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Department of Economics
Working Paper Series
Information Aggregation and
Equilibrium Multiplicity
Morris-Shin
Meets Grossman-Stiglitz
George-Marios Angeletos
Ivan Werning
Working Paper 04-1
February 2004
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4 2004
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Information Aggregation and
Equilibrium Multiplicity:
Morris-Shin Meets Grossman-Stiglitz*
George-Marios Angeletos
MIT and NBER
Ivan Werning
MIT, NBER and UTDT
angelet@mit edu
iwerning@mit edu
.
.
This draft: February 2004
Abstract
This paper argues that adding endogenous information aggregation to situations
where coordination is important - such as riots, self-ful filing currency crises, bank
runs, debt crises or Qiancial crashes - yields novel insights into multiplicity and characterization of equilibria.
Morris and Shin (1998) have highlighted the importance
They also show that, with exogenous
of the information structure for this question.
information, multiplicity collapses
enough idiosyncratic
when
individuals observe fundamentals with small
noise. In the spirit of
nize public information
Grossman and
by allowing individuals
Stiglitz (1976),
we endoge-
to observe Qiancial prices or other noisy
indicators of aggregate activity. In equilibrium these indicators imperfectly aggregate
disperse private information without ever inducing
common
knowledge. Importantly,
their informativeness increases with the precision of private information.
that multiplicity
Interestingly,
ensured
when
may
survive and characterize the conditions under which
We
it
show
obtains.
endogenous information typically reverses the limit result: multiplicity is
individuals observe fundamentals with small enough idiosyncratic noise.
JEL Codes: D8, E5, F3, Gl.
Keywords: Multiple
equilibria, coordination, self-ful filing expectations, speculative at-
tacks, currency crises,
bank
runs, financial crashes, rational-expectations, global games.
Introduction
1
It's
a love-hate relationship, economists are at once fascinated and uncomfortable with mul-
tiple equilibria.
*We thank
ballero.
On
helpful
the one hand, a variety of
comments and
discussion from
phenomena seem characterized by
large
Daron Acemoglu, Francisco Buera and Ricardo Ca-
and abrupt changes
outcomes not obviously triggered by commensurate changes in fun-
in
Commentators often attribute these changes
damentals.
sentiments' or 'animal
spirits'.
Models with multiple
Qiancial crashes, riots and political regime changes.
is
equilibria
may
formally capture these
Prominent examples include self-fulQling bank runs, currency attacks, debt
ideas.
arises
to arbitrary changes in 'market
1
In this class of models, multiplicity
due to a coordination problem: attacking a 'regime' -
beneCbial
On
if
and only
if
crises,
for instance, a
currency peg -
enough agents are expected to attack.
the other hand, models with multiple equilibria are sometimes viewed as incomplete
theories that should ultimately be extended in
and Shin (1998) 2 have contributed
Recently, Morris
information structure away from
survives
when
analysis
is
some dimension
common
to resolve the indeterminacy.
to this perspective
by enriching the
knowledge. They show that a unique equilibrium
individuals observe fundamentals with small enough idiosyncratic noise. Their
particularly attractive because
it
can be viewed as a small perturbation around
the original common-knowledge model.
More
generally, Morris
and Shin introduce a useful framework
tion heterogeneity affects the determinacy
for studying
and characterization of
equilibria.
how
The
informa-
dispersion
of valuable information about uncertain fundamentals plays a critical role for their unique-
ness result.
An
earlier literature dealt
with the transmission and aggregation of disperse
In particular, Green (1973) and
information in rational-expectations equilibria.
(1981) highlight that prices
ing that in
may be
some cases they can be
Grossman
excellent aggregators of dispersed information
fully revealing, yielding
common knowledge
by show-
of economic
fundamentals.
Morris-Shin abstracts from the role of Qiancial prices and other indicators as endogenous
aggregators of information. Taking the information structure as exogenous
step
and helps
isolate the critical role played
by disperse
beliefs.
However,
if
is
a useful Qst
Qiancial prices
convey information about the underlying fundamentals, then the dispersion of
beliefs is de-
termined endogenously in equilibrium. Information aggregation can thus play an important
role in determining
multiple equilibria
whether multiple equilibria
may
arise.
Indeed, Atkeson (2000) notes that
survive in the extreme case that Qiancial prices are fully revealing
and restore common knowledge. Moreover,
for
most of the applications of
interest, it
seems
unnatural to rule out endogenous information aggregators, such as Qiancial prices or other
indicators of economic activity.
In this paper,
we make the Qst attempt,
to the best of our knowledge, to incorporate
^or example, Diamond and Dybvig (1983), Obstfeld (1986, 1996), Velasco (1996), Calvo (1988), Cooper
and John (1988), Cole and Kehoe (1996).
2
Morris and Shin (1998) build on Carlson and van Damme (1993) who developed the Global Games
approach for the two-player two-action games. See also Morris and Shin (1998, 2000, 2001, 2003, 2004).
endogenous information aggregation in coordination economies with heterogenous informa-
Our model takes
tion.
as given the initially dispersed private information that is crucial
in the Morris-Shin framework.
nomic indicator
restoring
build on this by allowing individuals to observe an eco-
that, in equilibrium, aggregates disperse information. Importantly,
common knowledge by
Grossman and
mon
We
we
avoid
allowing enough 'noise' in the aggregation process, as in
Stiglitz (1976, 1980).
Thus, none of our results are driven by restoring com-
knowledge, the main theme in Atkeson's comments.
We
consider a variety of endogenous information structures, including indicators other
than Qiancial
prices.
Indeed,
to condition their behavior
we begin our
on a noisy
begin with this model for two reasons.
exploration with a model that allows agents
signal of the aggregate actions of other agents.
First, this situation
applications. For instance, during riots or
bank
runs,
We
seems directly relevant in many
an important part
of the story
may be
people are actively watching what others are doing. Ignoring this aspect
is
that
missing an
important piece of the puzzle. Second, this model parsimoniously highlights aspects of the
problem that recur when Qiancial prices are the instruments
We
study two versions of the observable actions case.
assumed
to observe a noisy signal of
for aggregating information.
In the
Qst
version, agents are
An
contemporaneous aggregate actions.
requires that agents choose optimally given the observed signal and, at the
signal
is
be generated by the aggregation of individual
choices.
equilibrium
same time,
this
Thus, our equilibrium concept
novel and unavoidably at the crossroads of rational-expectations and
game
theory.
The
second version avoids the simultaneity in the signal and actions by dividing the population
into
two groups,
'early'
and
'late'
agents. Early agents
to attack solely on their private information.
move Qst and base
their decisions
Late agents move second and can observe a
noisy signal about the aggregate activity of early agents. This non-simultaneous version only
requires standard game-theoretic equilibrium concepts.
second version converge to that of the Drst as the
We next
We
show that the
size of the early
equilibria of this
movers vanishes.
study environments where individuals cannot observe others actions directly but
instead can trade in a Qiancial asset market.
their private information
This market opens after they have received
but prior to choosing whether to attack the regime. The rational-
expectations equilibrium in the asset market generates imperfect public information that
agents use in addition to their private information
when they decide whether
to attack. This
framework opens up new modeling choices regarding the specittation of the
and the preferences over
its
risky return.
Indeed,
we
asset's payoff
consider four different specittations
that can be solved in closed-form.
A common insight emerges from all the cases we study:
information
is
the precision of endogenous public
increasing in the exogenous precision of private information.
We show that this
implies that introducing endogenous sources of information
the likelihood of uniqueness vs.
we
the six specifications
result:
multiplicity
important for understanding
Interestingly, for all
multiplicity of equilibria.
but one of
study, endogenous information reverses the Morris-Shin limiting
ensured
is
is
when
individuals observe fundamentals with small enough
idiosyncratic noise. Conversely, uniqueness
is
ensured
idiosyncratic noise
we view our paper
Despite the difference in the limiting result
theme emphasized by Morris-Shin, that
if
is
large enough.
as underscoring the general
multiplicity or uniqueness
may depend on
details of
the information structure and that these are worth exploring. This paper has explored the
importance of endogenous information aggregation. 3
The
rest of the
paper
is
organized as follows. Section 2 introduces the basic model and
reviews the Morris-Shin benchmark with exogenous public information. Section 3 introduces
endogenous public information with a signal on aggregate actions. Section 4 studies the
of Dnancial prices as
role
endogenous aggregators of information. Section 5 concludes.
The Basic Model
2
We
present an abstract general formulation of the basic model and then brieDy- discuss the
various interpretations available in the literature.
Actions,
Outcomes and
an alternative. There
is
Payoffs. There are two possible regimes, the status quo and
a measure-one continuum of agents, indexed by
can choose between an action that
favorable to the status quo.
All agents
We
move
e
{0, 1},
The
if
and
where
a;
1
=
G
{0, 1},
We
represents collapse.
where
is
is
abandoned and
R=
represents survival of the
similarly denote the action of an agent with
represents "not attack" and a;
payoff from not attacking
is
R
=
1
represents "attack".
normalized to zero. The payoff from attacking
—c <
Endogenizing the information structure
is
otherwise. Hence, the utility of agent
=
also the
They examine how the information conveyed by
how the
Each agent
these actions, respectively, "attack" and "not attack".
call
U{a i} R)
3
[0, 1].
simultaneously.
R=
the status quo
€
favorable to the alternative regime and an action
denote the regime outcome with
status quo
at
We
is
i
is
b
>
i is
<n(bR-c).
theme
in Angeletos,
Hellwig and Pavan (2003, 2004).
active policy interventions
may
result to multiplicity,
and
evolution of information over time affects the dynamics of regime change. Dasgupta (2002), on the
other hand, considers how social learning affects coordination.
quo
Finally, the status
abandoned (R
is
=
1) if
and only
if
A>9,
where
A = f atdi
G
denotes the mass of agents attacking and
[0, 1]
exogenous strength of the status quo
£
=
=
and
(or the quality of the
vidual to attack
if
and only
if
and only
is
the heart of the model.
parameterizes the
economic fundamentals).
Let
if
complements, since
strategic
it
pays for an indi-
the status quo collapses and, in turn, the status quo collapses
a sufficiently large fraction of the agents attacks. This coordination problem
Interpretations.
This simple model can capture the role of coordination and mul-
tiplicity of equilibria in
a variety of interesting applications.
self-fulQling currency crises (Obstfeld, 1986, 1996; Morris
bank
1R
1.
Note that the actions of the agents are
if
G
For instance, in models of
and Shin, 1998), there
and a large number
interested in maintaining a currency peg
is
a central
of speculators, with Unite
wealth, deciding whether to attack the currency or not. In this context, a "regime change"
when
occurs
bank
to
a sufficiently large mass of speculators attacks the currency, forcing the central
abandon the peg.
In models of self-fulling bank runs, a "regime change" occurs once a sufficiently large
number
of depositors decide to
to the system, forcing the
bank
withdraw
to
their deposits, relative to liquid resources available
suspend
its
payments. Similarly, in models of self-fulQling
debt crises (Calvo, 1988; Cole and Kehoe, 1996; Morris and Shin; 2003), a "regime change"
occurs
when
Finally,
may
or
a lender
fails
may
riots.
The
potential rioters
not overwhelm the police force in charge of containing social unrest depending
rioters
Information. Suppose
0,
sufficiently large fraction of its creditors.
Atkeson (2000) interprets the model as describing
on the number of the
<
by a
to obtain reQiancing
and the strength
for a
moment
of the police force.
that
were commonly known by
the fundamentals are so weak that the regime
unique equilibrium
is
>
every agent attacking. For
9,
is
doomed with
all
agents. For
certainty
and the
the fundamentals are so strong that
the regime can survive an attack of any size and the unique equilibrium
is
every agent not
attacking.
For intermediate values,
large attack
G
(0,0],
and there are multiple
the regime
is
sound but vulnerable to a
equilibria sustained
by self-fulQling expectations. In one
equilibrium, individuals expect everyone else to attack, they then Old
to attack, the status
quo
is
sufficiently
it
individually optimal
abandoned and expectations are vindicated. In another,
uals expect no one else to attack, they then
Dad
it
individ-
individually optimal not to attack, the
status quo
The
spared and expectations are again fulUled.
is
interval (0, 8] thus represents
the set of "critical fundamentals" for which multiple equilibria are possible under
common
knowledge.
by
Implicitly, each equilibrium is sustained
With common knowledge,
other agents do.
different
self-MQling expectations about what
can perfectly forecast
in equilibrium individuals
each other actions and coordinate on multiple courses of action. Following Morris and Shin
we assume
(1998),
that 6
private noisy information about
may
actions that
8.
have a
common
uniform over the entire real
where the idiosyncratic noise £ {
Xi is
that individuals instead have
Private information serves as an anchor for individual's
avoid the indeterminacy of expectations about others actions.
Initially agents
erate)
common knowledge and
never
is
about
prior
line.
Agent
is A/"(0, o~l)
8; for simplicity,
i
we
be (degen-
let this prior
then observes a private signal
with a x
>
and
is
independent of
The
8.
signal
thus a sufficient statistic for the private information of an agent.
Note that because there
economy,
(^i) ie
ji
r
,
is
is
a continuum of agents the information contained by the entire
enough to
dispersed throughout the population, which
Finally, agents
may
fundamental
infer the
is
However, this information
9.
the key feature of the Morris-Shin framework.
have access to some public information.
also
is
the Morris-Shin benchmark, where the public signal
is
exogenous.
We start by reviewing
We then consider the
endogenous public information generated by a noisy indicator of aggregate activity or prices
2.1
The Morris-Shin Benchmark: Exogenous Information
In this subsection,
we assume an exogenous
addition to the private signals Xi
=
8
+
public signal z
£ i; where
^ ~
= 8 + v,
A/"(0, <t£).
where v
The
~ _A/"(0, o^)
private noise £ and
the public noise v are distributed independently of each other and independently of
model then reduces
to that of Morris
in
8.
Our
and Shin (2000, 2001), with exogenous private and
public information (see also Hellwig, 2002).
In a
monotone equilibrium,
an agent attacks
decreasing in
if
and only
we
if
and only
so that there
8,
8
if
<
8*(z).
A
if
for
x
is
<
any realization of
x*(z).
By
z,
there
.
In step
3,
quo
also a threshold 9*(z) such that the status
monotone equilibrium
we
a threshold x*(z) such that
implication, the aggregate size of the attack
is
identiCed by x* and 8*
characterize the equilibrium 8* for given x*. In step 2,
for given 8*
is
we
.
is
is
abandoned
In step
1,
below,
characterize the equilibrium x*
characterize both conditions and examine equilibrium existence
and uniqueness.
6
Step
1.
For given realizations of 8 and
mass of agents who receive
signals
x
<
=
a~ 2
x*(z).
the aggregate size of the attack
That
so that regime change occurs
if
given by the
is,
the precision of private information. Note that A(8, z)
is
is
= $(^(x*(z)-d)),
A(d,z)
where a x
z,
and only
if
8
<
where 8*(z)
8*(z),
is
decreasing in
is
8,
the unique solution to
A(8*(z),z)=8*(z).
Rearranging we obtain:
x*{z)=d*(z)
+ -^=$-
1
{F{z)).
(1)
Jot*
Step
2.
Given that regime change occurs
if
and only
<
8
if
8*(z), the payoff of
an agent
is
= a(bPi[8 <
E[U(a,R{8,e))\x,z]
Let
a x = a~ 2 and a z = a~ 2
mation.
The
posterior of the agent
= a x /(a x + a z
)
is
x,z]-c).
\
denote, respectively, the precision of private and public infor-
8\x,z
where 5
8*(z)
is
~M(6x + (l—5)z
a' 1
,
)
,
the relative precision of private information and
a
= ax + az
the overall precision of information. Hence, the posterior probability of regime change
Pi[9<6*(z)\x,z] =
which
is
monotonic in
l-$ (^{5x +
x. It follows that
(1
the agent attacks
if
8)z
-
8*(z)))
and only
if
is
is
,
x <
x*(z),
where
x* (z) solves the indifference condition
bPv[8<8*{z)
|
x*(z),z] =c.
Substituting the expression for the posterior and the deQiition of 5 and a,
+ ^-z-8*( z )))
^(v^T^ (^r-x*(z)
a +a
\a + a
z
\
Step
if
3.
and only
Combining
if it
x
(1)
and
(2),
z
x
we conclude
z
JJ
we
b
= —^.
obtain:
(2)
b
that 8*{z) can be sustained in equilibrium
solves
G(8*(z),z)=g,
(3)
=
where g
+ a z /a x §
y/l
l
(1
-
and
c/b)
^
= -^(z~8) +
G(8,z)
1
(8)
'a.
With
6*{z) given
by
(3),
x*(z)
then given by
is
We
(1).
now
are
in a position to establish
existence and determinacy of the equilibrium by considering the properties of the function G.
that, for every
Note
zGE,
G(6,
z) is
which implies that there necessarily
This establishes existence;
continuous in
exists
we now turn
dG(8,
max^R <j){w) =
1/\/2tt then
to uniqueness.
z)
a z /^/a^ <
which implies a unique solution to
8,
and there
in 8
£
if
and only
We
(8, 8).
y^
(8))
(1)
G
we have that
\/27r
a z /^fcTx >
is
strictly increasing in
V2tt, then
G is non-monotonic
admits multiple solutions 9*(z) whenever
We conclude that monotone equilibrium is unique
~
results in the following proposition.
(Morris- Shin) Suppose agents observe an exogenous public and private
1
Let a x and a z denote the standard deviations of the private and the public noise,
respectively.
Monotone
equilibria exist
Finally, consider the limits as
ctz/^foLc
for all z.
—
>
and y/(a x
+ az
)
—
ax
/a x
and are unique
—
>
>
1.
for given
az
Condition
,
(3)
if
or
and only
az
—
»
if
a x /a z2 <
v2tt-
oo for given a x In either case,
.
then implies that 8*(z)
—
>
8
= 1 — c/b,
This proves the following result, which we refer to as the Morris-Shin limit
Proposition 2 (Morris-Shin limit) In
oo for given a x
3
oo,
a z /-sfa x < y2n.
if
Proposition
only
=
z)
az
1
instead
an interval (z,z) such that
summarize these
signal.
(3). If
and a unique solution otherwise.
z
(z, z)
is
and G{8,
Note that
1
0($"
if
= -co
z)
a solution and any solution satisCes 8*(z) e
88
Since
with G(8,
8,
,
there is a unique
ifd<6, where
8
= 1 - c/b
e
the limit as either
ax
—
monotone equilibrium in which
>
for given a z
,
result:
or
az
the regime changes if
—
>
and
(6, 6)
Endogenous Information on Actions
We now study the case where public information is
endogenous. Agents no longer receive the
public signal z as assumed in section 2.1. Instead, individuals are able to observe a public
noisy signal of the aggregate actions of others.
We
study two versions of such a model. In the dst version, contemporaneous actions
are observed with noise.
Thus, our equilibrium concept
crossroads of rational-expectations and
The second
game
is
novel and unavoidably at the
theory.
version, in Section 3.3, has non-simultaneous
and
lation into two groups, 'early'
moves by dividing the popu-
movers. Individuals in the early group
'late'
make
their
decisions to attack or not based solely on their private information. Individuals in the late
group move and are able to observe a noisy signal of the early group's aggregate action. This
non-simultaneous version only requires standard game-theoretic equilibrium concepts.
show that the
equilibria of this second version converge to that of the
Qst
We
as the size of the
early movers vanishes.
Equilibrium with Endogenous Information
3.1
We
assume that agents can condition
their behavior
on a noisy indicator
of the contempora-
neous aggregate attack:
y
where e
is
random
ously so that y
is
noise
and
s
:
[0, 1]
x
=
s(A,e)
R —* 1R.
All agents are
assumed
o-
move simultane-
a signal about contemporaneous actions.
For reasons of tractability we specify the signal function as s(A,e)
noise s as J\f(0,
to
2
e)
with a e
>
0.
The common
fundamentals 6 and the idiosyncratic noise
£.
noise e
As we
is
=
<£
-1
(A)
+e
and the
distributed independently of the
will see, the
above speciCbation allows
the equilibrium to preserves normality of the information structure, which in turn permits
closed-form solutions. This convenient specittation was introduced by Dasgupta (2001) in
a different setup.
The information
structure
is
parameterized by the pair of standard deviations (a x ,a £ ). In
any symmetric equilibrium agents are distinguished solely by their information, summarized
by their observation of the private signal
action chosen by such an agent.
%i
and the public
signal y. Let a(x{,y) denote the
A symmetric rational-expectations equilibrium is deQied as
follows.
A
symmetric equilibrium consists of an endogenous signal y
=
Y(9,e), an individual
attack strategy a(x,y), and an aggregate attack A(9,y), that satisfy:
=
a(x, y)
max E
arg
[
U{a, R(9, y))
x,
j
y
(4)
]
a£[0,l]
A(9,y)
y
for all (9, e, x, y)
Condition
change occurs
.
Where
and only
if
t
y)d^^^y
(5)
= $-\A(9,y))+s
=
i?(0, y)
means that a(x,y)
(4)
if
R4
e
= Ja(x
is
>
A(9, y)
1 if
A(0, y)
(6)
>
9
and E(0,
y)
=
otherwise.
the optimal strategy for the agent given that regime
whereas condition
9,
means that A(9,y)
(5)
is
simply
the aggregate across agents. Of course, the aggregate public signal y must be consistent with
individual actions which gives condition
(6).
This
is
the rational-expectations feature in our
equilibrium concept.
we
For tractability
focus on symmetric equilibria where the information structure
normally distributed and the strategy of the agents
we
is
monotone
shall see below, normality of the information structure
simultaneous model of section
3.2
3.3.
monotone
as
and the regime changes
with the
if
and
and only
mappings
triplet of
construct the set of
x*, 9*
(4)-(6). In
0*(y) such that
if
9
<
A
9*(y).
monotone
equilibria, for
an agent attacks
if
any realization
and only
monotone equilibrium
is
if
as given
optimal. In Step
monotone
and use condition
3,
<
x*(y)
thus identiCed
equilibria in four steps.
(5)
and
In Step
1,
we
start
(6) to characterize
(4) to
we study the Cked
compute the threshold
point x*
=
x**.
x** that
Finally, in Step 4,
with an
the implied
aggregate attack A, the resulting 9* and the possible public signals Y. In Step
Y
x
and Y.
by the agents and use conditions
arbitrary x* used
and
equilibria.
Equilibrium Analysis
of y, there exist thresholds x*(y)
9*
As
an implication of the non-
We refer to such equilibria simply
We now study the equilibrium conditions
We
is
in private information.
is
2,
we take
is
individually
we
consider the
deter minacy of Y.
Step
1.
function x*.
In a
monotone equilibrium, a(x,y)
The aggregate
attack
is
=
1
if
and only
if
x
<
x*(y), for
some
then
A(9,y)
=
<S>(^-x (x*(y)-9)),
10
(7)
.
=
where a x
that A{9,y)
a x 2 Note that A(9,y)
>
9
if
and only
if
9
<
decreasing in 9 so there exists a function 9*(y) such
is
.
The
9*{y).
threshold 9*(y) solves A(0*(y),y)
=
6*(y), or
equivalently
**(y)
= ^(y) + -4=*- 1 (^(y)).
(8)
\/CXt.
Equilibrium condition
that the signal signal must satisfy,
(6) implies
y
=
^faT
x [x*{y)
-8]+e,
or equivalently
x*(y)-a x y
For any (9,e) €
z.
R2
let z
,
=
Z(6,e)
=
9
= 9-ax e.
— a x e and
(9)
note that
a relation between y and
(9) is
DeQie the correspondence
y(z)
In Step
Now
we show
4,
is
non-empty and examine when
take any function Y(z) that
Y(z) £ y{z)
see
that y(z)
= {yeR\x*(y)-ax y = z}.
and
for all z,
let
is
2.
We now map
9*
=
the signal be Y(6,e)
and
Y
|
single- or multi-valued.
Y(Z(9,e))
i.e.,
= Y{9 - a x e). As we
to x**. Given that regime change occurs
x,y]
is
if
= a{6Pr
[
<
9
That
is, it is
as
if
x*(y)
- axy =
<t~
2
,
ae =
we conclude
9
- ax e =
the agents observe a public signal z about
that each agent also observes a private signal about
ax =
cr~
2
,
and a z
= a x ae =
(a x a £ )~
that the posterior of an agent
2
.
9,
= ax + a z
is
if
9*
(y)
\
x,
y
}
z
9,
with noise
A/"(0, cr^cr
with idiosyncratic noise
Combining the two sources
2
A/"(0,
).
<j
Recall
x ).
Let
of information
is
9\x,y ~ Af(5x + (l-S)Z(y)
where a
and only
\x,y]-c}.
9*{y)
[
=
shall
given by
We thus need to consider the determination of the posterior probability Pr 9 <
The observation of y = Y(9,e) = Y(z) is equivalent to the observation of
Z(y)
such that
of the information structure.
9 <-9*(y), the expected payoff for the agent
E[U(a,R(9,e))
it is
a selection from this correspondence,
any such selection preserves normality
Step
(10)
,
the total precision of information and S
11
a'
1
)
,
= a x /{a x + a z
)
is
the precision
of
x
relative to the total.
It follows
that
Pr
[6
<d*(y)\x,y]
Note that the above
is
monotonic
= l-$
in
(yfa (Sx
+
-
(1
Z(y)
5)
x so the agent attacks
if
-
6*(y)))
and only
if
x
.
<
x**(y),
where
x**(y) solves
bPv[9<6*(y) \x**(y),y]
= c.
Combining the above two conditions and substituting Z(y)
x**(y)
=
x*(y)
—
y/.y/ax~,
we did
that
must solve
$ (y/Z
Step
(5x**(y)
+ (1-5)
3. In equilibrium, x**
=
x*
and
~
- -j=y\ - P(y)X\ =
(x*(y)
(11)
(11) reduces to
1-5 \\ b-c
=
${^(x *(y)-6\y)-±-=y\\
Combining the above with
we
(8),
using 5
= a x /(a x + a z
and a
)
= ax + az
,
and rearranging,
obtain:
6 -(y)
= •
(-*-»
+ X^Z*-(^i)),
\a +a
b
a +a
x
**(y)
where a z
and
= a x aE
.
Hence, for
9*(y)
all
+ -^t>-
x
z
\
y.
1
(0*(y)),
(13)
a x and ae the equilibrium x* and
6* are
,
Finally, 9*(y) does not
e*(y)-+$(y) asa £
Step
is
4.
We
(12)
JJ
irrespectively of the selected equilibrium signal Y. Moreover,
increasing in
this
=
V
z
depend on a x
,
6*(y)
—
>
determined uniquely
both 0*(y) and x*(y) are
1
—
c/6 as
cr e
—
>
oo and
^0.
Qially need to consider the equilibrium correspondence y(z). Recall that
given by the set of solutions to
x*(y)
r=zy
12
=
z.
Using (12) and
the above reduces to
(13),
\a x + a z
where
as y
A=
—>
\Ja x j (a x
+ a z )Q~
+co, and F(y)
an equilibrium always
for all
z,.
which
true
is
Differentiating
F
—
+oo
>
exist.
1
(l
as
'
>
>
To examine
if
F
we ask whether y(z)
multiplicity,
monotonic
is
=
in y.
+A
--*=-(l--£=#Y-5-V
y/a x
\a x + a
x + a
\
a>
z
z
follows that the determinacy of equilibrium hinges on the ratio
Shin benchmark.
therefore y(z)
is
max^gK <j){w)
Since
{z,~z).
are, respectively,
=
single- valued for all z,
V2tt, there are thresholds z
values for z €
1/v2tt,
if
we have
and only
~
F
=
F(y) and
that the equilibrium
is
unique
if
=
is
like in
=
y,
and
a z / sfcTx >
instead
for z
~z
the Morris-
decreasing in
\f2ir. If
and z such that y(z) takes one value
the lowest and highest solution with F'(y)
,
that
'
These thresholds are given by z
a z = a £ a x we conclude
a z / y/oi^,
a z j yfa x <
if
Unlike the Morris-Shin benchmark, however, the ratio
The
single-valued
is
we obtain
fto)
It
a x + az
V
— c/b). Note that F(y) is continuous in y, and F(y) — — oo
y — — oo. Thus, the correspondence y(z) is non-empty and
and only
if
JoTx
J
'
(z,z) but three
^
F(y), where y and y
0.
a z / ^/a x
~
and only
if
is
endogenous.
a ey/a x <
'
Using
\f2ir.
following proposition summarizes these results.
A monotone
Proposition 3 (Morris-Shin meet Grossman-Stiglitz)
equilibrium
is
characterized by a triplet of mappings (Y,x*,6*) such that the endogenous signal satisDes
y
=
Y(9,e), a agent attacks
if
and only
if
x <
and regime change occurs
x*(y),
if
and only
tf0<9*(y).
A monotone
If
a\a x
<
equilibrium exists for
all
(a x ,a 6 ) and
l/\/27r, the equilibrium signal function
is
Y
unique
if
and only
if
o\o~ x
>
1/V27r.
indeterminate, but the equilibrium
is
threshold functions x* and 9* remain unique and independent of the selected Y.
We
conclude that the equilibrium
is
unique only
if
there
of information, the exogenous information of the agents
aggregate activity.
is
enough noise
in
both sources
and the endogenous signal about
Multiple equilibria survive as long as either source of information
is
sufficiently precise.
Interestingly,
when
multiplicity arises
with respect to individual behavior.
knowledge limit (a x
=
ae
—
0), in
it is
with respect to aggregate outcomes but not
To understand
which case x
13
—
9
this result,
and y
= ^~
1
consider the
common-
(A), so that the agent learns
to attack
if
and only
a(x,y) for the agent
A
values of
A=
and
A
by observing x and learns
9 perfectly
A >
if
is
by observing
or equivalently
9,
=
The
of indeterminacy remains.
any given observation x and
A =
is
uniquely determined for
A and
y for
e.
formation economy of Morris-Shin,
it is
economy
is
different
from the exogenous
'.
However, note that, in equilibrium, the
equivalent to the information generated by z
Our endogenous-information economy
is
in-
interesting that the determinacy of equilibrium in
both cases hinges on exactly the same ratio a z /-s/ax
is
both
(9,6],
but there can be multiple equilibrium values of
Finally, since our endogenous-information
information generated by y
E
o^ and a e are non-zero, the same nature
equilibrium behavior of the agent
y,
any given realization of 6 and
When
optimal
However, the equilibrium,
$(y).
for every 9
and y are not uniquely determined. Instead,
it
Here, the equilibrium strategy
$(y).
uniquely determined with x*(y)
4
can be sustained in equilibrium.
1
<
x
The agent then Qids
y.
=
Z(y)
= 6 — a x e.
thus related to an exogenous-information economy
= ay = ae a x
az
with precision of public information given by
.
Indeed, substituting this
expression into the criterion for multiplicity from proposition, ?? that o x ja\
the criterion for multiplicity in proposition
3,
<
that a^a x
1/\/2tt.
As the
>
v27r, yields
precision of private
information becomes inLnite so does the precision of public information.
Proposition 3 establishes that, for any given level of noise in the agents' private infor-
mation, multiple equilibria exist
if
and only
an increase
sufficiently small. Intuitively,
in
the noise in the macroeconomic indicator
if
is
a £ reduces the public information generated by
the observation of y and thus reduces the ability of the market to coordinate on multiple
courses of action. Indeed, the equilibrium conditions (12) and (13) imply that, for every y,
we have
9*(y)
—
>
1
—
c/b
=
9
and x*(y)
—
>
6
+
a x ^~ 1 (9)
Proposition 4 (Limit a £ —> oo) Fix a x and
if
and only
if
9
<
9,
where 9
The Morris-Shin outcome
becomes
=
1
is
arbitrarily large. This
—
c/b G
let
ay
—
»
as
at
—
>
oo.
oo. In the limit, the regime changes
(9, 9).
obtained as the noise in the observation of aggregate activity
is
no information
intuitive, for in this case
the observation of y and the endogeneity of public information
is
of
is
generated by
no importance.
Consider next the limit of the precision of agents' private information, for given level
of noise in y. Proposition (3) establishes that, for given
only
if o~ x is
sufficiently small.
The
a e multiple
,
equilibria exist
interval (z,z) represents the region of multiplicity
if
and
and a
4
1
If A — 0, then y = — oo and x*{y) = <5~ (— oo) = 8, in which case all agents attack whenever 6 < 6 and
no agent attacks whenever 8 > 9_. If instead A = 1, then y = +oo and x*(y) = $ _1 (+oo) = 9, in which case
all agents attack whenever 8 < 8 and no agent attacks whenever 8 > 8. In the former case, a regime change
is
triggered
if
and only
if
8
<
9;
in the latter,
if
and only
14
if
8
<
8.
>
.
reduction in a x reduces z and increases z making larger the multiplicity region. Indeed, as
a x —*
we can show
that any outcome
—
Proposition 5 (Limit a x
>
0)
Fix
possible for
is
and
o~ e
ax
let
—
G
9
all
(9, 9).
There exists an equilibrium with
0.
the probability of regime change converging to zero for all 9
G
as well as
(9, 9),
with the probability of regime change converging to one for every 9 G
an equilibrium
(#, 9)
— +oo as a x — 0. Next, note that both \o e u x — (j)(y)\
and \o\o x — <f)(y)\ vanish. Since lim^-oo <f>{y)y = lim^ +00 (j)(y)y = 0, the latter implies
and a x y — 0. Hence, z —> $(— oo) = 9 and z — $(+oo) = 9 as x — 0. Moreover,
xy —
for every 9 and e, 6 — a x e —
as x — 0. It follows that
Proof.
o~
First,
note that y
>
— — oo
>
>
let
Y(9,e)
— ax e G
=
cr
>
o"
>
Pr
that 9
>
>
1
Next,
J2
and y
[
9
- ax e £
(z,z).
>
(z,z)
|
— ^e) and
min3^(^
9
e
{9,6)
Y(9,e)
Note that y(^,e) < y
]
->
= max3
<y <
(12), 0*(y) is
$(+co)
=
9 as y
Pr[ 9
<
—
>
independent of a x
+oo.
It
;
]
->
and
Pr
[
<
(6
l
cj x
-»
0.
— a x e) and
consider (^,e) such
Y(9,e) and therefore
<&(— oo)
follows that, as long as 9
9* (Y(9,e))
which establishes the
—*
9*{y)
,
as
1
F(0,e)—>— oo and y(#,£)—>+oo
From
»
G
ox
as
=
^
—
>
0.
— — oo,
as y
>
and
6*{y)
(9, 9),
5* (F(d,e))
]
-
1
as
a x ->
0,
result.
This result stands in stark contrast to the Morris-Shin limit result in Proposition
With exogenous
reason
is
2.
public information a unique equilibrium survives as the noise in private
information vanishes.
present with
—
With endogenous
common knowledge
public information as modeled here the multiplicity
obtains as the noise in private information vanishes.
As the
once again the endogeneity of public information:
The
precision of private
information increases, the precision of public information also increases, and indeed at the
same
3.3
The
rate, so that
common knowledge
is
recovered in the limit.
Non-Simultaneous Signal
analysis so far has
assumed that agents can condition
noisy indicator of contemporaneous aggregate behavior.
15
their decision to attack
We now
on a
consider an alternative
.
that introducing
to break the simultaneity of signals
some simple dynamics
a result, the equilibrium concept in this model
is
There are two types of agents, "early" and
and
As
actions.
entirely game-theoretic.
"late"
.
Early agents move
[1st,
on the basis
Late agents move second, on the basis of their private
of their private information alone.
information as well as a noisy public signal about the aggregate activity of early agents.
Neither group can observe contemporaneous activity, but late agents can condition their
behavior on the activity of early agents.
Let
and
agents,
^ ii 4 1
is
+
e
fj,
(1
A2
A
denote the fraction of early agents,
(0, 1)
The regime changes
the aggregate activity of late agents.
- fi)A 2 >
The
6.
signal generated
by
the aggregate activity of early
Y
early agents
if
and observed only by
and only
if
late agents
given by
= $-\A + e,
y1
where e
In a
x\.
~ A/"(0, a^)
is
independent of 6 and
monotone equilibrium, an
The aggregate
£.
early agent attacks
attack of early agents
A
1
(15)
1)
is
if
and only
if
x
<
x\ for
,
some threshold
thus
= $( /^[9-x* ]).
(9)
y
(16)
1
Hence, in equilibrium
m = &The observation
of y
is
1
(A,
(9))
+e=
^T
x [x\
-
9}
+ e.
which
is
contingent on his private signal x and the public signal
y.
thus equivalent to the observation of a public signal
z,
dedied by
-y
=
- a x e.
y/ot x
The
strategy of a late agent
Since y and z has the
is
same informational
of a late agent as a function of
and only
if
x
attack of late agents
is
attacks
if
<
x and
x\(z), for
z.
content,
we can
equivalently express the strategy
Hence, in a monotone equilibrium, a late agent
some threshold function
x\. It follows that the
A2 (6,z) = $(^x-[6-xl(z)}).
Combining
(16)
and
A
(17),
(6, z)
=
we obtain the
ijl§
overall size of attack:
{^Tx [9-xl]) + (l- /i)$ (Vc£ [9 - x*2 {z]\)
16
aggregate
(17)
.
It
follows that the regime changes
and only
if
<
8
if
8*(z),
where 9*(z) solves
A (8*(z), z) =
9*(z), or equivalently
//$
[6*{z)
{yfe
-
x*])
+
(1
-
//)$
(Vo~ [8*(z) -
6Pr
[8
to the late agents but
<
8* {z)\x* {z) z]
2
,
=
where 5
&Pr
[9
<
cx.
x /{ol x
9*(y)\x{]
=
c,
$
(yfc(5x'2 (z)
+ az
—
c,
unknown
)
8*(z).
(18)
The
to the early agents.
is
threshold x 2 (z) thus solves
or equivalently
+ {l-6)z-
= ax + a
and a
z
The
.
8*(z)))
=
~,
(19)
threshold x\, on the other hand, solves
or equivalently
f$(^[d*(z) where we used the
=
Note that the realization of z
Next, consider the optimal behavior of the agents.
known
x*2 (z)})
x\])
fact that, conditional
^^(^[[z - x])dz =
on
x, z
is
^,
(20)
distributed normal with precision
ot\
=
a x a£ /{\ + a £ ). 5
Solving (18) for x*2 {z) gives
= 9*{z) - y^®- U(z) + -^—
1
x*2 {z)
Substituting the above into (19) and using 5
= a x /(a x + a z
T(8*(z),z,x*1
where g
=
y/l
+ a z /a x §
r(e,z, Xl ,n)
x
(1
—
For any x\
G
E
T(8,z,xl,fi)
=
—00 and T{9,z,x\,n) —
z
:
K —
>
[8,8],
f
and a
= a x + a z we
(21)
1
(9
+ -±-
[8
-$
(^rx [9 - Xl ])]\
:
>
continuous in
To
[8,8].
On
=
6
- ax e =
x
-
£
with
the other hand, for any given function
condition (20) determines a threshold x\ G K.
see this, note that z
8,
Hence, for any given threshold x\ G M, con-
00.
M—
is
An
equilibrium
solution to (20) and (21).
5
obtain:
,
i)=g,
G M, we have that Y(9,z,x\,n)
dition (21) determines a function 8*
8*
,
)
.
and
c/b)
= ^=(z-9) + &-
and any
-§(^TX \9*{z) - x\})]\
[8*{z)
- a x e,
so that z\x
17
~ M (0, a x + a\a^)
is
any
joint
>
.
Consider
now
the limit as
->
fi
Note
0.
6l
that, for all (0, z, x\)
r(M,s» - G(M)
= ^=(z-9) +
lOLn
^
2
,^0 implies
(0)
.
G is independent of x{ and is the same as in the Morris-Shin benchmark.
function 9* R — [0,0] such that, for all 2,
Consider
Note that
now
a
>
:
G(9*(z),z)=g.
earlier, 9* is
As shown
unique
and only
if
(22)
a z /^/a^ < v2n.
if
If
instead
a z /^/a^ >
V27T,
there are multiple 9* solving (22). Moreover, for any such 9*, (20) admits a unique solution
x\
We
e R.
conclude that, for
whenever a z /y/a^
as
>
OL e ^fOL^
>
fi
small enough, there are multiple solutions to (20) and (21)
= a£ ax
But a z
^/2rr.
so that multiple equilibria again
,
emerge
as long
v2tt.
Moreover, the equilibria of this economy approximate the equilibria of our benchmark
model
£^
in the following sense. Let
denote the "dynamic" economy of this section and S
the "static" economy of the previous section. Let x* and 9* denote the equilibrium thresholds
for £,
y
the correspondence deQied in (10), and
dence. For any
x l(
)(
z)
~~
u >
X *(Y( Z ))
0,
we can Qid
< u
f° r
an equilibrium of £^y To see
only
if
the composite 9* o
Y
->
is
is
is
Hence, in equilibrium,
and
0^
Y
Since y 2 conveys no
we
pM-p<y(*)) <
(
are the thresholds associated with
are part of an equilibrium for
random
variable y 2 deQied
$
S
if
and
= $-
1
by
(A 2 )+e,
is
a noisy indicator about the activity
unobservable, late agents continue to attack
_1
u and
a solution to (22).
(A2 )
+£ =
y 2 (z)
if
.
the same realization as the one in (15). y 2
of late agents. If y 2
changes
,
note that 9* and
y2
where £
a function selected from this correspon-
small enough such that
au z where x*2
this,
Finally, let us introduce a
\i
Y
[x 2 {z)
^JaZ
=
—
y/aZ{x*2 {z)
more information than
z
z]
-
and y 2
is
if
and only
if
<
x
x 2 {z).
a function of z alone:
z).
and thus no more information than
allow late agents to condition their behavior on y 2 in addition to
yi
y\.
,
nothing
That
is,
the equilibria of the economy where late agents observe both yi and y 2 coincide with the
equilibria of the
Now
economy where
late agents observe only y\
consider again the limit as
\i
—
>
0.
Since x*2 , Az)
18
—
>
x*(Y(z)),
we have y 2 (^)(z) —
—
^/o^ x*(Y(z))
V2(ii)( z )
~~>
.
By
Y( z )- That
an equilibrium
economy
z
is,
in the limit as
economy
in
by a random variable
£^
of
y-zfji)
\x
—
0,
>
is
economy
variable
=
t/2(/x)
<&
_1
—
z
.
that
{A{6
i
is
y))
Hence,
part of
+
e for
£ can be approximated
This indeed provides a justification for the
£( M ).
we made
which the
not a function of z alone,
is
random
the
^JaZ x*(Y(z))
part of an equilibrium for
equilibrium selection
signal y
=
solves the Dxed-point condition y
any Y(z) that
£. Conversely,
we have Y(z)
definition of Y(z),
in Section 3.1.
Any
equilibrium of the "static" economy in
exists,
if it
can not be approximated by an
equilibrium of a "dynamic" economy.
Financial Prices
4
The
analysis so far has
assumed that agents can condition
indicator of aggregate behavior.
that
is
We
determined
Instead,
we now
on a noisy
their decision to attack
allow agents to observe a Dnancial price
earlier in a competitively asset market.
modify the environment as
follows.
There are two stages and we
as the 'asset market'. All individuals begin with the
exogenous private signal
x,
=8+£
{
,
same endowment
^
where the noise
is
refer to stage 1
w
of wealth
and an
as before.
In the fist stage, agents trade a financial asset that has a dividend that depends on the
underlying fundamentals, either directly or indirectly.
whether to attack the regime or not as
In the second stage, agents decide
in the basic model.
The return
of the asset
and the
regime outcome are realized at the end of the second stage. Here, in deciding whether to
attack agents can no longer condition their choice on a direct signal of the aggregate attack
as in Section
3.
However, in addition to their private information, they can use information
revealed by the equilibrium price from the [1st stage.
This framework opens up new modeling choices regarding the specification of the Qiancial
asset's payoff
and the preferences over
risky payoffs. Following Grossman-Stiglitz,
our choice with an eye towards tractability.
The
four specifications
we
we guide
solve below are
designed so that they preserve normality of the information structure and are solvable in
closed-form.
We
Qst introduce some general notation and
by p the price
of the asset or
dividend paid by the asset
this asset.
For
all
cases
his portfolio choice
we
is
state the equilibrium conditions.
more generally some measure
represented by / and
consider
we can
k{
=
of the terms of trade. Let
be the investment agent
i
/ the
makes
in
express the indirect utility the agent enjoys from
by a function V(k, f,p) so that the
Ui
We denote
V(ki,f,p)
19
+
total payoff
U(a,i,R),
is
where the
utility
Let aggregate
from attacking
demand
U
is
just as in Section 2.
be
for the asset
One
the exogenous net supply of the asset.
results
it
and the private
A/^O,
k{di.
We
assume there
o~1)
This shock
'noisy' traders.
and independent
Grossman and
K=e
a shock e to
of
is
is
that
not observed
both the fundamentals
which determines an equilibrium price function P(8,e).
Stiglitz (1976, 1980), the role of the
shock e
the information revealed by Qiancial prices about fundamentals.
is
is
noise.
Market clearing requires
in
it is
J
interpretation of the net supply shock
from a shock to the demand of other
by individuals and we assume that
As
K=
is
to introduce noise in
Since the price function
a function of both 9 and e the observation of p does not reveal 9 perfectly, leading to a
signal-extraction problem.
A
rational- expectations equilibrium
is
a price function, p
=
P(9,e), individual strategies
l
investment and attacking, k(x ,p) and a(x\p), and aggregate investment and attack
for
functions, K(9,p)
and A(9,e), such that
k(x,p)
in the Crst stage:
= argmaxE[
K(0,p)
V(k,f,p) \x,p]
(23)
= Jk(x,p)d$(Z^\
(24)
K(9,P(9,e))=e
and
(25)
in the second stage
a(x,p)
= axgmaxR[U(a,R)\x,p],
a£[0,l]
(26)
v
;
'
A(6,p)= f a (x,p)d<S>(^^-j,
where
R=
1 if
A(8,p)
>
8
and
R=
The information an agent has
observed price
(27)
otherwise.
consists of the privately observed signal
x and the publicly
p. In this sense, the price p takes the place that the signal y
observable action model of Section
.
had
in the
Condition (25) imposes market clearing in the asset
market. Condition (23) and (27) requires that an agent's investment take into account the
information contained in their private information and prices.
The
four speciCcations
we
consider below generate the following information structure.
20
There
is
a strictly monotone function Z{y) and a random variable v that
and independent
=
and £ such that Z(P(9,e))
of 9
9
+
Thus, the observation of an equilibrium price realization p
=
observation of a public signal z
The
is A/"(0,
Q{a x a E )~ l
,
and
v for every realization of 9
is
)
e.
informationally equivalent to the
Z(p) on 9 with normal error and precision
agent's posterior conditional on his private information x
ap
= Q(a x ,a e
and the observed
price
).
p
is
\x,p
9
where
5
= a x /(a x +ap
)
and a
of equilibrium turns out to
~N(5x + {l-8)z
= a x +ap
,
a' 1
)
,
Like in the Morris-Shin benchmark, the determinacy
.
depend on the
ratio
ap
that
is,
the ratio of the precision of the public information generated by the price to the
square root of the precision of the exogenous private information.
the equilibrium
is
unique.
If
instead
ap j\/a x >
~
If
ap /\faZ < V^w, then
V2ty, then there are multiple equilibria.
Unlike Morris-Shin, however, the precision of public information ap and therefore the ratio
,
c^p/V^ri are endogenous and affected by a x
what
In
follows,
In each case,
Q.
we
we
.
consider four alternative speciCcations of the asset market (stage
solve for the equilibrium price function
We then examine the critical ratio
ol
~
p J sJ~oix to
and the associated mappings
Z
1).
and
study the determinacy of equilibrium as a
function of the exogenous information structure, (a x ,a £ ), or equivalently (a x ,a E ).
Risk Aversion — Fundamental Dividend
4.1
We start with an example that maps directly to the CARA-normal framework introduced by
Grossman and
Stiglitz (1976, 1980) [see also
either in a risky asset or a risk-less asset.
it
costs the agent
1
in the
dst stage and
Hellwig (1980)].
invest his wealth
We normalize the gross return of the risk-less
delivers 1 in the second stage.
the agent p in the Crst stage and delivers f(8)
asset
The agent can
=
9 in the second.
The
asset:
risky asset costs
Here the return of the
depends directly on the exogenous fundamental.
The agent
enjoys utility only from second-stage consumption and his preferences over
second-stage consumption exhibit constant absolute risk aversion (CARA).
utility
from
his portfolio choice
V(k,
is
The
indirect
thus given by
f,p)=u(w- pk + fk)
,
21
u(c)
= - exp(-2 7C )/27,
(28)
where k
c
is
the
amount invested
= w — pk + fk
We
[
x,p]
\
=
we
obtain,
then guess and verify that
E[
for
invested in the risk-less asset,
is
second-period consumption.
is
J^E V(k, f,p)
Setting
w — pk
and
in the risky asset
some
5
G
(0, 1)
6
1
|
x,p
=
]
and a >
0,
+
<fa
in
(1
-5)p
and
Var[
x,p
(9
|
= a"
]
1
,
which case the optimal demand reduces to
k(x,p)
=
— (x-p).
7
It
follows that the aggregate
demand
for the asset is
K(9,p)
= —(e- P
).
1
In equilibrium,
By
K = J kidi = e,
and therefore the equilibrium price
implication, the observation of
9 with precision (5a/^)
ap =
(5a/~f)
We now
precision
2
aE
2
a e That
.
p
= P(e,e)=e-^-e.
p
is
satis Ces
(29)
da
is,
equivalent to the observation of a public signal about
in this case
we have Z(p) =
p,
v
=
—(Sa/j)e, and
.
determine 5 and a. Note that x and Z(p)
a x and ap
,
E
respectively. It follows that
Oi X
[
a
= a x (l + a x a £ /^ 2
ap
Recall that in Section 3
|
x,p
),
5
=
(Sa/^y)
1/(1
}
—
2
+
8x
—
(1
5)p,
where
ae
+ ax a
= Q(ae ,ax = J-^L
we found that the
are independent signals of 9 with
Q>n
a
a x + ap
a = a x + ap = a x +
Solving the above gives
9
=p
,
£ / 'y
2
)
,
and
-
)
.
.
(30)
precision of endogenous information increased pro-
portionally with the precision of private information. Here the precision of public information
22
.
'
increases
more than proportionally with the precision
of private information. This will only
and
serve to reinforce our conclusions regarding the comparative static
To
verify this, consider stage
by thresholds x*(p) and
regime changes
A
monotone (continuation) equilibrium
6*(p) such that
and only
if
2.
if
<
9
9*(p).
an agent attacks
The threshold
if
ax
limit exercises for
and only
if
x*(p) solves frPr
x
[9
is
.
characterized
< x*(p) and the
< 9*(p)\x,p] = c,
or equivalently
h
*
- P<P))) = -^+ 7TT7rS(P)
(^+^ (-~-*>)
\a + a
a +a
P
V
The threshold
x
x
p
x *(p)=9*(p)
if
and only
'
'"
:
if it
-
(p
31 )
9*(p), or equivalently
+ -±=<l>-\9*(p)).
(32)
= p, we have that 9*(p)
Combining the above two conditions and using Z(p)
in equilibrium
=
on the other hand, solves A(9*(p),p)
9*(p),
(
b
JJ
p
can be sustained
solves
9*(p))
+
3>-\9* (p))
--
J?Z±**-i
(1
-
c /b)
(33)
OCt.
Similar arguments as those used for Proposition
only
if
aP l ^/ox~ >
V2tt,
given by (29) and
is
where ap
is
9*{p) if
As
given by (30).
V
is
given by (28) and f
always uniquely determined.
and only
in the
On
9*
(p) if
and
the other hand, the price function
is
uniquely determined.
Proposition 6 Suppose
P{9,e)
is
imply that there are multiple
1
< 7 2 (27r)~ 1/2
if g\<j\
=
The equilibrium price function
9.
There are multiple equilibrium thresholds x*(p) and
.
benchmark model, multiple
equilibria survive as long as either noise
is
small
enough. Indeed, the common-knowledge outcomes are once again obtained in the limit as
ax
—
>
4.2
for
any given a £
.
Risk Aversion — Speculative Dividend
We now modify the previous example by letting
of the asset
stage.
We
is
now a
earlier
/
=
f(A)
= — $ -1 (,4).
That
is,
the dividend
function of the equilibrium size of the attack realized in the second
showed
that, in equilibrium,
23
A = $
/c7x~[x*
[y
(p)
-
9})
.
It
follows that
=
/
—
y/a^[9
x*
and therefore
(p)]
o^E[
= v
/
k
9
|
x,p
}
ja x Ven[
-
y/a^x*(p)
9
\x,p]
-p
Let
1
=
P
zp
+ X*(p),
(34)
>OL n
and note
demand
that, for every p, the
above deQies a unique
as
= E[0
k
|
where 7
=
r
)y/a~x~.
The
rest
x,p
\
can then rewrite the optimal
-p
x,p]
7Var[ 9
We
p.
]
then as in the previous example, provided we replace 7 with
is
we have
7. In particular,
(35)
da
so that Z(p)
= p,
v
=
—(5a/^)e, and ap
= a £ a x2 j^(
precision of the information revealed by the price
ap
Once
.
is
=
Using 7
now
= Q(a £ ,a x =
7^/0^,
we conclude
that the
given by
(36)
)
/y2.
again, the precision of public information increases
more than proportionally with the
precision of private information, which only reinforces our results.
Indeed, consider stage
solve (31)
p
=
-7==P
and
The
(32).
+ x*(p).
As
2.
in the previous
difference
Hence, (??)
is
now
is
given by (36).
What may be
mined.
Using
(34), (35),
where
A = $ _1
(37),
=$
1
+ ap
(1
-
now
we have
ax +
p
—
>•
is
p
,
>
as
p
(37)
-P
and x*(p) are uniquely deter
the price function.
a7
+A +
a.
lot..
'
— +00
=
that p must solve
(-^) ^foTx /\/a x~+~ap and z
—00, and F(p)
given by Z(p)
a x + ap
'
=
9
—
—
>
ax +
(Sa/j)e.
equation (14) in the benchmark model. Note that F(p)
as
is
ar
c/b)
follows that the threshold 9*(p)
indeterminate
and
F{p)
It
signal
replaced by
z®-
y/a x
where ap
now the endogenous
'OL,
=$
9*{p)
that
is
example, the thresholds x*(p) and 9*{p)
is
p+A
(38)
a.
This equation
is
analogous to
continuous in p, and F{p)
+00, which implies that a solution always
24
— —00
>
exists.
.
Moreover,
=
**(p)
+a
-^[^-«M--^-p
a +a
a
\ a +a
x
so the solution
is
unique
\
p
for all z if
x
p
av j ^fal <
p
ap /^/a^ >
y/2n. If instead
v2tt, there are
thresholds z and z such that there exist multiple equilibrium prices whenever z G
V
Proposition 7 Suppose
olds x*(p)
and
= — <& -1 (A).
given by (28) and f
9*{p) are always uniquely determined.
functions P{6, e)
The
is
if
and only
if
(z, z)
The equilibrium
thresh-
There are multiple equilibrium price
a1a x < 7 2 j\phx.
results here are reminiscent of the ones in the
benchmark model. In equilibrium, the
we found
price plays the role of an anticipatory signal of the size of the attack. Indeed, as
there, the strategies of the agents are uniquely determined, although the equilibrium price
may
function
not be.
In contrast, in the previous example the price played the role of a
signal for the exogenous fundamental
and not
9.
Here indeterminacy
arises for individual strategies
for the price function.
In both cases endogeneity of public information implies that
with the precision of private information. In both cases
its
precision
this implies that multiplicity survives
with small noise and the common-knowledge outcomes obtain in the limit as a x
given a £
—
>
for
any
.
Risk Neutrality - Fundamental Dividend
4.3
We
increasing
is
modify the asset market and the preferences as
unit of the asset costs
utility
in the Crst period
1
from the portfolio choice
is
follows.
and pays / — p
There
in the
is
no
risk-less
bond. One
second period. The indirect
thus given by
V(kJ,p)=u 1 (w-k)+u 2 ((f-p)k)
where k
utility
is
the amount invested, u\
is
the utility from Crst-period consumption, u 2
from second-period consumption, and
Consider an agent
who
(39)
U
is
is
the
the payoff from attacking.
receives a private signal
x and observes a
price p. His optimal
investment k solves
u l (w-k)=E[(f-p)u'2 ((f-p)k) \x,p}.
/
We
assume that U\(c)
linear relation, ki
=
is
quadratic and
k {E
[/|x,p]
112(c) is linear, in
— p] +
A, for
which case
some constants k >
25
(40)
(40) reduces to
0,
AeM.
a simple
With out any
we normalize A
loss of generality,
=
we
0. Finally,
=
/
let
=
f(9)
That
9.
the return of
is,
the asset depends only on the exogenous fundamental.
The
analysis here
is
similar to that in the Crst example.
k
=
The optimal
individual
demand
for the asset is
We
k {E
/
[
|
x,p
- p} =
]
k {E
[
9
x,p
|
]
- p}
.
conjecture
B[d\x,p]=5x + (l-6)p
some
for
K(9,p)
6
5
=
k6
(0, 1)
to be determined. It follows that k
— p)
(9
.
In equilibrium,
P
By
p
implication,
Z(p)
=p
ensured
is
is
for
It
)
a x high enough. To complete
a
—
¥
oo.
=
see this, let
5
e
Using these
results,
'
a = ax j
(a e K
2
)
—
5)
<
sufficiently
\
— 5) —
as either
>
is
V
is
always uniquely determined.
and only
if
ax
is
a and
5,
in this
is,
example
with 5
—
*
a =
as
as
S
5 /(1
a —
and
>
—
5
5).
—
>
1
r2
{Ky/a £ )
/ Oirr
\ / LXn-
Kyfire )y/5(l-8).
ax —
ax
or
>
—
>
oo then implies that, given a £
therefore the equilibrium in unique
ae <
sufficient for multiplicity.
Proposition 8 Suppose
is
(
1/4 necessarily and therefore
high
.
and rewrite the above
=
On
either sufficiently small or sufficiently high.
5 (1
a E That
an increasing function of a u and a decreasing
K 1x2 CXr
Otrp.
we have that av j\fotx < v2tt and
is
.
we Old
=
fact that 5(1
is
ax + ae 52 K 2
+ ap
(0, 1) as
ap
The
therefore
the analysis, note that
Obviously, this gives a monotonic relation between
as
— p) and
bounded above by K 2 a e and therefore we immediately have that unique-
The above uniquely determines
.
k8(x
Hence, the equilibrium price
= P (M = *_-L e
ol.
a £ To
=
remains to pin down 8 and the function Q.
d
function of
e.
k(x,p)
a public signal about 9 with precision k 2 5
= — ^e.
and v
Note that ap
ness
is
K=
=
Sn/K,
We
if
and only
the other hand, for given
2
is
if
ax
a x we have
,
whereas
ae
The equilibrium price function
P
sufficient for uniqueness,
conclude:
given by (39) and f
=
9.
There are multiple equilibrium thresholds x*(p) and 8*(p)
either sufficiently small or sufficiently high relative to
26
,
a£
.
if
$
Thus, the Morris-Shin limit result
we have examined, the
is
preserved in this example. Like in
4.4
when a x
is
is
not strong enough to restore
sufficiently small.
Risk Neutrality — Speculative Dividend
We modify the previous
is
previous cases
precision of public information increases with the precision of private
information. Unlike the previous cases, however, this effect
multiplicity
all
example by
letting the return of the asset
the aggregate size of the attack occurring in the second stage.
to the second example. In particular,
but the price function in stage
1
we
will
show that the
be /
The
=—
_1
analysis
(A),
is
where
now
A
similar
strategies in stage 2 are unique
can be indeterminate.
Let x*(p) denote the threshold agents use in stage 2 in deciding whether to attack. In
equilibrium,
so that the asset return
k
= k{E[
=
/
is
f
y/a^[9
x,p
|
]
—
x*
(p)].
The demand
-p] = k{^/c^E[
9
\
for the asset is thus
x,p]-p-
y/a^x*(p)}
.
Let
+ x*(p)
P= —p=P
JOL*
and note
that, for every p, the
above deQies a unique
k
where k
=
Ky/a^.
We now
follows that
We
can thus write the demand as
= H{E[9 x,p]-p}
|
conjecture
E[9
It
p.
K = k5(9 —p)
|
x,p]
=6x +
(l
-5)p.
and therefore
p
= 9-±s.
ko
Hence, the observation of p
9
with precision ap
=
2
equivalent to the observation of p, which
is
2
K 5 ae
E[0
It
.
|
follows that
x,p]
=E[0
|
x,p]
27
=8x +
(l-8)p,
is
a public signal for
where
This
is
ct x
=
r
o
ax
-^3oa x + aE b k
=
a x + ap
the same as in the previous example, with k replacing k. Using k
=
n^/a^,
is
proportional to
we
infer
i
5
=
1
so that 5
ax
is
like in
,
decreasing in
a e but independent
increasing in both
and
is
We
conclude:
Proposition 9 Suppose
and
9*
functions P(6,e)
Hence, the
Final
5
of
'
a x This means
~2
c-2
b
ae
a £ and ax The rest
.
V
,
22
and only
if
o~ x
and/or a £
common- knowledge outcomes
now
ap
given by
—
of the analysis
given by (39) and f
is
is
N
is
= -$ _1
are always uniquely determined.
(p)
if
k
that
.
the benchmark model. Indeed, the critical ratio
av
olds x* (p)
+ a s 5 2 K?
similar to the second example.
(
The equilibrium
4).
J
thresh-
There are multiple equilibrium price
are sufficiently small.
are once again obtained as
a x —>
0.
Remarks
Building on Morris and Shin (1998) this paper introduced instruments that endogenized
the sources of public information in models where coordination
public information by either:
(i)
is
important.
a noisy signal of aggregate activity; or
price that reveals information in equilibrium.
cases
is
An
that the precision of public information
(ii)
We
modeled
a Qiancial asset's
important feature of the equilibrium in
is
endogenous and
rises
all
with the precision
of private information.
We
showed that
in all but
one of the
six
models considered
this effect is strong
enough
to reverse the limiting uniqueness result obtained with exogenous public information. Thus,
typically,
with endogenous public information multiplicity
serve fundamentals with small
if
idiosyncratic noise
We
is
enough idiosyncratic
is
ensured
when
individuals ob-
noise. Conversely, uniqueness
is
ensured
large enough.
view the main theme in Morris-Shin as emphasizing the importance of the details
of the information structures for the multiplicity or uniqueness of equilibria.
contributes to this
same theme by studying the importance
aggregation.
28
This paper
of endogeneous information
References
[1]
Angeletos, George-Marios, Christian Hellwig, and Alessandro Pavan (2003) "Coordination
[2]
and Policy Traps," MIT/UCLA/Northwestern working paper.
Angeletos, George-Marios, Christian Hellwig, and Alessandro Pavan (2004)
"On the Dy-
namics of Information, Coordination, and Regime Change," MIT/UCLA/Northwestern
working paper.
Andrew
on Morris and Shin,"
[3]
Atkeson,
[4]
Calvo, G. (1988), "Servicing the Public Debt:
(2000), "Discussion
Economic Review,
[5]
[6]
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