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Working Paper Series
COORDINATION AND POLICY TRAPS
George-Marios Angeletos
Christian Hellwig
Alessandro Pavan
Working Paper 03-18
May
2003
RoomE52-251
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Cambridge,
MA 021 42
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MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
JUN
5 2003
LIBRARIES
Coordination and Policy Traps
George-Marios Angeletos
MIT
and
:
Christian Hellwig
Alessandro Pavan
UCLA
Northwestern University
NBER
This draft:
May 2003
Abstract
This paper examines the ability of a policy maker to control equilibrium outcomes
in
an envi-
ronment where market participants play a coordination game with information heterogeneity.
We
consider defense policies against speculative currency attacks in a model where speculators
observe the fundamentals with idiosyncratic noise.
The
policy
maker
is
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 action, thus resulting in policy traps, where
the devaluation outcome and the shape of the optimal policy are dictated by
self-fulfilling
mar-
ket expectations. Despite equilibrium multiplicity, robust policy predictions can be made.
The
probability of devaluation
is
monotonic
in
the fundamentals, the policy maker adopts a costly
defense measure only for a small region of moderate fundamentals, and this region shrinks as
the information in the market becomes precise.
Key Words:
global games, coordination, signaling, speculative attacks, currency crises, multiple
equilibria.
JEL
Classification
Numbers: C72, D82, D84, E5, E6, F31.
'For encouragement and helpful comments, we are thankful to Daron Acemoglu,
Marco Bassetto,
Olivier Blanchard, Ricardo Caballero, V.V. Chari, Eddie Dekel,
Patrick Kehoe, Robert Lucas
Jean
Tirole.
Jaume Ventura,
Jr.,
SED
Glenn
Atkeson, Gadi Barlevy,
Ellison,
Paul Heidhues,
Narayana Kocherlakota, Kiminori Matsuyama, Stephen Morris, Nancy Stokey,
Ivan Werning, and seminar participants at
AUEB,
Lausanne, LSE, Mannheim, MIT, Northwestern, Stanford, Stony Brook, Toulouse.
2002
Andy
annual meeting, and the 2002
UPF
Carnegie-Mellon, Harvard, Iowa,
UCLA,
the Minneapolis Fed, the
workshop on coordination games. Email addresses: angelet@mit.edu.
chris@econ.ucla.edu, alepavan@northwestern edu.
.
1
G.M. Angeletos, C. Hellvvig 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
economic slowdown.
and cuts rates now
policy-making competence. But
to develop
much
heeds the clamour from
it
of Wall
risks being seen as panicky, jeopardizing its reputation for
it
...
If
if
it
waits until
even more powerful downward
March
momentum
20,
in
it
risks allowing the
economy
what could prove a crucial three
weeks. (Financial Times, February 27, 2001)
1
Introduction
Economic news anxiously concentrate on the information that
different policy choices
the intentions of the policy maker and the underlying economic fundamentals,
interpret
and react to
down animal
spirits
different policy measures,
and ease markets
investigates the ability of a policy
outcomes
in
how markets may
and whether government intervention can calm
to coordinate
maker to
convey about,
on desirable courses of
influence market expectations
environments where market participants play a coordination
actions.
and
This paper
control equilibrium
game with
information
heterogeneity.
A
large
be modeled
number
of economic interactions are characterized by strategic complementarities, can
as coordination games,
and often exhibit multiple equilibria sustained by
expectations. Prominent examples include self-fulfilling bank runs
self-fulfilling
(Diamond and Dybvig,
1983),
debt crises (Calvo, 1988; Cole and Kehoe, 1996), financial crashes (Freixas and Rochet, 1997;
Chari and Kehoe, 2003a,b), and currency crises (Flood and Garber, 1984; Obstfeld, 1986, 1996). J
Building on the global-games work of Carlsson and
Van Damme
(1993), Morris
and Shin (1998,
2000) have recently argued that equilibrium multiplicity in such coordination environments
"the unintended consequence" of assuming
common knowledge
which implies that agents can perfectly forecast each others'
When
is
merely
of the fundamentals of the game,
beliefs
and actions
in equilibrium.
instead different agents have different private information about the fundamentals, players
Strategic complementarities arise also in economies with restricted market participation (Azariadis, 1981), pro-
duction externalities (Benhabib and Farmer, 1994; Bryant. 1983),
demand
spillovers
(Murphy, Shleifer and Vishny,
1989), thick-market externalities (Diamond, 1982), credit constraints (Bernanke and Gertler, 1989; Kiyotaki
Moore, 1997), incomplete
grom and Roberts,
in
macroeconomics.
risk sharing (Angeletos
1990; Vives 1999).
The
and Calvet, 2003) and imperfect product market competition
and
(Mil-
reader can refer to Cooper (1999) for an overview of coordination games
Coordination and Policy Traps
are uncertain about others' beliefs
Iterated deletion of strictly
and one system
and
dominated
3
and thus perfect coordination
actions,
is
no longer possible.
strategies then eliminates all but one equilibrium
outcome
of beliefs.
This argument has been elegantly illustrated by Morris and Shin (1998) in the context of currency
crises.
When
it
common knowledge
is
speculative attack, two equilibria coexist:
that a currency
One
in
which
all
is
sound but vulnerable to a large
speculators anticipate a
crisis,
attack
the currency and cause a devaluation, which in turn confirms their expectations; another in which
speculators refrain from attacking, hence vindicating their beliefs that the currency
On
ger.
signals
the contrary, a unique equilibrium survives
about the willingness and
ability of the
when
speculators receive noisy idiosyncratic
monetary authority to defend the currency against a
speculative attack (the fundamentals). Devaluation then occurs
below a
critical state.
if
This unique threshold in turn depends on
affecting the payoffs of the speculators. Morris
equilibrium models,
and only
if
the fundamentals
all financial
and Shin hence argue
model allows analysis
[their]
not in dan-
is
fall
and policy variables
that, "in contrast to multiple
of policy proposals directed at curtailing currency
attacks." For example, raising domestic interest rates, imposing capital controls, or otherwise in-
creasing the opportunity cost of attacking the currency, reduces the set of fundamentals for which
devaluation occurs.
However, when the fundamentals are so weak that the collapse of the currency
there
is
no point
Similarly,
in raising
currency faces no threat, there
market can
information in turn
may
is
are so strong that the size of the attack
infer that the
permit coordination
and Shin
(1998).
move second,
The
we add a
central
minuscule and the
first
raises the
fundamentals are neither too weak nor too strong. This
maker
we analyze endogenous
particular environment,
is
no need to intervene. Therefore, whenever the bank
interfering with the ability of the policy
In this paper,
inevitable,
domestic interest rates or adopting other costly defense measures.
when the fundamentals
interest rate, the
is
in the
market on multiple courses of action, thus
to fashion equilibrium outcomes.
policy in a global coordination game.
To
settle in a
stage to the speculative currency attacks model of Morris
bank moves
first,
setting the domestic interest rate.
Speculators
taking a position in the exchange rate market on the basis of their idiosyncratic noisy
signals
about the fundamentals, as well as their observation of the
bank.
Finally, the
bank observes the
interest rate set
by the central
fraction of speculators attacking the currency
and decides
whether to defend the currency or devalue.
"See Morris and Shin (2001)
for
an extensive overview of
waite and Vives (1987) and Chamley (1999).
this literature. Earlier uniqueness results are in Postle-
G.M. Angeletos,
4
The main
result of the
paper (Theorem
Hellwig and A. Pavan
C.
establishes that the endogeneity of the policy leads
1)
to multiple equilibrium policies and multiple devaluation outcomes, even
same
coordination in the reaction of the market to the
interest rate.
(or the cost) of
As long
an attack
is
by
Different equilibria are sustained
are observed with idiosyncratic noise.
and r the
when the fundamentals
9,
modes
of
policy choice. Let 9 denote the fundamentals
peg
as the value of defending the
decreasing with
different
the central bank
is
is
increasing with 6
and the
size
willing to incur the cost of a high
interest rate in order to defend the currency only for intermediate values of 9.
The observation
of an
increase in the interest rate thus signals that the fundamentals are neither too low nor too high, in
which case multiple courses of action
may be sustained
the policy maker. If speculators coordinate on the
never finds
it
the market, creating different incentives for
same response
to any level of the policy, the
bank
optimal to raise the interest rate. Hence, there exists an inactive policy equilibrium,
which the bank sets the same low interest rate r
in
in
and speculators play the Morris-Shin
for all 9
continuation equilibrium in the foreign exchange market.
instead speculators coordinate on a
If
when
lenient continuation equilibrium (not attacking the currency)
the bank raises the interest
rate sufficiently high, and otherwise play an aggressive continuation equilibrium (attacking the
currency for sufficiently low private signals), the bank finds
for
and only
if
which the bank sets a high interest rate r* > r
9
<
9*
Which equilibrium
.
order to be spared from an attack, and
are
optimal to raise the interest rate
an intermediate range of fundamentals. Hence, there also exists a continuum, of active-policy
equilibria, in
if
it
all
determined by
traps: In her
self-fulfilling
is
played,
what
for 9
how high
is
€
[9*, 9**},
and devaluation occurs
the rate r* the bank needs to set in
the threshold 9* below which devaluation occurs,
is
market expectations. This result thus manifests a kind of policy
attempt to fashion the equilibrium outcome, the policy maker reveals information that
market participants can use to coordinate on multiple courses of action, and finds herself trapped
in
a position where both the effectiveness of any particular policy choice and the shape of the optimal
policy are dictated by arbitrary market sentiments.
The second
result of the
paper (Theorem
nificantly reduces the equilibrium set as
2) establishes
compared
to
that information heterogeneity sig-
common knowledge and
enables meaningful
policy predictions despite the existence of multiple equilibria. In all robust equilibria, the probability of
devaluation
is
monotonic
in the
by raising the interest rate only
fundamentals, the policy maker
for a small region of
region" vanishes as the information in the market
if
the fundamentals were
common
In the
benchmark model, the
policy
be made
is
"anxious to prove herself
moderate fundamentals, and the "anxiety
becomes
precise.
None
of these predictions could
knowledge.
maker
faces no uncertainty about the aggressiveness of
Coordination and Policy Traps
5
We
market expectations and the effectiveness of any particular policy choice.
in Section 5
relax this assumption
by introducing sunspots on which speculators may condition their response to the
The same equilibrium
interest rate
may now
lead to different devaluation outcomes.
These
policy.
results
thus reconcile the fact documented by Kraay (2003) and others that raising interest rates does
not systematically affect the outcome of a speculative attack - a fact that prima facie contradicts
Moreover, these results help make sense of the
the policy prediction of Morris and Shin (f998).
Financial Times quote:
The market may be
either as a signal of strength, in
signal of panic, in
equally likely to "interpret" a costly policy intervention
which case the most desirable outcome
which case the policy maker's attempt
may be
attained, or as a
market on the preferable
to coordinate the
course of action proves to be in vain.
On
a more theoretical ground, this paper represents a
global coordination
The
games.
first
attempt to introduce signaling in
receivers (speculators) use the signal (policy) as a coordination
device to switch between lenient and aggressive continuation equilibria in the global coordination
game, thus creating different incentives for the sender (policy maker) and resulting in different
equilibria in the
policy
game
(policy traps).
multiplicity result in standard signaling games.
Our
multiplicity result
It is
sustained by different
not by different systems of out-of-equilibrium beliefs, and
refinements, such as the intuitive criterion test of
paper
this
is
also different
(e.g.,
arises in
modes of coordination,
survives standard forward induction
Cho and Kreps
from the multiplicity that
with exogenous public signals
it
thus different from the
is
The
(1987).
multiplicity result in
standard global coordination games
Morris and Shin, 2001; Hellwig, 2002). In our environment,
the informational content of the public signal (the policy)
of the self-fulfilling expectations of the market.
is
endogenous and
Most importantly, our
it is
it is
effectiveness of the policy
in the policy
rather about
how endogenous coordination
in the
game
is
not
given any
market makes the
depend on arbitrary market sentiments and leads to multiple
equilibria
game.
Finally, our policy traps are different
Barro-Gordon environments
Albanesi, Chari
(e.g.,
from the expectation traps that
arise in
Kydland-Prescott-
Obstfeld, 1986, 1996; Chari, Christiano and Eichenbaum, 1998;
and Christiano, 2002). In these works, multiple
ment's lack of commitment and would disappear
rule.
the result
policy-traps result
merely about the possibility of multiple continuation equilibria in the coordination
realization of the policy;
itself
if
equilibria originate in the govern-
the policy maker could commit to a fixed policy
In our work, instead, equilibrium multiplicity originates in endogenous coordination and
orthogonal to the commitment problem; what
is
more, despite the risk of
falling into
is
a policy trap,
the government need not have any incentive ex ante to commit to a particular interest rate.
G.M. Angeletos,
6
The
rest of the
paper
is
Hellwig and A. Pavan
C.
organized as follows: Section 2 introduces the model and the equilibrium
Section 3 analyzes equilibria in the absence of uncertainty over the aggressiveness of
concept.
market expectations. Section 4 examines to what extent meaningful and robust policy predictions
made
can be
despite the presence of multiple equilibria. Section 5 introduces sunspots to examine
which the policy maker faces uncertainty over market reactions. Section 6 concludes.
equilibria in
The Model
2
Model Description
2.1
The economy
indexed by
is
populated by a continuum of speculators (market participants) of measure one,
and uniformly distributed over the
i
one unit of wealth denominated
in
asset or convert to foreign currency
who
banker (the policy maker),
peg, or
some kind
benefits the
of
domestic currency, which he
and invest
M.
managed exchange-rate system.
3
bank associates with maintaining the peg,
peg 9 and
We
let 9
fixes
three stages. In stage
1,
=
9
+
e£,
about
9.
The
scalar e 6
absolutely continuous
c.d.f.
\I/
is
noise,
and density
the entire real line (unbounded
hill
is
©
fixed
parametrize the cost and
or equivalently her willingness
is
a central
and
ability
private information to the
bank
For simplicity, we
uniform.''
the central banker learns the value of maintaining the
the domestic interest rate r £ 7Z
private information about 9 and £ {
We
G
refer to as "the fundamentals."
tp
(0,
=
[r,r].
In stage 2, speculators choose their
by the central bank and
portfolios after observing the interest rate r set
signals x,
either invest in a domestic
and the common prior shared by the speculators be a degenerate
The game has
endowed with
and seeks to maintain a
controls the domestic interest rate
and corresponds to what Morris and Shin (1998)
G=
may
is
in a foreign asset. In addition, there
it
to defend the currency against a potential speculative attack. 9
let
Each speculator
interval.
[0. 1]
after receiving private
oc) parametrizes the precision of the speculators'
i.i.d.
across speculators
strictly positive
and independent
and continuously
of
9,
with
differentiable over
support) or a closed interval [— l.+l] (bounded support).
adopt the convention of using female pronouns
for
the centra] banker and masculine pronouns for the specu-
lators.
That the fundamentals
fication.
Our
economy, and
for
results
We
economy coincide here with the type
if
C R and any
refer to
i
bank
is
clearly just a simpli-
receives about the information of the policy maker. Also, our results hold
strictly positive
Morris and Shin (2001)
priors in global coordination games.
of the central
one reinterprets 9 as the policy maker's perception of the fundamentals of the
x, as the signal speculator
any interval
"global."
of the
remain true
and continuous density over 0, provided that the game remains
for a discussion of the role of
degenerate uniform and general
common
Coordination and Policy Traps
Finally, in stage 3, the central
bank observes the aggregate demand
Note that stages
decides whether to maintain the peg.
global speculative
On
fixed.
game
if
and 3
2
the fundamentals were
and
for the foreign currency
our model correspond to the
of
and Shin (1998); our model reduces to
of Morris
the other hand,
7
theirs
common knowledge
if
=
(e
r
is
exogenously
and 3
0), stages 2
would correspond to the speculative game examined by Obstfeld (1986, 1996).
We
normalize the foreign interest rate to zero and
u(r,
where
G
a^
[0, 1]
That
abandoned.
is
— (l- a,i)r + a
>
ty
is
r
Dir,
the devaluation premium, and
who
a speculator
is,
whereas a speculator who attacks the peg
certainty,
the pay-off for a speculator be
the fraction of his wealth converted to foreign currency (equivalently, the
is
probability that he attacks the peg),
the peg
di,D)
let
does not to attack
=
(a^
1)
earns
tt if
(a,
D
is
=
0) enjoys r
the peg
the probability
with
abandoned and
is
zero otherwise.
We denote
speculators
with a the aggregate
who
attack the peg) and
U{r, a,
V(9 a)
:
is
demand
D, 9)
V
C
and
devalue, whereas for 9
x
let
=
are continuous.
>
9
it is
inf
{x
Pr(f?
:
[—1,-1-1],
unbounded
full
x
r.
V
is
a and C(r) the
increasing in 9 and decreasing in a,
=
V(9, 1)
=
0.
For 9
<
9
is
it
dominant
dominant to maintain the peg. The interval
range" of 9 for which the peg
support
D)V{9, a) - C{r).
<
9\x)
<
1}
=
9
—
e
and x
is
and x
=
cost of
C is increasing
5
Let 9 and 9 be defined by V(9,0)
"critical
= (I-
the net value of defending the currency against an attack of size
and both
r,
the foreign currency (equivalently, the measure of
the payoff for the central bank be
let
raising the domestic interest rate at level
in
for
sound but vulnerable to a
= sup{x
9
support. Next, note that r
+
>
e,
:
Pr(#
<
whereas x
9\x)
bank to
thus represents the
sufficiently large attack. Also,
> 0}
= — oo
[#, 9]
for the
;
if
the noise £ has bounded
and x
= +oo
if
the noise has
represents the efficient, or cost-minimizing, interest
We normalize C(r) = and, without any loss of generality, let the domain of the interest rate
11 = [r, r], where r solves C(r) = max# {V(9, 0) — V(9, 1)}; r represents the maximal interest
rate.
be
rate the
assume r
offset
bank
<
7T,
ever willing to set in order to deter an attack.
which ensures that
the devaluation premium.
follows that 9
J
is
=
=
C{r) and 9
it is
To make
1
=
we
too costly for the bank to raise the interest rate to totally
Finally, to simplify the exposition,
=
things interesting,
we
let
V(9, a)
—
C{f).
That the bank has no private information about the cost
of raising the interest rate
is
not essential.
9
—
a. It
G.M. Angeletos, C. Hellwig and A. Pavan
8
Equilibrium Definition
2.2
In the analysis that follows,
sitive to
we
restrict attention to perfect
whether the idiosyncratic noise
unbounded
We
support.
(full)
in
the observation of the fundamentals has bounded or
refer to equilibria that
the information structure as robust equilibria.
the intuitive criterion refinement,
nation
in
game described
the literature,
it is
in Section 2.1
We
can be sustained under both specifications of
will also verify
introduced
first
Bayesian equilibria that are not sen-
in
that
Cho and Kreps
all
robust equilibria satisfy
As the
(1987).
global coordi-
very different from the class of signaling games examined
is
useful to formalize the equilibrium concept
and the
intuitive criterion test for
the strategic context of this paper.
Definition
K ->
[0, 1],
1
a
:
A
perfect
R
x
K ->
Bayesian equilibrium
a
[0, 1],
x
:
K ->
[0, l]
is
a set of functions r
and
,
(cOxRxR-.
—
>
:
[0, 1]
,
1Z.
D
such
x
:
[0. 1]
x
that:
r{9) G argmaxt/(r,a(6>,r),.D(6*,o(0,r),7-),6l);
(1)
+oo
max J u(r, a, D
a(x,r) G arg
a€[0,l]
(#,
a(9,r),r))du(9\x,r)
anda{9,r)—
J
a(x, r)ip (^r - ) dx;
D(9,a,r) G are max U(r,a,D,9);
(3)
v
'
£>e[o,i]
u
(9\x, r)
where 0(x)
r(9)
is
=
=
for
{9
:
ip
{
all
9
£=^)
0(x) and
(£
>
is
0}
[i
[9\x. r)
satisfies
Bayes' rule for any r G r(0(x)),
(4)
the set of fundamentals 9 consistent with signal x.
the policy of the bank and D(9,a,r)
is
the probability of devaluation.
u(9\x.r)
a speculator's posterior belief about 9 conditional on private signal x and interest rate
is
(2)
e
r,
is
a(x,r)
the speculator's position in the foreign-exchange market, and a(9,r) the associated aggregate
demand
for foreign currency.
6
and the devaluation decision
(2)
means that the
beliefs u(9\x,r)
Conditions
(1)
and
9.
mean
that the interest rate set in stage
is
sequentially optimal for the speculators, given
Finally, condition (4) requires that beliefs
do not assign positive measure
to fundamentals 9 that are not compatible with the private signals x. and are pinned
Bayes' rule on the equilibrium path.
D{9)
=
That
To simplify
notation, in
what
follows
we
D(9,r(9),a{9,r(9)) as the equilibrium devaluation probability for type
a(8, r)
from the Law
of
=
/_
°°
a(x, r)ip (-
—
1
stage 3 are sequentially optimal for the bank, whereas condition
in
portfolio choice in stage 2
about
(3)
)
dx represents the fraction
Large Numbers when there are countable
down by
will also refer to
9.
of speculators attacking the currency follows directly
infinitely
many
agents; with a continuum, see
Judd
(1985).
Coordination and Policy Traps
Definition 2
A
perfect
Bayesian equilibrium
robust
is
if
and only
9
same
if the
policy r(9)
and
devaluation outcome D(9) can be sustained with both bounded and unbounded idiosyncratic noise
in the speculators' observation of
As
it
depend
will
become
critically
9.
clear in Section 4, this refinement simply eliminates strategic effects that
on whether the speculators' private information has bounded or unbounded
full
support over the fundamentals. Note that robustness imposes no restriction on the precision of the
signals.
Definition 3 Let U{9) denote the equilibrium payoff of the central bank when the fundamentals are
9,
and
whom
define 0(r) as the set of 9 for
the choicer
is
dominated in equilibrium by r (9)
any 9 £ 0(r) whenever Q(r) C Q(x). 7
as the set of posterior beliefs that assign zero measure to
perfect Bayesian equilibrium survives the intuitive criterion test if
and any
r
e
U (9) > U (r, a(6*,r), D(9,a(9,r),r),9)
TZ.
In words, a perfect Bayesian equilibrium
,
dominated
in equilibrium
by
r{9).
^
when
A
for any # £
<E
there
M{r).
a type
is
r(9) should speculators' reaction not
measure to types
beliefs that assign positive
Such
if,
for a(9,r) satisfying (2) with a
9 that would be better off by choosing an interest rate r
be sustained by out-of-equilibrium
and only
the intuitive criterion test
fails
andM{r)
beliefs are often highly implausible,
with the definition of perfect Bayesian equilibria. In our game,
all
for
whom
r
is
although consistent
robust equilibria pass the intuitive
criterion test.
Model Discussion
2.3
The
particular pay-off structure
and many of the
institutional details of the specific coordination
environment we consider in this paper are not essential
for our results.
For example, the cost of
the interest rate can be read as the cost of implementing a particular policy; this
may depend
also
on the fundamentals of the economy, as well as the aggregate response of the market. Similarly,
the value of maintaining the peg represents the value of coordinating the market on a particular
action; this
may depend,
but also on the policy
That stage
3
is
devaluation policy
7
Formally, 0(r)
=
not only on the underlying fundamentals and the reaction of the market,
itself.
strategic
is
is
sequentially optimal
{9 such that
H £ A4(r)}, wherejVI(r)
also not essential.
=
{fi
U
(9)
> U
satisfying (4)
if
Given the payoff function of the central bank, the
and only
if
D(9,
r,
a)
=
1
whenever V(9, a)
{r,a(9,r), D(9,a(9,r),r),9) for any a(9,r) satisfying (2)
and such that
fi
(9\x,r)
=
for
any 9 € 0(r)
if
Q(r)
C
<
and
G(i)}.
and
(4)
with
G.M. Angeletos, C. Hellwig and A. Pavan
10
D(9,
r,
=
a)
whenever V(9, a)
>
exogenously devalued for any 9 and
0, for
any
r.
a such that
to abandon the peg whenever the aggregate
All our results
V(9, a)
demand
<
would hold true
0, as in
if
the currency were
the case the bank has no choice but
amount
of foreign currency (a) exceeds the
Indeed, although described as a three-stage game, the model essentially
of foreign reserves (6).
reduces to a two-stage game, where in the
stage the policy
first
maker
about 6
signals information
to the market and in the second stage speculators play a global coordination
game
in response to
the action of the policy maker. Stage 3 serves only to introduce strategic complementarities in the
8
actions of market participants.
we suggest that our model may
In short,
well
fit
a broader class of environments in which a
stage of endogenous information transmission (signaling)
such
as, for
is
followed by a global coordination game,
example, in the case of a government offering an investment subsidy in an attempt
to stimulate the adoption of a
new
technology, or a telecommunications
company undertaking an
aggressive advertising campaign in attempt to persuade consumers to adopt her service.
Policy Traps
3
Exogenous versus Endogenous Policy
3.1
Suppose
for
a
moment
that the interest rate r
is
reduces to the model of Morris and Shin (1998).
=
exogenously fixed, say r
The
lack of
Our model then
r.
common knowledge
over the fun-
damentals eliminates any possibility of coordination and iterated deletion of strongly dominated
strategies selects a unique equilibrium profile
Proposition
exists a
and
1
(Morris-Shin) In
the speculative
game with exogenous
the bank devalues if
and only
if
6
< 9ms- The
are decreasing in
Note that,
if
^
thresholds
^ /S
=*
discretion
is
that the
x < xms
are defined by
(5)
)
r.
the bank could
and expectation
if
gMS
commit never
to devalue in stage 3, there
would be no speculation
the market would not play a coordination game. Such lack of commitment
this paper
and only
if
xms an d 9ms
;
(
beliefs.
interest rate policy, there
unique perfect Bayesian equilibrium, in which a speculator attacks
9ms =
and
and a unique system of equilibrium
traps,
game
but not for our
results.
What
is
is
essential for the kind of policy traps
the market plays in response to the action of the policy maker
game; the origin of strategic complementarities
is
irrelevant.
in stage 2,
essential for the literature
is
we
and
on policy
identify in
a global coordination
Coordination and Policy Traps
Proof.
Uniqueness
xms and 9ms
in
established in Morris
is
speculator with signal x finds
<
H(0
=
9 MS \x)
Given that D{9)
our setting.
By Bayes
fg <eMs dfi(0\x).
solves the indifference condition
1
it
—
\Im
we
and Shin
(1998).
—
< 9ms and D(9) =
1
optimal not to attack
it
11
rule, fi{9
XMS ~ MS
if
9
if
and only
M s\x) =
<
9
=
r.
The
Here,
if -k[x{9
1
- *
only characterize
otherwise, a
< 9ms\x) <
(s=tot*>)
L,
where
Thus, x MS
.
fraction of speculators attacking
J
is
=
then a{9,r)
Pr(x
the bank devalues,
(
bank
central
The
j
.
9
if
gives (5). It
solves
xms
interest rate, or
it is
for a speculator to take
are decreasing functions of
r,
V(9MS, a {&MSiL))
—
0,
that
is,
in r.
the risk of attacking the currency
more generally reducing the speculators'
if
follows
crisis
ability
by simply
raising the domestic
and incentives
to attack.
the policy did not convey any information to
the market. However, since raising the interest rate
willing to defend the currency
It
which suggests that the monetary authority should
Indeed, that would be the end of the story
is
costly,
a high interest rate signals that the
and hence that the fundamentals are not too weak.
other hand, as long as speculators do not attack
when
their private signal
is
On
the
sufficiently high, the
faces only a small attack
when
the fundamentals are sufficiently strong, in which case there
no need to raise the interest
rate.
Therefore, any attempt to defend the peg by increasing the
bank
is
and hence
an d 9ms are both decreasing
be able to reduce the likelihood and severity of a currency
is
0,
larger the return on the domestic asset, or the higher the cost of short-selling the domestic
xms and 9ms
bank
<
follows that V(9,a(6,r))
indifference condition for the speculators with that for the
then immediate that
is
it
< 9ms, where 9ms
Combining the
currency, the less attractive
that
= * Umjz=1\
x MS \9)
and only
if
XM5 ~ MS
&MS ~ ^
<
interest rate
speculators
is
9
interpreted by the market as a signal of intermediate fundamentals, in which case
may
coordinate on either an aggressive or a lenient course of action. Different modes of
coordination then create different incentives for the policy maker and result in different equilibria.
At the end, which equilibrium
in order to
be spared from a
a devaluation occurs, are
Theorem
perfect
'
1 (Policy
all
is
played,
crisis,
how
and what
determined by
Traps) In
high
is
e
>
the level of the policy the bank needs to set
the critical value of the fundamentals below which
self-fulfilling
the speculative
Bayesian equilibria for any
is
game
market expectations.
ivith
endogenous policy, there
0.
This interpretation of the information conveyed by a high interest rate follows from Bayes' rule
level of
the interest rate
is
exist multiple
if
the observed
on the equilibrium path, and from forward induction (the intuitive criterion) otherwise.
G.M. Angeletos, C. Hellwig and A. Pavan
12
There
(a)
r for
6
an inactive-policy equilibrium: The bank
and devaluation occurs
There
(b)
r*
9
all
is
r* only for 9 € [9* 9**],
,
and devalues
9*=C(r*)
is
9*
< 9 ms-
6
if
an equilibrium in which the bank sets either r or
{r,r\, there is
The threshold
and only
and
if
9**
and only
=
9*
+
if
<
9
_1
e
[*
9*
(l
,
r*, raises the interest rate at
where
^0*) - # _1
-
—
>
strategies for the speculators that are
sections.
Theorem
Theorem
(0*)]
(6)
0.
All the above equilibria are robust, satisfy the intuitive criterion,
of
For any
independent of e and can take any value in {9,9ms], whereas the threshold 6**
is
increasing in e and converges to 9* as e
The proof
— ^p;
continuum of active-policy equilibria: Let r solve C(r)
a
is
if
sets the cost-minimizing interest rate
1
monotonic
states that there
is
be
supported by
in x.
follows from Propositions 2
1
and can
and
3,
which we present
in the next
two
a continuum of equilibria, which contrasts with the unique-
ness result and the policy conjecture of Morris
and
Shin.
The
policy
maker
is
subject to policy traps:
In her attempt to use the interest rate so as to fashion the size of a currency attack, the monetary
authority reveals information that the fundamentals are neither too weak nor too strong. This
formation facilitates coordination in the market, sustains multiple
self-fulfilling
in-
market responses to
the same policy choice, and leads to a situation where the effectiveness of any particular policy (the
eventual devaluation outcome) and the shape of the optimal policy are dictated by the arbitrary
aggressiveness of market expectations.
bank never to
only
if
which
raise the interest rate
x < xms)
in
f° r
an Y
In the inactive-policy equilibrium, speculators expect the
and play the same continuation equilibrium (attacking
r Anticipating this, the
-
bank
finds
and
pointless to raise the interest rate,
turn vindicates market expectations. In an active-policy equilibrium instead, speculators
expect the bank to raise the interest rate at r* only for 9 G
response for any r
<
r*
and on a lenient one
simply conforming to the arbitrary
All equilibria in
Theorem
1
that are monotonic in x. That
and only
it
if
his private signal
for
self-fulfilling
any r >
r*.
[9*, 9**],
coordinate on an aggressive
Again, the bank can do no better than
expectations of the market.
can be supported by simple threshold strategies
is,
given any policy choice
about the fundamentals
falls
r,
for
the speculators
a speculator attacks the currency
if
below a threshold which depends on the
particular equilibrium played by the market.
Finally, note that all active-policy equilibria in
the devaluation outcome
is
monotonic
in
Theorem
1
share the following properties. First,
the fundamentals.
Second, the devaluation threshold
Coordination and Policy Traps
is
determined by
self-fulfilling
market expectations, but never exceeds the devaluation threshold
when the fundamentals
that would prevail under policy inaction. Third,
market can easily recognize
sufficiently strong, the
policy
maker to
raise the interest rate;
that the market
and
it is
likely to
is
13
this, in
are either very weak, or
which case there
no value
is
for the
then only for a small range of moderate fundamentals
it is
be "uncertain"
about eventual devaluation outcomes,
or "confused"
thus only for moderate fundamentals that the policy maker
"anxious to prove herself
is
by taking a costly policy action. Fourth, the "anxiety region" shrinks as the precision of market
information increases. In Section
4,
we
further discuss the robustness of these policy predictions.
Inactive Policy Equilibrium
3.2
When
the interest rate
is
endogenous, the Morris-Shin outcome survives as an inactive policy
equilibrium, in which the interest rate remains uninformative about the probability the currency
be devalued.
will
Proposition 2 (Perfect Pooling) There
a robust inactive policy equilibrium, in which the
is
central bank sets r for all 9, speculators attack if
rate chosen by the central bank,
and
9ms
and devaluation occurs
= r,
in equilibrium all 9 set r
the fundamentals 9 and hence the continuation
There
to the Morris-Shin game:
and only
if
< xms
x
is
game
>
r
is
dominated
dominated
is
<
where C(r)
C(r)
=
of-equilibrium beliefs
fi(9
<
beliefs,
if
if
< 9ms- The
9
interest
thresholds
xms
^A/s|x,r)
is
and only
if
Note that
r
9
is
by r
and only
if
it
by
-
r(9) for
for
For 9
C(r)
<
<
in
< 9ms-
=
r
is
isomorphic
Equilibrium beliefs
and
=
r,
10
let
equilibrium by r for
if
and only
h(9ms\ x
all
6
7^
,
0ms-
tO
On
<
any 9 such that 9
satisfies irn(9
=
9 ms, V[6,0)
V(6,a{9,r)).
any out-of-equilibrium
optimal to attack
Finally, for r
dominated
9
Note that r solves C(r)
r
that satisfy (4) and the intuitive criterion,
non-increasing in x
< 9ms-
if
for all types:
a(9,r) and thus V(6,Q)
in equilibrium
a speculator finds
sets r
rule.
9ms- Therefore,
/i
bank
a unique continuation equilibrium, in which a speculator attacks
in equilibrium
> 9Ms >
9 ms, C{r)
r G (r,r)
and only
starting after the
Next, consider out-of-equilibrium interest rates.
>
x < xms, independently of the
the observation of r conveys no information about
an d the central bank devalues
down by Bayes'
are then pinned
r
if
if
are defined as in (5).
Proof. Since
if
and only
^
i.e.
V(#mS)0). Any
—
C(r) <
0;
for
the other hand, any
C{r) or a(9,r)
<
C(r),
r,
one can construct out-
\±
G M{r), and such that
< 9Ms\ x r — r at x — X MS- For such
if x < xms, forcing devaluation to occur
,
=
1 for all
)
x such that 6ms £ ®{ x ), and
G.M. Angeletos, C. Hellwig and A. Pavan
14
fj,{0\x,r)
is
=
n{0\x) otherwise, so that again (4) and the intuitive criterion are satisfied. Then, there
game
a mixed-strategy equilibrium for the continuation
attacks
if
and only
if
x
< xms and
type
Given that speculators attack
bank
to set r{9)
6ms = ^—^
G
=
9ms
if
< xms
x
anY r
f° r
r/it.
it
i
which a speculator
r, in
i
optimal
s
for
the central
r for all 9. Finally, consider robustness in the sense of Definition 2.
(0,1),
is
(5)
any e >
satisfied for
unbounded support, by simply
and any
xms = 9ms+^^~ 1
letting
and the devaluation outcome D(9)
for all 9,
devalues with probability
and only
if
=
following r
—
{&ms)-
with either bounded or
\P,
It follows
< 9ms and
9
1 for
c.d.f.
Given
that the policy r{9)
=
D(9)
=
r
otherwise, can be
sustained as a robust equilibrium.
The
inactive policy equilibrium
is
illustrated in Figure
on the vertical one. The devaluation threshold 9ms
>
9 MS-
Consider a deviation to some
Note that C(r') > 9
r' is
=
U{9)
is
dominated
deviates to
7-',
in
if
and only 9 <
9'
equilibrium by r
if
r'
for all 9
G
(r, r)
if
<
and
and C(r') > a{9,
and only
9
the market "learns" that 6 £ [9',9"}.
is
on the horizontal
a and
axis,
C
the point of intersection between the value of
is
defending the peg 9 and the size of the attack a(9,r). Figure
Note that the equilibrium payoff
9
1.
9
1
also illustrates the intuitive criterion.
MS
let 9'
r)
if
=
and U(9)
and 9" solve
and only
if
Hence,
9
£
[#',#"].
}l
Furthermore, since
if
9",
G
a{6,r)
=
9'
>
6
-
6
C(r')
>
for all
=
a(9",r).
which implies that
[9',
6"]
9ms 6
and the bank
\®' -.9"},
one can
construct beliefs that are compatible with the intuitive criterion for which speculators continue to
attack whenever x
< xms,
in
which case
it is
pointless for the
Insert Figure
any system of
Clearly,
beliefs
1
bank
to raise the interest rate at
here
and strategies such that a(9,r) > a(9,r)
policy inaction as an equilibrium; the
r'
bank has then no choice but to
set r(9)
for all r
—
>
r sustains
r for all 6, confirm-
ing the expectations of the market. Note that a higher interest rate increases the opportunity cost
of attacking the currency and, other things equal, reduces the speculators' incentives to attack. In
an inactive-policy equilibrium, however, this portfolio
ulators attach to a final devaluation.
2 have the property that these
11
two
The
particular beliefs
effects just offset
Throughout the paper, the expression "the market
common p =
1
belief
among
and Kajii and Morris (1995).
effect
learns
the speculators given that
Y
is
is
offset
by the higher probability spec-
we consider
in
the proof of Proposition
each other, in which case speculators use the
X
by observing Y" means that the event
common
X
becomes
knowledge; see Monderer and Samet (1989)
Coordination and Policy Traps
same strategy on and
information.
the equilibrium path and condition their behavior only on their private
off
then as
It is
speculators do not pay attention to the policy of the bank. 12
if
Active Policy Equilibria
3.3
We now
prove the existence of robust active-policy equilibria in which the bank raises the interest
[9*, 9**].
rate at r* for every 9 G
Proposition 3 (Two-Threshold Equilibria) For any
librium in which the central bank sets r* for
and only
9
<
15
9*
rium
<
if either r
r*
x* solves itu(9
.
<
and only
exists if
<
x*
,
9*\x*, r)
—
r,
and x
if r*
G
or r
>
all
r*
9 G [9* ,9**]
and x
<
{r,r\, there is a two-threshold equi-
€
r*
and r otherwise; speculators attack
if
and only
if
devaluation occurs
x; finally,
whereas 9* and 9** are given by
{r,r\ or, equivalently, if
if 9*
and only
(6).
A
if
two-threshold equilib-
G {9,9ms\- All two-threshold
equilibria are robust.
Proof. Let 9 G
any r
<
r*
,
[9* ,9**)
solve V{9,a{9,r))
speculators trigger devaluation
Consider
0(x)
r
E
decreasing in x.
we
{r,r*),
x and
<
7Tfj,(9
9\x,r)
u(9 G [9*,9**]\x,r*)
>
let
=
r*
Q\ x r )
,
=
r at
i.e., if
=
x
C(r)
1 for all
and 0(x) n
—
=
1
When
and 0(x)
[x
=
is
vp
x*.
Furthermore,
,
if
<
0, or r
Given these
>
we
that, for
r, beliefs
are pinned
down by
Bayes'
Therefore,
=
-
<
<
jj,(9
C(r).
9\x,r)
let
£
restrict
When
r
=
9*\x*,r)
some 9 G Q{x),
[9*, 9**]
r*,
and note
,
9.
then we further
x such that 6(x) n
—
for
(~-)
=
is
r
is
For any
non-increasing in
not dominated in
to satisfy /j,(9\x,r)
/j,
r*
r/ir.
—
Bayes' rule implies
,
0. Finally, for any (x,r) such that
u(9
>
beliefs, (4) is satisfied,
9\x,r)
/j,
G
=
1
M{r)
for all
for
any
x
r,
> x and
and the
sequentially optimal.
how
to
behave
off the
equilibrium path; they simply continue to
strategy they "learned" to play in equilibrium.
the noise
=
r
=
x* be the unique solution to [i(9
'"Moreover, speculators do not need to "learn"
same
<
fj.{0<e*\x,r)
< min {a{9,r),9}
[9* ,9**}
otherwise.
strategy of the speculators
play the
9
When
G Q(x) such that 9 < C(r) or a(9,r)
for all 9
m(^
We
if
consider out-of-equilibrium beliefs a such that
equilibrium by r(9),
either r
and only
13
[9* ,9**] for all x.
= l\x,r)=n(6<6\x,r) =
(jL{D{9)
is
^
where a{9,r)
0,
the behavior of the speculators.
first
rule for all z, since (6) ensures
which
if
=
—
e,
is
x
unbounded,
+
e],
for
any
this
x.
is
immediate; when
it is
bounded,
it
follows from the fact that \6"
—
6*\
<
2e
G.M. Angeletos, C. Hellwig and A. Pavan
16
Consider next the central bank. Given the strategy of the speculators, the bank clearly prefers
r to any r 6
(r,
and
r*)
any
r* to
r*. 9* is indifferent
>
r
attacked) and setting r (and being forced to devalue)
=
0*
C{r*). Similarly, 9**
is
r (and being attacked without devaluing)
=
C(r*)
i.e.
a
(9*, 9**),
9 6
any 9 > 9**
for
> a
(9,r)
.
From
= C (r*)
(9**,r)
i^
= *
a(9**,r), where a(9",r)
11
and thus
r*
-
V(9**,Q)
if
For any 9
)-
preferred to
is
—
V[9*,0)
if
C(r*)
r* (and not being attacked)
and only
if
and only
if
between setting
indifferent
between setting r* (and not being
the indifference conditions for 9*, 9**
C{r*)
<
9*
,
r
i.e.
and setting
V(9*\ a{9**
is
optimal; for any
whereas the reverse
r,
0,
=
and x* we obtain x*
,
,
=
=
9**
r)),
true
is
+ e^~
l
{9*)
and
V
+£
It follows
that 9*
infer that a
<
9**
if
and only
<
9*
if
two threshold equilibrium
^-^**)-*-
1
[ix
- r)/ir = 9 MS
exists
and only
if
Finally, consider robustness in the sense of Definition
any
arbitrary 9**
>
For any
9*.
support R, we have that k
Hence,
any
for
9**
For r*
(0, oo).
—
>
r,
9*
9*
6
<
=
9 £
1
for
any
—
9**
= 9ms
[6>*,#**]
Using 9*
-
9*
and
r*
if
~
*)
^
9*
and D(9)
=
A
two-threshold equilibrium
(9,
=
C{r*) and 9 MS
=
C{r), we
9ms], or equivalently r* £
let
9*
—
{r,r\.
C(r*) and take
and
(
e *)} e (0, oo) since (1
=
- ^9*) e
can always be satisfied by letting e
k. condition (7)
r(#)
6
(7)
For any r* £ {r,r),
2.
(7) holds for
every e and every ^.
then clearly satisfied at x*
is
(**)]
with either bounded support [— l.+l] or unbounded
- ^Tr
(l
and
condition for the speculators
policy r{9)
V
=
^
c.d.f.
if
1
=
9**
+
The
=
(fi*, 1).
—^— €
indifference
e^>~ i (9*). It follows that the
and the devaluation outcome D{9)
r otherwise,
full
=
1 if
otherwise, can be sustained as a robust equilibrium.
is
illustrated in Figure
>
the observation of any interest rate r
r
is
2.
Like in the inactive policy equilibrium,
interpreted by market participants as a signal of
intermediate fundamentals. However, contrary to the inactive policy equilibrium, speculators switch
from playing aggressively (attacking
only
if
if
and only
if
x < x*) to playing leniently (attacking
x < x) whenever the policy meets market expectations
by market participants, the bank
finds
it
(r
>
r*).
unsuccessful attack
9** are
in
is
>
optimal to raise the interest rate to defend the currency
9*),
2.
lower than the cost of raising the interest rate (9
Finally, note that, in
become common knowledge among
offsets
but not so strong that the cost of facing a small and
determined by the indifference conditions
Figure
and
Anticipating this reaction
whenever the fundamentals are strong enough that the net value of defending the currency
the cost of raising the interest rate (9
if
9*
=
<
9**).
C{r*) and a(9**,r)
The thresholds
—
9*
and
C(r*), as illustrated
any two-threshold equilibrium, the exact fundamentals 9 never
speculators.
What
the market "learns" from equilibrium policy
Coordination and Policy Traps
observations
is
only whether
[6*, 9**]
6
or
but
[6*, 9**},
£
this
17
enough to
is
facilitate coordination.
Insert Figure 2 here
There are other out-of-equilibrium
and
beliefs
strategies for the speculators that also sustain
we could have assumed
the same two-threshold equilibria. For example,
the most aggressive continuation equilibrium (attack
short of market expectations (r
<
r
<
r*).
and only
if
x
if
<
speculators coordinate on
x)
whenever the policy
Nonetheless, the beliefs and strategies
we
the proof of Proposition 3 have the appealing property that the speculators' strategy
<
for all r
r*
.
It is
then as
short of market expectations and continue to play exactly as
For any r*, the devaluation threshold 9*
—
9**
>
—
9* as e
>
9*
an arbitrary devaluation threshold
that a two-threshold equilibrium exists
From
as follows.
r* raises
r*
6
if
and only
9*
which
x*|0**),
is
9*
whereas 9**
increasing in e and
is
bank needs to
raise the interest
limit, the interest rate policy
>
<
falls
has a spike
9ms- The intuition behind
— C (r*) = a (9** ,r) we
,
size of the attack at 9**
9**\x*). Therefore, Pr(0
>
.
also
this result
is
infer that a higher
The
latter
is
given
9**\x*) increases with
For a marginal speculator to be indifferent between attacking and non attacking conditional
.
=
on r
r, it
must be that
<
Pr(6>
0*|x*) also increases with r*
is
decreasing in r*. Obviously, A0(r*)
if
and only
A0(r)
=
if
0,
>
A9(r*)
0.
By
and A0(r*) >
is
also continuous in r*.
=
0**
exists
if
=
9ms-
It
and only
if
and only
if
follows that, r solves C(r)
if
r*
£
r*
is
=
<
It
A
follows that A0(r*)
=
9**
-
9*
two-threshold equilibrium exists
r.
But A0(r)
=
if
and only
(essentially) the Morris-Shin
V(0ms,
{r r\, or equivalently, if
7
.
the monotonicity of A9(r*), there exists at most one r such that
which case the continuation game following r
0*
if
and the
Pr(0
also equal to
e,
9ms] dictated by market expectations. Note
(9,
the bank's indifference conditions, 9*
both the devaluation threshold
by Pr(x <
the same
is
there had been no intervention.
precise, the
At the
rate only for a smaller measure of fundamentals.
at
if
independent of
is
As private information becomes more
0.
consider in
speculators simply "ignore" any attempt of the policy maker that
if
falls
0)
=
and only
if
0*
game and
=
0**, in
therefore
-^= and a two-threshold equilibrium
if
9*
6
(0,
9ms]- In Section 4,
we
will
establish that these properties extends to all robust equilibria of the game.
3.4
We
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
G.M. Angeletos, C. Hellwig and A. Pavan
18
to the policy maker, such as
when the bank
(sender) interacts with a single speculator (receiver). 14
In such environments, the policy can be non-monotonic
the receivers have access to exogenous
if
information that allows to separate very high from very low types of the sender (Feltovich, Harbaugh
Moreover, multiple equilibria could possibly be supported by different out-of-
and To, 2002).
equilibrium beliefs. However, a unique equilibrium would typically survive the intuitive criterion
or other proper refinements.
does not depend
in
any
To the
critical
contrary, the multiplicity
way on
is
sharp
if
we consider
e
—
>
0.
The
0*
and only
6
if
<
0.
environments with a single receiver.
in
limit of all equilibria of the
speculator would have the latter attacking the currency
if
identified in this paper
the specification of out-of-equilibrium beliefs, originates
merely in endogenous coordination, and would not arise
The comparison
we have
if
and only
Contrast this with our findings in Theorem
1,
if
x <
6,
game with
and the bank devaluing
where the devaluation threshold
can take any value in (0,6 ms]Second, consider environments where there
games. As shown
equilibria
if
in
is
no signaling, such as standard global coordination
different
may
Morris and Shin (2001) and Hellwig (2002), these games
market participants observe
sufficiently informative public signals
fundamentals. The multiplicity of equilibria documented in Theorem
from the kind of multiplicity
in that literature.
The
1,
exhibit multiple
about the underlying
however,
policy in our
it
depends on the particular equilibrium played
in
the market.
is
substantially
model does generate a
public signal about the fundamentals. Yet, the informational content of this signal
as
a single
is
endogenous,
Moreover, Theorem
1
is
not
about the possibility of multiple continuation equilibria in the coordination game that follows a
rather about
how endogenous
coordination in the market
given realization of the public signal;
it is
makes the
depend on arbitrary market sentiments and leads to multiple
effectiveness of the policy
equilibria in the signaling game.
Third, note that policy introduces a very specific kind of signal.
reveals that the fundamentals are neither too
of information that restores the ability of the
In this respect, an interesting extension
with noise.
small
It
bounded
idiosyncratic.
can be shown that
and
market to coordinate on
it
is
this particular kind
different courses of action.
to consider the possibility the policy
equilibria of
Theorem
1
itself is
observed
are robust to the introduction of
is
aggregate or
15
commitment. Note that policy traps
example, Drazen (2001).
'The proof
strong,
observation of active policy
noise in the speculators' observation of the policy, whether the noise
Lastly, consider the role of
'See, for
all
is
weak nor too
The
of this claim
is
available
upon
request.
arise in our
environment because
Coordination and Policy Traps
the policy maker moves
first,
19
thus revealing valuable information about the fundamentals, which
market participants use to coordinate 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
commitment
is
not necessarily optimal, suppose the noise £ has
the policy maker commits ex ante to
If
that devaluation will occur
e -»
Let
0.
16
if
and only
some
8
if
8
>
<
Uc — max r U(r)
and 8 C
—
8{r)
=
meaning the bank pays C(r*) only
ante value of discretion
is
Ud
—
8* G (9, 8 C ) necessarily leads to
that, even
when
long as she
is
Pr(#
Ud
.
is
As
e
—
for a negligible
> #*)E
> Uc A
commitment
perfect
If
=
all
>
Pr(0
>
8
>
>
0.
and ensures
6(r), a(8,r)
8{r))E[8\8
—
?(/)]
—
>
as
-C(r).
instead the policy maker retains the
the ex ante payoff depends on the particular equilibrium
#*,#**) the policy maker expects to be played. 17
8**,
U{r)
is
see that
support and consider e
^^^j moreover, for
min{8(r c )\r c G argmax r U(r)}.
8,
full
interest rate r, she incurs a cost C(r)
Hence, the ex ante payoff from committing to r
option to fashion the policy contingent on
(r*,
To
thus inducing a unique continuation equilibrium in the coordination game.
6,
[8\8
>
similar
8*}
>
0, 8**
>
8*
measure of
government
and a(8,r)
It
8.
But note that 9 C
.
argument holds
possible, the
—
>
—
for all
>
follows that the ex
8,
and therefore any
for arbitrary e.
We
will prefer discretion
conclude
ex ante as
not too pessimistic about future market sentiments.
Robust Policy Predictions
4
Propositions 2 and 3
classes of
Theorem
1
left
open the possibility that there also
exist other equilibria outside the
which would only strengthen our argument that policy endogeneity
,
coordination and leads to policy traps. Nonetheless,
we
two
facilitates
are also interested in identifying equilibria
that are not sensitive to the particular assumptions about the underlying information structure of
the game, in which case "robust" policy predictions can be made.
We
first
note that,
when the
noise has
unbounded
full
support, for any r* G (r,r\ one can con-
struct a one-threshold equilibrium, in which speculators threaten to attack the currency whenever
they observe any r
<
r*
:
the interest rate at r* for
This threat
is
no matter how high their private signal x, thus forcing the bank to raise
all
8 above the devaluation threshold 9* where 8* solves V(6*, 0)
,
=
C(r*).
sustained by beliefs such that speculators interpret any failure of the bank to raise
the interest rate as a signal of devaluation independently of their private information about the
I6
1
'
These results
follow
from Proposition
Note that the payoff
for the policy
1,
replacing r with an arbitrary
r,
maker associated with the pooling equilibrium
payoff associated with the two-threshold equilibrium in which 8*
=
0ms-
of Proposition 2 equals the
G.M. Angeletos, C. Hellwig and A. Pavan
20
fundamentals.
It
seems more plausible, however, that speculators remain confident that the peg
be maintained when they observe
will
rate, in
which case
bounded: For x
> x+£
On
>
all
x
bank
the
sufficiently high
6>!
= 9+e
faces
speculators find
it
2
<
2
+ 2e <
<
3
4
,
the bank
€
it is
is
it
belief
=
r'
for
any
G
may
this possibility critically
noise,
We
and
for all
U
[0 3
market to
some 01,62,03,84
4]
,
and r(0)
bounded, whenever
G
Theorem
^
r'
r' is
[0i,02] or
different subsets of types
who
However,
1.
depends on small bounded noise and disappears with large bounded or
noise.
vice versa.
It is
these considerations that motivated the refinement in Definition
2.
bounded
or
seek to identify equilibrium predictions that are robust to whether the noise
2
(Robust Equilibria) Every
two classes of Theorem.
1.
Theorem
is
likelihood ratio
and present the formal proof
2 here
set r,
If all
in the
we have
3.
Appendix.
perfect pooling.
let
0'
=
be, respectively, the lowest
uncertainty, the central
pay the
monotone
a two-threshold equilibrium as in Proposition
Consider any perfect Bayesian equilibrium of the game.
Otherwise,
1.
If the distribution of the noise satisfies the
sketch the intuition for
is
robust perfect Bayesian equilibrium belongs to one of
property, any robust active-policy equilibrium
We
is
conclude that unbounded noise introduces strategic effects that are not robust to bounded
Theorem
to
is
for
speculators whether
unbounded. The following then provides the converse to Theorem
the
2]
[0i,
lead to equilibria different from the ones in
unbounded supports, whatever the precision of the
We
among the
market to separate
In principle, the possibility for the
[03,04]-
the same interest rate
set
1
the noise
possible for the
For example, suppose that
the bank sets r(0)
common p =
when
r..
otherwise. Since [0i,02] and [03,04] are sufficiently apart and the noise
observed in equilibrium,
to raise the interest
necessarily true
is
bounded support,
noise has a
fails
dominant not to attack, which implies that
no attack and hence sets
when the
the other hand,
<
if
one-threshold equilibria disappear. This
separate types that set the same interest rate.
with
x even
inf{0
:
r(0)
r}
0"
and
and highest type who
=
sup{0
:
any equilibrium observation of r
>
>
r only
r(0)
>
r)
raise the interest rate. Since there
bank can perfectly anticipate whether she
cost of an interest rate r
if
this leads to
will devalue
is
no aggregate
and hence
no devaluation.
is
willing
It follows
that
r necessarily signals that there will be no devaluation and
induces any speculator not to attack.
in equilibrium,
>
If
there were
more than one
interest rates
above r played
and the noise were unbounded, the bank could always ensure no devaluation by
Coordination and Policy Traps
setting the lowest equilibrium interest rate above r.
that at most one interest rate above r
interest rate
<
rate for any 9
9*
9>x + e = 9 + 2e
now the
9*
and define
Theorem
Obviously,
1.
Hence, in any robust equilibrium,
.
would necessarily
strictly increasing,
is
played in any robust equilibrium.
is
9** as in
and
Since C(r)
21
Hence,
set r.
>
9'
Let r* denote this
never pays to raise the interest
it
9*. If
the noise were bounded,
any robust equilibrium, 9"
in
follows
it
<
oo.
all
Compare
strategy of the speculators in any such equilibrium with the strategy in the corresponding
Any
two-threshold equilibrium.
positive payoff
incentives to attack
exist types 9
=
by setting r
G
9
r*
.
>
9* necessarily does not devalue as
If
there also exist types 9
when observing
who do
[#*,#"]
when
r are lower than
informative of devaluation than in the case where
are
most aggressive
definition of 9**
when D(9)
possible
is
,
if
all
who do
9*
9
can guarantee herself a
<
not devalue, then the
6* devalue.
Similarly,
if
not raise the interest rate at r*, then the observation of r
less
at r
<
it
=
1 for all
and only
if
—
9"
9
<
9*
9**
.
9 G [#*,#"] set r*
all
—
and r(9)
.
there
=
r
is
Hence, speculators
r* for all [#*,#"], which,
by
Equivalently, the size of the attack in the
two-threshold equilibrium corresponding to r* represents an upper bound on the size of the attack
in
any active-policy equilibrium
9"
<
9**
which
.
r*
Since 9**
>
r.
On
<
9*
in
which
whenever r*
>
all
9
<
r,
<9' <
we have
9" <9**.
any robust equilibrium
corresponding two-threshold equilibrium.
can also show that
played. It follows that, in any robust equilibrium,
immediately rules out the possibility of equilibria in
<
the other hand, for any r*
follows that the anxiety region of
Theorem
is
r, this
9*
It
r*
By
bounded by the anxiety region of the
is
iterated deletion of strictly
9* necessarily devalue
and that
9'
—
9*
,
dominated
strategies,
which proves the
first
one
part of
2.
Observe next that the monotonicity of the devaluation policy implies monotonicity of the speculators' strategy as long as the posterior probability of devaluation
private signals.
The
latter
likelihood ratio property
is
necessarily true
(MLRP), that
is,
when
size of the attack a(9,r) is decreasing in 9,
at r*.
=
But then 9"
part of the theorem.
9**
and
When
r{9)
=
r*
if
when
monotonic
is
in
the speculators'
the noise distribution satisfies the
monotone
ip'(z)/ip(z) is decreasing in z. In this case, the
and therefore
and only
if
all
9 G
9 G [9*, 9"\ raise the interest rate
[9*, 9**],
instead the speculators' posteriors
fail
which completes the second
to be
monotonic
in x,
we can
not exclude the possibility there also exist active-policy equilibria different from the two-threshold
equilibria,
subset of
namely
[9*, 9**}.
equilibria in
which the policy maker raises the interest rate at
Nevertheless, in any such equilibrium,
it
r* only for a
remains true that devaluation occurs
if
G.M. Angeletos,
22
and only
if
<
9
9*
and the policy
is
C.
Hellwig and A. Pavan
we
[9*, 9**}. Finally,
active only for 9 £
noted that the
earlier
perfect-pooling and two-threshold equilibria can be supported by strategies for the speculators that
are monotonic in the private information x. If one restricts attention to such simple
strategies, the
MLRP
condition
in
the second part of
Recall that in an inactive-policy equilibrium r
In other words, r represents the
maximal
is
Theorem
is
bounded above by
can be dispensed.
dominated by r
interest rate that the
deviate to in the inactive-policy equilibrium. Following
equilibrium
2
Theorems
x— monotonic
for every 9
if
and only
if r.
>
bank would ever be tempted to
1
and
2,
the policy in any robust
Equivalently, the devaluation threshold with active policy
r.
r.
is
always lower than the one with inactive policy.
An immediate
corollary of
Theorem
2
is
that as private information becomes
more
precise, the
anxiety region in any robust active-policy equilibrium shrinks. In the limit, the policy converges to
a spike around the devaluation threshold dictated by market sentiments.
Corollary
r{9)
=
(Limit)
1
r for
The
9^9*,
all
any robust equilibrium as
limit of
for any arbitrary 9* £
{9, #a/s]>
—
£
>
is
such that the policy
an d devaluation occurs
if
and only
is
if
e<e*.
At
this point,
that would arise
Proposition 4
it is
if
interesting to
compare the above
fundamentals were
common
(Common Knowledge)
results with the set of equilibrium policies
knowledge.
Suppose e
=
0.
An
interest rate policy r
Q—
>
:
7Z can
C {r(9)) < V(9, 0) for 9 € [9, 9] and r{9) = r
G — [0,1] can be part of a subgame perfect
for 9 $ [9,9]. Similarly, a devaluation outcome D
= 1 for 9 < 9. and D{9) = for 9 > 9.
equilibrium if and only if D{9) €
1] for 9 G [9.9], D{9)
be part of a
subgame perfect equilibrium
if
and only
if
>
:
[0,
Proof. The second part
for
is
obvious, so consider the interest rate policy. For 9
the bank to set r and devalue and for speculators to attack. Similarly, for 9
devalues, speculators do not attack,
any 9 €
[9, 9].
The
continuation
game
and there
is
no need to
full
attack.
is
<
r{9). Clearly,
equivalently
it is
C {r{9)) <
optimal
9.
u
for
the bank to set r{9)
>
r
if
9,
a coordination
dominant
the bank never
Finally, take
game with two
Let r{9) be the minimal r for which
speculators coordinate on the no-attack continuation equilibrium,
r
>
9, it is
raise the interest rate.
following any interest rate r
(extreme) continuation equilibria, no attack and
<
i.e.
and only
they attack
if
V(9,0)
-
if
and only
C(r(9))
>
0,
if
or
Coordination and Policy Traps
That
is,
essentially
if
the fundamentals were
any shape
in
the critical range
and multiple non-monotonicities.
shape
in
[9, 6]
Theorem
is
8.
range vanishes as e
knowledge, the equilibrium policy r(6) could take
For example,
it
Similarly, the devaluation
When
1.
small but positive), the policy
We
[6, 9].
and need not be monotonic.
2 and Corollary
[6*, 6**], this
common
is
—
>
23
These
could have multiple discontinuities
outcome D(8) could
results contrast sharply
the information about fundamentals
is
also take
any
with our results in
very precise
(i.e.
e
active only for a small range of intermediate fundamentals
0,
and the devaluation outcome
is
necessarily monotonic in
conclude that introducing small idiosyncratic noise in the observation of the fundamentals
does reduce significantly the equilibrium set, as
compared to the common knowledge
case.
The
global-game methodology thus maintains a strong selection power even in our multiple-equilibria
environment.
Another interesting implication of Corollary
1 is that,
in the limit, all active-policy equilibria
are observationally equivalent to the inactive-policy equilibrium in terms of the interest rate policy,
although they are very different in terms of the devaluation outcome.
then
fail
to predict the probability of devaluation
An
and the policy of the monetary authority.
unobservable market sentiments arises
on the basis
An
of information
econometrician
may
on the fundamentals
even sharper dependence of observable outcomes on
one introduces sunspots, as we discuss next.
if
5
Uncertainty over the Aggressiveness of Market Expectations
The
analysis so far
assumed the policy maker was able to anticipate perfectly the aggressiveness
of market expectations, which
we
identify with the threshold r* at
an aggressive to a lenient response to the
to predict, even
which speculators switch from
policy. In reality, however,
when the underlying economic fundamentals
market expectations are hard
are perfectly
known
to the policy
maker. To capture this possibility, we introduce payoff-irrelevant sunspots, on which speculators
may
condition their responses to the actions of the policy maker. Instead of modelling explicitly
the sunspots,
we assume
TV C
1Z.
but
unknown
is
The
directly that r*
realization of the
to the
bank when
expectations then generates
leads to
random
random
it
random
is
a
random
variable r*
variable with
is
c.d.f.
5>
over a compact support
common knowledge among
the speculators,
sets the policy. Uncertainty over the aggressiveness of
market
variation in the effectiveness of any given policy choice and
variation in the devaluation outcome. Different sunspot equilibria are associated
with different distributions ($,Tl*) and result in different equilibrium policies r(9).
G.M. Angeletos, C. Hellwig and A. Pavan
24
Proposition 5 Take any random
<f>.
For
>
e
sufficiently small,
<£
[9* ,9**\,
and
certainty for 9
TV
r(9) G
<
9**
.
e {9,9 ms) an d
#**
and
£
than one and non-increasing in 9 for 9 e
—
Finally, 9* is independent of e, ivhereas 9**
The
maker
policy
is
a lenient one whenever r
—
>
The
0.
>
r*
When
.
<
9
9*
interval
TV
>
9* as e
—
r for
and
0.
represents the set of
raising the policy to
,
to the expected value of defending the peg, in
and devalue with
TV
=
[#*,#**],
thresholds r* such that speculators coordinate on an aggressive response whenever r
level in
r{9)
anxious to prove herself only for a small range of moderate
fundamentals, and this range vanishes as e
r
an d a
(#*>#)
equilibria of Proposition 5 are qualitatively similar to the two-threshold equilibria
of Proposition 3.
compared
distribution
with r(9) non- decreasing in 9 for 9 G [#*,#**]; devaluation occurs with
9*, with probability less
never occurs for 9 >
The sunspot
there exist thresholds 9*
(r,r)
The central bank follows a non-monotonic policy with
robust equilibrium such that:
9
TV C
variable r* with compact support
16
When
certainty.
instead 9 6
any
level in
which case the bank
[9*, 9**}, it
finds
TV
it
random
<
r*
is
too costly
and on
optimal to set
pays to raise the interest rate at some
so as to lower the probability of a speculative attack. Since the value from defending
the currency
is
increasing in
the size of the attack at r
is
9,
so
is
the optimal policy in the range
so small that the
bank
[9*, 9**].
prefers the cost of such
Finally, for 9
>
9**,
an attack to the cost of a
high interest rate. The thresholds 9* and 9** axe again given by the relevant indifference conditions
for
the bank, but
differ
from the ones we derived
in the
absence of sunspots. The definition of the
thresholds and the complete proof of the above proposition are provided in the Appendix. 19
With random
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 simply confirms market expectations, but also the equilibrium
spirits
and market sentiments.
outcome of any given policy action
is
determined by animal
Empirical evidence suggests that raising interest rates does not
systematically prevent an exchange-rate collapse. Kraay (1993), for example, studies the behavior
of interest rates
and other measures of monetary policy during 192 episodes of successful and un-
successful speculative attacks
between
interest rates
and
finds a "striking lack of
and the outcome
observable fundamentals. This evidence
The assumption
that e
is
sufficiently small
any systematic association whatsoever
of speculative attacks," even after controlling for various
is
is
hard to reconcile with Morris and Shin's (1998) predicnot essential;
it
ensures
8" <
8,
which we use only to simplify the
construction of these equilibria (see the Appendix).
If
8"
one takes a sequence of sunspot equilibria such that TV converges to a single point
in Proposition 5 converge to the ones given in (6).
read as the limit of sunspot equilibria.
That
is,
r°, the thresholds 9"
and
the two-threshold equilibria of Proposition 3 can be
Coordination and Policy Traps
tion that defense policies decrease the probability of devaluation, but
equilibria of Proposition 5,
where the same combination of
25
is
consistent with the sunspot
interest rates
and fundamentals may
lead in equilibrium to either no devaluation or a collapse of the currency.
Finally, the results of this section help
commonly appear
in the
popular press,
understand and formalize the kind of arguments that
the one we quoted from Financial Times in the beginning
like
Once the policy maker has taken a
of the paper.
favorable outcome, the market
may be
costly policy action in
an attempt to achieve a
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 in vain.
6
Concluding Remarks
we
In this paper
investigated the ability of a policy
maker to influence market expectations and
control equilibrium outcomes in economies where agents play a global coordination game.
that policy endogeneity leads to multiple
observed with idiosyncratic noise.
the policy maker
is
The
self-fulfilling equilibria,
even when the fundamentals are
multiple equilibria take the form of policy traps, where
forced to conform to the arbitrary expectations of the market instead of being
able to fashion the equilibrium outcome. There
is
an inactive-policy equilibrium in which market
participants coordinate on "ignoring" any attempt of the policy
well as a
We found
continuum of active-policy equilibria
in
maker to
affect
market behavior,
which market participants coordinate on the
as
level
of the policy
beyond which they "reward" the policy maker by playing a favorable continuation
equilibrium.
Despite equilibrium multiplicity, information heterogeneity significantly reduces the
equilibrium set as compared to the case of common knowledge and enables robust policy predictions.
Although
approach
may
this
paper focused on the particular example of
currency attacks, our
extend to other environments where policy can serve as a coordination device among
market participants. Monetary policy
in
self-fulfilling
in
economies with staggered pricing,
fiscal
and growth
policies
economies with investment complementarities, and stabilization policies or regulatory interven-
tion during financial or debt crises, are only a few examples where our results might be relevant.
G.M. Angeletos, C. Hellwig and A. Pavan
26
Appendix
Proof of Theorem
When
2.
pooling equilibrium in (a) of
>
r(9)
some 9 and we
r for
=
9'
We
no
interest rate other
Theorem
equilibrium and 9"
Any
Proof.
only
if
D(9)
<
inf {9
r{9)
:
>
>
r
who
types
0; all
unbounded, any equilibrium
=
{9
at
is
most one
D (9) =
0}
is
>
interest rate r
Since C(r)
r.
part of the lemma.
Next,
not to attack whenever x
r(9)
=
r for all 9
bounded
noise,
Given any
> x+
of the proof,
r*
Lemma
Proof.
2.
9**
it
r played in
> x =
is
>
r.
r results in a posterior /x(0o|x, r)
—
1 for
strictly increasing in
+ e,
9
i.e.
tt
j
&
d/j,
^
r will
If
r,
and thus a(9,
r)
=
for
any robust
be chosen by the central bank, which proves the
which a(9,r)
in
>
this also implies that, in
r,
would be dominant
it
=
for all r
Therefore, an equilibrium with 9"
e.
D (9)
the expected
if
{9\x, t)
the noise were bounded,
if
and only
if
i.e.
=
for a speculator
whenever 9 > x
+
e
and thus
oo could never be sustained with
which proves the second part of the lemma.
£
9*
and note that
>
the set of fundamentals for which devaluation does not occur.
equilibrium, at most one interest rate r*
first
r}.
leads to no devaluation,
Speculators attack
r.
Hence, no speculator ever attacks when he observes an equilibrium r
any equilibrium r >
>
interest rate r*
if it
higher than the interest rate differential,
is
:
=
devalue set r
the noise
where Oo
r{9)
:
played in equilibrium only
is
premium
all x,
sup{6>
lemmas.
five
devaluation
is
=
9"
and
r}
oo.
interest rate r
—
we have the
played in equilibrium,
Hence, in what follows, we consider equilibria in which
1.
In any robust equilibrium, there
1.
is
let
prove the result in a sequence of
Lemma
than r
(r,f], define
=
>
when
r*
<
suffices to consider
r, 9**
the other hand, any 9
>
<
9*
=
+e
9*
9"
<
9'
and
9*, r* is strictly
U>~ 1
when
unbounded
For any r* € (r,r], 9*
Since for any 9
=
9**
and
C{r*)
9*
the thresholds
r*
=
(l
r.
-
^9*)
and
9**
<
- *9*
:
(9*)
when
r*
.
>
9"
<
9**.
dominated by
r, it
immediately follows that
,
=
for all 9
>
For the rest
noise.
can always set r* face no attack, and ensure a payoff V(9.
Therefore, necessarily D(9)
r.
9*.
However, there
may
exist types 9
9'
>
9*
.
On
0)
— C(r*) >
<
9* that also
0.
Coordination and Policy Traps
do not devalue
D—
27
Define S(x) as the probability, conditional on x, that 9
in equilibrium.
Further, define p(x) as the probability, conditional on x, that 9 G
0.
Then, the probability of devaluation conditional on x and r
[9*
,
and
#"]
<
9*
r (9)
and
=
r.
given by
is
l-*(z=f)-6{x)
=
(J.(D
l|x,r)
-V (*=£}
I
whenever
Clearly, a speculator never attacks at r
F(x;9*,9")
is
jjl(D
—
<
l|x,r)
*
=
(*=£.
)
+
x-e"
1-*
(*^
+*
strictly decreasing in x,
and
let
x
Define
r/ir.
tf
1
_$
1
Note that F(x;9*,9")
^(^
+p{x) +
[9*, 9") solve
F(£;0*,0") =r/w.
Since
<5
>
(x)
and p(x) >
whenever x > x(0*,d")
speculator with x
which
fi{D=
,
> x(O*,0"}
(J,{D
=
=
1 for all
F(x;6»*,6 ") for
< F(x;6*,e") < F(x;9*,9") =
l|x,r)
most
<
9
9*
=0
and
r{9)
some
for
=
a positive probability that r(9)
=
<
9
That
G
r* for all 9
follows that,
It
r/w, and therefore any
any equilibrium in
in
is,
would be
as aggressive as they
This
[9*, 9"].
if it
were
intuitive for
is
(i)
9* reduces the incentives to attack for every x,
some 9 G
r for
all x.
l
does not attack in equilibrium.
a positive probability that D{9)
(ii)
<
l|x,r)
r* is played, speculators are necessarily at
the case that D(9)
and
we have
0,
(8)
[#*,#"] reduces the probability that the
observation of r signals devaluation and therefore also reduces the incentives to attack conditional
on r and
for 9",
From
9*
that for any
x. It follows
we have that a
a
9,
(9,
= C (r*)
(9",r)
.
Since
the indifference conditions of Proposition
=* —
'
'
(
e
—
)
.
x(9*,9")-9
r)<^
From the
C (r*) =
(3),
Hence, in any equilibrium
in
0*, it follows
that 9*
£(g *' e
< *
(
and using x(9*,9**)
which
indifference condition
r*
is
=
played, 9"
x*,
must
we
"
H
£
also have that
satisfy
x(9*,9**)-9**<x(9*,9")-9".
From
(8),
x
(#*, 9")
Recall that 9*
9"
<
9**
<
6*
<
9',
—
<
9"
decreasing in
is
9**
which
played, necessarily 9*
<
',
and hence we conclude that 9"
<
= C _1
if
and only
is
a contradiction, since by definition
robust equilibrium with r* G
is
9"
9'
(r, r].
<
9"
if
On
<
r*
r
(9ms)
For any r*
9'
<
9"
.
<
>
9**
.
r,
Lemma
2 implies
Therefore, there exists no
the other hand, in any robust equilibrium where r* G (r,r]
9**.
G.M. Angeletos, C. Hellwig and A. Pavan
28
Next,
we show, by
librium in which r*
6'
Proof.
9
and only
if
any robust equi-
strategies, that in
if
<
9
9*
,
which in turn implies that
= 6*.
Lemma
r
played, devaluation occurs
is
dominated
iterated deletion of strictly
=
r
1 for all
9
<
=
D{9)
9*,
r(9),
we
and
9*,
9'
=
9*.
consider the continuation
T
and construct the iterated deletion mapping
and
>
for all 9
Given an arbitrary equilibrium policy
[0, 9'}
<E
=
D{9)
3.
—
[9, 9']
:
>
[9,
9'}
game that
follows
Take any
as follows.
let
1- *(==*)
G{x;9)
1-*
(^f-) + p(z) + *
<
G(x;9) thus represents the probability of 9
=
\\m x ^ +00 p(x)
and therefore lim I __ 0O G(x;
for every 9, there
is
=*
unique solution to 8
attack,
r,
<
That
9.
which
is,
^
I
9
if all
<
x(9)
i.e.
.
)
= min{x
=
Note that G(x;9)
=
G(x;9)
|
r.
Note that lim x __ 00 p(x)
and lim z ^ +00 G(x; 9)
1
> r/n
9 are expected to devalue,
in turn implies that
which happens
—
9)
at least one solution to the equation G(x; 9)
lowest solution to this equation,
every 9
conditional on x and
9,
for
Then
every x
if
9
T
>
9'
(V)
.
The
< x
is
0f
follows that,
=
=
x
x(9) be the
9(9) as the
(^J
necessarily find
>
9 for
optimal to
it
playing r* rather than
iterated deletion operator
= nun [<?',
let
< x and *
any 9 < 9 necessarily devalues, unless 9
in equilibrium only
0. It
z/ 7^}, an d define 9
x
all
r/n.
=
=
T
thus defined by
is
•
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, 0'] Obviously, T is
Observe that
in
9.
also
.
bounded above by
ruled out neither
Next,
9'
Finally, note that 9*
9'.
< 9ms,
we compare
T
$>
nor
T
:
[9,9]
the unique solutions to
In general,
in
which case
1
G
is
<
9**
<
00 and 9*
< 9ms, but
with the iterated deletion operator of the Morris-Shin
-»
[9,9],
- $
is
^~
(
game
strictly decreasing in x,
0. All
so far
we have
=
t/tt
and 9
J
is
?,
where x
I
Continuity
implying that G{x,8)
we need, however,
=
= $ ^^
in 6.
game without
r when the pooling equilibrium
at
defined by F{0)
x and therefore 6 need not be continuous
continuously increasing in
9'
^MS-
signaling (or, equivalently, of the continuation
This operator,
<
—
t/tt
j
.
is
T
-
x{9) and 9
is
=
played).
9{9) are
has a unique fixed point at
ensured when
<I<
satisfies
the
MLRP,
has a unique solution and this solution
monotonicity of the lowest solution, which
is
true for any
ty.
is
Coordination and Policy Traps
=
9
=
9
MSi and
M s,6}-
whenever 9 £
{9
and therefore
0(6*)
T(0)
=
9',
or 9(9)
Finally,
>
<
< T(9) <
satisfies 9
9
Moreover, since G(x;
9(9).
9',
We
whenever 9 £
>
9)
-^
1
>
[9,9
(^f^) for
conclude that, for any 9 £
which case T(9)
in
Ms
29
M s)
x and
all
9,
we consider the sequence {0 }£l x where
fc
,
monotone and bounded above by
must be a
9
=
1
k+1
9 and 9
Suppose 9°° <
=
T{9°°).
This can be true only
9'.
>
would then have T(9°°)
Therefore,
for T(-).
necessarily converges to
it
,
is, 6>°°
T, that
fixed point of
0'
next prove that 9°°
>
^(9°°)
9°°,
We
= T(9 k
= 9' whenever 9' < 9 MS
= 9' = 9". Suppose that 9' >
9'.
If
9*. If
9 £ (9* ,9') would devalue in equilibrium. If instead
But
no devaluation and a positive payoff by setting
< 9ms an d
x(9)
<
9'
9',
or
limit 9°°
some
9'
=
r*.
Therefore,
therefore 9°°
=
=
9'
<
9 MS whenever
is
> 9 MS
9 MS
-
we
,
a fixed point
>
9'
9 MS
We
-
9'
>
and
all
> 0ms >
9*
an d again
all
impossible as any 9
>
> 9ms,
is
6>°°
=
& <
either case
>
6>°°
This
1.
This limit
.
were the case that 9°°
it
>
fe
Since this sequence
0.
prove that either 9°°
whereas
,
9'
Ms
which case
for all
)
which contradicts the assumption that 9°°
6>°°
This also implies that
first
<
9(9°°)
if
9 £ (9*,9°°) would devalue in equilibrium.
>
x(0)
in
0',
9
T(9).
sequence represents iterated deletion of dominated strategies starting from
is
>
>
JF(?)
we have
either 0(0)
[9,9'},
>
and 9
9 MS
then 9°°
,
then 9°°
it is
9*
9*
9*
can ensure
necessarily the case that
9'
=
which completes the proof of
,
0*
this
lemma.
We
next show that,
if
the posterior beliefs given r are monotonic in x, then speculators follow
a threshold strategy and therefore the size of an attack
in turns implies that the
Lemma
4.
decreasing in
Proof.
bank
If in
equilibrium [i(0
9, in
which case 9"
Following
Lemma
3,
£
sets r* for all
=
<
0*\x,r)
9**,
is
is
=
in x,
r* for all 9
then necessarily a(9,r)
= l\x,r)
the probability of devaluation conditional on x and r
with lmxr^-oo G(x\9*)
is
—
1 if
x
< x'
1
and
and a(x,r)
since 0" solves a(9" ,r)
x
=
lim.T
a unique x' such that G(x';9*)
a(x,r)
1
=ii(9<9*\x,r)
=
=
C(r*),
—
all
strictly
_
vj,
i'
is
given by
x-e'
G(x;9*)
=
0.
When
G(x,9*)
monotonic
is
in x. there
r/n and thus speculators follow a threshold strategy with
x
9 £
x(9*,9"), implying that a(9.r)
is
G(x;9*
^ +00
=
if
=
This
[9* ,9**).
£
i
fj,(D
9.
[0*. 0**].
monotonic
and r(9)
decreasing in the fundamentals
>
[9*
x'
,
.
that a(9,r)
It follows
9") necessarily set r*.
£(e
= *
(
g
*'f>~ )
.
is strictly
But
then,
Thus, 9" solves
*
p
decreasing in 9 and,
(x)
=
for
any
fiiil£pzil\
x,
and
= C (r*),
G.M. Angeletos, C. Hellwig and A. Pavan
:iO
and from Proposition
Lemma
established in
The above
set r, this
this
is
is
x
(3),
2, it
follows that 9"
be the case
Proof.
5.
If
^
=
x
—
9**
(6* ,9")
-
From the monotonicity
9".
satisfies
the
of
£
(0*, 9)
-
9
.
f.i(9
<
9*\x,r) in x. Since
9
all
<
9*
a wide range of noise structures. In the next lemma, we prove
for
when the
necessarily the case at least
Lemma
6**
presumed monotonicity of the posterior
result
likely to
-
{9*, 9**)
MLRP,
noise £ follows a distribution which satisfies the
then
fi(9
Let 1(9) be the probability that 9 sets
<
r.
monotonic
9*\x,r)
is
Using p
(x)
MLRP.
in x.
+ # (~-) =
jip
fjj?
(
s=s-) I
{9) dO,
we have that
-*H
X
c(x-n
l-G(x;P)- fifty
(£=*) 1(9)
M'
and therefore
G
d(
" l-G(x;9')
i;fl ')
^o
(
I
1
- *
x-e~
±{&^{^)no)de)
Ue~^(^ )ne)d9Y
L
It
follows that
dG(x;6*)/dx <
&
Using the fact that I
(9)
=
\
and only
if
<0.
Khl>{*T)nO)d9
&(*?)*>
for all 9
¥ (*?)
E
if
<
9<9\
9*
,
the above
equivalent to
*(*?)
En
x, r
is
,
x
,
L
<0,
*(H*)
which holds true
if ip' /ip is
Combining Lemmas
to either (a) or (b) in
any equilibrium
and
9**
£
<J>.
and
1, 2,
Theorem
in (b)
is
Proof of Proposition
distribution
monotone decreasing.
1.
If,
in addition,
^
that any robust equilibrium belongs necessarily
satisfies
5.
Take any random variable
r*
We want to show that for e sufficiently small,
(9* .9),
£
we conclude
a system of beliefs
\9*,9**\.
the
MLRP, Lemmas
4 and 5 imply that
a two-threshold equilibrium, which completes the proof of the theorem.
a non-monotonic policy with r(9)
in 6 for 9
3,
D
(ii)
=
Whenever
fi,
<
r*,
[9*. 9**],
TV C
there exist thresholds x* 9* £
,
and a robust equilibrium such that:
r for 9 £
r
with compact support
(i)
(r, r)
(9,
The bank
and
9ms),
follows
and r(9) £ TV with r(9) non-decreasing
speculators attack
if
and only
if
x
<
x*;
whenever
Coordination and Policy Traps
r €
[r* >?"*),
x <
x.
they attack
and only
if
<
x
if
x;
and non-increasing
The proof
is
€
in 9 for 9
in five steps:
D(9)
Let r*
=
min7?.*
>
r*
—
1
for 9
<
9*,
and with probability D(9)
Steps
and
1
= max TV <
r and f*
r.
beliefs
and only
if
<
1
9**
>
for 9
9**,
,
and
Step 3
x*\
and the strategy of the speculators;
2.
Define
if
9
<
9,
if
9
€
[0,9],
if
9
>
0,
,r"]
max {9$(r)-C{r)}
U(9)
if
with probability D(9)
—
max {-C(r)}
r£\r*
they attack
,
2 characterize the thresholds 9*
Step 5 establishes robustness in the sense of Definition
1.
>
[#*,#**],
examines the behavior of the bank; Step 4 examines the
Step
r
and whenever
(in) Devaluation occurs with probability
31
re[r*,?*]
max {9-
-
(l
- C(r)\
$( r ))# (i=2)
and r(9) as the corresponding argmax Note that maximizing over
€ TV
ing over 7Z* as necessarily r(9)
Therefore, there exists a unique 9*
< 9\ and 0(9) >
9
all
0.
Step
2.
also that U(9)
non-decreasing for
is
> -C(r*). Moreover, 0(9) = -C(r*) <
increasing whenever 0(9)
Oms ~ C(r) =
Note
''.
for all 9
>
equivalent to maximiz-
[r*, r*] is
.
and G(9 MS )
G (9,9 MS ) such that 0(9*)
all 9,
and
strictly
> 9 MS ~ C(r*) >
=
0(9)
0,
<
for
9*.
For any 9 >9*, define the functions x
(9; 9*)
and v(9; 9*) as follows
'"'
"
'
.fcn-'-t^'""'-'
and
'
'
>
Note that dx(6;9*)/d9 G (0,1) and therefore dv(9;9*)/d9
*
""*
:
(
w>
£
0.
—
as e
0.
>
To
J
This can be true only
and therefore lim £ _o
x
\P
if
~'~
'
'
)
which
is
x
a contradiction.
theorem, dU(9)/d9
is
(9; 9*)
1.
=
^
pWl-g' ^
.j.
Consider
now
bounded above by
8*
9, in
which case#
- 9 MS
-
Since 9*
>
also that for
^
X
(
)
e
any
=w
6>
>
6>*,
some
for
9* implies \im e ^ox(9;9*)
>
9*
But then
1
r
-/-
,
^(
>
1 for
(9;
o
=
+w
x{e-fi*)-o \
=
n
<
r
-,
tt'
J
—
the function g(9)
over this range. Moreover, by definition of x
therefore v(0*;9*)
>
i-«(aa^)
7
<
e^o|
2
— =
(
lim
see this, suppose instead that lim e _o
lim £ _o
(
Note
1.
any 9
9*)
,
<
—
that
\I/
(
^
From the envelope
U(9).
9 and hence g(9)
we have
< 9 M s and
v(9\9*)
is
strictly increasing in 9
'-^
J
0(9*)
=
0,
we conclude
= ^^ = 9ms
that g(9*)
<
0.
and
Next,
G.M. Angeletos, C. Hellwig and A. Pavan
32
<
note that for any
>
every 9
e
>
9"
and therefore
=
u(0;
(9*)
—^
>
We
0.
<
17(0; 0*)
fact that x(0;0*)
>
0**.
17(0; 0*)
<
u(0;0*)
v(0;0*)
=
0G
r.
G
[r*,r*].
We
now
<
prefers r to any r
r
>
for all
9*
<
and C(r(9))
>
<
e.
>
for
U{9;9*)
<
<
<
and
U{9)
=
whenever
for all 6
<
0**, (7(0; 0*)
0**. Finally, let
>0>
is
Step
only
=
=
and
—
for
»
follows that there exists
-
C(r(9))
>
0(9)
Moreover, note that, as e ->
Next,
0.
->
v(0;0*) for
=
0,
let
U(9**;9*)
all
0**
<
9
=
u(0**;0*), while the
and c/(0;0*) > f(0;0*)
we conclude that
,;(0;0*)
=
U{9)
=
at
0**,
be the unique solution to c/(0;0*)
and t>(0;0*)
=
>
and note that
t/(0*;0*).
r* to
=
>
t/(0)
U(9)
0(9) for
any r >
r*
>
if
.
Furthermore, by definition, r(0) dominates any
G
for all
>
all
[0,
(0*,0], t/(0)
0**. Therefore,
l)for0G
=
max{0,
f>(0; 0*)}.
(0*,0**). It follows that t/(0)
G
and r otherwise. The
$(f(0)) G
it
>
is
t/(0)
=
>
U(9)
indeed optimal
=
<
the bank to play r(9)
resulting probability of devaluation
[0*,0**],
and D(9)
=
>
for
0**.
[/(0) for all
G (0,0**), and
for all
for
=
Note that U(9)
is
7>(0)
=
1 for
Since? is non-decreasing,
non-increasing in the range [0*,0**].
Consider next the behavior of the speculators. For any r
4.
if
<
0.
r,
the probability of devaluation
=
ii
(6
<0*|x,r) -
<
9\x* ,r)
1-9
<
6\x,r)
construction, x* solves
the currency
if
and only
if
7r/i(0
x
<
<
—
In equilibrium, at r
fj,(9
By
as e
>
the behavior of the bank. Given the strategy of the speculators, the bank
r*,
[0*,0**]
G
<0*, D(9)
D
>
0(9)
—
j
thus need to compare only the payoff from playing r(0) with that from playing
f7(0)
>
0*, C/(0)
0, it
—
= 9-9
Playing f(0) yields 0(9), while playing r yields U(9)
if
9**.
(0**; 0*) == x* implies
implies U{9;9*)
<
e
But then u(0;0*) >
follows that 9** -> 0* as e
and
*'
(
there exists a unique 0** 6 {9*, 9) such that
e,
>
p(0)
\P
e,
this result with the properties of the function <?(0),
< U{9)
Consider
3.
it
for e
with v(0;0*). That £
(0*,0**), since l>(0**;0*)
Step
0*;
£(0**;0*)
Combining
> U{9)
>
increasing in
is
for all
< 9*% and
every
£/(0) for
>
whereas
of
C(?(9)) whenever e
conclude that,
for 9
x*
Compare now
<
]
[
g{9)
>
6»
*(W)- g
*
g{9)
0,
and ?() are independent
(See the argument above). Since 9
.
such that
5(0**)
0*
0,
x*
is
=
r,
and since
/.i(9
<
r*,
is
devaluation occurs
if
and
given by
(*=£)
0|x,r)
is
decreasing in x, attacking
indeed optimal. For any out-of-equilibrium r
<
r*,
we consider
Coordination and Policy Traps
beliefs
any r >
>
/u(#
<
such that n(9
[i
we
r*,
>
x
1 for
of-equilibrium
restrict
\i
r, if
r
=
(r)
{0
:
>
r*, fi(9
for
1
=
x
r at
=
9\x,r)
= r}
r{9)
=
0\x,r)
<
[x,x], /x(0
any r G r(0) with r
0, where 9~
(r) 7^
—
—
G ©(x) such that 9
for all 9
G A4(r)\ and given these
/./,
x G
for all
1
<
ivii{9
x*
<
a;
.
For
and
x,
G 0~ l (r)\x,r)
=
1
In addition, for any out-
.
not dominated in equilibrium by r(#) for some 9 G 0(x), then we further
is
to satisfy fi(9\x, r)
(4), as well as
—
x. In particular, for
D 9~
x such that 6(x)
for all
non-increasing in x and
is
G (9,9)\x,r)
let /j,(0
=
9\x,r)
6\x,r)
33
<
C(r)
U(9). These beliefs satisfy
the strategy of the speculators
beliefs,
sequentially
is
optimal for any r and x.
Step
Finally, consider robustness.
5.
^
Assume
has bounded support,
and construct the corresponding sunspot equilibrium
Let a*
as above.
e be sufficiently small,
let
=
=
a(9*,r)
~e
x
ty
(
j
and a**
£**
and
<
=
= ^
*(£**
for all £
We
= *
a{9**,r)
-
_1
<
fc)
1,
£**
and a
F(0 >
k],
<
distribution for
all
<
9
of F, the functions x
that the
U_{0) also
<
0,
9**
same
9. It follows
(9;
we
9*)
is
G
for all
=
Similarly, let
£<£**-
=
in a
and
when the
sustained
is
g{9)
remain the same
=
and construct
all
£
G [f*
follows that
F
so that
- k,C +
F
is
k],
proves that the
[#*,#**]
in
=
x
U(9)
is
and x*
9**. Since
[0* ,0**}
*(£)
+
noise distribution
is
ty,
A;.
can
U,r,x,v,g,U, and
F
and r(0)
same policy
£*
and
has bounded support
F(f) <
*(0
same
=
r{9)
0.
By
U
construction
a neighborhood of 9**
(0**; 9*)
.
same
follows
It
But then U{0;
.
also the
=
for all 9
r(0) can
and
all
for all
0*)
<
>
—
£<£**<!».
2k, £*
k,
We
+
is
be sustained with either
9** continue to
is
D{9)
=
$
9,
and
and
2k]
and
^
or F.
5>(f(0)). Therefore,
A
symmetric
an arbitrary k
satisfies
and F(f) > *(£)
also the
maintain the peg
or F.
£** defined as above, take
[£**
and D{9) as
Since r(0)
r otherwise.
the probability of devaluation
unbounded. With
sustains the
Proposition 5 are robust.
£*
the same devaluation probability D(9) can also be sustained with either
^
such that
U{9**). Like in Step 3 above, the latter ensures that the
G
r(0) for all
=
0* continue to devalue with certainty
if
>
>
F. Indeed, consider the functions
neighborhood of
U{9**;9*)
with certainty, while for 9 G
argument applies
f
for all £
and F(f) < #(£)
fc,
that the same 9* continues to solve U(9*)
v{9\9*),
,
[#*,#**], this
<
Similarly, all
r{9)
1.
with unbounded support such that F{£)
and x* continue to solve g{9**)
infer U{0**)
<
a*
= *- x (a*)
replaced with F. Note that U{9) and r(9) are independent of the noise
remain the same
optimal policy
same
ty is
F
tf (f) for all
is
<
a**
Hence, one can always find a k
£*.
c.d.f.
be sustained when the distribution
defined above, where
<
and note that
)
argue that the same r(#) and £>(#) that
also
0**
+
#(£*
- k,C +
G [r*
f
11
and note that
(q**)
<
k)
^
5
>
— ^(0 for
> C* - k. It
F(£)
for all
£
conclude that the sunspot equilibria of
G.M. Angeletos, C. Hellwig and A. Pavan
34
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Phenomenon,"
Figure
1
There exists an inactive policy equilibrium
attack
is
a(Q,r),
and devaluation occurs
equilibrium by r if and only
that, for
any 0e
if
C{r
r
)
in
if
which the
and only
> 9 or C{r r) >
[0',0"], if the central
bank deviates from r
bank
to raise the interest rate.
when
bank
sets r for all 8, the size
< M s- Any r'e(r,r]
if
a(Q,r), that
Speculators then coordinate on the same behavior as
the
central
is,
if
and only
to r\ the
r
=
r,
if
0g
is
dominated
[0',0"]. It
market "learns"
of the
that
0e
in
follows
[G',6"]-
thus eliminating any incentive for
Figure 2
For each
9 e [9
,
9
re
]
(r,r], there
and r(9)
central
bank
attack.
When
is
a two-threshold equilibrium in which the interest rate
= r otherwise, and
raises the interest rate at
instead the bank sets
which case the
size
of the attack
raise the interest rate at r if
is
r,
in
which devaluation occurs
speculators attack
decreasing in
and only
if
r, the market "learns" that 9 e
if
C(r')
9. It
< 9 and
if
and only
follows that
C(r')
<
and only
[9*, 9**]
if
is
9<
r(Q)
9*.
r" if
When
the
and coordinates on no
if their signal is sufficiently
it is
=
low, in
optimal for the central bank to
a(B,r), that
is,
if
and only
if
9 e
[9*, 9**].
32^
u
1
(
AUG
2 6 2003
Date Due
MIT LIBRARIES
i pfnNiHiiPii
3 9080 02524 4150