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DEWEY
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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
CRISES
AND
PRICES
Information Aggregation, Multiplicity
Volatility
George-Marios Angeletos
Ivan Werning
Working Paper 04-43
December 15, 2004
Room
E52-251
50 Memorial Drive
Cambridge,
MA 02142
This paper can be downloaded without c harge from the
Social Science Research Network Paper Collection at
http://ssrn.com/abstract=634881
and
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
FEB
1
2005
LIBRARIES
j
Crises and Prices
Information Aggregation, Multiplicity and Volatility*
Ivan Werning
George-Marios Angeletos
MIT and NBER
MIT,
angeletOmit edu
UTDT
and
iwerningOmit edu
.
First draft:
NBER
.
January 2004. This version: December 2004.
Abstract
Many
argue that crises
- such as currency attacks, bank runs and
described as times of non-fundamental
when endogenous
We
riots
- can be
argue that crises are also times
sources of information are closely monitored and thus an important
part of the phenomena.
volatility
volatility.
We
by introducing a
study the role of endogenous information in generating
financial
market
in
a coordination
game where
agents have
heterogeneous information about the fundamentals. The equilibrium price aggregates
common knowledge. In contrast to the case with exogenous information, we find that uniqueness may not be obtained as a perturbation
from common knowledge: multiplicity is ensured when individuals observe fundamentals with small idiosyncratic noise. Multiplicity may emerge also in the financial price.
When the equilibrium is unique, it becomes more sensitive to non-fundamental shocks
information without restoring
as private noise
JEL
is
reduced.
Codes: D8, E5, F3, Gl.
Keywords: Multiple
rency
crises,
bank
equilibria, coordination, global
games, speculative attacks, cur-
runs, financial crashes, rational expectations.
*For useful comments and suggestions,
we thank Daron Acemoglu, Manuel Amador, V.V. Chari, Christian
and seminar participants at MIT, Stanford, UCLA, UC Berkeley,
Duke, the Minneapolis and Chicago Federal Reserve Banks, the Minnesota Summer Workshop, and the
UTDT Summer Workshop.
Hellwig, Patrick Kehoe, Balazs Szentes,
Introduction
1
a love-hate relationship, economists are at once fascinated
It's
On
tiple equilibria.
and uncomfortable with mul-
the one hand, economic and political crises involve large and abrupt
Com-
changes in outcomes, but often lack obvious comparable changes in fundamentals.
mentators attribute an important role to more or
or 'animal spirits',
and models with multiple
less arbitrary shifts in
'market sentiments'
equilibria formalize these ideas.
On
the other
hand, models with multiple equilibria can also be viewed as incomplete theories that should
some dimension
ultimately be extended along
The
first
view
empirical side,
is
Kaminsky
crises is affected
On the theoretical side,
political
the
significant
amount
of volatility
models featuring multiple equilibria attempt to
Bank
address such non-fundamental volatility.
and
On
documents that the hkelihood of economic
(1999), for example,
by observable fundamentals, but that a
remains unexplained.
crashes, riots
to resolve the indeterminacy.
represented by a large empirical and theoretical literature.
runs, currency attacks, debt crises, financial
regime changes are modeled as a coordination game: attacking a
regime - such as a currency peg or the banking system -
is
worthwhile
if
and only
if
enough
agents are also expected to attack. 1
Morris and Shin (1998, 2000) contribute to the second view by showing that a unique
equilibrium survives in such coordination games
when individuals observe fundamentals with
small enough private noise. 2
striking
The
result
is
most
the original common-knowledge model, which
is
when seen
as a perturbation around
ridden with equilibria. Most importantly,
their contribution highlights the role of the information structure in determining
and char-
acterizing the equilibrium set.
The aim
two
of this paper
distinct forms of
is
to understand the role of information in crises.
non-fundamental
volatility:
the sensitivity of a unique equilibrium to non-fundamental disturbances.
considering endogenous information
Information
is
is
focus on
We
argue that
crucial for these questions.
typically treated exogenously in coordination models, but
dogenous in most situations of
We
the existence of multiple equilibria and
interest.
Financial prices
is
largely en-
and macroeconomic indicators
convey information regarding others actions and their information. During times of
crises
such indicators are monitored intensely and appear to be an important part of the phenomena.
As an example, consider the Argentine 2001-2002
of the peso, sovereign- debt default,
crisis
and suspension
of
crisis,
which included devaluation
bank payments. Leading up to the
throughout 2001, bank deposits and the peso-forward rate deteriorated
steadily.
Such
1
See. for example, Diamond and Dybvig (1983), Obstfeld (1986, 1996), Velasco (1996), Calvo (1988),
Cooper and John (1988), Cole and Kehoe (1996).
2
See Carlson and van Damme (1993) for an earlier contribution and Morris and Shin (2001) for a review.
variables were widely reported
by people making
crucial
by news media and investor reports, and were
closely
watched
economic decisions.
These observations lead us to introduce endogenous sources of public information
in a
coordination game. In our baseline model, individuals observe their private signals and the
price of a financial asset.
The
rational-expectations equilibrium price aggregates disperse
by introducing enough noise
private information, but avoids perfect revelation
in the aggre-
gation process, as in Grossman and Stiglitz (1976). Thus, none of our results are driven by
restoring
common-knowledge. 3
The main
insight to
emerge
is
that the precision of the endogenous public information
increases with the precision of the exogenous private information.
is
more
precise, individuals' asset
result, the equilibrium price reacts
more
demands
more
are
more
sensitively to
When private
information
sensitive to their information.
As a
fundamental variables, thus conveying
precise information.
This result has important implications for the determinacy of equilibria, as a horse-race
between private and public information emerges.
information makes coordination more
tinct information.
An
increase in the precision of private
difficult as individuals rely
more on
their
own
dis-
However, the consequent increase in the precision of endogenous public
information facilitates coordination. This indirect effect typically dominates, reversing the
limiting result:
enough private
multiplicity
is
ensured
when
individuals observe fundamentals with small
noise.
Uniqueness therefore can not be attained as a small perturbation around commonknowledge. To illustrate this point Figure
plicity in the
Multiplicity
1
displays the regions of uniqueness and multi-
space of exogenous levels of public and private noise
is
ensured when either noise
is
vate noise act symmetrically. In contrast, when information
facilitates multiplicity,
but
(<7 E
and a x
,
respectively).
sufficiently small. In this sense, public
less private noise contributes
is
and
pri-
exogenous, less public noise
towards uniqueness.
Moreover,
with endogenous information, equilibria outcomes are continuous with respect to information parameters. In contrast, with exogenous information, the limiting result discussed above
illustrates a
sharp discontinuity.
Interestingly, multiplicity
may
emerge, not only with respect to the probability of regime
change, but also in the asset price.
This occurs when the asset's dividend depends on
the endogenous aggregate activity in the coordination game, which one can interpret also
as a short of investment game. Different equilibrium prices are then sustained
by
different
expectations about the aggregate size of attack/investment, and therefore about the dividend
3
Atkeson (2000) has already pointed out that perfectly revealing asset markets would restore equilibrium
multiplicity.
uniqueness
multiplicity
Figure
1:
a x measures the exogenous
noise in private information
and
<r £
the exogenous
public noise in the aggregation of information.
of the asset, which in turn are possible only thanks to the coordinating role prices play in
the
first place.
In regions where the equilibrium
We
is
unique,
we
are able to perform comparative statics.
find that a reduction in exogenous noise increases the sensitivity of the regime
to non-fundamental disturbances.
noise
may
also increase the
When
the asset's dividend
non-fundamental
is
volatility in the financial price.
results parallel the determinants of equilibrium multiplicity,
outcome
endogenous, a reduction in
we conclude
Since these
that lower noise
increases volatility for both forms of volatility.
We
also
study a separate model motivated by the bank run and riots applications. In this
model there
is
no financial market. However, information
observe a public noisy signal of others actions.
An
is
endogenous because individuals
advantage
is
that this model allows us to
study endogenous information with a minimal modification of the Morris-Shin framework.
The model
shares the
main
insights
and
results for equilibrium multiplicity obtained
with
the financial price model.
This paper contributes to the real-expectations literature by examining the coordinating
role of financial prices
and
its
implications for volatility. In
where the dividend of the asset
is
is
Stiglitz (1981),
exogenous and there are no complementarities in real
vestment choices, the equilibrium price
information structure. This
Grossman and
is
unique and
is less
in-
volatile the lower the noise in the
because in that framework prices merely convey information
about the underlying fundamentals, which helps every individual figure out his best choice
without regard to what other agents are choosing.
In the presence of complementarities,
instead, financial prices serve also a coordinating role: they help agents coordinate their real
investment choices by improving the ability of each agent to predict what other agents do.
And
it is
this coordinating role that explains
emerge also in the
Our work
or uniqueness
is
prices.
why
multiple equilibria and high volatility
may
4
builds on Morris-Shin
and underscores
their general
theme that
multiplicity
Within such a framework, our aim
depends on the information structure.
to understand the effect of endogenous sources of public information such as financial
prices
and other indicators of aggregate
HeUwig and Pavan (2003, 2004)
activity. Angeletos,
also endogenize information, but not through information aggregation as here. Instead, they
consider the signaling effects of policy interventions and the flow of information in a
dynamic
global game.
Related
and allow
focus on
devalue
is
Mukherji, Tsyvinski and Hellwig (2004),
who
consider a currency-crises
financial prices to directly affect the coordination outcome.
how
multiplicity or uniqueness
game
In particular, they
depends on whether the central bank's decision to
triggered by large reserve losses or high interest rates. Tarashev (2003) also en-
is
dogenizes interest rates in a currency-crises model, but does not investigate the implications
for equilibrium multiplicity.
Finally,
we share with Chari and Kehoe
(2003) the motivation in examining the role of
information in generating non-fundmental volatility, but we focus on coordination rather
than herding. Indeed, we bring closer the global-games and social-learning approaches to
crises
by allowing endogenous public
cascades by assuming that no agent
how
coordination
may
signals
is
about others'
activity,
"big". Building again
but avoid informational
on Morris-Shin, we also show
help understand high non-fundamental volatility even with a unique
equilibrium.
The
rest of the
paper
is
organized as follows.
We
describe the basic model in Section 2
and review the exogenous information benchmark. Section 3 incorporates the financial asset
and examines multiplicity of
equilibria. Section 4
examines the determinants of
volatility as
a comparative static along a unique equilibrium. Section 5 studies the model with a direct
signal of aggregate activity. Section 6 concludes.
Barlevy and Veronessi (2004) consider a Grossman-Stiglitz-like economy which admits multiple rationalexpectations equilibria, but the source of multiplicity is very different. In their model, there are no complementarities in real investment choices (the dividend of the asset
not play any coordinating role
is
actually exogenous)
and the price does
our framework. Multiplicity instead originates in the non-linearity of
the inference problem faced by uninformed traders when they interact with informed and less risk-averse
agents.
like in
The Basic Model: Exogenous Information
2
we review the
Before endogenizing public information,
model with exogenous informa-
basic
tion.
Actions and Payoffs. There
indexed by
quo
(oj
=
i
£
[0, 1].
1) or
Each agent
not attack (a 2
payoff from attacking
c
6
(0, 1)
only
if yl
is 1
—
=
c
i
1).
>
is
a status quo and a measure-one continuum of agents,
can choose between two actions, either attack the status
The
if
payoff from not attacking
the status quo
The
parametrizes the cost of attacking.
>
6,
where
A
status quo
U(a ,A,6)
i
R (A, 9)
=
It follows
—c
otherwise, where
abandoned
in turn
if
denotes the regime outcome, with
R=
1 if
Interpretations. In models of self-fulfilling currency
attacks the currency, forcing the central
of foreign reserves or
more
maintain the peg. In models of
that the payoff of agent
a l (R(A,6)-c)
and Shin, 1998), a "regime change" occurs when a
amount
is
and
denotes the mass of agents attacking and 9 represents the exogenous
fundamentals, namely the strength of the status quo.
where
normalized to zero. The
is
abandoned and
is
(1)
A>
9
and
otherwise.
crises (Obstfeld, 1986, 1996;
bank to abandon the peg;
self-fulling
R=
sufficiently large
generally the ability
i is
Morris
mass of speculators
9 then parametrizes the
and willingness
of the central
bank
to
bank runs, on the other hand, a "regime change"
occurs once a sufficiently large number of depositors decide to withdraw their deposits,
forcing the
bank
to suspend its payments; 9 then parametrizes the liquid resources available
5
to the banking system.
Throughout the paper, we focus on
crises as the
applications, however, are also possible.
main
interpretation of the model. Other
For example, one could interpret the model as
an economy with investment complementarities, in which
investment,
A
In short, the key property of the payoff structure
coordination motive:
U (1, A, 9) — U (0, A, 9)
9
€
when
(#, #]
—
=
9
is
(0,1].
common
The
knowledge, both
we
1
represents undertake an
consider
increases with A,
incentive to take one action increases with the
Indeed,
a;
the aggregate level of investment, and 9 the exogenous productivity.
mass
A =
I
is
that
it
of other agents taking the
and
introduces a
meaning that the individual
A=
is
same
action.
an equilibrium whenever
interval (9,6] thus represents the set of "critical fundamentals" for
which agents can coordinate on multiple courses of action under
Information. Following Morris-Shin, we assume
6 is not
common
common
knowledge.
knowledge. In the
Other related interpretations include debt crises and financial crashes (Calvo, 1988; Cole and Kehoe,
Finally, Atkeson (2000) interprets the model as describing riots: the
potential rioters may or may not overwhelm the police force in charge of containing social unrest depending
on whether the number of rioters exceeds the strength of the police force.
1996; Morris and Shin, 2003, 2004).
,
beginning of the game, nature draws 6 from a given distribution, which constitutes the
agents'
common
prior about
where
o~ x
is
a positive scalar
independent of
positive scalar
9.
For simplicity, this prior
9.
uniform over the entire real
line.
and
6
Agent
assumed to be an (improper)
is
^ ~ AA(0, 1)
is
idiosyncratic noise,
Agents also observe an exogenous public signal
~ Af(0, 1)
and v
is
common
z
= 9 + a.v,
The information structure is thus parametrized parsimoniously by a x and a z
a x = o~ 2 and a z =
Equilibrium Analysis.
if
and only
We
if
monotone
focus on
his private signal
is
equilibria, that
is less
than a threshold x*(z). The
and
is
decreasing in
where 9*(z)
is
the unique solution to A(9, z)
of his private signal
<1>
/q^(x*(z)
(v
if
9
9*{z),
—
9))
x *( z )
ct x
=
posterior of the agent about 9
+
oc z
,
where a x
public information.
—
a~
2
and a z
It follows
=
6. It
9*{z)
is
+
Hence, an equilibrium
a~
x ^,
and
where a z
is
,
a
£.
the standard
the precisions of
,
[V^+cZ (e*(z)
is
2
o~ x
< a z ^/2n.
This assumption
is
—
9,
(See
is
which
that,
if
the realization x
then A(9,
if
=
z)
and only
or equivalently
{£)).
(2)
a*
x
a
-
[9
^
it
- ar
+
optimal to attack
<
6*(z)\x,
x*{z)
z]
unique
Appendix
=
c,
if
a
z
and precision
and only
if
x
<
x* (z)
or equivalently
^-z)) =
-
z
c.
(3)
:
simply identified with a solution to
and
and only
equilibria in
Thus suppose
denote, respectively, the precision of private and
that the agent finds
solution to (4) always exist
equivalently
we
2
size of the attack is
^-\9*
Substituting (2) into (3) gives a single equation in 6*(z)
A
0"
both 9 and
follows that regime change occurs
where x*(z) solve the indifference condition Pr
$
if
Normal with mean
2
is,
sufficiently low.'
given a realization z of the public signal, an agent attacks
The
-f-
and public information.
an agent attacks
<
o~~
9
across agents
i.i.d.
noise, distributed independently of
deviations of the two noises; or equivalently by
private
=
then receives a private signal Xj
i
for all z
if
(4).
and only
for a detailed derivation.)
without any serious loss of generality: by letting the
if
a z /^/a x < V2n,
~
We
common
or
conclude that the
prior be uninformative,
bias the results against multiplicity.
7
This is without serious
unique and the information
loss of generality for
is
two reasons.
First,
when the monotone equilibrium
is
exogenous, a standard argument of iterrated deletion of strongly dominated
show that there are no other non-monotone equilibria either. Second, to prove the
common-knowledge outcomes with either exogenous
or endogenous information, it suffices to look at monotone equilibria.
strategies can be used to
existence of multiple equilibria and the convergence to
Figure
2:
ax and a z parametrize the
noise in private
and public information; uniqueness
is
ensured for a x small enough.
equilibrium
is
Proposition
unique
if
unique
and only
if
the private noise
if
ax <
sufficiently small.
illustrated in Figure 2,
is
For any a z
is
ficiently small.
In the limit as a x
unique.
common-knowledge game,
incomplete-information
in
—
>
>
0,
which depicts the regions of
0,
the incomplete-information
which there are multiple
game remains unique and
non-fundamental
volatility
(o~ x
,az )
uniqueness in ensured by letting
the regime outcome becomes independent of
all
the equilibrium is
a^y/2/K.
the equilibrium
to matter, but
is
(Morris-Shin) In the game with exogenous information,
1
and only
This result
if
e.
o~ x
for
which
>
suf-
game approaches the
equilibria; yet, the equilibrium of the
actually asymptotes to a situation where
In other words, not only sunspots cease
- the
volatility of
A
or
R
conditional on 9 -
vanishes.
Corollary 1 In
<
6 and A{9,z) -*
This result
o~ x
the limit as
=
:
The key
behavior:
is
if 6
o~ x
>
intriguing, as
0,
it
—
>
0,
where
there is a unique equilibrium, in which
A
((9,
z)
—
>
1
if
6 = 1 - c.
manifests a sharp discontinuity of the equilibrium set around
a tiny perturbation away from perfect information obtains a unique equilibrium.
intuition
it
is
that disperse private information serves as an anchor for individual
limits the ability to forecast each others' actions
multiple equilibria.
and thereby to coordinate on
Indeed,
when
agents share the same information about the underlying fundamentals,
all
they can perfectly forecast each others' actions in equilibrium and can therefore perfectly
when
coordinate on either everybody or nobody attacking
6
G
(6, 6}
.
But when agents have
heterogeneous information about the fundamentals, each agent faces uncertainty regarding
other agents' beliefs about 9 and their actions. For any given precision of public information,
the higher the precision of the agents' private information, the more heavily agents condition
their actions
on
own
their
private information,
and thus the harder
it is
for other agents to
sufficiently small relative to a., this
anchoring effect of
predict their actions.
When a x
private information
strong enough that the ability to coordinate on multiple courses of
is
action totally brakes
signal
become
down and
is
a unique equilibrium survives. Finally, as
infinitely precise relative to the public signal, in
ax
—
>
0,
the private
which case agents cease to
condition on the public signal and by implication the sensitivity of equilibrium outcomes to
the shock e vanishes.
Endogenous Information
3
The
results reviewed
I:
Financial Prices
above presume that the precision of public information remains
ant while changing the precision of private information. This, however,
when
may
invari-
not be possible
the public information consists of financial prices that endogenously aggregate the
disperse private information in the population.
To capture the
role of prices as a channel of information aggregation,
environment as follows. There are
now two
stages. In stage
1,
we modify the
agents trade a financial asset
whose dividend depends on the underlying exogenous fundamentals and/or the endogenous
outcome
of Stage 2. Stage 2 in turn
is like
section, except for that the public signal
stage
the benchmark regime-change
is
now the
game
of the previous
price that cleared the asset
This framework opens up
new modeling
choices regarding the specification of the asset's
dividend and the preferences over risky payoffs.
We
guide our choices with an eye towards
tractability: to isolate the effects of information aggregation
from any
direct payoff linkages
between the two stages, we assume that preferences are separable between the
portfolio choice
and the second-stage attacking
information structure,
and
market in
1.
we model
Stiglitz (1976, 1980)
stage
and Hellwig
1
decision;
along the
8
first-stage
and to preserve normality
CARA-normal framework
of
of the
Grossman
(1980).
'However, we do allow the payoff of the asset to be depend on the aggregate activity of stage
2.
.
Model Set-up
3.1
Timing, information, and payoffs. The game
an improper uniform over R. Each agent
an exogenous private signal
£; is A/"(0, 1)
In stage
i.i.d.
,
1,
riskless asset
=
9 +- cXx^.
The
distribution of
and independent of
across agents,
with nature drawing 8 from
starts again
has a given endowment of wealth Wi and receives
w-i is
arbitrary
and the noise
9.
agents can invest their wealth either in a risk-less asset or a risky asset.
in infinitely elastic supply,
is
The
the second stage.
/
Xi
i
risky asset,
it
costs
and
in the first stage,
1
on the other hand, costs p
delivers
it
and
in the first stage
The
1
in
delivers
in the second.
The
agent enjoys utility from final-stage consumption and has constant absolute risk
(CARA). The payoff from the
aversion
portfolio choice
c
= w —
is
the coefficient of absolute risk aversion.
pk
+ fk
is
thus
V (c) = — exp(— 7c) /7,
consumption, k denotes investment in the risky
is
an unobserved random variable
K (e)
=
The
net supply of the risky asset
where £
cr s e,
~
jV(0, 1)
where
and 7 >
asset,
is
given by
independent of both
is
the fundamentals and the private noise and a £ parametrizes the exogenous noise in the
The unobserved shock
aggregation process.
'noise traders'
and
its role is
e
can also be interpreted as the demand of
to introduce noise in the information revealed by financial
prices about fundamentals.
In stage
=
(R
1) if
as in the
offs
agents chose whether to attack or not.
>
9
are realized at the
What
end of stage
2.
is
is
status quo
U(a, A,
asset's dividend,
9)
is
abandoned
= a(R (A, 6) —
c)
and the agents' pay-
the specification of the asset's dividend
are interested in two alternatives: the case that
9;
2
9
remains to close the model
economic fundamentals
The the
and the agent's payoff from stage
benchmark model. The regime outcome, the
Asset.
We
2,
and only if A
and the case that /
is
/
is
/.
determined merely by the underlying
a function of the endogenous activity
A
of
the coordination stage.
What we
game
is
wish to capture with the latter case
some short
of
an investment game which
is
environments where the coordination
also determines asset returns. For example,
imagine agents have the option to invest in a new sector (or adopt a new technology),
the dividend of the asset depends on the profitability of this sector, and the
profitable only
if
enough agents
Equilibrium.
asset
is
Because of the CARA-normal
specification, the
independent of wealth. The individual's demand k in stage
9
new
sector
is
invest.
1
demand
for the risky
and action a
in stage
Given the separability of payoff between the two stages, one could reinterpret the model as if the agents
in stage 1 were different than the agents who move in stage 2, provided of course that the latter
observe the price that cleared the asset market in stage 1
who move
,
.
2 are thus functions of (x,p) alone. Since there
the exogenous state
A
Definition 1
(0,
e)~.
We
price,
on the other hand,
is
a function of
thus define
rational- expectations equilibrium is a price function P(9, e), individual strate-
and a(x,p) for investment and
gies k(x,p)
a continuum of agents, the corresponding
is
The equilibrium
aggregates are functions of (9,p).
and A(9,p), such
attacking,
and corresponding aggregates K{Q,p)
that: (i) in stage 1,
k(x,p)
= argmaxE
K(9,p)
=
— p)
V((f
[
I k(x,p)d$
k)
\
x,p
(5)
]
(^—)
(6)
= K(e)
K(9,P(9,e))
(7)
in stage 2,
(ii)
a(x, p)
=
arg
max E
A(0,p)=
The
[
U(a, A(9, p),6)
x,p]
\
(8)
fa(x,p)d$(—-)
interpretation of these conditions
is
(9)
and
straightforward. Condition (5)
(9) requires
that an agent's investment take into account the information contained in their private
information and prices, whereas condition (7) imposes market clearing in the asset market.
Note that
(5)- (7) define
(8)-(9) define a
of notation,
we
a standard rational-expectations equilibrium
standard Perfect Bayesian equilibrium in stage
let
R (9, e) = R (9, A (9, P (9, £)))
a function of the exogenous state
for stage 1,
Finally,
whereas
with some abuse
denote the equilibrium regime outcome as
(9, e)
Exogenous dividend
3.2
We
2.
consider
fundamentals
We
first
9.
the case that the dividend / depends only on the exogenous economic
To
preserve Normality,
we
let
/
=
/
(9)
=
9.
start the characterization of the equilibrium with stage
Thanks to the CARA-Normal
specification, the
E{V(c)\x,p}
The FOC, which
is
expected
1
(the financial market).
utility of the
= V((E[f\x,p}-p)k-lV&r[f\x,p}k 2
both necessary and
sufficient, implies that
10
agent reduces to
)
the optimal
demand
for the
asset
is
k(x,p)
E[f\x,p]-p
=
10
r,
iVax\f\x,p]
where /
=
9.
We
and precision
propose that the posterior of 9 conditional on x,p has mean Sx
a, for
some
and therefore K(8,p)
=
(8a/j)(9
=
market-clearing price as p
9
—
—
p).
=
tion of p
(5a/-y)~
1
aE
is
It
0.
K (e)
=
+
(1
(Sa/j) (x
—
S)p
—
p)
which gives the
,
(Sa/j)~ a E e, or equivalently
= 9-op e
(11)
the standard deviation of the price conditional on
therefore equivalent to the observation of a
is
=
fohows that k(x,p)
In equilibrium, K(9,p)
P(9,e)
where ap
>
G (0,1) and a
8
Normal
9.
The
observa-
signal about 9 with precision
= a~ 2 Moreover, x and p are independent and therefore 9\x,p ~ JV (Sx + (1 — 5)p, a)
£ = & x /a, and a = ^
ap which verifies our initial guess. Solving for a, 8, and ap we get
a = a x (l + a x a E /^y 2 ), 8 = 1/(1 + ax ae /j 2 ), and ap = a e a^/7 2 That is, the precision of the
ap
.
,
-(-
,
,
.
public information revealed
by the price
is
an increasing function of both the precision the
aggregation process and the agents' primitive private information. Equivalently,
ap
= jae al,
(12)
which together with (11) completes the characterization of stage
Next, consider stage 2 (the coordination stage).
stage 2
signal
is
z
= p = P (9, e)
and hence a z
of our earlier analysis in Section
<
x
if
x* (p)
abandoned
if
,
and only
unique
if
if
and
if <J
o~
.o\
< 7
outcomes,
(27r)
-1 / 2
R (9,p)
.
,
9
<
= ap The
the agent's strategy
A (6,p) = $
is
(11), the continuation
is
(a/o^ (x* (p)
a(x,p)
—
6))
,
=
1 if
2
e
the
ax
in
and only
and the regime
is
9*(p); second, the thresholds x* (p) and 9* (p) are given by the
;
if
game
following are thus immediate implications
.
first,
Given
of Section 2, except for the fact that the public
once we replace z with p and a, with ap third, the solution
(4)
and only
2
if
and only
Proposition 2 In
2
2:
the aggregate attack
and only
solution to (2)
if
game
isomorphic to the benchmark
is
1.
< ap ^/2n.
Using then
u x > j 2 (2ir)~ 1 ^ 2
economy with f
(11),
we conclude
is
unique
that the equilibrium
is
.
=
f
There are then multiple
(9),
there are multiple equilibria if
strategies,
and only
if
a(x,p), attacks A(9,p) and regime
for the coordination stage, but the asset demands, k (x,p) and
K (9,p)
,
and the price function, P(9,e), are always uniquely determined.
Note how
this result contrasts
with Proposition
a sufficiently low a x ensures uniqueness; but
when
11
1:
when
public information
the public information
is
is
exogenous,
endogenous as
Figure
With endogenous
3:
ity is therefore
in the present
public information, as
ensured for sufficiently small ax
ax
decreases,
o~ z
also decreases; multiplic-
.
model, a sufficiently low a x ensures multiplicity. This
private information
is
because more precise
now implies more precise public information, indeed at
a rate high enough
that the agents' ability to predict each others' actions increases as their information improves.
Figure 3 illustrates the result in the (a x a.) space: unlike Figure
,
decreases and fast enough that the
economy
2,
as
ax
Uniqueness can therefore not be seen as a small perturbation away from
edge.
As
it
was
earlier illustrated in
az
decreases,
also
necessarily enters the region of multiplicity.
Figure
1,
precision of the endogenous public information
is
when
either
ax
or
ae
is
common
knowl-
small enough, the
sufficiently high that multiple courses of ac-
tion can be sustained in the coordination stage. Moreover, the
common-knowledge outcomes
can actually be recovered as as either type of noise vanishes.
Corollary 2 Consider
o~ x
.
the limit as
ax
—
>
for given
There exists an equilibrium in which R(9,s)
equilibrium in which
R (9, e) —
»
1
whenever 9 €
o~ E
—
,
or the limit as
whenever 9 £
»
(0, 9).
o~ s
{9,6),
—
>
for given
as well as
In every equilibrium, P(9,
e)
—
an
9 for
all (9,e).
3.3
Endogenous dividend
We now
consider the case that the payoff of the asset
coordination stage.
To preserve Normality, we now
thus pays more the lower the size of the attack, which
"attack"
means
refraining from
some short
is
let
determined by the outcome of the
/
we could
of investment.
12
=
f (A)
= —^~
1
(A).
The
asset
interpret as a situation where
We start
again the equilibrium analysis with stage
for
some threshold
s/a^[9
so
—
is /.
x* (p)
Hence,
we have
,
that
= 1 if and only if x < x* (p)
A (8,p) = $ (y^fx* (p) — #]) and therefore / =
(p)].
It
follows that the individual optimal
Substituting /
=
y/a^[9
—
we can rewrite the optimal demand
as k
=
section,
we then propose
=
Sx
K (9,p)
=
E[#|x,p]
— p) and
is
is
for d, a,
= 9-
Normal
and ap gives
(p),
— p)
(13)
/ (^fVax[9\x,p\). Like in the previous
=
5)p and Var[0|x,p]
a.
It
follows that
T^-=e.
signal
(14)
= (5a/^) 2 a e
a = a x + a p Solving
about 9 with precision ap
)
and
.
2.
The
= ja^l-
(15)
threshold 9*(p) solves 9
=
8*(p)
— A (9,p)
equivalently,
;
+ ^-\9*(p)).
on the other hand, solves Pr
$
[9
<
9*(p)\x,p]
(16)
=
(v^T^ (>(P) - 5fe^(p) " sfep))
These equations are the analogues of
substitute (13) and (16) into (17),
*(P)
which case
7^/0^ and
,
x*(p)
threshold x*
=
+ (1 — 5)p, a) with 5 = a x / (q x + a p
=
ap a E a x /^f = a e a x /'j 2 or equivalently
Next, consider stage
in
7
9\x,p~J\f (8x
ap
The
for the asset is again as in (10).
letting
a one-to-one mapping between p and p, the observation of p
equivalent to the observation of a
and therefore
verify later),
therefore market clearing implies
p
Hence, provided that there
—
(1
Normal (which we
+ x*(p)
(E[9\x,p]
+
is
demand
and
x* (p)] into the latter
p
(5a/ 7)) (9
(x, p)
the agents posterior about 9
x*
if
Guessing (and later verifying) that
1.
agents use monotone strategies in stage 2 such that a
-*
(16) implies also a
(2)
we
and
(4) in
c;
=c
unique x*
$_1
C
(p)
.
1
-
c
)
- S=fe?)
we
:
>
(
18 )
Hence, unlike either the benchmark game or
the case of the previous section, the strategy of the agents in the coordination
13
(17)
the benchmark game. However, once
get a unique solution for 9*(p)
(y^
equivalently,
game and the
.
regime outcome are always uniquely determined as functions of the realized price.
To complete the
analysis,
we need
mapping p
the equilibrium price
=
to determine the
mapping between p and
P(9,e). Using (13), the aggregate
demand
p,
for
that
is,
the asset
is
K(9,p)
Since x*
= j{6-^=p-x*(p)).
uniquely determined,
(p) is
k(x,p)). Moreover, for any
K{9,p) —* — oo
p —
as
a solution; that
>
9,
K (9,p)
K(9,p)
is
determined (and similarly for
also uniquely
continuous in p, K(9,p)
-f oo. It follows that
K (9,p)
a market- clearing price p
is,
is
(19)
—
e,
= P (9, e)
—
>
oo as p
—* — oo, and
or equivalently (13), always admits
exists for
any 9 and
e.
However, the
market-clearing price need not always be unique. Note that 9*(p) and x* (p) are decreasing
in p,
which in turn implies the
As a
result, the
demand
asset's dividend,
for the asset
=
/
yfaZ\9
— x*
(p)], is itself
increasing in p.
can be non-monotonic. Indeed, (16) and (19) imply
*»{^}--^-,(.-<0)}
and therefore
K
p)
(9,
a l a x > 7 2 /\/27r.
If
monotonic and there
whenever p ^
(pi
is
globally decreasing in
instead o\(j\
is
if
^fa^jav
<
>
demand
y2ft, or equivalently
for the asset is
non-
such that (13) admits a single solution
(p\,p~2)
(pi p%)
,
mappings between p and
p,
.
Different selections
that
is,
from these
different equilibrium price
P (9, e)
o\a\ < 7 2 (27r)
,
and only
p 2 ) but three solutions whenever p E
,
Proposition 3 In
k (x,p)
if
j /\/2x, the aggregate
a non-empty interval
solutions then sustain different
functions
p
2
-1 / 2
.
K (9,p)
,
the
economy with f
—
f (A), there are multiple
equilibria if
the strategy a (x,p)
,
and
the attack
A (9,p)
In the model of the previous section, the dividend
if
are always uniquely determined.
was a function
of 9 alone
the price played the role of a signal about the exogenous fundamental.
minacy emerged
and only
There are then multiple price functions P(9,e), but the asset demands
for the strategies
and the
size of
As a
and therefore
result, indeter-
the attack in the coordination stage, but
not for the price clearing the asset market. Here, instead, the dividend depends on
A
and
therefore the price plays the role of an anticipatory signal about the endogenous size of the
attack.
A
particular price realization coupled with an agent's private information then pins
down a unique
is
posterior about the size of the attack, in which case the agent's best response
of course unique. This explains
In stage
1,
on the other hand,
why
the strategies in stage 2 are uniquely determined.
different levels of the price are associated
with different
expectations about the size of the attack and therefore about the dividend of the asset: in
14
equilibrium, a higher price signals a higher dividend.
offset
the usual negative payoff
cost of obtaining the asset
When this effect is strong enough,
it
can
- namely that a higher price means a higher
effect of the price
- and may therefore induce the demand
for the asset to increase
with the price. This non-monotonicity of the asset demand introduces the possibility for
multiple market-clearing prices. Finally, the positive effect of the price
is
higher, for a higher
ap
is
when ap
stronger
increases the sensitivity of the agents' actions in stage 2 to the price
and therefore of the dividend to the
price.
why
This explains
multiple equilibrium prices
exist for for small levels of noise.
To
why
further understand
of the agents,
exogenous signal
=
then x
is
9
moment the
optimal to attack
strategy
p and
(p,A)
is
if
and only
if
A>
A
A are not uniquely determined. Instead, for
= (— oo, 1) can be sustained in equilibrium.
In conclusion, small noise
now
— —^~
in the strategy
=
=
a£
The
0).
(A) and therefore an
p. Clearly,
the agent then finds
< $(—p), which means
or equivalently x
9,
(a x
1
p
is
by observing
uniquely determined as a function of x and
game
no-noise
and the equilibrium price
agent learns perfectly 9 by observing x and
it
and not
multiplicity emerges in the price
helps to consider for a
it
that his
However, the equilibrium values of
p.
every 9 £
(9, 9],
both
(p,
=
A)
(oo, 0)
and
ensures multiplicity in both the asset price and the co-
ordination outcome: different equilibrium price functions result to different realizations of
the price and the size of the attack for the same 9 and e and therefore to different regime
outcomes as
well.
What
is
more, the no-noise (or common-knowledge) outcomes are once
again obtained as the noise vanishes.
Corollary 3 Consider the
ax
.
There
well as
3.4
is
limit as
o~ x
—
for given
>
an equilibrium in which R(9,
e)
—
1
an equilibrium in which R(9,e)
>
—*
1
<x £
or the limit as
,
and P(9,
and P(9,e)
e)
—
>
— — oo
>
ae
—
>
whenever 9 6
-co whenever
9
£
for given
(9, 9),
as
(9,9).
Discussion
Although the property that the precision of endogenous public information increases with
the precision of exogenous private information
effect is
and whether
it
is
can overturn the direct
likely to
be very robust, how strong
effect of private
on the details of the channel through which private information
result that multiplicity
is
ensured for small a x
relies
information
is
aggregated. Indeed, the
on the precision of public information
increasing at a rate higher than the square root of the precision of public information.
was the case
in the
two specifications we consider above
need not be true in general.
We now
this
may depend
(as evident in (12)
and
That
(15))
but
provide an example which illustrates this possibility.
15
To
this aim,
we modify
stage
The agent
as follows.
1
is
assumed to be
from the portfolio choice
utility
>
When f —
(i)
9, the
rium regime outcome
high relative to a £
(ii)
When f
=
c
=
w-pk-^k + fk
2
(20)
We
parametrizes the liquidity /transaction cost.
R
V
is
given by (20).
equilibrium price function
is
consider again two
and endogenous.
cases for the dividend, exogenous
Proposition 4 Suppose
indirect
thus given by
is
V(c)
where the scalar k
risk neutral
The
and face a quadratic liquidity/transaction cost for investing in the risky asset.
unique
if
and only
if
ax
P
is
is
either sufficiently small or sufficiently
always unique, whereas the equilib-
.
= — $ -1 (v4),
sufficiently small.
there are multiple equilibria if
and only
Multiplicity then emerges in both the regime
if
and/or a e are
o~ x
outcome
R
and the price
function P.
Here, unlike the earlier
CARA-normal
cases, the increase in the precision of public infor-
mation generated by an increase in the precision of private information
enough to
offset the direct effect of the private
not always strong
of the asset
because the slope of individual demand on the expected dividend
exogenous. This
is
dependent of the
risk in the dividend,
(less risk).
When
is
is in-
which implies that the precision of public information
increase less with private information than
information
is
when the payoff
information
when the
slope itself increases with
the dividend depends only on
9,
more
precise
the anchoring effect of private
information dominates, thus ensuring uniqueness for a x small enough.
When, however, the dividend
of the asset
depends on the
size of the attack, the fact that
the sensitivity of the size of the attack and therefore of the dividend
itself
to the fundamental
increases as the agents' information improves compensates for the lack of such a sensitivity
in the
demand
for the asset.
As a
result, coordination
once again becomes easier as private
information becomes more precise and multiplicity reemerges for small enough noise.
These examples highlight that the
ter.
They
details of the process of information aggregation
also suggest that the coordinating effect of prices
agents are risk averse and
when the dividend
coordination outcome. In Section
5,
we
or
o~ z
endogenous signal
are small enough. In
is
what
likely to
of the traded assets
will consider
mat-
be stronger when
depend
directly
on the
an alternative channel of information
aggregation, namely a direct signal about others' activity.
effect of this
is
We will find that the coordinating
once again strong to deliver multiplicity when either
follows,
we continue
16
to focus
on the more
o~ x
interesting case
that the precision of public information increases fast enough with the precision of private
information.
Noise and Volatility
4
We now investigate the role of the information structure for
the volatility of the regime outcome
is,
R
non-fundamental
(or the size of the attack
volatility, that
A) conditional on
9.
In general, there are two sources of non-fundamental volatility: volatility generated
the shock e
when the equilibrium
when
variables (sunspots)
is
by
unique; and volatility generated by payoff-irrelevant
there are multiple equilibria and agents use these sunspots to
coordinate their behavior.
With exogenous
is
information, multiplicity disappears
with small idiosyncratic noise.
tals
What
small enough.
is
more, as
By
—
o~x
—
o~ x ,
>
0.
On
Hence,
e.
as in this case the
common-knowledge outcomes
o~ x
and therefore introducing more sunspot
is
maximized when
or
is
What
is
when a x
volatility vanishes as
maximized when a z —*
is
o~ e
can increase
volatility.
for given
are obtained.
Corollaries 2
(9, 9]
as
,
quite different.
is
by ensuring multiplicity
volatility
either noise vanishes:
regime can either collapse or survive for any given 9 €
sunspot.
volatility
information, the impact of private noise on volatility
A sufficiently large reduction in either
sunspot volatility
agents observe the fundamen-
no sunspot
non-fundamental
all
the other hand, non-fundamental volatility
With endogenous
is
the size of the attack and by implication the
0,
>
regime outcome become independent of
ax
when
implication, there
and 3 indeed imply that
—
ax
a£
or
>
—
>
0,
the
purely as a function of the
more, sunspot volatility can show up in prices as
well:
when the dividend
endogenous, the equilibrium price can be arbitrarily low or arbitrarily high for any given
9e
{9,9}.
The property
that, with
endogenous information,
less noise
may
increase volatility does
not rely on the existence of multiple equilibria. As we show next, when the equilibrium
unique, a reduction in either
o~ x
or
o~ s
may
increase the sensitivity of the regime
the asset price to the exogenous shock e and
may
is
outcome and
therefore result to higher non-fundamental
volatility.
4.1
Regime
Volatility
In equilibrium, the regime
depends on
9,
since
p
is
abandoned (R
= P (6,e) To
of the exogenous variables 9
.
and
e,
=
1) if
and only
if
9
<
9*
[p)
,
but p in turn
express the equilibrium regime outcome as a function
note that, as long as the equilibrium
17
is
unique, 9* (p)
is
continuously decreasing in
R (9, e)
=
1 if
and only
if
9
p and
P (9, e)
is
9 (s)
where 9
(e) is
<
,
Consider
we
3.2,
follows that
(e) to e.
the case the dividend
first
where
is
exogenous (/
ip
=
+
y/l
1/
therefore, for
respect to
let £o
:
any
s\
e.
By
(^a^)^
1
(1
-
c). It
follows that
39
<t>{§-\9))
_
(9)).
Using the results of Section
any given
(22)
a reduction in a £ or a x increases the slope of 9
9,
(e)
implication, 9 satisfies a single-crossing property with respect to
<
such that e\
e^
non-fundamental
(2i)
2
-ya £ a x
=
be the unique value of e for which d9/da e
and
similarly 3|^(e 2 )
A
/
= $(ip + -2— £ }
de
and
=
get
e( £ )
a£
It
can thus examine the non-fundamental volatility of the regime outcome by examining
the sensitivity of 9
for
9.
the unique solution to
= 6*(P(9,e)).
e
We
continuously increasing in
—
9(ei)\/g e
<
0.
Eq
<
£2,
we
still
or equivalently
have that d\9(s2)
—
89/da x
9(e{)\/a £
<
with
a x or
=
0;
and
In this sense, a reduction in either type of noise increases
volatility.
similar result holds
(22) continue to hold
if
when
we
the dividend
replace
7 with 7
is
endogenous (/
= ja x We
.
=
/
{A)).
Indeed, (21) and
conclude that
Proposition 5 Less noise implies more non-fundamental
volatility
rium
increases the slope of 9 (e) with
unique: for any given
is
respect to
Our
9,
a reduction in
a e or a x
even when the equilib-
e.
earlier results regarding equilibrium multiplicity
reincarnation of the above result.
not only because the outcome
is
When
the noise
is
can thus be viewed as an extreme
sufficiently small, volatility
can be high,
very sensitive to the exogenous noise, but also because the
outcome can depend on arbitrary sunspots.
4.2
We
Price Volatility
next examine the comparative statics of the volatility of prices.
implications for volatility that do not derive from multiplicity,
that the equilibrium
is
unique or that there are no sunspots.
18
we
first
To emphasize the
focus again on case
When
the dividend
instead the dividend
=
and ap
7cr £ 0"^.
is
is
=
exogenous, we have /
=
endogenous, we have /
In either case,
we can
p
9,
-1
=
[A)
"3?
/ — ap e, and ap
=
(9
—
x*)/a x
write the equilibrium price as the
,
— ^a z a\. When
p=
sum
f
"—
(o-
p /a x ) e,
of the dividend
and the supply shock appropriately weighted:
V
Keeping / constant, the
or
.
the dividend
/ -
(l<y e o\) e
(23)
volatility of the price clearly decreases
a 6 But what about the
When
=
is
with a reduction in either a x
volatility of the dividend itself?
exogenous, /
is
independent of
exactly like in Grossman-Stiglitz: a reduction in either
o~ x
e.
The impact
or
a e implies lower
of noise
is
then
volatility in
equilibrium prices.
When
instead the dividend
is
endogenous, / depends on
e,
because agents' actions in
the second stage depend on the price, which in turn depends on the shock
same reason that a reduction
essentially the
outcome to the shock
e,
a reduction in noise increases the sensitivity of / on
the overall impact of noise on the volatility of prices
less noise
now ambiguous: on
is
The second
which depicts
low a x (In
.
o~ x
indeed dominates in some cases.
,
for different values of e
sensitivity of
P (9, e)
similar picture emerges
is
to e
if
the dividend
is
we
is
non-monotonic:
In the analysis so far,
is
illustrated in Figure ??,
solid line
corresponds to a
it is
highest
.
when a x
We
is
The
effect
either high
thus conclude:
exogenous, the volatility of the price conditional on 9
we have assumed
a x or
II:
o~ E
.
But when
may
the dividend depends
on
increase price volatility.
Observable Actions
that agents observe a signal generated from a
than the coordination game.
generated within the coordination game.
let
The
consider the impact of a e
Endogenous Information
market and
9.
high enough that the equilibrium in unique.)
the coordination outcome, a reduction in either noise
different stage
itself.
An example
and a given
necessarily decreases with a reduction in either
is
result,
,
however, a x
Proposition 6 When
5
As a
the one hand,
the dotted one to an intermediate a x and the dashed one to a relatively
all cases,
a x on the
A
effect
P (9, e)
relatively high
or low.
e.
reduces the volatility of the price for any given volatility of the dividend; on the
other hand, less noise increases the volatility of the dividend
of
Moreover, for
e.
in noise increases the sensitivity of the regime
We now
examine the case that the information
In particular,
we assume away the
financial
the agents observe a noisy public signal about the aggregate activity of other
agents in the coordination game.
19
We first
consider a model where the signal
this case, our equilibrium concept
game
expectations and
dynamic variant
about others' contemporaneous actions. In
novel and unavoidably at the crossroads of rational-
later
show that the same
of this model, in which the population
who have only
movers,
is
We
theory.
is
is
can be obtained in a
and
private information about the fundamentals,
can also observe a public signal about the early movers'
is
results
divided into two groups: 'early'
'late'
movers,
who
The equilibrium concept
activity.
then standard game-theoretic.
Model set-up
5.1
There
is
no asset market and,
except, for the
the benchmark model analyzed in Section
—9+
o~ x £,
s:[0,l]xR-^l and
e is noise,
=
s
~ jV(0, 1)
10
of standard deviations
is
identical to
after observing
The
is
s(A,e)
independent of 9 and
The exogenous information
.
game
signal about the size of the attack.
the information structure and obtain closed-form solution,
and
signal, the
move simultaneously
whereas the public signal
y
where
All agents
and a public
private signals about the fundamentals
private signals are again x
endogenous public
2.
structure
is
£.
we
To preserve Normality
let
s(A,e)
= §~
l
(A)
,
+
of
a£ e
then parameterized by the pair
(<x x ,cr £ ).
Since the information of each agent includes a signal about other agents' actions, the
equilibrium concept
we
define
is
a hybrid of a perfect Bayesian equilibrium and a rational-
expectations equilibrium.
Definition 2
A
rational- expectations equilibrium co?isists of an endogenous signaly
an individual attack strategy a{x,y), and an aggregate attack A[9,y),
a(x,y)
=
max E
arg
[
U(a,R(9,y))
\
x,
y
=
Y(9,e),
that satisfy:
(24)
}
a€[0,l]
for
all
(9,e,x,y) €
M4
,
A(6,y)
=
y
=
fa(x,y)d$(—)
(25)
s(A(9,y),e)
where R{9,y)
=
1 if
A(9,y)
(26)
>
9 and R{9,y)
=
otherwise.
This convenient specification was introduced by Dasgupta (2001) in a different set-up.
20
Condition (24) means that a(x, y)
means that A(9,
(25)
introduced by (26)
y) is
is
the optimal strategy for the agent, whereas condition
y.
Equilibrium analysis
5.2
focus again on
monotone
and the status quo
identifies
with a
is
(y/a^(x*(y)
—
and only
if
6)}
.
It follows
=
9,
and only
if
9
if
and
triplet of functions x*,9*,
9*{y) solves A(9, y)
which an agent attacks
equilibria, in
abandoned
Given that an agent attacks
$
fixed-point relation
the rational-expectations feature of our context: the signal y must be
is
generated by individual actions, which in turn are contingent on
We
The
simply the aggregate across agents.
9*(y).
A
if
and only
<
that the regime
is
x*(y), the aggregate attack
abandoned
if
and only
if
= 8*(y) +
^-
<
x*(y)
is
thus
9
<
is
A{9, y)
9*(y),
=
where
1
(9*(y)).
(27)
Condition (26), on the other hand, implies that the signal must satisfy y
=
y/a^ [x*(y)
—
9]-
or equivalently
-a x y = 9-
x*{y)
Note that
(28)
is
a mapping between y and z
y(z)
We will
Now
x
or equivalently
x*(y)
a E e,
if
monotone equilibrium
Y.
x
if
<
later
show that y(z)
is
all
z
9
aE e.
— ax ae e.
(28)
Define the correspondence
= {yeR\x*(y)-ax y = z}.
non-empty and examine when
take any function Y(z) that
that Y(z) G y(z) for
=
<rx
- and
is
let
(29)
it is
single- or multi- valued.
a selection from this correspondence - that
is,
such
e)
=
Y(9 — ax a E e). Any such
Y{9, e)
is
then equivalent to the observation of z
Y(9,
selection preserves
normality of the information structure.
Indeed, the observation of y
9
=
— o x o-s = Z (y) where Z (y) =
,
Normal public
signal
x*(y)
— a x y.
about 9 with precision
az
The
Therefore,
az = a x a E
—
ax a£
precision of endogenous public information
is
,
it is
as
if
=
the agents observe a
or equivalently
(30)
.
thus once again increasing in the precision
of exogenous private information.
For the individual agent in turn to find
it
optimal to attack
21
if
and only
if
x
<
x* (y),
it
must be that x*
(y) solves
Z (y) =
— a x y and
x*(y)
9*(y)
which together with
^x*
-
(v^T^I (<r(y)
$
Using
the indifference condition Pr (9
(y)
-
<
9*
-^-Z (y))) =
substituting x* (y) from (27),
= $ (-^r-y +
\a x + a z
J^hr*\ a + a
x
A (9, y)
we
1
(i
=
c,
or equivalently
c.
(31)
get
-
c))
(32)
,
J
z
unique 6*(y) and x*
(27) determines
agents a (x, y) and the aggregate attack
(y) \x,y)
(y).
Hence, the strategy of the
are uniquely determined.
We finally need to examine the equilibrium correspondence y(z). Recall that
set of solutions to x*(y) — x y = z. Using (27) and (32), this reduces to
by the
= ^(-^—y + A)+-
F(y)
=
where A
—
as y
this is given
o~
\Ja x j (a x
a 2 )$
-f
+oo, and F(y)
>
—
_1
—
(l
+co
c).
Note that F{y)
—
as y
=(--^—y + A\=z,
l
is
continuous in
-co, which ensures that
>
(33)
and F(y)
y,
3^(z) is
— -co
»
non-empty and
therefore that an equilibrium always exist. Next, note that
sign {F'(y)}
and therefore F(y)
is
=
-sign
globally monotonic
single valued. If instead
a z j sfa x >
~
'
takes three values whenever z
£
if
I 1
- -^=<p
and only
y/2n, there
(z, z)
.
is
if
(^^y + Aj
>
a z /\fa^ < v 27r,
in
which case y(z)
is
a non-empty interval (z,z) such that y(z)
Different selections then sustain different equilibrium
signal Y.
Using a z
noise
is
= a £ a x we
,
conclude that multiple equihbria survive as long as either source of
sufficiently small.
A monotone rational- expectations equilibrium exists for all (cr x ,a £
and only if o\o x > l/\/27r. // o\o~ x < l/\/27r, the equilibrium strategy a
Proposition 7
is
unique
if
aggregate attack
Interestingly,
)
A
(o~ x
attack
if
when
0), in
and only
and
remain unique, but there are multiple signal functions Y.
multiplicity arises,
it is
with respect to aggregate outcomes but not
with respect to individual behavior. To understand this
— aE =
and
which case x
if
x
<
3>(y),
=
9 and y
=
^
1
result, consider
the no-noise limit
(A). Clearly, the agent finds
it
optimal to
which uniquely determines the equilibrium strategy a(x,
the agent. However, for every 9 G {9,9}, both (y,A)
22
—
(— oo,0) and (y,A)
—
(+oo,
y) for
1)
can
be sustained as
equilibria.
When a x
remains: the behavior of the agent
can be multiple equilibrium value
sense, the multiplicity here
is
and
is
are non-zero, the
o~ £
same nature
of indeterminacy
uniquely determined for any given x and
for y
and
A
we encountered
similar to the one
y,
any given realization of 9 and
for
in Section 3.3,
but there
e.
when
In this
agents
traded a financial asset whose dividend depended on the size of the attack.
Finally, the
common-knowledge outcomes
are once again obtained as either source of
information becomes infinitely precise, and therefore sunspot volatility
On
the noise vanishes.
is
the other, the Morris-Shin limit can be obtained
public signal becomes infinitely imprecise, in which case
it is
as
if
maximized when
if
the endogenous
the endogenous signal were
not available.
Corollary 4
(i)
in which R(9,e)
whenever 9 G
(ii)
<
or
>
(9,9], as well as
—
o~ e
>
0.
There exists an equilibrium,
an equilibrium in which R(6,e)
—
>
1
(9, 9].
9
R (9, e)
and
—
—>
>
oo.
There
whenever
9
is
>
a unique equilibrium in which
9,
where
R (9, e) —
>
1
9 = 1 — c.
Non-simultaneous signal
5.3
The
whenever 9 G
>
Consider the limit as a £
whenever 9
—
Consider the limit as either a x
—
analysis above has
assumed that agents can condition
indicator of contemporaneous aggregate behavior.
their decision to attack
We now
show that our
results
on a noisy
extend to
a simple dynamic model in which no agent has information about contemporaneous actions
of other agents
and therefore a standard game-theoretic equilibrium concept (namely PBE)
can be used.
The population
first,
is
on the basis of
divided into two groups, 'early' and
'late'
their private information alone. Late agents
agents. Early agents
move
move second, on the
basis
of their private 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
ji
G
(0, 1)
denote the fraction of early agents, Ai the aggregate activity of early
and A2 the aggregate
agents,
and observed only by
activity of late agents.
late agents
is
The
y^Q-^AJ+e,
where
e
~
A/"(0, o~l) is
signal generated
by early agents
given by
independent of 9 and
£.
(34)
Early agents can condition their actions
only on their private information, whereas late agents can condition their actions also on
23
y\.
Finally, the regime changes
We
if
and only
+
fiAx
if
agents not, a monotone equilibrium
9*
R —
:
9.
>
can condition their behavior on
R
a scalar x\ 6
is
and a pair
such that: an early agent attacks
(0, 1)
<
if
abandoned
if
is
if
and only
follows that y x
of yx
is
=
if
x
_1
<1>
and the regime
x\(y\)\
[Ax (9))+e
=
—
[x\
yfa^
is
of functions
and only
In such an equilibrium, the aggregate attack of early agents
attacks
It
>
ji)A 2
look for perfect Bayesian equilibria in which the strategy of the agents in monotonic
in their private information. Since late agents
and
—
(1
if
<
x
and only
A\
(9)
if
9
x2
:
K— R
a late agent
x\;
=$
but early
yx
<
0*(y 1 ).
(s/q^
[x±
—
#])
.
9]+e. Hence, in equihbrium, the observation
equivalent to the observation of
1
a,
which
=
az
a public signal with precision
is
cr £
a x ),
like in
az =
ct E
ax
(equivalently, with standard deviation
the simultaneous-signal model.
Since yx and z have the
same informational
content,
we can
equivalently express the
strategy of a late agent as a function of z instead of yx and replace the functions x 2 (yx)
and
$
8*(yx) with
[s/a^
pi)A 2
[x 2 (z)
(9, z)
.
A(9*(z),z)
It
=
—
x 2 (z) and
6])
The aggregate
9*{z).
and the
overall attack
attack of late agents
from both groups
follows that the regime changes
if
and only
is
A
<
is
(9, z)
if
9
-
6*(z)\)
9*(z),
then
=
A 2 (9,z) =
+ (1 —
\iAx (9)
where 9*(z) solves
9*(z), or equivalently
-
M $ (Vc£[*J
9*(z)})
+
(1
-
/.)$
(yfe[z*2 (z)
=
9*{z).
Next, consider the optimal behavior of the agents. Since the realization of z
to late agents, their decision problem
solves
Pr
[9
<
9*(z)\x 2 (z),
$
where
and
5
=
c,
is
known
benchmark model: the threshold x 2 (z)
or equivalently
{s/^(5x*2 (z)
and a
in the
+ (l-S)z-
= a x + a..
9*(z)))
=l-c,
(36)
Early agents, on the other hand, do not observe z
therefore face a double forecast problem: they are uncertain about both the fundamental
and the
Pr
— a x /(a x + a~)
z]
is like
(35)
[9
<
signal
upon which
9*(y)\xl\
=
c,
late agents will condition their behavior.
The threshold
x* solves
or equivalently
f*
{^Tx [x\ -
9*(z)})
^l<t> (V5T
24
[x{
- A)
dz
= \-
c,
(37)
=
where a^
A
a x a e /(l
+ ae u
).
monotone equilibrium
x*2 (z)
=
+ -Lsr
9*(z)
We
therefore a joint solution to (35)-(37).
is
dimensionality of the system by solving (35) for x^(z)
1
+ -JL- {8\z) - $ (,/S [x* -
(V(z)
Substituting the above into (36) and using 5
=
r(r(z),x*)
can reduce the
:
a x /(a x
+ az
)
and a
6T
(*)])})
.
= a x + a z we
obtain:
,
= 5 (z),
(38)
where
r(<9,
xO =
+ Q z /a x $ _1
—%e + *-
2
+ -r— {e-§ (^
e
f
(z)
=
yjl
rium
is
a joint solution of (37) and (38) for a threshold x| G
g
(1
—
—
c)
(a z /y/a^)
z,
and a,
[
x*
-
0])}
= Q £ a x We conclude that an equilibR and a function 9* R — (0, 1).
.
»
:
Let C denote the set of piecewise continuous real functions with range a subset of
9*
For any given function
the other hand, (38)
(ftu
aE
,
fi)
.
may admit
[0, 1].
unique x\ G R. For given x\ G R, on
a unique or multiple solutions in 9* G C, depending on
Different solutions to (38) for given x\ represent different continuation equilibria
game between
for the
C, (37) always defines a
G
late agents defined
question of interest, however,
In the appendix
we show
the fixed point of (37)-(38)
solutions for every x\
is
is
by a
fixed strategy for the early agents.
The
the determinacy of equilibrium in the entire game.
that,
when
(38) admits a
also unique.
G R, we prove that
On
unique solution for every x\ G R,
the other hand, when (38) admits multiple
(37)-(38) also admits multiple fixed points. This
provides us with the following sufficient (but not necessary) conditions for uniqueness and
multiplicity:
Proposition 8
(i)
There
exists a
unique equilibrium
i
ai<7 x
(ii)
u To
on
x, z
There exist multiple equilibria
see this, note that z
is
=
8
1
,
,
(1
—
u)
if
— ax e = x —
distributed normal with precision
>
if
£
a.\
— a x e. so that z\x ~ TV (0, a\ +
= a x a e /{\ + a e ).
25
cr
x oV)
.
That
is.
conditional
<
For any a E and a x such that o\o x
1/\/2k, multiplicity in ensured for
game
In this sense, the multiplicity result of the simultaneous-move
fraction of informed (late) agents
—
vanishes as u
and therefore
»
is
it
<
more, for any u
can be shown that the equilibria of the simultaneousequilibria of this
multiple equilibria exist as long as
1,
a x and any
the other hand, for any
cr £
,
uniqueness
intuitive, since in this case the role of the
We
we
conclude that the insights
survives as long as the
Indeed, the dependence of (38) on x\
high enough.
model can be approximated by
signal
low enough.
fi
is
dynamic model as n
a £ and ax
—
>
0.
What is
On
are sufficiently low.
ensured by taking
\i
—
1,
which
is
also
informed agents vanishes.
derived in the simultaneous-signal model extend to
the present framework and do not hinge on the fixed-point nature of the hybrid equilibrium
concept
we used
Remarks
Final
6
there.
Building on Morris and Shin (1998, 2000) this paper introduced instruments that endogenized the sources of public information in models where coordination
modeled public information by
equilibrium, or
equilibrium in
(i)
cases
is
is
important.
We
a financial asset's price that reveals information in
An
a direct noisy signal of aggregate activity.
(ii)
all
either
important feature of the
that the precision of public information
endogenous and
is
rises
with the precision of private information.
We
result
showed that
this effect is typically strong
obtained with exogenous public information:
multiplicity
is
ensured
fundamentals or the
uniqueness
We
is
enough to reverse the limiting uniqueness
also
is
when
common
ensured
if
showed that
with endogenous public information,
either the idiosyncratic noise in the individuals' observation of
noise in the aggregation process
is
small enough; conversely,
large enough.
the noise
is
less noise
may have a
unique: a reduction in either source of noise
destabilizing effect even
may
when the
outcome and the price to exogenous shocks, thus leading to an increase
volatility.
Our
multiplicity result can thus
equilibrium
increase the sensitivity of the coordination
in
non-fundamental
be interpreted as an extreme version of
this
negative effect of on volatility.
In conclusion,
we view the main theme
in Morris-Shin as emphasizing the importance
of the details of the information structures for understanding the determinacy of equilibria
and the
volatility of
outcomes.
This paper contributes to this same theme by studying
the importance of endogenous information aggregation.
understand
crises
phenomena such
as currency attacks,
26
Our
bank
results
on
volatility
runs, or debt crises.
may
help
Appendix
7
Proof of Proposition
Rewrite
1.
(4) as
G(6*(z))=g(z),
(39)
= - (a.J^) 9 + §~ (6) and g(z) = yjl + a z /a x $- (1 - c) - (az /^) z.
every zGE, G(6) is continuous in 8, with G(0, z) = -co and G(l, 2) = 00, which implies
l
where G{9)
For
x
that there necessarily exists a solution and any solution satisfies 9*(z) G
that
G{&)
=
—% +
a,
(0, 1).
Next, note
1
1
(*))
and max^eR
=
0(w;)
a z /^/a^ <
\j\pht. If
which implies a unique solution to
and there
in 9
z G
if
a z j^oTx <
if
y/2n.
Proof of Proposition
.
In either case,
—
9*{z)
of
z.
»
=
5
—
1
=
6
1
+
cr x
az
/'
c for
/a x —
and
~
>
y
any
$
(^) if
2,
we
??.
assume that Ui(c)
linear relation,
G
increasing in
is strictly
y/2n, then
G
is
9,
non-monotonic
We conclude that monotone equilibrium is
unique
y{a x + a z /a x
)
—
—
for given a~, or
>
so that the regime-change threshold
—
>
instead consider the limit
Paz-i
=
where x
x,
az
9
if
—
we
00.
>
>
00 for given
is
miique and independent
consider the limit
ax —
>
0,
QED.
who
Consider an agent
(zj.
—
Condition (39) then implies that
1.
>
az
receives a private signal x
price p. His optimal investment k solves
u'l
We
'
Consider the limits as a x
1.
Proof of Proposition
and observes a
that
a z /^/a x >
QED
Similarly, condition (??) gives x*(z)
_1
and x
instead
an interval (z,z) such that (39) admits multiple solutions 9*{z) whenever
is
and a unique solution otherwise.
(z, z)
and only
ax
(39). If
we have
\Z2k
fcj
loss of generahty,
=
is
(w-k)=E[(f-p)u'2 ((f-p)k)
quadratic and
k {E [/|x,p]
112(c) is linear, in
— p} +
we normalize A
=
0.
A, for
\x,p].
which case (40) reduces to a simple
some constants K >
Finally,
we
let
/
(40)
=
f(9)
—
0,
9.
A G R. With out any
That
is,
the return of
the asset depends only on the exogenous fundamental.
The
for
analysis here
the asset
is
k
We
similar to that in the first example.
The optimal
is
= k{E[
f
x,p]
I
-p} = k{E[
9
I
x,p] -p}.
conjecture
9\x,p]
=
5x
27
+ (l-
S)p
individual
demand
some
for
By
G
8
=
K(9,p)
k8
(0, 1)
(9
implication, p
Z(p)
=p
and v
.
is
is
K=
In equilibrium,
e.
It
k(x,p)
=
—
k8(x
a £ That
.
and therefore
p)
is
in this
is,
example
remains to pin down 8 and the function Q.
bounded above by K2 a£ and therefore we immediately have that unique-
ensured for a x high enough. To complete the analysis, note that
is
ax +
The above uniquely determines 5 G
function of
a £ To
.
(0, 1)
a = ax /
see this, let
Q.
ax + a£ 82 n2
p
as an increasing function of
a—
oo.
»•
Using these
results,
we
au and
(a e n 2 ) and rewrite the above as
Obviously, this gives a monotonic relation between
as
—
k
Hence, the equilibrium price
2
a public signal about 9 with precision k 8
= — -^e.
Note that ap
ness
to be determined. It follows that
— p)
a and
<5,
with 5
—
>•
as
a decreasing
a =
a —
3
8 /(l
and
>
—
6
5).
—
>
1
find
= (K^)y/5(l-6).
The
fact that 5(1
—
6)
—
as either
we have that av j\fa x < \/2n and
'
is
6
a x —*
On
either sufficiently small or sufficiently high.
(1
—
8)
<
sufficiently high is sufficient for multiplicity.
Part
(ii).
a£
oo then imphes that, given
>
if
and only
the other hand, for given
and therefore a £ < 8tt/k 2
1/4 necessarily
—
a2
or
therefore the equilibrium in unique
is sufficient for
if
,
ax
a x we have
,
uniqueness, whereas
a£
QED
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
|
=
x,p
^/a^[9
}
—
x* (p)]
.
The demand
— p} — k{^/o^E[
9
|
x,p
}
for the asset is thus
— p — v/a^x*(p)}
.
Let
p
28
+ x*(p)
(41)
'
and note
that, for every p, the above defines a unique p.
k
where k
=
K^/a^.
We now
= K{E[6
follows that
can thus write the demand as
-p}
conjecture
E[9
It
x,p]
|
We
K — k5(6 — p)
\
x,p]
=
+
5x
-S)p.
(l
and therefore
t-»-y.
Hence, the observation of p
9 with precision
ap = H
5
2
is
a£
.
(42)
equivalent to the observation of p, which
It follows
is
a public signal for
that
E[9\x,p]=E[9\x,p] =8x +
(l-5)p,
where
.
This
oc x
ax
ax + ap
a x + a E 52 h?
_
the same as in the previous example, with H replacing
is
t
1
so that 5
a x hke
,
is
decreasing in
in the
a e but independent
benchmark model. Indeed, the
-%= =
and
is
increasing in both
The
9*
(p)
+ a£ 52 K?
a s and a x
of
Using k
=
ap
is
a x This means
.
now
that
x»
=
proportional to
(*%.) Vo-
(43)
second example. In particular, the thresholds
-^- $-
1
$fV V a x + &p
e*{p)
infer
.
rest of the analysis is similar to the
=
we
given by
and x* (p) are uniquely determined and are given by
9*{p)
K^/a^,
'
critical ratio is
~^~ =
At.
+ ~^-
1
(l-c/6)--^- P y
ce x
(9*{P )),
29
+ ap
J
>
.
with ap as in
ap /^/a^ <
If
Next, combining (41) and (42), we infer that the equilibrium price solves
(43).
the above has a unique solution.
2-7T,
Proof of Corollary
—
9*(y)
9
— ax e
Part
—
>
\a £ a x
—
ax y
-
cp(y)\
and a x y
>
Pr
let Y_(9,e)
that 9
-
[
-
<7z£
= min^^ —
(??), 0*(y) is
=
as y
Pr
[
9
—
<
>
+00.
9
9
—
<
1. It
the above has
>
(??)
we have
that, for every y,
Condition (28) then implies
oo.
unique signal function in the limit
0.
>
It
result.
8.
9
e,
—
— ax e —
9 as
|
e
(0,0)
=
and Y(0,e)
>
cr x
,
9*(y)
]
-»
and
Pr
[
>
is
-»
1
Next, note that both
and
6
—
z
e)
=
<
9*
E
9 as
such
(0, e)
and therefore
$(—00)
>
the
0,
=
d^O.
as
as
>
=
$(+00)
>
follows that
0. It
— +00
—
9
—
ax
Y(9,
follows that, as long as 9
]
=
0.
lim y ^ +00 <p(y)y
max^y(0 — a x e) and consider
<y <y <
e)
a x —>
$(—00)
>
>
and Y(9,e)
as
lim^.^ (p(y)y =
Hence, z
e (1,2)
<j x e)
+co
»
cr x
—
>•
0.
—
9 as y
>
-00, and 0*(y)
—
(9, 0),
(7(0, e))
]
->
1
as
a x ->
0,
QED
For any
/i
6
(0, 1)
and any x\ 6 R, r(0,£*)
ji)
follows that (38) always admits
a solution 0* (z)
given x\ 6 R, (38) defines at least one function 9*
We
cr £
= —00 and F(9,xl,/j.) = 00, where 9 = (x*, Q x ,/x)
respectively,
+ ^- {0 - $ (yaT^ [x\ - 9}) } = and = 1, and
with T(9,x\,
<
—
—
y
Since
vanish.
— —00
9* (Y(9, e))
Proof of Proposition
solves,
>
independent of
which establishes the
0,
-
OzS e (z,z). Note that Y(0,
Y.(0, e)
$(+00)
and
conditions (??)
— — oo and
Moreover, for every 9 and
Next,
2ir,
e.
2
and
(f>(y)\
0.
From
Prom
(i).
note that y
First,
(ii).
-
av j\Jol^ >
instead
l
latter implies
ax
Part
4.
— c/b = 9 and x*(y) —> 9 + a x §~ (9) = x as
= x*(y) — a x y —> x — a x y and therefore the
1
Y(9,e)-*(x-9)/a x +
\g\gx
If
QED
multiple solutions.
:
R—
>
-^= + A (9-x,, ax ,,)
30
9
(a;*,
a x ,jj)
therefore satisfy
6 (9,9). That
next examine under what conditions the function that solves (38)
=
continuous in
=
is,
for
<
any
(0, 0)
that
Te
is
and 9
is
unique.
Note
,
}
.
where
A (0; xl, a x
As
6
—
>
,
p)
=
^_ l(9+ _^
—
(equivalently, z
or
{
^
»
that, since
<f>
=
8 (x\
,
(0,
ax
,
/z)
G
(0, 0)
<
Combining, we conclude that,
{0
1-//
J
cj>
(^/ce^ [x\
($
—
_1
+ ^- {0 - $ (^/a^ [x^
= 1/2, and using again
[ (9
5]) }
we have
+
——
+ y/a£<j> yja x
~
-
j^l
\x\
-fljjj
>
+
1
I
—$
\ 1
1
J
be the solution to
l/\/27r,
M
v^i +
v
1-/
=
(fl;x*,o; g ,/xj
/x)
,
l/v27r],
0+ y^is
(j>
A (0; x\, a x
inf
1/V^l
l/v^7T, or equivalently the solution to
< A
=
/i)
g D]
+oo. Let
>
>
ax)
the fact that the maximal value of
A'(x*,a x ,/x.)
,
takes values in
A'^,^,
Moreover, letting 8
ax
+ j-^ [1 + v^^ (>/^ [»i ~
[l
_ e])}))
A (0; Xj, a^, fi) —
±oo),
A' (xj,
and note
(v/5 [i;
-PL
for all
K (ax
(x\,a x
>
,/j,)
/i)
,
K (xl,ax
,fi)
>
K(fj,)
where
K (a x
fj)
=
K(»)
=
,
V27T
Case(i):
^<
0,
for all
and
+
1-M
+
1
l-/x-
Note that, importantly, neither bound
In this case, -^==
{ 1
is
a function of x\.
K(p).
<
all x\. It
K_(fi)
<
inf x . A"
(x^,a x ,/x) and therefore
T
is
follows that (38) defines a unique function 8*
given x*. Moreover, since
T
is
decreasing in x* and g
decreasing in z and increasing in x^ Finally, 8*
.
31
is
is
decreasing in
strictly increasing in
:
R—
z,
(0, 0) for
the function
continuous in both z and x \
any
8* is
Next, consider (37). For any given 9*
Moreover, this solution
C
Let
be the
R— C
increasing
mapping
It is
exists.
>
and
>
(6_,
6)
,
(37) admits a unique solution Xj
continuous and increasing in
is
set of continuous
a mapping
R—
:
T R—
9*
(and bounded) functions
mapping
C—
>
:
R—
>
[9_,
T (— oo) > -co and T(+oo) < oo. Hence, a
arbitrary x\ and a > 0, let £** = x* + a and let
easy to check that
Moreover, for
and
solutions to (38) for x*
Then, (38) and
x\*, respectively.
.
#*
and
= i-c
v^ [Txr -#**(*)]) vW*(v^[^r--])^
= l-c
< y^1 we
,
(38)
is
fixed point always
/'<i>(v o~[Tx*-r( 2 )])vsr^(v/sr[^i-~]) d ~
/"$(
Then,
clearly have 8**
<
=
6
9*
+
0**
be the
(37) give
,
Since -7=
#)
R. Together, they define a continuous and
R.
>
:
a
is
(37)
G R.
9*.
a and, by implication,
Tx" <
X\,
where X\
solves
/
If Xi
=
$( v/o~[xi -0*(z) -a]) v^oT^Cv^i £ -2])^
Tx\-\-a (> Tx*), the above would have been positive, so
< Tx\ +
Therefore, Tx\*
Case
(ii):
<J
2
E
a x < -4=
In this case, -^=
empty region
z
G
K
T
(1
—
(/j.)
Z—
(z_,z)
fj,
>
—
fia
2
)
,
or equivalently -%=
K (x*,a: x
sup x ,
— Z (x*)
(Ty) be the associated mappings.
each mapping.
Let 9*L
(^1)
<
,/z)
G
Each
and T(+oo) <
(9*H )
Z and
of the
00.
T}j (x*) for
any
than one
>
K
(/1).
and therefore V necessarily has a nonany
x*, there
is
a non-
be the function denned by selecting the
the unique one whenever z
(z)
and the
x*. It follows that
<
9*
H
is
Z. Let Tj,
a fixed point (at least one) for
(z) for all z
fact that
Z
the fixed point of
32
G'
mappings Tl and T# are continuous and
Hence, there
Moreover, for any given x*, 9*L
implies (because of the monotonicity in (37)
Tl
is less
such that (38) admits three distinct solutions whenever
a unique one otherwise.
T (— 00) > -co
T
has a unique fixed point.
lowest (highest) solution whenever z
satisfy
must be that X\ < Tx^ + a.
of non-monotonicity in 9 for all x*. It follows that, for
interval
Z and
>
it
= l-c
which proves that the slope of the mapping
a,
for every x\. It follows that
empty
i
[
G Z, which
in turn
has positive measure) that
Tj, is
lower than the fixed
point of Th, which together with the fact that 6*L
associated x\ and 6* satisfy the
same
< 9*H
any given x\ implies that the
for
ordering.
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34
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