DUPL
MIT LIBRARIES
ii
him
i
nil
mil
3 9080 02617 8241
in
t
iiflnoiiiun
u u
f
ill
Hlflfisl
ill
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium
Member
Libraries
http://www.archive.org/details/coordinationpoli00ange2
!4
31
115
t>3 -
£
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
COORDINATION AND POLICY TRAPS
George-Marios Angeletos
Christian Hellwig
Alessandro Pavan
Working Paper
May
Rev.
03-1
2003
September 2003
Room
E52-251
50 Memorial Drive
Cambridge, MA 02142
This paper can be downloaded without charge from the
Social Science Research Network
Paper Collection
http://ssrn.com/abstract=41 01 00
at
1
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
Coordination and Policy Traps*
George-Marios Angeletos
Christian Hellwig
Alessandro Pavan
MIT and NBER
UCLA
Northwestern University
This draft: September 2003
Abstract
This paper examines the ability of a policy maker to fashion equilibrium outcomes in an envi-
ronment where market participants play a coordination game with heterogeneous information.
We
consider a simple model of regime change that
literature.
In equilibrium, the policy
maker
is
embeds many applications examined
in the
willing to take a costly policy action only for
moderate fundamentals. Market participants can use
this information to coordinate
on
different
responses to the same policy choice, thus inducing policy traps, where the optimal policy and the
resulting regime
outcome are dictated by
multiplicity, robust predictions
the fundamentals, the policy
and
self-fulfilling
can be made. The probability of regime change
maker intervenes only
this region shrinks as the information in the
Key Words:
market expectations. Despite equilibrium
in
is
monotonic
in
a region of intermediate fundamentals,
market becomes
precise.
strategic complementarities, global games, signaling, regime change,
market expec-
tations, policy.
JEL
Classification
Numbers: C72, D70, D82, D84, E52, E61, F31.
'For encouragement and helpful comments, we are thankful to Daron Acemoglu,
chetta, Gadi Barlevy,
lison,
Marco Bassetto,
Nancy
Stokey, Jean Tirole,
Philippe Bac-
Olivier Blanchard, Ricardo Caballero, V.V. Chari, Eddie Dekel,
Paul Heidhues, Patrick Kehoe, Robert Lucas
Morris,
Andy Atkeson,
Muhamet
Jr.,
Yildiz, Ivan
Glenn
El-
Narayana Kocherlakota, Kiminori Matsuyama, Stephen
Werning, and seminar participants at
AUEB,
Berkeley,
Carnegie-Mellon, Harvard, Iowa, Lausanne, LSE, Mannheim, MIT, Northwestern, Stanford, Stony Brook, Toulouse,
UCLA,
2003
the Minneapolis Fed, the 2002
CEPR
SED
annual meeting, the 2002
ESSIM, and the 2003 SITE workshop. Email
van@northwestern.edu.
UPF
workshop on coordination games, the
addresses: angelet@mit.edu, chris@econ.ucla.edu, alepa-
G.M. Angeletos, C. Hellwig and A. Pavan
As Mr.
Greenspan prepares
to give a critical
new assessment
of the monetary policy
outlook in testimony to Congress on Wednesday, the central bank faces a difficult choice
in grappling with the
Street
and cuts
rates
economic slowdown. If
now
...
it
heeds the clamour from
much
of Wall
risks being seen as panicky, jeopardizing its reputation for
policy-making competence. But
to develop
it
if it
waits until
even more powerful downward
March
momentum
20,
it
risks allowing the
economy
what could prove a crucial three
in
weeks. (Financial Times, February 27, 2001)
Introduction
1
Economic news anxiously concentrate on the information that
the competence and intensions of the policy maker,
policy measures,
how markets may
interpret
convey about
and react to
and whether government intervention can calm down animal
markets to coordinate on desirable courses of actions.
policy
different policy choices
spirits
different
and ease
This paper investigates the ability of a
maker to influence market expectations and control equilibrium outcomes
in
environments
where market participants play a coordination game with heterogeneous information.
A
large
number
of
economic interactions are characterized by strategic complementarities, can
be modeled as coordination games, and often exhibit multiple
expectations. Prominent examples include
equilibria sustained
by
self-fulfilling
bank runs (Diamond and Dybvig, 1983), currency
(Obstfeld, 1986, 1996; Velasco, 1996), debt crises (Calvo, 1988; Cole
and Kehoe, 1996),
crises
financial
crashes (Freixas and Rochet, 1997; Chari and Kehoe, 2003a,b), and regime switches (Chamley,
1999). Strategic complementarities also arise in economies with production externalities (Bryant,
1983;
Benhabib and Farmer, 1994), network externalities (Katz and Shapiro, 1985, 1986;
Farell
and Saloner, 1985), imperfect market competition (Milgrom and Roberts, 1990; Vives 1999), Keynesian frictions (Cooper and John, 1988; Kiyotaki, 1988), thick-market externalities (Diamond,
1982; Murphy, Shleifer and Vishny, 1989), restricted market participation (Azariadis, 1981), and
incomplete financial markets (Bernanke and Gertler, 1989; Kiyotaki and Moore, 1997; Angeletos,
2003).
l
Other examples include lobbying,
political reforms, revolutions, riots,
and social change
(Kuran, 1987; Atkeson, 2000; Battaglini and Benabou, 2003).
Building on the global-games work of Carlsson and
Van Damme
(1993), Morris
and Shin (1998,
2000, 2001) have recently argued that equilibrium multiplicity in such coordination environments
'An
excellent overview of coordination
games
in
macroeconomics
is
in
Cooper
(1999).
Coordination and Policy Traps
is
often "the unintended consequence" of assuming
of the
game
beliefs
and actions
3
common knowledge
of the payoff structure
(the "fundamentals"), which implies that agents can perfectly forecast each others'
in equilibrium.
When
instead different agents have different private information
about the underlying fundamentals, perfect coordination on multiple courses of action
possible.
2
no longer
This argument has been elegantly illustrated by Morris and Shin (1998) in the context
of self-fulfilling currency crises.
willingness
is
and
ability of the
When
speculators receive noisy idiosyncratic signals about the
monetary authority to defend the currency (the fundamentals), a
unique equilibrium survives, in which devaluation occurs
and only
if
a critical state. This unique threshold in turn depends on
all
if
the fundamentals
fall
below
policy variables affecting the payoffs
For example, raising domestic interest rates, imposing capital controls, or
of the speculators.
otherwise increasing the cost of attacking the currency, reduces the set of fundamentals for which
devaluation occurs. Morris and Shin hence argue that, "in contrast to multiple equilibrium models,
[their]
model allows analysis of policy proposals directed
at curtailing currency attacks."
However, policy choices also convey information. For example, when the fundamentals are so
weak that the
collapse of the currency
when
measures. Similarly,
and the currency
faces
inevitable, there
is
is
no point
no threat, there
may permit
is
no need to
act.
we analyze endogenous
participants),
game
of Morris
who may
sufficiently large.
A
The
policy in a global coordination game.
and Shin (1998). There
status quo
policy maker,
weak nor too
strong. This
maker to fashion equilibrium outcomes.
either attack the status
or abstain from attacking.
minuscule
Therefore, whenever the policy maker
model of regime change that embeds many applications examined
currency crises
is
coordination on multiple courses of action in the market, thus
interfering with the ability of the policy
In this paper,
adopting costly defense
the fundamentals are so strong that the size of the attack
intervenes, the market can infer that the fundamentals are neither too
information in turn
in
who
is
is
is
We consider a simple
in the literature, including the
a large number of private agents (market
quo (undertake actions that favor a regime change)
abandoned
if
and only
if
the mass of agents attacking
is
interested in influencing the probability of regime change,
controls a policy instrument that affects the agents' payoff from attacking. Let 9 parametrize the
policy maker's willingness and ability to defend the status quo (the fundamentals).
maker moves
first,
setting the policy on the basis of her knowledge of
9,
The
policy
and agents move second,
deciding whether to attack on the basis of their idiosyncratic noisy signals about 9 and their
observation of the policy choice. In the context of speculative currency attacks, the status quo
is
a
"See Morris and Shin (2001) for an extensive overview of the global-games literature. Earlier uniqueness results
are also in Postlewaite
and Vives (1987) and Chamley (1999).
G.M. Angeletos, C. Hellwig and A. Pavan
4
may attempt
currency peg, which the monetary authorities
to defend by raising domestic interest
rates or imposing capital controls; in industries with network externalities, a regime change can
be interpreted as the adoption of a new technology standard, which the government
encourage with appropriate subsidies or regulatory incentives; in the context of
may wish
to
political change,
the status quo corresponds to a dictator or some other sociopolitical establishment facing the risk
of collapse
if
attacked by a large fraction of the society.
The main
result of the
paper (Theorem
1)
establishes that the endogeneity of the policy leads
to multiple equilibrium policies and multiple coordination outcomes, even
when
the fundamentals
modes
Different equilibria are sustained by different
are observed with idiosyncratic noise.
coordination in the agents' response to the same policy choice, and such coordination
is
of
possible
only because a policy intervention signals that the fundamentals are neither too weak nor too
strong.
If
private agents coordinate on the
maker never
in
finds
it
same response
to any level of the policy, the policy
optimal to intervene. Hence, there always exists an inactive policy equilibrium,
which the policy maker
sets the
same
equilibrium in the continuation game.
If
policy for
9
all
and private agents play the Morris-Shin
instead private agents coordinate on a lenient continuation
equilibrium (not attacking the status quo)
when the
policy
maker
raises the policy sufficiently high,
and otherwise play an aggressive continuation equilibrium (attacking
signals), the policy
maker
finds
it
[9*, 9**),
below which a regime change occurs, are
all
maker
reveals information that
if
and only
if
9
<
9*.
raises the
Which equilibrium
is
spared from an attack, and the threshold 9*
is
determined by
result thus manifests a kind of policy traps:
which the policy maker
equilibria, in
and the regime collapses
played, the level of the policy above which the regime
the policy
low private
optimal to intervene for an intermediate range of fundamentals.
Hence, there also exists a continuum of active-policy
policy only for 9 6
for sufficiently
self-fulfilling
market expectations. This
In her attempt to fashion the equilibrium outcome,
market participants can use to coordinate on multiple
courses of action, and finds herself trapped in a position where both the optimal policy and the
regime outcome are dictated by arbitrary market sentiments.
The second
result of the
paper (Theorem
nificantly reduces the equilibrium set as
2) establishes that information heterogeneity sig-
compared
to
common knowledge and
predictions despite the existence of multiple equilibria.
change
is
monotonic
in the fundamentals, the policy
In particular, the probability of a regime
maker
is
"anxious to prove herself with a
costly policy intervention only for an intermediate region of fundamentals,
shrinks as the information in the market becomes precise.
equilibria.
What
is
enables meaningful
more, none of these predictions could be
and the "anxiety region"
In essence, there
made
if
is
a unique class of
the fundamentals were com-
Coordination and Policy Traps
mon
5
knowledge. Our results thus strike a delicate balance between equilibrium indeterminacy and
predictive ability.
In the
benchmark model, the policy maker
market expectations
any given equilibrium.
in
sunspots on which market participants
of the policy
may now
may
faces
no uncertainty about the aggressiveness of
We relax this
assumption in Section 5 by introducing
condition their response to the policy
lead to different coordination outcomes along the
to currency crises, these results help reconcile the fact
same
The same
level
equilibrium. Applied
documented by Kraay (2003) and others
that raising interest rates or otherwise increasing the cost of attacking the currency does not
systematically affect the likelihood or severity of a speculative attack - a fact that prima facie
contradicts the policy prediction of Morris and Shin (1998).
sense of the Financial Times quote:
The market may be
Moreover, these results help make
equally likely to "interpret" a costly
policy intervention either as a signal of strength, in which case the
most desirable outcome may be
attained, or as a signal of panic, in which case the policy maker's attempt to coordinate the market
on the preferred course of action proves to be
On
The
in vain.
a more theoretical ground, this paper introduces signaling in global coordination games.
receivers (private agents) use the signal (policy) as a coordination device to switch
lenient
and aggressive behavior
Our
in the signaling game.
in the global coordination
between
game, thus inducing multiple equilibria
policy traps are thus different from the multiplicity of equilibria in
standard signaling games, which
is
sustained by different systems of out-of-equilibrium beliefs and
is
not robust to proper forward induction refinements. They are also different from the multiplicity in
standard global coordination games with exogenous public signals (Morris and Shin, 2001; Hellwig,
2002).
The
fulfilling
informational content of the policy
expectations of the market.
equilibria in the policy (signaling)
is
endogenous and
Most importantly, our
game, not the multiplicity
is itself
result
is
the result of the
self-
about the multiplicity of
in the coordination
game.
Finally,
our policy traps are different from the expectation traps that originate in the government's lack
of
commitment
(Obstfeld, 1996; Chari, Christiano
Christiano, 2002). Indeed, the policy
particular policy, even
The
rest of the
if
maker need not have any incentive ex ante to commit to a
that would ensure a unique equilibrium.
paper
rium concept. Section
and Eichenbaum, 1998; Albanesi, Chari and
is
organized as follows: Section 2 introduces the model and the equilib-
3 analyzes equilibria
and Section 4 examines robust predictions. Section 5
introduces sunspots. Section 6 concludes. All formal proofs are confined to the Appendix.
G.M. Angeletos, C. Hellwig and A. Pavan
6
The Model
2
Model Description
2.1
Consider an economy populated by a continuum of measure-one of private agents (also referred to
market participants), indexed by
as
i
and uniformly distributed over the
two possible regimes, a status quo and an
Each agent can choose between two
alternative.
(take an action that favors the status quo). In addition, there
in defending the status
We
let
is
a policy maker
quo and controls a policy instrument that
Q
£
9
There are
interval.
actions,
quo (take an action that favors a regime change) or abstain from attacking
either attack the status
attacking. 3
[0, 1]
who
interested
is
affects the agents' payoff
from
parametrize the strength of the status quo, or equivalently the policy
maker's willingness and ability to defend the status quo against a potential attack.
9
is
private
information to the policy maker and corresponds to what Morris and Shin (1998) refer to as the
"fundamentals"
.
For simplicity, we
=R
let
and the
common
prior shared by the private agents
be a degenerate uniform. 4
The game has
on
In stage
9.
2,
three stages. In stage
1,
the policy maker learns 9 and sets the policy contingent
each agent decides whether to attack after observing the policy choice and after
receiving a noisy private signal X{
=
+ e^
9
about
precision of the agents' private information about 9
and independent
The
9.
and
of 9, with absolutely continuous c.d.f.
£,
$
scalar e G (0, oo) parametrizes the
is
Finally, in stage 3, the policy
i.i.d.
and density
continuously differentiable over the entire real line (unbounded
[— 1,+1] (bounded support).
noise,
full
tp
across private agents
strictly positive
and
support) or a closed interval
maker observes the aggregate
size of
the attack, denoted by A, and decides whether to maintain or abandon the status quo.
The
payoffs for a private agent
an agent who decides to attack
for
otherwise. Letting a
1
is
decides not to attack are normalized to zero.
>
b
in the
denote the probability agent
regime changes, agent
this
who
i's
payoff
is
=
a
u (ai,D)
=
given by Ui
z
r
We
4
the
=
c/(b
+
c)
G
(0, 1)
abandoned and — c <
attacks the regime and
i
[Db+ (1 — D)(-c)]. Up
D
the probability the
to a linear transformation,
results hold for
game remains
common
any interval
"global."
We
a
l
(D -r),
parametrizes the agent's cost of attacking. For simplicity, we assume
adopt the convention of female pronouns
Our
general
is
payoff
equivalent to
is
where
l
event the status quo
The
0CK and
refer to
for the policy
any
Morris and Shin (2001)
priors in global coordination games.
maker and masculine pronouns for the agents.
and continuous density over 0, provided that
strictly positive
for
a discussion of the role of degenerate uniform and
Coordination and Policy Traps
the policy maker controls r directly.
We
=
1Z
let
the cost associated with raising the policy at level
The
status quo against an attack of size A.
U(r,A,D,e)
V
is
increasing in 9
and decreasing
simplify the exposition,
abandoned
and x
=
x < x
for
in stage 3
sup {x
it is
:
if
and only
dominant
>
9\x)
for
if
0}
A
.
>
x
policy,
C(r)
and V(9, A) the net value of defending the
C is increasing in r,
exceeds
and
9.
<
For 9
it is
be the domain of the
thus given by
is
= (l-D)V(6,A)-C(r).
and both
V(9,A)
let
Define 9
=
0,
9
V
:
,
>
is
inevitable
9 no attack
dominant not to attack. The
is
To
are continuous.
= 9 — A. Hence, the status quo is
= 1, x = inf {x Pr(f9 < 9\x) < 1}
Similarly, for 9
the "critical range" of fundamentals for which the regime
C
and
9 the collapse of the regime
an agent to attack.
regime change and thus for x
r,
(0, 1)
policy maker's payoff
we normalize C(r) —
<
Pr(9
in A,
C
[r,f]
7
interval
and thus
can ever trigger a
[9, 9]
thus represents
sound hut vulnerable to a
sufficiently
large attack.
2.2
Model Discussion and Applications
Many
of the simplifying assumptions in the
the policy maker
may
model are not
For example,
essential for the results.
favor a regime change instead of the status quo.
Similarly, the cost of
the policy could depend on the fundamentals and the reaction of the market, and the value of
preserving the status quo could depend on the level of the policy. Moreover, the policy
have private information about
that the continuation
game
strategic complementarities
model
9,
but this information need not perfect. Finally,
following the policy choice
is
irrelevant.
essentially reduces to a two-stage
is
it
maker must
is
important
a coordination game, but the source of
Indeed, although described as a three-stage game, the
game, where in the
first
stage the policy
maker
information about 9 and in the second stage private agents play a global coordination
response to the action of the policy maker.
broad
class of coordination
Currency Crises.
6
In short,
we suggest
if
Note that x
8 were
6
=
common
8
—
e
may
game
well
fit
in
a
environments. Below we discuss two concrete examples.
Let the status quo be a currency peg, the private agents be speculators,
and the policy maker be a central bank. Each speculator
5
that our analysis
signals
and x
knowledge,
= 8 + e if £ has bounded
x = 8 and x = 8.
is
endowed with a unit
support, whereas x
= — oo
and x
= +00
if
of wealth.
Let
£ has full support;
Stage 3 in our game serves only to introduce strategic complementarities in the actions of the private agents. All
the results would hold true
if
the status quo were exogenously abandoned whenever
^4
exceeds
8.
G.M. Angeletos, C. Hellwig and A. Pavan
8
a. (
be the fraction of wealth that speculator
i
converts to foreign currency and invests abroad
(equivalently, the probability he attacks the peg)
with
the size of the speculative attack. Denote
D the probability of devaluation, 5 the domestic interest rate, 5*
devaluation premium, and
payoff of a speculator
Ui
Up
A
and
=
t
the foreign interest rate,
the transaction cost associated with investing abroad.
7r
the
The expected
then given by
is
-
(1
Oi){l
+
5)
to a linear transformation, this
monetary authority may increase
r
+
is
+ 6*
Oi[D(l
+7T
equivalent to Ui
- 1) +
=
a.i(D
-
(1
—
£>)(1
r),
+
5*
where r
-
=
£)].
(S
—
by raising domestic interest rates or transaction
+
t)/ir.
The
costs. Let
C{r)
5*
represent the cost of such policies, 9 the bank's willingness and ability to defend the currency, and
=
V(9,A)
the bank
—A
is
the net value of maintaining the peg against an attack of size A.
U=
thus
directly into the
(1
— D)V(9,A) —
We
C(r).
model of regime change of
reduce to that of Morris and Shin (1998);
payoff of
conclude that speculative currency attacks
this paper. If r
if
The
in addition 9
were exogenously
were
common
fixed, the
map
game would
knowledge, the game would
reduce to that of Obstfeld (1986, 1996).
Revolutions and Political Reforms.
lishment)
who can be overturned
the regime.
If
only
There
a dictator (or some other sociopolitical estab-
is
a sufficiently large fraction of the population raises against
if
the revolution succeeds, the citizens
who supported
enjoy the benefits associated
it
with democracy and freedom, as well as the moral satisfaction of having done the right thing. Those
who
instead supported the dictator
or face sanctions from the
and
may
s
>
new
may
regime;
the potential sanctions.
On
free ride
on some benefits but may also loose some
>
m>
let b
denote the benefits,
the other hand,
imprison, torture, and punish the rebels, and
if
may
the revolution
This
is
= D [(n{b + m) +
thus
-
a r ) (6
again strategically equivalent to u r
represent the cost of
V(9,A)
(1
=
9
—A
U — (1 —
=
more sever punishments
-
s))
+
(1
- D)
ai(D-r), where
i
We
is
repressed, the dictator
[a t
r
(-p)
=
(p
+
(1
- a*) q]
+ q)/{p + q + s + m).
Let C{r)
to the rebels or higher rewards to the supporters
The
and
payoff of the dictator
is
conclude that this game also maps into our model.
For a more detailed discussion about the value of maintaining the peg and the costs of defensive
Drazen (2001).
p
thus
is
the net value of repressing a revolution of size A.
D)V(9,A) — C(r).
the moral satisfaction,
favor and reward the supporters; let
denote the penalties and q the rewards. The payoff of citizen
uz
rights
policies, see
Coordination and Policy Traps
9
Equilibrium Definition
2.3
we
In the analysis that follows,
sitive to
restrict attention to perfect
Bayesian
equilibria that are not sen-
whether the idiosyncratic noise in the observation of the fundamentals has bounded or
unbounded
We
support.
(full)
refer to equilibria that
This refinement imposes no restriction on the
of the information structure as robust equilibria.
precision of private information and, as
we
can be sustained under both specifications
explain in Section
4,
only eliminates strategic effects
that depend critically on the support of the signals.
A
Definition 1
r(9)
perfect
Bayesian equilibrium
is
a set of functions r,a,A,D,fi, such that:
e &vgmaxU(r,A(9,r),D(9,A(9,r),r),9);
(1)
+oo
max f u(r, a, D
a6[0,l]
a(x,r) G arg
(9,A(9,r),r))du(9\x,r) and A(9,r)
=
J
D(9,A,r) earg max U(r,A,D,9)
and
D(9)
=
—
=
[9
Definition 2
A
where Q(x)
same
the
for
:
ip
all
6
(^7^)
perfect
(2)
(3)'
v
-
G(x) and u(9\x,r)
€"
>
0}
is
satisfies
Bayes' rule for any r G r(0(x)),
(4)
the set of fundamentals 9 consistent with signal x.
Bayesian equilibrium
probability of regime change
robust
is
D{9) can
idiosyncratic noise in the agents' observation of
r(8)
(^) dx;
D(9,r(9),A(9,r(9));
£>e[o,i)
fi(6\x,r)
a(x,r)ip
_ OQ
Q
if
and only
if
the
be sustained with both
same
policy r(0)
and
bounded and unbounded
9.
the equilibrium policy, u(9\x,r) the agent's posterior beliefs, a(x,r) the agent's best
is
response, A(9, r) the corresponding size of an attack, 8 and D{8) the equilibrium probability of
regime change.
Conditions
whereas condition
(1),
in
follows,
we
if
(3)
guarantee that strategies are sequentially optimal,
never assign positive measure to fundamentals 9 that
and are pinned down by Bayes'
(1987).
A
perfect Bayesian equilibrium satisfies the intuitive criterion test
no 9 would be better
off
by choosing r
^
beliefs that assign positive
dominated
and only
in equilibrium; that
That A(9,r)
=
°°
f_
is, if
—-) dx
a(x,r)il> (-
if
if
r{9) should private agents' reaction not
be sustained by out-of-equilibrium
8
rule along the equilibrium path. In
also verify that all robust equilibria satisfy the intuitive criterion, first introduced
Cho and Kreps
and only
and
(4) requires that beliefs
are not compatible with signal x
what
(2)
measure to types
for
U (9) > U (r,A(9,r), D(9,A(9,r),r),9)
whom
r
is
for all 9,
r,
represents the fraction of agents attacking the status quo follows directly
from the Law of Large Numbers when there are countable
infinitely
many
agents; with a continuum, see
Judd
(1985).
G.M. Angeletos, C. Hellwig and A. Pavan
10
and A(9,r) satisfying
6 for
to
whom
r
(2)
with
dominated
is
€ A4(r), where U(9)
/j,
and A4(r)
in equilibrium,
any 9 € 6(r) whenever 0(r)
is
is
the equilibrium payoff, Q(r)
is
the set of
the set of beliefs that assign zero measure
0(x). 9
C
Policy Traps
3
Exogenous versus Endogenous Policy
3.1
Suppose
moment
a
for
to that of Morris
that the policy
and Shin (1998),
in
exogenously
is
which case iterated deletion of
Proposition
(Morris-Shin) In
1
the global coordination
a unique perfect Bayesian equilibrium, in which
status quo
is
abandoned
and only
if
if
thresholds
dominated strategies
strictly
beliefs.
with exogenous policy, there exists
an agent attacks
< 6ms- The
9
game
Our model then reduces
r.
if
and only
xms and 6ms
if
x
< Xms an d
the
& re decreasing in r
are defined by
MS = l-r =
.
An
Thus,
agent finds
xms
lM g~
ty
=
unique equilibrium profile and a unique system of equilibrium
selects a
and
fixed, say at r
(
)
solves r
it
optimal to attack
=
1
—
I
MS
.
9(
and only
The
(5)
y
if
r
< n(D(6) =
l\x)
=
1
fraction of private agents attacking
- *
is
I
x - u ms
then A(6)
=
J
•
It
follows that the status
6ms = A(0ms) — ^ ( XM f~
The higher the
that
IA/g ~
\I/
if
XMS - 9Ms
xms and 9ms
e
)
.
quo
is
abandoned
if
and only
Combining the above conditions
cost r, the less attractive
it is
for
if
6
< 6ms, where 6ms
solves
gives (5).
an agent to attack the status quo.
are decreasing functions of r, which suggests that the policy
It
follows
maker should be
able to reduce the likelihood and severity of an attack simply by undertaking policies that reduce
the individual payoff from attacking the regime. Indeed, that would be the end of the story
policy did not convey any information to the market. However, since raising r
maker
will
not intervene
signals that 6
is
if
the collapse of the regime
not too low.
On
is
sufficiently high, in
which case there
is
Formally, 6(r)
M{r)}, wherejW(r)
=
=
{9 such that
{fi
U (0) > U
satisfying (4)
and hence any policy intervention
faces at
when
most a small attack when 6
no need to take costly policy measures. Therefore, any
attempt to defend the status quo by raising r
9
maker
the
costly, the policy
the other hand, as long as private agents do not attack
their private signals are sufficiently high, the policy
is
inevitable,
is
if
is
correctly interpreted by the market as a signal of
(r,A(9,r), D{6, A(0,r),r),0) for any A(0,r) satisfying (2) with
and such that fi{8\x,r)
=
for
any 9 e 6(r)
if
Q{r)
C
Q(x)}.
ji
G
Coordination and Policy Traps
11
intermediate fundamentals, 10 in which case coordination on either an aggressive or a lenient course
of action
becomes
possible. Different
maker and
policy
Theorem
1
modes
of coordination then create different incentives for the
result in different equilibrium policies.
(Policy Traps) In the global coordination game with endogenous
multiple perfect Bayesian equilibria for any e
(a)
There
minimizing
r*
G
€
9*
an equilibrium in which the policy maker
[6*
,
9**},
and the status quo
is
abandoned
and only
if
9
< 9ms-
sets either r or r*
if
=
Let r solve C(r)
and only
if
9
^
<
1
—
the policy
,
9*
,
for any
r;
is
raised
where
=
is
independent of e and can take any value in [9,9ms], whereas the threshold 9**
C(r*)
and
=
9**
All the above equilibria are robust,
strategies for the agents that are
policy
if
sets the policy at its cost-
9*
9*
increasing in e and converges to 9* as e —*
The
abandoned
is
a continuum, of active-policy equilibria:
is
(r,r\, there is
The threshold
is
and the status quo
level r for all 6
at r* only for 9
0.
an inactive-policy equilibrium: The policy maker
is
There
(b)
>
policy, there exist
maker
is
+e
U/' 1
(l
- ^0*) -
(0*)1
(6)
.
0.
and can
satisfy the intuitive criterion,
monotonic in
be
supported by
x.
thus subject to policy traps, which contrasts with the policy conjecture of
Morris and Shin. In her attempt to use the policy so as to fashion the equilibrium outcome, the
policy
maker
reveals information that facilitates coordination
market and hence
on multiple courses of action
in the
finds herself
trapped in a situation where the optimal policy and the eventual
fate of the regime are dictated
by the arbitrary aggressiveness of the market expectations. The
proof of Theorem
1
follows
from Propositions 2 and
With endogenous
policy, the
in the next
two
sections.
Morris-Shin outcome survives as a perfect-pooling equilibrium.
Proposition 2 (Perfect Pooling) There
policy
maker
9ms
is
a robust inactive policy equilibrium, in which the
sets r for all 9, private agents attack if
level of the policy,
if
which we present
Inactive Policy Equilibrium
3.2
10
3,
and
the status quo is
abandoned
if
and only
and only
if
if
9
x
<
xms-, independently of the
< 9ms- The
thresholds
xms an d
are defined as in (5).
This interpretation of the information conveyed by an active policy intervention r > r follows from Bayes' rule
the observed level of the policy
otherwise.
is
on the equilibrium path, and from forward induction (the
intuitive criterion)
G.M. Angeletos,
12
The construction
and
C
of this equilibrium
on the vertical one. Since
Hellwig and A. Pavan
C.
illustrated in Figure
is
1.
maker
in equilibrium the policy
8
is
on the horizontal
=
r
is
isomorphic to the Morris-Shin game. The threshold
between the value of defending the status quo
= 8 M s- The
r solve C(r)
for all 9
> 9ms-
deviation to
and only
Hence, any deviation to
some
<
9
8'
equilibrium by r
equilibrium payoff
r'
6
{r,r\
and C(r')
if
and
>
and only
if
the market "learns" that 8
r',
9
^
if
[0'
U{9)
is
r'
and the
>
r
is
=
and only
€ [6',8"}. n
Since
9
if
Hence,
,0"\.
for all 9
=
8'
if
>
8
9ms €
8ms
is
game
the point of intersection
U{8)
=
x
< xms
for
any r >
r, in
which case
it is
Consider a
A(9",r). Note that C{r')
which implies that
,
= 8- A(8,r) >
in equilibrium for all 8.
C(r')
9"
< 9 MS and
r' is
>
9
dominated
if
in
G [8\8"\ and the policy maker deviates to
[9',
6"],
one can construct
beliefs that are
compatible with the intuitive criterion and for which private agents continue to attack
if
starting
(cost of the) size of the attack A(9,r). Let
dominated
and 8" solve
let 8'
A(9,r)
8
A
sets r for all 8, the observation
of r conveys no information about the fundamentals, in which case the continuation
after r
axis,
pointless for the policy
maker
if
and only
to ever raise the policy.
(See Appendix for the specification of beliefs and the proof of robustness.)
Insert Figure
Clearly,
any system of
beliefs
and
1
here
strategies such that A(8,r)
policy inaction as an equilibrium: If the size of the attack
can do no better than setting r(8)
=
r for
all 9.
Note
is
>
A(0,r) for
non-decreasing in
r,
all
r
>
r sustains
the policy maker
also that a higher level of the policy increases
the agents' cost of attacking the status quo and, other things equal, reduces the agents' incentive
to attack.
In an inactive-policy equilibrium, however, this payoff effect
probability agents attach to a regime change.
The
particular beliefs
is
offset
we consider
by the higher
in
the proof of
Proposition 2 have the property that these two effects just offset each other, in which case private
agents use the same strategy on and off the equilibrium path and condition their behavior only on
their private information. It
is
then as
if
the market does not pay any attention to the actions of
the policy maker. 1 "
Throughout, the expression "the market learns
beliefs
among
the agents given that
Y
is
common
X
by observing
V""
means that the event
X
becomes common
knowledge; see Monderer and Samet (1989) and Kajii and Morris
(1995).
'"Moreover, private agents do not need to learn
how
to behave off the equilibrium path; they simply continue to
play the same strategy they learned to play in equilibrium.
Coordination and Policy Traps
Active Policy Equilibria
3.3
The
following proves the existence of robust active-policy equilibria in which the policy
r* for every 9
€
equilibrium in which the policy
attack
if
if 9
9*.
maker
r*
and x
The thresholds
9*
and
if
either r
two-threshold equilibrium exists
The construction
if
<
x*
or x
,
<
r*
G
ifr*
G
quo
by market participants as a
and only
if
<
x
if
and only
if
the cost of facing a small and unsuccessful attack
A(9,r)
C(r*)
=
>
9*
C{r*).
The
and C(r*)
=
is
i.e.,
Combining the three
beliefs that satisfy
optimal. (See
A{9**,r)
= *
1
f^
^
namely r
11
=
)-
On
if
it
(9,
9ms}-
G
(r, r\.
r*
interpreted
inactive-
x
<
sets r
>
and only
maker
if
optimal to raise
9
>
C(r*), but not so strong that
the other hand, the threshold x* solves the
u(D(9)
l\x*,r)
=
(i(0
< 9*\x\r)
=
l\x*,r),
!_#
indifference conditions gives (6).
-#
where
'
1*1=2-)
+
One can then
q.
^)"
construct out-of-equilibrium
the intuitive criterion and for which the strategy of the agents
Appendix
is
A
lower than the cost of a policy intervention,
1
H(D(9)
r
r.
thresholds 9* and 9** are thus determined by the indifference conditions
indifference condition of the agent,
=
G
and
the fundamentals are strong enough that the net value of
maintaining the status quo offsets the cost of the policy,
i.e.,
>
x) whenever the policy
Anticipating this reaction by market participants, the policy maker finds
the policy from r to r*
if 9*
if
=
9*\x*,r)
Take any
policy equilibrium, private agents switch from playing aggressively (attacking
if
<
But contrary to the
signal of intermediate fundamentals.
x*) to playing leniently (attacking
abandoned
and only
illustrated in Figure 2.
is
is
and x* solves n{9
(r, r\ or, equivalently, if
of a two-threshold equilibrium
a robust two-threshold
Like in the inactive policy equilibrium, the observation of any policy choice r
r*.
raised at
and r otherwise; private agents
x; finally, the status
9** are given by
(6),
and only
(r,r\, there is
[9*, 9**}
G
sets r* for all 9
<
and only
<
is
[9*, 9**}.
Proposition 3 (Two-Threshold Equilibria) For any
only
13
for the specification of beliefs
is
sequentially
and the proof of robustness.)
Insert Figure 2 here
In any two-threshold equilibrium, the exact fundamentals never
What
9
fi
the market "learns" from equilibrium policy observations
[9* ,9**],
but this
is
enough
is
become common knowledge.
only whether 9 G [#*,#**] or
to facilitate coordination on different courses of action.
G.M. Angeletos, C. Hellwig and A. Pavan
14
For any
whereas
the threshold 9* below which the status quo
r*,
9**
Note
C (r*) = A (9**,r)
at 9**.
The
>
we
,
latter
is
=
9**
—
is
equilibrium exists
if
and only
0**
,
in
and therefore
x*\9**),
which
and only
Obviously, A9(r*)
.
A9(r*) >
0.
and A9(r*) >
By
=
9**
if
=
9ms-
and only
It
if
9*
< 9ms- The
=
and only
£
(n, r],
will establish that these properties
or equivalently,
extend to
<
r.
between attacking and non
But A9(r)
.
A
two-threshold
if
=
if
and only
(essentially) the Morris-Shin
= 9ms =
follows that r solves C(r)
r*
9**\x*). Therefore,
the monotonicity of A9(r*), there exists at most
is
if
>
also continuous in r*
is
which case the continuation game following r
9*
if
also increases with r*. It follows that
r*
0,
9ms] dictated by market
also equal to Pr(0
is
if
equilibrium exists
we
=
if
At the
policy maker's indifference conditions, 9*
must be that Pr(# < 9*\x*)
it
decreasing in r*
one r such that A9(r)
4,
if
9**\x*) increases with r*. For a marginal agent to be indifferent
9*
=
<
given by Pr(z
= r,
becomes more
a smaller range of fundamentals.
(9,
e,
a higher r* raises both the threshold 9* and the size of the attack
infer that
attacking conditional on r
A9(r*)
From the
as follows.
is
independent of
is
private information
also that a two-threshold equilibrium exists
intuition behind this result
0*
for
As
0.
»
the equilibrium policy has a spike at an arbitrary threshold 9* €
expectations.
Pr(#
—
9* as e
»
maker needs to intervene only
precise, the policy
limit,
—
increasing in e and 9**
is
abandoned
is
and only
1
if
— £ and
9*
if
game
a two-threshold
G {9,9ms]-
In
Section
robust equilibria of the game.
all
Finally, note that there are other strategies for the private agents that also sustain the
same
two-threshold equilibria. For example, we could have assumed private agents coordinate on the most
aggressive continuation equilibrium (attack
market expectations
(r
^
r, r*).
if
and only
if
re
<
x)
whenever the policy does not meet
Nonetheless, the beliefs and strategies
3 have the appealing property that the private agents follow the
r
<
r*
and the same lenient behavior
"ignore" any attempt of the policy
play exactly as
3.4
We
if
there
had been no
for
any r >
maker that
falls
r*
.
It is
we
consider in Proposition
same aggressive behavior
then as
if
for
any
market participants simply
short of market expectations and continue to
intervention.
Discussion
conclude this section with a few remarks about the role of coordination, signaling, and com-
mitment
in
our environment.
First, consider
environments where the market does not play a coordination game in response to
the policy maker, such as
when
the policy maker (sender) interacts with a single representative agent
Coordination and Policy Traps
(receiver).
13
15
In such environments, the policy can be non-monotonic
if
the receivers have access to
exogenous information that allows to separate very high from very low types of the sender (Feltovich,
Harbaugh and To, 2002). Moreover, multiple
out-of-eqmlibrmm
equilibria could possibly be supported
paper does not depend
single receiver.
in
any
The comparison
and only
if
contrary, the multiplicity
way on the
critical
is
sharp
if
we consider
with a single agent would have the latter attacking
abandoned
To the
if
9
<
9.
if
e
—
>
The
0.
and only
if
Second, consider environments where there
is
no
limit of all equilibria of the
<
x
if
market participants observe
fundamentals.
The
1,
where the threshold
the kind of multiplicity in that literature.
signaling, such as standard global coordination
Theorem
The
1,
however,
policy in our
is
in the market.
Moreover, Theorem
of multiple continuation equilibria in the coordination
public signal;
it
is
rather about
game
about the underlying
substantially different from
is
endogenous, as
1 is
how endogenous market coordination
the existence of policy traps the essence of
it is
exhibit multiple
it
signal
depends
not about the possibility
that follows a given realization of the
endogeneity of the policy leads to multiple equilibria in the policy
other words,
may
model does generate a public
about the fundamentals. Yet, the informational content of this signal
on the particular equilibrium played
9*
(9, 9 ms]-
sufficiently informative public signals
multiplicity of equilibria of
game
and the status quo being
9,
games. As shown in Morris and Shin (2001) and Hellwig (2002), these games
equilibria
environments with a
arise in
Contrast this with the result in Theorem
below which the regime changes can take any value in
we have
specification of out-of-equilibrium
merely in endogenous coordination, and would not
beliefs, originates
different
However, a unique equilibrium would typically survive the intuitive
beliefs.
criterion or other proper forward induction refinements.
identified in this
by
game
Theorem
that
is
facilitated
(the signaling
by the
game)
.
In
1.
Third, note that endogenous policy choices introduce a very specific kind of signal.
The
ob-
servation of r
>
this particular
kind of information that restores the ability of the market to coordinate on different
r reveals that the
courses of action.
icy itself
is
fundamentals are neither too weak nor too strong, and
In this respect, an interesting extension
observed with noise.
It
can be shown that
all
is
it is
to consider the possibility the pol-
equilibria of
Theorem
1
are robust to
the introduction of small bounded noise in the agents' observation of the policy, independently of
whether the noise
is
aggregate or idiosyncratic.
Lastly, consider the role of commitment.
the policy maker moves
13
14
first,
See, for example, the seminal
The proof
of this claim
is
Note that policy traps
arise in our
environment because
revealing information which market participants use to coordinate
work by Spence (1973) or the currency-crises example by Drazen (2001).
upon request.
available
G.M. Angeletos, C. Hellwig and A. Pavan
16
their response to the policy choice. This raises the question of
whether the policy maker would be
better off committing to a certain level of the policy before observing
To
continuation equilibrium in the coordination game.
optimal, suppose the idiosyncratic noise £ has
commits ex ante to some
be abandoned
will
if
full
if
<
9
= 1—
9{r)
7-;
Hence, the ex ante payoff from committing to r
Uc = maxr [/(r)
and
9C
= mm{9(r c)\r c
{r*,9*,9**) the policy
9
>
9**,
e
(6,
9,
is
9 C ) necessarily leads to
when
long as she
is
perfect
=
Ud
Pv(9
Ud >
commitment
>
is
the policy maker
If
0.
=
all
Pr(0
If
9
>
>
?(r))E[0|0
9(r), A(9, r)
>
instead the policy
>
similar
0,
9** -> 9*
—
as e
>
-
9(r)}
maker
—
>
0.
15
C{r). Let
retains the
and A(9,r) ->
a negligible measure of
for
6>*)E [9\9
Uc A
.
*•
not necessarily
the ex ante payoff depends on the particular equilibrium
meaning the policy maker pays C(r*) only
that, even
U(r)
maker expects to be played. 16 As e ->
ex ante value of discretion
9*
moreover, for
is
—
is
a cost C(r) and ensures that the status quo
6 argmax r U(r)}.
option to fashion the policy contingent on
commitment
support and consider e
level of the policy r, she incurs
and only
see that
thus inducing a unique
9,
9*]
.
But note that
argument holds
possible, the policy
maker
not too pessimistic about future market sentiments. 1
6C
9. It
>
9
for all
follows that the
and therefore any
for arbitrary e.
We
will prefer discretion
conclude
ex ante as
'
Robust Predictions
4
Propositions 2 and 3
Theorem
1,
left
open the
possibility that there also exist equilibria outside the
which would only strengthen our argument that policy endogeneity
two
classes of
facilitates
coordination on multiple courses of action and leads to policy traps. Nonetheless,
we
market
are also
interested in identifying equilibrium predictions that are not unduly sensitive to the particular
assumptions about the underlying information structure of the game.
We
first
note that, when the noise has unbounded
full
support, for any r* G (r,r] one can
construct a one-threshold equilibrium, in which private agents threaten to attack the status quo
whenever they observe any r <
15
These
results follow
r*
from Proposition
Note that the payoff
for the policy
,
1,
no matter how high their private signal
replacing r with an arbitrary
The above argument assumes commitment
however,
is
true in the case of
r.
maker associated with the pooling equilibrium of Proposition
payoff associated with the two-threshold equilibrium in which 9'
11
thus forcing the
x,
commitment
=
9ms-
to a fixed uncontingent level of the policy.
to a contingent policy rule. If the policy
about the aggressiveness of market expectations, she
will
2 equals the
maker
continue to prefer discretion.
is
A
similar argument,
sufficiently optimistic
Moreover, a contingent
policy rule, unlike a fixed uncontingent policy, does not necessarily induce a unique continuation equilibrium in the
coordination game.
trap.
As a
result, the
optimal policy rule
itself
may be
indeterminate, resulting
in
another sort of policy
Coordination and Policy Traps
9*
>
=
9*
17
policy
maker
beliefs
such that market participants interpret any failure of the policy maker to raise the policy
to raise the policy at r* for all 9
as a signal that the status
the fundamentals.
quo
to intervene, in which case
the noise
implies that for
On
>
9
all
and
[#i,
2 ]U[0 3
6>
4]
same
and
>x = 9 + e
noise has
private agents find
^
r'
bounded support,
otherwise, where
and the noise
the market "learns" whether 9 € [#i,#2] or 9 €
to separate different subsets of types
Theorem
is
sustained by
the policy maker
if
it
necessarily true
dominant not to attack, which
may be
it
is
fails
r.
possible for the market to
=
For example, suppose that the policy maker sets r{9)
policy.
r(9)
[93,94] are sufficiently apart
the ones in
x even
the policy maker faces no attack and hence sets
when the
,
sufficiently high
one-threshold equilibria disappear. Indeed, this
all
+e
x
the other hand,
any 9 G
C(r*). This threat
be abandoned independently of their private information about
when they observe
bounded: For x
apart types that set the
tell
for
is
where
seems more plausible, however, that market participants remain confident
It
that the regime will survive
when
will
,
who
is
<
<
92
<
9 2 +2s
bounded, whenever
[#3, #4].
set the
1
93
<
4
.
Since
[9 1
,
r'
92]
observed in equilibrium,
r' is
In principle, the possibility for the market
same policy may lead to
equilibria different
from
However, this possibility criticaUy depends on small bounded noise and
1.
disappears with large bounded or unbounded supports, whatever the precision of the noise.
It
is
these considerations that motivated the refinement in Definition
2.
Robustness only
eliminates strategic effects that are unduly sensitive to whether the support of the signals
or
unbounded. The following then provides the converse to Theorem
Theorem
2
(Robust Equilibria) Every
the two classes of
The
game.
Theorem
intuition for
If all
9 set r,
we have
2
1.
robust perfect Bayesian equilibrium belongs to one of
is
as follows.
Consider any perfect Bayesian equilibrium of the
perfect pooling. Otherwise, let
= M{9
be, respectively, the lowest
bounded
1.
Theorem
9'
is
:
r(9)
>
r}
and
9"
=
sup{9
:
r(9)
>
and highest type who take a costly policy
r}
action.
Since there
is
no
aggregate uncertainty, the policy maker can perfectly anticipate whether the status quo will be
abandoned when she
to
no regime change.
sets r
It
and hence
is
willing to
pay the cost of a policy r
follows that any equilibrium observation of r
>
the status quo will be maintained and induces any agent not to attack.
>
r only
if
this leads
r necessarily signals that
If
there were
more than
one r above r played in equilibrium, and the noise were unbounded, the policy maker could always
save the status quo by setting the lowest equilibrium r above
r.
Since C(r)
is
strictly increasing,
it
G.M. Angeletos, C. Hellwig and A. Pavan
18
most one r above r
follows that at
the policy and define 9* and 9** as in
9
<
6*.
Hence,
in
played in any robust equilibrium. Let r* denote this level of
is
Theorem
any robust equilibrium,
would necessarily
set
9'
Obviously,
>
1.
6*
.
If
never pays to raise the policy for any
it
the noise were bounded,
<
Hence, in any robust equilibrium, 9"
r.
oo.
> x+e =
9
all
Compare now
+ 2e
9
the strategy of
the private agents in any such equilibrium with the strategy in the corresponding two-threshold
equilibrium. Note that a regime change never occurs for 9
by setting r
herself a positive payoff
then the incentives to attack
9
<
9*.
Similarly,
if
the observation of r
for all 9
€
and r(6)
=
[8*, 9"].
=
r*. If
when observing
>
the status quo
preserved also for
is
r
is
less
some
€
<
9
9
G
[9*, 6"],
which, by definition of 9**,
possible
is
=
1
then
[9*, 6"),
for all 9
and only
if
=
9"
if
9*
for all
informative of regime change than in the case where r{9)
Hence, private agents are most aggressive at r when D{9)
r* for all 9
some
than when a regime change occurs
r are lower
the policy maker does not raise the policy at r* for
=
maker can guarantee
9* as the policy
=
r*
<
9*
9**.
Equivalently, the size of the attack A(9,r) in the two-threshold equilibrium corresponding to r*
represents an upper
is
played.
this
r*
<
It
bound on the
size of the attack in
follows that, in any robust equilibrium, 9"
<
9**.
Since 9**
immediately rules out the possibility of equilibria in which r*
r,
>
r.
<
On
9*
in
which
r*
>
r,
whenever
r*
the other hand, for any
we have
9*
It
any active-policy equilibrium
follows that the anxiety region of
is
monotonic
2 permits us to
By
2.
9**.
is
abandoned
for all 9
to intervene;
<
9*
and that
Second, the probability of regime change
<
it is
9'
=
<
9*
9\js,
then only
is
lower in any active-
for
this, in
which case there
is
no value
a small range of moderate fundamentals that
the agents are likely to be "uncertain" or "confused" about the eventual fate of the regime, and
is
thus only for moderate fundamentals that the policy maker
policy action.
is
9ms)- Third, when the fundamentals are either
very weak or very strong, private agents can easily recognize
maker
of the
predictions: First, the probability of regime change
policy equilibrium than under inactive policy (9*
for the policy
bounded by the anxiety region
(See Appendix.)
make robust
in the fundamentals.
<
iterated deletion of strictly dominated strategies, one
necessarily
which completes the proof of Theorem
Theorem
9"
any robust equilibrium
corresponding two-threshold equilibrium.
can also show that the status quo
<9' <
is
it
"anxious" to undertake a costly
Fourth, the "anxiety region" shrinks as the precision of the agents' information
increases. In the limit, the policy converges to a spike
around the threshold below which a regime
Coordination and Policy Traps
19
change occurs. 18
Corollary
Q z£ 9*
The
limit of
any robust equilibrium as e
for any arbitrary 9* G
,
The above
(9,
9ms], and the status quo
Proposition 4
the fundamentals were
if
(Common Knowledge)
perfect equilibrium
if
and only
if
regime outcome D{9) of any shape over
shape
is,
if
9 were
[9, 9]
[9,9].
The equilibrium
These
[9,9]
can
For example,
and need not be monotonia
subset of
9 for 9
G
is
abandoned
if
is
r{9)
and only
if
=
9
r for all
<
9*
would have been impossible to make any
knowledge.
=
0.
[9, 9]
A
policy r(9) can be part of a subgame
and
be part of a
=
r(9)
r for 9 ^
[9, 9].
Similarly, a
subgame perfect equilibrium.
knowledge, the equilibrium policy r(9) could take essentially any
in the critical range [9,9].
non-monotonicities.
in
common
common
Suppose e
C (r(9)) <
it
such that the policy
is
>
predictions are intuitive. Nevertheless,
of these predictions
That
—
could have multiple discontinuities and multiple
it
probability of regime change D{9) could also take any shape
For example, the status quo could be abandoned
results contrast sharply
with our results in Theorem
2.
We
for
any
conclude that
introducing idiosyncratic noise in the observation of the fundamentals does reduce significantly
common knowledge
the equilibrium set as compared to the
The global-game methodology
case.
thus maintains a strong selection power even in our multiple-equilibria environment and enables
meaningful predictions that would have been impossible otherwise.
Remark
Theorem
2 leaves open the possibility that there exist active-policy equilibria different
from the two-threshold equilibria of Proposition
only for a subset of
[9*
,
9**].
3,
namely
equilibria in
which the policy
is
active
This possibility, however, does not affect any of the predictions men-
tioned above and can be ruled out by imposing monotonicity of the strategy of the agents in their
private information. Monotonicity
ratio property
18
An
(MLRP),
that
is,
is
guaranteed when the noise
when V '(C)/V'(0
J
interesting implication of the Corollary
is
is
decreasing in
that, in the limit,
all
satisfies
£.
the monotone likelihood
(See Appendix.)
active-policy equilibria are observationally
equivalent to the inactive-policy equilibrium in terms of the level of the policy, although they are very different in
terms of the regime outcome.
An
econometrician
basis of information on the fundamentals
unobservable market sentiments arises
if
and the
may then
policy.
fail
An
to predict the probability of regime change on the
even sharper dependence of observable outcomes on
one introduces sunspots, as we discuss
in the
next section.
G.M. Angeletos,
20
C.
Hellwig and A. Pavan
Uncertainty over the Aggressiveness of Market Expectations
5
The
assumed the policy maker was able
analysis so far
to anticipate perfectly the aggressiveness of
market expectations, which we identify with the threshold
r* at
which private agents switch from an
aggressive to a lenient response to the policy. In reality, however, market expectations are hard to
predict, even
To capture
may
when
known
the underlying economic fundamentals are perfectly
this kind of uncertainty,
we introduce
payoff-irrelevant sunspots,
to the policy maker.
on which private agents
condition their reaction to the policy maker. Instead of modelling explicitly the sunspots,
assume
directly that r*
realization of r*
a
random
variable with
common knowledge among
is
maker when she
is
sets the policy, in
$
c.d.f.
TV C
over a compact support
the private agents, but
which case random variation
is
unknown
The
Tt.
to the policy
random
in r* leads to
we
variation in
the probability of regime change for given policy. Different sunspot equilibria are associated with
different distributions (<P,TZ*)
and
result in different equilibrium policies r(9).
Proposition 5 (Sunspot Equilibria) Take any random
(r,r)
and distribution $. Fore
9**
{9*, 9)
£
and
r(9) £
TV
<
The sunspot
of Proposition
9*
,
>
3.
The
active only for
9**. Finally, 9* is
[9*, 9**];
independent of
=
sets r(9)
the status quo
£ (9,6ms) & n d
r for 9
e,
whereas 9**
—
»
^
[9*, 9**],
abandoned with
is
9* as e
[9*, 9**],
—
>
and
0.
in the fundamentals, the pol-
an intermediate range of fundamentals, and
this range vanishes as e —> 0.
represents the set of
response whenever r
is
<
random thresholds
r*
is
r* such that private agents coordinate
and on a lenient one whenever
r
>
r*
.
When
9
<
9*
,
on an aggressive
raising the policy to
too costly compared to the expected value of defending the status quo, in which
case the policy maker finds
[9* ,9*"\, it
of a regime change.
in the
it
optimal to set r and a regime change occurs with certainty.
pays to raise the policy at some
level in
TV
so as to lower the probability
Since the value from defending the status quo
range
[9*, 9**].
Finally, for 9
>
When
is
9**, the size of
increasing in 9, so
the attack at r
is
is
that e
is
sufficiently small
is
not essential;
construction of these equilibria (see the Appendix).
it
ensures
8" <
9,
the
so small
that the policy maker prefers the cost of such an attack to the cost of a policy intervention.
The assumption
is
monotonic
probability of regime change
TV
optimal policy
The policy maker
TV C
equilibria of Proposition 5 are qualitatively similar to the two-threshold equilibria
is
instead 9 £
there exist thresholds 9*
with probability less than one and non-increasing in 9 for 9 £
icy
TV
that:
13
with r(9) non- decreasing in 9 for 9 £
never abandoned for 9
level in
sufficiently small,
and a robust equilibrium such
certainty for 9
any
>
variable r* with compact support
The
which we use only to simplify the
Coordination and Policy Traps
21
thresholds 6* and 6** are again given by the relevant indifference conditions for the policy maker,
but
differ
from the ones we derived
in the absence of sunspots.
and the complete proof of the above proposition are provided
With random
The
in the
definition of the thresholds
Appendix. 20
variation in the aggressiveness of market expectations, policy traps take an even
stronger form. Not only the policy maker has to adopt a policy that confirms market expectations,
but also the coordination outcome that follows a policy intervention depends on market sentiments.
For example, commentaries in the media that provide no information about the fundamentals
and only second-guess the reaction of the market, may
interfere with the effectiveness of policy
intervention.
In the context of currency crises, empirical evidence suggests that raising the level of domestic
interest rates or taking other defense
collapse.
Kraay (1993),
for
measures does not systematically prevent an exchange-rate
example, studies the behavior of interest rates and other measures of
monetary policy during 192 episodes
and unsuccessful speculative currency attacks
of successful
and finds a "striking lack of any systematic association whatsoever between
interest rates
and
the outcome of speculative attacks," even after controlling for various observable fundamentals.
This evidence
is
hard to reconcile with Morris and Shin's (1998) prediction that defense
consistent with the predictions of Proposition
decrease the probability of devaluation, but
is
where the same combination of
and fundamentals may lead
interest rates
policies
5,
in equilibrium to either
no devaluation or a collapse of the currency.
Finally, the results of this section help
commonly appear
of the paper.
the market
in the
Once the
may be
popular press,
policy
like
understand and formalize the kind of arguments that
the one
we quoted from
maker has intervened
in
Financial
Times
in the beginning
an attempt to achieve a favorable outcome,
equally likely to "interpret" this action either as a signal of strength, in which
case market participants coordinate on the desirable course of action, or as a signal of panic, in
which case the policy maker's attempt proves
arbitrary reaction of the market, the policy
policy
2
If
in vain.
maker
And
finds
it
faced with the need to guess the largely
even harder to decide what the optimal
is.
one takes a sequence of sunspot equilibria such that
9** in Proposition 5 converge to the ones given in (6).
read as the limit of sunspot equilibria.
TV
That
is,
converges to a single point r*
,
the thresholds 9* and
the two-threshold equilibria of Proposition 3 can be
G.M. Angeletos, C. Hellwig and A. Pavan
22
6
Concluding Remarks
In this paper
we
maker
investigated the ability of a policy
control equilibrium outcomes in economies
to influence
where agents play a global coordination game.
that policy endogeneity leads to policy traps, where the policy maker
largely arbitrary expectations of the
come. There
is
affect
in
forced to conform to the
market behavior, as well as a continuum of active-policy
level of the policy
"reward" the policy maker by playing a favorable continuation equilibrium.
beyond which they
Despite equilibrium
multiplicity, information heterogeneity significantly reduces the equilibrium set as
case of
common knowledge and
found
which market participants coordinate on "ignoring"
which market participants coordinate on the
equilibria in
is
We
market instead of being able to fashion the equilibrium out-
an inactive-policy equilibrium
any attempt of the policy maker to
market expectations and
compared
to the
enables meaningful predictions.
Stabilization policy in economies with
macroeconomic complementarities, preemptive
policies
against speculative currency attacks, and regulatory intervention during financial crises, are only a
few examples where our results might be relevant.
of our results to
more complex environments
We
leave these applications
and the extension
for future research.
Appendix
Proof of Proposition
the main text. Uniqueness
Proof of Theorem
r conveys
sets r
=
is
characterization of the equilibrium in our setting
2.
is
Since in equilibrium the policy maker sets r for
isomorphic to the Morris-Shin game.
equilibrium, in which an agent attacks
and only
if
9
< 9ms-
if
and only
follows that there
It
if
x
< xms
all 6,
= 9ms =
1
— £. Any
r
>
r
is
dominated
is
the observation of
after the policy
by r
abandoned
is
if
rule.
Note that r (defined
in equilibrium
maker
a unique continuation
an d the status quo
Equilibrium beliefs are then pinned down by Bayes'
Next, consider out-of-equilibrium levels of the policy.
solves C(r)
presented in
3.
no information about 9 and hence the continuation game starting
r
is
established in Morris and Shin (1998).
Follows from Propositions 2 and
1.
Proof of Proposition
The
1.
for all 9:
in
Theorem
For 9
1)
< 9mS:
Coordination and Policy Traps
- C{r) <
V(6,0)
On
0; for
> 8 MS
8
the other hand, any r G (r,r)
<
or A(9,r)
can construct out-of-equilibrium
is
C(r)
=
beliefs
fi
non- increasing in x and satisfies
finds
optimal to attack
it
and only
if
< 9ms-
9
=
fi(9Ms\x,r)
let
again
€ M.{r) and
fi
game
continuation
=
following r
r, in
= 1—H
Given 9ms
=
set r{9)
G
r for
Q(x), and
and only
letting
by r
=
/j,(9\x,r)
<
one
8ms\x,t)
a agent
beliefs,
abandoned
is
for all 9
^ 8ms
if
an d
otherwise, so that
fJ-(9\x)
a mixed-strategy equilibrium for the
and only
if
< xms
x
if
any s
>
if
any
for
and any
xms — 9ms jt^~
=
and the probability of regime change D(9)
for all 8,
xms- For such
jjl(9
r,
x
< xms an d
type
9ms
r,
is
it
optimal
for
the
Finally, consider robustness in the sense of Definition 2.
(0, 1), (5) is satisfied for
unbounded support, by simply
^
r.
if
all 9.
=
in equilibrium
which an agent attacks
Given that private agents attack
maker to
dominated
is
and such that
which case the status quo
in
< C(r)
any 9 such that 9
for
(4)
x
r at
V(0,A(6,r)).
any out-of-equilibrium r
for
(4) is satisfied. It follows that there is
abandons the status quo with probability
policy
< xms,
9ms G
x such that
by r{9)
in equilibrium
G M(r) that satisfy
x
if
- C{r) <
A(6,r) and thus V(6,0)
9ms- Therefore,
Finally, note that r
for all
1
Ms >
< 0Ms\^,r) =
fj,(9
and only
if
9
dominated
is
<
C(r), where C(r)
>
C{r)
,
23
1
1
c.d.f.
with either bounded or
follows that the policy r(9)
It
{9ms)-
\1/,
< 9ms and D{9) =
for 9
=
r
otherwise, can
be sustained as a robust equilibrium.
Proof of Proposition
3.
and note that,
<
Consider
for
first
any r
fj.{D(9)
which
(r,r*),
fi(9
by
<
=
l\x,r)
Q(x)
=
<£ [9*, 9**]
ji{9
<
9\x,r)
we consider out-of-equilibrium
9\x, r)
=
i.e., if
such that 8
<
r at
C{r)
x
=
=
v{8
'When
< min {A(8, r), 9} we
,
C(r) or A(8,r)
the noise
=
[x
is
beliefs
x*. Furthermore,
x such that 0(x)
and Q(i)
When
for all x.
<
D
unbounded,
— e, x + e]
,
for
any
<
C(r).
[8*, 8**]
this
x.
is
^
if
for
r
21
=
r, beliefs
r
9*\x,r)
such that
=
r*,
it is
and only
are pinned
if
9
<
down by
fj,(9
ji{9
<
//
<
9\x,r)
r
is
=
9*\x*,r)
is
,
9.
Bayes' rule
r.
For any r €
non-increasing in x and
not dominated in equilibrium
to satisfy fi(0\x, r)
=
Bayes' rule implies p(9 G
Finally, for
immediate; when
if
(^)
l-y(3zf.Y + y(*=p.Y
some 9 6 0(x),
=
= V
where A{6,r)
Therefore,
further restrict
When
0.
fi
0,
a regime change
decreasing in x. Let x* be the unique solution to
is
r(8),
for all
r*, private agents trigger
the behavior of the agents.
for all x, since (6) ensures
=
Let 9 6 [0*,6**] solve V{9,A(9,r))
for all 8
[6*
,
6**}\x,r*)
any (x,r) such that either r
bounded,
it
G 0(x)
=
follows from the fact that |#**
—
=
1
r*
and
9*\
<2e
G.M. Angeletos, C. Hellwig and A. Pavan
24
0(x) n
=
[9*, 6**}
>
0, or r
we
r*,
otherwise. Given these beliefs, (4)
is
let
>
n(6
satisfied,
is
=
6\x,r)
G A4(r)
\x
1
for
any
> x and
x
for all
r,
(
>
i{9
=
9\x,r)
and the strategy of the agents
sequentially optimal.
Given the strategy of the agents, the policy maker clearly
Consider next the policy maker.
prefers r to any r G
the regime
V{9\
0)
(r,
r*)
and
any r
r* to
>
r*
9*
.
between setting
indifferent
is
not attacked) and setting r (in which case the status quo
is
- C{r*) =
= C (r*)
9*
0, i.e.
Similarly, 9**
.
abandoned)
is
which case
(in
if
and only
attacked) and setting r (and being attacked without being forced to abandon the status quo)
only
if
V{0**,0)
preferred to
6**,
and
9*
<
For any 9
z*,
- C{r*) = V(0** ,A{9**
,
r
optimal; for any 9 G {9*, 9**), A{9,r)
is
whereas the reverse
r,
we obtain x* =
9**
is
true for any 9
+ e*"
1
that 9*
<
9**
if
and only
if
6*
that a two threshold equilibrium exists
>
R, we have that k
any
9**
For r*
>
=
9*
the agents
is
9 G [9* ,9**}
D(9)
=0
and
r, 9*
For any
9*.
=
= *
any
for
9*"
if
and only
and r(9)
=
k,
A{9**,r)
A {9*\r) = *
= C (r*) and thus
2.
- * -1
o=7^*)
is
,
6>
A/5
if
Using
.
9*
G
(9,
5*
=
(7)
C(r*) and
M5 =
0ms]> or equivalently r* G
let 9*
For any r* G {r,r),
=
_1
G
{9*)
(0,
oo) since (1
C(r),
holds for every e and every
=
9**
+ e^~
l
\&.
The
{9*). It follows
we
infer
(r,r\. Finally,
C(r*) and take any
- -^9*) G (9\
condition (7) can always be satisfied by letting e
(7)
r*
(9*)]
with either bounded support [— 1, +1] or unbounded
- j^O*) - ^
and
and
the indifference conditions for 9*
From
.
where
=
full
1).
Hence,
~e
9
support
G
for
(0, oo).
indifference condition for
that the policy r(9)
r otherwise, and the probability of regime change D(9)
=
1
if
9
<
=
9*
r*
if
and
otherwise, can be sustained as a robust equilibrium.
9
and we
2.
When
Theorem
no r other than r
Hence,
1.
in
what
is
played in equilibrium, we have the pooling
follows,
we
consider equilibria in which r(9)
>
r
let
9'
We
(l
\I/
then clearly satisfied at x*
equilibrium in (a) of
some
c.d.f.
= 9ms
Proof of Theorem
for
_1
-
(l
< 1 -r =
consider robustness in the sense of Definition
arbitrary 9**
9**
>
>
,
if
^*) and
+ £ * _1
It follows
C (r*) = A (0**,r)
r)), i.e.
,
if
between setting r* (and not being
indifferent
is
7'*
=
inf{0
:
r{9)
>
r}
and
9"
=
sup{0
:
r{6)
>
r}.
prove the result in a sequence of three lemmas.
Lemma
1.
In any robust equilibrium, there
equilibrium and 9"
<
oo.
is
at
most one
level of the policy r*
>
r played in
Coordination and Policy Traps
Any
Proof.
all
>
r
r
who abandon
9
played in equilibrium only
is
change exceeds the cost of attacking,
>
equilibrium r
set of
r*
^
i.e.
f
>
Q
=
1
for all x,
which case A(9,
=
r)
would be dominant
whenever 9
for all r
=
an equilibrium with 9"
=
D(9)
if
0;
the probability of regime
the noise
If
r.
if
only
i.e.
=
where 9o
unbounded, any
is
{9
D (9) =
:
0}
the
is
maintained. Hence, no agent ever attacks when he
is
=
r)
for
any equilibrium r >
r will be chosen by the policy maker, which proves the
it
>
any robust equilibrium,
this also implies that, in
noise were bounded,
and only
if
D (9) dp (9\x, r)
p(@o\x,r)
and thus A(9,
r,
leads to no regime change,
if it
Agents attack
which the status quo
for
observes an equilibrium r
r,
r.
r results in a posterior
fundamentals
increasing in
=
the status quo set r
25
r.
Since C(r)
most one
at
level of the policy
part of the lemma. Next,
first
=
an agent not to attack whenever x > x
for
> x + e and
=
thus r(9)
oo could never be sustained with
strictly
is
bounded
r for all 9
noise,
> x+
e.
if
+
9
the
e,
in
Therefore,
which proves the second
part of the lemma.
For the rest of the proof,
it
suffices to consider
unbounded
Given any
noise.
r*
G
{r,f}, define
the thresholds
r = C{r*)
and note that
and
9**
<
Lemma
Proof.
9*
2.
9**
>
when
9*
r*
when
>
the other hand, any 9
>
and
D=
0.
<
-r =
9*
1
<
(r,r], 9*
9*
9*
9
$
M s,
he. for r*
l-r
<
r
= C~
,
r*
is
<
9'
and 9"
strictly
<
1
(9
(9
M s)
when
r*
>
9*
=
r,
9**.
dominated by
r, it
immediately follows that
,
=
for all 9
>
9*
However, there
.
may
exist 9
<
Further, define
p
(x) as the probability, conditional
on
1
M {D =
is
x, that 9
£
S(x)
9* that also
[9*
,
at r
whenever
=
<
1_$
p,
(D
l\x, r)
#
-)
1
1-* (.5
*
+
1
-*
*
do not
x, that 9
<
9*
9"}
and r
=
r.
Define
-,
r
F(x;
r.
0.
on
l|z,r)
an agent does not attacks
On
.
)
given by
- * {*=f) ~
9'
V (9, 0) — C(r* >
in equilibrium. Define 5(x) as the probability, conditional
Then, the probability of regime change conditional on x and r
Clearly,
//)*>
m~ 1
l
1
can always set r* face no attack, and ensure a payoff
Therefore, necessarily D(9)
abandon the status quo
9**
r.
For any r* G
Since for any 9
=
and
-1
(9)
G.M. Angeletos, C. Hellwig and A. Pavan
26
Note that F(x;
9* ,9")
and
strictly decreasing in x,
is
let
x
(9* 9") solve
,
F(x-9\9")=r.
>
Since 5 (x)
and p(x) >
whenever x > x
(9* ,9")
0,
we have
=
l\x,r)
>
and therefore any agent with x
be
if it
for
(i)
From
=
1
for all 9
any
\t
(
—
^
which
'-r^~
r* is played, 9"
That
most
9*
and r{9)
some
for
=
9
r for
<
=
r* for
all
9
is,
as aggressive as they
€ [9\9"\. This
is
in
any
would
intuitive
9* reduces the incentives to attack for
some 9 £
[9*, 9"]
makes the observation
of
9,
)
•
must
(9* 9**)
Using x
threshold equilibrium of Proposition 3
in
=
<
= C (r*)
the indifference condition for 9", we have that A[9",r)
<
=r
change and therefore also reduces the incentives to attack conditional
x. It follows that, for
follows that 9*
for all x. It follows that,
x(9*,9") does not attack in equilibrium.
a positive probability that r(6)
r less informative of regime
on r and
< F(x;9*,9")
<F(x;9*,6") <F(x;9*,9")
a positive probability that D(9)
(ii)
l\x,r)
played, private agents are necessarily at
is
were the case that D{6)
every x, and
(D =
,
fi{D
equilibrium in which r*
\±
(8)
,
^ ^-^
C (r*) =
Since
9*, it
x* and the indifference condition for the two-
=
-
(
=
.
C(r*)
=
9*
we have that
any equilibrium
in
J
satisfy
x(0*,9**)-9*' <x(9*,9") -9".
From
(8),
x
-
(9*, 9")
<
Recall that 9*
9"
<
9**
<
9*
<
9"
6**
which
9',
decreasing in 9", and hence
is
played, necessarily 9*
<
= C~
1
if
and only
is
a contradiction, since by definition
robust equilibrium with r* £ {r,r).
is
<
we conclude
9'
<
9"
if
On
<
r*
r
that 9"
For any r*
(9ms)
9'
<
<
>
9**.
r,
Lemma
2 implies
9". Therefore, there exists
no
the other hand, in any robust equilibrium where r* 6 {r,r\
9**.
Next, we show, by iterated deletion of strictly dominated strategies, that in any robust equilibrium in which r* G {r,r\
turn implies that
Lemma
3.
9'
D{9)
=
=
is
played, the status
quo
is
abandoned
if
and only
9*
1
for all 9
< 9\ D(0) =
for all 9
>
9*
,
and
9'
=
9*
if
9
<
9*
,
which in
Coordination and Policy Traps
Proof.
r
=
€
r
Given an arbitrary equilibrium policy
we consider the continuation game that
r(0),
T
and construct the iterated deletion mapping
[9, 9']
and
27
—
[9, 0']
:
>
Take any
as follows.
[0, 0']
follows
let
G(x;0)
1
=
- * fa=&\
l-*(~)+p(x) + *( x-e'
G(x;
0)
lim-x-t+oo p(x)
=
and therefore lim z _,_ 0O
for every 9, there
at least
is
=^
solution to
it
is, if
(
&=$
any 6
i.e.
x(9)
=
Note that G(x;
<
=
unless r(9)
9,
<
x
all
which
x,
r* rather
T is
iterated deletion operator
conditional on x and
=
C?(x; 0)
min{x
>
9)
G(x\
|
=
9)
x
r for every
9)
r},
<x
which happens
r,
Note that lim^-^-oo p(x)
=
r.
=
Then
and define 9
and
*
(^=1
for all 9
in turn implies that a
than
r.
and \im x ^ +oo G(x;0)
1
quo to be abandoned
private agents expect the status
optimal to attack for
for
.
]
0,
one solution to the equation G(x;
lowest solution to this equation,
That
<
thus represents the probability that
<
x
let
=
=
follows that,
x(0) be the
9(9) as the unique
>
j
9,
0. It
=
<
g for every
^.
necessarily they find
regime change necessarily occurs
in equilibrium only
if
9
>
9'
The
.
thus defined by
T (e)
jo}
EEmin
G is strictly increasing in 9, implying that x and therefore 9 are also strictly increasing
We conclude that the mapping T is weakly increasing for all 9 G [9, 9'] Obviously, T is
Observe that
in 9.
also
22
.
bounded above by
0'
ruled out neither
9'
.
< 9ms,
Next, we compare
T
nor
9'
T
:
unique solutions to
[9,9],
— Vf ^^
)
(
and
Moreover, since
We
2
in
=r
is
game
defined by
and 9
jr
J
G
at r
=
(?)
= ^ ^~
I
>
6
[0,0'],
either 0(0)
>
oo and 9*
< 9ms,
but so far we have
I
when
?,
the pooling equilibrium
where x
=
x{9) and
?
=
is
played).
0(0) are the
T has a unique fixed point at = 9 = 0ms,
.
0',
in
which case T(0)
=
9',
or 0(0)
<
9',
in
F(9).
In general, x and therefore 6 need not be continuous in
which case
<
^(0)
conclude that, for any
which case T(0)
9**
< 0ms whenever 9 € [9,9ms) and 9 > F(9) > 9ms whenever G (0ms,@\G(x; 0) > 1 - * (^) for all x and 9, we have x(0) > x(0) and therefore 9(9) > 0(0).
<
satisfies
<
with the iterated deletion operator of the Morris-Shin game without
[0,6] -*
1
9'
> 9ms-
signaling (or, equivalently, of the continuation
This operator,
<
Finally, note that 0*
is strictly
continuously increasing in
8.
9.
Continuity
decreasing in x, implying that G(x; 6)
All
we
need, however,
is
=
is
ensured when
^
satisfies
the
MLRP,
r has a unique solution and this solution
monotonicity of the lowest solution, which
is
true for any
"J.
is
G.M. Angeletos,
28
Hellwig and A. Pavan
C.
we consider the sequence {9 k }™=1 where
Finally,
9
,
and 9 k+1
=
1
= T(9 k
sequence represents iterated deletion of strictly dominated strategies starting from
sequence
monotonic and bounded above by
is
This limit must.be a fixed point of T, that
or 9°°
0°°
>
9
M s-
Suppose
9°°
<
0',
=
0°°
is,
T{6°°).
This can be true only
9'.
< 9 MS we would then have T(0°°) > ^(0°°) >
a fixed point for T(-).
0'
> 9 MS We
and
9°°
€
all
case
is
9*
and again
>
<
for all
1
0*
<x>
)
9*
9'
Suppose that
9*.
<
9'.
prove that either 0°°
=
9',
0(0°°)
if
<
0'.
were the case that
If it
< 9ms, whereas 0°° > 9ms whenever
0' > 0*. If 9' < 9
M s, then 0°° = 9' > 9*
in equilibrium.
> 9ms,
0'
instead
If
would abandon the status quo
But
in equilibrium.
then
either
can ensure no regime change and a positive payoff by setting
necessarily the case that 0'
it is
=
that D{9)
€ (0*,0
all
impossible as any
Therefore,
=
0'
whenever
0'
would abandon the status quo
(0*, 9')
> 9ms >
=
next prove that 0°°
-
=
Therefore, 0°°
first
Since this
9.
which contradicts the assumption that 0°°
0°°,
,
is
We
This
1.
limit 9°°
some
necessarily converges to
it
>
k
for all
)
and D{9)
=
=
0*
>
for all
< 9ms
9*
and since
=
then 0°°
9'
=
9*
It
.
r*.
follows
9*.
Combining the above three lemmas, we conclude that any robust equilibrium belongs necessarily to either (a) or (b) in
Theorem
which completes the proof of the theorem.
1.
Proof of Remark on Monotone Strategies/MLR P
prove that
if
a(x,r)
which
fundamentals
prove that
if
monotonic
in x.
Step
If in
1.
Proof.
If
monotonic (decreasing)
is
\I/
in
satisfies
is
condition A(9",r)
the
MLRP,
monotonic
=
in x, the size of
C(r*),
is
in x,
i.e.
iitp'/ip is
monotonic
in x,
then 0"
then necessarily
follows that
it
fi(D=
l\x,r)
exists a
>
unique x
=
x
strategy with a(x,r)
Thus, 0" solves
(0*, 9")
=
1 if
x
\i
(D =
such that [i(D
<
=
x and a(x,r)
* f ^',op~e" \ = c ^^ &nd
\j>
l|.r,r)
=
=
r
l|x, r)
=
€
if
is
=
r* for all
decreasing in
0.
From
and
Next, we
[0*,0**].
r{9)
0**,
first
decreasing in the
.4(0, r) is
for all
We
steps.
necessarily
G
[0*, 0**].
the indifference
In this case, the
.
given by
is
=n{9<9*\x,r)
decreasing in x. Since lim x _ _ 00
two
[0*,0"] necessarily set r*
_
is
=
.4(0, r) is
€
all
7'*
in
is
monotonic decreasing, then a(x,r)
probability of regime change conditional on x and r
which
an attack
turns implies that the policy maker sets
equilibrium a(x,r)
a(x,r)
The proof
1
i-f
(
]
+
\j>
/
j-0"
and lim x _» +oc
fx
(D =
l\x,r)
=
0,
there
and thus private agents follow a threshold
x > x implying that A(0,r)
= ^ —
from Proposition ( 3 ^ %{9* ,9**)-9**
f
=
x
.
£
(9*, 9")
J
-0".
Coordination and Policy Traps
From
0"
=
—
the monotonicity of x(9*,9)
Theorem
9 established in
(2)
29
- Lemma
2 -
it
follows that
9**.
Step
2.
If
MLRP,
the
satisfies
\l/
then
fi(0
<
0*\x,r)
and thus a(x,r) are necessarily decreasing
in x.
Proof.
Let 1(9) be the probability that 9 sets
=
Using the fact that 1(0)
r.
1 for all
9
<
9*
,
the
probability of regime change conditional on x and r can be written as
1-f
M(0<0*|x,r
Using
[i(6
M(x)
1-tf
<0*\x,r)
l-»(0<9*\x,r)
(*=<>)
1(9)40'
^ («=2)
/ (0) d0]
f~ lip
we have that
dM
<*=
It
follows that
"
^
/<?
<2/u(0
<
(*=*) I
0?)
<
9*\x,r)/dx
«»
if
[/~
and only
f-l^'i^dd
I-l
Using the fact that I
=
(0)
W
E<?
(*=*)
<
9
<9*,
9*
if
f-^>(^)l(9)d9
& {*?) dO
1 for all
Ir
,
JV- (*f*) ' (0)
the above
®9
x, r
*{*?)
which holds true
if ip' /if) is
From the monotonicity
cutoff x' given by
/j,
(9
<
Proof of Proposition
status quo
and
for private
[0, 9}.
w
equivalent to
m
>
9
,
<0,
x, r
monotonic decreasing.
of fi(0
For 9
<
=
<
0*\x,r),
it
follows immediately that a(x,r)
is
monotonic with
r.
0, it is
dominant
for the policy
agents to attack. Similarly, for 9
private agents do not attack,
take any 9 6
is
<0.
dO
+(*?)
9*\x',r)
4.
2
and there
is
The continuation game
>
9,
maker to
set r
the status quo
is
and abandon the
never abandoned,
no need to undertake any costly policy measure. Finally,
following any level of the policy r
with two (extreme) continuation equilibria, no attack and
full
attack.
is
a coordination
game
Let r(0) be the minimal r
G.M. Angeletos,
30
C.
Hellwig and A. Pavan
which private agents coordinate on the no-attack continuation equilibrium,
for
and only
if
<
r
Clearly,
r{9).
V(8,0) - C(r(9))
>
and
9**
£
sets r(8)
=
<
x
if
r*
<
<£
r* with
private agents attack
,
and whenever r
x;
9 for 9 £ [9*,6**],
is
ji,
and a robust equilibrium such
=
>
r*
if
,
1 for
and only
<
and with probability D{8)
in five steps:
Steps
1
if
and
9*
,
=
x
<
if
and only
x*;
whenever r £
with probability D(9)
<
>
x
Let r*
=
min7£*
>
= max TV <
r and r*
examines the
r.
0(9)
=
£
(9,
policy
9ms),
maker
[9*, 9**}.
£
(ii)
they attack
if
status quo
is
The
and non-increasing
in
and the strategy of the
beliefs
2.
if
<
0,
if
£
[0,0],
if
>
0,
,r*]
max
I
8*
The
[r*,r*),
(in)
1
,
and
(r,r)
Define
max {-C{r)}
r£\r*
if
the thresholds 0*, 0**, and x*; Step 3
private agents; Step 5 establishes robustness in the sense of Definition
1.
and only
if
0**.
2 characterize
4
if
(i)
for 9
in
x.
for
examines the behavior of the policy maker; Step
Step
that:
<
they attack
9
if
compact support TV C
and r(8) £ TV with r{0) non-decreasing
[8*, 8**},
abandoned with probability D(9)
The proof
r
they attack
8.
Take any random variable
5.
a system of beliefs
r for 9
r
C (r(8)) <
>
We want to show that for e sufficiently small, there exist thresholds x*
(9*, 9),
Whenever
and only
<£.
optimal for the policy maker to set r(9)
is
or equivalently
0,
Proof of Proposition
distribution
it
i.e.
{0$(r)
-
C{r)}
r£\r*,r"}
max {0-(l-$(r))*(s=£)-C(r)}
7-S[ r *,r*]
and
r(0) as the corresponding
argmax Note
.
that maximizing over [r*,f*]
ing over TV, since necessarily f(0) £ TV. Note also that U(9)
>
increasing whenever U{9)
8ms ~
C(r)
<
all
Step
0*,
2.
=
0.
Therefore, there exists a unique 9* £
>
for all
For any
>
0*, define
_^
$
f
-^
£
J
9
M s)
<
all 0,
and
strictly
and U(9 M s) > Oms - C{r*) >
such that 0(9*)
=
0,
0(9) <
for
0*.
the functions x(0;0*) and v(8;8*) as follows
(^) +* (Sjfi)
Note that dx(8;8*)/d8 £
'
>
(0,
equivalent to maximiz-
non-decreasing for
-C(r*). Moreover, U{9) = -C(r*)
and 0(9)
1
is
is
as e
—
>
(0,1)
0.
To
and therefore dv(9:9*)/d9
>
1.
Note
see this, suppose instead that lim t-_o
also that for
x
^
'
(
£
)
>
any
=w
for
9*,
some
Coordination and Policy Traps
uj
>
This can be true only
0.
and therefore lim £ ^o
^
'
'".
theorem, dU(9)/d9
is
(9; 9*)
,
T|
^
( x(B-fi*)-0* \
now
bounded above by
=
Next, note that for any 9
>
every ^
e
>
9*
*
/
u(6»;
<
0, p(0)
0*)-+0>
U{9)
0.
We
fact that
x
for all 9
>
9**
U{9;9*)
<
v{9-9*)
9
e
(6»;
9*)
is
x
ee
U{9)
< U{9)
for all 9
>
(6*,9**), since U(9;9*)
Step
3.
9**,
9*)
<
9 MS
,
>
>
=
0*;
it
v{9;9*).
6»
lim^o %
for e
<
9
=
\I/
=
0,
and U(0*)
e,
—
v(9;9*)
whereas
(
^z— = 1 — r = 9ms
—
L
1
we conclude that
—
£
(
0, it
)
That x
U(9; 9*)
=
<
9**,
U(9;9*)
9**. Finally, let 9
>
=
9**.
(9*
9
>9- C{?{6))
0.
for
> U{9)
Moreover, note that, as e -»
as e ->
0.
Next,
v{9; 9*) for all 9
=
v(9;9*)
=
let
=
<
9**
v(9**;9*), while the
and U(9;
U{9) at 9
=
>
9*)
>
v{9; 9*)
we conclude that
0**,
be the unique solution to U(9;9*)
>
0,
l
- V
x* implies U(9**;9*)
and U(9**;9*)
increasing in
—
as e
>
<
there exists a unique 9** G {9*, 9) such that
e,
{9**; 9*)
g(9*)
follows that there exists
this result with the properties of the function g{9),
is
9*
strictly increasing in 9
is
'
\I>
and C(r(9)) >
follows that 0** -*
and
<
>
From the envelope
U(9).
and hence g{9)
for 9
implies U{9; 9*)
for all 9
(9; 9*)
=0<r,
+ LO
we have that
9*
and g{9) >
(0**; 0*)
increasing in
9* implies
C{?{9)) whenever e <e. But then v(9;9*)
>
for every
Combining
.
<
j
<
Compare now U{9;6*) with
>
(9;
independent of
conclude that,
for 9
x*
v(9;9*)
9, f(-) is
n^'H
and therefore g(6) >
=
<
Since 9*
.
<
any 9
for
1
(See the argument above). Since 9
.
such that
0(0**)
- 9 MS
9*
,'
„x
x(O;O*)-0 \
T, (
the function g(9)
over this range. Moreover, by definition of x
and therefore v(9*;9*)
which case 9 >
9, in
+*
,
,
*(
Consider
>
But then
1-
,~
I
a contradiction.
is
J~— ) =
'
(
e-»0
which
lim e ^o x
if
31
>
and U(6;6*)
=
and note that
U(9*;9*).
Consider now the behavior of the policy maker. Given the strategy of the private agents,
the policy maker prefers r to any r
dominates any r £
from playing
U(9)
=
all
<
0*,
U_{9)
=
U{9)
if
r.
9
[r*,r*].
We
<
r*,
and
any r
r* to
>
f*
.
Furthermore, by definition, r(9)
thus need to compare only the payoff from playing r{9) with that
Playing r(9) yields U(9), while playing r yields U_(9)
<
U{9)
9
and U{9)
>
> U{9)
=
=
U{9)
U{9) for
for all 9
r(0) whenever 9 G [6>*,0**]
>
all
>
if
9
>
9 e
=
max{0, U{9;
(9*, 9**). It follows that
9 G (9*, ?], U{9)
9**. Therefore, it is
> U{9) = U{9) >
U{9)
for all 9
9*)}.
<
e
Note that
=
U{9)
{9,9**),
for
and
indeed optimal for the policy maker to play
and r otherwise. The resulting probability of regime change
is
D(9)
=
1
G.M. Angeletos, C. Hellwig and A. Pavan
32
<
for 9
non-decreasing,
Step
=
D{9)
6*,
D
- $(f(0)) G
1
abandoned
and only
if
if
<
9
=
In equilibrium, at r
9.
<
For any r
Consider next the behavior of the private agents.
4.
>
9**.
Since r
is
the status quo
is
for 9
[9*, 9**].
non-increasing in the range
is
=
and D{9)
[9*, 9**],
G
[0,1) for 6
r*
,
the probability of regime change
r,
is
given
by
-4>
l
=
<9\x,r)
fi(9
=
fi{9 <9*\x,r)
1
By
construction, x* solves
the regime
beliefs
\x
any r
>
>
/z(0
if
and only
such that
we
r*,
0|x,r)
of-equilibrium
restrict
pi
let fi(9
any r and
Step
/j,
<
C* < C-
for all ^
that
is
=
^
(r)
9\x,r)
=
decreasing in x, attacking
is
[x,x], /i(0
l
0, where 9~ (r)
for all 9
<
9\x,r)
<
0|a:,r)
>
=
{0
:
r(0)
G 0(x) such that 9
=
r*,
r}
— C(r) <
,
we
consider
=
x*. For
r at x
x <
for
1
_1
G
/j.(9
9
and
x,
=
(r)|x,r)
1
G 0(x), then we further
U(9). These beliefs satisfy
the strategy of the agents
beliefs,
r*
In addition, for any out-
.
some
=
=
<
sequentially optimal
is
x.
9*\x,r)
Assume
first
=
and
G
r
=
.4**
has bounded support,
= * (^f1 ) and
4'" (.4*) and
<f =
A{9**,r)
>
with unbounded support such that F(f)
=
< C* -
T+
fe,
and F(f) < #(£)
when
<
such that
>
for all C
the noise distribution
is 4',
$(£)
fc.
0.
By
same
=
and x*
=
x
of 9**. Since U{9)
is
also the
(9**;9*)
.
But then U(9;
same
for all 9
<
< A* <
1
A**
4>(£*
G [C* -k,£*
and note that
+
<
.
9*)
9,
1,
argue that the same r{9) and
U_ defined
above, where
follows that the
and U{9)
and
<
and a
L>(6>)
can also be sustained when the distribution
and
It
k)
+ k], F(0 > *(0
4^ is
<
9**
<
9,
also
we
same
9**
9. It
follows that the
infer U{9**)
Like in Step 3 above, the latter ensures that the optimal policy
is
r{9)
=
,
v(9;9*),
and x* continue
remain the same
=
is
replaced with F.
construction of F, the functions x(9;9*)
a neighborhood of 9**
g{9) remain the
in
=
=
<
r(9) are independent of the noise distribution for all 9
9* continues to solve U(9*)
Let A*
= ^^{A**)
4>(£** - k)
for all ^
We
£**
e be
let
as above.
note that
1
(0,1). Similarly, let
Hence, one can always find a k
sustained
^
that
and construct the corresponding sunspot equilibrium
Note that U(9) and
g(9**)
<
any r G r(0) with r
F. Indeed, consider the functions U,r,x,v,g,U,
same
x 6
for all
1
x. In particular, for
_1
fi(9
not dominated in equilibrium by r(9) for
is
= %(?£=£.}
A(9*,r)
and since
r,
+$
non-increasing in x and \i{9
G M(r); and given these
sufficiently small,
F
is
Finally, consider robustness.
5.
as fi(9
r
r, if
=
£=<L]
indeed optimal. For any out-of-equilibrium r
is
G {9,9)\x,r)
>
x
1 for
9\x*,r)
9\x,r)
to satisfy fj,(9\x,r)
(4), as well as
c.d.f.
<
x such that Q(x) n
for all
for
=
x < x*
if
[i{9
<
/j,(9
_^
in a
to solve
neighborhood
U(9**;9*)
r{9) for
and
all
=
9 G
U{9**).
[9*,
9**
Coordination and Policy Traps
and r(9)
=
r otherwise. Since r(9)
is
also the
policy r(9) can be sustained with either
quo with certainty and
6
9
[9*
,
9**]
>
9
all
\I>
same
or F. Similarly,
9** continue to
the probability of regime change
is
unbounded. With £* and
F
has bounded support
D{9)
^(0 < *(0
same
r(9)
for a11 C
and D{9)
- 2k,C +
< T* -
as
^P.
=
fc,
We
2fc]
and F(C)
9
<
proves that the
9* continue to
and
$(f(0)). Therefore, the
$
£** defined as above, take
[£**
all
[9*, 6**}, this
same
abandon the status
maintain the status quo with certainty, while for
regime changeD(6>) can also be sustained with either
is
£
for all 6
33
A
symmetric argument applies
an arbitrary k
satisfies
> *(0
or F.
F(£)
for all C
=
same probability of
>
and construct
#(f) for
> T* ~
fc-
It
all
£ G
*
[<f
follows that
F
F
if
^
so that
-*,£*
+
fc],
sustains the
conclude that the sunspot equilibria of Proposition 5 are robust.
References
[1]
Albanesi, Stefania, V.V. Chari, and Lawrence Christiano, (2002), "Expectation Traps and
Monetary
[2]
Policy,"
NBER
working paper 8912.
Angeletos, George-Marios (2003), "Idiosyncratic Investment Risk in a Neoclassical
Economy,"
NBER
working paper.
NBER
[3]
Atkeson, Andrew (2000), "Discussion on Morris and Shin,"
[4]
Azariadis, Costas (1981), "Self-fulfilling Prophecies," Journal of
[5]
Battaglini, Marco,
Macro Annual.
Economic Theory
25, 380-96.
and Roland Benabou, (2003), "Trust, Coordination, and the Industrial
Organization of Political Activism," Journal of the European Economic Association
[6]
Benhabib,
of
[7]
1.
and Roger Farmer (1994), "Indeterminacy and Increasing Returns," Journal
Economic Theory
Bernanke, Ben, and
tions,"
[8]
Jess,
Growth
63, 19-41.
Mark
Gertler (1989), "Agency Costs, Net Worth, and Business Fluctua-
American Economic Review
79, 14-31.
Bryant, John (1983), "A Simple Rational Expectations Keynes- Type Model," Quarterly Journal of Economics 98,
3,
525-28.
G.M. Angeletos, C. Hellwig and A. Pavan
34
[9]
Calvo, G. (1988), "Servicing the Public Debt: the Role of Expectations." American Economic
Review, 78(4), 647-661.
[10]
Econometnca
[11]
"Global
Games and Equilibrium
Selection,"
61, 5, 989-1018.
114,
3,
869-905.
Chari, V.V., Lawrence Christiano, and Martin
Discretion," Journal of
[13]
(1993),
Chamley, Christophe (1999), "Coordinating Regime Switches," Quarterly Journal of Eco-
nomics
[12]
Damme
Carlsson, Hans, and Eric van
Economic Theory
Chari, V.V., and Patrick
Eichenbaum
(1998), "Expectation Traps
and
81, 2, 462-92.
Bank
Kehoe
(2003a), "Hot Money," Federal Reserve
Kehoe
(2003b), "Financial Crises as Herds," Federal Reserve
of Minneapolis
staff report 228.
[14]
Chari, V.V., and Patrick
Bank
of Minneapolis working paper 600.
[15]
Cho, In-Koo, and David Kreps (1987), "Signaling Games and Stable Equilibria," Quarterly
Journal of Economics 102,
[16]
Cole, Harold, and
2,
179-221.
Timothy Kehoe
(1996), "Self-fulfilling
Debt
Crises," Federal Reserve
Bank
of Minneapolis staff report 211.
[17]
Cooper, Russell (1998), Coordination Games, Complementarities and Macroeconomics, Cambridge University Press.
[18]
Cooper, Russell, and Andrew John (1988), "Coordinating Coordination Failures
in
Keynesian
"Bank Runs, Deposit Insurance, and
Liquidity,"
Models," Quarterly Journal of Economics 103, 441-463.
[19]
Diamond, Douglas, and Philip Dybvig
(1983),
Journal of Political Economy 91, 401-19.
[20]
Diamond, Peter
Political
[21]
(1982), "Aggregate
Economy
Demand Management
in Search Equilibrium,"
Journal of
90, 5, 881-94.
Drazen, Alan (2001),
"Interest
Rate Defense Against Speculative Attack as a Signal:
Primer," University of Maryland mimeo.
A
Coordination and Policy Traps
[22]
Farrell, J.
35
and G. Saloner (1985), "Standardization, Compatibility, and Innovation," Rand
Journal of Economics 16, 70-83.
[23]
Feltovich, Nick, Rick
Countersignaling,"
[24]
Freixas, Xavier,
London:
[25]
MIT
Harbaugh, and Ted To (2002), "Too Cool
Rand Journal
and Jean-Charles Rochet (1997), Microeconomics of Banking, Cambridge and
Press.
Hellwig, Christian (2002), "Public Information, Private Information and the Multiplicity of
Judd, Kenneth, (1985) "The
ables," Journal of
[27]
Signaling and
School?
of Economics 33, 630-649.
Equilibria in Coordination Games," Journal of
[26]
for
Law
of Large
Economic Theory
Kajii A. and S. Morris (1997),
Economic Theory,
Numbers with
a
107, 191-222.
Continuum
of IID
Random
Vari-
35, 19-25.
"Common
p-beliefs:
The General
Case,"
Games and Economic
Behavior, 18, 73-82.
[28]
Katz, M. and C. Shapiro (1985), "Network Externalities, Competition, and Compatibility,"
American Economic Review
[29]
Katz,
ities,"
[30]
M. and
C. Shapiro (1986), "Technology Adoption in the Presence of
Journal of Political
Economy
Network External-
94, 822-841.
Kiyotaki, Nobuhiro (1988), "Multiple Expectational Equilibria under Monopolistic Competition," Quarterly
[31]
75, 424-440.
Journal of Economics 103, 695-713.
Kiyotaki, Nobuhiro, and John
Moore
(1997), "Credit Cycles," Journal of Political
Economy
105, 211-248.
[32]
Kraay, Aart (2003), "Do High Interest Rates Defend Currencies During Speculative Attacks?",
Journal of International Economics 29, 297-321.
[33]
Kuran, Timur (1987), "Preference
vatism,"
[34]
Economic Journal
Falsification,
Policy Continuity and Collective Conser-
97, 642-665.
Milgrom, Paul, and John Roberts (1990), "Rationalizability, Learning and Equilibrium in
Games with
Strategic Complementarities," Econometrica, 58, 1255-1277.
G.M. Angeletos, C. Hellwig and A. Pavan
36
[35]
Monderer, D. and D. Samet (1989), 'Approximating
liefs,"
[36]
Games and Economic
Behavior,
Hyun Song Shin
Morris, Stephen and
1,
Morris, Stephen and
(1998), "Unique Equilibrium in a
nomics,"
[38]
NBER
Common
Be-
3,
Model
of Self-Fulfilling
587-597.
(2000), "Rethinking Multiple Equilibria in Macroeco-
Macro Annual.
Morris, Stephen and
in
Hyun Song Shin
with
170-190.
Currency Attacks", American Economic Review, 88,
[37]
Common Knowledge
Hyun Song Shin
(2001), "Global
Games
-
Theory and Applications,"
Advances in Economics and Econometrics, 8th World Congress of the Econometric Society
(M. Dewatripont, L. Hansen, and
S.
Turnovsky,
eds.),
Cambridge University Press, Cambridge,
UK.
[39]
Murphy, Kevin M., Andrei
Shleifer,
Push," Journal of Political
[40]
76,
1,
American
72-81.
Postlewaite,
40, 3-5, 1037-47.
Andrew, and Xavier Vives (1987), "Bank Runs
Journal of Political
[43]
97, 5, 1003-26.
Obstfeld, Maurice (1996), "Models of Currency Crises with Self-Fulfilling Features," European
Economic Review
[42]
Economy
Obstfeld, Maurice (1986), "Rational and Self-Fulfilling Balance-of-Payments Crises",
Economic Review
[41]
and Robert Vishny (1989), "Industrialization and the Big
Economy
as
an Equilibrium Phenomenon,"
95, 3, 485-91.
Spence, Michael (1973), "Job Market Signaling," Quarterly Journal of Economics 87,
3,
355-
374.
[44]
Velasco, Andres (1996),
"Fixed Exchange Rates: Credibility, Flexibility and Multiplicity,"
European Economic Review
[45]
40, 3-5, 1023-36.
Vives, Xavier (1999), Oligopoly Pricing: Old Ideas
and
New
Tools,
MIT
Press.
Figure
1
ffere exists
attack
is
an
equilibrium
that,
0e
for
inactive policy equilibrium in
A(Q,r), and the status quo
by r
if
and only
any 0e[0',0"],
[0',0"]- flvate
if
is
C(r0>6
if the
which
abandoned
maker
the policy
and only
if
or C(r0>,4(8,r), that
maker
< 0^
is,
if
to intervene.
same behavior
as
sets r for all 9, the size
.
r'e
riy
and only
policy maker deviates from r to
agents then coordinate on the
incentives for the policy
if
r',
when
if
(r,
8g
r]
is
of the
dominated
[0',0"]. I
in
follows
the market "learns" that
r
=
r,
thus eliminating any
Figure 2
For each r e fcf], there exists a two-threshold equilibrium
9 e [9
When
,
9
]
and r(9)
the policy
=
r otherwise, and in
maker
coordinate on no attack.
signal
is
which the
raises the policy at r, the
When
sufficiently low, in
instead the policy
which case the
size
optimal for the policy maker to raise the policy
if
and only
if
9 e
[9*, 9**].
at
status
in
quo
is
which the policy
abandoned
market "learns" that 9 e
maker
sets
r,
of the attack
if
[9*, 9**]
is
r(9)
and only
if
=
9
r
if
< 9
.
and private agents
private agents attack if and only if their
is
decreasing in
r if and only if C{r*)
< 9 and
9.
It
C(r')
follows that
< A(Q,r),
it
that
is
is,
3837
(
39
Date Due
MIT LIBRARIES
3 9080 02617 8241
"1111
ii