn
We may
probabilities
Jn
imagine that the central bank and individual agents are able to form precise estimates over the
aggregate behavior of shocks from observing past data.
4.2
Collective risk
management and wasted
collatercd
Consider a central bank that alters agents' decisions by increasing
decreasing
collateral,
C2,u>
by the same amount.
This can be implemented,
by requiring agents to hold two more units of r and one
Ci,„
by an infinitesimal amount, and
amount
of intermediaries'
The value
of the reallocation
for a given
less unit of
s.
is:
V^''
=
^ [0S^(l)«'(ci,J
In the fully robust equilibrium, the
first
-
0^^(2)«'(c2,j]
order condition for agents
dio.
is,
W'(Cl,a;)
The budget
constraint for every agent
that,
is
p{r)r
Since the central bank knows that
conclude that Ci^^
=
ci for all w.
all
wq.
agents face the same prices and have the same wealth
Then, we can
v^''
+ p{s)s =
= «'(co/j€^(i)-€^(2)]rfw
=
^«'(ci)(</.(l)-0(2))
>
0.
u'jco,,^)
^
4>{1)
Ci for all uj.
Then
is,
1
^
Again, given that agents face the same prices and have the same wealth
=
can
write,
Similarly, in the partially robust case, the first order condition for agents
that Ci^u
level, it
substituting the agents'
first
17
level,
the central bank can conclude
order condition into the central bank's objective,
we
find,
y«^
'(1)
=
=
We
summarize these
Proposition 2 For any
1
1
2
2
1
+
K
>
0,
is
Thus by
Z).
many
The
tie
up
as
much
about individual B^s. In
important point
The somewhat
at
some
^" If
may
is
much
is
where the more
likely
away from
collateral as safe claims
do
toward more risky claims and
central
this sense, the central
shock (the
important when aggregate collateral
(i.e.
the collateral constraint
less safe claims,
may
is
is
intermediaries
never occur.
bank reaches these conclusions requiring only knowledge
that the central
Lender of
last resort
maximizes the expected
reallocating agent insurance
of their resources to cover a second-shock that
As we have remarked, the
probabilities.
differ
central bank that
safe claims in a scenairio
portfolio of insurance
shifting agents' decisions
Finally, note that the central
may
A
insurance
valuable to the central bank because from the central bank's perspective agents are
is
insurable with risky claims.
do not have to
4.3
first.
economy can improve outcomes by
limited because risk}' claims lock only half as
R/2+S <
much
and toward risky claims.
reallocation
one)
A'
agent decisions are collectively biased. Agents choose too
wasting aggregate collateral by using too
first
A'
results in the following proposition:
(ex-post) utility of agents in the
The
dw
+
0.
against receiving shocks second relative to receiving shocks
safe claims
1
1
<^(iy(ci)
>
0(2)
<j){l)
n
1
<Pt''m
0(i)«'(ci)
of aggregate
bank may be much more confused than individual agents
bank may be the
bank does not
suffer
from
least
informed agent of the economy. The
collective bias.^^
last resort
abstract reallocation experiment discussed above highlights
how
central
bank policy choices
from robust agents' policy choices. More concretely, we now focus on the value of a lender of
(LLR)
which
cost,
the central
it
(i.e.
in
assume that the central bank obtains resources ex-post,
can credibly pledge to agents
bank were
imagine a situation
liquidity crises
We
in the robust equilibrium.
to be uncertain
about the values oi <t>{l) and
which the central bank
is
However,
two-shock event. ^'^ In practice, this pledge
<^(2),
then we could overturn our
uncertain about these probabilities, and
the incidence of both shocks occurring).
•'overinsurance" bias of agents.
in the
this "bias"
is
its
In this case, the biased central
of a different nature than the one
may
result. In particular,
we
objective function overweights
bank
will also
we emphasize
be subject to the
as
it
would not be
collectively inconsistent with conditional probabilities.
^^As
in
at least,
it
collateral.
Woodford (1990) and Holmstrom and
Tirole (1998), the
LLR
has access to collateral that intermediaries do not (or
has access at a lower cost). Woodford and Holmstrom and Tirole focus on the direct value of intervening using this
Our
analysis focuses on the gains
beyond the
direct value of the intervention.
18
.
be supported by costly ex-post
and carried out by guaranteeing, against
inflation or taxation
liabilities of financial intermediairies
who have
We
sold financial claims to both markets.
default, the
analyze the impact
and marginal benefit of such a guarantee.
Formally, the central bank credibly expands the collateral of the financial sector in the two-shock event
by an amount Z'^
Since
we
ai^e
.
Thus, the collateral constraints on intermediaries are altered to
computing the marginal benefit
interested in
of intervention,
we study an
infinitesimal inter-
vention of Z'^
If
in
the central bank offers more insurance against the two-shock event, this insurance has a direct benefit
terms of reducing the disutility of an adverse outcome. The direct benefit of the
where the
2 in front reflects the fact that
consumption
for the
group
(of
an extra unit of collateral
measure one
half) that
is
for the
LLR
is,
second shock yields two units of
by the second wave.
hit
However, the anticipation of the central bank's second-shock insurance leads agents to
insurance decisions.
reoptiiTiize their
Agents reduce insurance against the second-shock (reduce safe claims) and increase
their first-shock claims (increase risky claims).
The
total benefit of the intervention includes
both the direct
benefit as well as any benefit from portfolio reoptimization:
duj.
Since Ci,„
+
ct.w
= {R +
S)
+S=
2{Z
+
Z'^), the reoptimization of
each agents' portfolios must
sum up
to
twice the increase in total resources:
As we have shown, a reoptimization
dci^u
dc2,u
dZ^
dZ(^
of agents' portfolio
_
~
r,
"'
away from
safe claims
and towards risky claims
has a first-order benefit in the robust equilibrium. The following proposition confirms this additional value
in the context of the lender of last resort;
Proposition 3 For any
K > 0,
the total value of the lender of last resort policy exceeds its direct value:
total .^ jrdirect
V za
> v^Q
Proof.
We
rewrite the total value of intervention as.
V'S"'
=
/^ [C^(l)«'(ci,.)^
=
0(2)u'(c2)+/ i0^^(l)u'(ci,„)-0^^(2)u'(c2.^)|^cL;.
19
+
</>5^(2)k'(c2,J
(2
dZ°
duj
We
can now substitute using the
first
order condition for agents' decision to arrive
Tlie last step follows since agents substitute
some
at,
of the second-shock insurance of the
LLR toward purchasing
first-shock insurance.
Welfare equivalence and paternalism
4.4
We
conclude this section by revisiting the non-paternalistic case, where we compute welfare using agents'
subjective and conservative probabilities.
beyond
its
objective direct value. There
but the direct benefit
rises since the
is
It
turns out that there
is
still
an important value
to the
LLR
no indirect benefit from reallocation (by the envelope theorem),
agents collectively exaggerate the likelihood of the extreme two-shocks
event
Jn
That
is,
the
LLR
=
^(2)u'(C2)
>
0(2)li'(c2)
has an extra benefit which comes from reducing the "anxiety" of robust agents.
Moreover, in the fully robust case we note that
to imply:
,,airect,int
>di7-ect,inf
^zo
_
-
0(1) +(j>{2)
'VK^J
'i^\^l „,li„
^,^
'
W
7)
\
ICoj
__ -Trtotal
-
l'2G
2
This means that in the fully robust case the value of anxiety reduction (from the agents' perspective)
to the value of insurance reallocation (from the central bank's perspective).
is
equal
While the exact equivalence
result carries over to the partially robust case only for the case of log-preferences,
case there
is
even in the partially robust
an extra benefit due to anxiety reduction.
Comparing the
paternalistic
and non-paternalistic perspectives, we note that agents value the intervention
more because they exaggerate the probability
of the event that
is
being subsidized, while the central bank
values the intervention because the subsidy reduces private collateral waste.
the econom}' were to be
drawn
infinite times, all agents
the direct ex-post benefit of intervention.
The agent
The
latter effect
would receive higher average ex-post
means that
utility
if
beyond
perceives this additional benefit ex-ante, although for
the "wrong" reasons.
20
Price of Liquidity and Collateral Risk
5
Up
now we have analyzed
to
of liquidity insurance
(i.e.,
/?
a model where intermediaries factor zero actuarial cost
=
For this case, the intermediaries'
0).
setting the prices
order condition pins
a particularly straightforward algebraic analysis,
fixing relative prices allows for
down
the
be constant and independent of the agents' portfolio decisions.
relative price of safe to riskless claims to
Although
first
when
it
hides
some
of
the mechanisms underlying the model.
When P >
agents' robustness concerns have important price effects. In the
0,
we elaborate on
these price effects to further clarify our model. In the second part
The
considerations exacerbate the costs of collateral reductions.
latter point
is
first
part of this section,
we show that robustness
the main substantive result
in this section.
As
in Kiyotaki
and Moore (1997), reductions
financial accelerator.
Our model
lateral lead to a fiight-to-quality
We
the
set
first
5.1
/?
=
1,
in collateral (or collateral values in their
illustrates a further channel for amplification.
which locks already scarce
now
so that intermediaries
Reductions in aggregate
and amplifies the
collateral
model) trigger a
col-
financial accelerator.
factor in the actuarial cost of disbursing insurance resources in
and second shocks.
Abundant
collatereJ
Z
Starting with a benchmark, suppose that
financial sector
is
enough so
large
is
that, at equilibrium levels of
demand, the
unconstrained in selling insurance. In this case the prices of the two financial claims just
reflect probabihties:
Pi-r)
=
-^
and
p{s)
=
.
Facing these prices, robust agents make their insurance decisions to
Given p{r) and p{s) we can rewrite the
At
K=
0,
agents choose ci
=
C2.
cP{2)u'{c2)
_p{s)
-0(l)u'(cx)
p{r)
first
It is
order condition for agents
easy to verify that ior
This choice remains optimal because by choosing
probabilities,
When
concerns.
and the robust decision
there
is
rule
still
abundant intermediary
Agents know that intermediaries
satisfy:
ci
=
C2,
K
>
as.
agents continue to choose
agents' decisions are robust to the
ci
=
C2.
unknown
does best.
collateral, the equilibrium
will
is
unaffected by agents' robustness
not default on their insurance claims.
21
Thus, they are not
concerned about their shock being
or second relative to other agents. Their uncertainty about the sliock
first
^^
ordering becomes irrelevant.
Using the budget constraint
for agents,
we can calculate that facing the actuarially
fair prices p{r)
and
p{s), agents will choose,
=
r
s
0;
=
=
Ci
=
C2
In order for intermediaries to not be collateral constrained in equilibrium,'^
Price of liquidity and aggregate shortage
5.2
Z
Suppose next that
falls
below
Then, at actuarially
Z_.
fair prices, agents'
demand
saturates the collateral
constraint of the intermediaries and prices rise to reflect the collateral Umitation:
where
>
fi
We
Q
is
M
4>{l)
^
,
J
,
0(1)
X
+
0(2)
the Lagrange multiplier on the intermediaries' collateral constraint.
can compute these prices explicitly using the budget constraint
and the intermediary
collateral constraint {^
Proposition 4 When
Z<
=
Pir)
Z_,
+s=
the prices of risky
5+
(0(1)
for the agent {p(r)r
and
safe claims as a function of Ci
- 0(2))|| and
p{s)
=
^-
(0(1)
-
fall in Ci
and equal
rise in co causes the prices of both risky
and safe claims
2.
A
fall in Ci
and equal
rise in c^ causes the relative price ratio,
^W —
3.
The economy has an aggregate liquidity shortage.
thereby decreasing
claims (point
cj
1), as
'"The irrelevance
previous section,
and increasing
Ct.
when
and
\,
to rise.
to rise.
agents' uncertaintj' rises they purchase
These actions lead
co are:
0(2))^^-
A
in the
wq)
Z):
1.
As we have shown
+ p[s)s =
more
to a rise in the equilibrium price of
safe claims,
both financial
well as to a rise in the relative price of safe claims over risky claims (point 2).
result in the case of
abundant
collateral also relates to discussions in the asset pricing literature as to
when
the actions of "rational" agents will eliminate any effects stemming from the biases of "irrational" agents. In our model, the
intermediaries are Bayesian and risk-neutral, but are subject to a constraint whereby they have to hold collateral to back-up
any short-sales of financial
and prices
securities.
When
there
is
sufficient
reflect the rational intermediaries' valuations.
It is
intermediary collateral, the short-sale constraint does not bind
a feature of our
model that agents' insurance decisions are not
collectively biased at the actuarially fair prices.
'^In the
insurance
/3
=
case
we have studied
earlier
Z
is
infinite
because intermediaries are always at a corner solution
sales.
22
in their
Point
3,
regarding the aggregate liquidity shortage,
A
shocks.
rise in
K
who
also are subject to first
and second shocks and wish to insure against these
increases the price of liquidity insurance for
Savage agents. In this sense,
our model
flight to quality in
clarify this
Suppose we introduce a small number of Savage (Bayesian)
point, consider the following thought experiment.
agents into the economy
an important aspect of our model. To
is
is
all
agents in the economy, both robust and
not just a problem affecting the decision rules
of robust agents. It has a larger cost because the actions of robust agents raise insurance prices and thereby
distort the insurance of all agents in the economy.
The mechanism behind
ularly transparent
have shown
if
earlier,
the increase in prices
we examine the
is
the collateral lock-up
The
risky claim.
is
indirect.
This
demand
free to
R/2
(i.e.
The
back risky claims. That
-H
5 <
is,
Agents increase their demand
since safe claims lock
up twice the amount of
amount
Z
is
used
inefficiently,
leading to a smaller effective
Z
we
Thus
causing
for safe claims,
of collateral that
collateral as risky claims
Z), as robustness concerns rise, agents force the intermediaries to lock
intermediaries' limited
partic-
for risky claims as robustness rises.
intermediaries to lock-up collateral to cover their shocks and thereby decreasing the
is
becomes
effect
risky claim price rises as robustness rises; but, as
robust agents actually decrease their
the increase in price of the risky claim
eiTect.
of the
up more
collateral.
economy and higher
insurance prices.
We
aries
can think of claim prices as the marginal cost of liquidity provision
have a limited amount of collateral to back
economy
the capacity of the
liquiditj' provision, at
the aggregate level,
Our model shows
to provide liquidity to agents.
economy. Since intermedi-
in the
Z
parameterizes
that the actions of robust
agents decreases the effective amount of liquidity provided to the entire economy.
Reductions
liquidity
in aggregate liquidity provision
premia are two central ingredients
market structure -
would also be
5.3
We
e.g.
if
each
w was
next study
how
a
fall in
in
most descriptions of
flight to quality events.
rise in financial
With a
richer
a separate asset market - the rise in the cost of liquidity provision
reflected as higher liquidity
Collateral reduction
by financial intermediaries and a corresponding
and
premia
flight
in the asset
markets where the agents are
active.
to quality
Z, for fixed K, affects the equilibrium of the economy.
described a flight to quality event in terms of increasing K.
We now
Thus
far
we have
show that our model generates
similar
implications from a decrease in Z.
The
first
order condition for an agent's decision
0(2) u'{c2)
'^</.(l)u'(ci)
^
is:
2wa21.0 +
p{s)
p(r)
where the second equality follows from Proposition
4.
23
(0(1)
(0(1)
-
0(2))ci
^'
0(2))C2
^
Additionally, from the collateral constraint of inter-
know
mediaries we
that
1 + 1=^.
Equations (12) and (13)
in
examining how
c\
jointlj' define
and
the equiUbrium values of ci and c^ as a function of Z.
Z
change as
cn
falls
below
and study the function a{Z). As we have remarked
Proposition 5 Suppose
comparing
Z
is,
Z
as
=
earlier, for
Z>
Moreover, suppose that
0.
= 2a{Z)Z
Let us wTite ct
Z_.
economies with different
equilibria across
That
falls.
K
that
(13)
levels of
Z
Z_,
ci is
— c ",.'^?
and
We are interested
c\
=
2(1
—
equal to co so that a(Z)
is
weakly decreasing in
the (unique) equilibrium, value of
,
q:(Z))Z,
=
Then
c.
a
|.
falls as
the ratio of insurance devoted to the first shock over that to the second shock
falls,
rises.
When
Proof.
K equals zero,
-
F(^"' 7\
=
6'^
From equation
1.
^~^(l)w'(2(l-a)Z)
The
first
order condition in (12)
sign{^) equals the negative
is
satisfied
of sign
(
-
1
0(2) u'{co) (
Z
<p{l)u'{ci)\
when F[a[Z), Z) =
da
This term
is
+
u"{c2)
T
C2—7
-
(0(l)-0(2))(l-a)
Uo-(0(l)-0(2))(l-a)Z
on
o^ 0(2)^'(C2) (U"{C2)
=
This term
is
is
negative. Therefore sign{j^)
negative for
The
falls,
(0(1)
K=
u'(ci)
to the first shock
first
We
+ (0(1)-0(2))qZ
;
also note that the equilibrium
is
unique since
^
is
more
likely
we have derived
in the
previous sections. As
reflects the limited availability of
Z
intermediary
then the second shock, the central bank allocates more resources
than to the second shock.
affected by Z. For
Z>
agents act as
this will leave
positive.
and second shock insurance
Consider next the robust economy where
If
u-o
>v
case corresponds to the central bank's solution
the allocation between
below Z.
^;o-(0(l)-0(2))(l-a)Z \
/
all a.
resources. Since the first shock
is
- 0(2))a
\
- 0(2))aZy'
(0(1)
u"{c^)
- 0(2))qZ
is
+
^
(0(1)-0(2))Z
+
(0(1)
t^o
u{c).
0(1) u'(ci) \u'{c2)
Wo
find that
u'(ci)
- 0(2))qZ
da
we
u"(ci)
positive given the regularity condition
9^
Implicitly differentiating F,
Ci
u'{c2)
(0(1)
0.
'
J
7i;o-(0(l)-0(2))(l-Q)Z /
u;o
define,
-g-
\
OF
dZ
we
^° ~ ('^(^^ ~ '^^^^^(^ ~ "^^
wo + (0(1) - 0(2))aZ
^'(^^'^)
'^(^^
(12)
agents set
Z^,
if
S^
them exposed
=
1,
they
ci
K>
0.
equal to
will
We begin with a heuristic description of how equilibrium
co.
Starting from this point, suppose that
decrease C2 and increase
to the possibility that
9^^ is
24
high.
ci slightly (see
Taking
Z
Proposition
falls slightly
5).
However,
this into account, agents will consider
when making
6'"
a larger
decisions. For
Z
close to
As
Z
falls sufficiently
the fully robust insurance decisions.
a small increase in 0"
Z_,
is
choose
sufficient for agents to
below Z, 6" reaches
maximum \+K,
its
with
c\
larger than C2. In this case, agent decisions are partially robust.
Proposition 6 Suppose
— c l^,,7
that
weakly decreasing in
is
Z
across economies with different levels of
agents
make
equilibrium decisions as
we have
,
Then comparing
c.
(the unique) equilibria
For a given value of Z, robust
the following results.
if,
=
eZ
1
+ min{K, K(Z -
Z)),
whCTG
— Z)
Since K{Z_
•
For
Z
such that 0"
•
For
Z
small enough that
We
Proof.
increases as
=
a =
An
for
\.
1
choose
falls.
fully robust.
agent decisions are partially robust.
,
61;::;
=
1
+
A'
if
a <
=
i;
61;;;;
economy
-
1
is
/"ST
if
-
u^p
q >
a
a value
i;
-
(0(1)
+
w;o
'
equilibrium in the robust
and
+ K) =0. d
to F{a, Z;9'^
=
—
is
1
For a value of
a
value of
Z
>
\
g'^'
•^'
First,
and
^(2))(1
- a)Z
(0(l)-</.(2))aZ
'
indifferent over
is
e
61^
=
that satisfies F[a, Z; 0")
is
we note that the proof of Proposition
negative for any 0"
uniquely determined since
— K) =
Z
satisfies
for
0.
Since F{a, Z\
which d
<
,
a, the equilibrium
is
0,
-
[1
where
+ K]
A', 1
if
optimal
is
0'^,
first
sets 6*"
order conditions
|.
F{a =
^,
Z=
Nature
is
Nature sets
at d.
order conditions
Nature
(i.e.
as the solution to
we
define
we have that d >
increasing in 0",
is
d
a. Similarly,
all
=
—
5 directly generalizes so that
Next, we define
0.
negative for
first
at a.
=
at
is
9'^) is
a
is
>
^, the equilibrium
the equilibrium
market clearing and agents'
>
^
market clearing and agents'
which Q > ^
for
satisfies
Oi
= 1+K
Z
nature at that a.
positive
of
9'^
4>{2))Z
{4>il)-4>i2))Z-
K{Z_ — Z), agent decisions are
+
-
increases (weakly) as
0(2)
v^^^-flc.
"^"
"4'{l)u'{2[l-a)Z)
'
We proceed as follows.
is
falls, 6'^
u'{2aZ)
^
will
Z
(.^(1)
define,
F(
Nature
1
v^
^-
I
•
_ 0(1) S^'o ^- ^{2)2wo +
ry(^
.
(i.e.
—
1
F() =
A'
0).
0'^
F(')
if
=
=
a >
K
=
as the solution
and
,
Finall}', for
indifferent over the choice of 9'^
\i
a <
\,
and
this
Likewise, for a value
0).
^
F[a,Z]9'^
'^
a.
+
1
a
g'^'
at this value of
a value of
and
Z
for
a
which
sets 9'^ such that
F{a^\,Z-9"J=Q.
We know
in a,
that at
Z =
Z,
we must have that d >
5 at
Z=
Z_;9'^
=
1)
=
0.
Since
F
is
Z. Moreover since decreases in
25
increasing in
Z
6*"
and
F
is
decreasing
decrease the function F(-),
d must
as
fall
Z
below \
The
Two
falls.
.
conclusions follow:
The statements
d'^
of an agent
is
It is
not possible that
a function of both
The
agents directly increaes &^.
this sense, our
5.4
Risky
K and
d'^
Z.
in
agents' response to a
model generates
collateral risk, the
risk in
for
Z
sufficiently
low a must
fall
falls,
fall in
their
Z
is
robust agents grow increasingly concerned
own
similar to
from either a
flight to quality
rise in
how
K
K
Likewise as
decisions.
rises,
agents respond to a
or a
fall in
the
rise in
Z.
and policy
collateral
(as in Kiyotaki
Z
As
making
In practice, the collateral of financial intermediaries
economy
and
of the proposition follow immediately.
about receiving a shock second and increase
K. In
a>\\
risky
is
and Moore, 1997). Although
comparative static with respect to
it
Z
is
and varies with the underlying state of the
beyond the scope of
this
paper to endogenize
provides some insight into the effects of collateral
our economy.
Starting from a state of the world where
suppose there
financial claims,
become
safe claims
is
Z
is
plentiful
and agents are confident that they hold
a shock that reduces Z. As collateral
falls,
safe
agents recognize that previously
risky as they lose their collateral backing. In response, agents act aggressively to shield
themselves from risk by shedding risky claims and purchasing safe claims, in a manner consistent with a
flight to quality.
These actions lock-up the
the effective collateral of the economy.
collateral of financial intermediaries,
Our model shows
and thereby further reduce
economy exacerbates the
that the robust
collateral
shortage caused by reduced aggregate collateral, amplifying the standard financial accelerator highlighted
by Bernanke and Gertler (1989) and Kiyotaki and Moore (1997), among
We
to
i.e.
is
can also think of the lender of
increased collateral
To a
The
will deliver
on
The problem
last resort policy in these terms.
that agents respond
large extent, private sector collateral will always be perceived as risky
-
A
is
intermediaries can never insulate themselves against default risk.
"riskless."
6
risk.
others.
central
bank promise,
if
credible,
lender of last resort policy derives benefits because agents are certain that the central bank
promise.
its
The
value of such riskless promises rises
when
private sector collateral
is
risky.
Moral Hazard and Irrelevance Critiques
In this section
central
bank
we
discuss our lender of last resort
intervention.
The moral hazard
recommendation
critique
is
in light of
that the anticipation of an explicit or implicit
insurance policy leads the private sector to underinsure against shocks and
to backfire.
The
irrelevance critique
is
two prominent critiques of
may
lead the central
bank policy
that the events warranting insurance, such as the Great Depression,
are so rare in economies with deep and sound financial markets, that while in principle the central
insurance
is
useful, in practice
it
has minimal ex-ante welfare implications.
26
bank
In discussing tliese critiques,
Moral hazard
6.1
The moral hazard
we
also extend our
model
to tire case oi
critique
In our model, in keeping with the moral hazard critique, agents
activities.
the reallocation improves (ex-post) outcomes on average.^®
We
waves of aggregate shocks.
predicated on agents responding to the provision of public insurance by cutting
is
from the publicly insured shock. However, when
reallocate insurance awa}'
in
2
critique
back on their own insurance
complements
N>
flight to quality is
Public and private provision of insurance are
our model.
note that the central bank can achieve the same distribution of insurance
intervene in the
the concern,
first
shock.
However the expected
intervention, since the central
is
bank rather than the
is
much
instead
it
commits
to
larger than the extreme event
private sector bears the cost of insurance against the
Agents would reallocate the expected resources from the central bank to the
(likely) single-shock event.
two-shock event, which
cost of this policy
if
exactly the opposite of what the central bank wants to achieve.
In this sense,
interventions in intermediate events are subject to the moral hazard critique.
More
^=
formally, let us return to the simpler
shock
in the first
model and note that the
direct benefit of intervention
is;
V^r^''^'^^'
=
2
/
0^^(l)u'(ci,^)do.
=
d{l)u'{C2).
Jn
The
Kjc^'i remains unchanged because the relevant constraint remains the constraint on
total benefit,
two-shock insurance;
V -total
^S^(l)-'(=i,
=
J^
=
J§f +
g^)
+
[05^(l)«'(c:,J
-
[0^^(l)^/(c,,J (2
4>il)n'ic,)
-
-
0^^(2)^.'(C3,
J
(2 -
€^(2)u'(c2,.)
<^^«(2)k'(c2,„)]
j^
•I
=
The
r direct, first
""-^
-— "(^^^l^r^\^)
f/
\
^
( t
l
+
\
min(A-,A-)
1
dC2.ui
du)
^d^
"^2,cj
^ xr direct, first
jdZ^^^^-
Agents reduce insurance against the first-shock (reduce risky claims) and increase their second-
shock claims (increase safe claims). This private insurance reallocation
policy.
has to be a
^^Note that
That
last
if
is,
the lender of last resort
facility, to
be
effective
offsets
some
of the benefits of the
and improve private financial markets,
not an intermediate resort.
the direct effect of intervention
inconsistent. This result
the policy
dw
)
anticipation of the central bank's first-shock insurance leads agents to reoptimize their insurance
decisions.
LLR
^
is
is
insufficient to justify intervention,
then the lender of
last resort policy
is
time
not surprising as the benefit of the policy comes precisely from the private sector reaction, not from
itself.
27
Multiple shocks
6.2
It is
clear that the
for late shocks;
LLR
but
if
should not intervene during early shocks and instead should only pledge resources
we move away from our two-shock model
potential waves of aggregate shocks,
To answer
experience
The
[T
<
A''
central
this question
how
injects 1/(A''
—T+
into the model.
The LLR
1/A^tli of the agents.
1) units of liquidity for all
N). For simplicity we focus on the
fully
more
realistic context
with multiple
late?
is
we introduce multiple shocks
waves of shocks, each affecting
bank
The value
late
to a
We
assume the economy may
policy takes the following form:
shocks after (and including) the Tth wave
robust case.
of the intervention as a function of
T
is,
N
i=T
=
where we used the
insurance.
As
fact that
T
one unit of additional collateral allows intermediaries to
sell A''
units of additional
before, the anticipation of the central bank's insurance leads agents to reoptimize their private
insurance decisions. Agents reduce insurance against the publicly insured shocks and increase their private
insurance for the rest of the shocks.
well as
The
total benefit of the intervention includes
any benefit from portfolio reoptimization:
N
'^
i=l
with:
V^^-A^
^ dZG
i
=l
so that,
In the fully robust case,
Ci
and
j^
are the
same
for all
i.
Then,
7
=1
= «'(ci)^E^(^)
1=1
28
both the direct benefit as
Note that
this expression
is
is
decreasing with respect to
is
strictly greater
than one
Of course, the above
it
independent of the intervention rule T. In contrast,
T
since the
T>
for all
1
(/i(i)'s
and
is
are monotonically decreasing.
amount
of resources available for intervention, the
—
1st shock,
it
LLR
should
first
completely replaces private insurance,
it
pledge
should
and so on.
Irrelevance critique
6.3
The
the ratio:
increasing with respect to T.
resources to the A''th shock and continue to do so until
N
Then
V^]^^<^^
result does not suggest that intervention should occur only in the A''th shock. Instead
suggests that for any given
then move on to the
apparent that
it is
multiple shock model also clarifies another benefit of late intervention.
insured by the
LLR become
resources for the central
increasingly less
bank
is
likely.
If
we take the
As
T rises,
case where the
constant, the expected cost of the
LLR
policy
events that are being
shadow
falls
as
T
cost of the
rises,
LLR
while the
expected benefit remains constant.
In other words, as
T
rises, it is
more
private resources to early and
quality
the private sector that increasingly improves the allocation of scarce
likely aggregate shocks,
phenomenon. In contrast, the central bank plays a decreasingly small
value of resources actually disbursed, as
Thus, while a well designed
(hence the irrelevance critique),
benefits
7
thereby reducing the extent of the flight-to-
come from the impact
LLR
its
T
terms of the expected
increases.
policy
main
role in
may
indeed ha\'e a direct effect only in highly unlikely events
benefits are
during more likely and
felt
less
extreme events. These
of the policy in unlocking private collateral.
Final remarks
a pervasive
phenomenon
that exacerbates the impact of recessionary shocks and finan-
Flight to quality
is
cial accelerators.
In this paper we present a model of this
financial specialists.
We show
that
when aggregate
phenomenon based on robust
intermediation collateral
is
decision
plentiful, robustness consider-
ations do not interfere with the functioning of private insurance markets (credit lines). However,
think that aggregate intermediation collateral
is
making by
when agents
scarce, they take a set of protective actions to guarantee
themselves safety, but which leave the aggregate economy overexposed to recessionary shocks.
In this context, a Lender of Last Resort policy
events.
The main
benefit of this policy
is
useful
if
used to support extreme rather than intermediate
comes not so much from the
29
direct effect of the policy during
extreme
events,
more
which are very
rare,
but from
its
abiHty to unlock private sector collateral during milder, and far
frequent, contractions.
The
implications of the framework extend bej'ond the particular interpretation
and policymakers. For example,
the polic3'maker as the
IMF
in
we have given
to agents
the international context one could think of our agents as countries and
or other IFFs. Then, our model prescribes that the
IMF
not support the
first
country hit by a sudden stop, but to commit to intervene once contagion takes place. The 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]
Barro, R.J., 2005, "Rare Events and the Equity Premium," Mimeo, Harvard University.
[2]
Bernanke, B., M. Gertler, and
S. Gilchrist,
1999,
"The Financial Accelerator
in a Quantitative Business
Cycle Framework," Handbook of Macroeconomics, Taylor and Woodford, eds.
[3]
WP #
[4]
[5]
11996.
Diamond, D.W. and P.H. Dybvig, 1983, "Bank Runs, Deposit Insurance, and
Political
Economy
Dow,
and
J.
Portfolio,"
[6]
NBER
Caballero, R.J. and A. Krishnamurthy, 2005, "Financial System Risk and Flight to Quality,"
91, pp. 401-419.
Werlang, 1992, "Uncertainty Aversion, Risk Avei'sion, and the Optimal Choice of
S.
Econometrica
60, p. 197-204.
Friedman, B., and Kuttner, K., 1993,
Stock,
J.,
Liquidity," Journal of
Watson, M. (Eds.),
New
"Why
does the paper-bill spread predict real economy activity."
In;
research in Business Cycles: Indicators and ForecEisting. University
of Chicago Press, Chicago.
[7]
Easley, David
and Maureen O'Hara, 2005, "Regulation and Return: The Role
of Ambiguity," working
paper, Cornell University.
[8]
Epstein, L., 2001, "Sharing Ambiguit}',"
[9]
Epstein, L. and
American Economic Review,
M. Schneider, 2004, "Recursive Multiple
91(2), 45-50.
Priors," Journal of
Economic Theory,
113(1),
1-31.
[10]
Epstein, L. and T.
Wang,
1994, "Intertemporal Asset Pricing
Under Knightian
Uncertaintj','
Economet-
rica 62, 283-322.
[11]
Gatcv, Evan, and Phil Strahan, 2005, "Banks' Advantage
in
Hedging Liquidity Risk:
Theory and
Evidence from the Commercial Paper Market," forthcoming. Journal of Finance.
[12]
Gilboa,
I.
and D. Schmeidler, 1989, "Maxmin Expected Utihty with Non-unique
Mathematical Economics,
[13]
Priors," Journal of
18, 141-153.
Greenspan, A, 2004, "Risk and Uncertainty
in
Monetary
Policj',"
Remarks
for the
AEA
Meetings,
January.
[14]
Hansen, L. and T. Sargent, 1995, "Discounted Linear Exponential Quadratic Gaussian Control,'
Transactions on Automatic Control 40, 968-971.
31
IEEE
[15]
Hansen, L.P. and T.J. Sargent, 2003, "Robust Control of Forward-looking Models," Journal of Monetary
Economics
[16]
50(3), 581-604.
Hansen, L.P., T.J. Sargent, G.A. Turmuhambetova, N. Williams, 2004, "Robust Control and Model
Misspecification," April,
[17]
Mimeo.
Holmstrom, Bengt and Jean
Economy, 106(1), 2/98, pp.
[18]
Tirole,
"Private and Public Supplj' of Liquidity,"
Journal of Political
1-40.
Kashyap, Anil, Jeremy Stein and David Wilcox, "Monetary Policy and Credit Conditions: Evidence
from the Composition of External Finance," Am.erican Economic Review, 83(1), 3/93, pp 78-98.
Moore, 1997: "Credit Cycles," Journal of Political Economy, 105
[19]
Kiyotaki, N. and
[20]
Krishnamurthy, Arvind, 2003, "Collateral Constraints and the Amplification Mechanism," Journal of
J.
Economic Theory,
[21]
(2),
211-248.
111(2), 277-292.
Krishnamurthy, Arvind, 2002, "The Bond/Old-Bond Spread," Journal of Financial Economics 66(2),
463-506, 2002.
W.
Xiong, 2001, "Contagion as a Wealth Effect", Journal of Finance 56, 1401-1440.
[22]
Kyle, A. and
[23]
Longstaff, R-ancis, 2004, "The Flight-to-Liquidity
Premium
in U.S.
Treasury Bond Prices," Journal of
Business 77(3), 511-526.
[24]
Routledge, Bryan and Stanley Zin, 2004, "Model Uncertainty and Liquidity," working paper, Carnegie
Mellon
[25]
Universit}'.
Sims, Cliristopher A., 2001, "Pitfalls of a
Minimax Approach
to
Model Uncertainty," working paper,
Princeton University.
[26]
Skiadas, Costis, 2003, "Robust Control and Recursive Utihty," Finance and Stochastics
[27]
Stock,
J.
and Watson, M., 1989.
New
7,
475-489.
indexes of coincident and leading economic indicators.
NBER
Macroeconomics Annual, 351-394.
[28]
Woodford, M., "Public Debt as Private Liquidity," American Economic Review, Papers and Proceedings
80,
May
1990, pp. 382-88.
32
52
7